In Chapter 10 of your textbook, panel data estimation was introduced. Panel data consist of observations on the same n entities at two or more time periods T. For two variables, you have
(X_it, Y_it), i = 1,..., n and t = 1,..., T
where n could be the U.S. states. The example in Chapter 10 used annual data from 1982 to 1988 for the fatality rate and beer taxes. Estimation by OLS, in essence, involved "stacking" the data.
(a)What would the variance-covariance matrix of the errors look like in this case if you allowed for homoskedasticity-only standard errors? What is its order? Use an example of a linear regression with one regressor of 4 U.S. states and 3 time periods.
(b)Does it make sense that errors in New Hampshire, say, are uncorrelated with errors in Massachusetts during the same time period ("contemporaneously")? Give examples why this correlation might not be zero.
(c)If this correlation was known, could you find an estimator which was more efficient than OLS?

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The Answer of In Chapter 10 of your textbook, panel data estimation was introduced. Panel data consist of observations on the same n entities at two or more time periods T. For two variables, you have (X_it, Y_it), i = 1,..., n and t = 1,..., T where n could be the U.S. states. The example in Chapter 10 used annual data from 1982 to 1988 for the fatality rate and beer taxes. Estimation by OLS, in essence, involved "stacking" the data. (a)What would the variance-covariance matrix of the errors look like in this case if you allowed for homoskedasticity-only standard errors? What is its order? Use an example of a linear regression with one regressor of 4 U.S. states and 3 time periods. (b)Does it make sense that errors in New Hampshire, say, are uncorrelated with errors in Massachusetts during the same time period ("contemporaneously")? Give examples why this correlation might not be zero. (c)If this correlation was known, could you find an estimator which was more efficient than OLS?

In Chapter 10 of Your Textbook, Panel Data Estimation Was