Heteroscedasticity

Heteroscedasticity

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  Muhammad Ali Lecturer inStatistics GPGC Mardan. 1 Heteroscedasticity Definition One of the assumption of the classical linear regression model that the error ( iε ) term having the same variance i.e. δ2 . But in most practical situation this assumption did not fulfill, and we have the problem of heteroscedasticity. Heteroscedasticity does not destroy the unbiased and consistency property of the ordinary least square estimators, but these estimators have not the property of minimum variance. Recall that OLS makes the assumption that V (εi ) =σ2 for al i. That is, the variance of the error term is constant. (Homoscedasticity). If the error terms do not have constant variance, they are said to be heteroscedasticity. The term means “differing variance” and comes from the Greek “hetero” ('different') and “scedasis” ('dispersion').] When heteroscedasticity might occur/causes of heteroscedasticity 1. Errors may increase as the value of an independent variable increases. For example, consider a model in which annual family income is the independent variable and annual family expenditures on vacations is the dependent variable. Families with low incomes will spend relatively little on vacations, and the
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  Muhammad Ali Lecturer inStatistics GPGC Mardan. 2 variations in expenditures across such families will be small. But for families with large incomes, the amount of discretionary income will be higher. The mean amount spent on vacations will be higher, and there will also be greater variability among such families, resulting in heteroscedasticity. Note that, in this example, a high family income is a necessary but not sufficient condition for large vacation expenditures. Any time a high value for an independent variable is a necessary but not sufficient condition for an observation to have a high value on a dependent variable, heteroscedasticity is likely. 2. Other model misspecifications can produce heteroscedasticity. For example, it may be that instead of using Y, you should be using the log of Y. Instead of using X, maybe you should be using X2 , or both X and X2 . Important variables may be omitted from the model. If the model were correctly specified, you might find that the patterns of heteroscedasticity disappeared. 3. As data Collection techniques improve, δ2 i is likely to decrease. Thus banks that have sophisticated data processing equipment are likely to commit fewer errors in the monthly or quarterly statements of their customers than banks without such facilities. 4. Heteroscedasticity can also arise as a result of the presence of outliers. An outlying observation is an observation that is much different in relation to the observations in the sample.
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  Muhammad Ali Lecturer inStatistics GPGC Mardan. 3 5. Error learning models, as people learn, their errors of behavior become smaller over time. In this case, δ2 i is expected to decrease. As an example, the number of typing speed errors decreases as the number of typing practice increases, the average number of typing errors as well as their variances decreases. Consequences of heteroscedasticity Following are the consequences of the heteroscedasticity: 1. Heteroscedasticity does not result in biased parameter estimates. However, OLS estimates are no longer BLUE. That is, among all the unbiased estimators, OLS does not provide the estimate with the smallest variance. Depending on the nature of the heteroscedasticity, significance tests can be too high or too low. 2. In addition, the standard errors are biased when heteroscedasticity is present. This in turn leads to bias in test statistics and confidence intervals. 3. Fortunately, unless heteroscedasticity is “marked,” significance tests are virtually unaffected, and thus OLS estimation can be used without concern of serious distortion. But, severe heteroscedasticity can sometimes be a problem. Warning: Note that heteroscedasticity can be very problematic with methods besides OLS. For example, in logistic regression heteroscedasticity can produce biased and misleading parameter estimates.
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  Muhammad Ali Lecturer inStatistics GPGC Mardan. 4 OLS estimation in presence of heteroscedasticity If we introduce heteroscedasticity by letting that E( 22 ) ii δε = but retain all other assumptions of the classical model the OLS estimates are still unbiased. Consider the two variable regression model. iii XY εββ ++= 10 We know that the ordinary least square estimate of β1 is: A x x x Xx x xXx x xXxx x Xx x xY x Yx xYYx x yx i ii i ii i iiii i iiiii i iii i i i ii iii i ii −−−−−− ∑ ∑ + ∑ ∑ = ∑ ∑+∑ = ∑ ∑+∑+∑ = ∑ ++∑ = ∑ ∑ − ∑ ∑ = ∑−∑= ∑ ∑ = 22 1 2 1 2 10 1 2 10 1 221 2 1 21 ˆ )(ˆ ˆ /)(ˆ ˆ εβ εβ εββ β εββ β β β β Now 1 )()( )( )()( )( 2 = −∑−−∑ −∑ = −∑−∑ −∑ = ∑ ∑ XXXXXX XXX XXXX XXX x Xx iii ii ii ii i ii
