Problem Solving Exercises

Note

Answers to exercises will only be provided during class time. If you cannot make it to class, you will need to see me during consultation times and we will work through the exercises together. (When you see me during consultation times, I expect you to be prepared. I will never merely provide answers to exercises. Instead, I want to see good faith effort on your part in which case I will be more than happy to help you work throught the exercises.)

Week 1

None

Week 2

  1. Properly define the individual causal effect in the more general model with \(k\) independent variables.
  2. What does the individual causal effect in the model with \(k\) independent variables boil down to, once you impose linearity?
  3. Let \(Y_i \sim \text{i.i.d.}(\mu,\sigma^2)\). You know from EMET2007 that \(\bar{Y}\) is an unbiased and consistent estimator for the population mean \(\mu\). Are the following estimators also unbiased or consistent for \(\mu\)? Discuss!
    1. \(\hat{\mu}_2 := 42 \qquad\) (‘the answer to everything’ estimator)
    2. \(\hat{\mu}_3 := \bar{Y}_n + 3/n\)
    3. \(\hat{\mu}_4 := (Y_1 + Y_2 + Y_3 + Y_4 + Y_5)/5\)

Week 3

  1. Derive the classical measurement error bias.

    (Hint: Frame the problem of classical measurement error as an omitted variable bias problem and apply the ovb results.)

  2. In the model \(Y_i = \beta_0 + \beta_1 X_i + u_i\) the dependent variable \(Y_i\) is measured with error. Instead of \(Y_i\) you observe \(\tilde{Y}_i= Y_i+w_i\) where \(w_i\) is purely random.

    1. Derive \(E \left[ \hat{\beta}_1 | X_i \right]\)
    2. Derive \(Var \left[ \hat{\beta}_1 | X_i \right]\)

    Modify the proofs of bias and variance from lecture 5 of last semester’s EMET2007 to derive the results.

Week 4

  1. A sociologist is interested in the causal effect of a mother’s smoking behavior during pregnancy on the violence of her child during the child’s teenage years.

    The sociologist has available household survey data that collects information on mothers and their children. The sociologist wants to run the following regression:

    \[\text{Fights}_i = \beta_0 + \beta_1 \text{Smoking}_i +\beta_2 \text{Age}_i + u_i,\]

    where

    • \(\text{Fights}_i\) is the number of fights teenager i was involved in during the month before the survey interview
    • \(\text{Smoking}_i\) is a dummy variable indicating if the mother of teenager i smoked during pregnancy
    • \(\text{Age}_i\) is the age of teenager i

    Does \(\hat{\beta}_1\) estimate the causal effect of a mother’s smoking during pregnancy on her child’s violence later in life?

  2. Prove that the TSLS estimator is consistent.

    Use the regression equation \(Y_i = \beta_0 + \beta_1 \hat{X}_i + w_i\) from this week’s lecture and show that the sample covariance between \(\hat{X}_i\) and \(w_i\) converges to zero (in probability).

Week 5

None

Week 6

  1. Prove that \(s_{\hat{X} Y} = \hat{\pi}_1 s_{Z Y}\) and that \(s_{\hat{X}}^2 = \hat{\pi}_1^2 s_Z^2\).

  2. Prove that

    \begin{align*} \hat{\beta}_{1,TSLS} &:= \frac{\sum_{i=1}^{n}(Z_i-\bar{Z})(Y_i-\bar{Y})} {\sum_{i=1}^{n}(Z_i-\bar{Z})(X_i-\bar{X})} = \frac{s_{Z Y}}{s_{Z X}}\\ \end{align*}

Weeks 7 - 13

None