Pairs of random variables (Xi; Yi) were observed. They were assumed to follow a linear regression with E(Yi|Xi) = θ1+θ2Xi but with t-distributed noise, rather than the usual normally distributed noise. More specifically, the assumed model was that conditionally, given Xi, Yi is t-distributed with mean θ1 + θ2Xi, standard deviation θ3, and degrees of freedom θ4. Also, the pairs (X1; Y1), . . . , (Xn, Yn) are mutually independent. The model could also be expressed as
Yi = θ1 + θ2Xi + ∈i
where ∈1, . . . . ∈n are i.i.d. t with mean 0 and standard deviation θ3 and degrees of freedom θ4. The model was fit by maximum likelihood. The R code and output are
#(code to input x and y)
library(fGarch)
start = c(lmfit$coef,sd(lmfit$resid),4)
loglik = function(theta)
{
-sum(log(dstd(y,mean=theta[1]+theta[2]*x,sd=theta[3],
nu=theta[4])))
}
mle = optim(start, loglik, hessian=T)
FishInfo = solve(mle$hessian)
mle$par
mle$value
mle$convergence
sqrt(diag(FishInfo))
qnorm(.975)
> mle$par
[1] 0.511 1.042 0.152 4.133
> mle$value
[1] -188
> mle$convergence
[1] 0
> sqrt(diag(FishInfo))
[1] 0.00697 0.11522 0.01209 0.93492
>
> qnorm(.975)
[1] 1.96
>
(a) What is the MLE of the slope of Yi on Xi?
(b) What is the standard error of the MLE of the degrees-of-freedom parameter?
(c) Find a 95% confidence interval for the standard deviation of the noise.
(d) Did optim converge? Why or why not?