邹至庄 检验，在分组 回归中有所应用。方法的提出之初用于时间点 breakpoint 前后 回归方程的系数（整体等于0，whole test，结构变动；至于单个系数 在两组的检验，参考其他资料）变动情况。在计算机实现方面，我们可以使用STATA软件来做，具体来说，两种方案 都可以计算F值。如果P<0.05，表明结构变动了。

参考资料：

http://www.stata.com/support/faqs/statistics/computing-chow-statistic/

 STATA code

方案一：手工计算

sysuse auto

reg  rep78 headroom turn if  foreign==0

reg  rep78 headroom turn if  foreign==1

reg  rep78 headroom turn

 

 

方案二：计算机自动

①首先确定分组标识；

gen g2=( foreign==1)

gen g2headroom=g2* headroom

gen g2turn=g2* turn

reg  rep78  headroom turn g2 g2headroom g2turn

②最后F检验，如果P-value is less than 0.05，可以判断 发生结构性变动。

test  g2 g2headroom g2turn

SAS CODE:

http://support.sas.com/rnd/app/examples/ets/chow/

http://www.rci.rutgers.edu/~roos/Courses/grstat502/chowformula205.pdf

proc sort data=dummy;
by age;
run;

proc autoreg;
model hight= weight /chow=(88);
run;

and this:
proc sort data=dummy;
by sex;
run;

proc autoreg;
model hight= weight /chow=(88);
run;

Perhaps the most important assumption of any time series model is that the underlying process is the same across all observations in the sample. It is, therefore, necessary to analyze carefully time series data that include periods of violent change. A tool that is particularly useful in this regard is the Chow test.

The Chow test is commonly used to test for structural change in some or all of the parameters of a model in cases where the disturbance term is assumed to be the same in both periods.

 

The Chow test is an application of the F-test, and it requires the sum of squared errors from three regressions - one for each sample period and one for the pooled data.

Analysis

In an investigation of the demand for food in the United States, researchers may want to determine whether the structure of the demand equation changes after World War II.

Exploring the Data Set

The data for this study include yearly observations on per capita food consumption, the price of food, and per capita income for the years 1927-1941 and 1948-1962 (Maddala 1992). There are no observations for the war years between 1942 and 1947. The DATA step creates a SAS data set named FOOD, reads data values into the variables YEAR, Q, P, and Y, and creates the constant term ONE and the log transformations LNQ, LNP, LNY. 

   data food;

      input year q p y @@;

      retain one 1;

      lnq = log(q);

      lnp = log(p);

      lny = log(y);

      datalines;

   27  88.9   91.7   57.7

   28  88.9   92.0   59.3

   29  89.1   93.1   62.0

    ...

   ;

   run;

Once the FOOD data set is created, the interactive data analysis feature of SAS/INSIGHT software can be used to check the data for errors and to explore graphically possible relationships among the variables.

In this case, a 3-D Rotating Plot of the variables LNQ, LNP, and LNY show a break between the observations before 15 and after 16 corresponding to the years 1927-1941 and 1948-1962. This evidence suggests that a test for a structural break in a model for the demand for food may be appropriate.

Computing the Chow Test

The AUTOREG procedure specifies a linear regression of the log of per capita food consumption on the log price of food, the log of per capita income, and a constant term (automatically included). The CHOW= option in the model statement performs Chow tests at the specified breakpoints. The breakpoint candidates 15, 16, and 17 corresponding to the years 1941, 1948, and 1949 are chosen from the preceding analysis. 

   proc autoreg data=food;

      model lnq = lnp lny / chow=(15 16 17);

   run;

CHOW Test

The AUTOREG Procedure

Dependent Variable

lnq

 

Ordinary Least Squares Estimates
SSE	0.00286947	DFE	27
MSE	0.0001063	Root MSE	0.01031
SBC	-182.30489	AIC	-186.50848
Regress R-Square	0.9731	Total R-Square	0.9731
Durbin-Watson	1.2647

 

Structural Change Test
Test	Break Point	Num DF	Den DF	F Value	Pr > F
Chow	15	3	24	5.07	0.0074
Chow	16	3	24	5.54	0.0049
Chow	17	3	24	1.29	0.2992

 

Variable	DF	Estimate	Standard Error	t Value	Approx Pr > |t|
Intercept	1	4.0473	0.1360	29.76	<.0001
lnp	1	-0.1189	0.0404	-2.95	0.0066
lny	1	0.2412	0.0134	17.95	<.0001

 

 R2 is a measure for the appropriateness of the model. The value of 0.973 implies that LNP and LNY together explain more than 90% of the variation in LNQ.

The coefficient estimates are highly significant, while the negative sign on LNP and the positive sign on LNY satisfy the intuition that price elasticity has an inverse relationship to quantity demanded and income elasticity has a direct relationship.

Notice that the Chow test is highly significant for break points 15 and 16, which correspond to the years 1941 and 1948. This is not a very surprising result given the state of the world at that time.

References

Chow, G.C. (1960), "Tests of Equality between Sets of Coefficients in Two Linear Regressions," Econometrica, 28, 591-605.

Fisher, F.M. (1970), "Tests of Equality between Sets of Coefficients in Two Linear Regressions: An Expository Note," Econometrica, 38, 361-366.

Greene, W.H. (1993), Econometric Analysis, Second Edition, New York: Macmillan Publishing Company.

Maddala, G.S. (1992), Introduction to Econometrics, Second Edition, New York: Macmillan Publishing Company.