CHAPTER 3

1. CHAPTER 3 QUALITATIVE EXPLANATORY VARIABLES REGRESSION MODELS Damodar Gujarati Econometrics by Example, second edition

2. QUALITATIVE VARIABLES • Qualitative variables are nominalscale 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, second edition

3. DUMMY VARIABLE TRAP • If an intercept is included in the model andif 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 self-reported health as a choice among excellent, good, and poor, we can have at most two dummy variables to represent the three categories. • If we do not follow this rule, we will fall into what is called the dummy variable trap, the situation of perfect collinearity. Damodar Gujarati Econometrics by Example, second edition

4. 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, second edition

5. 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. Damodar Gujarati Econometrics by Example, second edition

6. 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. Damodar Gujarati Econometrics by Example, second edition

7. 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). Damodar Gujarati Econometrics by Example, second edition

8. 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. Damodar Gujarati Econometrics by Example, second edition

9. 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. Damodar Gujarati Econometrics by Example, second edition

10. 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, second edition

11. PIECEWISE LINEAR REGRESSION • A piecewise linear regression is made up of pieces of two linear regression lines joined at the knot or the threshold value • Could also have multiple piecewise linear regressions (with multiple knots) • Type of spline function • Regression represented as: • where X* represents the threshold valueand • D=1 if X*>X and D=0 if X*<X. Damodar Gujarati Econometrics by Example, second edition