Summary Chapter12-Econometrics-TestsforStructuralChangeandStability

Chapter 12
Tests for Structural Change and Stability


A fundamental assumption in regression modeling is that the pattern of data on dependent and independent
variables remains the same throughout the period over which the data is collected. Under such an
assumption, a single linear regression model is fitted over the entire data set. The regression model is
estimated and used for prediction assuming that the parameters remain same over the entire time period of
estimation and prediction. When it is suspected that there exists a change in the pattern of data, then the
fitting of single linear regression model may not be appropriate, and more than one regression models may
be required to be fitted. Before taking such a decision to fit a single or more than one regression models, a
question arises how to test and decide if there is a change in the structure or pattern of data. Such changes
can be characterized by the change in the parameters of the model and are termed as structural change.


Now we consider some examples to understand the problem of structural change in the data. Suppose the
data on the consumption pattern is available for several years and suppose there was a war in between the
years over which the consumption data is available. Obviously, the consumption pattern before and after the
war does not remain the same as the economy of the country gets disturbed. So if a model
yi   0  1 X i1  ...   k X ik   i , i  1, 2,..., n

is fitted then the regression coefficients before and after the war period will change. Such a change is
referred to as a structural break or structural change in the data. A better option, in this case, would be to fit
two different linear regression models- one for the data before the war and another for the data after the war.


In another example, suppose the study variable is the salary of a person, and the explanatory variable is the
number of years of schooling. Suppose the objective is to find if there is any discrimination in the salaries of
males and females. To know this, two different regression models can be fitted-one for male employees and
another for females employees. By calculating and comparing the regression coefficients of both the models,
one can check the presence of sex discrimination in the salaries of male and female employees.


Consider another example of structural change. Suppose an experiment is conducted to study certain
objectives and data is collected in the USA and India. Then a question arises whether the data sets from both
the countries can be pooled together or not. The data sets can be pooled if they originate from the same
model in the sense that there is no structural change present in the data. In such case, the presence of

Econometrics | Chapter 12 | Tests for Structural Change and Stability | Shalabh, IIT Kanpur
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, structural change in the data can be tested and if there is no change, then both the data sets can be merged
and single regression model can be fitted. If structural change is present, then two models are needed to be
fitted.


The objective is now how to test for the presence of a structural change in the data and stability of regression
coefficients. In other words, we want to test the hypothesis that some of or all the regression coefficients
differ in different subsets of data.


Analysis
We consider here a situation where only one structural change is present in the data. The data, in this case, be
divided into two parts. Suppose we have a data set of n observations which is divided into two parts
consisting of n1 and n2 observations such that

n1  n2  n.

Consider the model
y    X   

where  is a  n 1 vector with all elements unity,  is a scalar denoting the intercept term, X is a  n  k 

matrix of observations on k explanatory variables,  is a  k 1 vector of regression coefficients and  is a

 n 1 vector of disturbances.


Now partition , X and  into two subgroups based on n1 and n2 observation as

  X   
   1 , X   1 ,    1 
 2   X2  2 
where the orders of 1 is  n1 1 ,  2 is  n2 1 , X 1 is  n1  k  , X 2 is  n2  k  , 1 is  n1 1 and

 2 is  n2  1 .


Based on this partitions, the two models corresponding to two subgroups are
y1   1  X 1  1
y2    2  X 2    2 .




Econometrics | Chapter 12 | Tests for Structural Change and Stability | Shalabh, IIT Kanpur
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