Re: st: how do we jointly test coefficients from different regressions?

[David Hoaglin <dchoaglin@gmail.com>]

Arthur,

One simple approach would be to test a = 0, b = 0, and c = 0
separately, each within its own analysis, and apply the Bonferroni
correction, so that significance on the individual test requires a
P-value < alpha/3.

You didn't give the details of the analysis, but it doesn't look as if
you're considering three models: unconstrained a, b, and c; a = b = c,
but not further constrained; and a = b = c = 0.

The suggestion of stacking y1, y2, and y3 into a column vector seems
to be headed toward a multiple regression (in which the "design"
matrix also stacks x1 and x2 for each of the y's) and then perhaps a
likelihood-ratio test.  It may be appropriate (or necessary) to take
into account correlation among y1, y2, and y3; that would turn the
analysis into a multivariate regression with (y1, y2, y3) as the
vector dependent variable.  Even without correlation, y1, y2, and y3
may not have the same variance.

The fact that "3" is actually 25 makes all of this more "interesting."
 And maybe your asset-pricing model involves other factors besides x2.

The result of using the Bonferroni correction might be that x1 makes a
significant contribution in some portfolios and not others.  I would
think that is worth knowing about.

I'm not familiar with portfolio analysis, so I'm not sure what it
means to say that "x1 is not a priced factor."  Remember that the
coefficient of x1 tells you about the contribution of x1 after
adjusting for the contribution of x2 (and the contributions of the
other predictors, if any, in the model).  Thus, a model with 0 as the
coefficient of x1 corresponds to saying that x1 makes no additional
contribution.

I hope this discussion helps.

David Hoaglin

On Mon, Mar 18, 2013 at 11:55 PM, Arthur Boman <boman@berkeley.edu> wrote:
> Hello,
>
> I am working on a joint test.  The test is NOT of the standard f-test
> form:
>
> y = a*x1 + b*x2+ c*x3 +d*x4, and then testing the null whether a=b=c=0.
>
> The test is of the form:
>
> y1= a*x1 + f*x2  and  y2= b*x1 + g*x2 and  y3= c*x1 + h*x2 and testing the
> null whether a=b=c=0
>
> I want to allow the constant.
>
> I have looked a lot and cannot figure out how to do in Stata.
>
> y1, y2, y3, x1, x2 are time series data by year... one value per year.  I
> have data for all five of those variables for each of 68 consecutive years.
> I don't have data for any of them for any other years.
>
> Someone suggested I stack (y1, y2, y3) into a column vector.  I dont get
> how that would work and cannot ask the person.
>
> Thanks,
> Arthur
>
> (More background: y1, y2, and y3 are portfolio returns by year.  I want to
> test the hypothesis that x1 is not a priced factor in ANY of the portfolios
> (i.e. that the coefficient on x1 is zero for ALL portfolios).  x2 is just
> another factor in my asset-pricing model.  There are actually 25
> portfolios, not just three.  I will be testing whether we can reject the
> null hypothesis that all of the 25 coefficients are zero.)
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