A Smart Guide to Dummy Variables: Four Applications and a ...

1 A Smart Guide TO Dummy VARIABLES: FOUR Applications AND A MACRO. Susan Garavaglia and Asha Sharma Dun & Bradstreet Murray Hill, New Jersey 07974. Abstract: Dummy variables are variables that take qualitative concepts such as season, male or the values of only 0 or 1. They may be explanatory female, smoker or non-smoker, etc., is required for or outcome variables; however, the focus of this many models to make sense. article is explanatory or independent variable In a regression model, a Dummy variable with a construction and usage. Typically, Dummy variables value of 0 will cause its coefficient to disappear from are used in the following Applications : time series the equation. Conversely, the value of 1 causes analysis with seasonality or regime switching ; the coefficient to function as a supplemental analysis of qualitative data, such as survey intercept, because of the identity property of responses; categorical representation, and multiplication by 1.

2 This type of specification in a representation of value levels. Target domains may linear regression model is useful to define subsets be economic forecasting, bio-medical research, of observations that have different intercepts and/or credit scoring, response modeling, and other fields. slopes without the creation of separate models. In Dummy variables may serve as inputs in traditional logistic regression models, encoding all of the regression methods or new modeling paradigms, independent variables as Dummy variables allows such as genetic algorithms, neural networks, or easy interpretation and calculation of the odds Boolean network models. Coding techniques include ratios, and increases the stability and significance of "1-of-N" and "thermometer" encoding. Statistical the coefficients. Examples of these results are in properties of Dummy variables in each of the Section 3.

3 In addition to the direct benefits to traditional usage and application contexts are statistical analysis, representing information in the discussed, and a more detailed introduction of a form of Dummy variables is makes it easier to turn Boolean network model is presented. Because the model into a decision tool. Consider a risk conversion of categorical data to Dummy variables manager who needs to assign credit limits to often requires time-consuming and tedious re- businesses. The age of the business is almost coding, a SAS macro is offered to facilitate the always significant in assessing risk. If the risk creation of Dummy variables and improve manager has to assign a different credit limit for productivity. each year in business, it becomes extremely complicated and difficult to use because some 1. Introduction to Dummy Variables businesses are several hundred years old.

4 Bi- variate analysis of the relationship between age of Dummy variables are independent variables business and default usually yields a small number which take the value of either 0 or 1. Just as a of groups that are far more statistically significant " Dummy " is a stand-in for a real person, in than each year evaluated separately. quantitative analysis, a Dummy variable is a Synonyms for Dummy variables are design numeric stand-in for a qualitative fact or a logical variables [Hosmer and Lemeshow, 1989] , Boolean proposition. For example, a model to estimate indicators, and proxies [Kennedy, 1981]. Related demand for electricity in a geographical area might concepts are binning [Tukey, 1977] or ranking, include the average temperature, the average because belonging to a bin or rank could be number of daylight hours, the total number of formulated into a Dummy variable.

5 Bins or ranks structure square feet, numbers of businesses, can also function as sets and Dummy variables can numbers of residences, and so forth. It might be represent non-probabilistic set membership. Set more useful, however, if the model could produce theory is usually explained in texts on computer appropriate results for each month or each season. science or symbolic logic. See [Arbib, et. al., 1981]. Using the number of the month, such as 12 for or [MacLane, 1986]. December, would be silly, because that implies that Dummy variables based on set membership can the demand for electricity is going to be very help when there are too few observations, and thus, different between December and January, which is degrees of freedom, to have a Dummy variable for month 1. It also implies that Winter occurs during every category or some categories are too rare to the same months everywhere, which would preclude be statistically significant.

6 Dummy variables can the use of the model for the opposite polar represent mixed or combined categories using hemisphere. Thus, another way to represent logical operations, such as: a. a business in the wholesale or retail trades; it is false. The creation of each variable requires b. a business in the retail trades and that is considerable pre-processing, with TOPQUICK. less than 3 years old; requiring the most complicated processing, c. a business that has had a prior bankruptcy or because, at some point, population norms for the payments placed for collection, but not both quick ratio would have to be established to (exclusive or). determine quartile breaks. Variable BUSAGE2 just needs the current year and the year the business In a., two categories from the same variable, started; BNKRPIND needs bankruptcy history on the industry group are combined, using a logical or.

7 Case; CHILDAPP needs the SIC (Standard Industrial Two categories, industry and age of business are Classification) code; and NEREGN needs the state. combined using the logical and operator. The The impact of these variables on further analysis exclusive or (XOR) operator is not part of many depends on the application. For example, programming languages, including the SAS BUSAGE2 might be a derogatory indicator for credit lanaguage. However, the discussion of Boolean risk but a positive indicator for mail-order response. networks in Section 4 includes a programmable The information value of these variables definition of XOR. depends on the overall proportion of observations The four Applications of Dummy variables having these Dummy variables containing ones. discussed here are: 1) regime switching (or The mean, d ,of a Dummy variable is always in the seasonality); 2) categorical representation; 3) interval [0,1], and represents the proportion, or interval level representation, and, 4) Boolean percentage of cases that have a value of 1 for that operators.

8 The rest of this article is organized variable. Therefore, it is also the probability that a around answering the following questions: (Section 1 will be observed. It is possible to calculate the 2) What is the information content of Dummy variance and standard deviation, d , of a Dummy variables and how is it measured?; (Section 3) How variable, but these moments do not have the same can Dummy variables add predictive power and meaning as those for continuous-valued data. This stability to traditional regression analysis?; (Section is because, if d is known for a Dummy variable, so 4) How are Dummy variables used in non- is d because there are only two possible (x - d ). parametric analysis and dynamic systems?; and, values. The distribution of any Dummy variable can (Section 5) How can developers use the SAS be classified as a Binomial distribution of n Bernoulli language to make Dummy variable coding easy?

9 Trials. Some helpful tables on distributions and their Section 6 is a summary. moments are in Appendix B of [Mood, et. al., 1977] . The long expression for calculating the Standard 2. An Information Theoretic Interpretation of Deviation is (( d (1 - d )2)+ (1 - d)(0 - d)2)) . the Statistical Properties of Dummy Sometimes statistics texts refer to (1- p) as q, and Variables the standard deviation reduces to (pq) . What is the information content of a Dummy Any definition of any Dummy variable implies a variable? If d = 1 or d = 0, there is no logical proposition with a value of true or false, a uncertainty - an observer will know what to expect statement of fact, and the respective information 100% of the time. Therefore, there is no benefit in value of that fact. Here are some typical examples further observation nor will this variable be of facts about businesses, followed by hypothetical significant in prediction, estimation, or detection of variable names, that can be represented by Dummy any other information.

10 As d moves up or down to variables: , the information content increases, because there is less certainty about the value of the a. Business is at least 3 years old and less than 8 variable. This is discussed further with more years old. (BUSAGE2); examples in [Garavaglia 1994]. b. Business has experienced a prior bankruptcy A set of Dummy variables can also be thought of (BNKRPIND); as a string of bits (a common name for binary digits c. Business is in the top quartile in its industry with in computer science). One of the roles of basic respect to its Quick Ratio (TOPQUICK); Information Theory [Shannon and Weaver, 1948] is d. Business is a retailer of Children's Apparel to provide a methodology for determining how many (CHILDAPP); bits are needed to represent specific information e. Business is located in the Northeast Region which will be transmitted over a channel where (NEREGN).