7.11 Question bank

No answers are given for this section, by design.

Multiple choice

QB 7.1 The OLS slope estimator equals

  1. the correlation between \(x\) and \(y\)
  2. the sample covariance of \(x\) and \(y\) divided by the sample variance of \(y\)
  3. the sample covariance of \(x\) and \(y\) divided by the sample variance of \(x\)
  4. the ratio of the two sample means

QB 7.2 Which of the following hold by construction in every OLS regression? (select all that apply)

  1. \(\sum \hat{u}_i = 0\)
  2. \(\sum x_i \hat{u}_i = 0\)
  3. \(E(u \mid x) = 0\)
  4. The line passes through \((\bar{x}, \bar{y})\)

QB 7.3 In a simple regression, \(\hat{\sigma}^2\) divides the sum of squared residuals by

  1. \(n\)
  2. \(n - 1\)
  3. \(n - 2\)
  4. \(n - 3\)

QB 7.4 \(\mathrm{Var}(\hat{\beta}_1)\) decreases when

  1. \(\sigma^2\) increases
  2. the spread of \(x\) increases
  3. the spread of \(y\) increases
  4. \(R^2\) decreases

QB 7.5 A regression of wages on schooling gives \(R^2 = 0.11\). This means

  1. the model is misspecified
  2. schooling accounts for 11% of the variation in wages
  3. 11% of the coefficient is reliable
  4. schooling causes 11% of wage differences

QB 7.6 Omitted variable bias arises when the omitted variable

  1. is correlated with \(y\) only
  2. is correlated with \(x\) only
  3. is correlated with both \(x\) and \(y\)
  4. has a large variance

Short reasoning

QB 7.7 Explain why \(\sum x_i\hat{u}_i = 0\) is not evidence that \(E(u \mid x) = 0\).

QB 7.8 A student reports that their regression “explains 95% of the variation, so the model must be correct.” Evaluate.

QB 7.9 Why does the divisor in \(\hat{\sigma}^2\) change from \(n-1\) in Unit 3 to \(n-2\) here?

QB 7.10 In the fertiliser example the slope was 1.25 quintals per kilogram. Explain what would happen to that number, and to its standard error, if fertiliser were measured in grams instead.

QB 7.11 Explain why a regression can have a correctly computed standard error, a valid \(p\)-value, and a completely misleading interpretation.

QB 7.12 For six observations, \(\sum(x_i - \bar{x})(y_i - \bar{y}) = 84\), \(\sum(x_i - \bar{x})^2 = 56\), \(\bar{x} = 9\), \(\bar{y} = 21\) and \(\sum \hat{u}_i^2 = 12\).

  1. Compute \(\hat{\beta}_1\) and \(\hat{\beta}_0\).
  2. Compute \(\hat{\sigma}^2\) and \(\mathrm{se}(\hat{\beta}_1)\).
  3. Test \(H_0: \beta_1 = 0\) at the 5% level.
  4. Construct a 95% confidence interval for \(\beta_1\).

Numerical

QB 7.13 A regression on 30 observations gives \(\hat{\beta}_1 = 4.8\), \(\mathrm{se}(\hat{\beta}_1) = 1.5\), \(\text{SST} = 900\) and \(\text{SSR} = 400\).

  1. Compute \(R^2\).
  2. Test \(H_0: \beta_1 = 0\) at the 1% level.
  3. Test \(H_0: \beta_1 = 8\) at the 5% level.
  4. Compute \(\hat{\sigma}^2\).

QB 7.14 Yields and rainfall for five districts:

Rainfall (cm) 60 80 100 120 140
Yield (quintals) 18 23 24 30 35
  1. Compute the OLS estimates.
  2. Compute all five residuals and verify they sum to zero.
  3. Compute \(R^2\).
  4. Predict yield at 110 cm, and say why predicting at 300 cm would be unwise.