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Thomas Andren
Econometrics
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Econometrics
Contents
Contents
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Basics of probability and statistics
Random variables and probability distributions
Properties of probabilities
The probability function – the discrete case
The cumulative probability function – the discrete case
The probability function – the continuous case
The cumulative probability function – the continuous case
The multivariate probability distribution function
Characteristics of probability distributions
Measures of central tendency
Measures of dispersion
Measures of linear relationship
Skewness and kurtosis
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The simple regression model
The population regression model
The economic model
The econometric model
The assumptions of the simple regression model
Estimation of population parameters
33
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36
37
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Specification
Choosing the functional form
The linear specification
The log-linear specification
The linear-log specification
The log-log specification
Omission of a relevant variable
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73
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The method of ordinary least squares
Properties of the least squares estimator
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5
Econometrics
Contents
Inclusion of an irrelevant variable
Measurement errors
76
77
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3
8
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5
Dummy variables
Intercept dummy variables
Slope dummy variables
Qualitative variables with several categories
Piecewise linear regression
Test for structural differences
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85
87
89
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1
9
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2
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2
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3
9
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1
Heteroskedasticity and diagnostics
Consequences of using OLS
Detecting heteroskedasticity
Graphical methods
Statistical tests
Remedial measures
Heteroskedasticity-robust standard errors
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4
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106
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110
113
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115
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Discover the truth at www
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ca/careers
© Deloitte & Touche LLP and affiliated entities
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deloitte
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Simultaneous equation models
Introduction
The structural and reduced form equation
Identification
The order condition of identification
The rank condition of identification
Estimation methods
Indirect Least Squares (ILS)
Two Stage Least Squares (2SLS)
125
125
127
129
130
132
133
134
135
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7
Econometrics
Basics of probability and statistics
1
...
If these concepts are new to you, you should
make sure that you have an intuitive feeling of their meaning before you move on to the following chapters in
this book
...
1 Random variables and probability distributions
The first important concept of statistics is that of a random experiment
...
That is, the outcome of the experiment can not be predicted with certainty
...
The set of all possible outcomes of on experiment is called the sample space of the experiment
...
If the experiment was to pick a card from
a deck of cards, the sample space would be all the different cards in a particular deck
...
An event is a collection of outcomes that resulted from a repeated experiment under the same condition
...
Alternatively, two events that have no outcomes in common are mutually exclusive
...
These two events are therefore not mutually exclusive
...
For
example, when rolling a die, the outcomes 1, 2, 3, 4, 5, and 6 are collectively exhaustive, because they
encompass the entire range of possible outcomes
...
The outcomes 1 and 3 are mutually exclusive but not collectively
exhaustive, and the outcomes even and not-6 are collectively exhaustive but not mutually exclusive
...
For that purpose we introduce
the concept of a random variable
...
By convention, random variables are denoted by capital letters, such as X, Y, Z, etc
...
A random variable from an
experiment can either be discrete or continuous
...
That is, the result in a test with 10 questions can be 0, 1, 2, …, 10
...
Other examples could be the number of
household members, or the number of sold copy machines a given day
...
However, when the number of unites can be
very large, the distinction between a discrete and a continuous variable become vague, and it can be unclear
whether it is discrete or continuous
...
com
8
Econometrics
Basics of probability and statistics
A random variable is said to be continuous when it can assume any value in an interval
...
But in practice that does not work out
...
Variables
related to time, such as age is therefore also considered to be a continuous variable
...
However, the values are usually very large so counting each
Euro or dollar would serve no purpose
...
Since the value of a random variable is unknown until the experiment has taken place, a probability of its
occurrence can be attached to it
...
1)
This formula is valid if an experiment can result in n mutually exclusive and equally likely outcomes, and if
m of these outcomes are favorable to event A
...
This formula follows the classical definition of a
probability
...
1
You would like to know the probability of receiving a 6 when you toss a die
...
You are interested in one of them, namely 6
...
Example 1
...
First we have to find the total
number of unique outcomes using two dice
...
How many of them sum to 7? We have (1,6), (2,5),
(3,4), (4,3), (5,2), (6,1): which sums to 6 combinations
...
The classical definition requires that the sample space is finite and that the each outcome in the sample space
is equally likely to appear
...
We therefore need a
more flexible definition that handles those cases
...
Formally, if in n trials, m of them are favorable to the
event A, then P(A) is the ratio m/n as n goes to infinity or in practice we say that it has to be sufficiently large
...
3
Let us say that we would like to know the probability to receive 7 when rolling two dice, but we do not know
if our two dice are fair
...
We could then
perform an experiment where we toss two dice repeatedly, and calculate the relative frequency
...
1
we report the results for the sum from 2 to 7 for different number of trials
...
com
9
Econometrics
Basics of probability and statistics
Table 1
...
1
0
...
2
0
...
2
100
0
...
02
0
...
12
0
...
17
1000
0
...
046
0
...
114
0
...
15
Number of trials
10000
100000
0
...
0283
0
...
0565
0
...
0831
0
...
1105
0
...
1359
0
...
1658
1000000
0
...
0555
0
...
1114
0
...
1669
0
...
05556
0
...
11111
0
...
16667
From Table 1
...
For this particular experiment 1 million trials would be sufficient to receive a
correct measure to the third decimal point
...
1
...
1 Properties of probabilities
When working with probabilities it is important to understand some of its most basic properties
...
1
...
2
...
)
P( A) P( B)
...
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10
Econometrics
Basics of probability and statistics
Example 1
...
The event A represents receiving a club, and event B
represents receiving a spade
...
Therefore the probability of the event
C = A + B that represent receiving a black card can be formed by P ( A B ) P ( A) P( B )
3
...
) P ( A) P ( B)
...
5
Assume picking a card from a deck of cards
...
These two events are mutually exclusive and collectively exhaustive
...
4
...
5
...
6
Assume that we carry out a survey asking people if they have read two newspapers (A and B) a given day
...
In order to
calculate the probability that a randomly chosen individual has read newspaper A and/or B we must
understand that the two events are not mutually exclusive since some individuals have read both papers
...
Only if it had been an impossibility to have read both papers the
two events would have been mutually exclusive
...
We must then ask if event B has any influence on event A or if event A and B are independent
...
The
conditional probability of event A given event B is computed using the formula:
P( AB)
(1
...
7
We are interested in smoking habits in a population and carry out the following survey
...
The results are shown in Table 1
...
Table 1
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com
11
Econometrics
Basics of probability and statistics
Using the information in the survey we may now answer the following questions:
i) What is the probability of a randomly selected individual being a male who smokes?
This is just the joint probability
...
Thereafter we have to find the number of smoking males: 19
...
19
...
We can therefore say that we condition on smokers when we ask for the
probability of being a male in that group
...
2)
...
That turned out to be 0
...
Secondly, we have to find the probability of being a smoker
...
31
...
We have 0
...
31=0
...
Hence there is 61 % chance that a randomly selected smoker is a man
...
1
...
Using the probability function we may form the corresponding
probability distribution
...
Let us take an example to illustrate
the meaning of those concepts
...
8
Consider a simple experiment where we toss a coin three times
...
The following 8 outcomes represent the sample space for this experiment: (HHH), (HHT), (HTH),
(HTT), (THH), (THT), (TTH), (TTT)
...
The random variable we are interested in is the number of heads received on one trial
...
X can therefore take the following values 0, 1, 2, 3, and the probabilities of occurrence differ
among the alternatives
...
Using the classical definition of probabilities we receive the following probability
distribution
...
3 Probability distribution for X
X
P(X)
0
1/8
1
3/8
2
3/8
3
1/8
From Table 1
...
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1
...
It is defined in the following way:
F(X )
P ( X d c)
(1
...
9
Consider the random variable and the probability distribution given in Example 1
...
Using that information
we may form the cumulative distribution for X:
Table 1
...
3 are mutually exclusive
...
As an example:
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P( X d 2)
P( X
0) P ( X
1) P ( X
2)
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Econometrics
Basics of probability and statistics
1
...
4 The probability function – the continuous case
When the random variable is continuous it is no longer interesting to measure the probability of a specific
value since its corresponding probability is zero
...
Formally we
may express the probability in the following way:
b
P ( a d X d b)
³ f ( x)dx
(1
...
There exist a number of standard
probability functions, but the single most common one is related to the standard normal random variable
...
10
Assume that X is a continuous random variable with the following probability function:
3e 3 X X ! 0
°
f (X ) ®
° 0
else
¯
Find the probability P(0 d X d 0
...
Using integral calculus we find that
0
...
5)
³ 3e
3 x
dx
> e @ > e
3 x 0
...
5
@ > e @
3u0
e 1
...
777
0
1
...
5 The cumulative probability function – the continuous case
Associated with the probability density function of a continuous random variable X is its cumulative
distribution function (CDF)
...
However, for
the continuous random variable we have to integrate from minus infinity up to the chosen value, that is:
c
F (c )
P( X d c)
³ f ( X )dX
(1
...
2) P( X t a ) 1 F ( a)
3) P( a d X d b)
F (b) F (a )
In order to evaluate this kind of problems we typically use standard tables, which are located in the appendix
...
com
14
Econometrics
Basics of probability and statistics
1
...
Often we may be interested in probability statements for several random
variables jointly
...
In the discrete case we talk about the joint probability mass function expressed as
f ( X ,Y )
P( X
x, Y
y)
Example 1
...
We form the random variables X = “number of heads obtained by
A”, and Y = “number of heads obtained by B”
...
The sample space for person A and B is the same and
equals {(H,H), (H,T), (T,H), (T,T)} for each of them
...
Counting the different combinations, we end up with the results presented in Table 1
...
Table 1
...
00
1) 2 / 16 1 / 8
...
When that is done using a joint distribution function we call it the marginal probability function
...
The
marginal probability functions for X and Y is
f (X )
¦ f ( X , Y ) for all X
(1
...
7)
y
f (Y )
x
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Find the marginal probability functions for the random variables X
...
Two random variables X and Y are said to be statistically independent if and only if their
joint probability mass function equals the product of their marginal probability functions for all combinations
of X and Y:
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3 Characteristics of probability distributions
Even though the probability function for a random variable is informative and gives you all information you
need about a random variable, it is sometime too much and too detailed
...
Below we will shortly describe
the most basic summary statistics for random variables and their probability distribution
...
3
...
The expected value of a discrete random variable is denoted E[X], and defined as
follows:
E >X @
n
¦ xi f ( xi )
PX
(1
...
It is simply a weighted average of all
X-values that exist for the random variable where the corresponding probabilities work as weights
...
13
Use the marginal probability function in Example 1
...
E >X @ 0 u P( X
0) 1u P( X
1) 2 u P( X
2) 0
...
25 1
When working with the expectation operator it is important to know some of its basic properties:
1) The expected value of a constant equals the constant, E >c @ c
2) If c is a constant and X is a random variable then: E >cX @ cE> X @
3) If a, b, and c are constants and X, and Y random variables then: E >aX bY c @ = aE >X @ bE >Y @ c
4) If X and Y are statistically independent then and only then: E >X , Y @ E > X @E >Y @
The concept of expectation can easily be extended to the multivariate case
...
10)
Y
Example 1
...
5
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17
Econometrics
Basics of probability and statistics
1
...
2 Measures of dispersion
It is sometimes very important to know how much the random variable deviates from the expected value on
average in the population
...
The variance of X is defined as
2
Var > X @ V X
>
E X P X 2
@ ¦ X P
X
2 f ( X )
(1
...
The most important properties of the variance is
1) The variance of a constant is zero
...
2) If a and b are constants then Var (aX b) Var (aX )
a 2Var ( X )
3) Alternatively we have that Var(X) = E[X 2] - E[X]2
4) E[X 2] =
¦ x2 f ( X )
x
Example 1
...
6 Probability distribution for X
X
P(X)
1
1/10
2
2/10
3
3/10
4
4/10
In order to find the variance for X it is easiest to use the formula according to property 4 given above
...
1
2
3
4
2 u 3u 4 u
3
10
10
10
10
1
2
3
4
E[X 2] = 12 u
22 u 32 u 42 u
10
10
10
10
10
E[X] = 1 u
Var[X] = 10 – 32 = 1
1
...
3 Measures of linear relationship
A very important measure for a linear relationship between two random variables is the measure of
covariance
...
12)
The covariance is the measure of how much two random variables vary together
...
If they tend to vary in opposite direction, that is, when
one tends to be above the expected value when the other is below its expected value, we have a negative
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If the covariance equals zero we say that there is no linear relationship between the two random
variables
...
That makes it
very hard to compare two covariances between different pairs of variables
...
One such standardization gives us the correlation between the two random variables
...
13)
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19
Econometrics
Basics of probability and statistics
Example 1
...
7
...
7 The joint probability mass function for X and Y
1
0
0
...
3
1
2
3
P(Y)
X
Y
2
0
...
2
0
...
6
3
0
0
...
1
P(X)
0
...
6
0
...
0
We will start with the covariance
...
We have
E >X @ 1 u 0
...
6 3 u 0
...
2
E >Y @ 1 u 0
...
6 3 u 0
...
8
E >XY @ 1 u 1 u 0 1 u 2 u 0
...
3 2 u 2 u 0
...
1
3 u 1 u 0 3 u 2 u 0
...
2 u 1
...
04 ! 0
We will now calculate the correlation coefficient
...
> @ 1 u 0
...
6 3 u 0
...
2
E >Y @ 1 u 0
...
6 3 u 0
...
6
V >X @ E >X @ E >X @ 5
...
2 0
...
6 1
...
36
E X2
2
2
2
2
2
2
2
2
2
2
2
2
2
Using these calculations we may finally calculate the correlation using (1
...
04
0
...
36
0
...
3
...
The Skewness of a distribution function is defined in the following way:
S
E >X P X @3
V3
X
(1
...
If it is not skewed we say that the distribution is
symmetric
...
1 give two examples for a continuous distribution function
...
com
20
Econometrics
Basics of probability and statistics
a) Skewed to the right
Figure 1
...
Formally it is
defined in the following way:
K
E >X P X @4
>E>X P @ @
(1
...
A distribution that are
long tailed compared with the standard normal distribution has a kurtosis greater than 3 and if it is short tailed
compared to the standard normal distribution it has a kurtosis that is less than three
...
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Econometrics
Basics of probability distribution in econometrics
2
...
There exist a number of different probability distributions for discrete and continuous random
variables, but some are more commonly used than others
...
For that matter we need to
know something about the most basic probability functions related to continuous random variables
...
Having knowledge about their properties we will be able to construct most of
the tests required to make statistical inference using regression analysis
...
1 The normal distribution
The single most important probability function for a continuous random variable in statistics and
econometrics is the so called normal distribution function
...
Its Probability Density Function (PDF) and the corresponding Cumulative Distribution Function
(CDF) are pictured in Figure 2
...
1
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
3,
3
2,
9
2,
1
2,
5
1,
7
1,
3
0,
5
0,
9
0,
1
-0
,3
-0
,7
-1
,5
-1
,1
-2
,3
-1
,9
-2
,7
-3
,1
-3
,5
2,
9
3,
3
2,
1
2,
5
1,
3
1,
7
0,
5
0,
9
0,
1
-0
,3
-0
,7
-1
,1
-1
,5
-1
,9
-2
,3
-2
,7
-3
,1
-3
,5
0
X
X
a) Normal Probability Density Function
Figure 2
...
