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B.1 OLS and 2SLS In this problem, you are to analyze the relationship between two variables, x1 (explanatory variable) and y (dependent variable). You run the following four regressions on Stata with a random sample of 2000 observations and obtain corresponding estimated coefficients: (1) reg logy x1 x2 x2sq ?log(y) = 4.590 + 0.085 x x1 + 0.092 x x2 – 0.003 x x2^2, R^2 = 0.156 (0.093) (0.005) (0.010) (0.001) (2) reg logy x1 ?log(y) = 5.782 + 0.040 x x1, R^2 = 0.047 (0.056) (0.004) (3) ivregress 2sls logy (x1 = z1) x2 x2sq ?log(y) = 1.531 + 0.256 x x1 + 0.196 x x2 – 0.005 x x2^2 (1.019) (0.057) (0.037) (0.001) (4) ivregress 2sls logy (x1 = z1 z2) x2 x2sq ?log(y) = 0.976 + 0.287 x x1 + 0.215 x x2 – 0.005 x x2^2 (1.050) (0.058) (0.038) (0.001) (a) (5 points) You want to estimate the effect on y (in terms of percentage change) from an extra unit of x1. You assume that assumptions A1-A5 of linear regression model hold and there are no other relevant factors. Under these conditions, what estimate from the above regressions would you use? What does this estimate tell you about the effect of x1 on y? Be sure to indicate which of the above regressions (by number) and exactly what estimate from that regression you are interested in. (b) (5 points) You think x2 and x2^2 should also be included in the regression. You want to convince yourself with a joint hypothesis test. State your null hypothesis and calculate the corresponding test statistic using the information provided in the above regressions. Hint: Recall that the sample size is 2000 and R^2 = 1 - SSE/SST. (c) (5 points) A consultant to your project is worried that the OLS estimator is problematic as x1 is very likely to be endogenous. If this is true, which assumption of linear regression model is violated? What is wrong with using OLS if x1 is indeed endogenous? (d) (5 points) The consultant suggests that you should adopt 2SLS rather than OLS. In particular, he proposes two instrumental variables, z1 and z2, for the endogenous variable x1. What conditions must hold for z1 and z2 to be valid instrumental variables? (e) (5 points) Now assume that both z1 and z2 are valid instruments. Which estimate from which of the above regressions do you think is the best estimate of the effect on y (in terms of percentage change) from an extra unit of x1? Be sure to give the regression number and particular coefficient and briefly justify your choice. (f) (5 points) Describe how to assess the strength of the instruments used in Regression (4).

          B.1 OLS and 2SLS

In this problem, you are to analyze the relationship between two variables, x1 (explanatory variable) and y (dependent variable). You run the following four regressions on Stata with a random sample of 2000 observations and obtain corresponding estimated coefficients:

(1) reg logy x1 x2 x2sq
?log(y) = 4.590 + 0.085 x x1 + 0.092 x x2 – 0.003 x x2^2, R^2 = 0.156
(0.093) (0.005) (0.010) (0.001)

(2) reg logy x1
?log(y) = 5.782 + 0.040 x x1, R^2 = 0.047
(0.056) (0.004)

(3) ivregress 2sls logy (x1 = z1) x2 x2sq
?log(y) = 1.531 + 0.256 x x1 + 0.196 x x2 – 0.005 x x2^2
(1.019) (0.057) (0.037) (0.001)

(4) ivregress 2sls logy (x1 = z1 z2) x2 x2sq
?log(y) = 0.976 + 0.287 x x1 + 0.215 x x2 – 0.005 x x2^2
(1.050) (0.058) (0.038) (0.001)

(a) (5 points) You want to estimate the effect on y (in terms of percentage change) from an extra unit of x1. You assume that assumptions A1-A5 of linear regression model hold and there are no other relevant factors. Under these conditions, what estimate from the above regressions would you use? What does this estimate tell you about the effect of x1 on y? Be sure to indicate which of the above regressions (by number) and exactly what estimate from that regression you are interested in.

(b) (5 points) You think x2 and x2^2 should also be included in the regression. You want to convince yourself with a joint hypothesis test. State your null hypothesis and calculate the corresponding test statistic using the information provided in the above regressions. Hint: Recall that the sample size is 2000 and R^2 = 1 - SSE/SST.

(c) (5 points) A consultant to your project is worried that the OLS estimator is problematic as x1 is very likely to be endogenous. If this is true, which assumption of linear regression model is violated? What is wrong with using OLS if x1 is indeed endogenous?

(d) (5 points) The consultant suggests that you should adopt 2SLS rather than OLS. In particular, he proposes two instrumental variables, z1 and z2, for the endogenous variable x1. What conditions must hold for z1 and z2 to be valid instrumental variables?

(e) (5 points) Now assume that both z1 and z2 are valid instruments. Which estimate from which of the above regressions do you think is the best estimate of the effect on y (in terms of percentage change) from an extra unit of x1? Be sure to give the regression number and particular coefficient and briefly justify your choice.

(f) (5 points) Describe how to assess the strength of the instruments used in Regression (4).

B.1 OLS and 2SLS

In this problem, you are to analyze the relationship between two variables, x1 (explanatory variable) and y (dependent variable). You run the following four regressions on Stata with a random sample of 2000 observations and obtain corresponding estimated coefficients:

(1) reg logy x1 x2 x2sq
?log(y) = 4.590 + 0.085 x x1 + 0.092 x x2 – 0.003 x x2^2, R^2 = 0.156
(0.093) (0.005) (0.010) (0.001)

(2) reg logy x1
?log(y) = 5.782 + 0.040 x x1, R^2 = 0.047
(0.056) (0.004)

(3) ivregress 2sls logy (x1 = z1) x2 x2sq
?log(y) = 1.531 + 0.256 x x1 + 0.196 x x2 – 0.005 x x2^2
(1.019) (0.057) (0.037) (0.001)

(4) ivregress 2sls logy (x1 = z1 z2) x2 x2sq
?log(y) = 0.976 + 0.287 x x1 + 0.215 x x2 – 0.005 x x2^2
(1.050) (0.058) (0.038) (0.001)

(a) (5 points) You want to estimate the effect on y (in terms of percentage change) from an extra unit of x1. You assume that assumptions A1-A5 of linear regression model hold and there are no other relevant factors. Under these conditions, what estimate from the above regressions would you use? What does this estimate tell you about the effect of x1 on y? Be sure to indicate which of the above regressions (by number) and exactly what estimate from that regression you are interested in.

(b) (5 points) You think x2 and x2^2 should also be included in the regression. You want to convince yourself with a joint hypothesis test. State your null hypothesis and calculate the corresponding test statistic using the information provided in the above regressions. Hint: Recall that the sample size is 2000 and R^2 = 1 - SSE/SST.

(c) (5 points) A consultant to your project is worried that the OLS estimator is problematic as x1 is very likely to be endogenous. If this is true, which assumption of linear regression model is violated? What is wrong with using OLS if x1 is indeed endogenous?

(d) (5 points) The consultant suggests that you should adopt 2SLS rather than OLS. In particular, he proposes two instrumental variables, z1 and z2, for the endogenous variable x1. What conditions must hold for z1 and z2 to be valid instrumental variables?

(e) (5 points) Now assume that both z1 and z2 are valid instruments. Which estimate from which of the above regressions do you think is the best estimate of the effect on y (in terms of percentage change) from an extra unit of x1? Be sure to give the regression number and particular coefficient and briefly justify your choice.

(f) (5 points) Describe how to assess the strength of the instruments used in Regression (4).

Added by Sebastian M.

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