Exam 12: Instrumental Variables Regression

Consider a competitive market where the demand and the supply depend on the current price of the good.Then fitting a line through the quantity-price outcomes will

The conditions for a valid instruments do not include the following: a. each instrument must be uncorrelated with the error term. b. each one of the instrumental variables must be normally distributed. c. at least one of the instruments must enter the population regression of $X$ on the $Z$ 's and the $W$ 's. d. perfect multicollinearity between the predicted endogenous variables and the exogenous variables must be ruled out.

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

The two conditions for instrument validity are $\operatorname { corr } \left( Z _ { i } , X _ { i } \right) \neq 0 \text { and } \operatorname { corr } \left( Z _ { i } , u _ { i } \right) = 0$ The reason for the inconsistency of OLS is that $\operatorname { corr } \left( X _ { i } , u _ { i } \right) \neq 0$ But if X and Z are correlated, and X and u are also correlated, then how can Z and u not be correlated? Explain.

Estimation of the IV regression model

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?

Instrument relevance a. means that the instrument is one of the determinants of the dependent variable. b. is the same as instrument exogeneity. c. means that some of the variance in the regressor is related to variation in the instrument. d. is not possible since $X$ and $u$ are correlated and $Z$ and $u$ are not correlated.

The two conditions for a valid instrument are a. $\operatorname { corr } \left( Z _ { i } , X _ { i } \right) = 0$ and $\operatorname { corr } \left( Z _ { i } , u _ { i } \right) \neq 0$ . b. $\operatorname { corr } \left( Z _ { i } , X _ { i } \right) = 0$ and $\operatorname { corr } \left( Z _ { i } , u _ { i } \right) = 0$ . c. $\operatorname { corr } \left( Z _ { i } , X _ { i } \right) \neq 0$ and $\operatorname { corr } \left( Z _ { i } , u _ { i } \right) = 0$ . d. $\operatorname { corr } \left( Z _ { i } , X _ { i } \right) \neq 0$ and $\operatorname { corr } \left( Z _ { i } , u _ { i } \right) \neq 0$ .

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.

Here are some examples of the instrumental variables regression model.In each case you are given the number of instruments and the J-statistic.Find the relevant value from the $\chi _ { m - k } ^ { 2 }$ distribution, using a 1% and 5% significance level, and make a decision whether or not to reject the null hypothesis. (a) $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { 1 i } + u _ { i } , i = 1 , \ldots , n ; Z _ { 1 i } , Z _ { 2 i } \text { are valid instruments, } J = 2.58 \text {. }$

Consider the following model of demand and supply of coffee: Demand: $Q _ { i } ^ { \text {Coffee } } = \beta _ { 1 } P _ { i } ^ { \text {Coffee } } + \beta _ { 2 } P _ { i } ^ { T e a } + u _ { i }$ Supply: $Q _ { i } ^ { \text {Coffee } } = \beta _ { 3 } P _ { i } ^ { \text {Coffee } } + \beta _ { 4 } P _ { i } ^ { \text {Tea } } + \beta _ { 5 }$ Weather $+ v _ { i }$ (variables are measure in deviations from means, so that the constant is omitted). What are the expected signs of the various coefficients this model? Assume that the price of tea and Weather are exogenous variables.Are the coefficients in the supply equation identified? Are the coefficients in the demand equation identified? Are they overidentified? Is this result surprising given that there are more exogenous regressors in the second equation?

The reduced form equation for X

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

The TSLS estimator is

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.

(requires Appendix material) The relationship between the TSLS slope and the corresponding population parameter is:

The IV regression assumptions include all of the following with the exception of a. the error terms must be normally distributed. b. $E \left( u _ { i } \mid W _ { 1 i } , \ldots , W _ { r i } \right) = 0$ . c. Large outliers are unlikely: the $X$ 's, $W$ 's, $Z$ 's, and $Y$ 's all have nonzero, finite fourth moments. d. $\left( X _ { 1 i } , \ldots , X _ { k i } , W _ { 1 i } , \ldots , W _ { r i } , Z _ { 1 i } , \ldots Z _ { m i } , Y _ { i } \right)$ are i.i.d. draws from their joint distribution.

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): = 15.63+0.80\times;=0.65,SER=9.63 (0.10) (a)Calculate the predicted temperature for the current year if the temperature in the previous year was 400F, 780F, and 1000F.How does this compare with you prior expectation? Sketch the regression line and compare it to the 45 degree line.What are the implications?

When there is a single instrument and single regressor, the TSLS estimator for the slope can be calculated as follows a. $\widehat { \beta } _ { 1 } ^ { T S L S } = \frac { s _ { Z Y } } { s _ { Z X } }$ . b. $\widehat { \beta } _ { 1 } ^ { \text {TSLS } } = \frac { s _ { X Y } } { s _ { X } ^ { 2 } }$ . c. $\widehat { \beta } _ { 1 } ^ { \text {TSLS } } = \frac { s _ { Z X } } { s _ { Z Y } }$ . d. $\hat { \beta } _ { 1 } ^ { T L L S } = \frac { s _ { Z Y } } { s _ { Z } ^ { 2 } }$ .

To study the determinants of growth between the countries of the world, researchers have used panels of countries and observations spanning over long periods of time (e.g.1965- 1975, 1975-1985, 1985-1990).Some of these studies have focused on the effect that inflation has on growth and found that although the effect is small for a given time period, it accumulates over time and therefore has an important negative effect. (a)Explain why the OLS estimator may be biased in this case.