Mock Exam 2025

An example exam based on the materials covered in 2025
Warning

The contents of the final exam may include any of the topics covered in the course (except for those marked as non-examinable), not just those that appear in this mock exam. This mock exam is intended as a reference in terms of exam format.

Duration: 90 minutes
Maximum points: 15 points 

Short Questions

Each question in this section is worth 1 point. These questions do not require any proofs, only brief direct answers.

1. Define consistency of an estimator \(\hat{\bbeta}\) for some parameter \(\bbeta\).

2. Suppose that \(X_N\xrightarrow{p} X\) and \(Y_N\xrightarrow{p} Y\). Is it always true that \(X_N+Y_N\xrightarrow{p} X+Y\)? (yes/no).

3. Suppose that \(\sqrt{N}(\hat{\beta}-\beta) \xrightarrow{d} N(0, \sigma^2)\), where \(\beta\) is scalar. Let \(f(\cdot)\) be a continuous function with \(f'(\beta)\neq 0\). What is the asymptotic distribution of \(f(\hat{\bbeta})\) according to the delta method?

4. Suppose that we observe \(N\) units over two periods \(t=1, 2\). Some treatment affects all units between \(t=1\) and \(t=2\). The average outcome in \(t=2\) is \(\bar{Y}_2 = 30\). The average outcome in \(t=1\) is \(\bar{Y}_1 =10\). Compute the event study estimator for the average treatment effects.

5. In the context of supervised prediction with label \(Y\) and predictors \(\bX\), consider the hypothesis class \(\Hcal= \curl{h(\bx)= \bx'\bbeta: \bbeta \in \R^{\dim(\bbeta)}}\). Write down the optimization problem that defines the optimal \(L^1\)-penalized linear predictor (LASSO).

Problem 1: Testing a Scalar Hypothesis

Let the outcome \(Y_i\), the scalar covariate \(X_i\), and an unobserved component \(U_i\) be linked through the linear potential outcomes model \(Y_i^{x} = \beta x + U_i\). We have some estimator \(\hat{\beta}\) that satisfies \[ \sqrt{N}(\hat{\beta}- \beta)\xrightarrow{d} N(0, \avar(\hat{\beta})). \] There is a consistent estimator \(\widehat{\avar}(\hat{\beta})\) for \(\avar(\hat{\beta})\).

We are interested in testing \[ H_0: \beta = c \quad \text{vs.} \quad H_1: \beta\neq c. \]

  1. (1 points) Propose a consistent test for \(H_0\) vs. \(H_1\) that has asymptotic size \(\alpha\).
  2. (2 points) Show that your test has asymptotic size \(\alpha\).
  3. (2 points) Show that your test is consistent.

Problem 2: One-Way FE Estimators Under Heterogeneous Effects

Let the outcome \(Y_i\), the scalar covariate \(X_i\) and unobserved components \((\alpha_i, \beta_i, U_i)\) be linked through the linear potential outcomes panel data model \[ Y_{it}^x = \alpha_i + \beta_i x + U_{it}. \tag{1}\] We assume that \(\E[U_{it}|X_{i1}, \dots, X_{iT}]=0\).

  1. (1 point): Define the one-way within transformation of \((Y_{it}, X_{it})\) (recall: the transformation designed to eliminate the \(\alpha_i\)). Write down the relationship sastisfied by the within-transformed realized values \((\tilde{Y}_{it}, \tilde{X}_{it})\) of \((Y_{it}, X_{it})\).
  2. (2 points) Write down the one-way fixed effects estimator in terms of \((\tilde{Y}_{it}, \tilde{X}_{it})\) (the within estimator, not the LSDV one). Find its probability limit under model (1). State explicitly any moment conditions you assume.

In the next two questions, you may be informal and use only words. Correct intuition is sufficient

  1. (1 point) True or false: under model (1), the probability limit of the one-way FE estimator is always the same as the probability limit of the two-way FE estimator. Why or why not?
  2. (1 point) Suppose that the model now has homogeneous effects: \[ Y_{it}^x = \alpha_i + \beta x + U_{it}. \tag{2}\] True or false: under model (2) the probability limit of the one-way FE estimator is always the same as the probability limit of the two-way FE estimator. Why or why not?