In a Monte Carlo study, econometricians generate multiple sample regression functions from a known population regression function. For example, the population regression function could be Y_i = ?₀ + ?₁X_i = 100 - 0.5 X_i. The Xs could be generated randomly or, for simplicity, be nonrandom ("fixed over repeated samples"). If we had ten of these Xs, say, and generated twenty Ys, we would obviously always have all observations on a straight line, and the least squares formulae would always return values of 100 and 0.5 numerically. However, if we added an error term, where the errors would be drawn randomly from a normal distribution, say, then the OLS formulae would give us estimates that differed from the population regression function values. Assume you did just that and recorded the values for the slope and the intercept. Then you did the same experiment again (each one of these is called a "replication"). And so forth. After 1,000 replications, you plot the 1,000 intercepts and slopes, and list their summary statistics.

Here are the corresponding graphs: Using the means listed next to the graphs, you see that the averages are not exactly 100 and -0.5. However, they are "close." Test for the difference of these averages from the population values to be statistically significant.

Question

In a Monte Carlo study, econometricians generate multiple sample regression functions from a known population regression function. For example, the population regression function could be Y_i = ?₀ + ?₁X_i = 100 - 0.5 X_i. The Xs could be generated randomly or, for simplicity, be nonrandom ("fixed over repeated samples"). If we had ten of these Xs, say, and generated twenty Ys, we would obviously always have all observations on a straight line, and the least squares formulae would always return values of 100 and 0.5 numerically. However, if we added an error term, where the errors would be drawn randomly from a normal distribution, say, then the OLS formulae would give us estimates that differed from the population regression function values. Assume you did just that and recorded the values for the slope and the intercept. Then you did the same experiment again (each one of these is called a "replication"). And so forth. After 1,000 replications, you plot the 1,000 intercepts and slopes, and list their summary statistics.

Here are the corresponding graphs:

Using the means listed next to the graphs, you see that the averages are not exactly 100 and -0.5. However, they are "close." Test for the difference of these averages from the population values to be statistically significant.

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The Answer of In a Monte Carlo study, econometricians generate multiple sample regression functions from a known population regression function. For example, the population regression function could be Y_i = ?₀ + ?₁X_i = 100 - 0.5 X_i. The Xs could be generated randomly or, for simplicity, be nonrandom ("fixed over repeated samples"). If we had ten of these Xs, say, and generated twenty Ys, we would obviously always have all observations on a straight line, and the least squares formulae would always return values of 100 and 0.5 numerically. However, if we added an error term, where the errors would be drawn randomly from a normal distribution, say, then the OLS formulae would give us estimates that differed from the population regression function values. Assume you did just that and recorded the values for the slope and the intercept. Then you did the same experiment again (each one of these is called a "replication"). And so forth. After 1,000 replications, you plot the 1,000 intercepts and slopes, and list their summary statistics. Here are the corresponding graphs: Using the means listed next to the graphs, you see that the averages are not exactly 100 and -0.5. However, they are "close." Test for the difference of these averages from the population values to be statistically significant.

In a Monte Carlo Study, Econometricians Generate Multiple Sample Regression