# Regression Analysis : Correlation Between A Response Variable And Another Set Of Variables

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Regression analysis is the analysis of the relationship between a response variable and another set of variables. The relationship is expressed through a statistical model equation that predicts a response variable from a function of regressor variables and parameters. In a linear regression model the predictor function is linear in the parameters. The parameters are estimated so that a measure of fit is optimized. For example, the equation for the observation might be: where Y_i is the response variable, Χ_i is a regressor variable, β_0 and β_1 are unknown parameters to be estimated, and ε_i is an error term. This model is termed the simple linear regression model, because it is linear in β_0 and β_1 and contains only a single regressor…show more content…
The equation for the i^th observation might be:
There are many cases where the dependent variable is restricted to take on a limited range of values, for example only values 0 or 1 (binary logistic regression). In this case the regression model is slightly different.
We introduce the idea of Random Utility Model. Assume that an individual has to make a choice between two alternatives. Let U_ij be the utility that individual i (for i=1... N) gets if alternative j (for j=0, 1) is chosen. The individual makes choice 1 if U_1i≥U_0i and makes choice 0 otherwise. Since U_1i≥U_0i is equivalent to U_1i-U_0i≥0, the choice can be seen to depend on the difference in utilities across the two alternatives and we define this difference as
Y_i^* should depend on an individual’s characteristics (in our case LoanValue, EmpStatus, Age, HouseholdStatus etc.). Idea that a variable depends on some characteristics (explanatory variables) is a multiple regression model:
To make the derivations easier, we write some of the formulae below in terms of the simple regression model
The problem with this regression is that we do not observe individual’s utility and, thus, Y_i^* is unobservable. The logistic regression model can be interpreted as this regression, where the errors are assumed to satisfy all the classical assumptions except one. The exception is that the errors are assumed to have a logistic distribution.
Y_i^* is unobservable,