Exam 12: Instrumental Variables Regression

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.

To calculate the J-statistic you regress the a. squared values of the TSLS residuals on all exogenous variables and the instruments. The statistic is then the number of observations times the regression $R ^ { 2 }$ b. TSLS residuals on all exogenous variables and the instruments. You then multiply the homoskedasticity-only $F$ -statistic from that regression by the number of instruments. c. OLS residuals from the reduced form on the instruments. The $F$ -statistic from this regression is the $J$ -statistic. d. TSLS residuals on all exogenous variables and the instruments. You then multiply the heteroskedasticity-robust $F$ -statistic from that regression by the number of instruments.

When calculating the TSLS standard errors

In the case of the simple regression model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i } , i = 1 , \ldots , n$ when X and u are correlated, then

Two Stage Least Squares is calculated as follows;in the first stage

The following will not cause correlation between X and u in the simple regression model:

If the instruments are not exogenous,

Weak instruments are a problem because

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?

In the case of exact identification

Consider the following population regression model relating the dependent variable $Y _ { i }$ and regressor $X _ { i }$ , =++,i=1,\ldots,n. \equiv+ where Z is a valid instrument for X. (a) $\text { Exlain why you should not use OLS to estimate } \beta _ { 1 } \text {. }$

The rule-of-thumb for checking for weak instruments is as follows: for the case of a single endogenous regressor, a. a first stage $F$ must be statistically significant to indicate a strong instrument. b. a first stage $F > 1.96$ indicates that the instruments are weak. c. the $t$ -statistic on each of the instruments must exceed at least 1.64. d. a first stage $F < 10$ indicates that the instruments are weak.

Having more relevant instruments

The J-statistic a. tells you if the instruments are exogenous. b. provides you with a test of the hypothesis that the instruments are exogenous for the case of exact identification. c. is distributed $\chi _ { m - k } ^ { 2 }$ where $m - k$ is the degree of overidentification. d. Is distributed $\chi _ { m - k } ^ { 2 }$ where $m - k$ is the number of instruments minus the number of regressors.

You started your econometrics course by studying the OLS estimator extensively, first for the simple regression case and then for extensions of it.You have now learned about the instrumental variable estimator.Under what situation would you prefer one to the other? Be specific in explaining under which situations one estimation method generates superior results.

In practice, the most difficult aspect of IV estimation is

The distinction between endogenous and exogenous variables is

Your textbook gave an example of attempting to estimate the demand for a good in a market, but being unable to do so because the demand function was not identified.Is this the case for every market? Consider, for example, the demand for sports events.One of your peers estimated the following demand function after collecting data over two years for every one of the 162 home games of the 2000 and 2001 season for the Los Angeles Dodgers. =15,005+201\times Temperat +465\times DodgNetWin +82\times OppNetWin (8,770)(121)(169)(26) +9647\times DFSaSu +1328\times Drain +1609\times D 150m+271\times DDiv -978\times D2001; (1505) (3355) (1819) (1,184)(1,143) =0.416,SER=6983 Where Attend is announced stadium attendance, Temperat it the average temperature on game day, DodgNetWin are the net wins of the Dodgers before the game (wins-losses), OppNetWin is the opposing team's net wins at the end of the previous season, and DFSaSu, Drain, D150m, Ddiv, and D2001 are binary variables, taking a value of 1 if the game was played on a weekend, it rained during that day, the opposing team was within a 150 mile radius, plays in the same division as the Dodgers, and during 2001, respectively. Numbers in parenthesis are heteroskedasticity- robust standard errors. Even if there is no identification problem, is it likely that all regressors are uncorrelated with the error term? If not, what are the consequences?

