Suppose that our data comprise a set of pairs (Yi, Xi), for i = 1, . . . , n. Here, each Yi is a random variable and each Xi is a known constant. Suppose that we use a simple linear regression model in which E(Yi) = β0 + β1Xi . Let stand for the least squares estimator of β1. Suppose, however, that the Yi’s are actually random variables with translated and scaled t distributions. In particular, suppose that (Yi − β0 − β1Xi)/σ are i.i.d. having the t distribution with k ≥ 5 degrees of freedom for i = 1, . . . , n. We can use simulation to estimate the standard deviation of the sampling distribution of .
a. Prove that the variance of the sampling distribution of does not depend on the values of the parameters β0 and β1.
b. Prove that the variance of the sampling distribution of is equal to vσ2, where v does not depend on any of the parameters β0, β1, and σ.
c. Describe a simulation scheme to estimate the value v from part (b).