Re: Linear regression with functional heteroscedasticity

On Tue, 10 Jul 2007 12:26:47 -0700, mikko.kauppila@xxxxxxxxx wrote:

Hello fellows,

The basic homoscedastic linear regression problem is as follows:

Y | X ~ N(g(X), v)

where g(x) = kx + b is the regression function. Here the parameters
(k,b,v) can be estimated quite easily.

Now, suppose we modify the problem a little by introducing functional
heteroscedasticity, i.e.:

Y | X ~ N(g(X), f(X))

where g(x) = kx + b is the regression function and f(x) = wx + v is
the
variance function, assumed linear in x for simplicity.

Is it possible to estimate all parameters (k,b,w,v) with a closed
form solution if the variance function f is simple enough (e.g. linear
as above)?
Or can anyone suggest pointers in the literature? I worked out the
gradient
of the log-likelihood function on paper, but I have no idea how to
find
the zeros of the gradient as it looks quite non-linear.

Thanks for your time fellows,
- Mikko

While not a closed-form solution, there is an easy way to do this.
First, estimate g(x) by least squares. Then regress the squared
residuals on x to estimate the f(x). If my memory is correct, this
gives consistent estimate of both g(x) and f(x). I believe if you use
estimated f(x) to correct for heteroskedasticity, you then get an
efficient estimate of g(x).

-Dick Startz
.

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