Chapter 9 Dummy Variables Undergraduated Econometrics Page 1

9.2 The Use of Intercept Dummy Variables 9.3 Slope Dummy Variables Chapter Contents 9.1 Introduction 9.2 The Use of Intercept Dummy Variables 9.3 Slope Dummy Variables 9.4 An Example: The University Effect on House Price 9.5 Common Applications of Dummy Variables 9.6 Testing the Existence of Qualitative Effects 9.7 Testing the Equivalence of Two Regression Using Dummy Variables Undergraduated Econometrics Page 2 Chapter 9: Dummy Variables

9.1 Introduction Undergraduated Econometrics Page 3 Chapter 9: Dummy Variables

The multiple regression model is 9.1 Introduction The multiple regression model is We explore the variables ways, dummy variables can be included in a model and the different interpretations that they bring. Undergraduated Econometrics Page 5 Chapter 9: Dummy Variables

Assumptions of the multiple regression model 9.1 Introduction Assumptions of the multiple regression model Undergraduated Econometrics Page 6 Chapter 9: Dummy Variables

The Use of Intercept Dummy Variables 9.2 The Use of Intercept Dummy Variables Undergraduated Econometrics Page 7 Chapter 9: Dummy Variables

9.2 The Use of Intercept Dummy Variables Dummy variables allow us to construct models in which some or all regression model parameters, including the intercept, change for some observations in the sample Undergraduated Econometrics Page 7 Chapter 9: Dummy Variables

9.2 The Use of Intercept Dummy Variables Consider a hedonic model to predict the value of a house as a function of its characteristics: size Location number of bedrooms age Undergraduated Econometrics Page 8 Chapter 9: Dummy Variables

Consider the square footage at first: 9.2 The Use of Intercept Dummy Variables Consider the square footage at first: β2 is the value of an additional square foot of living area and β1 is the value of the land alone Undergraduated Econometrics Page 9 Chapter 9: Dummy Variables

How do we account for location, which is a qualitative variable? 9.2 The Use of Intercept Dummy Variables How do we account for location, which is a qualitative variable? Dummy variables are used to account for qualitative factors in econometric models They are often called binary or dichotomous variables, because they take just two values, usually one or zero, to indicate the presence or absence of a characteristic or to indicate whether a condition is true or false They are also called dummy variables, to indicate that we are creating a numeric variable for a qualitative, non-numeric characteristic We use the terms indicator variable and dummy variable interchangeably Undergraduated Econometrics Page 10 Chapter 9: Dummy Variables

Generally, we define an indicator variable D as: 9.2 The Use of Intercept Dummy Variables Generally, we define an indicator variable D as: So, to account for location, a qualitative variable, we would have: Undergraduated Econometrics Page 11 Chapter 9: Dummy Variables

Adding our indicator variable to our model: 9.2 The Use of Intercept Dummy Variables Adding our indicator variable to our model: If our model is correctly specified, then: Undergraduated Econometrics Page 12 Chapter 9: Dummy Variables

9.2 The Use of Intercept Dummy Variables Adding the dummy variable causes a parallel shift in the relationship by the amount δ An indicator variable like D that is incorporated into a regression model to capture a shift in the intercept as the result of some qualitative factor is called an intercept indicator variable, or an intercept dummy variable Undergraduated Econometrics Page 13 Chapter 9: Dummy Variables

FIGURE 9.1 An intercept dummy variable 9.2 The Use of Intercept Dummy Variables FIGURE 9.1 An intercept dummy variable Undergraduated Econometrics Page 14 Chapter 9: Dummy Variables

9.3 Slope Dummy Variables Undergraduated Econometrics Page 15 Chapter 9: Dummy Variables

Suppose we specify our model as: 9.3 Slope Dummy Variables Suppose we specify our model as: The new variable (S×D) is the product of house size and the indicator variable It is called an interaction variable, as it captures the interaction effect of location and size on house price Alternatively, it is called a slope-indicator variable or a slope dummy variable, because it allows for a change in the slope of the relationship Undergraduated Econometrics Page 16 Chapter 9: Dummy Variables

Now we can write: 9.3 Slope Dummy Variables Undergraduated Econometrics Page 17 Chapter 9: Dummy Variables

FIGURE 9.2 (a) A slope dummy variable 9.3 Slope Dummy Variables FIGURE 9.2 (a) A slope dummy variable (b) Slope- and intercept dummy variables Undergraduated Econometrics Page 18 Chapter 9: Dummy Variables

The slope can be expressed as: 9.3 Slope Dummy Variables The slope can be expressed as: Undergraduated Econometrics Page 19 Chapter 9: Dummy Variables

