Quantitative Business Analysis for Decision Making Multiple Linear RegressionAnalysis

Outlines n n Multiple Regression Model n n Estimation n n Testing Significance of Predictors n n Multicollinearity n n Selection of Predictors n n Diagnostic Plots

Multiple Regression Model Multiple linear regression model: are slope coefficients of X 1, X 2,…,X k. quantifies the amount of change in response Y for a unit change in X i when all other predictors are held fixed.

Multiple Regression Model (con’t) In the model, is the mean of Y. – – Contributes to the variation in Y values from their mean, and – – is assumed normally distributed with mean 0 and standard deviation

Sampling A random sample of n units is taken. Then for each unit k+1 measurements are made: Y, X 1, X 2, …., X k

Estimated Model Estimated multiple regression model is: Expressions for b i are cumbersome to write. is an estimate of

Standard Error Sample standard deviation around the mean (estimated regression model) is: It is an estimate of Standard error of (for specified values of predictors) is denoted by

Testing Significance of a Predictor For comparing with a reference,test statistic is: and for estimating by a confidence interval, compute

Coefficient of Determination Coefficient of determination R 2 quantifies the % of variation in the Y-distribution that is accounted by the predictors in the model. If – –R 2 = 80%, then 20% variation in the Y- distribution is due to factors other than those in the model. – –R 2 increases as predictors are added in the model but at the cost of complicating it.

Testing the Model for Significance Null hypothesis = predictors in the relationship have no predictive power to explain the variation in Y- distribution Test statistic: F =. It has F- distribution with k and (n-k-1) degrees of freedoms for the numerator and denominator.

Multicollinearity and Selection of Predictors n n Multicollinearity - occurs when predictors are highly correlated among themselves. In its presence R 2 may be high, but individual coefficients are less reliable. n n Screening process (e.g. stepwise regression) can eliminate multicollinearity by selecting only those predictors that are not strongly correlated among themselves.

Diagnostic Plots n n Residuals are used to diagnose the validity of the model assumptions. n n A scatter plot of the residuals against the predicted values can serve as a diagnostic tool. n n A diagnostic plot can identify outliers, unequal variability, and need for transformation to achieve homogeneity etc.

Indicator Variables n n Indicator variables (also called dummy variables) are numerical codes that are used to represent qualitative variables. n n For example, 0 for men and 1 for women. n n For a qualitative variable with c categories, (c-1) indicator variables need to be defined.