8.10 Question bank

Multiple choice

QB 8.1 In the model \(y = \beta_0 + \beta_1x_1 + \beta_2x_2 + u\), the coefficient \(\beta_1\) measures

  1. the total change in \(y\) when \(x_1\) rises by one unit
  2. the change in average \(y\) when \(x_1\) rises by one unit, holding \(x_2\) fixed
  3. the correlation between \(x_1\) and \(y\)
  4. the share of variation in \(y\) explained by \(x_1\)

QB 8.2 The degrees of freedom for a \(t\)-test on a single coefficient in a regression with \(n = 150\) and four explanatory variables is

  1. 146    b. 145    c. 148    d. 149

QB 8.3 Perfect collinearity between two explanatory variables means

  1. the OLS estimates are biased
  2. the OLS estimates have large standard errors
  3. the OLS estimates cannot be computed at all
  4. the \(R^2\) will be exactly one

QB 8.4 An omitted variable causes no bias if

  1. it is uncorrelated with the dependent variable
  2. it is uncorrelated with the included explanatory variables
  3. its coefficient is large
  4. the sample is large

QB 8.5 The \(F\) test for exclusion restrictions is always

  1. two-tailed
  2. left-tailed
  3. right-tailed
  4. tailed according to the alternative hypothesis

Short reasoning

QB 8.6 Explain, without using the phrase “holding constant”, what the Frisch–Waugh–Lovell result says a multiple regression coefficient is computed from.

QB 8.7 A colleague argues that since adding controls always reduces precision, one should include as few as possible. Assess this.

QB 8.8 Why does the numerator of the \(F\) statistic divide by \(q\) and the denominator by \(n-k-1\)? What would go wrong if neither division were performed?

QB 8.9 In a study of the effect of an irrigation scheme on farm incomes, the researcher controls for landholding, soil quality and district. Name two variables likely still omitted, and use the bias formula to argue in which direction each would push the estimate.

QB 8.10 A regression of infant mortality on the number of doctors per thousand people across Indian districts returns a positive coefficient. Explain using the omitted variable bias formula, and suggest a control.

QB 8.11 The overall \(F\) statistic in a regression is significant but no individual coefficient is. What does this suggest about the explanatory variables, and what would you do next?

QB 8.12 Explain why an omitted variable that strongly affects \(y\) but is unrelated to the explanatory variable of interest costs precision but causes no bias.

Numerical

QB 8.13 A researcher estimates \(\widehat{\text{wage}} = 4200 + 380\,educ\) and then \(\widehat{\text{wage}} = 3100 + 240\,educ + 95\,exper\). Compute the implied regression slope of experience on education, and comment on its sign.

QB 8.14 An unrestricted model with \(n = 120\), \(k = 6\) has \(R^2_{ur} = 0.54\). Imposing four exclusion restrictions gives \(R^2_r = 0.41\). Compute the \(F\) statistic and state its degrees of freedom.

QB 8.15 A regression with \(n = 500\) and \(k = 3\) reports \(\sum \hat{u}_i^2 = 892\). Compute \(\hat{\sigma}^2\) and \(\hat{\sigma}\), and state why the divisor is not \(n\).

QB 8.16 For a regression of yield on fertiliser, irrigation and labour, the \(R^2\) from regressing fertiliser on the other two is 0.76. By what percentage is the standard error on fertiliser inflated relative to a hypothetical case in which fertiliser were uncorrelated with the other regressors?