Linear Regression Exam Q

Question 1

Gauss-Markov Theorem

Answer 1

• under certain conditions, the OLS estimator of the coefficients of a linear regression model is the best linear unbiased estimator

Question 2

Gauss-Markov Assumptions

Answer 2

• Linearity of the model in parameters
• no perfect multicollinearity
• strict exogeneity
• homoscedasticity
• no serial correlation

Question 3

no perfect multicollinearity

Answer 3

none of the explanatory variables are exact linear combinations of the others

Question 4

Homoscedasticity

Answer 4

• the error terms have constant variance

Question 5

no serial correlation

Answer 5

• errors at different time points should be uncorrelated
• Cov(ϵ_t,ϵ_t-k) = 0 (∀ k != 0)

Question 6

Explain how to test a null hypothesis e.g. H₀ = 0

Answer 6

• state the hypotheses: H₀, H_a
• calculate the t-statistic: t = (a - 0) / SE(a)
• If t-stat > 1.96 => statistically different from zero
• reject null hypothesis

Question 7

Durbin-Watson Procedure

Answer 7

• Fit the regression and obtain the residuals (rearrange)
• Calculate the DW statistic
• Interpret the statistic
• Compare to critical values based on number of observations and number of explanatory variables

Question 8

How to interpret Durbin-watson t-statistic

Answer 8

• DW = 2: no autocorrelation
• 2 > DW > 0: positive autocorrelation
• 4 > DW > 2: negative autocorrelation

Question 9

How to interpret DW critical values

Answer 9

• DW > upper critical: fail to reject
• upper critical > DW > lower critical: inconclusive
• DW < lower critical: reject null

Question 10

DW Hypotheses

Answer 10

• H₀: no first order autocorrelation
• H_a: first order autocorrelation exists

Question 11

DW statistic

Answer 11

DW = (∑^T_t=2(ε^{^}_t - ε^{^}_t-1)²) / (∑^T_t=1 ε^{^2}_t)
where:
- ε^{^}_t are the regression residuals
- T is the number of time periods

Question 12

Consequences of serial correlation for OLS estimators

Answer 12

• The OLS estimators remain unbiased but are no longer efficient
• The standard errors are now biased, leading to invalid standard inference
• To correct for this, use robust standard errors such as newey-west to ensure reliable statistical inference

Question 13

T-test procedure

Answer 13

• State the hypotheses
• Calculate the t-statistic
• Rule of thumb, reject null if |t-stat| > 1.96