The
mathematical expression for the normal density function is given by:
f (X )
1
VX
1§ X P
°
X
exp® ¨
2¨ VX
2S
©
°
¯
·
¸
¸
¹
2½
°
¾
°
¿
which should be used in order to determine the corresponding CDF:
c
P( X d c)
³ f ( X )dX
f
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...
For that reason
most basic textbooks in statistics and econometrics has statistical tables in their appendix giving the
probability values for different values of c
...
The normal distribution curve is symmetric around its mean, P X , as shown in Figure 2
...
2
...
3
...
4
...
7 % of the area below the normal curve is covered by the interval of plus minus
three standard deviations around its mean: P X r 3 u V X
...
A linear combination of two or more normal random variables is also normal
...
1
If X and Y are normally distributed variables, then Z
variable, where a and b are constants
...
The skewness of a normal random variable is zero
...
The kurtosis of a normal random variable equals three
...
A standard normal random variable has a mean equal to zero and a standard deviation equal to one
...
Any normal random variable X with mean P X and standard deviation V X can be transformed into a
X PX
standard normal random variable Z using the formula Z
...
2
Assume a random variable X with expected value equal to 4 and a standard deviation equal to 8
...
It is now easy to show that Z has a mean equal to 0 and a variance equal to 1
...
com
23
Econometrics
Basics of probability distribution in econometrics
Since any normally distributed random variable can be transformed into a standard normal random variable
we do not need an infinite number of tables for all combinations of means and variances, but just one table
that corresponds to the standard normal random variable
...
3
Assume that you have a normal random variable X with mean 4 and variance 9
...
5
...
That is:
3
...
5 P¨ Z d
¸
3 ¹
©
P Z d 0
...
We therefore need to transform
our problem so that it adapts to the table we have access to
...
That implies
that P Z d 0
...
167 and that P Z t 0
...
167
...
Hence, the solution is:
P X d 3
...
167 1 0
...
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Econometrics
Basics of probability distribution in econometrics
Example 2
...
5 d X d 4
...
Whenever dealing with intervals we need to split up the probability expression in two parts
using the same logic as in the previous example
...
5 4 ·
3
...
5 d X d 4
...
5 P X d 3
...
167 P Z d 0
...
The sampling distribution of the sample mean
Another very important concept in statistics and econometrics is the idea of a distribution of an estimator,
such as the mean or the variance
...
This issue will
discussed substantially in later chapters and then in relation to estimators of the regression parameters
...
Whenever using a sample when estimating a population parameter we receive
different estimate for each sample we use
...
Since we are using
different observations in each sample it is unlikely that the sample mean will be exactly the same for each
sample taken
...
The question is whether it is possible to say something about this distribution
without having to take a large number of samples and calculate their means
...
In statistics we have a very important theorem that goes under the name The Central Limit Theorem
...
A basic rule of thumb says that if the sample is larger than 30 the shape of the distribution will be sufficiently
close, and if the sample size is 100 or larger it will be more or less exactly normal
...
Basics steps in hypothesis testing
Assume that we would like to know if the sample mean of a random variable has changed from one year to
another
...
In the following
year we would like to carry out a statistical test using a sample to see if the population mean has changed, as
an alternative to collect the whole population yet another time
...
com
25
Econometrics
Basics of probability distribution in econometrics
1) Set up the hypothesis
In this step we have to form a null hypothesis that correspond to the situation of no change, and an alternative
hypothesis, that correspond to a situation of a change
...
If we
do that we will be able to say something with a statistical certainty
...
The
hypothesis given above is a so called a two sided test, since the alternative hypothesis is expressed with an
inequality
...
In most cases, you should prefer to use a two sided test since it
is more restrictive
...
Since we have taken a sample
and calculated a mean we know that the mean can be seen as a random variable that is normally distributed
...
That will
give us a new random variable, our test function Z, that is distributed according to the standard normal
distribution
...
We will discuss this issue further
below
...
The fewer number of observations we have, the less we know about the distribution of
Z, and the more likely it is to make a mistake when performing the test
...
Since we know the distribution of Z, we also know that realizations of Z take values between -1
...
96
in 95 % of the cases (You should confirm this using Table A1 in the appendix)
...
This knowledge will
now be used using only one sample
...
com
26
Econometrics
Basics of probability distribution in econometrics
If we take a sample and calculate a test value and find that the test value appear outside the interval, we say
that this event is so unlikely to appear (less than 5 percent in the example above) that it cannot possible come
from the distribution according to the null hypothesis (it cannot have the mean stated in the null hypothesis)
...
In this discussion we have chosen the interval [-1
...
96] which cover 95 % of the probability distribution
...
Alternatively, with a significance level of 5 % there is a 5 % chance that we
will receive a value that is located outside the interval
...
If
we believe this is a large probability, we may choose a lower significance level such as 1 % or 0
...
It is our
choice as a test maker
...
5
Assume that you have taken a random sample of 10 observations from a normally distributed population and
found that the sample mean equals 6
...
You would
like to know if the mean value of the population equals 5, or if it is different from 5
...
For this example we have:
H0 : P 5
H1 : P z 5
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27
Econometrics
Basics of probability distribution in econometrics
You know that according to the central limit theorem the sampling distribution of sample means has a normal
distribution
...
236
We know that our test function follows the standard normal distribution (has a mean equal to zero) if the null
hypothesis is true
...
A significance level of 1 % means that
there is a 1 % chance that we will reject the null hypothesis even though the null hypothesis is correct
...
576; 2
...
Since our test value is located
within this interval we cannot reject the null hypothesis
...
We cannot say that it is significantly different from 5
...
2 The t-distribution
The probability distribution that will be used most of the time in this book is the so called t-distribution
...
In large
samples the t-distribution converges to the normal distribution
...
The t-distribution is symmetric around its mean
...
The mean equals zero just as for the standard normal distribution
...
The variance equals k/(k-2), with k being the degrees of freedom
...
That was under condition that we knew the
values of the population parameters
...
The transformation formula would then have a distribution that is different from the
normal in small samples
...
Example 2
...
You would like to know if the population mean is different from 6
...
com
28
Econometrics
Basics of probability distribution in econometrics
t
X PX
~ t( n 1)
S
n
Observe that the expression for the standard deviation contains an S
...
Since it is based on a sample it is a random variable, just as the mean
...
That implies more variation, and therefore a distribution that deviates from
the standard normal
...
Hence in our case the test value equals
t
X PX
S
n
56
3
60
2
...
If we choose a significance level of 5 % the critical
values according to the t-distribution would be [-2
...
0]
...
That we have no
information about the population mean is of no problem, because we assume that the population mean takes a
value according to the null hypothesis
...
That is part
of the test procedure
...
3 The Chi-square distribution
Until now we have talked about the population mean and performed tests related to the mean
...
For that purpose we are going to work
with another distribution, the Chi-square distribution
...
It turns out that the sum of squared
independent standard normal variables also is Chi-squared distributed
...
Z k ~ F (2k )
Properties of the Chi-squared distribution
1
...
It is skewed to the right in small samples, and converges to the normal distribution as the
degrees of freedom goes to infinity
3
...
In this case we may rely on statistical
theory that shows that the following function would work:
Download free ebooks at bookboon
...
How could this function be used to perform a test related to the population
variance?
Example 2
...
Some
years later we suspect that the population variance has increase and would like test if that is the case
...
Using this information we set up the test
function and calculate the test value:
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Test Function
(n 1) S 2
V
2
(25 1) u 600
400
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30
Econometrics
Basics of probability distribution in econometrics
We decided to have a significance level of 5 % and found a critical value in Table A3 equal to 36
...
Since
the test value is lower than the critical value we cannot reject the null hypothesis
...
2
...
In shape it is very similar to the Chisquare distribution, but is a construction of a ratio of two independent Chi-squared distributed random
variables
...
That is:
2
Fm
~ Fm,l
F l2
Properties of the F-distribution
1
...
The F-distribution converges to the normal distribution when the degrees of freedom
become large
3
...
It is especially interesting when we would like to
know if the variances from two different populations differ from each other
...
8
Assume that we have two independent populations and we would like to know if their variances are different
from each other
...
38 and S 2 13
...
Under the null hypothesis we know that the ratio of the two sample variances is F-
distributed with 25 and 29 degrees of freedom
...
com
31
Econometrics
Basics of probability distribution in econometrics
S12
2
S2
8
...
14
0
...
Assume that we choose a significance level of 5 %
...
Since the area outside
the interval should sum up to 5 %, we must find the upper critical point that corresponds to 2
...
If we look
for that value in the table we find 2
...
We call this upper point F0
...
In order to find the lover point we
can use the following formula:
F0
...
025
1
2
...
464
Please click the advert
We have therefore received the following interval: [0
...
154]
...
It is therefore quite possible that the two
population variances are the same
...
com
32
Econometrics
The simple regression model
3
...
When looking at a single variable we could describe its behavior by using any summary
statistic described in the previous chapters
...
The mean
value would be a description of the central tendency, and the variance or the standard deviation a measure of
how the average observation deviates from the mean
...
But we can say nothing about the factors that
make single observations deviate from the mean
...
The initial discussion will be related to models that use one single explanatory factor or
variable X that explains why observations from the variable Y deviate from its mean
...
A simple regression model
is seldom used in practice because economic variables are seldom explained by just one variable
...
It is
therefore important to have a good understanding of the simple model before moving on to more complicated
models
...
1 The population regression model
In regression analysis, just as in the analysis with a single variable, we make the distinction between the
sample and the population
...
Using this sample, we try to make inference on the population, that is, we
try to find the value of the parameters that correspond to the population
...
3
...
1 The economic model
The econometric model, as appose to models in statistics in general, is connected to an economic model that
motivate and explains the rational for the possible relation between the variables included in the analysis
...
In order to
confirm that the made assumptions are in accordance with the reality, it is important to specify a statistical
model, based on the formulation of the economic model, and statistically test the hypothesis that the
economic model propose using empirical data
...
It is therefore very important to remember that all
econometric work has to start from an economic model
...
Economic theory claims that there is a relationship between food
consumption and disposable income
...
That means that if the household
disposable income increases, the food expenditure will increase as well
...
Since we talk about averages we may express the economic
model in terms of an expectation:
Download free ebooks at bookboon
...
1)
The conditional expectation given by (3
...
We have imposed the assumption that the relationship between Y and X1 is linear
...
The parameters of interest are B0 and B1
...
B0 will represent the average food expenditure by households when the disposable
income is zero ( X 1
0) and is usually referred to as the intercept or just the constant
...
Furthermore, the slope coefficient will represent the marginal propensity to
spend on food:
B1
dE >Y | X 1 @
dX 1
3
...
2 The econometric model
We now have an economic model and we know how to interpret its parameters
...
The economic model is linear so we will be able to use linear regression analysis
...
1) represents an average individual
...
We might have households with the same disposable income, but
with different level of food expenditures
...
This is something that we have to deal with
...
In statistical analysis we therefore control for
the individual deviation from the regression line by adding a stochastic term (U) to (3
...
The econometric model is therefore:
Yi
B0 B1 X 1i U i
(3
...
That is explicitly
denoted by the subscript i, that appear on Y, X1 and U but not on the parameters
...
2) the
population regression equation
...
In the literature the name for the stochastic term differ from book to
book and are called error term, residual term, disturbance term etc
...
Download free ebooks at bookboon
...
2)
for all observations
...
It is quite reasonable to believe that many other variables are important determinants of the household food
expenditure, such as family size, age composition of the household, education etc
...
To be general we may say that:
Y
f ( X 1 , X 2 ,
...
Hence, having access to only one explanatory variable we may write the complete
model in the following way for a given household:
Y
B0 B1 X 1 f ( X 2 , X 3 ,
...
This way of thinking of the error term is very useful
...
It is
seldom the ambition of the researcher to include everything that accounts but just the most relevant
...
The model should be a simplistic version
of the reality
...
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35
Econometrics
The simple regression model
Sometimes it might be the case that you have received data that has been rounded off, which will make the
observations for the variable less precise
...
If these measurements errors are made
randomly over the sample, it is of minor problem
...
In chapter 7 we will discuss this issue thoroughly
...
1
...
It is therefore important to have a sound understanding of what
the assumptions are and why they are important
...
That is very important to remember! The
assumptions must hold for each observation
...
This assumption
also impose that the model is complete in the sense that all relevant variables has been included in the model
...
Furthermore, there must not be any relation between the
residual term and the X variable, which is to say that they are uncorrelated
...
Assumption 3:
V >Y @ V >U @ V 2
The variance of the error term is homoscedastic, that is the variance is constant over different observations
...
Assumption 4:
Cov(U i ,U j ) Cov(Yi , Y j ) 0
iz j
The covariance between any pairs of error terms are equal to zero
...
Assumption 5:
X need to vary in the sample
...
Furthermore, it is a mathematical necessity that X takes at least two different values in the sample
...
That means that the expected value of X
is the variable itself, and the variance of X must be zero when working with the regression model
...
This assumption is often imposed to make the mathematics easier to deal
with in introductory texts, and fortunately it has no affect on the nice properties of the OLS estimators that
will be discussed at the end of this chapter
...
Download free ebooks at bookboon
...
The assumption affects the distribution of the estimated
parameters
...
When the sample is larger then 100
the distribution of the estimated parameters converges to the normal distribution
...
Remember that when we are dealing with a sample, the error term is not observable
...
Furthermore, these assumptions need to hold true for each single observation, and hence using only one
observation to compute a mean and a variance is impossible
...
2 Estimation of population parameters
We have specified an economic model, and the corresponding population regression equation
...
For that purpose we need a sample regression equation,
expressed as this:
Yi
b0 b1 X 1i ei
(3
...
In the population regression equation the parameters are fixed
constants
...
In the sample regression equation the parameters are random variables with a
distribution
...
The error term is also an estimate and corresponds to the population
error term
...
In this text we will call the estimated error term the residual term
...
2
...
The most
common ones are the method of maximum likelihood, the method of moment and the method of Ordinary
Least Squares (OLS)
...
100,00
80,00
Y
60,00
40,00
20,00
0,00
0,00
2,00
4,00
6,00
8,00
10,00
X
Figure 3
...
com
37
Econometrics
The simple regression model
The OLS relies on the idea to select a line that represents an average relationship of the observed data
similarly to the way the economic model is expressed
...
1 we have a random sample of 10
observations
...
In mathematical terms using equation (3
...
4)
In order to be more general we assume a sample size of n observations
...
4) with respect to b0 and b1
...
We have:
wRSS
wb0
wRSS
wb1
n
2
¦ Yi b0 b1 X1i
(3
...
6)
i 1
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...
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38
Econometrics
The simple regression model
By rearranging these two equations we obtain the equation system in normal form:
nb0 b1
n
¦
n
¦ Yi
X 1i
i 1
n
b0
¦
i 1
n
X i b1
i 1
¦
n
X 12
i
i 1
¦ X1iYi
i 1
Solving for b0 and b1 gives us:
b0
Y b1 X 1
(3
...
8)
i 1
i 1
The slope coefficient b1 is simply a standardized covariance, with respect to the variation in X1
...
Remember that b0
and b1 are random variables, and hence it is important to know how their expected values and variances look
like
...
