Financial analysts often need to be able to explain the outcomes of a qualitative dependent variable. Qualitative dependent variables are dummy variables used as dependent variables instead of as independent variables.
For example, to predict whether or not a company will go bankrupt, we need to use a qualitative dependent variable (bankrupt or not) as the dependent variable and use data on the company's financial performance (e.g., return on equity, debt-to-equity ratio, or debt rating) as independent variables. Unfortunately, linear regression is not the best statistical method to use for estimating such a model. If we use the qualitative dependent variable bankrupt (1) or not bankrupt (0) as the dependent variable in a regression with financial variables as the independent variables, the predicted value of the dependent variable could be much greater than 1 or much lower than 0. Of course, these results would be invalid. The probability of bankruptcy (or of anything, for that matter) cannot be greater than 100 percent or less than 0 percent. Instead of a linear regression model, we should use probit, logit, or discriminant analysis for this kind of estimation.
Probit and logit models estimate the probability of a discrete outcome given the values of the independent variables used to explain that outcome. The probit model, which is based on the normal distribution, estimates the probability that Y = 1 (a condition is fulfilled) given the value of the independent variable X. The logit model is identical, except that it is based on the logistic distribution rather than the normal distribution.74 Both models must be estimated using maximum likelihood methods.75
Another technique to handle qualitative dependent variables is discriminant analysis. In his Z-score and Zeta analysis, Altman (1968, 1977) reported on the results of discriminant analysis. Altman uses financial ratios to predict the qualitative dependent variable bankruptcy. Discriminant analysis yields a linear function, similar to a regression equation, which can then be used to create an overall score. Based on the score, an observation can be classified into the bankrupt or not bankrupt category.
Qualitative dependent variable models can be useful not only for portfolio management but also for business management. For example, we might want to predict whether a client is likely to continue investing in a company or to withdraw assets from the company. We might also want to explain how particular demographic characteristics might affect the probability that a potential investor will sign on as a new client, or evaluate the effectiveness of a particular direct-mail advertising campaign based on the demographic characteristics of the target audience. These issues can be analyzed with either probit or logit models.
74 The logistic distribution e{hi)+b,x)l[ 1 + e{ba+b[X)] is easier to compute than the cumulative normal distribution. Consequently, logit models gained popularity when computing power was expensive.
75 For more on probit and logit models, see Greene (2003).
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