tailieunhanh - Class Notes in Statistics and Econometrics Part 20

CHAPTER 39 Random Regressors. Until now we always assumed that X was nonrandom, ., the hypothetical repetitions of the experiment used the same X matrix. In the nonexperimental sciences, such as economics, this assumption is clearly inappropriate. | CHAPTER 39 Random Regressors Until now we always assumed that X was nonrandom . the hypothetical repetitions of the experiment used the same X matrix. In the nonexperimental sciences such as economics this assumption is clearly inappropriate. It is only justified because most results valid for nonrandom regressors can be generalized to the case of random regressors. To indicate that the regressors are random we will write them as X. 891 892 39. RANDOM REGRESSORS . Strongest Assumption Error Term Well Behaved Conditionally on Explanatory Variables The assumption which we will discuss first is that X is random but the classical assumptions hold conditionally on X . the conditional expectation E e X o and the conditional variance-covariance matrix V e X ct21. In this situation the least squares estimator has all the classical properties conditional ly on X for instance E 3 X ft V 3 X ct2 XtX -1 E s2 X a2 etc. Moreover certain properties of the Least Squares estimator remain valid unconditionally. An application of the law of iterated expectations shows that the least squares estimator is still unbiased. Start with 3 - XTX -1Xe E 3 - X E XTX -1XTe X XTX -1XT E e X o. E 3 - E E - X o. PROBLEM 408. 1 point In the model with random explanatory variables X you are considering an estimator of . Which statement is stronger E or E 3 X . Justify your answer. ANSWER. The second statement is stronger. The first statement follows from the second by the law of iterated expectations. . STRONGEST ASSUMPTION ERROR TERM WELL BEHAVED CONDITIONALLY ON EXPLA PROBLEM 409. 2 points Assume the regressors X are random and the classical assumptions hold conditionally on X . E e X o and V e X a21. Show that s2 is an unbiased estimate of a2. ANSWER. From the theory with nonrandom explanatory variables follows E s2 X 72. Therefore E s2 E E s2 X E a2 72. In words if the expectation conditional on X does not depend on X then it is also the unconditional