The
variance of the intercept is slightly more involved, but since text books in general avoid showing how it could
be done we will do it here, even though the slope coefficient is the estimate of primary interest
...
Since the intercept is expressed as a function of the slope coefficient we will start with the slope estimator:
n
b1
n
i 1
i 1
n
¦ X1i X1 Yi Y ¦ X1i X1 Yi
n
¦ X i1 X1
i 1
2
¦ X i1 X1
i 1
2
ª
º
«
»
n
« X 1i X 1 »Y
« n
» i
2
i 1«
X i1 X1 »
«i 1
»
¬
¼
¦
¦
n
¦WiYi
i 1
Wi
n
b1
¦WiYi
(3
...
com
39
Econometrics
The simple regression model
For the intercept we do the following:
1
n
Y b1 X 1
b0
n
¦
n
Yi X 1
i 1
n
¦
WiYi
i
1
§1
·
¨ X 1Wi ¸Yi
n
¹
1©
¦
i
n
§1
·
¦ ¨ n X1Wi ¸ B0 B1 X1i U i
©
¹
i 1
( 3
...
10)
i 1
Hence, the OLS estimators are weighted averages of the dependent variable, holding in mind that Wi is to be
treated as a constant
...
11)
i 1
º
ª
E « Wi B0 B1 X 1i U i »
»
«i 1
¼
¬
º
ª
E >b1 @ E « WiYi »
»
«i 1
¼
¬
n
n
¦
¦
º
ª
ª
º
»
« n
« n
»
º
ª n
E « B0 Wi » E « B1 Wi X 1i » E « WiU i »
« i1 »
« i1
»
»
«i 1
¼
¬
»
«
« 1 »
0 ¼
¬
¼
¬
¦
¦
¦
and
E >b1 @ B1
n
¦Wi E>U i @
i 1
B1
(3
...
You should confirm these
steps by your self
...
Also remember
that the population parameter is a constant and that the expected value of a constant is the constant itself
...
The variance of the OLS estimators
When deriving the variance for the intercept, we utilize the definition of the variance that is expressed in
terms of expectations
...
com
40
Econometrics
The simple regression model
Square the expression and take the expectation and end up with
V >b0 @
§
¨
¨1
¨
¨n
¨
©
·
¸
2
¸ 2
X1
¸V
n
2¸
X1i X1 ¸
i 1
¹
¦
n
V 2 ¦ X 12i
i 1
n
(3
...
¦ X1i X1
2
i 1
ª
º
2
« ( X 1i X 1 ) »
«i 1
»
¬
¼
n
2
an
¦
d therefore
V >b1 @
V2
(3
...
com
41
Econometrics
The simple regression model
The covariance between the two OLS estimators can be received using the covariance operator together with
expressions (3
...
10)
...
The covariance is given by the following expression:
Cov(b0 , b1 )
X 1V 2
n
¦ X1i X1
2
i 1
In order to understand all the steps made above you have to make sure you remember how the variance
operator works
...
Also remember that the variance of the
population error term is constant and the same over observations
...
Observe that the variance of the OLS estimators is a function of the variance of the error term of the model
...
This is true
for the variance of the intercept, variance of the slope coefficient and for the covariance between slope and
the intercept
...
Also note that the larger the variation in X is, the smaller become the variance of the slope
coefficient
...
Increased variation in Y has of course the opposite effect, since the variance in Y
is the same as the variance of the error term
...
We therefore need to replace it by an
estimate, using sample information
...
We start by forming the residual term
ei
Yi b0 b1 X 1i
We observe that it takes two estimates to calculate its value which implies a loss of two degrees of freedom
...
That is:
n
ˆ
V2
¦ ei2
i 1
n2
Observe that we have to divide by n-2, which referees to the degrees of freedom, which is the number of
observations reduced with the number of estimated parameters used in order to create the residual
...
3
...
2 Properties of the least squares estimator
The OLS estimator is attached to a number of good properties that is connected to the assumptions made on
the regression model which is stated by a very important theorem; the Gauss Markov theorem
...
com
42
Econometrics
The simple regression model
The Gauss Markov Theorem
When the first 5 assumptions of the simple regression model are satisfied the parameter
estimates are unbiased and have the smallest variance among other linear unbiased estimators
...
The OLS estimators will have the following properties when the assumptions of the regression function are
fulfilled:
1) The estimators are unbiased
That the estimators are unbiased means that the expected value of the parameter equals the true
population value
...
Hence, on average we would be correct but it is not very likely that we will
be exactly right for a given sample and a given set of parameters
...
It requires that the variance is homoscedastic and
that it is not autocorrelated over time
...
3) Consistency
Consistency is another important property of the OLS estimator
...
An estimator can be biased and still consistent but it is not
possible for an estimator to be unbiased and inconsistent
...
According to the central limit theorem, the distribution of means is normally distributed
...
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...
Statistical inference
Statistical inference is concerned with the issue of using a sample to say something about the corresponding
population
...
In order to find a
plausible answer to these questions we need to perform statistical test on the parameters of our statistical
model
...
In the previous chapter we saw that the estimators for the population parameters were nothing more than
weighted averages of the observe values of the dependent variable
...
Furthermore, the distribution of the dependent variable coincides with the error term
...
According to statistical theory we know that a linear combination of normally distributed variables is also
normally distributed
...
In the previous chapter we derived the expected value and the corresponding variance
for the estimators, which implies that we have all the information we need about the sampling distribution for
the two estimators
...
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44
Econometrics
Statistical inference
§
b0 ~ N ¨ B0 ,
¨
¨
©
§
¨
b1 ~ N ¨ B1 ,
¨
©
V 2 ¦ X 12
i
·
¸
2¸
X1i X1 ¸
¹
(4
...
2)
¦
V2
¦
Just as for a single variable, the OLS estimators works under the central limit theorem since they can be
treated as means (weighted averages) calculated from a sample
...
However, in regression analysis we call them
standard errors of the estimator instead of standard deviations
...
Since we use samples in our estimations, we will never
receive estimates that exactly equal the corresponding population parameter
...
The important point to recognize is that this error on average will be smaller the larger the
sample become, and converge to zero when the sample size goes to infinity
...
4
...
The testing procedure will therefore be described by an example
...
1
Assume the following population regression equation:
Y
B0 B1 X U
(4
...
233 0
...
26) (0
...
4)
The regression results in (4
...
It is now time to ask some questions: Has X any effect on Y?
In order to answer that question we would like to know if the parameter estimate for the slope coefficient is
significantly different from zero or not
...
5)
H1 : B1 z 0
In order to test this hypothesis we need to form the test function relevant for the case
...
We may therefore transform the
estimated parameter according to the null hypothesis and use that transformation as a test function
...
com
45
Econometrics
Statistical inference
Test function:
t
b1 B1
~ tnk
se(b1 )
(4
...
It takes a t-distribution
since the standard error of the estimated parameter is unknown and replaced by an estimate of the standard
error
...
Had the standard error been known, the test function would have been normal
...
If
the null hypothesis is true the mean of the test function will be zero
...
Let us calculate the test value using the test function:
Test value:
t
0
...
213
2
...
7)
The final step in the test procedure is to find the critical value that the test value will be compared with
...
Otherwise, we just
accept the null hypothesis and say that it is possible that the population parameter is equal to zero
...
In this
example we choose the significance level to be at the 5 % level
...
In this particular case we receive:
Critical value: tc
1
...
8)
Since the test value is larger than the critical value we reject the null hypothesis and claim that there is a
positive relation between X and Y
...
2 Confidence interval
An alternative approach to the test of significance approach described by the example in the previous section
is the so called confidence interval approach
...
The
idea is to create an interval estimate for the population parameter instead of working with a point estimate
...
All tests need to start from some hypothesis and we will use (4
...
By choosing a 5 % significance level and the corresponding critical values from the t-distribution
table we may form the following interval:
P( 1
...
96) 95%
(4
...
In order to form a confidence interval for our case we substitute the test function
(4
...
9)
...
com
46
Econometrics
Statistical inference
§
·
b B1
P¨ 1
...
96 ¸ 0
...
96 u se(b1 ) d B1 d b1 1
...
95
which provides a 95 % confidence interval for B1
...
Alternatively, in repeated sampling the interval will cover the
population parameter in 95 cases out of 100 on average
...
By plugging in the values we receive a confidence interval that may be
expressed in the following way:
b1 r t c u se(b1 )
which in this case equals
0
...
96 u 0
...
And that’s
just what we are looking for
...
All rights reserved
...
One should therefore not forget that it is the interval that with a
certain probability will cover the true population parameter, and not the other way around
...
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ubs
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47
Econometrics
Statistical inference
Two important concepts to remember and distinguish in these circumstances are the confidence level and
significance level
...
When it is expressed as a percent, it is sometimes called the confidence coefficient
...
Hence, before being able to construct a confidence interval we have to pick a significance level, which is
usually set to 5 percent
...
The significance level is often denoted with the Greek letter Į,
which implies that the confidence level equals 1-Į
...
2
...
To investigate the p-value is a fast way to reach the conclusion that we otherwise would receive
by carrying out all the steps in the test of significance approach or the confidence interval approach
...
The P-value for sample outcome
The P-value for a sample outcome is the probability that the sample outcome could have been
more extreme than the observed one
...
If the P-value is less than the specified significance level: H1 is concluded
...
1 Regression output from Excel
Standard
Coefficients
Error
Intercept
9
...
0973966
X
7
...
4661782
t Stat
1
...
8569893
P-value
0
...
001260
Lower 95%
-11
...
7401968
Upper 95%
30
...
502227
In Table 4
...
This particular output is generated
using MS Excel, but most statistical software offers this information in their output
...
Download free ebooks at bookboon
...
That has implications on the P-value
...
0259
...
0259
...
0259
...
3349
...
0259
...
3349
...
1675
Since the P-value for the intercept is larger than any conventional significance levels, say 5 percent, we can
not reject the null hypothesis that the intercept is different from zero
...
4
...
We could accept a false null hypothesis and we could reject a correct null hypothesis
...
For a given sample size this is a difficult task, since any attempt to minimize one of them results in
increasing the other kind
...
The two types of errors mentioned above are referred to as the type I error and the type II error
...
P Reject H 0 | H 0 is true D
The type II error
The probability to accept a false null hypothesis
P Accept H 0 | H 0 is false E
An additional concept related to the two types of errors is the so called power of the test
...
It is always the case that we would like the power of
the test to be as large as possible
...
But for
a given sample we must remember that the smaller we choose the significance level, the larger become the
type II error and the smaller become the power of the test
...
com
49
Econometrics
Statistical inference
The power of a test
The probability to reject a false null hypothesis
P Reject H 0 | H 0 is false 1 P Accept H 0 | H 0 is false 1 E
Table 4
...
2
To better understand the mechanism of the two types of error we consider an example were we calculate the
probabilities for each cell given in Table 4
...
Let us use the regression results from Example 4
...
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D
Econometrics
Statistical inference
In that example we had a significance level of 5 percent
...
Given the significance level we can
calculate the interval that will be used as a decision rule for our test
...
In Example 4
...
Together with the
estimated standard error we know the distribution of the estimated parameter when the null hypothesis is
correct
...
For this particular case we have the following interval:
> 1
...
96 u se(b1 )@ > 0
...
41748@
(4
...
The interval can therefore be seen as a decision rule
...
Since we decided about the significance level we know the probability of the
type I error, since they always coincide
...
In order to be able to calculate the type II error we need to know the true value, that is, we need to know the
population value of the parameter
...
In this example we will assume that we
know the value and that it equals 0
...
Given this value we have a new distribution for our estimator that we
will use when calculating the probability of a type II error, namely:
(4
...
25,V (b1 )
This is the distribution related to the alternative hypothesis
...
10)
...
11):
P( 0
...
41748)
E
In order to calculate this probability we have to use the t-value transformation and use the table for the tdistribution to find the probability
...
41748 0
...
41748 0
...
213
0
...
783 0
...
7821
P 3
...
786 P(t d 0
...
134)
Hence, with this setup there is a 78 percent chance of committing a type II error
...
If that is not possible you are stuck with a
problem
...
com
51
Econometrics
Statistical inference
by (4
...
When that happens, a larger portion of the true distribution will be covered, and
hence increase the type II error
...
In this case
the power of the test would be (1 E ) =1-0
...
2179
...
4
...
Let us start with a definition:
Prediction and Forecasting
To make a statement about an event before the event occurs
...
The words prediction and forecasting are going to be used interchangeably
...
Since the literature does not show any consensus on this part
we will treat them synonymously in this text
...
, T
and we would like to make predictions about the future, that is we would like to know the value of Y in period
T+1
...
Whether we have exact information about X or not will affect the variance for the predicted value
...
The exact value of the population parameters is never an issue, and it is therefore obvious that
they have to be estimated
...
Since the sample estimators are
the same for all t, it is the value of Xt that generates the forecast for Yt
...
com
52
Econometrics
Statistical inference
This is often called a point prediction
...
The forecast error will help us say something about
how good the prediction is
...
Assuming that X is
known, the variance of the forecast error is given by:
>
ˆ
E YT 1 YT 1
@
2
E > B0 b0 B1 b1 X T 1 U T 1 @2
E >B0 b0 @2 E > B1 b1 X T 1 @2 E >U T 1 @2 E >2 B0 b0 B1 b1 X T 1 @
2
V b0 V b1 X T 1 V U 2Cov b0 , b1 X T 1
assuming that X is constant in repeated sampling
...
12)
Observe that the forecast error variance is smallest when the future value of X equals the mean value of X
...
That is often not the case and hence the formula has to
be elaborated accordingly
...
That is, assume that
*
X T 1
2
X T 1 H T 1 , H ~ N 0, V H
With this assumption we may form an expression for the error variance that takes the extra variation from the
uncertainty into account:
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...
13)
The important point to notice here is that this variance is impossible to estimate unless we know the exact
value of the variance for the uncertainty
...
Furthermore, the expression involves
the population parameter multiplied with the variance of the uncertainty
...
12) is often use,
but one should hold in mind that it most likely is an understatement of the true forecast error variance
...
12) or (4
...
With this
standard error it is possible to calculate confidence interval around the predicted values using the usual
formula for a confidence interval, that is:
Confidence interval of a forecast
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Model measures
In the previous chapters we have developed the basics of the simple regression model, describing how to
estimate the population parameters using sample information and how to perform inference on the population
...
The two most popular measures for model
fit are the so called coefficient of determination and the adjusted coefficient of determination
...
1 The coefficient of determination (R2)
In the simple regression model we explain the variation of one variable with help of another
...
Had they not been correlated there would be no explanatory power in our X
variable
...
Furthermore, the correlation coefficient can only be used
between pairs of variables, while the coefficient of determination can connect a group of variable with the
dependent variable
...
But the attempt of this chapter is to put the correlation coefficient in a context of the regression
model and show under what conditions it is appropriate to interpret the correlation coefficient as a measure of
strength of a causal relationship
...
It is therefore natural to start the derivation of the measure from the deviation
from mean expression and then introduce the predicted value that comes from the regression model
...
1)
Unexplained
We have to remember that we try to explain the deviation from the mean value of Y, using the regression
ˆ
model
...
The remaining part will therefore be denoted the
unexplained part
...
We must now transform (5
...
