[转载]转帖：邹⾄庄检验的stata操作
努⼒学习中
原⽂地址：转帖：邹⾄庄检验的stata操作作者：⾄臻思想
How can I compute the Chow test statistic?
Title Computing the Chow statistic
Author William Gould, StataCorp
Date January 1999; minor revisions July 2005
You can include the dummy variables in a regression of the full model and then u the test command on tho dummies. You could also run each of the models and then write down the appropriate
numbers and calculate the statistic by hand—you also have access to functions to get appropriate p-values.
Here is a longer answer:
Let’s start with the Chow test to which many refer. Consider the model
y = a + b*x1 + c*x2 + u
and say that we have two groups of data. We could estimate that model on the two groups parately:
y = a1 + b1*x1 + c1*x2 + u for group == 1
y = a2 + b2*x1 + c2*x2 + u for group == 2
and we could estimate a single, pooled regression
y = a + b*x1 + c*x2 + u for both groups
In the last regression, we are asrting that a1==a2, b1==b2, and c1==c2. The formula for the “Chow test” of this constraint is
and this is the formula to which people refer.ess_1 and ess_2 are the error sum of squares from the parate regressions, ess_c is the error sum of squares from the pooled (constrained) regression, k is the number or estim The resulting test statistic is distributed F(k, N_1+N_2-2*k).
Let’s try this. I have created small datats:
The models are different in the two groups, the residual variances are different, and so are the number of obrvations. With this datat, I can carry forth the Chow test. First, I run the parate
regressions:
and then I run the combined regression:
For the Chow test,
The Chow test is F(k,N_1+N_2-2*k) = F(3,174), so our test statistic is F(3,174) = 8.8730491.
Now, I will do the same problem by running one regression and using test to test certain coefficients equal to zero. What I want to do is estimate the model
y = a3 + b3*x1 + c3*x2 + a3'*g2 + b3'*g2*x1 + c3'*g2*x2 + u
where g2=1 if group==2 and g2=0 otherwi. I can do this by typing
Some of you may be concerned that in the pooled model (the one estimating a3, b3, etc.), we are co
nstraining the var(u) to be the same for each group, whereas, in the parate-equation model, we
estimate different variances for group 1 and group 2. This does not matter, becau the model is fully interacted. That is probably not convincing, but what should be convincing is that I am about to
obtain the same F(3,174) = 8.87 answer and, in my concocted data, I have different variances in each group.
So, here is the result of the alternative test coeffiecients against 0 in a pooled specification:
Same answer.
This definition of the “Chow test” is equivalent to pooling the data, estimating the fully interacted model, and then testing the group 2 coefficients against 0.
That is why I said, “Chow Test is a term I have heard ud by economists in the context of testing a t of regression coefficients being equal to 0.”
Admittedly, that leaves a lot unsaid.
The issue of the variance of u being equal in the two groups is subtle, but I do not want that to get in the way of understanding that the Chow test is equivalent to the “pool the data, interact, and test”
procedure. They are equivalent.
Concerning variances, the Chow test itlf is testing against a pooled, uninteracted model and so has buried in it an assumption of equal variances. It is really a test that the coefficients are equal and
variance(u) in the groups are equal. It is, however, a weak test of the equality of variances becau that assumption manifests itlf only in how the pooled coefficient estimates are manufactured. Since
the Chow test and the “pool the data, interact, and test” procedure are the same, the same is true of both procedures.
Your cond concern might be that in the “pool the data, interact, and test” procedure there is an extra assumption of equality of variances becau everything comes from the pooled model. As shown,
that is not true. It is not true becau the model is fully interacted and so the assumption of equal var
iances never makes a difference in the calculation of the coefficients.
/support/faqs/stat/chow.html