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  Muhammad Ali Lecturer inStatistics GPGC Mardan. 5 Put this value in equation (A) Similarly 00 )ˆ( ββ =E It is shown that in the presence of heteroscedasticity the OLS estimators are unbiased. Variance of OLS estimator in the presence of heteroscedasticity Since [ ] [ ] [ ] 2 2 1 2 22 2 22 222 2 2 2 2 1 2 1 222 2 2 2 2 1 2 1 i 112121 222 2 2 2 2 1 2 11 2i 2 i 2 121 2 11 )ˆ( )( ...w )(...)()(w 0)E(thatknowwebecausezerotoequalsrmproduct tecrossThe ......)ˆVar( wAswE resultpreviousUsing ˆ)ˆ( i i i i i ii nn nn j nnnnnn i i i i ii x Var x x w ww EwEwE wwwwwwwE x x x x E EVar ∑ = ∑ ∑ =∑= ++= ++= = ++++++= ∑ =∑=         − ∑ ∑ += −= −− δ β δδ δδδ εεε εε εεεεεεεβ ε β ε β βββ
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  Muhammad Ali Lecturer inStatistics GPGC Mardan. 6 Which is different when Homoscedasticity is present in the model. Tests for Detection of Heteroscedasticity The following tests to be used for detection of multicollinearity: 1. Park Test Park test suggest that δ2 i is some function of the explanatory variable Xi. i.e. iiXXu as iX eX iiiii iii ii i −−−−−−−−−−++=++= −−−−−−−−−−−++= = υβαυβδ δ υβδδ δδ υβ lnlnlnˆln .regressionfollowingtherunningandproxyauˆusingsuggestpark,unknownisSince lnlnln 22 2 i 2 i 22 22 If β found statistically significant in the above equation then it means that heteroscedasticity is present in the data, otherwise we may accept the assumption of Homoscedasticity. The Park test is thus a two-stage procedure. In the first stage we run the OLS regression disregarding the heteroscedasticity question. We obtain iuˆ from this regression, and then in the second stage we run the regression (ii).
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  Muhammad Ali Lecturer inStatistics GPGC Mardan. 7 2. Glejsar Test Glejsar test is much similar to Park test. After obtaining residuals iuˆ from the OLs regression Glejsar suggest regressing the absolute of the iuˆ on the X variable that is thought to be closely associated with δ2 i . Glejsar used the following functional form: ˆ ˆ 1 ˆ 1 ˆ ˆ υXββuˆ 2 21 21 21 21 21 ii21i iii iii i i i ii iii Xu Xu X u X u Xu υββ υββ υββ υββ υββ ++= ++= ++= ++= ++= ++= Where υi is the error term.
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  Muhammad Ali Lecturer inStatistics GPGC Mardan. 8 Goldfeld and Quandt point out that the error term vi has some problems in the above expressions. • Its expected value is not equal to zero. • It is serially correlated. • The last two expression are not linear in parameters and therefore cannot be estimated with the usual OLS procedure. 3. Spearman's Rank Correlation Test. The well known spearman's rank correlation coefficient is given by the following formula. ( )        − ∑ −= 1 61 2 2 nn d r i s Where d= difference between two rankings and n= number of individuals. The above spearman's rank correlation coefficient can be used to detect heteroscedasticity. The procedure for Spearman's rank correlation coefficient is as follows: i. Fit the regression line on Y and X and find the residuals. ii. Rank the residuals by ignoring their sign. iii. Rank either the value of X or Y. iv. Find difference between two rankings(di). v. Apply the following test statistic to test the hypothesis that the population rank correlation coefficient ρi = 0 and n > 8 i.e.
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  Muhammad Ali Lecturer inStatistics GPGC Mardan. 9 freedomofdegree2-n'with 1 2 2 s s r n rt − − = If the computed value of t exceeds than the tabulated value then we may accept the hypothesis of heteroscedasticity; otherwise we may reject it. 4. Goldfeld-Quandt Test This test is suggested if the heteroscedasticitic variance δ2 i is positively related to one of the predictor variables in the regression model. Consider the two-variable regression model: iii XY εββ ++= 21 Suppose that δ2 i is positively related to X as: δ2 i=δ2 Xi 2 Now to test the hypothesis that there is no heteroscedasticity we will follow the following steps. Step#1. Rank the observations beginning with the lowest value of X. Step#2. Omit 'c' central observations where 'c' is fixed in advance, and then divide the remaining observation into two groups. Step#3. Fit the OLS regression model to both groups and obtain sum of square of regression i.e. RSS1 and RSS2. RSS1 representing the RSS to the smaller
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  Muhammad Ali Lecturer inStatistics GPGC Mardan. 10 variance groups and RSS2 representing the RSS to the larger variance group. Both RSS1 and RSS2 having the same degrees of freedom. i.e. ( ) 2 2k-c-n or 2       − − k cn Where k is the number of parameters to be estimated. In two variable case k=2 Step#4 Compute the ratio dfRSS dfRSS / / 1 2 =λ If the error term ε is normally distributed i.e. ε~N(0,δ2 ) then λ follows the F distribution with 2/2and2/2 21 kcnkcn −−=−−= υυ degrees of freedom. If the computed value of λ is greater than the tabulated value of F then we can reject the hypothesis of Homoscedasticity.