We do that by squaring and summing over all n observations:
¦ Yi Y ¦
n
i 1
2
n
ˆ
ˆ
Yi Y Yi Yi
i 1
2
¦ ª Yˆi Y 2 Yi Yˆi 2 2 Yˆi Y Yi Yˆi º
«
»
¬
¼
i 1
n
It is possible to show that the sum of the last expression on the right hand side equals zero
...
com
55
Econometrics
Model measures
2
2
2
¦ Yi Y ¦ Yˆi Y ¦ Yi Yˆi
n
n
i 1
TSS
n
i 1
ESS
i 1
(5
...
On the left hand side we have the total
sum of squares (TSS) which represents the total variation of the model
...
Caution: be careful when using different text books
...
The identity we have found may now be expressed as:
TSS ESS RSS
which may be rewritten in the following way:
TSS
TSS
ESS RSS
TSS TSS
1
Hence, by dividing by the total variation on both sides we may express the explained and unexplained
variation as shares of the total variation, and since the right hand side sum to one, the two shares can be
expressed in percentage form
...
com
56
Econometrics
Model measures
Example 5
...
65
...
In the simple regression model there is a nice relationship among the measures of sample correlation
coefficient, the OLS estimator of the slope coefficient, and the coefficient of determination
...
3)
i 1
where S X and SY represents the sample standard deviation for X and Y respectively
...
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4)
where S XY represents the sample covariance between X and Y, and r the sample correlation coefficient for X
and Y
...
4) into (5
...
R
2
§ SX
¨ b1
¨ S
Y
©
·
¸
¸
¹
2
§ S X SY ·
¨r u
¸
u
¨
SY S X ¸
©
¹
2
r 2
(5
...
6)
This means that the smaller the correlation between X and Y, the smaller is the explained share of the
variation by the model, which is the same as to say that the larger is the unexplained share of the variation
...
This leads to an important conclusion about the importance of the coefficient of
determination:
R2 and the significance of the OLS estimators
An increased variation in Y, with an unchanged variation in X, will directly reduce the size of the
coefficient of determination
...
From (5
...
6) it is clear that an increased variation in Y will reduce the size of the coefficient of
determination of the regression model
...
It is therefore not obvious that the
significance of the parameter will be unchanged
...
The expression for the standard error of the OLS
estimator was derived in the previous chapter
...
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...
We should
therefore draw the conclusion that the coefficient of determination is just a measure of linear strength of the
model and nothing else
...
5
...
But its
size is dependent on the degrees of freedom
...
A solution to this problem is to control for the degrees of
freedom and adjust the coefficient of determination accordingly
...
7)
where R 2 denotes the adjusted coefficient of determination
...
Using equation (5
...
Rearranging (5
...
Another interesting
feature of the adjusted R2 is that it can be negative, an event impossible for the unadjusted R2
...
06
...
If you have Y in one model and ln(Y) in another, the
dependent variable is transformed and should not be treated as the same
...
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...
3 The analysis of variance table (ANOVA)
Almost all econometric software generates an ANOVA table together with the regression results
...
1 The ANOVA Table
Source of Variation
Explained
Unexplained
Total
Degrees of freedom
1
n-2
n-1
Sum of Squares
ESS
RSS
TSS
Mean Squares
ESS/1
RSS/(n-2)F=MSE=S2
The decomposition of the sample variation in Y can be used as an alternative approach of performing test
within the regression model
...
In the
multiple regression case we have an even more important use, which will be described in chapter 6, and is
related to simultaneous test on sub sets of parameters
...
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Hence
if the explained part increases sufficiently by including X, we would be able to say that the alternative
hypothesis is true
...
8)
In the numerator of equation (5
...
Since this is a simple
regression model the explained part goes from zero since no other variables are included and therefore the
degrees of freedom equals one
...
In the
denominator we have the variance of the residual
...
That is:
F
ESS / 1
~ F(1,n 2)
RSS /(n 2)
(5
...
Example 5
...
In order to answer this question we form a simple regression
model, and form the following hypothesis: H 0 : B1 0 vs
...
Use the following information to
perform the test:
ESS
51190, RSS
5232
In order to carry out the test, we form the test function and calculate the corresponding test value
...
9)
we receive:
51190 / 1
F
1399
...
025 (1,143) 5
...
Hence we can reject the null hypothesis and conclude that X has
a significant effect on Y
...
In the simple regression model case it involves just one single parameter, but in the multiple
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...
We will speak more about this in the next chapter
...
But
how are these two test functions connected
...
Hence, the outcomes of the two procedures are always consistent
...
Every single day
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...
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The multiple regression model
From now on the discussion will concern multiple regression analysis
...
That has consequences on the interpretation of the estimated
parameters, and violations to this condition will have consequences that will be discussed in chapter 7
...
6
...
The population regression function would now be expressed in the following way:
Y
B0 B1 X 1 B2 X 2 U
(6
...
Hence coefficient B1 represents the unique effect that comes from X1, controlling for X2, which
means that, any common variation between X1 and X2 will be excluded
...
Example 6
...
You set up the following regression model:
P
B0 B1 A B2 K U
(6
...
3)
B2
(6
...
3) represents the effect from a unit change in the age of the car on the conditional
expected value of sales prices
...
It is reasonable to believe that the age of the car is
correlated with the number of kilometers the car has gone
...
That common variation is excluded from
the estimated coefficients
...
Accordingly, the second marginal effect (6
...
The way the model is specified here, imply that the unique
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...
If this is implausible, one could
adjust for it
...
The extended model would then be:
P
B0 B1 A B2 A2 B3 K B4 K 2 B5 A u K U
(6
...
6)
wP
wK
B3 2 B4 K B5 A
(6
...
6) is the marginal effect on sales price from a unit increase in age
...
In order to receive a specific vale for the marginal effect we
need to specify values for A and K
...
The marginal effects given by (6
...
7) consist of three
parameter estimates, which individually can be interpreted
...
6) the first parameter estimate is B1
...
Strictly speaking it represents the marginal effect, when A and K both are zero, which would
be when the car was new
...
To include a squared
term is therefore a way to test if the relation is non-linear
...
Failure
to control for it, would lead to a biased marginal effect since it would be assumed to be constant, when it in
fact vary with the level of A
...
It is not obvious that such effect would exist in the Volvo S40 example
...
For instance in the US wage equation literature: being black and being
a woman are usually two factors that have negative effects on the wage rate
...
This would be an example of a negative
synergy effect
...
2 Estimation of partial regression coefficients
The mathematics behind the estimation of the OLS estimators in the multiple regression case is very similar
to the simple model, and the idea is the same
...
The sample estimators for model (6
...
com
64
Econometrics
The multiple regression model
b0
b1
b2
Y b1 X 1 b2 X 2
(6
...
9)
2
2
S1 (1 r12 )
SY 2 r12 rY 1SY S 2
(6
...
Observe the similarity between the sample estimators of the multiple-regression
model and the simple regression model
...
The two partial regression slope coefficients are
slightly more involved but possess an interesting property
...
9) we have that
SY 1 r12 rY 2 SY S1
SY 1
2
2
S1 (1 r12 )
S12
if r12
0
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...
However, if the correlation between X1
and X2 equals one (or minus one), the estimators are not defined, since that would lead to a division by zero,
which is meaningless
...
Equation (6
...
10) can be generalized further to include more
parameters
...
The measure of fit in the multiple regression case follows the same definition as for the simple regression
model, with the exception that the coefficient of determination no longer is the square of the simple
correlation coefficient, but instead something that is called the multiple-correlation coefficient
...
that is used to explain the variability of
the dependent variable Y
...
The square root of the coefficient of multiple determination is the coefficient of
multiple correlation, R, sometimes just called the multiple R
...
In practice this
statistics has very little importance, even though it is reported in output generated by softwares such as Excel
...
3 The joint hypothesis test
An important application of the multiple regression analysis is the possibility to test several parameters
simultaneously
...
11)
Using this model we may test the following hypothesis:
vs
...
H1: H0 not true
c) H 0 : B1
B2
B3
0 vs
...
We will therefore not go through these steps again but instead focus on the
simultaneous tests given by hypothesis b and c
...
3
...
In this example we choose to test B1 and B2 but it
could of course be any other combination of pairs of coefficients included in the model
...
com
66
Econometrics
The multiple regression model
H 0 : B1
B2
0
H1 : B1 z 0 and / or B2 z 0
It is often believed that in order to reject the null hypothesis, both (all) coefficients need to be different from
zero
...
It is important to understand that the complement of the null hypothesis in this
situation is represented by the case where at least one of the coefficients is different from zero
...
An F-test is based on a test statistic that follows the F-distribution
...
So, we are basically testing two specifications against each
other, which are given by:
Model according to the null hypothesis:
Y
B0 B3 X 3 U
(6
...
13)
A way to compare these two models is to see how different their RSS (Residual Sum of Squares) are from
each other
...
When looking at
specification (6
...
13) since two
of the parameters are forced to zero
...
13) on the other hand, the two parameters are free to take any value
the data allows them to take
...
12) and an Unrestricted RSS RSSU received from (6
...
In practices this means that you have to run
each model separately using the same data set and collect RSS-values from each regression and then calculate
the test value
...
14)
(D , df1 , df 2 )
RSSU / df 2
where df1 and df 2 refers to the degrees of freedom for the numerator and denominator respectively
...
Hence, df1 = (n-k1) – (n-k2) = k2-k1
...
In this case we have that k2-k1=2
...
However, if the fit differ extensively, the F-value will be
large
...
14) has a know distribution (if the null hypothesis is true) we will be
able to say when the difference is sufficiently large to say that the null hypothesis should be rejected
...
2
Consider the two specifications given by (6
...
13), and assume that we have a sample of 1000
observations
...
Running the
two specifications on our sample we received the following information given in Table 6
...
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...
1 Summary results from the two regressions
The Restricted Model
k1 = 2
RSS = 17632
The Unrestricted Model
k2 = 4
RSS = 9324
Using the information in Table 6
...
F
RSS R RSSU / df1 17632 9324 /(4 2)
RSSU / df 2
9324 /(1000 4)
4154
9
...
73
The calculated test value has to be compared with a critical value
...
We choose the standard level of 5 percentage and find the following value in the
table: FC 4
...
Observe that the hypothesis that we are dealing with here is one sided since the restricted RSS never can be
lower than the unrestricted RSS
...
That is, the parameters involved in the test
have a simultaneous effect on the dependent variable
...
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3
...
Alternatively, we ask if the
population coefficients (excluding the intercept) are simultaneously equal to zero, or at least one of them are
different from zero
...
15)
Model according to the alternative hypothesis:
Y
B0 B1 X 1 B2 X 2 B3 X 3 U
(6
...
To see this we can rewrite the RSS R in the following way:
2
2
¦ Yi Yˆi ¦ Yi b0 2 ¦ Yi Y
i 1
i 1
i 1
n
RSS R
n
n
TSSU
Hence the test function can be expressed in sums of squares that could be found in the ANOVA table of the
unrestricted model
...
3
Assume that we have access to a sample of 1000 observations and that we would like to estimate the
parameters in (6
...
Running the regression using our sample
we received the following ANOVA table:
Table 6
...
33
1
...
2 we can calculate the test value:
F
ESS /( k 1)
RSS /(n k )
1394
...
424
979
...
We can therefore conclude that the included parameters explains a significant
part of the variation of the dependent variable
...
com
69
Econometrics
Specification
7
...
Sometimes the underlying theory of the model gives us guidelines on how it should be specified, but in
other cases we have to rely on statistical tests
...
7
...
When formulating the model we need to know how the coefficients are to be
interpreted, and how the marginal effect and elasticity looks like
...
7
...
1 The linear specification
When talking about a linear specification we have to remember that all models that we are talking about in
this text are linear in their parameters
...
The linear specification is appropriate when Y and X has a linear relation
...
1)
For simplicity reasons we express the model as the simple regression model
...
2)
This means that when X increases by 1 unit, Y will change by B1 units
...
If Y represents the yearly disposable income
expressed in thousands of Euro and X represents age given in years, we have to understand that a unit change
in X represents a year and the corresponding effect in the yearly disposable income is in thousands of Euros
...
That is, the effect of a unit change in X may be different whether the level of X is 10 or if it
is 10,000
...
3)
which usually is expressed using mean values of X and Y
...
A 1 percent increase in X will result in e percent
change in Y
...
com
70
Econometrics
Specification
Example 7
...
1 received from a
sample of data using model (7
...
Table 7
...
066
4
...
557
0
...
5
48
...
In this example we receive
dY
dX
8
...
6 units
...
6 u
5
...
983
48
...
98 percent
...
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1
...
4)
Several factors could motivate this specification
...
This could be motivated in the following way: assume that the rate of return
to an extra year of education is denoted by r
...
After s years of schooling we would have an earnings
(1 r ) s w0
...
5)
which is a log-linear relationship between years of schooling and earnings
...
By including an error term we form a statistical model in the
form of (7
...
How do we interpret the slope parameter? It is important to remember that we are primarily
intereted in the effect on earnings not the logarithm of the earnings
...
Taking the derivative of
earnings with respect to schooling gives us the following expression:
d ln(ws )
ds
1 dws
ws ds
dws / ws
ds
B1
(7
...
6) shows us that the slope coefficient should be interpreted as the relative change in earnings as
a ratio of the absolute change in schoolings
...
Using (7
...
7)
Hence, the marginal effect is an increasing function of earnings itself
...
Hence the response on the dependent variable will change in
terms of unit, but is constant in relative terms
...
7) we can derive the earnings elasticity with respect to years of schooling:
e
dws s
ds ws
B1ws
s
ws
B1 u s
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...
Hence the longer you
have studied the larger is the earnings elasticity
...
1
...
8)
Taking the derivative of Y with respect to X we receive:
dY
dX
1
B1
X
dY
dX / X
<=>
B1
(7
...
Using the expression for the coefficient we may write the elasticity as follows:
e
dY X
dX Y
1
X
B1
X
Y
1
B1
Y
Hence the elasticity is a function of the dependent variable, and the larger the dependent variable is the
smaller become the elasticity, everything else equal
...
1
...
The so called Cobb-Douglass functions are often used as production functions in economic
theories
...
10)
(7
...
This model is multiplicative and non linear in nature which makes it difficult to use
...
Doing that and
adding an error term we receive:
ln Q ln A B1 ln L B2 ln K U
B0 B1 ln L B2 ln K U
(7
...
11)
...
How do we interpret these parameters? Let us focus on B1 when
answering this question
...
com
73
Econometrics
Specification
w ln Q
w ln L
wQ / Q
wL / L
wQ L
wL Q
B1
(7
...
So the elasticity and the
coeffieints coincide
...
The marginal effect of the log-log model can be received from (7
...
13)
which is a function of both Q and L
...
7
...
We should also remember that the first assumption related
to the regression model concerns the fact that all that is relevant should be included in the model
...
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Unfortunately it has
several meaning and we usually make the distinction between statistical and economic relevance
...
That is, if we are able to
reject the null hypothesis
...
The economic relevance is related to the underlying theory that the model is based on
...
That some of the variables are not
significantly different from zero is not a criterion for exclusion
...
To see this consider the following two specifications:
The correct economic model:
(7
...
