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Concepts

Description

Covariance

Correlation and Regression
Covariance : measure the linear relationship between 2 random variables

𝐶𝑜𝑣

,

=

∑

𝑋 −𝑋 × 𝑌 −𝑌
𝑛−1

Correlation coefficient

Correlation coefficient : strength of the linear relationship
𝐶𝑜𝑣 ,
𝑟 =
𝜎 ×𝜎
−1 ≤ 𝑟 ≤ 1

Scatter plot

Scatter plot : collection of point on the graph , each represents the value of 2 variables (X and Y)
- Upward scatter plot : positive correlation
- Downward scatter plot : negative correlation

Limitation to correlation analysis

1
...
Spurious Correlation : may appear to have a relationship when there is none
3
...
Linear regression exists between dependent and independent variables
2
...
Expected value of residual term E(ε) = 0
4
...
Residual term is independently distributed (residual for observation A is not correlated with residual for observation B)
6
...


𝑌 = 𝑏 +𝑏 ×𝑋
Confidence intervals for predicted value (Y)

𝑌± 𝑡 ×𝑠

→𝑌− 𝑡 ×𝑠

≤ 𝑌 ≤ 𝑌+ 𝑡 ×𝑠

In which :
𝑡 = 2 𝑡𝑎𝑖𝑙𝑒𝑑 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑡 𝑣𝑎𝑙𝑢𝑒 , 𝑑𝑓 = 𝑛 − 2
𝑠 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑟𝑟𝑜𝑟 𝑜𝑓 𝑓𝑜𝑟𝑒𝑐𝑎𝑠𝑡
1
𝑋−𝑋
𝑠 = 𝑆𝐸𝐸 × 1 + +
𝑛
𝑛−1 ×𝜎
Analysis of variance (ANOVA)

Analysis of variance (ANOVA) : analyse total variabiility of dependent variable
1
...
Regression sum of squares (RSS) : variation in the dependent variable that is explained by independent variable

𝑅𝑆𝑆 =

𝑌 −𝑌

3
...
Parameter instability : Linear relationships can change overtime → es mated equa on based on data from a specific me period may not be relevant for forecasts / predic on in another
time period
2
...
Assumptions underlying regression analysis do not hold → interpreta on and tests of hypothesis may not be valid
- Heteroskedastic : nonn-constant variance of error terms
- Autocorrelation : error terms are not independent

Concepts

Description

Multiple regression

Multiple Regression and Issues in Regression Analysis
Multiple regression : regression analysis with more than 1 independent variable

𝑌 = 𝑏 +𝑏 ×𝑋 +𝑏 ×𝑋 +⋯+ 𝑏 ×𝑋 +𝜀
Intercept term : value of dependent variable when all independent variables = 0
Slope coefficient : estimated change in dependent variable for 1 unit change in that independent variable, holding other independent variable constant (partial slope coefficient)

Hypothesis testing of Regression
Coefficient

Hypothesis testing of Regression Coefficient : to determine if that independent variable makes a significant contribution to explaining the variation in dependent variable
t-test
(*) 2 tailed test

𝑏 −𝑏
𝑠
𝑑𝑓 = 𝑛 − 𝑘 − 1
𝑘 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑖𝑛 𝑡ℎ𝑒 𝑟𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛
p-value : smallest significant level for which the null hypothesis can be rejected (compared -value straigth with significant level, no conversion table required)
- p-value < Significant level → Reject null hypothesis
- p-value > Significant level → Cannot reject null hypothesis
𝑡=

Confidence intervals for a
Regression Coefficient

Confidence intervals for a Regression Coefficient

𝑏 ± 𝑠 ×𝑡
𝑑𝑓 = 𝑛 − 𝑘 − 1

Assumptions in multiple regression 1
...
Independent variables are not random ; no exact linear relation between any group of independent variables
3
...
Varianc of error terms : constant for all observations
5
...
Error term is normally distributed
F-statistic

