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In statistics and econometrics , particularly in regression analysis , a dummy variable also known as an indicator variable , design variable , Boolean indicator , binary variable , or qualitative variable   is one that takes the value 0 or 1 to indicate the absence or presence of some categorical effect that may be expected to shift the outcome. A dummy variable can thus be thought of as a truth value represented as a numerical value 0 or 1 as is sometimes done in computer programming.
Dummy variables are "proxy" variables or numeric stand-ins for qualitative facts in a regression model. In regression analysis, the dependent variables may be influenced not only by quantitative variables income, output, prices, etc. A dummy independent variable also called a dummy explanatory variable which for some observation has a value of 0 will cause that variable's coefficient to have no role in influencing the dependent variable , while when the dummy takes on a value 1 its coefficient acts to alter the intercept.
For example, suppose membership in a group is one of the qualitative variables relevant to a regression. If group membership is arbitrarily assigned the value of 1, then all others would get the value 0. Then the intercept the value of the dependent variable if all other explanatory variables hypothetically took on the value zero would be the constant term for non-members but would be the constant term plus the coefficient of the membership dummy in the case of group members.
Dummy variables are used frequently in time series analysis with regime switching, seasonal analysis and qualitative data applications. Dummy variables are involved in studies for economic forecasting , bio-medical studies, credit scoring , response modelling, etc. Dummy variables may be incorporated in traditional regression methods or newly developed modeling paradigms. Dummy variables are incorporated in the same way as quantitative variables are included as explanatory variables in regression models.
For example, if we consider a Mincer-type regression model of wage determination, wherein wages are dependent on gender qualitative and years of education quantitative:. Note that the coefficients attached to the dummy variables are called differential intercept coefficients.
The model can be depicted graphically as an intercept shift between females and males. Dummy variables may be extended to more complex cases. For example, seasonal effects may be captured by creating dummy variables for each of the seasons: In the panel data , fixed effects estimator dummies are created for each of the units in cross-sectional data e. However in such regressions either the constant term has to be removed or one of the dummies has to be removed, with its associated category becoming the base category against which the others are assessed in order to avoid the dummy variable trap:.
The constant term in all regression equations is a coefficient multiplied by a regressor equal to one. When the regression is expressed as a matrix equation, the matrix of regressors then consists of a column of ones the constant term , vectors of zeros and ones the dummies , and possibly other regressors. If one includes both male and female dummies, say, the sum of these vectors is a vector of ones, since every observation is categorized as either male or female.
This sum is thus equal to the constant term's regressor, the first vector of ones. As result, the regression equation will be unsolvable, even by the typical pseudoinverse method. This is referred to as the dummy variable trap. The trap can be avoided by removing either the constant term or one of the offending dummies. The removed dummy then becomes the base category against which the other categories are compared.
A regression model in which the dependent variable is quantitative in nature but all the explanatory variables are dummies qualitative in nature is called an Analysis of Variance ANOVA model.
Suppose we want to run a regression to find out if the average annual salary of public school teachers differs among three geographical regions in Country A with 51 states: Say that the simple arithmetic average salaries are as follows: The arithmetic averages are different, but are they statistically different from each other?
To compare the mean values, Analysis of Variance techniques can be used. The regression model can be defined as:. In this model, we have only qualitative regressors, taking the value of 1 if the observation belongs to a specific category and 0 if it belongs to any other category. Now, taking the expectation of both sides, we obtain the following:. The error term does not get included in the expectation values as it is assumed that it satisfies the usual OLS conditions, i.
The expected values can be interpreted as follows: Thus, the mean salaries of teachers in the North and South is compared against the mean salary of the teachers in the West. Hence, the West Region becomes the base group or the benchmark group ,i. The omitted category , i. The regression result can be interpreted as: To find out if the mean salaries of the teachers in the North and South are statistically different from that of the teachers in the West the comparison category , we have to find out if the slope coefficients of the regression result are statistically significant.
For this, we need to consider the p values. The model is diagrammatically shown in Figure 2. Here, Marital Status and Geographical Region are the two explanatory dummy variables. In this model, a single dummy is assigned to each qualitative variable, one less than the number of categories included in each.
Here, the base group is the omitted category: Unmarried, Non-North region Unmarried people who do not live in the North region.