15)
From chapter 3 we know that the sample estimator for the slope coefficient in the simple regression model is
given by:
n
n
i 1
n
n
i 1
¦ X1i X1 Yi Y ¦ X1i X1 Yi ¦ X 1i X 1 B0 B1 X1i B2 X 2i U i
b1
i 1
n
¦ X1i X1
n
2
2
i 1
i 1
i 1
(7
...
17)
2
i 1
Simplify and take the expectation of the estimator:
n
E >b1 @ B1 B2
¦ X1i X1 X 2i
i 1
n
¦ X 1i X1
2
B1 B2
Cov( X 1 , X 2 )
Var ( X 1 )
(7
...
The expected value of the estimator is a function of the true
population parameter of B1 and the true population parameter B2 times a weight that persist even if the
number of observations goes to infinity
...
However, if the excluded variable is
statistically independent of the included variable, that is if the covariance between X1 and X2 is zero,
exclusion will not be a problem, since the second component of (7
...
com
75
Econometrics
Specification
be unbiased
...
A common example of this kind of bias appears in the human capital literature when they try to estimate the
return to education on earnings without including a variable for scholastic ability
...
Since scholastic ability is believed to be positively correlated with
schooling as well as with the earnings, the rates of returns to education are usually overestimated, due to the
second component in (7
...
7
...
The researcher might be keen on avoiding the problem of excluding any relevant variables, and
therefore include variables on the basis of their statistical relevance
...
The important
question to ask is what those consequences are
...
19)
The estimated model:
Y
b0 b1 X 1 b2 X 2 e
(7
...
20) includes two variables, and X2 is assumed to be economically irrelevant, which
means that its coefficient is of minor interest
...
21)
2
Substitute the (7
...
However, the standard error of the estimator is larger when
including extra irrelevant variables, compared to the model where only the relevant variables are included,
since more variation is added to the model
...
com
76
Econometrics
Specification
efficiency and the estimator is no longer BLUE
...
Therefore, when one is unsure about a model specification, one is better
off including too many variables, than too few
...
7
...
That is seldom the case and therefore it is important to understand the consequences it has on the
OLS estimator
...
In order to analyze the consequences of the first case we have to assume a structure of the error
...
22)
where Y* represent the observed variable, Y the true, and İ the random measurement error that is independent
2
of Y, with a mean equal to zero and a fixed variance V H
...
22) with Y:
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77
Econometrics
Specification
Y
B0 B1 X U
Y * H
B0 B1 X U
Y * B0 B1 X (U H )
B0 B1 X U *
(7
...
That is, we have
Cov X ,U * Cov X , H U Cov X Cov X 0
H
,
,U
0
0
2
2
However, the new error would have a variance that are larger than otherwise, that is, V H U V U V H
...
Hence the two variances only add to a larger total variance, which affects the
standard errors of the estimates as well
...
In the second case the measurement error is attached to the independent variable, still under the assumption
that the error is random
...
24)
2
with an error component that is independent of X, has a mean zero and a fixed variance, V H
...
The model we would like to study is defined as
Y
B0 B1 X U
but we only observe X*, which implies that the model become
Y
B0 B1 ( X * H ) U
Y
B0 B1 X * (U B1H )
B0 B1 X * U *
(7
...
That is, V (U *) V U B1H V U B1 V H
...
The measurement error creates a correlation that
is different from zero, that bias the OLS estimators
...
The only way to
void this problem is to force the variance of the measurement error to zero
...
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...
This case add nothing new to the discussion since the effect will be the same as when just the
explanatory variables contains measurement errors
...
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...
Dummy variables
Until now all variables have been assumed to be quantitative in nature, which is to say that they have been
continuous
...
These qualitative measures have to
be transformed into some proxy so that it could be represented and used in a regression
...
They are artificial variables that work as proxies for
qualitative variables and since they are discrete we need to be careful when working with then and how to
interpret them
...
Gender is a typical example of a qualitative variable that need to be transformed into a numerical form so that
it could be used in a regression
...
We therefore need to decide what category the dummy should represent and what category that
should be used as a reference
...
It is therefore important to be sure
that all other observations really represent what you want it to represent
...
1)
When running the regression you can treat the dummy variable D as any other variables included in the
model
...
However, the interpretation is easiest when using 1 and 0, which is the
reason why you should follow the structure of (8
...
8
...
Using the
categorical variable defined by (8
...
Y
B0 B1D B2 X U
(8
...
1) D takes only two values
...
3)
(8
...
When
D=1 we see that the conditional expectation in (8
...
However, when D=0, the conditional expectation will be given by (8
...
Hence, the model as a whole contains two intercepts B0 and B0+B1
...
com
80
Econometrics
Dummy variables
If we take the difference between the two conditional expectations we receive:
E >Y | D 1, X @ E >Y | D 0, X @ B1
(8
...
Since our binary variable D is discrete, we can not take
the derivative of Y with respect to D, since a derivative requires a continues variable, and therefore is
undefined here
...
5) and conclude that when D moves from 0 to 1, the conditional expectation of Y change by B1
units, which represents the marginal effect for the linear model
...
But with other functional forms it makes a difference
...
1
Assume the following regression result from a model given by (8
...
The dependent variable is expressed in Swedish
kronor (SEK)
...
9 21
...
4 X
8
...
30 0
...
First we
have to check if the coefficient for the male dummy is significant
...
The marginal effect measured with
this regression says that men earn 21
...
In the empirical human capital literature the functional form most often used is the log-linear, which means
that our model would look like this:
ln Y
B0 B1D B2 X U
(8
...
Therefore, the first step must be to transform the regression equation using the anti log and form the
conditional expectation of Y
...
7)
2
where V U represents the population variance of the error term
...
7) we have to assume that U is a normally distributed variable, with mean zero, and
> @
2
variance equal V U
...
2
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...
Since it
is not continuous, we have to form the relative change in Y using the conditional expectation given by (8
...
Doing that we receive:
Marginal effect:
E >Y | D 1, X @ E >Y | D 0, X @
E >Y | D 0, X @
e B0 B1 B2 X V U / 2 e B0 B2 X V U / 2
2
2
e B B X V
0
2
2
U
/2
e B1 1
(8
...
In order to find the corresponding standard error of the
marginal effect we simply apply a linear approximation to the non linear expression
...
9)
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82
Econometrics
Dummy variables
Example 8
...
6), with Y being the hourly wage rate, D a
dummy for men, and X a variable for years of schooling
...
Standard errors are given within parenthesis:
ˆ
ln Y
4
...
18 D 0
...
03 0
...
01
Since we are interested in the marginal effect of D on Y, we have to calculate it using the regression results
...
8) and (8
...
18 1 0
...
18 u 0
...
024
The t-value for the marginal effect equals 8
...
This implies a positive and significant marginal effect of 19
...
That is, men earns on
average 19
...
Observe that the estimated value is very close to the calculated relative change given by (8
...
It turns out that
when the estimated coefficient is lower than 0
...
8), and is therefore often used directly as such
...
3
therefore researcher often use b1 directly instead of the calculated value given by (8
...
8
...
Sometimes it
is reasonable to believe that the shift should take place in the slope coefficient instead of the intercept
...
This would mean that men and women have
slope coefficients that are different in size
...
10)
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...
Hence, a way to test if the
return to education differs between men and women would be to test if B2 is different from zero, which
should be tested before going on to test if B1+B2 is different from zero
...
Since D is binary, DX is only
active when D=1, and the corresponding effect is therefore related to the category specified by D=1
...
3
Use the same data set as in Example 8
...
10)
...
11 0
...
014 DX
0
...
003 0
...
11)
Use the regression results to investigate if there is a difference in the return to education between men and
women
...
Doing that,
we receive a t-value of 10
...
Hence, if
this specification is correct, we can conclude that the returns to education differ between men and women
...
2
...
If it is the case that D in itself has a positive
effect on the dependent variable, that unique effect will be part of the cross effect otherwise
...
12)
When we include the two variables, X and D separately and together with their product we allow for changes
in both the intercept and the slope
...
10), but not otherwise
...
4
Extend the specification of (8
...
That is, estimate the parameters of the model
given by (8
...
Doing that, we received the following results, with standard errors
within parenthesis
...
006 0
...
210 D 0
...
12)
0
...
004) 0
...
005
By investigating the t-values we see that b1 and b2 are statistically significant from zero
...
Since D alone has a significant effect on the dependent variable,
there is little effect left from the cross product, and hence we conclude that there is no difference in the return
to education between men and women
...
com
84
Econometrics
Dummy variables
Example 8
...
4 should convince you that it is very important to include the variables that appear in a
cross product separately, since they might stand for the main effect
...
3 we did not include D
even though it was relevant
...
In this case it made us to draw the wrong conclusion about the return to
education for men and women
...
3 Qualitative variables with several categories
The human capital model described above includes a continuous variable for the number of years of
schooling
...
An alternative approach would be to argue that it is the level of
schooling, the received diploma, that matters in the determination of the wage rate
...
For instance:
D
0 Primary schooling
°
®1 Secondary schooling
°2 Post secondary schooling
¯
(8
...
If that is not the case we have to allow for differences in
these two effects
...
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85
Econometrics
Dummy variables
The first and most basic approach is to create three binary variables; one for each educational level, in the
following way:
D1
1 Primary schooling
D2
®
¯0 otherwise
1 Secondary schooling
D3
®
¯0 otherwise
1 Post secondary schooling
®
¯0 otherwise
We can now treat D1, D2 and D3 as three explanatory variables, and include them in the regression model
...
The dummy variable trap appears
when the analyst tries to specify and estimate the following model:
ln Y
B0 B1D1 B2 D2 B3 D3 B4 X U
(8
...
14) since there is no variation in the sum of
the three dummy variables, since D1+D2+D3=1 for all observations in the data set
...
The easiest way
to solve this is to exclude one of them and treat the excluded category as a reference category
...
15)
That is, if D1 is excluded, the other categories will have D1 as reference
...
In order to determine the relative effects you may use the transformation described by (8
...
An alternative to exclude one of the categories is to exclude the constant term, which would give us a model
that looks like this:
ln Y
C1D1 C2 D2 C3 D3 B4 X U
(8
...
The coefficients can therefore not be interpreted as relative changes in this case
...
5
Estimate the parameters of (8
...
16) and compare and interpret the results
...
15)
Specification II: (8
...
929 0
...
295 D3 0
...
17)
3
...
083D2 4
...
009 X
(8
...
043 0
...
043
0
...
001
0
...
036 0
...
com
86
Econometrics
Dummy variables
The three dummy variables represent three educational levels, and X represents the age of the individual
...
Hence, the two specifications are very much
related
...
With help from specification II, we can derive the effect of
going from a high school diploma to a college diploma by taking the difference between C3 and C2 which
turns out to be equal to 0
...
e
...
However, that effect could also have been received by
taking the difference between B3 and B2
...
8
...
In Figure 8
...
A typical example of such a relationship would be related to the income tax,
which often is progressive, that is, the more you earn the larger share of your income should be paid in tax
...
19)
In order to transform (8
...
We define:
D1
1 Gross income is in the interval X 1 d X d X 2
®
¯0 otherwise
D2
1 Gross income is greater than X 2
®
¯0 otherwise
Y
X1
X2
Figure 8
...
com
87
Econometrics
Dummy variables
Next re-specify the intercept and the slope coefficient in (8
...
20)
B
B0 B1D1 B2 D2
(8
...
20) and (8
...
19) and receive:
Y
( A0 A1D1 A2 D2 ) ( B0 B1D1 B2 D2 ) X U
After multiply out the parenthesis we receive the following specification that could be used in estimation:
Y A0 A1D1 A2 D2 B0 X B1 ( D1 X ) B2 ( D2 X ) U
(8
...
com
88
Econometrics
Dummy variables
8
...
For instance, assume that we have the
following wage equation expressed with a semi logarithmic (log-linear) functional form:
ln Y
B0 B1 X 1 B2 X 2 U
(8
...
We would like to know if B1 and B2 differ between men (m) and women (w) simultaneously
...
24)
Equation (8
...
23) will be representing the restricted case where men and women have
the same coefficients
...
25)
If the RSSR is very different from RSSU we will reject the null hypothesis in favor of the alternative hypothesis
...
Example 8
...
23) differ between men and women in the
population
...
23) and one unrestricted model given by (8
...
Using a sample of 1483 randomly selected
individuals we received the following results:
Table 8
...
603
1483 – 3 = 1480
Unrestricted Model
140
...
1 we may calculate the test value using (8
...
The degrees of freedom for
the numerator is calculated as the difference between degrees of freedom for the RSS from the restricted and
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...
That is 1480 – 1477 = 3
...
The unrestricted model has 6 parameters, while the restricted model has only 3, which means
that three parameters have been set to zero in the restricted model
...
The test value using the test statistic is therefore equal to:
F
( RSS R RSSU ) / df1
RSSU / df 2
145
...
265 / 3
140
...
7793
18
...
095
The test value has to be compared with a critical value
...
6
...
We can therefore conclude that the coefficient of the regression model differ for men and
women
...
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90
Econometrics
Heteroskedasticity and diagnostics
9
...
This is referred to as a homoskedastic error term
...
This chapter will discuss the consequences of violating the homoskedasticity assumption,
how to detect any deviations from the assumption and how to solve the problem when present
...
1 Consequences of using OLS
The classical assumptions made on the error terms are that they are uncorrelated, with mean zero and
2
constant variance V U
...
1)
2
V >U i @ V U
(9
...
3)
0
Assumptions (9
...
3) are in use to make the OLS estimators unbiased and consistent
...
2)
is important for the OLS estimator to be efficient
...
2) is ignored we can no longer claim that our
estimator is the best estimator among linear unbiased estimators
...
Heteroskedasticity implies that
x
The OLS estimators of the population parameters are still unbiased and consistent
...
It is important to understand that the violation of (9
...
Therefore tests of hypothesis are no longer valid, since
the standard errors are wrong
...
4)
¦
The expression given by (9
...
Unfortunately it involves
the unknown population variance of the error term which is different for different observations
...
com
91
Econometrics
Heteroskedasticity and diagnostics
Since the error term is heteroskedastic, each observation will have a different error variance
...
5)
i 1
As can be seen in (9
...
An important use of the regression equation is that of making predictions and forecasts of the future
...
However, since the estimators are
inefficient, the uncertainty of the forecasts will increase, and the confidence interval of the forecast will be
biased and inconsistent
...
2 Detecting heteroskedasticity
Since we know that heteroskedasticity invalidate test results it is very important to investigate whether our
empirical model is homoskedastic
...
Below the most commonly used test will be
discussed
...
2
...
Since we
are interested in the behavior of the error term and its variation, two obvious scatter plots are given in Figure
9
...
In Figure 9
...
Here we can see a clear pattern of heteroskedastidity which is driven by the
explanatory variable
...
5,00
3,00
4,00
2,00
3,00
e1
Y
1,00
2,00
0,00
1,00
-1,00
0,00
-2,00
-1,00
0,000
0,200
0,400
0,600
0,800
0,000
1,000
0,200
0,400
0,600
0,800
1,000
X
X
a) Scatter plot of Y against X
b) Scatter plot of e against X
Figure 9
...
com
92
Econometrics
Heteroskedasticity and diagnostics
As an alternative to Figure 9
...
1b
...
However, when using a multiple regression models the picture might be different, since the residual
is a linear combination of all variables included in the model
...
If it is possible to find a systematic pattern that give indications of differences of the variances
over the observations, one should be concerned
...