9.3 Slope Dummy Variables Assume that house location affects both the intercept and the slope, then both effects can be incorporated into a single model: The variable (SQFTD) is the product of house size and the indicator variable, and is called an interaction variable Alternatively, it is called a slope-indicator variable or a slope dummy variable Undergraduated Econometrics Page 20 Chapter 9: Dummy Variables

Now we can see that: 9.3 Slope Dummy Variables Undergraduated Econometrics Page 21 Chapter 9: Dummy Variables

An Example: The University Effect on House Prices 9.4 An Example: The University Effect on House Prices Undergraduated Econometrics Page 22 Chapter 9: Dummy Variables

9.4 Slope Dummy Variables Suppose an economist specifies a regression equation for house prices as: Undergraduated Econometrics Page 23 Chapter 9: Dummy Variables

9.4 Slope Dummy Variables Suppose an economist specifies a regression equation for house prices as: Undergraduated Econometrics Page 24 Chapter 9: Dummy Variables

Table 9.1 Representative Real Estate Data Values 9.4 Slope Dummy Variables Table 9.1 Representative Real Estate Data Values Undergraduated Econometrics Page 25 Chapter 9: Dummy Variables

Table 9.2 House Price Equation Estimates 9.4 Slope Dummy Variables Table 9.2 House Price Equation Estimates Undergraduated Econometrics Page 26 Chapter 9: Dummy Variables

For a house in another area: 9.4 Slope Dummy Variables The estimated regression equation is for a house near the university is: For a house in another area: Undergraduated Econometrics Page 27 Chapter 9: Dummy Variables

We therefore estimate that: 9.4 Slope Dummy Variables We therefore estimate that: The location premium for lots near the university is $27,453 The change in expected price per additional square foot is $89.12 for houses near the university and $76.12 for houses in other areas Houses depreciate $190.10 per year A pool increases the value of a home by $4,377.20 A fireplace increases the value of a home by $1,649.20 Undergraduated Econometrics Page 28 Chapter 9: Dummy Variables

Common Application of Dummy Variables 9.5 Common Application of Dummy Variables Undergraduated Econometrics Page 29 Chapter 9: Dummy Variables

Consider the wage equation: 9.5 Common Application of Dummy Variables 9.5.1 Interactions Between Qualitative Factors Consider the wage equation: The expected value is: Undergraduated Econometrics Page 30 Chapter 9: Dummy Variables

Many qualitative factors have more than two categories. To illustrate: 9.5 Common Application of Dummy Variables 9.5.2 Qualitative Factors with Several Categories Many qualitative factors have more than two categories. To illustrate: Undergraduated Econometrics Page 31 Chapter 9: Dummy Variables

The omitted dummy variable identifies the reference 9.5 Common Application of Dummy Variables 9.5.2 Qualitative Factors with Several Categories Omitting one dummy variable defines a reference group so our equation is: The omitted dummy variable identifies the reference Undergraduated Econometrics Page 32 Chapter 9: Dummy Variables

We may want to include an effect for different seasons of the year 9.5 Common Application of Dummy Variables Indicator variables are also used in regressions using time-series data We may want to include an effect for different seasons of the year 9.5.3 Controlling for Time 9.5.3a Seasonal Dummies Undergraduated Econometrics Page 33 Chapter 9: Dummy Variables

9.5 Common Application of Dummy Variables In the same spirit as seasonal dummies, annual dummies are used to capture year effects not otherwise measured in a model An economic regime is a set of structural economic conditions that exist for a certain period The idea is that economic relations may behave one way during one regime, but may behave differently during another 9.5.3a Annual Dummies 9.5.3c Regime Effects Undergraduated Econometrics Page 34 Chapter 9: Dummy Variables

An example of a regime effect: the investment tax credit: 9.5 Common Application of Dummy Variables An example of a regime effect: the investment tax credit: The model is then: If the tax credit was successful, then δ > 0 9.5.3c Regime Effects Undergraduated Econometrics Page 35 Chapter 9: Dummy Variables

Testing the Existence of Qualitative Effects 9.6 Testing the Existence of Qualitative Effects Undergraduated Econometrics Page 36 Chapter 9: Dummy Variables

9.6 Testing the Existence of Qualitative Effects If the regression model assumptions hold, and the error e are normally distributed, or if the errors are mot normal but the sample is large, then the testing procedures outlined in Chapters 7.5, 8.1 and 8.2 may be used to test for the presence of qualitative effects. Tests for the presence of a single qualitative effect can be based on the t-distribution. 9.6.1 Testing For a Single Qualitative Effect Undergraduated Econometrics Page 37 Chapter 9: Dummy Variables

