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Re: st: Can we use the standard binary choice model?

From   Maarten buis <[email protected]>
To   [email protected]
Subject   Re: st: Can we use the standard binary choice model?
Date   Mon, 24 Jan 2011 13:08:06 +0000 (GMT)

--- On Mon, 24/1/11, Quang Nguyen wrote:
> We have a theoretical model predicing the relationship between 
> a binary variable "y" and a continous variable in the following
> pattern: y=1 if  y is in the range of [x1,x2] and y=0 if y is
> smaller than x1 or greater than x2. Where x1 and x2 are some
> threshold determined by the theoretical model's parameters.
> Can you suggest an empirical model to verify the above
> model's prediction?

Let me fix some terminology and notation:

y* is a latent continuous dependent/explained variable.
y  is the observed binary dependent/explained variable.
xb is the linear predictor: b0 + b1 x1 + b2 x2 + b3 x3, the xs
   are independent/explanatory variables not your theshholds.
a1 and a2 are your thresholds, (x1 x2 in your notation).

Your problem would be:
y* = xb

y  = 1 if a1 <= y* <= a2
y  = 0 if y* < a1 | y* > a2

This would be relatively easy to solve if a1 = - a2:

y*^2 = xb^2

y  = 1 if y*^2 <= a2^2
y  = 0 if y*^2 >  a2^2

y  = 1 if y*^2 - a2^2 <= 0
y  = 0 if y*^2 - a2^2 >  0

So now it is just a regular binary choice model whith a
lot of extra variables (squares and products of variables),
and a hard to interpret constant.

Hope this helps,

Maarten L. Buis
Institut fuer Soziologie
Universitaet Tuebingen
Wilhelmstrasse 36
72074 Tuebingen


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