CHAPTER

QUALITATIVE EXPLANATORY VARIABLES REGRESSION MODELS

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

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.

REFERENCE CATEGORYThe 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.

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.

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.

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).

USE OF DUMMY VARIABLESIN 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:

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.

1 2 2 3 3 4 4t t t t tSales A A D A D A D u

USE OF DUMMY VARIABLESIN 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.