Re: st: RE: how do we jointly test coefficients (fuller specification) from diff

[Arthur Boman <boman@berkeley.edu>]

Jorge and David,

Thanks again for your replies.  My understanding is that SUR is the same
as OLS if the error terms are uncorrelated - or if the equations have the
same RHS variables.  Don't we have both of those here? 

The variances of the error terms should be allowed to vary between
equations, but there is no serial correlation and there is no correlation
between the error terms of two equations.  

As to variance changing over time (heteroscedastic), it would be nice but
not important.  I repasted the fuller specification below.

Arthur

Here is my model:

 y1= a*x1 + f*x2 + e1
 y2= b*x1 + g*x2 + e2
 y3= c*x1 + h*x2 + e3

 e1, e2, e3 are independent, normal, and mean-zero.  They have different
 variances, but it would be okay to assume the variances do not change
 with
 time.

 Then the null is a=b=c=0.

 ( When I say independent I mean both cross-sectionally (e1 at any time is
 independent of e2 at any time) and independent across time as well, no
 serial correlation.

 Constant variance across time is okay, i.e. not heteroscedastic. If it is
 easy to allow heteroscedasticity and correct for this, then okay. )







On Tue, 19 Mar 2013 16:20:18 -0400, Jorge Eduardo Pérez Pérez
<perez.jorge@ur.edu.co> wrote:
> Since you are assuming the error terms have different variances, the
> stacking approach does not work here, since, as David points out, the
> stacking approach assumes the same variance and no correlation. A
> proper approach here is seemingly unrelated estimation (SUR), which is
> just a kind of multivariate regression. This will allow for different
> variance of the error terms as well as correlation. Here's how that
> would work:
> 
> clear all
> version 12
> * This just generates some random data. You do not need to do this.
> set obs 68
> set seed 135
> * x variables
> gen x1=rnormal()
> gen x2=rnormal()
> * time
> gen time=_n
> * 3 y variables (this could be any number of variables, i.e. 25)
> glo nvar=3
> * Coefficients. I am setting 0 for x1 and 1 for x2. Constants are set to
> 1,2,3
> forv i=1(1)$nvar {
> glo b`i'0=`i'
> glo b`i'1=0
> glo b`i'2=1
> }
> * Error terms are multivariate normal. Covariance matrix
> matrix C=[1,0.2,1\0.2,2,0.3\1,0.3,3]
> * Generate error terms
> drawnorm e1 e2 e3, cov(C)
> * Generate y variables,
> forv i=1(1)$nvar {
> gen y`i'=${b`i'0}+${b`i'1}*x1+${b`i'2}*x2+e`i'
> }
> * Now I have a data set like yours.
> * Run SUR. For 3 variables I would write sureg (y1 x1 x2) (y2 x1 x2)
> (y3 x1 x2). Following code loops for many variables.
> glo cmd ""
> forv i=1(1)$nvar {
> glo cmd " $cmd (y`i' x1 x2)"
> }
> sureg $cmd
> * Test. For 3 variables this could be test [y1]x1 =[y2]x1 =[y3]x1=0
> test [y1]x1=0
> forv i=2(1)$nvar {
> test [y`i']x1=[y1]x1, accum
> }
> test
> _______________________
> Jorge Eduardo Pérez Pérez
> 
> 
> On Tue, Mar 19, 2013 at 3:41 PM, Arthur Boman <boman@berkeley.edu>
wrote:
>> David (Jorge can check first part and think if it jives with the code
he
>> sent),
>> ____
>>
>> Thank you.  This got me thinking more about the model.  Here is my
model:
>>
>> y1= a*x1 + f*x2 + e1
>> y2= b*x1 + g*x2 + e2
>> y3= c*x1 + h*x2 + e3
>>
>> e1, e2, e3 are independent, normal, and mean-zero.  They have different
>> variances, but it would be okay to assume the variances do not change
>> with
>> time.
>>
>> Then the null is a=b=c=0.
>>
>> ( When I say independent I mean both cross-sectionally (e1 at any time
is
>> independent of e2 at any time) and independent across time as well, no
>> serial correlation.
>>
>> Constant variance across time is okay, i.e. not heteroscedastic. If it
is
>> easy to allow heteroscedasticity and correct for this, then okay. )
>>
>> ____
>>
>> Bonferroni:  Yes I had thought of this but I am not sure whether it is
>> accurate.  It seems like it should not be far off, or maybe it is
>> accurate.
>> What I wondered is whether the x's can be considered predetermined for
>> subsequent models, as they are the same for all.  I also wonder if I
did
>> the tests separately and allowed for heteroscedasticity, would
Bonferroni
>> work same way?
>>
>> ____
>>
>> The fact that "3" is actually 25 makes all of this more "interesting."
>>
>> ( -:
>>
>> ____
>>
>> Don't worry about the priced factor thing.  Testing if coeff on x1's
are
>> all zero with the other x's in there.
>>
>> Yes, there are x2, x3, and x4.
>>
>> ____
>>
>> Sounds complicated:
>>
>>> 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.
>>>
>> *
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>>
>>
> 
> 
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