9.13 Question bank

Multiple choice

QB 9.1 In the model \(\ln y = \beta_0 + \beta_1 x + u\), a coefficient of 0.045 means a one-unit rise in \(x\) is associated with

  1. a 0.045 unit rise in \(y\)
  2. a 4.5% rise in \(y\)
  3. a 0.045% rise in \(y\)
  4. a 4.5 unit rise in \(y\)

QB 9.2 A log–log coefficient is

  1. always between 0 and 1
  2. an elasticity, and unit-free
  3. equal to the correlation between the variables
  4. interpretable only if both variables are positive integers

QB 9.3 Adding an explanatory variable to a regression

  1. can raise or lower \(R^2\)
  2. can never lower \(R^2\)
  3. can never lower adjusted \(R^2\)
  4. always raises both

QB 9.4 Controlling for a variable that lies on the causal path between \(x\) and \(y\)

  1. removes omitted variable bias
  2. estimates the direct rather than the total effect
  3. has no effect on the coefficient
  4. always increases \(R^2\) without changing the estimate

QB 9.5 Including an irrelevant variable uncorrelated with the other regressors

  1. biases the coefficients downward
  2. leaves the coefficients unbiased and barely changes their standard errors
  3. leaves the coefficients unbiased but sharply raises their standard errors
  4. makes the model unidentifiable

Short reasoning

QB 9.6 In the model \(y = \beta_0 + \beta_1 x + \beta_2 x^2 + u\) with \(\beta_1 > 0\) and \(\beta_2 < 0\), explain what \(\beta_1\) measures and why quoting it as “the effect of \(x\)” is wrong. Where would you have to evaluate the marginal effect for \(\beta_1\) to be the right answer?

QB 9.7 A researcher takes logs of household land holdings and reports that the sample fell from 4,200 to 3,100 observations. What happened, what does it imply about the population the results describe, and what would \(\ln(x+1)\) have done instead?

QB 9.8 Explain why the constant in \(\ln(x+1)\) is not innocuous. If a colleague argues that adding 1 is standard practice and therefore safe, how would you respond?

QB 9.9 A study of fertiliser and yield reports a level–level slope of 0.83. A colleague asks whether that is a large effect. Explain what further information is needed, and what functional form would have avoided the question.

QB 9.10 Explain why \(R^2\) is a reasonable thing to examine when forecasting next year’s electricity demand, but a poor guide when estimating the effect of a school meals programme on attendance.

QB 9.11 A researcher studying the effect of a job training programme on earnings controls for whether the participant is currently employed. Which of the three cases in Section 9.10 is this, and what will it do to the estimate?

QB 9.12 Why can adjusted \(R^2\) be negative? What does a negative value tell you about the model?

QB 9.13 In a regression of infant mortality on household income across districts, a researcher adds “number of hospitals per capita” as a control and the income coefficient shrinks by half. Give two different causal stories consistent with that result, one in which the control belongs and one in which it does not.

QB 9.14 A dataset records earnings for people who are employed. Explain why regressing earnings on education using only these observations is a version of the collider problem, and what it implies for interpreting the coefficient.

QB 9.15 Explain why the choice between level–log and log–log cannot be made by comparing \(R^2\), but the choice between two different sets of controls in a log–log model could in principle be informed by it.

Numerical

QB 9.16 A log–level regression of earnings on years of experience gives a coefficient of 0.026. Give both the approximate and exact percentage returns to one more year, and to ten more years.

QB 9.17 A model with \(n = 150\), \(k = 4\) has \(\text{SSR} = 240\) and \(\text{SST} = 600\). Compute \(R^2\) and adjusted \(R^2\).

QB 9.18 Two nested models on the same 300 observations have \(R^2\) of 0.38 with 3 regressors and 0.41 with 7. Compute both adjusted \(R^2\) values, and the \(F\) statistic for the four added variables. Do the two criteria agree?

QB 9.19 A demand equation estimated in log–log form gives a price elasticity of \(-1.4\) with a standard error of 0.15. Test whether demand is unit-elastic, and state what the sign and magnitude imply about total revenue when price rises.