Question

Why is the use of OLS regression inappropriate when the dependent variable is dichotomous? explain one technique you can use instead, clearly stating how to interpret the estimated regression coefficients.   

Answer 1

• The linear regression model is based on an assumption that the outcome Y is continuous, with errors which are normally distributed. If the outcome variable is binary this assumption is clearly violated.
• Here we can use Logistic Regression. A logistic regression predicts the probability that an observation falls into one of two categories of a dichotomous dependent variable based on one or more independent variables that can be either continuous or categorical.
  The "logit" model is:
  where:
  ln is the natural logarithm, logexp, where exp=2.71828…
• p is the probability that the event Y occurs, p(Y=1)
• p/(1-p) is the "odds ratio"
• ln[p/(1-p)] is the log odds ratio, or "logit"
• all other components of the model are the same.
  The logistic regression model is simply a non-linear transformation of the linear regression. The "logistic" distribution is an S-shaped distribution function which is similar to the standard-normal distribution (which results in a probit regression model) but easier to work with in most applications (the probabilities are easier to calculate). The logit distribution constrains the estimated probabilities to lie between 0 and 1.

The estimated probability is:

p = 1/[1 + exp(-a - BX)]

• Interpreting logit coefficients:
  Instead of the slope coefficients (B) being the rate of change in Y (the dependent variables) as X changes (as in the LP model or OLS regression), now the slope coefficient is interpreted as the rate of change in the "log odds" as X changes.
  An interpretation of the logit coefficient which is usually more intuitive (especially for dummy independent variables) is the "odds ratio"-- expB is the effect of the independent variable on the "odds ratio" [the odds ratio is the probability of the event divided by the probability of the nonevent]. For example, if expB =5, then a one unit change in X would make the event 5 times as likely to occur. Odds ratios equal to 1 mean that there is a 50/50 chance that the event will occur with a small change in the independent variable. Negative coefficients lead to odds ratios less than one: if expB2 =.67, then a one unit change in X2 leads to the event being less likely to occur.