Correlation And Regression
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LOS a
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Sample
covariance
Sample
correlation
Measures how two variables move
together
Measures strength of linear relationship
between two variables
Captures the linear relationship between
two variables
Cov(x,y) =
Standardized measure of covariance
∑ (X − X) (Y − Y)
Cov(x,y)
r=
n−1
Cov(x,y) = r × Sx × Sy
Sx × Sy
Unit = No unit
2
Range = −1 to +1
Range = −∞ to +∞
r = 1 means perfectly +ve correlation
+ve covariance = Variables tend to
move together
r = 0 means no linear relationship
e
Unit = %
r = −1 means perfectly −ve correlation
re
−ve covariance = Variables tend to
move in opposite directions
−ve covariance
−ve correlation
−ve slope
+ve covariance
+ve correlation
+ve slope
Scatter plot: Graph that shows the relationship between values of two variables
LOS b
nT
Limitations to correlation analysis
Nonlinear relationship
Outliers
Spurious correlation
Measures only linear
relationships, not non linear
ones
Extremely large or small
values may influence the
estimate of correlation
Appearance of causal linear
relationship but no economic
relationship exists
Fi
LOS c Test of the hypothesis that the population correlation coefficient equals zero
Eg.
r = 0.4
n = 62
Confidence level = 95%
Step 1:
Define hypothesis
Step 2:
Calculate test statistic
Step 3:
Calculate critical values
Perform a test of significance
H0: r = 0, Ha: r ≠ 0
r × √n − 2
√1 − r
2
0.4 × √62 − 2
√1 − 0.42
3.2
t-distribution, DoF = 60
−2
+2
Since calculated test statistic lies outside the range, conclusion is ‘Reject the null hypothesis’
‘r’ is statistically significant, which means that population ‘r’ would be different than zero
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LOS d
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Dependent variable
Independent variable
Variable you are seeking to
explain
Variable you are using to explain
changes in the dependent variable
Also referred to as explained
variable/endogenous
variable/predicted variable
Also referred to as explanatory
variable/exogenous
variable/predicting variable
y
Dependent
variable
RFR
+
β
(Rm − RFR)
Independent
variable
Intercept
LOS e
Dependent
variable
Rp =
Slope
Independent
variable
x
Assumptions underlying linear regression
Œ

Ž

Sum of squared errors (SSE):
Regression line:
Sum of the squared vertical distances between the estimated
and actual Y-values
Line that minimizes the SSE
Describes change in ‘y’ for one unit change in ‘x’
nT
Slope coefficient (beta):
re
e
Relationship between dependent and independent variable is linear
Independent variable is uncorrelated with the error term
Expected value of the error term is zero
Variance of the error term is constant (NOT ZERO). The economic relationship
b/w variables is intact for the entire time period (eg. change in political regime)
 Error term is uncorrelated with other observations (eg. seasonality)
‘ Error term is normally distributed
Cov (x,y)
Variance (x)
LOS f
‘x’
10
15
20
30
Actual ‘y’
17
19
35
45
Predicted ‘y’
15.81
23.36
30.91
46.01
Errors
1.19
−4.36
4.09
−1.01
Squared errors
1.416
19
16.73
1.02
Fi
Eg.
Standard error of estimate, coefficient of determination
and confidence interval for regression coefficient
Standard error of estimate (SEE) =
Coefficient of determination (R2):
√
SSE
n−2
=
Sum of squared errors
(SSE)
38.166
38.166
2
√
= 4.36
% variation of dependent variable explained by % variation of
the independent variable
For simple linear equation, R2 = r2
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Confidence interval for regression coefficient
^
b1 ± (tc × SE)
Slope
Standard
error
Critical value
(t-value)
^
b1 = 0.48
Eg.
SE = 0.35
n = 42
Confidence interval:
LOS g
0.48 ± (1.684 × 0.35)
−0.109 to 1.069
Hypothesis testing for population value of a regression coefficient
^
b1 = 0.48
SE = 0.35
n = 42
Confidence interval = 90%
^
Step 1:
Define hypothesis
Step 2:
Calculate test statistic
Step 3:
Calculate critical values
Perform a test of significance
^
H0: b1 = 0, Ha: b1 ≠ 0
Sample stat. − HV
0.48 − 0
e
Eg.
