CHAPTER 3
QUALITATIVE EXPLANATORY
VARIABLES REGRESSION MODELS
Damodar Gujarati
Econometrics by Example
QUALITATIVE VARIABLES
 Qualitative variables are nominal scale variables which have
no particular numerical values.
 We can “quantify” them by creating the so-called dummy
variables, which take values of 0 and 1
 0 indicates the absence of an attribute
 1 indicates the presence of the attribute
 For example, a variable denoting gender can be quantified as
female = 1 and male = 0 or vice versa.
 Dummy variables are also called indicator variables,
categorical variables, and qualitative variables.
 Examples: gender, race, color, religion, nationality, geographical region,
party affiliation, and political upheavals
Damodar Gujarati
Econometrics by Example
DUMMY VARIABLE TRAP
 If an intercept is included in the model and if a qualitative
variable has m categories, then introduce only (m – 1) dummy
variables.
 For example, gender has only two categories; hence we introduce only
one dummy variable for gender.
 This is because if a female gets a value of 1, ipso facto a male gets a
value of zero.
 If we consider political affiliation as choice among Democratic,
Republican and Independent parties, we can have at most two
dummy variables to represent the three parties.
 If we do not follow this rule, we will fall into what is called the
dummy variable trap, the situation of perfect collinearity.
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Econometrics by Example
REFERENCE CATEGORY
The category that gets the value of 0 is called
the reference, benchmark, or comparison
category.
All comparisons are made in relation to the
reference category.
If there are several dummy variables, you must
keep track of the reference category; otherwise,
it will be difficult to interpret the results.
Damodar Gujarati
Econometrics by Example
POINTS TO KEEP IN MIND
 If there is an intercept in the regression model, the number of dummy
variables must be one less than the number of classifications of each
qualitative variable.
 If you drop the (common) intercept from the model, you can have as
many dummy variables as the number of categories of the dummy
variable.
 The coefficient of a dummy variable must always be interpreted in
relation to the reference category.
 Dummy variables can interact with quantitative regressors as well as
with qualitative regressors. If a model has several qualitative variables
with several categories, introduction of dummies for all the
combinations can consume a large number of degrees of freedom.
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Econometrics by Example
INTERPRETATION OF DUMMY VARIABLES
Dummy coefficients are often called
differential intercept dummies, for they show
the differences in the intercept values of the
category that gets the value of 1 as compared to
the reference category.
The common intercept value refers to all those
categories that take a value of 0.
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Econometrics by Example
INTERPRETATION OF DUMMY VARIABLES
 If we have: Yi = B1 + B2 Fi
where Y = wage and F = female dummy variable
 Then, on average, females earn a wage of (B1 + B2) and males
earn a wage of B1. (Note that B2 can be negative.)
 Thus females earn a wage that is B2 higher than males.
 Since wages tend to be skewed to the right, we might
instead model the wage function as: lnYi = B1 + B2 Fi
 In this case, females earn (eB2 – 1)*100% more than males on
average.
 On average, male wages are equal to eB1, and female wages are
equal to e(B1+B2).
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Econometrics by Example
USE OF DUMMY VARIABLES
IN SEASONAL DATA
 The process of removing the seasonal component from a
time series is called deseasonlization or seasonal
adjustment
 The resulting time series is called deseasonalized or seasonally
adjusted time series.
 Consider the following model predicting the sales of
fashion clothing:
Salest  A1  A2 D2t  A3 D3t  A4 D4t  ut
where D2 =1 for second quarter, D3 =1 for third quarter, D4= 1 for
4th quarter, Sales = real sales per thousand square feet of retail
space.
Damodar Gujarati
Econometrics by Example
USE OF DUMMY VARIABLES
IN SEASONAL DATA
 In order to deseasonalize the sales time series, we
proceed as follows:
 1. From the estimated model we obtain the estimated sales
volume.
 2. Subtract the estimated sales value from the actual sales
volume and obtain the residuals.
 3. To the estimated residuals, we add the (sample) mean
value of sales. The resulting values are the deseasonalized
sales values.
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Econometrics by Example
FRISCH-WAUGH THEOREM
 By introducing the seasonal dummies in the model we
deseasonalize all the time series used in the model
 If variables are subject to prior adjustment by
ordinary least squares and the residuals are
subsequently used in a regression equation, then the
resulting estimates are identical to those from a
regression which uses unadjusted data but uses the
adjustment variables explicitly.
Damodar Gujarati
Econometrics by Example