# [R] Restricted Least Squares

Mike Lonergan mel at mcs.st-and.ac.uk
Tue Apr 9 14:36:30 CEST 2002

```
> -----Original Message-----
> From: owner-r-help at stat.math.ethz.ch
> [mailto:owner-r-help at stat.math.ethz.ch]On Behalf Of Philippe Grosjean
> Sent: 09 April 2002 09:23
> Cc: R-help at stat.math.ethz.ch
> Subject: RE: [R] Restricted Least Squares
>
>
>  > Hi,
>
> > I need help regarding estimating a linear model where
> restrictions are
> imposed on the coefficients. An example is as follows:
>
> > Y_{t+2}=a1Y_{t+1} + a2 Y_t + b x_t + e_t
>
> > restriction
> > a1+ a2 =1
>
> > Is there a function or a package that can estimate the
> coefficient of a
> model like this? I want to estimate the coefficients rather
> than test them.
>
> > Thank you for your help
>
>
> You don't need something special. Just consider that
>
> a2 = 1 - a1 and replace it in your equation, which gives:
>
> Y_{t+2}=a1Y_{t+1} + (1 - a1) Y_t + b x_t + e_t
>
> Best,
>
> Philippe Grosjean

& it becomes really nice when Y_{t+1} is subtracted from both sides:

Y_{t+2} - Y_{t+1} = a1Y_{t+1} - Y_{t+1} + (1 - a1) Y_t + b x_t + e_t

Substituting

Z_{t+1} = Y_{t+1} - Y_{t}

then gives:

Z_{t+1} = -a2Z_{t} + bx_t + e_t

Which is unconstrained and much easier, unless I've entirely missed the
point.

Cheers,

Mike.

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