Calculate 90% confidence interval
Std. error
0.35
1.371
t-distribution, DoF = 40
1.684
re
−1.684
Since calculated test statistic lies inside the range, conclusion is ‘Failed to reject the null hypothesis’
Slope is not significantly different from zero
LOS h & i
Confidence interval for the predicted
value of dependent variable
nT
Predicted value of
dependent variable
^
Y
^
^
b0 + b1 × Xp
=
Intercept
Predicted
value (y)
Forecasted
value (x)
± (tc × SE)
Predicted
value (y)
Slope
Fi
Eg.
^
Y
Forecasted return (x) = 12%
n = 32
Intercept = −4%
Critical value
(t-value)
Slope = 0.75
Confidence interval
^
^
=
^
Standard error = 2.68
Calculate predicted value (y) and 95% confidence interval
Predicted value
Y
Standard
error
^
b0 + b1 × Xp
Y
± (tc × SE)
5 ± (2.042 × 2.68)
Y = −4 + 0.75 × 12 = 5%
−0.472 to 10.472
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LOS j
Analysis of variance (ANOVA)
Y: Mean
Yi: Actual value
Sum of squared errors
(SSE)
Measures unexplained
variation
aka sum of squared
residuals
^
Yi: Predicted value
Regression sum of
squares (RSS)
Total sum of squares
(SST)
Measures explained
variation
Measures total
variation
^
∑ (Yi − Yi)2
∑ (Yi − Yi)2
^
∑ (Yi − Yi)2
ª Higher the RSS, better the quality of regression
ª R2 = RSS/SST
ª R2 = Explained variation/Total variation
e
ANOVA Table
DoF
Sum of squares
Mean sum of squares
Regression
(explained)
k
RSS
MSR = RSS/k
Error
(unexplained)
Total
re
Source of variation
n−k−1
SSE
n−1
SST
MSE = SSE/n − k − 1
F-statistic = MSR/MSE with ‘k’ and ‘n − k − 1' DoF
nT
When to use F-test and t-test
F-test
Y = b 0 + b 1 x1 + b 2 x2 + ε
t-test
Limitations of regression analysis
Fi
LOS k
t-test
Linear relationships can change over time (parameter instability)
Public knowledge of regression relationship may make their future usefulness ineffective
If the regression assumptions are violated, hypothesis tests will not be valid
(heteroscedasticity and autocorrelation)
Multiple Regression And Issues In
Regression Analysis
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LOS a
Multiple regression equation
Y
b0 + b1 X1 + b2 X2 + …. + bk Xk + ε
=
Intercept
Dependent
variable
LOS b
Independent
variable
Slope
Interpreting estimated regression coefficients
Slope
coefficient
Value of dependent variable
when all independent
variables are equal to zero
Measures how much
dependent variable changes
when independent variable
changes by one unit,
holding other independent
variables constant
e
Intercept
term
re
LOS c & d
Hypothesis testing for population value of a regression coefficient
b1 = 0.15 SE1 = 0.38
Eg.
Error
term
b2 = 0.28 SE2 = 0.043
Confidence interval = 90%
Step 2:
Calculate test statistic
Calculate critical values
Perform a test of significance
H0: b1 = 0, Ha: b1 ≠ 0
H0: b2 = 0, Ha: b2 ≠ 0
Sample stat. − HV
0.15 − 0
Std. error
0.38
Sample stat. − HV
0.28 − 0
Std. error
0.043
0.394
6.511
t-distribution, DoF = 40
Fi
Step 3:
Define hypothesis
nT
Step 1:
n = 43
−1.684
1.684
Since calculated test statistic (b1) lies inside the range, conclusion is ‘Failed to reject the null hypothesis’
And test statistic (b2) lies outside the range, conclusion is ‘Reject the null hypothesis’
Variable with slope ‘b1’ is not significantly different from zero
and variable with slope ‘b2’ is significantly different from zero
Solution is to drop the variable with slope ‘b1’
DoF = n − k − 1
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P-value
Reject
FTR
5 ft.
P-value
FTR
Reject
4.5 ft.
5 ft.
FTR
3.8 ft.
4 ft.
6 ft.
