bbivariateivariateand andmultiplemultiple regression...
Post on 16-Feb-2019
219 Views
Preview:
TRANSCRIPT
BBIVARIATEIVARIATE ANDAND MULTIPLEMULTIPLE
REGRESSIONREGRESSION
LEZIONI IN LABORATORIO
Corso di MARKETING
L. Baldi
Università degli Studi di Milano
1
REGRESSIONREGRESSION
Estratto dal Cap. 8 di:“Statistics for Marketing and Consumer Research”,
M. Mazzocchi, ed. SAGE, 2008.
BIVARIATE LINEAR REGRESSION
i i iy xα β ε= + +Dependent variable
Intercept(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 (αααα and ββββ) need to be estimated (usually on sample data). We refer to the sample parameter estimates as a and b
2
Regression coefficientExplanatory variable
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.
§
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
The least squares parameter estimates are those who
i i ie y a bx= − −� The least squares parameter estimates are those who
minimize the sum of squared errors:
4
2 2
1 1
( )n n
i i ii i
SSE y a bx e= =
= − − =∑ ∑
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
Example: change in price x, what happens in
consumption y?
5
ˆ j jy a bx= +
MODEL 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
ii yy ˆ−
�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
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<0.05reject the null hyp.)
• This means that significant coefficient is when t-value is about greater than 2
7
COEFFICIENT OF DETERMINATION R2
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)
10 2 ≤≤ R
8
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.
10 ≤≤ R
MULTIPLE REGRESSION
The principle is identical to bivariate
regression, but there are more
explanatory variables
9
0 1 1 2 2 ...i i i k ki iy x x xα α α α ε= + + + + +
ADDITIONAL ISSUES:
Collinearity (or multicollinearity) problem:
� The independent variables must be also independent of each other.
� Otherwise we could run into some double-� 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
GOODNESS-OF-FIT
�The coefficient of determination R2 always
increases with the inclusion of additional
regressors
�Thus, a proper indicator is the adjusted adjusted
RRRR22 which accounts for the number of
explanatory variables (k) in relation to the
number of observations (n)
11
2 2 -11 (1 )
- -1
nR R
n k= − − 10 2 ≤≤ R
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
_______________________________________________________ applicazione della regressione multivariata con EXCEL
FILE:esregress.xls
obs. cons_elett Tmax Tmin velvento nuvole1 45 87 68 1 2,02 73 90 70 1 1,03 43 88 68 1 1,04 61 88 69 1 1,55 52 86 69 1 2,06 56 91 75 1 2,07 70 91 76 1 1,58 69 90 73 1 2,09 53 79 72 0 3,010 51 76 63 0 0,011 39 83 57 0 0,012 55 86 61 1 1,013 55 85 70 1 2,014 57 89 69 0 2,015 68 88 72 1 1,516 73 85 73 0 3,017 57 84 68 1 3,018 51 83 69 0 2,019 55 81 70 0 1,020 56 89 70 1 1,521 72 88 69 1 0,0
12
21 72 88 69 1 0,022 73 88 76 1 2,523 69 77 66 1 3,024 38 75 65 1 2,525 50 72 64 1 3,026 37 68 65 1 3,027 43 71 67 0 3,028 42 75 66 1 3,029 25 74 52 1 0,030 31 77 51 0 0,031 31 79 50 0 0,032 32 80 50 0 0,033 35 80 53 0 0,034 32 81 53 1 0,035 34 80 53 0 0,036 35 81 54 1 2,037 41 83 67 0 2,038 51 84 67 1 1,539 34 80 63 1 3,040 19 73 53 1 1,041 19 71 49 0 0,042 30 72 56 1 3,043 23 72 53 1 0,044 35 79 48 1 0,045 29 84 63 1 1,046 55 74 62 0 3,047 56 83 72 1 2,5
OUTPUT RIEPILOGO
Statistica della regressioneR multiplo 0,856R al quadrato 0,732R al quadrato corretto 0,707Errore standard 8,341Osservazioni 47 ANALISI VARIANZA
gdl SQ MQ F Significatività FRegressione 4 7997,08 1999,27 28,74 0,00000Residuo 42 2921,90 69,57Totale 46 10918,98
13
CoefficientiErrore
standard Stat tValore di
significativitàInferiore
95%Superiore
95%Inferiore 95,0%
Superiore 95,0%
Intercetta -85,05 16,56 -5,14 0,0000 -118,46 -51,64 -118,46 -51,64Tmax 0,62 0,32 1,95 0,0574 -0,02 1,25 -0,02 1,25Tmin 1,31 0,30 4,45 0,0001 0,72 1,91 0,72 1,91velvento -1,96 2,71 -0,72 0,4735 -7,42 3,51 -7,42 3,51nuvole -0,19 1,75 -0,11 0,9160 -3,71 3,34 -3,71 3,34
Confronto tra valori reali e stimati con il modello di regressione multipla
40
50
60
70
80
14
0
10
20
30
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47
Previsto cons_elett cons_elett
top related