Consider the Model Y_i - β₁X_i + U_i, Where the X_i

question 31

Essay

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.

---

Definitions:

Pelvic Girdle

A ring-like bony structure located at the base of the spine and connected to the thigh bones, supporting the weight of the upper body.

Hip Bone

A large, irregularly shaped bone forming the upper part of each half of the pelvis, composing of three parts - the ilium, the ischium, and the pubis.

Exoskeletons

Hard outer structures that provide support and protection to the bodies of certain organisms, such as insects and crustaceans.

Invertebrates

Animals without a backbone, comprising a large variety of species including insects, worms, and mollusks.