Exam 12: Instrumental Variables Regression

Consider the following model of demand and supply of coffee: $\text { Demand: } Q_{i}^{\text {Coffee }}=\beta_{1} P_{i}^{\text {Coffee }}+\beta_{2} P_{i}^{\text {Tea }}+u_{i}$ $\text { Supply: } Q_{i}^{\text {Coffee }}=\beta_{3} P{ }_{i}^{\text {Coffee }}+\beta_{4} P_{i}^{\text {Tea }}+\beta_{5} \text { Weather }+v_{i}$ (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?

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 = β₀ + β₁X_1i + u_i, i = 1, ..., n; Z_1i, Z_2i are valid instruments, J = 2.58. (b)Y_i = β₀ + β₁X_1i + β₂X_2i + β₃W_1i + u_i, i = 1, ..., n; Z_1i, Z_2i, Z_3i, Z_4i are valid instruments, J = 9.63. (c)Y_i = β₀ + β₁X_1i + β₂W_1i + β₃W_2i + β₄W_3i + u_i, i = 1, ..., n; Z_1i, Z_2i, Z_3i, Z_4i are valid instruments, J = 11.86.

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. (b)The individual's IQ or testscore on a work related exam. (c)Years of education for the individual's mother or father. (d)Number of siblings the individual has.

(Requires Matrix Algebra)The population multiple regression model can be written in matrix form as Y = Xβ + U where Y = $\left(\begin{array} { l } Y _ { 1 } \\Y _ { 2 } \\O \\Y _ { n }\end{array}\right)$ , U = $\left( \begin{array} { c } u _ { 1 } \\u _ { 2 } \\\mathrm { O } \\u _ { n }\end{array} \right)$ , X = $\left(\begin{array} { l } 1 &X_{11}& \mathrm{~N} &X_{k 1}& W_{11} &\mathrm{~N} &W_{r 1}\\1 &X_{12}& \mathrm{~N}& X_{k 2} &W_{12}& \mathrm{~N}& W_{r 2}\\O&O&R&O&O&R&O\\1 &X_{1 n}& \mathrm{~N}& X_{k n} &W_{1 n} &\mathrm{~N}& W_{m}\\\end{array}\right)$ , and β = $\left( \begin{array} { l } \beta _ { 0 } \\\beta _ { 1 } \\\mathrm { O } \\\beta _ { k }\end{array} \right)$ 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 $\hat { \beta } ^ { I V } = \left[ X ^ { \prime } Z \left( Z ^ { \prime } Z \right) ^ { - 1 } Z ^ { \prime } X \right] ^ { - 1 } X ^ { \prime } Z \left( Z ^ { \prime } Z \right) ^ { - 1 } Z ^ { \prime } Y,$ where 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 = $\left(\begin{array} { l } 1 &Z_{11} &\mathrm{~N}& Z_{m 1}& W_{11}& \mathrm{~N}& W_{r 1}\\1 &Z_{12}& \mathrm{~N}& Z_{m 2} &W_{12}& \mathrm{~N} &W_{r 2}\\O&O&R&O &O& R &O \\1 &Z_{1 n }&\mathrm{~N}& Z_{mn }& W_{1 n}& \mathrm{~N} &W_{m}\end{array}\right)$ It is of order n × (m+r+1). For this estimator to exist, both ( $Z ^ { \prime }$ Z)and [ $X ^ { \prime }$ Z( $Z ^ { \prime }$ Z)^-1 $Z ^ { \prime }$ X] must be invertible. State the conditions under which this will be the case and relate them to the degree of overidentification.

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. (b)Explain how methods using panel data could potentially alleviate the problem. (c)Some authors have suggested using an index of central bank independence as an instrumental. Discuss whether or not such an index would be a valid instrument.

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.

The two conditions for instrument validity are corr(Z_i, X_i)≠ 0 and corr(Z_i, u_i)= 0. The reason for the inconsistency of OLS is that corr(X_i, u_i)≠ 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.

Consider the a model of the U.S. labor market where the demand for labor depends on the real wage, while the supply of labor is vertical and does not depend on the real wage. You could argue that the supply of labor by households (think of hours supplied by two adults and two children)has not changed much over the last 60 years or so in the U.S. while real wages more than doubled over the same time span. At first that seems strange given the higher participation rate of females over that period, but that increase has been countered by a lower male participation rate (resulting from earlier retirement), an increase in legal holidays, and an increase in vacation days. a. Write down two equations representing the labor supply and labor demand function, allowing for an error term in each of the demand and supply equation. In addition, assume that the labor market clears. b. How would you estimate the labor supply equation? c. Assuming that the error terms are mutually independent i.i.d. random variables, both with mean zero, show that the real wage and the error term of the labor demand equation are correlated. d. If you find a non-zero correlation, should you estimate the labor demand equation using OLS? If so, what are the consequences? e. Estimating the labor demand equation by IV estimation, which instrument suggests itself immediately?

Consider the following population regression model relating the dependent variable Y_i and regressor X_i, Y_i = β₀ + β₁X_i + u_i, i = 1, …, n. X_i ≡ Y_i + Z_i where Z is a valid instrument for X. (a)Explain why you should not use OLS to estimate β₁. (b)To generate a consistent estimator for β₁, what should you do? (c)The two equations above make up a system of equations in two unknowns. Specify the two reduced form equations in terms of the original coefficients. (Hint: substitute the identity into the first equation and solve for Y. Similarly, substitute Y into the identity and solve for X.) (d)Do the two reduced form equations satisfy the OLS assumptions? If so, can you find consistent estimators of the two slopes? What is the ratio of the two estimated slopes? This estimator is called "Indirect Least Squares." How does it compare to the TSLS in this example?

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. $\widehat{\text { Attend }}=15,005+201 \times \text { Temperat }+465 \times \text { DodgNetWin }+82 \times \text { OppNetWin }$ $(8,770) \quad(121)\quad\quad(169) \quad\quad(26)$ $+9647 \times \text { DFSaSu }+1328 \times \text { Drain }+1609 \times \text { D } 150 m+271 \times \text { DDiv }-978 \times \text { D2001 }$ (1505) (3355) (1819) (1,184) (1,143) $R^{2}=0.416, S E R=6983$ 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?