Problem 1. The (Weak) Law of Large Numbers states (WLLN): Let $x_1,..., x_n$ be i.i.d. random variables with finite mean (i.e. $E[x_i] = \mu < \infty$), then $\bar{x} \xrightarrow{p} \mu$. This “law” is just a theorem that can be show with Chebyshev's inequality and the formal definition of convergence in probability.
Slusky's Theorem (proven by the same guy who made Slusky decomposition; he was both a statistician and an economist) shows that if $X_n \xrightarrow{p} a$ and $Y_n \xrightarrow{p} b$, then $X_n + Y_n \xrightarrow{p} a + b$ and $X_n Y_n \xrightarrow{p} ab$.
In the simple regression model under MLR.1, MLR.2, MLR.3 and MLR.4, we argued that the slope estimator, $\beta_1$, is consistent for $\beta_1$. Using $\hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}$, show that $\hat{\beta}_0 = \beta_0$. [You need to use the consistency of $\hat{\beta}_1$, WLLN, and Slutsky's Theorem, along with the fact that $\beta_0 = E[y] - \beta_1 E[x]$.]
