Microeconometrics Exercises and Questions -

Assignment 1 - Regression fundamental

Exercise 1.1 - Show the following related usefull results:

Β_k = Cov(Y^~_ki, X^~_ki)/V(X^~_ki)

Where Y^~_ki is a residual from a regression of Yi on every covariates except X_ki.

Application 1 - A usefull hand-on parenthesis

Take the dataset CPS91, assume it is your entire population.

We are interested in explaining Wages in log (lwage) by Schooling (educ)

Estimate the linear regression of y on x (command reg in STATA)

Compute y^-(X_i) = E( Y_i|X_i) (since X_i takes 18 different values, you are left with 18 values for y)

Compare the results and comments.

In which sense properties of the CEF and PRF are relevant for your finding?

Could these results be usefull when you lack individual data?

Using the definition of the PRF and the LIE show that the PRF amount regressing X_i on E(Y_i|X_i)

Give the definition of the PRF, i.e. the constraints on the parameters such that the MMSE problem is solved.

NB: above, we should rather read Using the definition of the PRF and the LIE show that the PRF amount regressing E(Yij|Xi ) on Xi.

Question: To what extent the Sample Regression Function (SRF) (i.e. an estimate of the PRF) is a good approximation of the Population Regression Function (PRF)?

Exercise 1.2 - What is needed to generate an estimate ("the cake")?

An estimator : this a rule or "a recipe"

A random sample : this is the ingredient

Define the cake, the sample and the recipe for E (wage) where wage is the wage of an employed worker in your country.

I choose half of men and half of women to put in my sample. Do you think I will get a god estimator? Yes or No, and in what sense?

Question: How best the SRF (i.e. the sample fitted line) can approach the PRF (i.e. population fitted line: the best we can get)?

Exercise 1.3 - Assuming the vector W_i ≡ [Y_i X'_i]' is iid in a sample of size N. The MME of β is:

β^{^} = [∑_iX_iX'_i]^-1 * ∑_iX_iY_i

For the case of one explanatory variable and using the FOC. We end up with two moment conditions that need to be satisfied by the PRF slope and intercept parameters. Derive the MM estimator for this simple case.

Note: the estimator β^{^} is a random variable (Y_i is random), it has a sampling distribution (i.e. takes different values in repeated sample from the population).

Exercise 1.4 - The PRF minimizes prediction errors in the sample, we can show that β^{^} achieve the same results on the sample.

Show that β^{^} minimizes the sum of square residual between observed Y and predicted Y^{^} in the sample.

The "maximum likelihood" is another recipe to derive β^{^} under the additional assumption that Y_i is normally distributed (see Wooldridge 2010).

Question: Consider the case of a regression of wages on a full set of dummies for every years of schoolings. Could X contains a constant?

Assignment 2 - A General Framework to Think About Causality

Exercise 2.1 - Give some examples in political sciences, education and health.

Question: How interesting do you think are these effects? Are we making steps toward solving the Fundamental Problem of Causal Inference?

Question: Why this matters - Causality in regression framework?

 Exercise 2.2 - Add a graph showing the CEF for the treated and non-treated.

Question: The effect of treatment on random person may not be very meaningful from an economic point of view (what about policy makers?)

Question: Show that the error term is correlated with Di,and thus that OLS estimates of ATT are biased.

Question: Would randomization helps for estimating ATT?

Exercise 2.3 - Show that comparing earnings of those who went to college and those we don.t is a bias measure of return to college for college graduate. What is the likely sign of the bias?

Question: Why does CIA hold in an experimental study?

Exercise 2.4 - Assume we have 2 demographic groups of workers, young and prime aged, among men and women. With how many causal effect do we end up?

Exercise 2.5 - Write down the overall causal effect of graduating from high school by invocating the LIE. Write down the causal effect for high school graduates.

Answer in word document only, do not use pictures or hand written attachment.