It is therefore necessary to use statistical test
...
Example 9
...
6)
You’re full of energy
and ideas
...
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...
ln Y
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Both explanatory variables are also squared to control
for any non linear relation between the dependent variable and the two explanatory variables
...
593 0
...
001ED 2 0
...
000 year 2
40
...
2
2
...
8
2
...
7)
ˆ
ln Y should be interpreted as the predicted value of lnY
...
The squared terms have very small coefficients, even though their t-values are
sufficiently large to make them significant
...
Since the value is very small and we only report the first three decimals it appears to be zero
...
We suspect that our residual might be heteroskedastic and we would like to investigate this by looking at a
graph between the residual and the two explanatory variables
...
If we do that we receive the
graphs given in Figure 9
...
14,00
12,00
12,00
10,00
10,00
8,00
8,00
E2
E2
14,00
6,00
6,00
4,00
4,00
2,00
2,00
0,00
0,00
5
10
15
20
25
30
0,000000000
35
10,000000000
20,000000000
30,000000000
40,000000000
50,000000000
60,000000000
70,000000000
Years of working experiance
Education in years
a) Plot between e2 and ED
Figure 9
...
2a we see the squared error term against the number of years of schooling, and a pattern can be
identified
...
This is of course just an indication that we need to investigate further using formal statistical test
...
1b the picture is less obvious
...
Since it is an unclear case, we still need to investigate the issue further, holding in mind that
hypothesis testing is meaningless if the standard errors of the parameter estimates are wrong
...
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...
2
...
Below we will shortly describe the logic of
the tests and how they are implemented
...
When this is true, the variance of one part
of the sample must be the same as the variance of another part of the sample independent on how the sample
is sorted
...
The following basic
steps complete the GQ-test:
1) Sort the sample according to a variable that you believe drives the size of the variance
...
If the sample size is very small (i
...
each group is less than 100 observations), it is enough to divide the sample into two groups without
omitting any observations
...
8)
i n1 1
i 1
3) Form the hypothesis that is to be tested:
H 0 : V i2
V2
H1 : V i2
V 2 X 1i
(9
...
10)
F
~ F( n1 k , n2 k )
2
S 2 RSS 2 /(n2 k )
As a rule of thumb, one should always put the larger variance in the numerator
...
If the test value is larger than the critical value you
choose to reject the null hypothesis
...
2
We are going to investigate model (9
...
1 to see if we can identify any heteroskedasticity
using the GQ-test
...
Therefore, we sort the data set in an increasing order of years of schooling and delete
the 33 percent in the middle
...
Using the results from these regressions we calculate the corresponding variance for each regression:
2
S1
RSS1
n1 k
59
...
1223756
2
S2
RSS 2
n2 k
43
...
08921339
(9
...
com
95
Econometrics
Heteroskedasticity and diagnostics
Using the estimated variances for the two sub samples we can calculate the test value:
F
2
S1
2
S2
0
...
3717
0
...
12)
Choosing a significance level of 5 percent we found a critical value equal to 1
...
Most statistical tables do
not offer information on critical values for the degrees of freedom we have in this example
...
Therefore we have been using Excel to calculate the critical value valid for the degrees of
freedom in our case
...
A general problem with this test is
that it tends to reject the null hypothesis very often
...
360°
thinking
...
However, we will not go through that here, and leaves that to the reader
...
360°
thinking
...
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...
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...
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...
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...
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...
D
Econometrics
Heteroskedasticity and diagnostics
The Breusch-Pagan test (BP) is also a popular test procedure presented in most econometric text books
...
The starting point is a set of explanatory variables that we believe drives the size of the variance of the
error term
...
Ah X h
(9
...
13) could be just a sub set of the explanatory variables of the model or it could
be all of them
...
1 we could not be conclusive about whether just one or if both of our variables
were driving the size of the variance
...
13)
...
13) as
stated, but we are going to use a linear specification, just as for the model we use
...
Ah 0
The hypothesis of this test is:
(9
...
, h
In order to test the hypothesis we have to go through the following basic steps:
1) Run the regression for the model you believe suffers from heteroskedasticity using OLS
...
Use the squared residual and run the following auxiliary
regression:
e2
A0 A1 X 1 A2 X 2
...
15)
Equation (9
...
13) with a linear specification
...
Instead the following test statistic could be used to test the null hypothesis:
LM
2
2
nRe ~ F h
(9
...
15) and Re is the coefficient of
determination received from (9
...
It turns out that the product of those two terms is chi-squared
distributed with h degrees of freedom, where h is the number of restrictions, which in this case
corresponds to the number of variables included in (9
...
The test value should therefore be
compared with a critical value received from the Chi-square table for a suitable level of significance
...
3
In this example we will use the same data set and the same model as in Example 9
...
But this time the test
will involve both the variables included in the model
...
Following the basic procedure of the BP-test we specify and
estimate the variance function with standard errors given within parenthesis:
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...
063 0
...
002 year
0
...
004 0
...
0028 n 1483
Using this information we are able to calculate the test value:
LM
2
nRe
1483 u 0
...
1524
Choosing a significance at the 5 percent level, the Chi-square table with 2 degrees of freedom shows a critical
value of 5
...
Hence, the test value is smaller than the critical value and we are unable to reject the null
hypothesis
...
Since the
GQ-test is very sensitive to small differences, we believe that the result of this test is more useful
...
Since we have more than 1000 observations we believe that our sample is sufficiently large, but
in order to be sure we will move on with yet another common test called the White’s test
...
Therefore, it is also a large sample test but it does not depend on any normality assumption
...
The basic steps in the procedure are as follows for a
model with two explanatory variables, where (9
...
18) the variance
function that contains all the variables of the main function and their squares and cross products:
Y
V2
B0 B1 X 1 B2 X 2 U
(9
...
18)
1) Estimate the parameters of equation (9
...
2) Square the residual and run the auxiliary regression model given by (9
...
3) Using the results from the auxiliary regression you can calculate the test value using (9
...
If the test
value is larger than the critical value chosen, you reject the null hypothesis of homoskedasticity
...
4
We repeat the test executed in Example 9
...
Observe that the only difference
is in the specification of the variance function
...
230 0
...
001 year 0
...
000 year 2 0
...
196 0
...
007 0
...
000
0
...
0039 n 1483
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...
Their t-values are definitely
different from zero
...
0039 u 1483 5
...
07, which is larger than the test value
...
That is, we have no statistical
material that points in the direction of heteroskedasticity
...
com
99
Econometrics
Heteroskedasticity and diagnostics
9
...
However, if we have followed the suggestion by the graphical
inspection and the GQ-test we would have believed that the heteroskedasticity could have been driven by one
of the explanatory variables, which is one example of how heteroskedasticty could look like
...
There is of course a number of different ways heteroskedasticty could be
expressed
...
When the nature of the heteroskedasticity is known, one can use Generalized Least Squares (GLS) to estimate
the unknown population parameters
...
To run a regression using GLS instead of OLS is in practical terms the same thing, but we call it GLS when
we have transformed the variables in the model so that the error term become homoskedastic
...
19)
V i2 V 2 X 1i
The first thing to note is that we still assume that the expected value of the error term equals zero, which
> @
means that the variance of the error term may be expressed as E U i2
V 2 X 1i
...
If we accomplish
this with just some transformations of the involved variables we are home free
...
> @
We know that the variance of the error term is V U i
V 2 X 1i
...
To see this:
Yi
X 1i
B0
X
X
Ui
1
B1 1i B2 2i
X 1i
X 1i
X 1i
X 1i
(9
...
Hence when running this specification in a computer you have to
ask the software to run the regression through the origin, since we now have a model specific constant that
moves with X1
...
When you transform the variables in this way, you
X 1i
X 1i
X 1i
X 1i
X 1i
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...
Once that is done, we
have a homoskedastic error term
...
21)
Observe that nothing happens with the parameter estimates
...
Case 2:
V i2 V 2 X 12
i
This case is very similar to the previous case with the exception that the variable X1 is squared, which means
that the variance increases exponentially with X1
...
Hence instead of dividing by the square root of X1 we simply
divide by X1 it self
...
22)
Case 3: Two different variances
In this case we have an error term that takes only two values
...
If these two groups are known, we can sort the data set with
respect to these groups
...
In order to solve the heteroskedasticity problem here, we need to estimate the two variances, by
splitting the sample in two parts and estimate the regression variance separately for the two groups
...
com
101
Econometrics
Heteroskedasticity and diagnostics
By scaling the error term for each group using their standard deviation, the new transformed error term will
have a variance that equals 1 in both sub samples
...
To see this
§U
V¨ i
¨V
© 1
·
¸
¸
¹
§U
V U i 1 for i=1, …n1 and V ¨ i
¨V
V 12
© 2
1
·
¸
¸
¹
1
2
V2
V U i 1 for i=n1+1, …,n
Example 9
...
Since we know how the
structure of the heteroskedasticity, we apply GLS according to case 1
...
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102
Econometrics
Heteroskedasticity and diagnostics
Table 9
...
005
0
...
996
0
...
425
0
...
475
0
...
578
0
...
497
0
...
210
0
...
956
0
...
3 we compare the residual plots before and after correcting for heteroskedasticity to see if the
problem is fully solved
...
3b the picture looks satisfying
...
3 Estimated residual plots before and after correction for heteroscedasticity
Table 9
...
As can be seen the
estimated coefficient does not deviate that much which is what we expected since heteroskedasticity has no
effect on the unbiasedness and consistency of the OLS estimator
...
The standard errors of the OLS estimated slope coefficients are
twice as large as those for the corrected model
...
That is due to the relatively large sample that was used
...
However, these results are sample specific
...
So it is impossible to say something in
advance with out knowing something about the exact nature of the heteroskedasticity
...
As can be seen it increased
substantially
...
Unfortunately it just
means that after a transformation of the variables of the kind we did here, the coefficient of determination is
of no use, since it is simply wrong
...
com
103
Econometrics
Heteroskedasticity and diagnostics
9
...
1 Heteroskedasticity-robust standard errors
The approach of treating heteroskedasticity that has been described until now is what you usually find in
basic text books in econometrics
...
We know how the variance of the OLS estimator should look like for the simple linear regression model:
n
V b1
¦ X i X V i2
2
i 1
§ n
·
¨ X i X 2 ¸
¨
¸
©i 1
¹
(9
...
By doing that one would receive consistent estimates of
the true standard errors which provide a basis for inference in large samples
...
24)
¦
Since (9
...
Fortunately there exist a small
sample adjustment factor that could improve the precision considerably by multiplying the variance estimator
given by n/(n-k)
...
Fortunately most econometric software
such as STATA and SAS, includes the option of receiving robust standard errors together with the parameter
estimates when running the regression
...
Example 9
...
We
are not sure whether we have a problem of heteroskedasticity and we therefore estimate the parameters with
and without robust standard errors, to see how the estimates of the standard errors change
...
com
104
Econometrics
Heteroskedasticity and diagnostics
Table 9
...
E
...
E
...
E
...
S
...
P
...
Inctercept
3
...
087
3
...
105
3
...
063
0
...
063
0
...
037
Years of Education 2
-0
...
0004
-0
...
0006
-
Male (dummy)
0
...
017
0
...
017
0
...
0
...
001
0
...
008
0
...
3079
R
...
E
...
041
0
...
0167
0
...
008
0
...
E
...
E
...
S
...
stands for Robust Standard Errors
...
Table 9
...
These results should be compared with the second column of estimates that use
robust standard errors, which are heteroskedasticity consistent standard errors
...
Including
irrelevant variables in the regression makes the estimates less efficient
...
In the third column, we re-estimate the model with out the squared term using
robust standard errors
...
If we had included the squared education term, the marginal effects of education on
earnings would be different and wrong
...
2
...
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...
Autocorrelation and diagnostics
Autocorrelation or serial correlation often appears when working with time series data
...
In statistical terms this could be expressed as:
>
@
Cov U i ,U j z 0
iz j
(10
...
This means that it is meaningless to look for autocorrelation when working with cross sectional
data which usually are based on random samples from a population, at a given point in time
...
If correlation is
found anyway, one can be sure that it is a fluke and has nothing to do with any underlying process
...
To find a correlation between two randomly chosen individuals in this sample is not very likely
...
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1 Definition and the nature of autocorrelation
An autocorrelated error term can take a range of different specifications to manifest a correlation between
pair wise observations
...
2)
where U refer to the error term of the population regression function
...
2) the error
term at period t is a function of it self in the previous time period t-1 times the coefficient,U, which is referred
to as the first order autocorrelation coefficient (This is the Greek letter rho, pronounced “row”)
...
It is often assume to be
standard normal
...
Since U is a function of it self one period back only, as appose to several periods, we call it the first order
autoregression error scheme, which is denoted AR(1)
...
We would then refer to it as the nth order of autocorrelation and it would be specified like this:
Ut
UU t 1 U 2U t 2
...
3)
The first order autocorrelation is maybe the most common type of autocorrelations and is for that reason the
main target of our discussion
...
2)
...
e( t )
3
e( t )
3
2
2
1
1
0
0
-1
-1
-2
-3
-2
-2
-1
0
1
2
3
-3
a) Positive autocorrelation, (+0
...
1 Scatter plots between et and et-1
-2
-1
0
1
2
3
e( t - 1)
e( t - 1)
b) Negative autocorrelation (-0
...
1 present two plots that are two examples of how the plots could look like when the error term is
autocorrelated
...
3
...
Sometimes you are exposed to plots where the dependent variable or the residual term is
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...
However, when the correlation is below 0
...
10
...
In short we have that:
1) The estimated slope coefficients are unbiased and consistent
...
3) With negative autocorrelation the standard errors are biased and too large
...
That is, the property of unbiasedness and consistency does not require uncorrelated error
terms
...
The efficiency property of the OLS estimator does, however, depend on the assumption of no autocorrelation
...
Assume the following set up:
Yt
B0 B1 X t U t
(10
...
5)
Vt ~ N (0,1)
(10
...
7)
With this setup we observe that the residual term, U, is autoregressive of order one
...
8)
When generalizing this expression to an arbitrary distance between two error terms it is possible to show that
it equals
Cov U t ,U t j
U jV 2
(10
...
The variance of the slope
coefficient can be expressed in the following way
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...
¸
2 ¨
¸
§ n
t 1
t 1
¹
2· ©t 1
¨ X t X ¸
¨
¸
©t 1
¹
V2
¦
¦
§
¨
2
¨
V
¨1 2 U
n
2¨
X t X ¨
t 1
©
¦
¦
¦
n
¦ X t X X t 1 X
t 1
n
¦ X t X
n
2U
¦ X t X X t 2 X
2
2
t 1
If the autocorrelation coefficient were zero i
...
U
t 1
n
¦ X t X
2
t 1
·
¸
¸
...
10)
0 , the infinite series within the parenthesis in (10
...
However, if ignoring the autocorrelation when present, we disregard this term which bias
the variance of the slope coefficient
...
In order to do that, we need to impose some assumptions on the
2
behavior of X
...
This implies that Cov ( X t , X t j ) r jV X , with r
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being the correlation coefficient for Xt and Xt-1
...
10) we receive
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109
Econometrics
V b1
Autocorrelation and diagnostics
2
VU
1 2 Ur 2 U 2 r 2 2 U 3r 3
...
