Re: st: missing standard error with nonlinear least squares

["Rodrigo Alfaro" <ralfaro76@hotmail.com>]

Dear Giulio,

Following the discussion of Nick, it could be a good idea to run the linear
regression model and deal with the estimation of the parameters with Minimum
Distance Estimation (MDE). The standard error can be computed with the delta
method.

In practice after -reg y x1 x2 x3 x4 x5- you will get 5 coefficients (let's
call them c's) then you have 6 equations c0=b0/(1-b3),..., etc and 6 unknown
b0 to b5. It is a non-linear system that can be solved in any software.

But you can use the information of your model to simply your task in
particular c1=b1 and c2=b2. Note that c4=b3*b2/(1-b3), replacing b2 by c2
you have c4/c2 = b3/(1-b3) assumming that b3 is between 0 and 1 you will get
that b3=(c4/c2)/[1+(c4/c2)], with this value you can solve the rest of the
coefficient.

Rodrigo.
PS: Sorry if you receive this email twice.

----- Original Message ----- 
From: "n j cox" <n.j.cox@durham.ac.uk>
To: <statalist@hsphsun2.harvard.edu>
Sent: Thursday, March 30, 2006 10:22 AM
Subject: Re: st: missing standard error with nonlinear least squares



> The principle is easy to illustrate at a much lower level, e.g.
>
> . set obs 1
> obs was 0, now 1
>
> . gen y = 42
>
> . su y
>
>     Variable |       Obs        Mean    Std. Dev.       Min        Max
> -------------+--------------------------------------------------------
>            y |         1          42           .         42         42
>
> Here, Stata following statistical logic believes that it can
> calculate a mean but not a standard deviation. Stata won't
> provide a regression for one just one observation, but with
> a small tweak it's the same story:
>
> . expand 2
> (1 observation created)
>
> . reg y
>
>       Source |       SS       df       MS              Number of obs = 2
> -------------+------------------------------           F(  0,     1) = 
> 0.00
>        Model |           0     0           .           Prob > F      = .
>     Residual |           0     1           0           R-squared     = .
> -------------+------------------------------           Adj R-squared = .
>        Total |           0     1           0           Root MSE      = 0
>
> ------------------------------------------------------------------------------
>            y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. 
> Interval]
> -------------+----------------------------------------------------------------
>        _cons |         42          .        .       .            . .
> ------------------------------------------------------------------------------
>
> This is not estimating a slope, except on unity, but again a minimal 
> example is a small step away:
>
> . gen x = _n
>
> . regress y x
>
>       Source |       SS       df       MS              Number of obs = 2
> -------------+------------------------------           F(  1,     0) = .
>        Model |           0     1           0           Prob > F      = .
>     Residual |           0     0           .           R-squared     = .
> -------------+------------------------------           Adj R-squared = .
>        Total |           0     1           0           Root MSE      = 0
>
> ------------------------------------------------------------------------------
>            y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. 
> Interval]
> -------------+----------------------------------------------------------------
>            x |          0          .        .       .            . .
>        _cons |         42          .        .       .            . .
> ------------------------------------------------------------------------------
>
> Your example is a much more complicated model, but the principle is
> a beast of the same kind. The error message is best thought of as a 
> warning, that the data don't provide enough information to assess the 
> model in that respect.
>
> On a different note, I note that the nonlinearity here is entirely
> a matter of complicated relationships between the parameters:
>
> Y = b0/(1-b3) + b1*X1 + b2*X2 + [(b3*b1+b4)/(1-b3)]*X3 + [b3*b2/(1-b3)]*X4 
> + [b5/(1-b3)]*X5 + e
>
> Very crudely, modulo my inability to count,
>
> b0 appears once,
> b1 twice,
> b2 twice,
> b3 six times,
> b4 once,
> b5 once.
>
> It doesn't seem especially surprising that b3 turns out to be hardest
> to determine, as it is so tangled up with most everything else. I can't 
> say more about the topology or geometry of this parameter space, but hands 
> up those who guess it's a mess.
>
> That suggests to me a comparison with a simple regression
> on X1 ... X5. No doubt that ignores your substantive or theoretical
> grounds for using this parameterisation, but _in practice_ does
> it give similar or different predictions for the response?
>
> Giulio Zanella
>
> I estimated a nonlinear model using command nl. Stata produced estimates 
> for
> all of the coefficients. However, the standard error of one such 
> coefficient
> (and this only) was not computed: a dot replaced the standard error and
> related statistics cells. Defining dep var Y coefficients b0 to b5 and
> regressors X1 to X5 (plus constant), the model is:
>
> Y = b0/(1-b3) + b1*X1 + b2*X2 + [(b3*b1+b4)/(1-b3)]*X3 + [b3*b2/(1-b3)]*X4 
> +
> [b5/(1-b3)]*X5 + e
>
> I don't understand why Stata can compute the mean of b3 but not its 
> standard
> error, while - for instance - standard errors associated with b1 and b2 
> are produced. Any clue?
>
>
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