[R] simultaneous estimation

[Duncan Murdoch]

On 31/08/2010 6:58 AM, Murali.Menon at avivainvestors.com wrote:
> Hi folks,
> 
> Not sure what this sort of estimation is called. I have a 2-column time-series x(i,t) [with (i=1,2; t=1,...T)], and I want to do the following 'simultaneous' regressions:
> 
> x(1,t) = (d - 1)(x(1, t-1) - mu(1))
> x(2,t) = (d - 1)(x(2, t-1) - mu(2))
> 
> And I want to determine the coefficients d, mu(1), mu(2). 
> 
> Note that the d should be the same for both estimations, whereas the coefficients mu will have two values mu(1), mu(2), one for each estimation.
> 
> Is this possible to do in R?
> 
> What would be the corresponding syntax in, say, lm?

Your specification is not complete: you haven't said what the errors 
will be, or how x(1,1) and x(2,1) are determined.  I assume you mean 
independent normal errors, but are you willing to assume the variance is 
the same in both series?  If so, then your model is almost equivalent to 
a linear model with concatenated x(1,t) and x(2,t) values.  (This would 
be the "partial likelihood" version of the model, where you don't try to 
fit x(i, 1), but you fit the rest of the values conditional on earlier 
ones.)

If you want the full likelihood or you want separate variances for the 
two series, you probably need to write out the likelihood explicitly and 
maximize it.

Duncan Murdoch