Help on Constraint Linear Regression

[FelixHugh]

Hi, I wanted to set a constraint on a linear regression model (y = b0+b1*x1+b2*x2+e), such that the the predicted value (b0+b1*x1+b2*x2) is positive. Is there a SAS procedure that can be used for the purpose. Any help is greatly appreciated!

[SteveDenham]

Easy way: Fit log(y) and back transform.

More difficult way: Use PROC MODEL with a RESTRICT statement.  I know it can be done, but I have never tried it myself.

Steve Denham

[FelixHugh]

Thank you Steve for your suggestions!

I did try PROC MODEL. I found it difficult to set up a RESTRICT statement for the purpose of bounding all the predicted values of y as positive.

I tried to use log(y) as the dependent variable, but the parameter estimates are not close to the values from the paper, which I was trying to replicate.

[Ksharp]

Doc Steve,

If I want all the coefficient ( b0 b1 b2)  greater than zero. what I am going to do ?

Best.

Xia Keshan

[FelixHugh]

Hi, sorry to just jump in. I think you may try to set multiple RESTRICT statements (restrict b0>0, b1>0, b2>0).

[Rick_SAS]

The RESTRICT statement in PROC REG only handles equality constraints, but you can use the BOUNDS statement in PROC NLIN to restrict the range of the parameters.

[Rick_SAS]

As stated, your problem is impossible unless b1=b2=0 and b0>0.  Otherwise there will always be a value of (x1, x2) for which the predicted value will be negative.

If you know that (x1,x2) are restricted to some domain (like the unit square), then it is possible.

[FelixHugh]

Thank you Rick for your reply!

The dependent variable is variance in stock market returns, which is why I want to make sure the predicted variance is positive.

The independent variables include variance and market return from last period. Market return can be negative.

[Rick_SAS]

In survival analysis, logistic regression, Poisson regression, and other models, the linear portion of the model is transformed by a "link function" to ensure that the result is positive.

[lvm]

Because your response variable is a variance, I recommend that you model it as a gamma distribution, with a log link. That is, use GENMOD or GLIMMIX, and choose dist=gamma and link=log. The gamma often works very well as an approximation for the true distribution of a variance at small (finite) sample sizes. You can still get predictions for the original scale in an output file. For instance,

proc glimmix ;

model var = .... / dist=gamma link=log s ;

output out=pred pred(blup ilink)=predicted;

run;

There are many other options for the output file.

Schabenberger and Pierce (2002 textbook) give an example of a regression analysis of a variance dependent variable using this idea (but with GENMOD).

[FelixHugh]

Thank you very much!

I will try that.

[Ksharp]

Doc Rick.

What about survival analysis ? in Survival Analysis , dependent variable(survival time) is always  > 0.