Q: What is the formula for the variance of the OLS estimator $$\hat{\beta}_1 in $$the presence of autocorrelation, as given in the study materials?

$$\displaystyle \text{var}(\hat{\beta}_1) = \frac{1}{T} \frac{\sigma_v^2$}{(\sigma_X^2$)^2$$} \times f_T$

Question 1

Which of the following best describes the purpose of the $$f_T$$ factor in time series econometrics?

Accepted Answer

It adjusts for the presence of heteroskedasticity and autocorrelation in the error terms.

Question 2

What is the formula for the variance of the OLS estimator $$\hat{\beta}_1 in $$the presence of autocorrelation, as given in the study materials?

Accepted Answer

$$\displaystyle 	ext{var}(\hat{\beta}_1) = \frac{1}{T} \frac{\sigma_v^2$}{(\sigma_X^2$)^2$$} 	imes f_T$

Question 3

Which estimator is most commonly used for $$f_T in $$the context of HAC standard errors?

Accepted Answer

Newey-West estimator

Question 4

What is the formula for the Newey-West estimator $$\hat{f}_T$$?

Accepted Answer

$$\displaystyle \hat{f}_T = 1 + 2 \sum_{j=1}^{m-1} \frac{m-j}{m} \hat{\rho}_j$$

Question 5

In the Newey-West estimator, what does the parameter $$m$$ represent?

Accepted Answer

The truncation parameter, which determines the number of autocorrelations included.

Question 6

Which rule of thumb is suggested for choosing the truncation parameter $$m in $$the Newey-West estimator?

Accepted Answer

$$m = 0.75 (T$$^{1/3}$$)$$

Question 7

Why are HAC (Heteroskedasticity- and Autocorrelation-Consistent) standard errors important in time series analysis?

Accepted Answer

They provide valid inference when errors are heteroskedastic and/or autocorrelated.

Question 8

Which of the following is a key assumption for using Generalized Least Squares (GLS) or Autoregressive Distributed Lag (ADL) estimation for dynamic causal effects?

Accepted Answer

The regressors are strictly exogenous.

Question 9

Why is the assumption of strict exogeneity considered very strong and rarely plausible in practice?

Accepted Answer

Future values of the regressors can be correlated with current errors.

Question 10

Suppose you are analyzing the effect of weather on orange juice sales using a distributed lag model. Why might the strict exogeneity assumption fail in this context?

Accepted Answer

Weather patterns may influence both current and future orange juice sales, violating strict exogeneity.

Question 11

Which of the following statements about clustered standard errors is correct?

Accepted Answer

They are appropriate when the number of clusters ($$n) is $$large.

Question 12

If you use standard errors that do not account for autocorrelation in time series data, what is the likely consequence?

Accepted Answer

The standard errors will be biased, leading to invalid inference.

Question 13

Which of the following best describes the difference between HAC and HAR standard errors?

Accepted Answer

There is no difference; they are two names for the same concept.

Question 14

A researcher uses the Newey-West estimator with $$m = 1 on a $$time series with $$T = 100. $$What is the main risk of choosing $$m$$ too small?

Accepted Answer

Important autocorrelation may be ignored, leading to underestimated standard errors.

Question 15

Which of the following is NOT a reason why GLS and ADL approaches are rarely used in practice?

Accepted Answer

They are less efficient than OLS in the presence of autocorrelation.

Heteroskedasticity and Autocorrelation Consistent (HAC) Standard Errors and Strict Exogeneity in Time Series Regression

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