Exam 17: The Theory of Linear Regression With One Regressor

Suppose that the conditional variance is var(u_i|X_i)= λh(X_i)where λ is a constant and h is a known function. The WLS estimator is

If the variance of u is quadratic in X, then it can be expressed as

Discuss the properties of the OLS estimator when the regression errors are homoskedastic and normally distributed. What can you say about the distribution of the OLS estimator when these features are absent?

Consider the model Y_i = β₁X_i + u_i, where u_i = c $x _ { i } ^ { 2 }$ e_i and all of the X's and e's are i.i.d. and distributed N(0,1). (a)Which of the Extended Least Squares Assumptions are satisfied here? Prove your assertions. (b)Would an OLS estimator of β₁ be efficient here? (c)How would you estimate β₁ by WLS?

In order to use the t-statistic for hypothesis testing and constructing a 95% confidence interval as $\pm$ 1.96 standard errors, the following three assumptions have to hold:

One of the earlier textbooks in econometrics, first published in 1971, compared "estimation of a parameter to shooting at a target with a rifle. The bull's-eye can be taken to represent the true value of the parameter, the rifle the estimator, and each shot a particular estimate." Use this analogy to discuss small and large sample properties of estimators. How do you think the author approached the n → ∞ condition? (Dependent on your view of the world, feel free to substitute guns with bow and arrow, or missile.)

What does the Gauss-Markov theorem prove? Without giving mathematical details, explain how the proof proceeds. What is its importance?

Consider the simple regression model Yi = β₀ + β₁X_i + u_i where X_i > 0 for all i, and the conditional variance is var(u_i | X_i)= θX 2 i where θ is a known constant with θ > 0. (a)Write the weighted regression as $\bar Y$ _i = β₀ $\bar X$ _0i + β₁ $\bar X$ _1i + $\bar u$ _i. How would you construct $\bar Y$ _i, $\bar X$ _0i and $\bar X$ _1i? (b)Prove that the variance of is $\bar u$ _i homoskedastic. (c)Which coefficient is the intercept in the modified regression model? Which is the slope? (d)When interpreting the regression results, which of the two equations should you use, the original or the modified model?

The following is not part of the extended least squares assumptions for regression with a single regressor:

The Gauss-Markov Theorem proves that

Consider the model Y_i - β₁X_i + u_i, where the X_i and u_i the are mutually independent i.i.d. random variables with finite fourth moment and E(u_i)= 0. (a)Let $\hat { \beta }$ ₁ denote the OLS estimator of β₁. Show that $\sqrt { n }$ ( $\hat { \beta }$ ₁- β₁)= $\frac { \frac { \sum _ { i = 1 } ^ { n } X _ { i } t _ { i } } { \sqrt { n } } } { \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } }$ (b)What is the mean and the variance of $\frac { \sum _ { i = 1 } ^ { n } X _ { i } u _ { i } } { \sqrt { n } }$ ? Assuming that the Central Limit Theorem holds, what is its limiting distribution? (c)Deduce the limiting distribution of $\sqrt { n }$ ( $\hat { \beta }$ ₁ - β₁)? State what theorems are necessary for your deduction.

All of the following are good reasons for an applied econometrician to learn some econometric theory, with the exception of

When the errors are heteroskedastic, then

(Requires Appendix material)This question requires you to work with Chebychev's Inequality. (a)State Chebychev's Inequality. (b)Chebychev's Inequality is sometimes stated in the form "The probability that a random variable is further than k standard deviations from its mean is less than 1/k²." Deduce this form. (Hint: choose ? artfully.) (c)If X is distributed N(0,1), what is the probability that X is two standard deviations from its mean? Three? What is the Chebychev bound for these values? (d)It is sometimes said that the Chebychev inequality is not "sharp." What does that mean?

The WLS estimator is called infeasible WLS estimator when

In practice, the most difficult aspect of feasible WLS estimation is

(Requires Appendix material)State and prove the Cauchy-Schwarz Inequality.

The link between the variance of $\bar { Y }$ and the probability that $\bar { Y }$ is within (± ? of ${ } ^ { \mu } \gamma$ is provided by

Homoskedasticity means that

If the functional form of the conditional variance function is incorrect, then