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Week 3&amp;4

# Week 3&amp;4 - II Testing for Multicollinearity When...

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II. Testing for Multicollinearity When two or more independent variables in a regression model are highly correlated with one another (or collinear), they will contribute “ redundant ” explanatory information. Hence, not all of those independent variables need to be in the model. Multi-collinear relations amongst regressors can cause a host of problems… making it difficult, if not impossible, to isolate the effects or impacts of individual regressors on the dependent variable.

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Consequences of “Multicollinearity” in Multiple Regression: Estimated OLS coefficients may test statistically insignificant (when they should not… given that they are known to be significant when running simple Bi- variate models)… and may even have the wrong sign. All (or almost all) of the estimated beta coefficients may test as “not significantly different” from zero, while the overall regression model yields very high explanatory power in terms of adjusted R-square-- a contradiction. Why? The presence of multicollinear regressors tends to inflate the standard errors of the models… leading to a greater likelihood that the beta’s are found to be not significantly different from zero. In short, multicollinearity “inefficient estimators”.
Y i = β 0 + β 1 X1 i + β 2 X2 i + ε i Consider a classic example from econometrics (temporal model): where Y = value of imports \$ X1 = GNP (Gross National Product) index of growth X2 = CPI (Consumer Price Index w.r.t. some base year) ε = stochastic disturbance/error term (i) = 1,…,n time periods (years) Expectations : X1 and X2 are possibly collinear. Suppose that r X1,X2 = .9972 (approx.)… indicating a very strong positive correlation between regressors.

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Database: Year Y X1 X2 base yr (4) millions millions inflation index 1 28.4 635.7 92.9 2 32.0 688.1 94.5 3 37.7 753.0 97.2 4 40.6 796.3 100.0 5 47.7 868.5 104.2 6 52.9 935.5 109.8 7 58.5 982.3 116.3 8 64.0 1,063.4 121.3 9 75.9 1,171.1 125.3 10 94.4 1,306.6 133.1 11 131.9 1,412.9 147.7 12 126.9 1,528.8 161.2 13 155.4 1,702.2 170.5 14 185.8 1,899.5 181.5 15 217.5 2,127.6 195.4 16 260.9 2.368.5 217.4 n=16 years (1991-2006) Base year
Y i = β 0 + β 1 X1 i + β 2 X2 i + ε i = -101.49 + 0.08 X1 i + .76 X2 i + ε i se(β’s) .0571428 .760 t-stats (1.40) (1.01) Note: t α/2 (critical) = 2.16 at α = .05 and 13 d.f. In both cases we “ fail to reject ” H o : β j = 0, (j=1,2); yet the overall model is said to explain 95.8% of the variation in Y as the adjusted R-square = .958, and an R-square of .970. Note also that F = 199.0, with prob(F) = .001 (showing significance at 99%). insignificant variables and significant model => inherent contradiction!

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error 0 -10 +10 Predicted values of Y Note that error terms do not appear to be random 100 200
But when we run the bi-variate models we get entirely different results... Y i = β 0 + β 1 X1 i + ε i = -69.03 + 0.13 X1 i + ε i se(β) .0004079 t-stat (31.87) R-square = .986 Y i = β 0 + β 1 X2 i + ε i = -149.52 + 1.82 X2 i + ε i se(β) .05911 t-stat (30.79) R-square = .985 Note that we now “ reject ” H o : β j = 0, (j=1). And the estimated slope coefficients are found to be significantly different (greater) than zero at both the 95% and 99% confidence levels.

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Week 3&amp;4 - II Testing for Multicollinearity When...

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