1. Chapter 9 Dummy Variables Undergraduated Econometrics Page 1

2. 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
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Chapter 9: Dummy Variables

3. 9.1 Introduction Undergraduated Econometrics Page 3
Chapter 9: Dummy Variables

4. 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
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5. Assumptions of the multiple regression model
9.1
Introduction
Assumptions of the multiple regression model
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6. The Use of Intercept Dummy Variables
9.2
The Use of Intercept Dummy Variables
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7. 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
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8. 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
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9. 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
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10. 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
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11. 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
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12. 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:
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13. 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
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14. FIGURE 9.1 An intercept dummy variable
9.2
The Use of Intercept Dummy Variables
FIGURE 9.1 An intercept dummy variable
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15. 9.3 Slope Dummy Variables Undergraduated Econometrics Page 15
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16. 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
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17. Now we can write: 9.3 Slope Dummy Variables
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18. 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
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19. The slope can be expressed as:
9.3
Slope Dummy Variables
The slope can be expressed as:
Undergraduated Econometrics
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20. 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
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21. Now we can see that: 9.3 Slope Dummy Variables
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22. An Example: The University Effect on House Prices
9.4
An Example: The University Effect on House Prices
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23. 9.4
Slope Dummy Variables
Suppose an economist specifies a regression equation for house prices as:
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24. 9.4
Slope Dummy Variables
Suppose an economist specifies a regression equation for house prices as:
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25. Table 9.1 Representative Real Estate Data Values
9.4
Slope Dummy Variables
Table 9.1 Representative Real Estate Data Values
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26. Table 9.2 House Price Equation Estimates
9.4
Slope Dummy Variables
Table 9.2 House Price Equation Estimates
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27. 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:
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28. 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 $ 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
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29. Common Application of Dummy Variables
9.5
Common Application of Dummy Variables
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30. 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:
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31. 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:
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32. 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
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33. 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
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34. 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
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35. 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
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36. Testing the Existence of Qualitative Effects
9.6
Testing the Existence of Qualitative Effects
Undergraduated Econometrics
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37. 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
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38. 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
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39. 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
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40. Testing the Equivalence of Two Regressions Using Dummy Variables
9.7
Testing the Equivalence of Two Regressions Using Dummy Variables
Undergraduated Econometrics
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Chapter 9: Dummy Variables

41. 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
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42. 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
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43. 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
Undergraduated Econometrics
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44. 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
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45. 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
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46. 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
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47. 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
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48. 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
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49. 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
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