Exercise list 1
Analytical problems
If \(\mathbb{E} \left[ Y | X \right] = a + bX\), find \(\mathbb{E}\left[ XY \right]\) as a function of moments of \(X\).
Show that in the linear regression model (see slides) for any function \(h(x)\) of the covariates, \(\mathbb{E} \left[ h(X)e \right] = 0\) as long as it is finite.
True of false. If \(Y = X\beta + e\), \(X \in \mathbb{R}\) and \(\mathbb{E} \left[ Xe \right] = 0\), then \(\mathbb{E} \left[ X^2e \right] = 0\)
Consider \(X\) and \(Y\) such that their joint density is \(f(x,y) = 3/2(x^2 + y^2)\) on \(\left[ 0, 1 \right]^2\). Compute the coefficients of the best linear predictor \(Y = \alpha + \beta X + e\). Compute the conditional expectation function \(m(x) = \mathbb{E}\left[Y | X = x \right]\). Are they different?
Consider the long and short projections \(Y = X_1 \gamma_1 + e\) and \(Y = X_1 \beta_1 + X_1^2 \beta_2 + u\). When is \(\gamma_1 = \beta_1\)? What if we consider \(Y = X_1 \theta_1 + X_1^3 \theta_2 + v\), is there a situation when \(\theta_1 = \gamma_1\)?
Consider the estimation of the (sample-wide) linear regression \(\mathbf{Y} = \mathbf{X}\beta + \mathbf{e}\). Now change the regressors to \(\mathbf{Z}=\mathbf{XC}\), where \(\mathbf{C}\) is a \(k\times k\) non-singular matrix. How does this affect (i) the OLS estimates, and (ii) residuals of this regression?
Consider \(\widetilde{\mathbf{Y}} = \mathbf{X}\left(\mathbf{X}^{\prime}\mathbf{X} \right)^{-1}\mathbf{X}^{\prime}\mathbf{Y}\). Find the OLS coefficient of a regression of \(\mathbf{\widetilde{Y}}\) on \(\mathbf{X}\).
Show that \(\mathbf{M}\) is idempotent.
Show that if \(\mathbf{X} = \left[ \mathbf{X_1} \mathbf{X_2} \right]\), and \(\mathbf{X_1}^{\prime} \mathbf{X_2} = 0\), then \(\mathbf{P} = \mathbf{P_1} + \mathbf{P_2}\)
A friend suggests that the assumption that observations \((Y_i, X_i)\) are i.i.d. implies that the errors \(e\) of the regression \(Y = X^{\prime}\beta + e\) are homoskedastic. Do you agree? Explain why.
Consider the model \(Y = X^{\prime}\beta + e\) and the (very important nowadays) ridge regression estimator: \[\widehat{\beta}_{\text{RIDGE}} = \left( \sum_{i=1}^n X_iX_i^{\prime} + \lambda I_k \right)^{-1} \left( \sum_{i=1}^{n} X_i Y_i \ \right)\]
Is \(\widehat{\beta}_{\text{RIDGE}}\) unbiased estimator of \(\beta\)? Is it consistent?
- Take a regression model with i.i.d. observations \((Y_i, X_i)_i\) and \(X \in \mathbb{R}\), such that \(Y = X\beta + e\), \(\mathbb{E}\left[ e | X \right] = 0\) and define \(\Omega = \mathbb{E} \left[ X^2 e^2 \right]\). If \(\widehat{\beta}\) is the OLS estimator and \(\widehat{e}_i\) the OLS residuals, find the assymptotic distribution \(\sqrt{n} ( \widehat{\Omega} - \Omega)\) of the estimators:
\[\widehat{\Omega} = \frac{1}{n} \sum_{i=1}^n X_i^2 e_i^2\] \[\widetilde{\Omega} = \frac{1}{n} \sum_{i=1}^n X_i^2 \widehat{e}_i^2\]
Computational/interpretative problems
For this list, we will use the cps09mar.dta file
provided by the textbook author. You should provide the
code and the results together.
