Chapter 8 Dummy Variables and Truncated Variables
What is in this Chapter? • This chapter relaxes the assumption made in Chapter 4 that the variables in the regression are observed as continuous variables. – Differences in intercepts and/or slope coefficients – The linear probability model and the logit and probit models. – Truncated variables, Tobit models
8. 2 Dummy Variables for Changes in the Intercept Term • Note that the slopes of the regression lines for both groups are roughly the same but the intercepts are different. • Hence the regression equations we fit will be
8. 2 Dummy Variables for Changes in the Intercept Term • These equations can be combined into a single equation where • The variable D is the dummy variable. • The coefficient of the dummy variable measures the difference in the two intercept terms
8. 2 Dummy Variables for Changes in the Intercept Term
8. 2 Dummy Variables for Changes in the Intercept Term • If there are more groups, we have to introduce more dummies. • For three groups we have • These can be written as where
8. 2 Dummy Variables for Changes in the Intercept Term • • 1. 2. 3. As yet another example, suppose that we have data on consumption C and income Y for a number of households. In addition, we have data on S: the sex of the head of the household. A: the age of the head of the household, which is given in three categories, <25 years, 25 to 50 year, and >50 years. E: the education of the head of the household, also in three categories, <high school, ≧high school but < college degree, ≧ college degree.
8. 2 Dummy Variables for Changes in the Intercept Term • We include these qualitative variable in the form of dummy variables
8. 2 Dummy Variables for Changes in the Intercept Term • For each category the number of dummy variables is one less than the number of classifications. • Then we run the regression equation • The assumption made in the dummy variable method is that it is the intercept that changes for each group but not the slope coefficients (i. e. coefficients of Y).
8. 2 Dummy Variables for Changes in the Intercept Term • The dummy variable method is also used if one has to take care of seasonal factors. • For example, if we have quarterly data on C and Y, we fit the regression equation
8. 2 Dummy Variables for Changes in the Intercept Term • If we have monthly data, we use 11 seasonal dummies • If we feel that, say, December (because of Christmas shopping) is the only moth with strong seasonal effect, we use only one dummy variable
8. 2 Dummy Variables for Changes in the Intercept Term • Two More Illustrative Examples • We will discuss two more examples using dummy variables. • They are meant to illustrate two points worth noting, which are as follows: – 1. In some studies with a large number of dummy variables it becomes somewhat difficult to interpret the signs of the coefficients because they seem to have the wrong signs. (The first example) – 2. Sometimes the introduction of dummy variables produces a drastic change in the slope coefficient. (The second example)
8. 2 Dummy Variables for Changes in the Intercept Term • The first example is a study of the determinants of automobile prices. • Griliches regressed the logarithm of new passenger car prices on various specifications. The results are shown in Table 8. 1 • Since the dependent variable is the logarithm of price, the regression coefficients can be interpreted as the estimated percentage change in the price for a unit change in a particular quality, holding other qualities constant • For example, the coefficient of H indicates that an increase in 10 units of horsepower, results in a 1. 2 increase in price
8. 2 Dummy Variables for Changes in the Intercept Term • As another example consider the estimates of liquid-asset demand by manufacturing corporations • Vogel and Maddala computed regressions of the form log C =α +ß log S, where C is the cash and S the sales, on the basis of data from the Internal Revenue Service, "Statistics of Income, " for the year 1960 -1961. • The data consisted of 16 industry subgroups and 14 size classes, size being measured by total assets.
8. 2 Dummy Variables for Changes in the Intercept Term • The equations were estimated separately for each industry, the estimates of β ranged from 0. 929 to 1. 077. • The R 2’s were uniformly high, ranging from 0. 985 to 0. 998. • Thus one might conclude that the sales elasticity of demand for cash is close to 1. • Also, when the data were pooled and a single equation estimated for the entire set of 224 observations, the estimate of β was 0. 992 and R 2=0. 897.
8. 2 Dummy Variables for Changes in the Intercept Term • When industry dummies were added, the estimate of β was 0. 995 and R 2=0. 992. • From the high R 2’s and relatively constant estimate of β one might be reassured that the sales elasticity is very close to 1. • However, when asset-size dummies were introduced, the estimate of β fell to 0. 334 with R 2 of 0. 996. • Also, all asset-size dummies were highly significant.
8. 2 Dummy Variables for Changes in the Intercept Term • The situation is described in Figure 8. 2. • That the sales elasticity is significantly less than 1 is also confirmed by other evidence. • This example illustrates how one can be very easily misled by high R 2’s and apparent constancy of the coefficients.
