Semester 2, 2015/2016

Transcription

1 ECN 3202 APPLIED ECONOMETRICS 5. HETEROSKEDASTICITY Mr. Sydney Armstrong Lecturer 1 The University of Guyana 1 Semester 2, 2015/2016

2 WHAT IS HETEROSKEDASTICITY? The multiple linear regression model can be written: MR5 the xik are not collinear, and for all k. MR6 the values of e are normally distributed (optional) In this lecture we consider models where assumption MR3 is violated. 2

3 WHAT IS HETEROSKEDASTICITY? cont. Assumption MR3 says the errors have equal variances, or equal (homo) spread or dispersion (skedasticity). An alternative and much more general assumption is Assumption MR7 says the errors are heteroskedastic. Heteroskedasticity is often encountered in cross-section studies (i.e., studies of different observations for the same period of time), different individuals may have very different characteristics, which is often reflected in different variances across individuals. It is less common in time-series studies 3

4 WHAT IS HETEROSKEDASTICITY? cont. 4

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6 PROPERTIES OF THE OLS ESTIMATOR If the errors are heteroskedastic (but all the other assumptions of MR1-MR5 hold) then OLS is still unbiased and consistent the variances of the OLS estimators are no longer given by the formulas we discussed in earlier lectures. Thus, the confidence intervals and hypothesis tests based on those variances are no longer valid! OLS would also be inefficient (i.e., it is no longer BLUE!). What can we do to improve the estimation? Need to account for heteroskedasticity! How? 6

7 Variances of the OLS Estimators The variances of the OLS estimators depend on σi2 (rather than σ2). So, in the case of the simple linear model for example, we will have where This is different from var(b2) in homoskedastic case! If we replace σi2 with ei2 we obtain White s heteroskedasticityconsistent estimator of variances of the OLS estimates of β1,β2, Standard errors calculated in this robust standard errors. 7

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9 GENERALISED LEAST SQUARES ESTIMATION If σi2 is known then we can weight the original data (including the constant term) and then perform OLS on the transformed model : Or Note, the transformed model satisfies all the assumptions MR1-MR5 (including homoskedasticity!), e.g.: 9

10 GENERALISED LEAST SQUARES ESTIMATION and, because the transformed model satisfies all the assumptions of the multiple regression model (including homoskedasticity) applying OLS to the transformed model yields best linear unbiased estimates Such estimator is known as Generalised Least Squares (GLS), Because it is the same Least Squares principle but for a more general model or Weighted Least Squares (WLS), because we weight observations by the variances of noise associated with those observations and then apply the same Least Squares principle 10

11 GLS cont. Sometimes σi2 is only known up to a factor of proportionality. In this case, we can still transform the original model in such a way that the transformed model satisfies MR1-MR5. So, its estimates would be BLUE! Popular heteroskedastic specifications (assumptions) are: Note, if our assumptions about the form of heteroskedasticity are incorrect then GLS will yield biased & inconsistent estimates. So, one must be careful in how to transform the model!... 11

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13 Estimating the Variance Function If σi2 is unknown then it must be estimated The resulting estimator is known as Feasible Generalised Least Squares (GLS). A popular in practice specification for the variance is where zik,,zis are some variables we suspect influence σi2. In this case, we then estimate auxiliary model: ei is the OLS estimated error (residual) obtained from the original model of yi on 1,xi2,,xiK then use the variance estimator (or fitted/predicted variance):

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15 A Heteroskedastic Partition Sometimes the structure of variance is such that the variance is the same within a group of observations, but may be different between the different groups group-wise heteroskedasticity. E.g., in case of two groups (call them A and B), we may have: Solution: Estimate σj2 for each group j by applying OLS to each group of observations separately from other groups Use these estimated variances for each group to transform the model (data) for each group separately. Apply OLS to the transformed model (data) in the usual way, using all N observations (all groups together). 15

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19 See tutorial for details 19

20 DETECTING HETEROSKEDASTICITY How can we know if we should account for heteroskedasticity? Use various methods for detecting a presence of heteroskedasticity: Residual analysis If only one variable is suspected to influence variance, use plots to see if those residuals vary in a systematic way relative to the suspected variable: the least squares residuals vs. that variable squared residuals vs. that variable If more than one explanatory variable, one can plot the least squares residuals against each explanatory variable, or, at least, against the fitted values of the dependent variable White s General test Goldfeld-Quandt test Breusch-Pagan test 20

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23 White s General Test To test H0 σi2=σ2 for all i. vs. H1 not H0 we use WG=N R2 ~ χs 12 where R2 is the coefficient of determination in the regression of the e i2 on a constant and all unique explanatory variables contained in X1i, XKi, their squares and their cross-products, while S is the number of coefficients in this equation. Advantage: The test doesn t require any specific assumptions about the form of heteroskedasticity. Disadvantages: It may have low power in rejecting H0 when H0 is false If it rejects H0, it is non-constructive, in the sense that it 23 doesn t tell us what to do next (which variance specification to try ).

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25 Goldfeld-Quandt Test To test H0 σj2=σ2 for all j. vs. H1 not H0 Procedure for the case of two groups: (H0 σ12=σ22) (1) we split the sample into two approximately equal subsamples of size N1 and N2 so that: observations with potentially high variances are in subsample 1, those with potentially low variances are in subsample 2. (2) we then estimate the model separately using each subsample and calculate σ 12 and σ 22, and use them to get test statistic: Remark: The power of the test can be increased by omitting about one third of the observations in the middle (if the original sample size permits so by being relatively large) 25

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27 Breusch-Pagan Test Here we hypothesise that σi2=h(α1+α2zi2+ +αszis) And so testing for heteroskedastisity H0 σi2=σ2 for all i vs. H1 not H0. We choose a functional form for h(α1+α2zi2+ +αszis) and estimate it in a usual way, via OLS, and then compute: where R2 is the coefficient of determination in the regression of the e i2 on a constant and the variables that we suspect may influence the variance, which we denoted by z2, zs these could be some or all of explanatory variables, their interactions or other variables 27

28 Breusch-Pagan Test cont. Remark: the BP test for heteroskedasticity is essentially the test of significance of the model of regression of squared residuals on the variables we suspect may influence the variance. Disadvantage of the BP Test: One must specify a particular functional form to implement it, A common choice is linear form of h, i.e., Advantage of the BP Test: For large samples, it usually has good power in rejecting H0 when it is false If it rejects H0, it is constructive, in the sense that it does suggest which variance specification to try for FGLS The BP test statistic for the linear specification of h is 28 valid (approximately, for large samples) even if h is not linear!...

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