The Chow test (), proposed by
econometrician
Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8†...
Gregory Chow
Gregory Chi-Chong Chow (; born December 25, 1930) is a Chinese-American economist at Princeton University and Xiamen University. The Chow test, commonly used in econometrics to test for structural breaks, was invented by him. He has also been in ...
in 1960, is a test of whether the true coefficients in two
linear regression
In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is call ...
s on different data sets are equal. In econometrics, it is most commonly used in
time series analysis
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
to test for the presence of a
structural break
In econometrics and statistics, a structural break is an unexpected change over time in the parameters of regression models, which can lead to huge forecasting errors and unreliability of the model in general. This issue was popularised by David ...
at a period which can be assumed to be known ''a priori'' (for instance, a major historical event such as a war). In
program evaluation
Program evaluation is a systematic method for collecting, analyzing, and using information to answer questions about projects, policies and programs, particularly about their effectiveness and efficiency. In both the public and private sectors, ...
, the Chow test is often used to determine whether the independent variables have different impacts on different subgroups of the population.
Illustrations
First Chow Test
Suppose that we model our data as
:
If we split our data into two groups, then we have
:
and
:
The
null hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...
of the Chow test asserts that
,
, and
, and there is the assumption that the
model errors are
independent and identically distributed
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usua ...
from a
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
with unknown
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
.
Let
be the sum of squared
residuals from the combined data,
be the sum of squared residuals from the first group, and
be the sum of squared residuals from the second group.
and
are the number of observations in each group and
is the total number of parameters (in this case 3, i.e. 2 independent variables coefficients + intercept). Then the Chow test statistic is
:
The test statistic follows the
''F''-distribution with
and
degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
.
The same result can be achieved via dummy variables.
Consider the two data sets which are being compared. Firstly there is the 'primary' data set i= and the 'secondary' data set i=. Then there is the union of these two sets: i=. If there is no structural change between the primary and secondary data sets a regression can be run over the union without the issue of biased estimators arising.
Consider the regression:
Which is run over i=.
D is a dummy variable taking a value of 1 for i= and 0 otherwise.
If both data sets can be explained fully by
then there is no use in the dummy variable as the data set is explained fully by the restricted equation. That is, under the assumption of no structural change we have a null and alternative hypothesis of:
The null hypothesis of joint insignificance of D can be run as an F-test with
degrees of freedom (DoF). That is:
.
Remarks
* The global sum of squares (SSE) is often called the Restricted Sum of Squares (RSSM) as we basically test a constrained model where we have
assumptions (with
the number of regressors).
* Some software like SAS will use a predictive Chow test when the size of a subsample is less than the number of regressors.
References
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External links
{{commonscat, Chow test
Computing the Chow statistic Series of FAQ explanations from the
Stata
Stata (, , alternatively , occasionally stylized as STATA) is a general-purpose statistical software package developed by StataCorp for data manipulation, visualization, statistics, and automated reporting. It is used by researchers in many fie ...
Corporation at https://www.stata.com/support/faqs/
Series of FAQ explanations from the
SAS Institute, SAS Corporation
Time series statistical tests
Regression diagnostics