library(haven)
library(kableExtra)
library(knitr)
# setwd() <- this might help
# you can download the file directly in R, or just do it manually
# url <- "https://www.ssc.wisc.edu/~bhansen/econometrics/Econometrics%20Data.zip"
# download.file(url, "./econ_data.zip")
# unzip("./econ_data.zip")
cps09mar <- read_dta("./cps09mar.dta")
knitr::kable(head(cps09mar, 10))| age | female | hisp | education | earnings | hours | week | union | uncov | region | race | marital |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 52 | 0 | 0 | 12 | 146000 | 45 | 52 | 0 | 0 | 1 | 1 | 1 |
| 38 | 0 | 0 | 18 | 50000 | 45 | 52 | 0 | 0 | 1 | 1 | 1 |
| 38 | 0 | 0 | 14 | 32000 | 40 | 51 | 0 | 0 | 1 | 1 | 1 |
| 41 | 1 | 0 | 13 | 47000 | 40 | 52 | 0 | 0 | 1 | 1 | 1 |
| 42 | 0 | 0 | 13 | 161525 | 50 | 52 | 1 | 0 | 1 | 1 | 1 |
| 66 | 1 | 0 | 13 | 33000 | 40 | 52 | 0 | 0 | 1 | 1 | 5 |
| 51 | 0 | 0 | 16 | 37000 | 44 | 52 | 0 | 0 | 1 | 1 | 1 |
| 49 | 1 | 0 | 16 | 37000 | 44 | 52 | 0 | 0 | 1 | 1 | 1 |
| 33 | 0 | 0 | 16 | 80000 | 40 | 52 | 0 | 0 | 1 | 1 | 1 |
| 52 | 1 | 0 | 14 | 32000 | 40 | 52 | 0 | 0 | 1 | 1 | 1 |
We are interested in running (linear) Mincerian regressions of the type \(\ln (\text{wage}) = X^{\prime}\beta + e\), where \(X\) is a vector of worker characteristics, the most important of which (for this kind of regression) is education.
Run a linear regression of log earnings on age, age squared, sex and education. What is the expected log earnings of a 20 years old woman as a function of her education? What is the average partial effect of another year of age on log earnings?
In the job market, an important predictor of wages is experience. Unfortunately, that is a variable almost universally missing from data sets, which at most have tenure at current work. So many applied economists proxy for experience as age - 15 (minimum working age at the data). Add this variable to the regression. What happens? Calculate \((\mathbf{X}^{\prime}\mathbf{X})\) and explain.
For what folllows, remove age variables and leave only experience and experience squared. (Also: now that you are already half-way there, also calculate \(\mathbf{X}^{\prime}\mathbf{Y}\) and calculate by hand both \(\widehat{\beta}\) and \(\text{Var}(\widehat{\beta})\))
Compute homoskedastic and heteroskedastic-robust standard errors of the estimators. Do they differ? Which one do you prefer? Now estimate cluster-robust standard errors at the level of the region. Discuss whether this is a reasonable approach in this case.
Now use the Frisch-Waugh-Lovell theorem to estimate the effect of education on wages, while partialling out the controls in (1). Is the estimated coefficient the same? What about its standard deviation?
Consider a 18 years-old prospective college student deciding whether to study or work. He wants to know the ratio \(\theta\) between returns of education and returns to experience, given his age. Write \(\theta\) as a function of the parameters \(\beta\) and estimate it.
Write out the formula for the asymptotic variance of \(\widehat{\theta}\) as a function of the variance-covariance matrix and find the standard deviation of the estimator.
Construct a 90% confidence interval for \(\widehat{\theta}\).
Test (jointly) whether \(\beta_{\text{educ}}\) equals \(\beta_{\text{exp}} + 6\beta_{\text{exp}^2}\) and \(\beta_{\text{educ}}\) for men equals \(\beta_{\text{educ}}\) for women using a Wald statistic. Interpret.