Question

Provide comp te explanations with relevant equations and terivo . Questi! 1 (15p): Consider the population model );-po +ax, + β2W, + u, a. What does it imply about the population that u, has a zero conditional mean? b. Let u be correlated with W, but not with X,. What can we say about the estimate d, in the case where Wi is included and when it is not included? c. What are the assumptions made for OLS estimiation? What are the main results that the OLS estimator satislies? a. It in stated in the test that the instrumenal variables etdinator i biased bvt consistent: What b. Show how a two stage lest squure extimator can estmate if X, is cotrlated with u, and we Question 2 (15p): Instrumental variables does this mean? have an instrument Z, What are the conditions for an instrunent to be valid? What bappens if the conditions do not hoid?

Answer 1

Question 1. Given a population model $Y = \beta_{0} + \beta_{1}X + \beta_{2}W + u$

a. u has a zero conditional mean. This is actually an assumption of OLS estimation (the error term should have a zero conditional mean). Now, why is this necessary? We can see this using graphs below. In the first graph, the zero conditional mean assumption is violated, while it is followed in the second graph.

b. This question indicates a violation of another OLS assumption: All independent variables are uncorrelated with the error term. This assumption is called exogenity. If it is correlated, we can use the variable to predict the error term. Violating this assumption biases the coefficient estimate  $\hat{\beta}$  when it is included.

c. OLS assumptions:

1: The population regression model is linear in its parameters and correctly specified as: $Y = \beta_{0} + \beta_{1}X + \beta_{2}W + u$
2: The observed data: (yi , xi, wi) for i = 1, ..., n represent an i.i.d. random sample of size $\mathbb{N}$ following the population model.
3: The observed data: xi for i = 1, ..., n are not all the same value (same for wi for i = 1....n).
4: The expected value of the error term is zero conditional on any value of the explanatory variable: E[u|X] = 0

Given OLS assumptions:
E[$\beta_{0}$] = $\beta_{0}$ and E[$\beta_{1}$] = $\beta_{1}$ and E[$\beta_{2}$] = $\beta_{2}$
The sampling distributions of the estimators $\beta_{2}$, $\beta_{1}$ and $\beta_{0}$ are centered about the true population parameter values $\beta_{2}$,  $\beta_{1}$and $\beta_{0}$.

Question 2.

a. Instrumental variables estimator is "biased but consistent". Instrumental variable methods provide consistent estimation when the explanatory variables are correlated with the error terms in a regression model. OLS estimators are biased but inconsistent in such cases. However, if an instrument is available, consistent estimates may still be obtained.

b. 2 Stage Least Squares is numerically similar to Instrument Variable with one instrument. Hence, we can find ${\beta_{1}}$ since X is correlated.

c. Instrument validity is defined as E[Z|u]=0 where Z is the instrumental variable and u is the error term of a univariate regression model.

Requirements for Z to be a valid instrument for X are:

• Relevance = Z needs to highly correlated with X
• Exogenous = Z is correlated with Y solely through its correlation with X; so Z is uncorrelated with the error in the outcome equation

The main idea behind the technique of instrument variable estimator is that when Z changes, it should also alter X, but not the part of X that is correlated with the error.

Hope this helped!