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Linearity in parameters Fixed X i  values Variation in the X i   values E(U i |X 2i , X 3i ,... X ki )=0 No autocorrelation Homoscedasticity The sample size is greater than the number of parameters No multicollinearity No specification Bias U i  is normally distributed U i ~N(0, σ 2 )

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Nature of  Multicollinearity Causes Consequences Detection Remedial measures

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Multicollinearity  means the existence of a perfect linear relationship among some or all explanatory variables Multicollinearity  does not violate the assumptions of CLRM Estimators are still BLUE Multicollinearity  is a sample phenomenon

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data collection methods Constraints on the model Model specification Common factors among explanatory variables The real problem with multicollinearity is in the standard error of the estimators

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VIF  shows the speed with which variances and  covariances  increase In general ,

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OLSE have large variances and covariances Wider CI Insignificant t ratio High R 2  accompanied by insignificant t ratio Sensitivity to small changes in the data

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OLSE have large variances and covariances Wider CI Insignificant t ratio

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High R 2  accompanied by insignificant t ratio

expenditure Income wealth 70 80 810 65 100 1009 90 120 1273 95 140 1425 110 160 1633 115 180 1876 120 200 2025 140 220 2201 155 240 2435 150 260 2686

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y x 2 x 3 1 2 4 2 0 2 3 4 12 4 6 0 5 8 16

y x 2 x 3 1 2 4 2 0 2 3 4 0 4 6 12 5 8 16

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High R 2  accompanied by insignificant t ratio High pair-wise correlations among explanatory variables  		Correlation coefficients Not suitable  for more than 2 explanatory variables if

Income Wealth Income 1 0.999 Wealth 0.999 1

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Partial correlations High                   and  low VIF:   if VIF of a variable exceeds 10 it is said to be highly   correlated with the other explanatory variables Eigenvalues and condition index CI

Tolerance VIF income .002  460.618  wealth .002  460.618

Eigenvalues Condition index 2.930   1 0.06965   6.486   .0001109   162.553

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Dropping a variable Transformation of variables Combining variables Combining data Increase sample size Statistical Techniques Factor analysis and principal components Ridge regression

Violation of Assumptions