Exercises: Asymptotic Inference

Theoretical and applied exercises on inference: hypothesis testing, nonlinear functions of parameters, consistency of HC0 asymptotic variance estimator.

Theoretical Exercises

Testing Scalar Restrictions

Consider the hypotheses $H_0: \beta_k = c$ and $H_1: \beta_k\neq c$, where $\beta_k$ is the $k$th coordinate of the $\bbeta$ vector. Propose a consistent test for $H_0$ vs $H_1$ that has asymptotic size $\alpha$.
Now let $\ba\neq 0$ be some known constant vector of the same dimension as $\bbeta$. Consider the hypotheses $H_0: \ba'\bbeta = c$ and $H_1: \ba'\bbeta\neq c$. Propose a consistent $t$-test for $H_0$ vs $H_1$ that has asymptotic size $\alpha$.
Why do we require that $\ba\neq 0$ in the previous question?

In both cases remember to show that your test is consistent and has the desired asymptotic size.

Testing Several Linear Restrictions

Let $\bbeta = (\beta_1, \beta_2, \dots, \beta_p)$ with $p\geq 4$. Consider the following two hypotheses on $\bbeta$: \[ H_0: \begin{cases} \beta_1 = 0, \\ \beta_2 - \beta_3 = 1, \\ \beta_2 = 4\beta_4 + 5, \end{cases} \quad H_1: \text{at least one equality in $H_0$ fails} \] Propose a consistent test for $H_0$ vs. $H_1$ with asymptotic size $\alpha$. Show that the test possesses these properties.

Inference on a Nonlinear Function of Parameters

Let the outcome $Y_i$, the covariates $\bX_i$, and an unobserved component $U_i$ be linked through the linear potential outcomes model \[ Y_i^{\bx} = \bx'\bbeta + U_i. \] Suppose that we observe an IID sample of data on $Y_i, \bX_i$, that $\E[U_i|\bX_i]=0$, that $\E[\bX_i\bX_i']$ is invertible, and that $\E[U_i^2\bX_i\bX_i']$ has maximal rank. Also suppose that $\bbeta$ has $p\geq 2$ components, that $\beta_1>0$ and $\beta_2>0$, and that you are interested in \[ \gamma = \sqrt{\beta_1\beta_2}. \]

Construct a confidence interval for $\gamma$ with asymptotic coverage $(1-\alpha)$.
Consider $H_0: \gamma=1$ vs. $H_1:\gamma\neq 1$. Construct a consistent test for $H_0$ vs. $H_1$ with asymptotic size $\alpha$.

Remember to prove coverage, consistency, and size properties.

Consistency of the HC0 Asymptotic Variance Estimator

Let the outcome $Y_i$, the scalar covariate $X_i$, and an unobserved component $U_i$ be linked through the linear potential outcomes model \[ Y_i^{\bx} = \beta X_i + U_i \tag{1}\] Suppose that we observe an IID sample of data on $Y_i, X_i$, that $\E[U_i|X_i]=0$, that $\E[X_i^2]\neq 0$, and that $\E[U_i^2 X_i^2]$ exists. Let $\hat{\beta}$ be the OLS estimator obtained by regressing $Y_i$ on $X_i$.

Recall the HC0 (White 1980) estimator for $\avar(\hat{\beta})$. In the scalar model (1) it is given by \[ \begin{aligned} \widehat{\avar}(\hat{\beta}) & = \dfrac{ N^{-1} \sum_{i=1}^N \hat{U_i}^2 X_i^2 }{ \left( N^{-1}\sum_{i=1}^N X_i^2 \right)^2 }. \end{aligned} \] Show that \[ \widehat{\avar}(\hat{\beta}) \xrightarrow{p} \avar(\hat{\beta}) \equiv \dfrac{ \E[U_i^2X_i^2] }{\left(\E[X_i^2] \right)^2 }. \] State explicitly any additional moment assumptions you make.

Applied Exercises

Applied exercises are from Wooldridge (2020). In all cases, use asymptotic $t$- and Wald tests with robust standard errors:

C9 in chapter 4,
C4 and C6 in chapter 5,
C8 in chapter 7.

For some more code examples and discussion, look at chapters 4, 5, 7 in Heiss and Brunner (2024)

References

Heiss, Florian, and Daniel Brunner. 2024. Using Python for Introductory Econometrics. 2nd edition. New York: Independently Published.

White, Halbert. 1980. “A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity.” Econometrica 48 (4): 817. https://doi.org/10.2307/1912934.

Wooldridge, Jeffrey M. 2020. Introductory Econometrics: A Modern Approach. Seventh edition. Boston, MA: Cengage.