Significance level
P-value is the lowest level of significance at which null hypothesis is rejected
LOS e
Predicted value of
dependent variable
Confidence interval for
regression coefficient
^
^
Y
Slope
^
^
^
^
^
Predicted
value (y)
Forecasted
value (x)
re
Slope
Assumptions of a multiple regression model
nT
ΠRelationship between dependent and independent variable is linear
 Independent variables are uncorrelated with the error term and there is no
exact linear relation between two or more independent variables
Ž Expected value of the error term is zero
 Variance of the error term is constant (NOT ZERO). The economic relationship
b/w variables is intact for the entire time period (eg. change in political regime)
 Error term is uncorrelated with other observations (eg. seasonality)
‘ Error term is normally distributed
LOS g
F-statistic
ª F-statistic = MSR/MSE with ‘k’ and ‘n − k − 1' DoF
Fi
^
^
b0 + b1 X1 + b2 X2 + …. + bk Xk
Intercept
Standard
error
Critical value
(t-value)
LOS f
=
e
b1 ± (tc × SE)
ª It is used to check the quality of entire regression model
ª One-tailed test, rejection region is on right side
ª If the result of F-test is significant, at least one of the independent variable is
able to explain variation in dependent variable
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n = 48
Eg.
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SST = 430
k=6
SSE = 190
Significance level = 2.5% and 5%
Perform an F-test
RSS =
SST − SSE
430 − 190
MSR =
RSS
k
240
6
40
MSE =
SSE
n−k−1
190
41
4.634
MSR
MSE
40
4.634
8.631
F-statistic =
240
Critical value (F-table) at 2.5% significance level (DoF 6,41) = 2.74
Calculated test statistic is on the right of critical value, therefore the conclusion is ‘Reject the null hypothesis’
Since the conclusion at 2.5% significance is ‘Reject’, the conclusion at 5% significance is also ‘Reject’
All the variables are significantly different from zero
2
LOS h
2
R and adjusted R
e
R2: % variation of dependent variable explained by % variation of all the independent variables
R2 = RSS/SST
re
R2 = Explained variation/Total variation
Adjusted R2 =
1−
])
)
n−1
n−k−1
]
× (1 − R2)
Adjusted R2 < R2 in multiple regression
n = 30
k=6
nT
Eg.
n = 30
k=8
R2 = 73%
R2 = 75%
Adjusted R21 =
1−
])
)
]
41.1%
Adjusted R22 =
1−
])
)
]
39.58%
30 − 1
× (1 − 0.732)
30 − 6 − 1
30 − 1
× (1 − 0.752)
30 − 8 − 1
Fi
Adding two more variables is not justified because adjusted R22 < adjusted R21
LOS i
ANOVA table
Source of variation
DoF
Sum of squares
Mean sum of squares
Regression
(explained)
k
RSS
MSR = RSS/k
Error
(unexplained)
n−k−1
SSE
MSE = SSE/n − k − 1
Total
n−1
SST
F-statistic = MSR/MSE with ‘k’ and ‘n − k − 1' DoF
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LOS j
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Multiple regression equation by using dummy variables
Y
b0 + b1 X1 + b2 X2 + …. + bk Xk + ε
=
Intercept
Dependent
variable
Independent
variable
Slope
Error
term
Dummy variables: Independent variables that are binary in nature (i.e. in the form of yes/no)
They are qualitative variables
Values: If true = 1, if false = 0
Use n – 1 dummy variables in the model
LOS k & l
Types of heteroskedasticity
Unconditional
e
Conditional
nT
Causes problems for
statistical inference
re
Occurs when
heteroskedasticity of
the error variance is
correlated with the
independent variables
Occurs when
heteroskedasticity of
the error variance is
not correlated with the
independent variables
Does not cause major
problems for statistical
inference
Conditional
heteroskedasticity
Positive serial
correlation
Negative serial
correlation
Multicollinearity
Meaning
Variance not
constatnt
Errors are
correlated
Errors are
correlated
Two or more
independent variables
are correlated
Effect
Type I errors
Type I errors
Type II errors
Type II errors
Detection
Examining scatter
plots or BreuschPagan test
Durbin-Watson
test
Durbin-Watson
test
F - significant
t - not significant
Correction
White-corrected
standard errors
Hansen method
Hansen method
Drop one of the
variables
Fi
Violations
ª
ª
ª
ª
ª
Breusch-Pagan test: n × R2
White-corrected standard errors is also known as robust standard error
Durbin-Watson test ≈ 2(1 − r).
