Exam 12: Instrumental Variables Regression

Let W be the included exogenous variables in a regression function that also has endogenous regressors (X). The W variables can

Write a short essay about the Overidentifying Restrictions Test. What is meant exactly by "overidentification?" State the null hypothesis. Describe how to calculate the J-statistic and what its distribution is. Use an example of two instruments and one endogenous variable to explain under what situation the test will be likely to reject the null hypothesis. What does this example tell you about the exactly identified case? If your variables pass the test, is this sufficient for these variables to be good instruments?

To calculate the J-statistic you regress the

Describe the consequences of estimating an equation by OLS in the presence of an endogenous regressor. How can you overcome these obstacles? Present an alternative estimator and state its properties.

You have estimated a government reaction function, i.e., a multiple regression equation, where a government instrument, say the federal funds rate, depends on past government target variables, such as inflation and unemployment rates. In addition, you added the previous period's popularity deficit of the government, e.g. the (approval rating of the president - 50%), as one of the regressors. Your idea is that the Federal Reserve, although formally independent, will try to expand the economy if the president is unpopular. One of your peers, a political science student, points out that approval ratings depend on the state of the economy and thereby indirectly on government instruments. It is therefore endogenous and should be estimated along with the reaction function. Initially you want to reply by using a phrase that includes the words "money neutrality" but are worried about a lengthy debate. Instead you state that as an economist, you are not concerned about government approval ratings, and that government approval ratings are determined outside your (the economic)model. Does your whim make the regressor exogenous? Why or why not?

Write an essay about where valid instruments come from. Part of your explorations must deal with checking the validity of instruments and what the consequences of weak instruments are.

In practice, the most difficult aspect of IV estimation is

For W to be an effective control variable in IV estimation, the following condition must hold

Instrumental Variables regression uses instruments to

The IV estimator can be used to potentially eliminate bias resulting from

The logic of control variables in IV regressions

Studies of the effect of minimum wages on teenage employment typically regress the teenage employment to population ratio on the real minimum wage or the minimum wage relative to average hourly earnings using OLS. Assume that you have a cross section of United States for two years. Do you think that there are problems with simultaneous equation bias?

You have been hired as a consultant to estimate the demand for various brands of coffee in the market. You are provided with annual price data for two years by U.S. state and the quantities sold. You want to estimate a demand function for coffee using this data. What problems do you think you will encounter if you estimated the demand equation by OLS?

Endogenous variables

Consider a model with one endogenous regressor and two instruments. Then the J-statistic will be large

(Requires Chapter 8)When using panel data and in the presence of endogenous regressors

Consider the following two equations to describe labor markets in various sectors of the economy $N d=\beta_{0}+\beta_{1} \frac{W}{P}+u$ $N s=\gamma 0+\gamma 1 \frac{W}{P}+v$ $N d=N s=N$

To analyze the year-to-year variation in temperature data for a given city, you regress the daily high temperature (Temp)for 100 randomly selected days in two consecutive years (1997 and 1998)for Phoenix. The results are (heteroskedastic-robust standard errors in parenthesis): $\widehat{\text { Temp } \begin{array} { l } \text { PHX } \\1998\end{array}}$ = 15.63 + 0.80 × ${\text { Temp } \begin{array} { l } \text { PHX } \\1998\end{array}}$ ; R²= 0.65, SER = 9.63 (0.10) (a)Calculate the predicted temperature for the current year if the temperature in the previous year was 40°F, 78°F, and 100°F. How does this compare with you prior expectation? Sketch the regression line and compare it to the 45 degree line. What are the implications? (b)You recall having studied errors-in-variables before. Although the web site you received your data from seems quite reliable in measuring data accurately, what if the temperature contained measurement error in the following sense: for any given day, say January 28, there is a true underlying seasonal temperature (X), but each year there are different temporary weather patterns (v, w)which result in a temperature $\hat { X }$ different from X. For the two years in your data set, the situation can be described as follows: $\mathrm{X}_{1997}=\mathrm{X}+v_{1997} \text { and } \mathrm{X}_{1998}=\mathrm{X}+w_{1998}$ $\text { Subtracting } X_{1997} \text { from } X_{1998} \text {, you get } X_{1998}=X_{1997}+w_{1998}-v_{1997}$ ^{Hence the population parameter} for the intercept and slope are zero and one, as expected. It is not difficult to show that the OLS estimator for the slope is inconsistent, where $\hat { \beta } _ { 1 } \stackrel { p } { \longrightarrow } 1 - \frac { \sigma _ { v } ^ { 2 } } { \sigma _ { x } ^ { 2 } + \sigma _ { v } ^ { 2 } }$ As a result you consider estimating the slope and intercept by TSLS. You think about an instrument and consider the temperature one month ahead of the observation in the previous year. Discuss instrument validity for this case. (c)The TSLS estimation result is as follows: $\widehat{\text { Temp } \begin{array} { l } \text { PHX } \\1998\end{array}}$ = -6.24 + 1.07× ${\text { Temp } \begin{array} { l } \text { PHX } \\1997\end{array}}$ ; (0.06) Perform a t-test on whether or not the slope is now significantly different from one.

In the case of exact identification

Using some of the examples from your textbook, describe econometric studies which required instrumental variable techniques. In each case emphasize why the need for instrumental variables arises and how authors have approached the problem. Make sure to include a discussion of overidentification, the validity of instruments, and testing procedures in your essay.