Multivariate Lineare Modelle
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1 0-1 TALEB AHMAD CASE - Center for Applied Statistics and Economics Humboldt-Universität zu Berlin
2 Motivation 1-1 Motivation Multivariate regression models can accommodate many explanatory which simultaneously affect the dependent variables. The hypothesis being tested is that there is a joint linear effect of the set of predictor variables on the set of response variables.
3 Motivation 2-1 Motivation The basic assumptions of multivariate regression model are - multivariate normality of the residuals - homogenous variances of residuals conditional on predictors - common covariance structure across observations - independent observations
4 Motivation 3-1 Motivation Heteroscedasticity where there is unequal variances for the predictor variables. Multicollinearity explaining variables coefficients are highly correlated. Autocorrelation when the assumption of zero correlation of the error term is violated.
5 Motivation 3-2 Outline 1. Motivation 2. The Boston housing data 3. für Boston housing data 4. References
6 The Boston housing data 4-1 Aim: explain price variation X 14 by the variations of all other 13 variables in the Boston Housing data. variables: see variables description
7 The Boston housing data 4-2 Explorative Zusammenhangsanalyse visual inspection skewness, kurtosis and outliers distribution Q-Q plots and Kolmogorov-Smirnov transformation, Z and log transformations to improve normality
8 Boxplots: original variables for Boston Housing data Boxplots: transformed variables for Boston Housing data Boxplots for variables
9 Q-Q plots: (left-right)original variables
10 Scatterplot matrix (X 1 tox 6 ) and (X 7 to X 14 ) with X 14 respectively
11 Scatterplot matrix for all variables
12 The Boston housing data 4-7 descriptive statistics Variable Mean Median Stdd. Skewn. Kurt. X X X X X X X X X X X X X X Table 1: Summary statistics: Boston data
13 Variable transformation 5-1 Transformations X 1 = log(x 1 ) X 2 = log(x 2/10) X 3 = X 3 (normal) X 4 = X 4 (binary) X 5 = log(x 5 ) X 6 = log(x 6 ) X 7 = ( X 7 ) 2 X 8 = log(x 8) 10 2 X 9 = log(x 9 ) X 10 = log(x 10 ) X 11 = exp(0.4x 11) X = X X 13 = (X 13 ) X 14 = log(x 14 )
14 Q-Q plots: (left-right) transformed variables
15 MLR Modelle 6-1 Multivariate Regression Lineare Modelle X = α 0 + α 1 X 1 + α 2 X α k X k + ε Estimate: 13 X 14 = α 0 + α j Xj + ε j=1 X j are transformed variables X 1 to X 14
16 MLR Modelle 6-2 Regressionschätzung: Methode Forward selection Step Multiple R R 2 F SigF Variable(s) X X X X X X X X X X X X2 Table 2: Forward Selection
17 MLR Modelle 6-3 Regressionschätzung: Methode Forward selection ANOVA SS df MSS F-test P-value Regression Residuals e Total Variation Multip. R = 0.87 R 2 = 0.76 Adj. R 2 = 0.75 Std. Error = 0.49 Table 3: Forward Selection
18 MLR Modelle 6-4 Regressionschätzung PARAMETERS Beta SE StandB t-test P-value Variable α Constant α X 1 α X 2 α X 3 α X 4 α X 5 α X 6 α X 8 α X 9 α X 10 α X 11 α X 12 α X 13 Table 4: Forward Selection
19 MLR Modelle 6-5 Regressionschätzung R 2 = 0.76 indicates 75% of variation of X 14 is explained by the model 13 X 14 = α 0 + α j Xj + ε P-values table 4 indicates that variables X 1, X 2, X 3 and X 8 have little influence on changes in X 14, the log price of the Houses. j=1
20 The Boston housing data: comprise 506 observations for each census district of the Boston metropolitan area.
21 References 7-1 References A. Handl Multivariate Analysemethoden. Springer, J. Schira Statistische Methoden der BWL und VWL. Pearson, H. Joe Multivariate Models and Dependence Concepts Chapman & Hall, London, W. Härdle und L. Simar Applied Multivariate Statistical Analysis. Springer, 2003.
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