$($Computer Simulation Programming$)$

This question asks you to conduct a simulation experiment to examine the finite$-$sample distributions of $t$ ratios using the standard errors based on widehat$(V{)}_{\text{w}\text{i}\text{d}\text{e}\text{h}\text{a}\text{t}()}^0($classical t ratios$)$ and widehat$(V{)}_{\text{w}\text{i}\text{d}\text{e}\text{h}\text{a}\text{t}()}^{HC0}($White t ratios$)$ under homoskedasticity and heteroskedasticity.

Take the data generating process:

$Y_i={}_0+x_i{}_1+e_i,i=1,$dots,$n$

$e_i|x_{i}N(0,{}^2(x_i))|$

where $x_i$inR and $({}_0,{}_1)=(1,0).$ Consider two specifications for $x_i$ : $x_i{}^{i.i.d.}(\frac{1}{4},1)$ and Lognormal$(0,4).$ Consider homoskedastic errors with ${}^2(x_i)=1$ and heteroskedastic errors with ${}^2(x_i)=1+x_i^2.$

The testing task is $H_0$:${}_1=0$ against $H_1$:${}_{1}0.$ Construct the classical and White $t$ ratios $T$ for ${}_1$ and calculate the finite$-$sample size of the twosided tests based on nominal $5\%t_{n-2}$ critical values. Set the sample sizes $n=\{30,60,120,500\}$ and the number of simulation replications $R=5000.$

$($a$)$ Fill in the rejection rates of the $t$ ratios in Table $1$ below.

$($b$)$ Are your simulation results consistent with the theoretical results you learned in class? Discuss the findings from your simulation results in Table $1.$

$($c$)$ Provide your simulation code.

Note: You can work with any statistical software$/$languages$.$ But instead of using the built$-$in functions such as $lm()$ and vcov$()$ in $R,$ you have to code from scratch using the formulae for widehat$(),$widehat$(V{)}_{\text{w}\text{i}\text{d}\text{e}\text{h}\text{a}\text{t}()}^0,$widehat$(V{)}_{\text{w}\text{i}\text{d}\text{e}\text{h}\text{a}\text{t}()}^{HC},$ and t ratios.

$1$

Table $1$: Rejection rates of the classical and White t ratios using nominal $5\%t_{n-2}$ critical values ${?}_{(})$

$\backslash$table$[[n,$Homoskedastic Error,Heteroskedastic Error$],[(\frac{1}{4},1),$LogNormal$(0,4),(\frac{1}{4},1),$LogNormal$(0,4)$