Multicollinearity: The question is never a yes or no, it is how much
None of the assumption violations have any impact on slope coefficients.
The impact is on standard errors and therefore on t-test
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LOS m
Model
specifications
Model
misspecifications
Model should have strong
economic reasoning
Omitting a variable
Variable should be transformed
Functional form of the variables
should be appropriate
Incorrectly pooling data
The model should be
parsimonious (concise/brief)
Using lagged dependent variable
as an independent variable
The model should be examined
for violations of assumptions
Forecasting the past
Measuring independent variables
with error
Model should be tested on out of
sample data
LOS n
Models with qualitative dependent variables
Logit
Based on the logistic
distribution
nT
Based on the normal
distribution
re
Probit
LOS o
e
Model misspecifications might have impact on both slope coefficient and error terms
Discriminant
Similar to probit and logit
but uses financial ratios as
independent variables
Interpretation of multiple regression model
Values of slope coefficients suggest that there is economic relationship
between the independent and dependent variables
But it may also be possible for a regression to have statistical
significance even when there is no economic relationship
Fi
This statistical significance must also be factored into the analysis
Time-series Analysis
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LOS a
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Predicted trend value for a time series
Time series:
Set of observations on a variable’s outcomes in different time periods
Used to explain the past and make predictions about the future
Linear trend
models
Log-linear
trend models
Log-linear trend is a trend in
which the dependent variable
changes at an exponential
rate with time
Linear trend is a trend in
which the dependent variable
changes at a constant rate
with time
Used for financial time series
Has a straight line
Has a curve
Upward-sloping line:
+ve trend
e
Convex curve:
+ve trend
Downward-sloping line:
−ve trend
LOS b
re
Equation:
yt = b0 + b1t + εt
Concave curve:
−ve trend
Equation:
ln yt = b0 + b1t + εt
How to determine which model to use
nT
Plot the data
Fi
y
Linear trend
model
y
x
x
Log-linear
trend model
Limitation of trend models is that they are not useful if the error terms are serially correlated
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LOS c
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Requirement for a time series to be covariance stationary
A time series is covariance stationary if it satisfies the following three conditions:
Constant and finite mean
Constant and finite variance (same as homoskedasticity)
Constant and finite covariance of time series with itself
Eg.
Xt =
b0
+
b1 Xt−1
Xt =
5
+ 0.5 Xt−1
Xt = 8
Xt − 1 = 20
Xt = 15
Xt − 1 = 8
Xt = 9
Xt − 1 = 15
Xt = 12.5
Xt − 1 = 9
Xt = 9.5
e
Xt − 1 = 6
re
Xt − 1 = 12.5
Xt − 1 = 10
Xt = 11.25
Xt = 10
If Xt − 1 = 10, then Xt = 10, Xt + 1 = 10, Xt + 2 = 10 and so on
This is called constant and finite mean
b0
1 − b1
=
5
1 − 0.5
=
10
nT
Mean of the time series =
For a model to be valid, time series must be covariance stationary
Most economic and financial time series relationships are not stationary
The model can be used if the degree of nonstationarity is not significant
Autoregressive (AR) model
Fi
LOS d
AR model: A time series regressed on its own past values
Equation AR(1): Xt = b0 + b1Xt − 1 + εt
Equation AR(2): Xt = b0 + b1Xt − 1 + b2Xt − 2 + εt
Chain rule of forecasting: Calculating successive forecasts
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LOS e
Autocorrelations of the error terms
If the error terms have significant serial correlation (autocorrelation), the AR
model used is not the best model to analyze the time series
Procedure to test if the AR model is correct:
Step 1: Calculate the intercept and slope using linear regression
Step 2: Calculate the predicted values
Step 3: Calculate the error terms
Step 4: Calculate the autocorrelations of the error terms
Step 5: Test whether the autocorrelations are significantly different from zero
If the autocrrelations are not statistically
significant from zero (if the decision is FTR):
Model fits the time series
If the autocrrelations are statistically
significant from zero (if the decision is reject):
Model does not fit the time series
Test used to know if the autocorrelations are significantly different from zero: t-test
Autocorrelation
Standard error
t statistic =
LOS f
e
Mean reversion
It means tendency of time series to move toward its mean
LOS g
Eg.