F-test : how well a set of independent variables - as a group - explains the variation in the dependent variable
(*) 1 tailed test

𝐻 ∶ 𝑏 = 𝑏 = ⋯ = 𝑏 = 0 ; 𝐻 ∶ 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 1 𝐻 ≠ 0
𝐹=
𝑑𝑓
𝑑𝑓

𝑀𝑆𝑅
𝑅𝑆𝑆⁄𝑘
=
𝑀𝑆𝐸 𝑆𝑆𝐸⁄ 𝑛 − 𝑘 − 1
=𝑘
= 𝑛−𝑘−1

𝑅𝑒𝑗𝑒𝑐𝑡 𝐻 𝑖𝑓 𝐹 > 𝐹
Coefficient of Determination /
Adjusted Coefficient of
Determination

1
...
Adjusted Coefficient of Determination : Similar with normal Coefficient of Determination, but overcome the weakness

𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛 = 𝑅
Dummy variable

=1−

𝑛−1
× 1−𝑅
𝑛−𝑘−1

Dummy variable : independent variable with binary nature , used to quantified impact of qualitative events
(*) Important consideration : distinguish between n-class → use n-1 variables
Example : regression equation explaining quarterly EPS

𝐸𝑃𝑆 = 𝑏 + 𝑏 × 𝑄 + 𝑏 × 𝑄 + 𝑏 × 𝑄 + 𝜀
𝑃𝑒𝑟𝑖𝑜𝑑 𝑡 𝑖𝑠 𝑄1 → 𝑄
𝑃𝑒𝑟𝑖𝑜𝑑 𝑡 𝑖𝑠 𝑄2 → 𝑄
𝑃𝑒𝑟𝑖𝑜𝑑 𝑡 𝑖𝑠 𝑄3 → 𝑄
𝑃𝑒𝑟𝑖𝑜𝑑 𝑡 𝑖𝑠 𝑄4 → 𝑄
Primary assumption violations

=1;
=1;
=1;
=𝑄

𝑄 =𝑄 =0
𝑄 =𝑄 =0
𝑄 =𝑄 =0
=𝑄 =0

1
...
Serial correlation (autocorrelation)
3
...

𝐴𝑣𝑔
...

𝐴𝑣𝑔
...
Definition : variance of residuals is not the same across all observations in the sample
- Unconditional heteroskedacity : heteroskedacity is not related to the level of independent variables → cause no major problems with the regression
- Conditional heteroskedacity : heteroskedacity is related to the level of independent variables → significant problems for sta s cal inference
2
...
Detection
- Method #1 : Examine scatter plot of residuals
- Method #2 : apply Breusch-Pagan chi-square test

𝐵𝑃 𝑐ℎ𝑖 − 𝑠𝑞𝑢𝑎𝑟𝑒 𝑡𝑒𝑠𝑡 = 𝑛 × 𝑅
𝑑𝑓 = 𝑘 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠 𝑡𝑜 𝑏𝑒 𝑡𝑒𝑠𝑡𝑒𝑑
4
...
White-corrected standard errors are used when only heteroskedacity appears
Serial correlation

1
...
Effect on Regression Analysis :
- Underestimate coefficient standard errors → Overes mate t-sta s c → Type 1 error : rejec on of null hypothesis when it is actually true
- Underestimate MSE → unreliable F-test → Type 1 error
3
...
Correcting serial correlation
- Adjust the coeffcient standard error using Hansen method (recommended by CFA, use Hansen method whenever there is serial correlation problem)
- Improve the specification of the model by incorprating the time-series nature of the data
Multicollinearity

1
...
Effect on Regression Analysis :
- Unreliable slope coefficients
- Artificially inflated standard errors → greater probability for incorrect conclusion that variable is significant
3
...
Correction : omit one or more of correlated independent variables

