Tutorial 1

Question 1

What are the implications of strict exogeneity for the relationship between explanatory variables and the error term in a time-series regression?

Answer 1

• mean independent of the error term in all time periods i.e. uncorrelated

Question 2

What assumption is needed to derive the OLS estimator and why?

Answer 2

• no perfect multicollinearity
• OLS estimation involves inverting the X’X matrix, if there is perfect multicollineartiy then it is singular (non-invertible)

Question 3

Conditions for Unbiased OLS Estimator

Answer 3

• linearity
• strict exogeneity
• no perfect multicollinearity
• random sampling

Question 4

What is the finite sampling distribution of an estimator?

Answer 4

the probability distribution of the estimator’s values across different random samples drawn from the same population

Question 5

Why is the sampling distribution of interest?

Answer 5

• allows us to draw statistical inference (conduct hypothesis tests about the unknown population parameter)

Question 6

What assumptions are needed about error terms to derive the finite sampling distribution?

Answer 6

• error terms are normally distributed with mean zero and constant variance

Question 7

Consequence of weak exogeneity

Answer 7

• the OLS estimator is no longer unbiased as unbiasedness requires holding for all possible realisations of X, not just conditional on a specific X

Question 8

What is meant by consistency and large sample (asymptotic) sampling distribution?

Answer 8

• consistency ensures that the estimator converges to the true parameter as the sample size increases. Similar to unbiasedness in finite samples.
• asymptotic sampling distribution of an estimator is the probability distribution it approaches as the sample size goes to infinity, serving as an approximation of the true sampling distribution.

Question 9

Assumptions for consistency and normal asymptotic sampling distribution?

Answer 9

for consistency:
- linearity
- weak exogeneity
- no perfect multicollinearity
in addition, for asymptotic normality:
- homoscedasticity
- no autocorrelation

Question 10

What is residual autocorrelation?

Answer 10

• the error terms are correlated

Question 11

Test for residual autocorrelation?

Answer 11

• durbin-watson test
• breusch-godfrey test

Question 12

How might you deal with residual autocorrelation?

Answer 12

• if the error terms are serially correlated, we can still apply OLS (still unbiased) but need to use newey-west standard errors as they are robust to heteroscedasticity and autocorrelation

Question 13

What is heteroscedasticity?

Answer 13

• the variance of the error term is not constant, conditional on X

Question 14

What are the consequences of heteroscedasticity?

Answer 14

• OLS estimator remains unbiased but is no longer efficient (BLUE)
• Standard errors are biased, standard inference is invalid
• To resolve this, apply robust standard errors

Question 15

Heteroscedasticity tests

Answer 15

• breusch-pagan test
• white test

Question 16

How might you deal with heteroscedasticity?

Answer 16

• if the error terms are heteroscedastic we can still apply OLS (still unbiased) but need to use robust standard errors

Question 17

what does weak exogeneity mean?

Answer 17

• if they are weakly exogenous, the error term is mean independent of current explanatory variables

Question 18

Breusch-Pagan Procedure

Answer 18

• Fit the regression and obtain the residuals
• Square the residuals and regress them on the explanatory variables
• calculate the test statistic
• compare with critical value from chi-squared distribution, based on degrees of freedom

Question 19

Breusch-Pagan Hypotheses

Answer 19

• H₀: Homoscedasticity
• H₁: Heteroscedasticity

Question 20

Breusch-Pagan Decision Rule

Answer 20

• If LM > critical value => reject null
• Otherwise, fail to reject

Question 21

Breusch-Pagan Test Statistic

Answer 21

LM = n * R²
- n = sample size
- R² = coefficient of determination