## **Logit Model - Binary Outcome and Forecasting linear regression**

A logit model is a binary model, which is used in marketing, and it uses a logistic function to model a binary dependent variable.

The dependent variable Y is a binary choice: either 0 or 1. For example, whether or not a customer purchases a widget, where 1 if yes, 0 if no.

= b0 + b1*x

where,

P (Y=1) is the probability when Y equals 1

is the odds of an event

Odds = probability / (1 – probability) = it is the ratio of the probability that an event happens to the probability that it does not happen

Probability = odds / (1 + odds)

Plot of Odds versus Probability

Plot of Log odds versus odds

The Logistic function

Odds ratio is the ratio od odds of an event occurring in one group to the odds of it occurring in another group.

If OR = 1, the odds of the event happening in both groups are equal.

If OR > 1, the odds of the event happening in group 1 are higher than the event happening in group 2.

If OR < 1, the odds of the event happening in group 1 are lower than the event happening in group 2.

Regression analysis is a model to examine the relationship between a variable Y and other variables X1 through Xp.

Y à response variable/dependent variable

X1,….Xp à predictors/independent variables

For a linear regression, we use a linear equation to build the relationship. Yt denotes the value of the response at time ‘t’. Thus,

Yt = b0 + b1X1t + … + bkXkt + et = b0 + + et

b0 = intercept

e = error terms

The dependent variable Y is a binary choice: either 0 or 1. For example, whether or not a customer purchases a widget, where 1 if yes, 0 if no.

= b0 + b1*x

where,

P (Y=1) is the probability when Y equals 1

is the odds of an event

Odds = probability / (1 – probability) = it is the ratio of the probability that an event happens to the probability that it does not happen

Probability = odds / (1 + odds)

Plot of Odds versus Probability

Plot of Log odds versus odds

The Logistic function

Odds ratio is the ratio od odds of an event occurring in one group to the odds of it occurring in another group.

If OR = 1, the odds of the event happening in both groups are equal.

If OR > 1, the odds of the event happening in group 1 are higher than the event happening in group 2.

If OR < 1, the odds of the event happening in group 1 are lower than the event happening in group 2.

**Linear Regression as a forecasting model**Regression analysis is a model to examine the relationship between a variable Y and other variables X1 through Xp.

Y à response variable/dependent variable

X1,….Xp à predictors/independent variables

For a linear regression, we use a linear equation to build the relationship. Yt denotes the value of the response at time ‘t’. Thus,

Yt = b0 + b1X1t + … + bkXkt + et = b0 + + et

b0 = intercept

e = error terms

__Assumptions of linear regression:__- Response and predictors have a linear relationship
- Independent variables are multivariate normal
- Independent variables have no or little multicollinearity
- No auto correlation. The error terms are independent à covariance between the error term at time i and j is equal to zero no matter what those time periods are.
- The error terms are distributed normal with a zero mean and the same variance