11)
In order to receive the compact expression given by (10
...
If you do not know that, do not worry! The important thing here is to see how the sign of the
autocorrelation coefficient and the correlation between of Xt and Xt-j affect the size of the variance and induce
a bias when ignoring autocorrelation
...
With this set up we can analyze the size of the adjustment factor due to
autocorrelation
...
With a fixed value
of r, the adjustment factor is increasing with the size of the autocorrelation coefficient, which increases the
bias
...
Most macro economic time series has an r value that is different from zero, and hence the case would in
general not appear
...
With a fixed value of r, and an
increasing value of the autocorrelation coefficient in absolute terms, the adjustment factor will be smaller,
and increase the bias
...
Furthermore, the coefficient of determination and the usual estimator for the error
variance of the model will be bias as well
...
10
...
Below we will describe the most common procedures found in the text book
literature
...
In the
introduction of the chapter we gave some examples on how graphical methods could be used
...
However, we will not discuss these methods here
...
3
...
Since first order autocorrelation is most likely to appear in time
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...
The Durbin-Watson test statistic for first order autocorrelation is given by:
T
¦ et et 1 2
t 2
DW
T
(10
...
To see that this test statistic is related to
the first order autocorrelation case we may rewrite (10
...
13)
2
t 1
ˆ
where U on the right hand side is the autocorrelation coefficient from an first order autoregression scheme
...
The larger the value of T the better is the approximation
...
13) it is possible to see that the DW test statistic only takes values between 0 and 4 since the
autocorrelation coefficient only takes values between -1 and 1
...
If DW > 2 we have an indication of a negative autocorrelation, and if
DW < 2 we would have an indication of a positive autocorrelation
...
So the standard question is how much it is allowed to deviate? Could we use some critical values to
help us interpret the estimated value of DW
...
For that reason
it is not possible to establish a precise critical value for the DW test statistic
...
Table 10
...
1 show five different regions where the DW-test value potentially could end up
...
However, if the DW-value is between 0 and the lower
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...
In the statistical table, with upper
and lower values for the DW-test, you will only find the values that refer to the section below 2
...
1
...
1
Assume that you have a time series with 150 observations, and two explanatory variables that will be used to
explain the dependent variable
...
63
...
71 and U=1
...
Since the test value
is outside the inconclusive interval and below the lower value we have to draw the conclusion that our model
suffer from positive autocorrelation
...
2
We have the same set up as in the previous example but with a DW-test value equal to 2
...
In the table we
found the value for L=1
...
76
...
Using the information in Table 10
...
76)
= 2
...
71) = 2
...
Since the test value of 2
...
29 we must conclude that our model suffer from negative autocorrelation
...
com
112
Econometrics
Autocorrelation and diagnostics
10
...
2 The Durbins h test statistic
As been described above, the DW-test is made for the purpose of testing for first order autocorrelation
...
When that is the case the DW-test has a tendency to be close
to 2 even though the error terms are serially correlated
...
14)
Fortunately there is an easy alternative to the DW-test that could be seen as a modified version of it and for
that reason is called the Durbins h statistic
...
15)
where DW is the standard DW-test, T the number of observations and Var (b1 ) the square of the standard error
of the estimated parameter for the lagged dependent variable
...
The presence of autocorrelation in models that include lagged dependent variables is even more affected than
the standard model
...
It is therefore very important to correct for the problem before using the estimates
for anything
...
3
Assume that we have estimated the parameters of a dynamic model and received the following results with
standard errors within parenthesis:
ˆ
Yt
1
...
789Yt 1 0
...
321 0
...
032
DW
1
...
Since our model includes a lagged
variable we lose one observation in the estimation
...
529 ·
¨1
¸
2 ¹ 1 119 u 0
...
925
Using a one sided test at the 5 percent significance level we receive a critical value of 1
...
Since the test
value is much larger than the critical value, we must conclude that our error terms are serially correlated
...
com
113
Econometrics
Autocorrelation and diagnostics
10
...
3 The LM-test
The LM test is a more general test of autocorrelation compared to the two previous tests
...
However, the LM test is a large sample test which means that it should
be treated as an approximation when using small samples, compared to the DW-test that could be seen as an
exact test
...
16)
2) Create the residual term using the estimated parameters and lag it
...
17)
Yt B0 B1 X t Uet 1 Vt
4) Test the null hypothesis H 0 : U 0 using a simple t-test
...
The equation given by (10
...
Example 10
...
Yt
B0 B1 X 1t B2 X 2t U t
(10
...
19)
Lag the estimated residuals from (10
...
18) and receive:
Yt
B0 B1 X 1t B2 X 2t U1et 1 U 2et 2 Vt
(10
...
20) and received the
following estimates with standard errors within parenthesis:
ˆ
Yt
3
...
378 X 1t 0
...
324et 1 0
...
301 0
...
192
0
...
101
(10
...
com
114
Econometrics
Autocorrelation and diagnostics
By investigating the significance of the coefficients of the two residual terms, we see that the first one is
significantly different from zero, while the other is not
...
Since we included lagged variables in the specification we lose some observations, and when including two
estimated residuals as in (10
...
If we have a large sample, losing two observations is not a big
deal
...
One
way to deal with this problem is to impose some values for e0 and e1
...
Running the regression with
and without the imposed values will give you an indication if the two missing observations are important
...
10
...
As for the case of heteroskedasticity, we need to transform the
involved variables and therefore use generalized least square
...
We will therefore look at the two most frequently used error structures, AR(1)
and AR(2), and show how it should be done for those two cases
...
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115
Econometrics
Autocorrelation and diagnostics
10
...
1 GLS when AR(1)
The transformation will be explained by an example
...
22)
For simplicity reasons we use the specification of the simple regression model
...
The objective is to transform the autocorrelated Ut with
something that free from autocorrelation Vt
...
23)
If we substitute (10
...
22) we receive:
Yt
B0 B1 X t UU t 1 Vt
(10
...
22):
UU t 1
UYt 1 UB0 UB1 X t 1
(10
...
25) into (10
...
26)
Equation (10
...
The error term of the original model is now
replaced by Vt that is free from autocorrelation and we can estimate the regression equation using OLS
...
10
...
2 GLS when AR(2)
The corresponding transformation in the AR(2) case is very similar
...
27)
With that in mind we can extend equation (10
...
28)
The whole description above is based on the idea that the autocorrelation coefficient has been known
...
The estimated value is often received when you test for
autocorrelation
...
29)
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...
29)
...
The coefficients in front of the
lagged residual terms in (10
...
27)
...
28)
...
However, statistical software, such as STATA and SPSS, will do most of
the job for you
...
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117
Econometrics
Multicollinearity and diagnostics
11
...
For the obvious reason it could never appear in the simple regression model, since
it only has one explanatory variable
...
We referred to
that as to fall in the dummy variable trap
...
When that happens we have what is
called perfect multicollinearity
...
Multicollinearity
The lack of independence among the explanatory variables in a data set
...
11
...
Assume that we would like to estimate the parameters of the following model:
Y
B0 B1 X1 B2 X 2 U
(11
...
2)
and where a and b are two arbitrary constants
...
2) into (11
...
3)
Since (11
...
2) implies (11
...
But since
these two expressions contain three unknown parameters there is no way we can receive estimates for all
three parameters in (11
...
We simply need more information, which is not available
...
This was an example of the extreme case of perfect multicollinearity, which is not very likely to happen in
practice, other than when we end up in a dummy variable trap or a similar situation
...
We
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...
1) under the assumption
that X1 and X2 is highly correlated but not perfect
...
The sample estimator for B1 is given by:
b1
rY 1 r12 rY 2 SY
2
(1 r12 )
(11
...
The first thing to observe is that r12 appears in both the numerator and the denominator, but that it is squared
in the denominator and makes the denominator zero in case of perfect correlation
...
However, it can be shown that the OLS estimators
remain unbiased and consistent, which means that estimated coefficients in repeated sampling still will center
around the population coefficient
...
Therefore we will go through an example in order to shed some light on this
issue
...
1
Consider the following regression model:
Y
B0 B1 X 1 U
We would like to know how the estimate of B1 changes when we include another variable X2 that is highly
correlated with X1
...
SY 5
...
843
S1 5
...
878
r12
0
...
843 u
5
...
0
0
...
For the multiple regression case when including both X1 and X2 we receive:
*
b1
0
...
924 u 0
...
1
u
5
...
9242
0
...
Hence, when including an additional variable the estimated coefficient decreased in size as a result of the
correlation between the two variables
...
com
119
Econometrics
Multicollinearity and diagnostics
size in absolute terms? Well, consider the case where X2 is even more correlated with X1, lets say that
r12=0
...
That would generate a negative estimate and the small number in the denominator will make the
estimate larger in absolute terms
...
Hence, the estimated slope coefficient could move in any direction as a result of
multicollinearity
...
The variance of (11
...
5)
2
1i
i 1
When the correlation between X1 and X2 equals zero, will the variance of the multiple regression coefficient
coincide with the variance for the coefficient of the simple regression model
...
5) will be undefined just as the estimated slope coefficient
...
But make no mistakes;
collinearity does not destroy the nice property of minimum variance among linear unbiased estimator
...
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120
Econometrics
Multicollinearity and diagnostics
It seems like the level of both the estimated parameter and its standard error are affected by multicollinearity
...
It can be shown that the computed t-value in
general will decrease since the standard error is affected more strongly compared to the coefficient
...
Another problem with multicollinearity is that the estimates will be very sensitive to changes in specification
...
Hence, the
parameter estimates are very unstable and sometimes it can even result in wrong signs for the regression
coefficient, despite the fact that it is unbiased
...
However, sometimes we
are dealing with inferior goods which means that we have to be careful with what we call “wrong” sign
...
11
...
All statistical measures have their
limitations, and therefore it is always useful to use several measures when investigation statistical properties
of a data set
...
6)
We suspect that the variables are highly correlated and would like to investigate the matter
...
That is easily done in any statistical software
...
1 A correlation matrix for the explanatory variables in (11
...
924
X2
1
X3
X3
0
...
085
1
As can be seen from Table 11
...
X1 is also correlated with X3 but to a much lower degree and the correlation between X2 and X3 is basically
zero
...
To further analyze the multicollinearity we turn to the next measure which is called the Variance Inflation
Factor (VIF)
...
com
121
Econometrics
Multicollinearity and diagnostics
VIF (bi )
1
1 Ri2
(11
...
The squared multiple-correlation coefficient for a
specific parameter is a measure of the linear strength between a variable Xi and the rest of the variables
included in the model
...
That is
2
R1 is received from:
X1
C10 C11 X 2 C12 X 3 U ,
VIF (b1 )
1
1 R12
2
R2 is received from:
X2
C20 C21 X 1 C22 X 3 U ,
VIF (b2 )
1
2
1 R2
2
R3 is received from:
X3
C30 C31 X 1 C32 X 2 U ,
VIF (b3 )
1
2
1 R3
When the model contains only two explanatory variables the squared multiple correlation will coincide with
the squared bivariate correlation coefficient between the two variables in the model
...
5) you
will see that the variance inflation factor is included in that expression, and is the factor that is multiplied
with the variance of the coefficient of the simple regression model
...
The expression for the
variance in the case of more than two explanatory variables has a similar expression but with the squared
multiple correlation coefficient instead
...
The closer the multiple-correlation coefficient is to one, the
larger the value of VIF
...
The tolerance measure is the denominator of the VIF expression
...
That means that we have
a measure of how large share of the variation in one variable that is explained by a group of other variables
...
When are VIF and the tolerance an indication of a multicollinearity problem? We can shed some lights on
that question by an example
...
6) and check the VIF and tolerance condition for
the variables in that case
...
Using one such routine in SPSS we received the following regression
results and collinearity statistics:
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...
2 Regression results for (11
...
E
...
E
...
085
99
...
334
3
...
857
2
...
186
1
...
801
R2
Test of over all significance: F =21
...
0001
VIF
708
...
343
104
...
00141
0
...
00956
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From Table 11
...
We also observe
that the coefficient of determination is above 80 percent
...
Furthermore, the
test of overall significance of the model is highly significant which is in line with the measure of fit
...
It will blow up the standard errors of the model even though the model as such has explanatory
power
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In the table we see that VIF take large values for all variables
if we compare to the case of no correlation that results in VIF=1
...
Even so, VIF for X3 is 104 and the
corresponding tolerance is as low as 0
...
9 percent unique
variation left of X3s total variation that can be used in explaining the variation in Y
...
One should observe that even
though the par wise correlations are relatively low, their multiple-regression correlation is much higher,
which emphasize the shortcomings of only looking at pair wise correlations
...
3 Remedial measures
With a given specification and data set there is not much one can do about the multicollinearity problem
...
Doing nothing is most often not a very attractive alternative
...
It would not solve the multicollinearity problem, but the small unique
variation that exist, will be based on more data and if the increase in the number of observations is large
enough, it could help increasing the precision of the estimators
...
Another alternative would be to change the variable specification
...
If we, in the first place, had an economic relevant specification we know that the
estimated parameters will be biased and inconsistent if dropping a relevant variable
...
An alternative approach would be to rethink the model so that it could be expressed in an alternative way
...
In our example discussed
above it was problematic to include X1 and X2 in the same regression
...
However, that would mean a slightly different model, and
we have to be willing to accept that
...
We will therefore not go into any of those more
advanced techniques here, since they would require more statistical knowledge that is beyond the scope of
this text
...
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...
Simultaneous equation models
One important assumption of the basic linear regression model is that the error term has to be uncorrelated
with the explanatory variables
...
In this chapter we will relax this assumption by
including additional equations to the model that explains where the correlation is coming from, and discuss
the conditions that need to be fulfilled to receive consistent estimate
...
1 Introduction
This chapter will only scratch the surface of the issues involved in estimating simultaneously equations and
should therefore be seen as an introduction to the subject
...
The assumption of random explanatory variables does not change anything related to the property
of the OLS estimators but it allows for the possibility of being correlated with the error term
...
To see this, consider the following simple macro
economic model of income determination:
Yt
Ct I t
(12
...
2)
with Yt being the national income, It investments, Ct the consumption expenditure and Ut a stochastic term
...
1) is an identity and an equilibrium condition
...
The second, equation given by (12
...
Equations with stochastic error terms are to be
considered behavioral
...
We therefore say that Yt and Ct are endogenous variables
...
Since it is a right hand variable in the consumption
function we say that it is an exogenous variable, which is to say that the value of investments are determined
outside the model, it is pre determined
...
The system of equations can be solved with respect to the two endogenous variables in order to receive their
long run expressions
...
2) into (12
...
That results in the following expression:
Yt
B0
1
1
It
Ut
1 B1 1 B1
1 B1
(12
...
1) into (12
...
com
125
Econometrics
Simultaneous equation models
Ct
B0
B
1
1 It
Ut
1 B1 1 B1
1 B1
(12
...
2) ignoring the fact that it is part of a
system
...
3) we can see that Yt is a function of Ut which means it is correlated with Ut
...
2) without bias
...
But
when that is not the case we see from (12
...
It should know be obvious that the OLS estimators are biased in small samples due to the correlation between
Yt and Ut
...
5)
2
t 1
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t 1
2
T
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...
12))
...
6)
¦
The problem with the expectation on the right hand side is that Yt is a random variable and correlated with Ut,
and for that reason we can not proceed as in chapter 3
...
