[R] simultaneous estimation

[David Winsemius]

On Aug 31, 2010, at 10:35 AM, <Murali.Menon at avivainvestors.com> <Murali.Menon at avivainvestors.com 
 > wrote:

> Hi Duncan,
>
> Thanks for your response.
>
> Indeed, independent normal errors were what I had in mind. As for  
> variances, if I assume they are the same, would a 'pooled model'  
> apply in this case? Is that equivalent to your suggestion of  
> concatenating x(1,t) and x(2,t)?
>

Wouldn't this be equivalent to a segmented regression analysis that  
would estimate the slopes in the two periods as mu(1) and mu(2), throw- 
away any level shift estimate at the join-point,  and which then  
estimated the residual one-lag autocorrelation (again omitting the  
join point) and assigned that value to "d"?

-- 
David.

> Cheers,
> Murali
>
> -----Original Message-----
> From: Duncan Murdoch [mailto:murdoch.duncan at gmail.com]
> Sent: 31 August 2010 12:31
> To: Menon Murali
> Cc: r-help at r-project.org
> Subject: Re: [R] simultaneous estimation
>
> 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
>
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David Winsemius, MD
West Hartford, CT