Model misspecification

Categories of model mispecification :
1
...
Independent variables are correlated with error term in time series model
- Model misspecification #4 : Using lagged dependent variables as independent variable
- Model misspecification #5 : Forecasting the past
- Model misspecification #6 : Measuring independent variables with error
3
...
Defintion : fail to include an important variable in the regression
2
...
Probit and logit models :
variables
- Probit : based on normal distribution
- Logit : based on logistic distribution
- Maximum likelihood methodology is used to estimate coefficients for probit and logit models
2
...
Constant and finite expected value
2
...
Constant and finite covariance between values at any given lag

Autoregressive model of order p - Autoregressive model of order p - AR(p) : p = number of lagged value
AR(p)
𝑥 = 𝑏 +𝑏 ×𝑥
+𝑏 ×𝑥
+⋯+𝑏 ×𝑥
+𝜀
Estimate one-period-ahead and
two-period-ahead

One-period-ahead forecast of AR(1) model

𝑥

=𝑏 +𝑏 ×𝑥

Two-period-ahead forecast of AR(1) model

𝑥

Test the autocorrelation of AR

=𝑏 +𝑏 ×𝑥

Step 1 : Estimate AR model being evaluated using linear regression
Step 2 : Calculate autocorrelations of the model's residuals
Step 3 : Test whether the autocorrelations are significant different from 0
𝜌 ,
𝑡
=
1⁄ 𝑇
𝜌 ,
= 𝑎𝑢𝑡𝑜𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛
𝑑𝑓 = 𝑇 − 2

𝑡
𝑡

<𝑡
>𝑡

→ 𝐴𝑅 𝑛𝑜𝑡 𝑠𝑒𝑟𝑖𝑎𝑙𝑙𝑦 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑒𝑑 → 𝐴𝑅 𝑚𝑜𝑑𝑒𝑙 𝑖𝑠 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑒𝑑 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑙𝑦
→ 𝐴𝑅 𝑖𝑠 𝑠𝑒𝑟𝑖𝑎𝑙𝑙𝑦 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑒𝑑 → 𝐴𝑅 𝑚𝑜𝑑𝑒𝑙 𝑖𝑠 𝑛𝑜𝑡 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑒𝑑 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑙𝑦

Mean reversion

Mean reversion : time-series that has tendency to move toward its mean
𝑏
𝑀𝑒𝑎𝑛 𝑟𝑒𝑣𝑒𝑟𝑡𝑖𝑛𝑔 𝑙𝑒𝑣𝑒𝑙 =
1−𝑏

In-sample forecasts /
Out-of-sample forecasts /
Root mean squared error

In-sample forecasts : forecasts made within the range of the sample period → measure the accuracy of the model in forecas ng the actual data used to develop the model
Out-of-sample forecasts : forecasts made outside of the sample period → measure the accuracy of the model in forecas ng the dependent variable for me period outsude the period
used to develop the model
Root mean squared error (RMSE) : compare the accuracy of AR models in forecasting out-of-sample values
Lower RMSE for out-of-sample data → lower forecast error → be er predic ve power in the future

Instability / Nonstationarity

Instability : due to financial and economic conditions are dynamic
- Shorter time series → more stable ; Longer me series → less stable (increase the chance that the underlying economic process has changed)
→ Trade off between increased reliability when using longer me periods vs stability when using shorter period

Random walk

1
...
Random walk with Drift

𝑥 =𝑏 +𝑥

+𝜀

3
...
Defintion : time-series pattern that tend to repeat year to year
2
...
Correction : add an additional lag as another independent variable (e
...
: seasonal lag = 4 for quarterly data ; seasonal lag = 12 for monthly data)

Autoregressive conditional
heteroskedacity (ARCH)

1
...
Testing ARCH(1) : using ARCH(1) regression model

𝜀 =𝑎 +𝑎 ×𝜀
+𝜇
𝑎 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑓𝑟𝑜𝑚 0 → 𝐴𝑅𝐶𝐻 1 𝑒𝑥𝑖𝑠𝑡𝑠
Cointegration

