Quantitative Econometrics Problems

In OLS simple regression y = β0 + β1 x + u, which formula gives the estimator (\hat{\beta}_1)?

Given residual standard error $s = 2$, sample size $n=30$, and $\sum (x_i-\bar x)^2 = 50$, what is the standard error of $\hat{\beta}_1$?

You estimated $\hat{\beta}_1=1.5$ with $SE(\hat{\beta}_1)=0.5$. What is the two-sided t-statistic and its interpretation at the 5% level?

Regression yields SSR = 120, SSE = 30, and SST = 150. What is R-squared and adjusted R-squared if n=25 and k=2 (including intercept)?

When constructing a 95% prediction interval for a new observation at $x_0$, which of the following terms appears in the interval width but not in the confidence interval for the conditional mean $E[y|x_0]$?

You run OLS with heteroskedastic errors. Which inference is most robust without modeling the heteroskedasticity?

If the t-statistic equals 2.5 and df = 20, the p-value for a two-sided test is less than 0.05.

A high R-squared always implies the estimated coefficients are statistically significant.

Using cluster-robust standard errors can change t-test conclusions even if point estimates are unchanged.

Explain how to compute a 95% prediction interval for a new observation at $x_0$ using OLS estimates and what each component represents.

Match each term with its definition: