Use the data in MATHPNL.RAW for this exercise. You will do a fixed effects version of the first differencing done in Computer Exercise C13.11. The model of interest is

where the first available year (the base year) is 1993 because of the lagged spending variable.
(i) Estimate the model by pooled OLS and report the usual standard errors. You should include an intercept along with the year dummies to allow a. to have a nonzero expected value. What are the estimated effects of the spending variables? Obtain the OLS residuals, it.
(ii) Is the sign of the lunchit, coefficient what you expected? Interpret the magnitude of the coefficient. Would you say that the district poverty rate has a big effect on test pass rates?
(iii) Compute a test for AR(1) serial correlation using the regression it on i.t-1. You should use the years 1994 through 1998 in the regression. Verify that there is strong positive serial correlation and discuss why.
(iv) Now, estimate the equation by fixed effects. Is the lagged spending variable still significant?
(v) Why do you think, in the fixed effects estimation, the enrollment and lunch program variables are jointly insignificant?
(vi) Define the total, or long-run, effect of spending as θ1 = y1 + y2. Use the substitution y1 = θ1 - y2 to obtain a standard error for θ1.

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