Quantitative Econometrics Problems

In an OLS regression $y = \beta_0 + \beta_1 x + u$, which formula gives the OLS estimator $\hat{\beta}_1$?

Given estimated coefficient $\hat{\beta}_1 = 2.5$ and its standard error $se(\hat{\beta}_1)=0.5$, what is the two-sided $t$-statistic for testing $H_0:\beta_1=0$?

Which expression gives the estimated variance of the OLS residuals $\hat{u}_i$ used to compute standard errors (homoskedastic case)?

An estimated simple regression gives $R^2=0.64$. Which interpretation is most accurate?

You estimate $\hat{y}_0$ at $x_0$ from OLS. Which formula gives the standard error of the prediction for the mean response $E[y|x_0]$ (assuming homoskedasticity)?

In presence of heteroskedasticity, which standard error is consistent for inference on $\beta$ without modeling the heteroskedasticity form?

A $t$-test for $H_0:\beta_1=0$ rejects at level $\alpha$ if $|t|>t_{n-k,1-\alpha/2}$.

Increasing the sample size always increases the $R^2$ of an OLS regression.

When constructing a prediction interval for a new observation at $x_0$, the interval is wider than the confidence interval for the mean response at $x_0$.

Explain why heteroskedasticity-robust standard errors are used and one limitation of using them.

Match each term with its definition: