Estratto dal Cap. 8 di: Statistics for Marketing and Consumer Research, M. Mazzocchi, ed. SAGE, 2008.

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

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

3 IL MODELLO DI REGRESSIONE LINEARE SEMPLICE Metodo dei minimi quadrati ordinari (OLS): Tecnica per individuare l equazione della retta che minimizza la somma totale dei quadrati delle deviazioni (errori) tra dati osservati e punti sulla retta. 3

4 TO STUDY IN DETAIL: LEAST SQUARES ESTIMATION OF THE UNKNOWN PARAMETERS For a given value of the parameters, the error (residual) term for each observation is e = y a bx i i i The least squares parameter estimates are those who minimize the sum of squared errors: n n 2 2 ( i i ) i i= 1 i= 1 = = SSE y a bx e 4

5 PREDICTION 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 ˆ j y = a + bx j Example: change in price x, what happens in consumption y? 5

6 MODEL EVALUATION An evaluation of the model performance can be based on the residuals ( yi yˆ i ), which provide information on the capability of the model predictions to fit the original data (goodness-offit) 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

7 HYPOTHESIS 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.) This means that significant coefficient is when t-value is about greater than 2 7

8 COEFFICIENT OF DETERMINATION R 2 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) 0 R 2 11 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. 8

9 MULTIPLE REGRESSION The principle is identical to bivariate regression, but there are more explanatory variables y x x x = i 0 1 1i 2 2 i k ki i α α α α ε 9

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

11 GOODNESS-OF-FIT The coefficient of determination R 2 always increases with the inclusion of additional regressors 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) 2 2 n -1 = 0 R 2 1 R 1 (1 R ) n - k -1 11

12 applicazione della regressione multivariata con EXCEL FILE:esregress.xls obs. cons_elett Tmax Tmin velvento nuvole , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,5 con: cons_elett= consumi di energia per condizionamento. Tmax= temperatura massima registrata Tmin= temperatura minima registrata velvento= velocità del vento (maggiore o minore di 6 nodi) nuvole=grado di copertura delle nuvole 12

13 OUTPUT RIEPILOGO Statistica della regressione R multiplo 0,856 R al quadrato 0,732 R al quadrato corretto 0,707 Errore standard 8,341 Osservazioni 47 ANALISI VARIANZA gdl SQ MQ F Significatività F Regressione , ,27 28,74 0,00000 Residuo ,90 69,57 Totale ,98 Coefficienti Errore standard Stat t Valore di significatività Inferiore 95% Superiore 95% Inferiore 95,0% Superiore 95,0% Intercetta -85,05 16,56-5,14 0, ,46-51,64-118,46-51,64 Tmax 0,62 0,32 1,95 0,0574-0,02 1,25-0,02 1,25 Tmin 1,31 0,30 4,45 0,0001 0,72 1,91 0,72 1,91 velvento -1,96 2,71-0,72 0,4735-7,42 3,51-7,42 3,51 nuvole -0,19 1,75-0,11 0,9160-3,71 3,34-3,71 3,34 13

14 Confronto tra valori reali e stimati con il modello di regressione multipla Previsto cons_elett cons_elett 14