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First, see the FAQ How can I compute a Chow test statistic?. The point of that FAQ is that you can do Chow tests using Stata’s test command and, in fact, Chow tests are what the test command reports.
Well, that’s not exactly right. test uses the estimated variance–covariance matrix of the estimators, and test performs Wald tests,
W = (Rb-r)'(RVR')-1 (Rb-r)
where V is the estimated variance–covariance matrix of the estimators.
For linear regression with the conventionally estimated V, the Wald test is the Chow test and vice versa.
You might say that you are performing a Chow test, but I say that you are performing a Wald test. That distinction is important, because the Wald test generalizes to different variance estimates of V, whereas the Chow test does not. After regress, vce(robust), for instance, test uses the V matrix estimated by the robust method because that is what regress, vce(robust) left behind.
Thus the short answer is that you estimate your model using regress, vce(robust) and then use Stata’s test command. You then call the result a Wald test.
If you are bothered that a Wald test produces F rather than chi-squared statistics, also see the FAQ Why does test sometimes produce chi-squared and other times F statistics?
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How can I compute the Chow test statistic?
Title Computing the Chow statistic
Author William Gould, StataCorp
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Note:This FAQ has been updated for Stata 14. The data are simulated, so results are different from previous versions because of the new 64-bit Mersenne Twister pseudorandom numbers.
You can include the dummy variables in a regression of the full model and then use the test command on those 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.
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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 we have two groups of data. We could fit that model on the two groups separately,
y = a1 + b1*x1 + c1*x2 + u for group == 1 y = a2 + b2*x1 + c2*x2 + u for group == 2
and we could fit a single, pooled regression
y = a + b*x1 + c*x2 + u for both groups
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In the last regression, we are asserting that a1==a2, b1==b2, and c1==c2. The formula for the “Chow test” of this constraint is
ess_c - (ess_1+ess_2) --------------------- k --------------------------------- ess_1 + ess_2 --------------- N_1 + N_2 - 2*k
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The models are different in the two groups, the residual variances are different, and so are the number of observations. With this dataset, I can carry forth the Chow test. First, I run the separate regressions:
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Same answer.
This definition of the “Chow test” is equivalent to pooling the data, fitting 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 used by economists in the context of testing a set of regression coefficients being equal to 0.”
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Why does the test command sometimes produce chi-squared and other times F statistics?How are the chi-squared and F distributions related?
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F and chi-squared statistics are really the same thing in that, after a normalization, chi-squared is the limiting distribution of the F as the denominator degrees of freedom goes to infinity. The normalization is
chi-squared = (numerator degrees of freedom) * F
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As the denominator degrees of freedom (the 71) get larger and larger, the F() value will go toward the value reported by chi2() of .1287:
F(2, 150) = 2.05 p = .1323 F(2, 250) = 2.05 p = .1309 F(2, 500) = 2.05 p = .1298 F(2, 1000) = 2.05 p = .1293 F(2, 5000) = 2.05 p = .1288 F(2, 10000) = 2.05 p = .1288 F(2,100000) = 2.05 p = .1287
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