8. 2 Dummy Variables for Changes in the Intercept Term
8. 3 Dummy Variables for Changes in Slope Coefficients and • We can write these equations together as or
8. 3 Dummy Variables for Changes in Slope Coefficients where for all observations in the first group for all observations in the second group for all observations in the first group i. e. , the respective value of x for the second group • The coefficient of D 1 measures the difference in the intercept terms and coefficient of D 2 measures the difference in the slope.
8. 3 Dummy Variables for Changes in Slope Coefficients • Suitable dummy variables can be defined when there are change in slopes and intercepts at different times. • Suppose that we have data for three periods and in the second period only the intercept changed ( there was a parallel shift). • In the third period the intercept and the slope changed.
8. 3 Dummy Variables for Changes in Slope Coefficients • Then we write • Then we can combine these equations and write the model as
8. 3 Dummy Variables for Changes in Slope Coefficients
8. 3 Dummy Variables for Changes in Slope Coefficients • An alternative way of writing the equations (8. 5), which is very general, is to stack the y variables and the error terms in columns. • Then write all the parameters α 1, α 2 , α 3 , β 1 , β 2 down with their multiplicative factors stacked in columns as follows:
8. 3 Dummy Variables for Changes in Slope Coefficients • What this says is where ( ) is used for multiplication, e. g. , α 3(0)=α 3× 0.
8. 3 Dummy Variables for Changes in Slope Coefficients where the definitions of D 1, D 2, D 3, D 4, D 5 are clear from equation(8. 7). • For instance,
8. 3 Dummy Variables for Changes in Slope Coefficients • Note that equation (8. 8) has to be estimated without a constant term. • In this method we define as many dummy variables as there are parameters to estimate and we estimate the regression equation with no constant term. • Note that equations (8. 6) and (8. 8) are equivalent.
8. 7 Dummy Dependent Variables • Until now we have been considering models where the explanatory variables are dummy variables. • We now discuss models where the explained variable is a dummy variable. • This dummy variable can take on two or more values but we consider here the case where it takes on only two values, zero or 1. • The linear probability model, logit and probit models
8. 8 The Linear Probability Model and the Linear Discriminant Function The Linear Probability Model • Similarly, in an analysis of bankruptcy of firms, we define • We write the model in the usual regression framework as with E(ui)=0.
8. 8 The Linear Probability Model and the Linear Discriminant Function • The condition expectation is equal to • This has to be interpreted in this case as the probability that the even will occur given the xi. • The calculated value if y from the regression equation (i. e. , ) will then give the .
8. 8 The Linear Probability Model and the Linear Discriminant Function • Since yi takes the value 1 or zero, the errors in equation (8. 11) can take only two values, (1 -βxi) and (-βxi). • Also, with the interpretation we have given equation (8. 11), and the requirement that E(ui)=0, the respective probabilities of these events are βxi and (1 -βxi). • Thus we have
8. 8 The Linear Probability Model and the Linear Discriminant Function • Hence
8. 8 The Linear Probability Model and the Linear Discriminant Function • Because of this heteroskedasticity problem the OLS estimates of β from equation (8. 11) will not be efficient. • We use the following two-step procedure: • First estimate (8. 11) by least squares. • Net compute and use weighted least squares, that is, defining • We regress
8. 8 The Linear Probability Model and the Linear Discriminant Function The problems with this procedure are 1. in practice may be negative, although in large samples this will be so with a very small probability since is a consistent estimator for.
8. 8 The Linear Probability Model and the Linear Discriminant Function 2. The most important criticism is with the formulation itself: that the conditional expectation be interpreted as the probability that the even will occur. In many cases lie outside the limits (0, 1).
8. 8 The Linear Probability Model and the Linear Discriminant Function
8. 8 The Linear Probability Model and the Linear Discriminant Function The Linear Discriminant Function • Suppose that we have n individuals for whom we have observations on k explanatory variables and we observe that n 1 of them belong to a second group where n 1+n 2=n. • We want to construct a linear function of the k variables that we can use to predict that a new observation belongs to one of the twp groups. • This linear function is called the linear discriminant function.
8. 8 The Linear Probability Model and the Linear Discriminant Function • As an example suppose that we have data on a number of loan applicants and we observe that n 1 of them were granted loans and n 2 of them were denied loans. • We also have the socioeconomic characteristics on the applicants
8. 8 The Linear Probability Model and the Linear Discriminant Function • Let us define a linear function • Then it is intuitively clear that to get the best discrimination between the two groups, we would want to choose that the ratio
8. 8 The Linear Probability Model and the Linear Discriminant Function • Fisher suggested an analogy between this problem and multiple regression analysis. • He suggested that we define a dummy variable
8. 8 The Linear Probability Model and the Linear Discriminant Function • Now estimate the multiple regression equation • Get the residual sum of squares RSS. • Then • Thus, once we have the regression coefficients and residual sum of squares from the dummy dependent variable regression, we can very easily obtain the discriminant function coefficients.