b0
1 − b1
re
Mean reverting level =
In-sample and out-of-sample forecasts and RMSE criterion
Xt − 1
Predicted
value
Error
Squared
errors
-
-
-
-
200
216.5
3.5
12.25
215
220
227.8
−12.8
163.84
205
215
225
−20
400
235
205
219.4
15.6
243.36
250
235
236.4
13.6
184.96
Sample
value (Xt)
200
Fi
nT
220
In-sample root mean
squared error (RMSE)
√
1004.41
SSE
n
1004.41
5
√
=
14.17
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Eg.
Actual
value
Predicted
value
Error
Squared
errors
215
-
-
-
235
225
10
100
220
236.4
−16.4
268.96
240
227.9
12.1
146.41
250
239.2
10.8
116.64
632
Out-of-sample root mean
squared error (RMSE)
√
SSE
n
√
632
4
=
12.57
Select the time series with lowest out-of-sample RMSE
LOS h
Instability of coefficients of time-series models
e
One of the important issues in time series is the sample period to use
Shorter sample period → More stability but less statistical reliability
Longer sample period → Less stability but more statistical reliability
Random walk
Random walk
with a drift
A time series in which
predicted value of a dependent
variable in one period is equal
to the value of dependent
variable in previous period
plus an error term
A time series in which
predicted value of a dependent
variable in one period is equal
to the value of dependent
variable in previous period
plus or minus a constant
amount and an error term
Equation:
Xt = Xt − 1 + εt
Equation:
Xt = b0 + Xt − 1 + εt
nT
LOS i
re
Data must also be covariance stationary for model to be valid
Fi
ª Both of the above equations have a slope (b1) of 1
ª Such time series are said to have ‘unit root’
ª They are not covariance stationary because they do not have a finite mean
ª To use standard regression analysis, we must convert this data to covariance stationary.
This conversion is called ‘first differencing’
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LOS j & k
Unit root test of nonstationarity
Autocorrelation
approach
Dickey-Fuller
test
If autocorrelations do not exhibit
these characteristics, it is said to
be a nonstationary time series:
More definitive than
autocorrelation approach
Xt − Xt − 1 = b0 + b1Xt − 1 − Xt − 1 + εt
Autocorrelations at all lags are
statistically insignificant from zero
Xt − Xt − 1 = b0 + (b1 − 1)Xt − 1 + εt
or
If null (b1 − 1 (g) = 0) can not be
rejected, the time series has a unit
root
As the no. of lags increase, the
autocorrelations drop down to zero
First differencing
Lag 1
-
-
230
270
290
∆ sales
∆ sales
(current year) (previous year)
-
-
-
230
40
-
270
20
40
290
20
20
30
20
nT
310
First difference
e
Sales
re
Eg.
310
340
^
Equation: y = 30 − 0.25x
^
Equation: y = 30 − 0.25(340)
^
y = (55)
Forecasted sales: 340 − 55 = 285
If time series is a random walk then we must convert this data to covariance stationary.