All comparisons would be made in relation to this base group or omitted category. Thus, if more than one qualitative variable is included in the regression, it is important to note that the omitted category should be chosen as the benchmark category and all comparisons will be made in relation to that category. The intercept term will show the expectation of the benchmark category and the slope coefficients will show by how much the other categories differ from the benchmark omitted category.
They statistically control for the effects of quantitative explanatory variables also called covariates or control variables. If we include a quantitative variable, State Government expenditure on public schools per pupil , in this regression, we get the following model:.
Figure 3 depicts this model diagrammatically. The average salary lines are parallel to each other by the assumption of the model that the coefficient of expenditure does not vary by state. The trade off shown separately in the graph for each category is between the two quantitative variables: Quantitative regressors in regression models often have an interaction among each other. In the same way, qualitative regressors, or dummies, can also have interaction effects between each other, and these interactions can be depicted in the regression model.
For example, in a regression involving determination of wages, if two qualitative variables are considered, namely, gender and marital status, there could be an interaction between marital status and gender. With the two qualitative variables being gender and marital status and with the quantitative explanator being years of education, a regression that is purely linear in the explanators would be.
This specification does not allow for the possibility that there may be an interaction that occurs between the two qualitative variables, D 2 and D 3. For example, a female who is married may earn wages that differ from those of an unmarried male by an amount that is not the same as the sum of the differentials for solely being female and solely being married. Then the effect of the interacting dummies on the mean of Y is not simply additive as in the case of the above specification, but multiplicative also, and the determination of wages can be specified as:.
Thus, an interaction dummy product of two dummies can alter the dependent variable from the value that it gets when the two dummies are considered individually. However, the use of products of dummy variables to capture interactions can be avoided by using a different scheme for categorizing the data—one that specifies categories in terms of combinations of characteristics. This specification involves the same number of right-side variables as does the previous specification with an interaction term, and the regression results for the predicted value of the dependent variable contingent on X i , for any combination of qualitative traits, are identical between this specification and the interaction specification.
A model with a dummy dependent variable also known as a qualitative dependent variable is one in which the dependent variable, as influenced by the explanatory variables, is qualitative in nature. Some decisions regarding 'how much' of an act must be performed involve a prior decision making on whether to perform the act or not. For example, the amount of output to produce, the cost to be incurred, etc.
Such "prior decisions" become dependent dummies in the regression model. For example, the decision of a worker to be a part of the labour force becomes a dummy dependent variable. The decision is dichotomous , i. So the dependent dummy variable Participation would take on the value 1 if participating, 0 if not participating.
Affiliation to a Political Party. When the qualitative dependent dummy variable has more than two values such as affiliation to many political parties , it becomes a multiresponse or a multinomial or polychotomous model. Analysis of dependent dummy variable models can be done through different methods. One such method is the usual OLS method, which in this context is called the linear probability model.
This is the underlying concept of the logit and probit models. These models are discussed in brief below. An ordinary least squares model in which the dependent variable Y is a dichotomous dummy, taking the values of 0 and 1, is the linear probability model LPM.
The model is called the linear probability model because, the regression is linear. Thus the relationship between the independent and dependent variables is necessarily non-linear. For this purpose, a cumulative distribution function CDF can be used to estimate the dependent dummy variable regression. Figure 4 shows an 'S'-shaped curve, which resembles the CDF of a random variable. In this model, the probability is between 0 and 1 and the non-linearity has been captured.
The choice of the CDF to be used is now the question. Two alternative CDFs can be used: The shortcomings of the LPM led to the development of a more refined and improved model called the logit model. In the logit model, the cumulative distribution of the error term in the regression equation is logistic. The logit model is estimated using the maximum likelihood approach. The model is then expressed in the form of the odds ratio: Taking the natural log of the odds, the logit L i is expressed as.
This relationship shows that L i is linear in relation to X i , but the probabilities are not linear in terms of X i. Another model that was developed to offset the disadvantages of the LPM is the probit model. The probit model uses the same approach to non-linearity as does the logit model; however, it uses the normal CDF instead of the logistic CDF.
From Wikipedia, the free encyclopedia. Journal of the American Statistical Association. Dummy Dependent Variable Models".
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