(Requires Matrix Algebra)The population multiple regression model can be written in matrix form as $\boldsymbol { Y } = \boldsymbol { X } \boldsymbol { \beta } + \boldsymbol { U }$ Where $\boldsymbol { Y } = \left( \begin{array} { l } Y _ { 1 } \\Y _ { 2 } \\\vdots \\Y _ { n }\end{array} \right) , \boldsymbol { U } = \left( \begin{array} { l } u _ { 1 } \\u _ { 2 } \\\vdots \\u _ { n }\end{array} \right) , \boldsymbol { X } = \left( \begin{array} { c c c c c c c } 1 & X _ { 11 } & \cdots & X _ { k 1 } & W _ { 11 } & \cdots & W _ { r 1 } \\1 & X _ { 12 } & \cdots & X _ { k 2 } & W _ { 12 } & \cdots & W _ { r 2 } \\\vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\1 & X _ { 1 n } & \cdots & X _ { k n } & W _ { 1 n } & \cdots & W _ { r n }\end{array} \right) \text { and } \beta = \left( \begin{array} { l } \beta _ { 0 } \\\beta _ { 1 } \\\vdots \\\beta _ { k }\end{array} \right)$ Note that the X matrix contains both k endogenous regressors and (r +1)included exogenous regressors (the constant is obviously exogenous). The instrumental variable estimator for the overidentified case is $\hat { \beta } ^ { V } = \left[ X ^ { \prime } Z \left( Z ^ { \prime } Z \right) ^ { - 1 } Z ^ { \prime } X \right] ^ { - 1 } X ^ { \prime } Z \left( Z ^ { \prime } Z \right) ^ { - 1 } Z ^ { \prime } Y ,$ where $\boldsymbol { Z }$ is a matrix, which contains two types of variables: first the $r$ included exogenous regressors plus the constant, and second, $m$ instrumental variables. $Z = \left( \begin{array} { c c c c c c c } 1 & Z _ { 11 } & \cdots & Z _ { m 1 } & W _ { 11 } & \cdots & W _ { r 1 } \\1 & Z _ { 12 } & \cdots & Z _ { m 2 } & W _ { 12 } & \cdots & W _ { r 2 } \\\vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\1 & Z _ { 1 n } & \cdots & Z _ { m n } & W _ { 1 n } & \cdots & W _ { m }\end{array} \right)$ It is of order $\mathrm { n } \times ( \mathrm { m } + \mathrm { r } + 1 )$ . For this estimator to exist, both $\left( Z ^ { \prime } Z \right)$ and $\left[ X ^ { \prime } Z \left( Z ^ { \prime } Z \right) ^ { - 1 } Z ^ { \prime } X \right]$ must be invertible. State the conditions under which this will be the case and relate them to the degree of overidentification.

Earnings functions, whereby the log of earnings is regressed on years of education, years of on the job training, and individual characteristics, have been studied for a variety of reasons.Some studies have focused on the returns to education, others on discrimination, union non-union differentials, etc.For all these studies, a major concern has been the fact that ability should enter as a determinant of earnings, but that it is close to impossible to measure and therefore represents an omitted variable. Assume that the coefficient on years of education is the parameter of interest.Given that education is positively correlated to ability, since, for example, more able students attract scholarships and hence receive more years of education, the OLS estimator for the returns to education could be upward biased.To overcome this problem, various authors have used instrumental variable estimation techniques.For each of the instruments potential instruments listed below briefly discuss instrument validity. (a)The individual's postal zip code. Answer Instrumental validity has two components, instrument relevance $\left( \operatorname { corr } \left( Z _ { i } , X _ { i } \right) \neq 0 \right)$ , and instrument exogeneity $\left( \operatorname { corr } \left( Z _ { i } , u _ { i } \right) = 0 \right)$ . The individual's postal zip code will certainly be uncorrelated with the omitted variable, ability, even though some zip codes may attract more able individuals. However, this is an example of a weak instrument, since it is also uncorrelated with years of education. (b)The individual's IQ or testscore on a work related exam. Answer: There is instrument relevance in this case, since, on average, individuals who do well in intelligence scores or other work related test scores, will have more years of education.Unfortunately there is bound to be a high correlation with the omitted variable ability, since this is what these tests are supposed to measure. (c)Years of education for the individual's mother or father. Answer: A non-zero correlation between the mother's or father's years of education and the individual's years of education can be expected.Hence this is a relevant instrument.However, it is not clear that the parent's years of education are uncorrelated with parent's ability, which in turn, can be a major determinant of the individual's ability.If this is the case, then years of education of the mother or father is not a valid instrument. (d)Number of siblings the individual has.