I see logit equations always written with a single dependent variable, however I am running a logit model which outputs the coefficients for three explanatory variables (X1, X2, and X3) with respect to a binary dependent variable (Y). X3 is income so I am using the log of X3.
If I were running an OLS I would write my model equation as Y = β0 + β1 X1 + β X2 + β3 log X3 + ε. What is the equivalent for my logit model?
I then estimate the odds ratio based on my logit model. How does one write this when there are multiple dependent variables?
There is no problem to include any $p$ (such that $p+1 \le n$) number of explanatory variables in the logistic regression, $$ P(Y=1|X_1=x_1, X_2=x_2, \log X_3=x_3) = \frac{1}{1+\exp\{- ( \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3)\}}. $$
And the OR can be defined for any subsets of the explanatatory variables, (e.g., for $X_2$) in the same way as for one explanatory variable $$ OR(X_2)= \frac{P(Y=1|x_1, x_2+1, x_3)/P(Y=0|x_1, x_2+1, x_3)}{P(Y=1|x_1, x_2, x_3/P(Y=0|x_1, x_2, x_3)} = \exp\{\beta_2\}. $$