Nội dung

Chapter 9 Dummy (Binary) Variables 9.1 Introduction The multiple regression model yt = β1 + β2xt2 + β3xt3 +…+ βKxtK + et (9.1.1) The assumptions of the multiple regression model are Assumptions of the Multiple Regression Model MR1. yt = β1 + β2xt2 + β3xt3 +…+ βKxtK + et, t = 1,…,T MR2. E(yt) = β1 + β2xt2 + β3xt3 +…+ βKxtK ⇔ E(et) = 0 MR3. var(yt) = var(et) = σ2 Slide 9.1 Undergraduate Econometrics, 2nd Edition –Chapter 9 MR4. cov(yt, ys) = cov(et, es) = 0 MR5. The values of xtK are not random and are not exact linear functions of the other explanatory variables. MR6. yt ~ N[(β1 + β2xt2 + β3xt3 +…+ βKxtK), σ2] ⇔ et ~ N(0, σ2) • Assumption MR1 defines the statistical model that we assume is appropriate for all T of the observations in our sample. One part of the assertion is that the parameters of the model, βK, are the same for each and every observation. Recall that βK = the change in E(yt) when xtK is increased by one unit, and all other variables are held constant = ∆E ( yt ) ∂E ( yt ) = ∆xtk (other variables held constant) ∂xtk Slide 9.2 Undergraduate Econometrics, 2nd Edition –Chapter 9 • Assumption 1 implies that for each of the observations t = 1,...,T the effect of a one unit change in xtK on E(yt) is exactly the same. If this assumption does not hold, and if the parameters are not the same for all the observations, then the meaning of the least squares estimates of the parameters in Equation (9.1.1) is not clear. • In this chapter we extend the multiple regression model of Chapter 8 to situations in which the regression parameters are different for some of the observations in a sample. We use dummy variables, which are explanatory variables that take one of two values, usually 0 or 1. These simple variables are a very powerful tool for capturing qualitative characteristics of individuals, such as gender, race, and geographic region of residence. In general, we use dummy variables to describe any event that has only two possible outcomes. We explain how to use dummy variable to account for such features in our model. • As a second tool for capturing parameter variation, we make use of interaction variables. These are variables formed by multiplying two or more explanatory Slide 9.3 Undergraduate Econometrics, 2nd Edition –Chapter 9 variables together. When using either dummy variables or interaction variables, some changes in model interpretation are required. We will discuss each of these scenarios. Slide 9.4 Undergraduate Econometrics, 2nd Edition –Chapter 9 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. To make matters specific, we consider an example from real estate economics. Buyers and sellers of homes, tax assessors, real estate appraisers, and mortgage bankers are interested in predicting the current market value of a house. A common way to predict the value of a house is to use a “hedonic” model, in which the price of the house is explained as a function of its characteristics, such as its size, location, number of bedrooms, age, etc. • For the present, let us assume that the size of the house, S, is the only relevant variable in determining house price, P. Specify the regression model as Pt = β1 + β2St + et (9.2.1) Slide 9.5 Undergraduate Econometrics, 2nd Edition –Chapter 9 In this model β2 is the value of an additional square foot of living area, and β1 is the value of the land alone. • Dummy variables are used to account for qualitative factors in econometric models. They are often called binary or dichotomous variables as they take just two values, usually 1 or 0, to indicate the presence or absence of a characteristic. That is, a dummy variable D is 1 D= 0 if characteristic is present if characteristic is not present (9.2.2) Thus, for the house price model, we can define a dummy variable to account for a desirable neighborhood, as Slide 9.6 Undergraduate Econometrics, 2nd Edition –Chapter 9 1 Dt =  0 if property is in the desirable neighborhood if property is not in the desirable neighborhood (9.2.3) • Adding this variable to the regression model, along with a new parameter δ, we obtain Pt = β1 + δDt + β2St + et (9.2.4) • The effect of the inclusion of a dummy variable Dt into the regression model is best seen by examining the regression function, E(Pt), in the two locations. If the model in (9.2.4) is correctly specified, then E(et) = 0 and (β + δ) + β2 St E ( Pt ) =  1  β1 + β2 St when Dt = 1 when Dt = 0 (9.2.5) Slide 9.7 Undergraduate Econometrics, 2nd Edition –Chapter 9 • In the desirable neighborhood, Dt = 1, and the intercept of the regression function is (β1 + δ). In other areas the regression function intercept is simply β1. This difference is depicted in Figure 9.1, assuming that δ > 0. • Adding the dummy variable Dt to the regression model creates a parallel shift in the relationship by the amount δ. In the context of the house price model the interpretation of the parameters δ is that it is a “location premium,” the difference in house price due to being located in the desirable neighborhood. • A dummy variable like Dt that is incorporated into a regression model to capture a shift in the intercept as the result of some qualitative factor is an intercept dummy variable. In the house price example we expect the price to be higher in a desirable location, and thus we anticipate that δ will be positive. • The least squares estimator’s properties are not affected by the fact that one of the explanatory variables consists only of zeros and ones. Dt is treated as any other explanatory variable. We can construct an interval estimate for δ, or we can test the Slide 9.8 Undergraduate Econometrics, 2nd Edition –Chapter 9 significance of its least squares estimate. Such a test is a statistical test of whether the neighborhood effect on house price is “statistically significant.” If δ = 0, then there is no location premium for the neighborhood in question. Slide 9.9 Undergraduate Econometrics, 2nd Edition –Chapter 9 9.3 Slope Dummy Variables • We can allow for a change in a slope by including in the model an additional explanatory variable that is equal to the product of a dummy variable and a continuous variable. In our model the slope of the relationship is the value of an additional square foot of living area. If we assume this is one value for homes in the desirable neighborhood, and another value for homes in other neighborhoods, we can specify Pt = β1 + β2St + γ(StDt) + et (9.3.1) • The new variable (StDt) is the product of house size and the dummy variable, and is called an interaction variable, as it captures the interaction effect of location and size on house price. Alternatively, it is called a slope dummy variable, because it allows for a change in the slope of the relationship. Slide 9.10 Undergraduate Econometrics, 2nd Edition –Chapter 9