1 B IVARIATE AND MULTIPLE REGRESSION Estratto dal Cap. 8 di: “Statistics for Marketing and Consumer Research”, M. Mazzocchi, ed. SAGE, LEZIONI IN LABORATORIO Corso di MARKETING L. Baldi Università degli Studi di Milano

B IVARIATE LINEAR REGRESSION Causality (from x to y ) is assumed The error term embodies anything which is not accounted for by the linear relationship The unknown parameters (  and  ) need to be estimated (usually on sample data). We refer to the sample parameter estimates as a and b 2 Dependent variable Intercept Regression coefficient (Random) error term Explanatory variable

T O STUDY IN DETAIL : L EAST SQUARES ESTIMATION OF THE UNKNOWN PARAMETERS For a given value of the parameters, the error (residual) term for each observation is The least squares parameter estimates are those who minimize the sum of squared errors: 3

T O STUDY IN DETAIL : A SSUMPTIONS ON THE ERROR TERM 1. The error term has a zero mean 2. The variance of the error term does not vary across cases ( homoskedasticity) 3. The error term for each case is independent of the error term for other cases 4. The error term is also independent of the values of the explanatory (independent) variable 5. The error term is normally distributed 4

P REDICTION Once a and b have been estimated, it is possible to predict the value of the dependent variable for any given value of the explanatory variable Example: change in price x, what happens in consumption y? 5

M ODEL EVALUATION An evaluation of the model performance can be based on the residuals ( ), which provide information on the capability of the model predictions to fit the original data ( goodness-of- fit ) Since the parameters a and b are estimated on the sample, just like a mean, they are accompanied by the standard error of the parameters, which measures the precision of these estimates and depends on the sampling size. Knowledge of the standard errors opens the way to run hypothesis testing. 6

H YPOTHESIS TESTING ON REGRESSION COEFFICIENTS T-test on each of the individual coefficients Null hypothesis: the corresponding population coefficient is zero. The p -value allows one to decide whether to reject or not the null hypothesis that coeff. =zero, (usually p<0.05 reject the null hyp.) F-test (multiple independent variables, as discussed later) It is run jointly on all coefficients of the regression model Null hypothesis: all coefficients are zero 7

8 C OEFFICIENT OF D ETERMINATION R 2 Definition: A statistical measure of the ‘goodness of fit’ in a regression equation. It gives the proportion of the total variance of the forecasted variable that is explained by the fitted regression equation, i.e. the independent explanatory variables. The natural candidate for measuring how well the model fits the data is the coefficient of determination, which varies between zero (when the model does not explain any of the variability of the dependent variable) and 1 (when the model fits the data perfectly)

M ULTIPLE REGRESSION The principle is identical to bivariate regression, but there are more explanatory variables 9

A DDITIONAL ISSUES : Collinearity (or multicollinearity ) problem: The independent variables must be also independent of each other. Otherwise we could run into some double- counting problem and it would become very difficult to separate the meaning. Inefficient estimates Apparently good model but poor forecasts 10

G OODNESS - OF - FIT The coefficient of determination R 2 always increases with the inclusion of additional regressors adjusted R 2 Thus, a proper indicator is the adjusted R 2 which accounts for the number of explanatory variables (k ) in relation to the number of observations (n) 11