1
...
Testing : residuals are test for unit root using Dickey Fuller test

𝑦 = 𝑏 +𝑏 ×𝑥 +𝜀 → 𝑦 −𝑥 =𝑏 + 𝑏 −1 ×𝑥 +𝜀
𝑡
<𝑡
→ 𝑐𝑎𝑛𝑛𝑜𝑡 𝑟𝑒𝑗𝑒𝑐𝑡 𝐻 → 2 𝑠𝑒𝑟𝑖𝑒𝑠 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑐𝑜𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑒𝑑
𝑡
>𝑡
→ 𝑟𝑒𝑗𝑒𝑐𝑡 𝐻 → 2 𝑠𝑒𝑟𝑖𝑒𝑠 𝑎𝑟𝑒 𝑐𝑜𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑒𝑑
𝑡
𝑖𝑠 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑢𝑠𝑖𝑛𝑔 𝐸𝑛𝑔𝑙𝑒 𝑎𝑛𝑑 𝐺𝑟𝑎𝑛𝑔𝑒𝑟
Regression using 2 time-series

𝑦 =𝑏 +𝑏 ×𝑥 +𝜀
Situations when working with 2 time series in a regression :
(1) Independent time series and Dependent time series has no unit root → Both are covariance sta onary → regression is valid ; coefficients are reliable
(2) Independent time series or Dependent time series has unit root → Only 1 me series is covariance sta onary → regression is invalid
(3) Independent time series and Dependent time series has unit root and are cointegrated → regression is valid
(4) Independent time series and Dependent time series has unit roor and are not cointegrated → regression is invalid

Method to determine the
appropriate model

Step 1 : Determine the goal
- to model the relationship of variables to other variables
- to model the variable over time
Step 2 : If using time-series analysis, lool for indicator of nonstationarity, e
...
:
- Heteroskedacity (non-constant variance)
- Non-constant mean
- Seasonality
- Structural change (shift in plotted data → divide data into 2 or mode dis nct pa erns)
Step 3 : If no seasonality or structural change → use trend model (linear or log-linear)
Step 4 : Run trend analysis, compute residuals, test for serial correlation using Durbin Watson test
- No serial correlation → use the model
- Yes serial correlation → Step 5
Step 5 : If data has serial correlation → reexamine the data for sta onarity
...
)
- 3 approaches to specify a distribution :
+ Historical data : assume future values of variables will be similar to its past
+ Cross-sectional data : estimate the distribution of variables based on peers' data
+ picking a distribution and estimate the parameters : when both historical data and cross-sectional data are not appropriate
Step 3 - Check for correlations among variables : use historical data to determine whether any input variables are systematically related
...
Number of simulations is driven by:
- Number of uncertain variables : More probabilistic inputs → more simula ons needed
- Types of distributions : More variability in types of distributions → more simula ons needed
- Range of the outcomes : wider range of outcomes of uncertain variables → more simula on needed

Common constraints of simulations 1
...
Earnings and CF constraints
- Imposed internally to meet analyst expectations / to achieve bonus targets
...
g
...
Market value constraints : to minimise the likelihood of financial distress / bankruptcy for the firm
Advantages of simulations

1
...
Provide distribution of expected value rather than point estimate → indicate risk in the investment
(*) If required rate of return is incorporated in the underlying rate via discount rate → projects should only be compared based on higher mean NPV, not double counted

Limitations of simulations

1
...
Inappropriate statistical distributions : improperly specified distribution of an input → poor quality of that input
3
...
Dynamic correlations : Correlations between input variables may not be stable → flawed output of the correla on changes

Scenario analysis / Decision trees / Method
Simulations
Simulations

Distribution of risk

Sequential

Correlated variables

Continuous

Does not matter

Can be built into the simulations

Scenario analysis

Discrete

No

Decision trees

Discrete

Yes

Note

Can be incorporated into scenario
analysis
- Compute value of investment under finite set of scenarios
- Full spectrum is not covered → total probability of outcomes <
1
Can not incorporated into decision - Decision trees consider all possible states of the outcome →
trees
sum of probability = 1



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