Even though this makes it clear
that the estimator no longer is unbiased, we do not know how the second component on the right hand side of
(12
...
It can be shown that the limit of the OLS estimator is given by the following
expression:
lim b1
t of
B1
2
(1 B1 )V U
2
V I2 V U
(12
...
7) show that in the limit the sample estimator still deviate from the population parameter, which means
that the bias remains in large samples
...
2 The structural and reduced form equation
From the previous discussion we learned that when an equation belongs to a system of equation, estimating
them separately using OLS would lead to biased and inconsistent estimates
...
Before going into issues of estimation we need to define some more concepts
...
8)
It
B0 B1Rt U 2t
(12
...
10)
It is a macro economic model that extends the example from the previous section and is based on three
equations
...
com
127
Econometrics
Simultaneous equation models
expenditure Ct, and one for net-investments It
...
The income equation that specifies the equilibrium
condition is a function of consumption, investment and government spending Gt
...
The system of equations given by (12
...
10) describes the structure of the economy that we would like to
investigate
...
The coefficients of the structural
equations represent the direct effect of a change in one of the explanatory variables
...
9) as an
example, B1 represents the marginal propensity to invest as a result from a change in the interest rate
...
Assume that we increase the interest rate
...
The income in its
term will affect the consumption level, and since income is endogenous it will have an effect the error term
U1 since they are correlated
...
We can therefore talk about two types of effect; the short run effect and the long run effect
...
To solve the system for Yt we simply substitute (12
...
9) into (12
...
If we do that we receive:
Yt
A0 B0
U U 2t
B
1
1 Rt
Gt 1t
1 A1 1 A1
1 A1
1 A1
(12
...
12)
(12
...
The coefficients of the reduced form
equations represent the full effect when the system is in equilibrium
...
It is also called the interest rate multiplier on income
...
Observe that the reduced form equation for investments only is a function of interest rate
...
The nice thing with the reduced form equations is that they may be estimated separately using OLS
...
Since the structural
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...
For that to be possible, certain
requirements need to be fulfilled
...
12
...
So, what do we
mean by that? To give an intuitive feeling for its meaning we will give an example before going into any
formal and mechanical tests
...
14)
Q
B0 B1P U 2
(Demand)
(12
...
These two
equations represent a demand and supply system for a given market
...
That is to ask if the parameters of the two equations can be estimated consistently
...
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To see this, consider
Figure 12
...
In order to identify the demand function we need some exogenous variation that could help us
trace out the function
...
The supply function contains an
exogenous variable X1 and the supply function takes a new position for each value of X1
...
But in the demand function we have nothing unique that does not appear in the
supply function so it is impossible to move the demand function while holding the supply function fixed
...
If X1 had been included in both equations there would have been no unique variation in
any of the equations and hence no equation had been identified
...
In that case both equations would have been identified
...
1 Demand and supply system
Q
The process of identifying equations can be formalized in a decision rule that specify the conditions that have
to be fulfilled in order to identify one or several equations in a system
...
12
...
1 The order condition of identification
The first decision rule for identification is the so called order condition
...
Unfortunately
it is not a sufficient rule, which means that it is possible that the equation is undefined even though the order
condition says it is identified
...
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...
The order condition states that:
1)
2)
3)
If K M 1
If K ! M 1
If K M 1
=> The equation is exactly identified
=> The equation is over identified
=> The equation is under identified
When checking the order condition you have to do it for each equation in the system
...
1
Consider the following system:
Y1
A0 A1Y2 A2 X 1 U1
(12
...
17)
Use the order condition to check if the equations are identified
...
This system contains two endogenous variables and the total number of variables,
endogenous as well as exogenous, is 4
...
16)
...
For the second equation we have M-1=1 and K=1 since X1 is excluded from (12
...
Since M-1=K we have
that also the second equation is exactly identified
...
Example 12
...
18)
Y2
B0 B1Y1 B2 X 2 U 2
(12
...
M-1 will in this example equal 1 as before since we still have only two endogenous variables
...
Since M1=K the equation is exactly identified
...
com
131
Econometrics
Simultaneous equation models
The second equation includes three variables which mean that two variables have been excluded
...
That means that K=2, which means that M-1
...
3
...
The rank condition is a necessary and sufficient
condition, which means that if we can identify the equations using the rank condition we can be sure that the
equation really is identified
...
If that is the case it is impossible to identify all structural parameters
...
Example 12
...
20)
Y2
B0 B1Y1 B2 X 2 B3 X 3 U 2
(12
...
22)
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132
Econometrics
Simultaneous equation models
This system contains three endogenous variables (Y1, Y2, Y3) and three exogenous variables (X1, X2, X3),
which means that we in total has six variables
...
For our system we receive the following matrix:
Matrix for the rank condition
Equation 1
Equation 2
Equation 3
Y1
1
1
0
Y2
0
1
1
Y3
1
0
1
X1
1
0
1
X2
0
1
1
X3
1
1
0
In order to check the rank condition for the first equation we have to proceed as follows: Delete the first row
and collect the columns for those variables of the first equation that were marked with zero
...
M refers to the number of equation just as in the order condition, which means that M-1=2
...
For equation 2 we proceed in the same way
...
For equation 2 that was the case for Y3 and X1, which is
to say that these two variables was not included in equation 2
...
The same procedure should be done for the third equation and if you do that you will see that it is
identified as well
...
When that happens it might still be possible to
generate estimates, but those estimates will not have any economic meaning since they will represent
averages of those equations that are linear combinations of each other
...
When using systems of more than two equations you should also confirm the
identification using the rank condition
...
4 Estimation methods
Once we have confirmed that our model is identified we can proceed with the estimation of the parameters of
the structural coefficients
...
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...
4
...
It is done by the following three steps:
1) Form the reduced form equations
2) Estimate the coefficients of the reduced form using OLS
3) Use the estimated coefficients of the reduced form to derive the structural coefficients
...
4 (ILS)
Consider the following simple macro economic model:
Yt
Ct I t
(12
...
24)
This model has two endogenous variables (Yt and Ct) and one exogenous variable (It), and we would like to
estimate the coefficients of the behavioral equation
...
The two structural equations could
be used to form the reduced form equations for consumption
...
25)
(13
...
4) show how the reduced form coefficients are related to the structural coefficients
...
We have:
S0
B0
1 B1
(12
...
27)
(12
...
27) can now be used to solve for B0 and B1
...
27) is an equation with only one
unknown we solve for B1 first (remember that S1 is an estimate and therefore a number in this expression)
...
26) to solve for B0
...
It can be shown that
the corresponding variance for B0 and B1 is:
2
2
V B0 | a 2V 0 b 2V 1 2abV 01
2
V B1 | b 2V 1
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...
ILS will result in consistent estimates but will still be biased in small samples
...
For that reason ILS is not used very
often in practice
...
12
...
2 Two Stage Least Squares (2SLS)
The procedure of 2SLS is a method that allows you to receive consistent estimates of the structural
coefficient when the equations are exactly identified as well as over identified
...
Consider the following model
A0 A1Y2 A3 X 1 A4 X 2 U1
(12
...
29)
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Y1
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...
The first equation (12
...
That means that it is over
identified
...
29), which means that the model is identified
...
We will now focus the discussion on the estimation of the first equation
...
28):
Step 1
Derive the reduced form equation for Y2 and estimate the predicted value of
ˆ
Y2 (Y ) on the reduced form using OLS
...
28) with its predicted value from the reduced
form and estimate the coefficient of the model using OLS
...
28)
...
ˆ
Hence, the problem is solved
...
A group of exogenous variables are by necessity uncorrelated with the random term
...
This correlation will not be perfect unless X3
and X4 also is included in the structural model of the first equation and is therefore nothing to worry about
...
There is one additional complication to be aware of when working with 2SLS
...
To see this we will consider a
simplified version of a model to make it clear where the problem appear
...
30)
In order to receive consistent estimates of B1 we replace Y2 with its predicted value and estimate the
parameters of following regression model using OLS:
Y1i
ˆ
B1Y2i (U1i B1U 2i )
ˆ
B1Y2i V1
(12
...
com
136
Econometrics
Simultaneous equation models
n
§ Y Y ·Y
¨ ˆ2i ˆ2 ¸ 1i
©
¹
1
¦
b1
i
n
§Y Y ·
¨ ˆ2i ˆ2 ¸
©
¹
1
¦
i
n
n
§ Y Y · B Y U
¨ ˆ2i ˆ2 ¸ 1 2i
1i
©
¹
1
¦
i
n
2
§Y Y ·
¨ ˆ2i ˆ2 ¸
©
¹
1
¦
i
2
B1
¨
¸
¦ § Yˆ2i Yˆ2 ·U1i
©
¹
i 1
n
§Y Y ·
¨ ˆ2i ˆ2 ¸
©
¹
1
¦
i
2
(12
...
Since it is consistent we
need to compare it with its asymptotic variance, that is, the formula of the variance when the number of
observation is very large (has gone to infinity)
...
33)
2
This is good, because it is very similar to the variance given by the standard OLS
...
When running the regression using (12
...
34)
i 1
Whereas the estimated residual should be given by the following expression
ˆ2
VU
1
n
n
¦ Y1i b1Y2i 2
(12
...
34) is based on the observed variable Y2 multiplied with the sample estimator b1 given by (12
...
Hence in order to receive consistent estimates of the standard errors,
one has to use (12
...
When using commercial software with routines for 2SLS they automatically make the
correction
...
In sum, the 2SLS has the following properties:
x
x
x
It generates biased but consistent estimates
The distribution of the estimators are normally distributed only in large samples
The variance is biased but consistent when using (12
...
com
137
Econometrics
Statistical tables
A
...
01
0
...
03
0
...
05
0
...
07
0
...
09
0
...
5
0
...
50798
0
...
51595
0
...
52392
0
...
53188
0
...
53983
0
...
54776
0
...
55567
0
...
56356
0
...
57142
0
...
2
0
...
58317
0
...
59095
0
...
59871
0
...
60642
0
...
61409
0
...
61791
0
...
62552
0
...
63307
0
...
64058
0
...
64803
0
...
4
0
...
6591
0
...
6664
0
...
67364
0
...
68082
0
...
68793
0
...
69146
0
...
69847
0
...
7054
0
...
71226
0
...
71904
0
...
6
0
...
72907
0
...
73565
0
...
74215
0
...
74857
0
...
7549
0
...
75804
0
...
76424
0
...
77035
0
...
77637
0
...
7823
0
...
8
0
...
79103
0
...
79673
0
...
80234
0
...
80785
0
...
81327
0
...
81594
0
...
82121
0
...
82639
0
...
83147
0
...
83646
0
...
84134
0
...
84614
0
...
85083
0
...
85543
0
...
85993
0
...
1
0
...
8665
0
...
87076
0
...
87493
0
...
879
0
...
88298
1
...
88493
0
...
88877
0
...
89251
0
...
89617
0
...
89973
0
...
91774
1
...
9032
0
...
90658
0
...
90988
0
...
91309
0
...
91621
1
...
91924
0
...
9222
0
...
92507
0
...
92785
0
...
93056
0
...
5
0
...
93448
0
...
93699
0
...
93943
0
...
94179
0
...
94408
1
...
9452
0
...
94738
0
...
9495
0
...
95154
0
...
95352
0
...
7
0
...
95637
0
...
95818
0
...
95994
0
...
96164
0
...
96327
1
...
96407
0
...
96562
0
...
96712
0
...
96856
0
...
96995
0
...
9
0
...
97193
0
...
9732
0
...
97441
0
...
97558
0
...
9767
2
0
...
97778
0
...
97882
0
...
97982
0
...
98077
0
...
98169
2
...
98214
0
...
983
0
...
98382
0
...
98461
0
...
98537
0
...
2
0
...
98645
0
...
98713
0
...
98778
0
...
9884
0
...
98899
2
...
98928
0
...
98983
0
...
99036
0
...
99086
0
...
99134
0
...
4
0
...
99202
0
...
99245
0
...
99286
0
...
99324
0
...
99361
2
...
99379
0
...
99413
0
...
99446
0
...
99477
0
...
99506
0
...
6
0
...
99547
0
...
99573
0
...
99598
0
...
99621
0
...
99643
2
...
99653
0
...
99674
0
...
99693
0
...
99711
0
...
99728
0
...
8
0
...
99752
0
...
99767
0
...
99781
0
...
99795
0
...
99807
0
...
9
0
...
99819
0
...
99831
0
...
99841
0
...
99851
0
...
99865
0
...
99874
0
...
99882
0
...
99889
0
...
99896
0
...
1
0
...
99906
0
...
99913
0
...
99918
0
...
99924
0
...
99929
3
...
99931
0
...
99936
0
...
9994
0
...
99944
0
...
99948
0
...
3
0
...
99953
0
...
99957
0
...
9996
0
...
99962
0
...
99965
3
...
99966
0
...
99969
0
...
99971
0
...
99973
0
...
99975
0
...
33: P( Z d 1
...
90824
Download free ebooks at bookboon
...
25
0
...
000001
0
...
764892
0
...
726687
0
...
711142
0
...
702722
0
...
697445
0
...
69383
0
...
691197
0
...
689195
0
...
687621
0
...
686352
0
...
685307
0
...
68443
0
...
683685
0
...
683044
0
...
681564
0
...
679981
0
...
678601
0
...
676951
0
...
10
0
...
077685
1
...
637745
1
...
475885
1
...
414924
1
...
383029
1
...
36343
1
...
350172
1
...
340605
1
...
333379
1
...
327728
1
...
323187
1
...
319461
1
...
316346
1
...
313704
1
...
311435
1
...
306212
1
...
30065
1
...
295821
1
...
290075
1
...
05
0
...
313749
2
...
353363
2
...
015049
1
...
894578
1
...
833114
1
...
795884
1
...
770932
1
...
753051
1
...
739606
1
...
729131
1
...
720744
1
...
71387
1
...
70814
1
...
703288
1
...
699127
1
...
689573
1
...
679427
1
...
670649
1
...
660235
1
...
025
0
...
70615
4
...
182449
2
...
570578
2
...
364623
2
...
262159
2
...
200986
2
...
160368
2
...
131451
2
...
109819
2
...
093025
2
...
079614
2
...
068655
2
...
059537
2
...
051829
2
...
045231
2
...
03011
2
...
014103
2
...
000297
1
...
983972
1
...
01
0
...
82096
6
...
540707
3
...
36493
3
...
997949
2
...
821434
2
...
718079
2
...
650304
2
...
602483
2
...
56694
2
...
539482
2
...
517645
2
...
499874
2
...
485103
2
...
472661
2
...
46202
2
...
437719
2
...
412116
2
...
390116
2
...
364213
2
...
005
0
...
6559
9
...
840848
4
...
032117
3
...
499481
3
...
249843
3
...
105815
3
...
012283
2
...
946726
2
...
898232
2
...
860943
2
...
831366
2
...
807337
2
...
787438
2
...
770685
2
...
756387
2
...
723809
2
...
689594
2
...
660272
2
...
625893
2
...
001
0
...
2888
22
...
21428
7
...
893526
5
...
785252
4
...
29689
4
...
024769
3
...
852037
3
...
732857
3
...
645764
3
...
579335
3
...
527093
3
...
484965
3
...
450186
3
...
42101
3
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