This conversion is called first differencing
How to test and correct for seasonality
Fi
LOS l
Seasonality can be detected by plotting the values on a graph or calculating autocorrelations
Seasonality is present if the autocorrelation of error term is significantly different from zero
Correction: Adding a lag of dependent variable (corresponding to the same period in previous year)
to the model as another independent variable
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LOS m
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Autoregressive conditional heteroskedasticity (ARCH)
ARCH exists if the variance of error terms in one period is
dependent on the variance of error terms in previous period
Testing: Squared errors from the model are regressed on the first
lag of the squared residuals
Equation:
^2
εt
=
^2
Intercept
Predicted
error term of
current period
LOS n
μt
a0 + a1 εt − 1 +
Predicted
error term of
last period
Slope
Error
term
How time-series variables should be analyzed for
nonstationarity and/or cointegration
e
To test whether the two time series have unit roots, a Dickey-Fuller test is used
Possible scenarios:
nT
re
ΠBoth time series are covariance stationary (linear regression can be used)
 Only the dependent variable time series is covariance stationary (linear regression
should not be used)
Ž Only the independent variable time series is covariance stationary (linear regression
should not be used)
 Neither time series is covariance stationary and the two series are not cointegrated
(linear regression should not be used)
 Neither time series is covariance stationary and the two series are cointegrated
(linear regression can be used)
Cointegration: Long term economic or financial relationship between two time series
LOS o
Appropriate time-series model to analyze a given investment problem
ª Understand the investment problem you have and make a choice of model
Fi
ª If you have decided to use a time-series model plot the values to see whether the time series
looks covariance stationary
ª Use a trend model, if there is no seasonality or structural shift
ª If you find significant serial correlation in the error terms, use a complex model such as AR model
ª If the data has serial correlation, reexamine the data for stationarity before running an AR model
ª If you find significant serial correlation in the residuals, use an AR(2) model
ª Check for seasonality
ª Test whether error terms have ARCH
ª Perform tests of model's out-of-sample forecasting performance (RMSE)
Probabilistic Approaches: Scenario Analysis,
Decision Trees And Simulations
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LOS a, b & c
Step 1
Steps in running a simulation
Determine probabilistic variables:
No constraint on number of input variables that can be
allowed to vary.
Focus on a few variables that have significant impact on
value.
Step 2
Define probability distributions
for these variables:
Three ways to define probability distribution
Historical data: Useful when past data is available and
reliable. Estimate the distribution based on past values.
Cross-sectional data: Useful when past data is
unavailable or unreliable. Estimate the distribution
based on the values of similar variables.
Check for correlation across
variables:
Step 4
Run the simulation:
If the correlation is strong, either allow only one of the
variables to vary (focus on the variable that has the
highest impact on value) or build the correlation into
the simulation
re
Step 3
e
Statistical distribution and parameters: Useful when
historical and cross sectional data is insufficient or
unreliable. Estimate the distribution and its parameters.
It means to draw an outcome from each distribution
and compute the value based on these outcomes
Types of distributions: Greater the diversity of
distributions, greater the number of simulations
required.
Range of outcomes: Greater the range of outcomes,
greater the number of simulations required.
Advantages of using simulations in decision making
Fi
LOS d
nT
Number of probabilistic inputs: Higher the number of
probabilistic inputs, greater the number of simulations
required.
Better input estimation
Provides a distribution of expected
value rather than a point estimate
An analyst will usually examine both
historical and cross-sectional data
to select a proper distribution and
its parameters, instead of relying on
single best estimates. This results in
better quality of inputs
Simulations provide distribution of
expected value however they do not
provide better estimates
Simulations do not always lead to better decisions
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LOS e
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Common constraints introduced into simulations
Book value constraints
Earnings and CF constraints
Imposed internally:
Analyst’s expectations
Regulatory capital
restrictions
Likelihood of financial
distress
Imposed externally:
Loan covenants
Negative equity
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Market value constraints
Indirect bankruptcy costs
Issues in using simulations in risk assessment
ª Garbage in, garbage out: Inputs should be based on analysis and data, rather than guesswork
ª Inappropriate probability distributions: Using probability distributions that have no
resemblance to the true distribution of an input variable will provide misleading results
e
ª Non-stationary distributions: Distributions may change over time due to change in market
structure. There can be a change the form of distribution or the parameters of the distribution
re
ª Dynamic correlations: Correlation across input variables can be modeled into simulations only
when they are stable. If they are not it becomes far more difficult to model them
Risk-adjusted value
Cash flows from simulations are not risk-adjusted and should not be discounted at RFR
Asset
Risk-adjusted
discount rate
Expected value
using simulation
σ from
simulation
A
15%
$100
17%
18%
$100
21%
nT
Eg.
B
ª We have already accounted for B’s greater risk by using a higher discount rate
ª
If we choose A over B on the basis of A’s lower standard deviation, we would
be penalizing Asset B twice
Fi
ª An investor should be indifferent between the two investments
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Selecting appropriate probabilistic approach
Type of risk
Correlated?
Sequential?
Appropriate approach
Continuous
Yes
Doesn’t matter
Simulation
Discrete
Yes
No
Scenario analysis
Discrete
No
Yes
Decision tree