Do a Monte Carlo simulation Take the model Y Xe with E Xe 0 where the parameter of interest is exp() Your data generating process (DGP) for the simulation is X isU 0,1 , e N(0,1) is independent of X, and n 50 Set 0 and 1 Generate B 1000 independent samples with On each, estimate the regression by least squares, calculate the covariance matrix using a standard (heteroskedasticity robust) formula, and similarly estimate and its standard error For each replication, store b, b, T b s b , and T b s b (a) Does the value of matter Explain why the described statistics are invariant to and thus setting 0 is irrelevant (b) From the 1000 replications estimate E b and E b Discuss if you see evidence if either estimator is biased or unbiased (c) From the 1000 replications estimate P T 1 645 and P T 1 645 What does asymptotic theory predict these probabilities should be in large samples What do your simulation results indicate

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Question: Do a Monte Carlo simulation. Take the model Y Xe with E[Xe] 0 where the parameter of interest is exp().

Do a Monte Carlo simulation. Take the model Y Æ ®Å X¯Åe with E[Xe] Æ 0 where the parameter of interest is µ Æ exp(¯). Your data generating process (DGP) for the simulation is: X isU[0,1], e » N(0,1) is independent of X, and n Æ 50. Set ® Æ 0 and ¯ Æ 1. Generate B Æ 1000 independent samples with ®. On each, estimate the regression by least squares, calculate the covariance matrix using a standard (heteroskedasticity-robust) formula, and similarly estimate µ and its standard error. For each replication, store b¯, bµ, T¯ Æ

¡ b¯¡¯

¡ b¯

, and Tµ Æ

¡bµ¡µ

¡bµ

(a) Does the value of ® matter? Explain why the described statistics are invariant to ® and thus setting

® Æ 0 is irrelevant.

(b) From the 1000 replications estimate E

£ b¯

and E

£bµ

. Discuss if you see evidence if either estimator is biased or unbiased.

(c) From the 1000 replications estimate P

T¯ È 1.645

and P[Tµ È 1.645]. What does asymptotic theory predict these probabilities should be in large samples? What do your simulation results indicate?

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