Discriminant Analysis • Discriminant analysis attempts to classify customers into two groups: – those that will default – those that will not • It does this by assigning a score to each customer • The score is the weighted sum of the customer data:
Discriminant Analysis • Here, wi is the weight on data type i, and Xi, c, is one piece of customer data. • The values for the weights are chosen to maximize the difference between the average score of the customers that later defaulted and the average score of the customers who did not default
Discriminant Analysis • The actual optimization process to find the weights is quite complex • The most famous discriminant scorecard is Altman's Z Score. • For publicly owned manufacturing firms, the Z Score was found to be as follows:
Discriminant Analysis
Discriminant Analysis • A company scoring less than 1. 81 was "very likely" to go bankrupt later • A company scoring more than 2. 99 was "unlikely" to go bankrupt. • The scores in between were considered inconclusive
8. 9 The Probit and Logit Models • An alternative approach is to assume that we have a regression model where is not observed. • It is commonly called a “latent” variable. • What we observe is a dummy variable yi defined by
8. 9 The Probit and Logit Models • For instance, if the observed dummy variable is whether or not the person is employed, would be defined as “propensity or ability to find employment. ” • Similarly, if the observed dummy variable is whether or not the person has bought a car, then would be defined as “desire or ability to buy a car. ” • Note that in both the examples we have given, there is “desire” and “ability” involved. • Thus the explanatory variables in (8. 12) would contain variables that explain both these elements.
8. 9 The Probit and Logit Models • The probit and logit model differ in the specification of the distribution of the error term u in equation (8. 12). • There are now several computer programs available for probit and logit analysis, and these programs are very inexpensive to run. • The difference between the specification (8. 12) and the linear probability model – the existence of an underlying latent variable for which we observe a dichotomous realization.
8. 9 The Probit and Logit Models Illustrative Example • As an illustration, we consider data on a sample of 750 mortgage applications in the Columbia, SC, metropolitan area. • There were 500 loan applications accepted and 250 loan applications rejected. • We define
8. 9 The Probit and Logit Models • Three model were estimated: the linear probability model, the logit model, and the probit model. • The explanatory variables were: AI =applicant’s and coapplicant’s income (103 dollars) XMD=debt minus mortgage payment (103 dollars) DF=dummy variable, 1 for female, 0 for male DR=dummy variable, 1 for nonwhite, 0 for white DS=dummy variable, 1 for single, 0 for otherwise DA=age of house (102 dollars)
8. 9 The Probit and Logit Models NNWP= percent nonwhite in the neighborhood (× 103) NMFI=neighborhood mean family income (105 dollars) NA=neighborhood average of house (102 years) The results are presented in Table 8. 3.
8. 9 The Probit and Logit Models
8. 9 The Probit and Logit Models Measure Goodness of Fit • There is a problem with the use of conventional R 2 -type measures when the explained variable y takes on only two values. • The predicted values are probabilities and the actual values y are either 0 or 1. • We can also think of R 2 in term of the proportion of correct predictions.
8. 9 The Probit and Logit Models • Since the dependent variables is a zero or 1 variable, after we computer the we classify the i-th observation as belonging to group 1 if <0. 5 and group 2 if >0. 5. • We can then count the number of correct predictions. • We can define a predicted value , which is also a zero-one variable such that
8. 9 The Probit and Logit Models • (Provided that we calculate yi to enough decimals, ties will be very unlikely. ) • Now define
Type I error vs. type II error • Limitation of the above count R 2 • Default prediction – The costs of a type I error: classifying a subsequently failing firm as non-failed – The type II error: classifying a subsequently non-failed firm as failed
Type I error vs. type II error • In particular, in the first case, the lender can lose up to 100% of the loan amount while, in the latter case, the loss is just the opportunity cost of not lending to that firm • Accordingly, in assessing the practical utility of failure prediction models, banks pay more attention to the misclassification costs involved in type I rather than type II errors.
8. 11 Truncated Variables: The Tobit Model • In our discussion of the logit and probit models we talked about a latent variable which was not observed, for which we could specify the regression model • In the logit and probit models, what we observe is a dummy variable
8. 11 Truncated Variables: The Tobit Model • Suppose, however, that is observed if and is not observed if ≦ 0. • Then the observed yi will be defined as >0
8. 11 Truncated Variables: The Tobit Model • This is known as the tobit model (Tobin’s probit) and was first analyzed in the econometrics literature by Tobit. • It is also known as a censored normal regression model because some observations on y* (those for which y* ≦ 0) are censored (we are not allowed to see them). • Our objective is to estimate the parameters β and σ.
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