6.10 Question bank

No answers are given for this section, by design.

Multiple choice

QB 6.1 In the model \(y = \beta_0 + \beta_1 x + u\), the term \(u\) represents

  1. measurement error in \(y\)
  2. all factors other than \(x\) that affect \(y\)
  3. the residual from the fitted line
  4. the difference between \(\hat{\beta}_1\) and \(\beta_1\)

QB 6.2 The population regression function is

  1. the line fitted by ordinary least squares
  2. the conditional mean \(E(Y \mid X = x)\)
  3. the average of \(y\) in the sample
  4. the line that minimises squared residuals

QB 6.3 \(E(u \mid x) = 0\) says that

  1. the residuals sum to zero
  2. the unobserved factors average zero at every value of \(x\)
  3. \(x\) and \(y\) are uncorrelated
  4. the model is linear

QB 6.4 Which of the following are unobservable? (select all that apply)

  1. \(u_i\)
  2. \(\hat{u}_i\)
  3. \(\beta_1\)
  4. \(\hat{\beta}_1\)

QB 6.5 Minimising \(\sum \hat{u}_i\) rather than \(\sum \hat{u}_i^2\) fails because

  1. the sum is always negative
  2. positive and negative residuals cancel
  3. it cannot be differentiated
  4. it requires a larger sample

QB 6.6 Treating the conditional expectation function as linear is

  1. implied by the definition of the PRF
  2. a modelling choice
  3. required for \(E(u) = 0\)
  4. a consequence of random sampling

Short reasoning

QB 6.7 Explain the difference between the population regression function and the sample regression function, and say which one changes when a new sample is drawn.

QB 6.8 A colleague says the error term captures “measurement error in the data”. Explain what is wrong with that, and give three examples of what \(u\) actually contains in a regression of wages on years of schooling.

QB 6.9 Why is \(E(u) = 0\) described as costing nothing, while \(E(u \mid x) = 0\) is the assumption on which everything rests?

QB 6.10 State the zero conditional mean assumption in words for a regression of crop yield on fertiliser use, and give one concrete reason it might fail.

QB 6.11 A researcher regresses district crop yield on rainfall and finds a strong positive slope. Give two reasons the slope may not measure the effect of rainfall on yield.

QB 6.12 Explain why the least squares criterion penalises one residual of 8 more heavily than eight residuals of 1.

Numerical

QB 6.13 A population’s conditional means are \(E(Y \mid X = x) = 12 - 0.5x\).

  1. State \(\beta_0\) and \(\beta_1\).
  2. What is the average \(y\) among individuals with \(x = 10\)?
  3. An individual with \(x = 10\) has \(y = 9\). Find \(u_i\).
  4. Interpret \(\beta_1\) in words.

QB 6.14 Four individuals with \(x = 6\) have \(y\) values 14, 18, 11 and 21.

  1. Estimate \(E(Y \mid X = 6)\).
  2. Compute the four implied values of \(u_i\).
  3. Verify that they sum to zero, and explain why that had to happen.
  4. Explain why this does not demonstrate that \(E(u \mid x) = 0\) in the population.