Reading 12: Multiple Regression

Question 1

Multiple Regression Equation

Answer 1

This shows several Betas for several independent variables that “predict” the dependent variable.

Question 2

What type of test needs to be performed in order to test the significance of one of the Betas? And what is the formula?

Answer 2

A t-statistic is done to check the significance

H0: Beta = 0

If t > t-critical : Reject H0

t-statistic =

Question 3

How can the significance be tested of individual components in the multiple regression?

Answer 3

Perform a t-statistic.

if t > t-critical reject H0

In the formula Beta tested agains the idea that the real beta = 0

Question 4

How can the significance be tested of a regression analysis as a whole?

Answer 4

An F-Test is to be performed. An F-test is performed as following:

F= MSR / MSE

MSR= RSS / k

MSE = SSE / [n-(k+1)]

Question 5

How is the Sum of Squared Errors (Residuals) calculated?

Answer 5

(SSE) with df = k

= Actual observation - Expected Observantion

Question 6

How is the Regression Sum of Squares calculated?

Answer 6

RSS is the explained variation by the independent variables.

= Expected Y - average observation Y

Question 7

F - Test is done how?

Answer 7

Question 8

What is R²?

Answer 8

R² is a measure of goodness of fit of the estimated regression of the data.

R² = Explained Varation / Total Variation

R² = RSS / SSE

Question 9

What is the R² - Adjusted?

Answer 9

This adjusted version of the R2 is done as this statistic is not automatically increasing when another variablle is added and is adjusted for degrees of freedom.

Question 10

A. What is Hetroskedasticity?

B. What are the consecquences?

C. How can we test for it?

D. How can it be corrected?

Answer 10

A. Hetroskedasticity is a non-constant variance of the errors accross the data set.

B. The Regression coefficients are not affected. ONLY the standard error of regression. -> in general lower erros lead to higher t-values -> leads to lower p-values.

C. Tested by the Breusch-Pagan test.

D. Corrected by:

• White corrected standard erros -> adjusts the standard errors of the LRM.
• Generalized least squares

Question 11

How and why is the Breusch - Pagan Test performed?

Answer 11

Done to test for conditional Hetroskedasticity in the regression.

dit is an Chi-statistic

X² = n R²

with k degrees of freedom (independant variables)

Question 12

What is Serial Correlation (Autocorrelation)?

Answer 12

This means that there is correlation among the error terms. This mostly happens in a time-series regression.

Question 13

Serial Correlation (Autoregression) A:

• Consequences?

Answer 13

There is Positive Serial Correlation:

• A positive error for one observation increases the chance of a positive error for another observation.
• We assume that the sign of the error term tends to persist from one period to the other.

Consequences of serial correlation: is an incorrect standard error of regression.

F-test may be inflated by underestimated MSE

T-test will be inflated and p-values show significance where not appropriate.

Question 14

Serial Correlation (Autoregression) B:

How to test for this?

Answer 14

We test with the Durbin and Watson test.

DW can be estimated by

DW = 2 (1-r)

H₀ = NO serial correlation

Question 15

Serial Correlation (Autocorrelation) C:

When do we reject the H_⁰: No serial correlation?

Answer 15

DW > D_u -> Fail to reject the null hypothesis

DW < D_l -> Reject the null hyphotesis

D_l u -> inconclusive test.

IF WE REJECT: UNDERESTIMATED ST. ERRORS

Question 16

How to correct for Serial Correlation (Autocorrelation)?

Answer 16

Two Ways:

1. Adjust the coefficient standard error (Hansen Method)
2. Modify the regression equation itself.

Hansen’ Method corrects for both Serial Correlation & conditional Hetroskedasticity

Question 17

Multicollinearity (A)

What is it?

Answer 17

Multicollinearity occurs when two or more independent variables are highly correlated

If the independent variables are perfectly correlated regression is impossible and is called perfect collinearity.

Question 18

Multicollinearity (B)

What is the problem generated by multicollinearity in a regression analysis?

Answer 18

• Presence does not affect the consistency of the OLS estimates / regression Coefficients
• Due to multicollinearity its impossible to distinguish the individual impacts of the independent variables on the dependent variable.
• Inflated OLS standard errors for the regression coefficients. Hence T-statistic very small -> no ability to reject H₀

Question 19

Multicollinearity (C)

How to detect Multicollinearity?

How to correct?

Answer 19

No statistic to measure it however:

If F-test is significant and High R² and t-tests are insignificant

Correct by ommitting one variable.

Question 20

Summary of Heteroskedasticity, Serial Correlation, Multicollinearity:

Answer 20

Question 21

Model Misspecification?

Answer 21

1. One or more important variables could be omitted from the regression
2. One or more of the regression variables may need to be transformed before estimating the regression
3. The regression model pools data from different samples that should not be pooled
4. Independent variables are correlated with error terms (violation)