To test the joint significance of all the qualitative factors 9.6 Testing the Existence of Qualitative Effects If a model has more than one dummy variable, representing several qualitative characteristic, the significance of each, apart from the others, can be tested using the t-test outlined in the previous section. To test the joint significance of all the qualitative factors 9.6.2 Testing Jointly For the Presence of Several Qualitative Effect Undergraduated Econometrics Page 38 Chapter 9: Dummy Variables

We do it by testing the joint null hypothesis 9.6 Testing the Existence of Qualitative Effects We do it by testing the joint null hypothesis against the alternative that at least one of the indicated parameters is not zero. Use the F-test procedure We reject the null hypothesis if F≥ Fc, where Fc is the critical value. Illustrated in Figure 8.1, for the level of significance α. 9.6.2 Testing Jointly For the Presence of Several Qualitative Effect Undergraduated Econometrics Page 39 Chapter 9: Dummy Variables

Testing the Equivalence of Two Regressions Using Dummy Variables 9.7 Testing the Equivalence of Two Regressions Using Dummy Variables Undergraduated Econometrics Page 40 Chapter 9: Dummy Variables

Suppose we have: and for two locations: Eq. 9.7.2 9.7 Testing the Equivalence of Two Regressions Using Dummy Variables Suppose we have: and for two locations: Eq. 9.7.2 Undergraduated Econometrics Page 41 Chapter 9: Dummy Variables

The Chow test is an F-test for the equivalence of two regressions 9.7 Testing the Equivalence of Two Regressions Using Dummy Variables 9.7.1 The Chow Test The Chow test is an F-test for the equivalence of two regressions Are there differences between the hedonic regressions for the two neighborhood or not?’’ If there are no differences, then the data from the two neighborhoods can be pooled into one sample, with no allowance made for differing slope or intercept Undergraduated Econometrics Page 42 Chapter 9: Dummy Variables

9.7 Testing the Equivalence of Two Regressions Using Dummy Variables 9.7.1 The Chow Test From(9.7.2), by testing , we are testing the equivalence of the two regressions If we reject either or both of these hypotheses, then the equalities are not true, in which case pooling the data together would be equivalent to imposing constraints, or restrictions, which are not true on the parameters of (9.7.3). Eq. 9.7.3 Undergraduated Econometrics Page 43 Chapter 9: Dummy Variables

The Chow test is an F-test for the equivalence of two regressions 9.7 Testing the Equivalence of Two Regressions Using Dummy Variables 9.7.1 The Chow Test The Chow test is an F-test for the equivalence of two regressions Are there differences between the hedonic regressions for the two neighborhood or not?’’ If there are no differences, then the data from the two neighborhoods can be pooled into one sample, with no allowance made for differing slope or intercept Undergraduated Econometrics Page 44 Chapter 9: Dummy Variables

In such a case, the usual F-test is not valid 9.7 Testing the Equivalence of Two Regressions Using Dummy Variables 9.7.1 The Chow Test Remark: The usual F-test of a joint hypothesis relies on the assumptions MR1–MR6 of the linear regression model Of particular relevance for testing the equivalence of two regressions is assumption MR3, that the variance of the error term, var(ei ) = σ2, is the same for all observations If we are considering possibly different slopes and intercepts for parts of the data, it might also be true that the error variances are different in the two parts of the data In such a case, the usual F-test is not valid Undergraduated Econometrics Page 45 Chapter 9: Dummy Variables

Table 9.3 Time Series Data on Real INV, V and K 9.7 Testing the Equivalence of Two Regressions Using Dummy Variables Table 9.3 Time Series Data on Real INV, V and K 9.7.2 An Empirical Example Of The Chow Test Undergraduated Econometrics Page 46 Chapter 9: Dummy Variables

The variables for each firm, in 1947 dollars, are 9.7 Testing the Equivalence of Two Regressions Using Dummy Variables 9.7.2 An Empirical Example Of The Chow Test The variables for each firm, in 1947 dollars, are A simple investment function is Include an intercept dummy variable and a complete set of slope dummy variables Undergraduated Econometrics Page 47 Chapter 9: Dummy Variables

The estimated restricted model with t-statistics in parameters 9.7 Testing the Equivalence of Two Regressions Using Dummy Variables 9.7.2 An Empirical Example Of The Chow Test The estimated restricted model with t-statistics in parameters Unrestricted Undergraduated Econometrics Page 48 Chapter 9: Dummy Variables

Since F≥ Fc, we cannot reject the null hypothesis. 9.7 Testing the Equivalence of Two Regressions Using Dummy Variables 9.7.2 An Empirical Example Of The Chow Test Constructing the F-statistic Since F≥ Fc, we cannot reject the null hypothesis. The advantage of this approach to the Chow test is that it does not require the construction of the dummy and interaction variables. Undergraduated Econometrics Page 49 Chapter 9: Dummy Variables