NPTEL Project. Econometric Modelling. Module 14: Heteroscedasticity Problem. Module 16: Heteroscedasticity Problem. Vinod Gupta School of Management
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1 1 P age NPTEL Project Econometric Modelling Vinod Gupta School of Management Module 14: Heteroscedasticity Problem Module 16: Heteroscedasticity Problem Rudra P. Pradhan Vinod Gupta School of Management Indian Institute of Technology Kharagpur, India rudrap@vgsom.iitkgp.ernet 1
2 2 P age 1. MODULE OBJECTIVE This module attempts to explore the possibilities of volatility of error variables. The estimated model by the application of OLS, discussed earlier, is based on the assumption that error variance should be uniform over time/ cross sectional units. That is covariance between two errors variables should equal to a constant [i.e. Cov (U i, U j ) = 0 for i = j]. If this assumption is violated, then it is heteroskedasticity; otherwise, it is the situation of homoskedasticity (i.e. equal error variance). In this module, we deal with the followings: 1. WHAT IS HETEROSKEDASTICITY AND HOW IS ITS NATURE? 2. WHAT ARE ITS CONSEQUENCES? 3. DOES IT REALLY A PROBLEM? 4. DETECTION CRITERIA 5. CAUSES OF HETEROSKEDASTICITY 6. REMEDIAL MEASURES 2
3 3 P age WHAT IS HETEROSKEDASTICITY? In general, heteroskedasticity means unequal error variance. If it is present in the estimated model, it is the violation of OLS technique and hence, the estimated model cannot be used for prediction and forecasting. The structure of heteroskedasticity is as follows: Y t = β 0 + β 1 X t + U t and where CONSEQUENCES OF HETEROSKEDASTICITY The estimation process requires that OLS applications of estimated parameters should follow the BLUE theorem. If not, there is problem on model reliability. In specific, the presence of heteroskedasticity makes the estimated parameters unbiased and their standard errors are relatively high. So, it drastically affects the minimum variance property and hence, affects the model reliability. DOES IT REALLY PROBLEM? On the first instance, any estimated parameters whose value does not follow BLUE theorem means it is really a problem. However, in the case of heteroskedasticity, it depends upon the objective specification. If the objective is model reliability, then it is serious issue, even if it is at the minor level. 3
4 4 P age DETECTION OF HETEROSCEDASTICITY The detection of heteroskedasticity can be done only after the estimation process. So, first we should have estimated model and then we can have the error term. Once we get the error term, the process of detecting heteroskedasticity is feasible. The residuals in case of heteroskedasticity can be calculated by plotting them in the time sequence plot or alternatively we can plot the standardized residuals against time. Apart from these there are several quantitative tests that one can apply in order to supplement the pure qualitative approach. These are as follows: PARK TEST GLESJER TEST SPEARMAN S RANK CORRELATION TEST GOLDFELD QUANDT TEST BREUSCH PAGAN GODFREY TEST WHITE GENERAL TEST Among them, Goldfeld- Quandt test and Spearman Rank Correlation is very popular. So, we briefly highlight these two tests here. 4
5 5 P age GOLDFELD QUANDT TEST In this method we do the following steps: rank the observations from lowest value Omit the central c observations and divide the rest of the observations in two halves Find the RSS of the two sets of observations and RSS of the original model Compute F-stat, which is as follows: It can be further shown that the follows the F distribution and if it is significant, then there is presence of heteroskedasticity; otherwise there is no heteroskedasticity in the system. Spearman Rank Correlation TEST In this test, we first assign rank to error term and any of the variables and then find out the rank correlation. If the correlation coefficient is statistically significant then there is presence of heteroskedasticity; otherwise there is no heteroskedasticity in the system. CAUSES OF HETEROSKEDASTICITY Error improvement model Growth and trend factors Misspecification of the random term An over determined model An under-determined model 5
6 6 P age Omission of variables Natural tendency of the variables Outliers problem Skewness in the distribution Wrong functional form Data improvement REMEDIAL MEASURES As we have seen, heteroscedasticity does not destroy the unbiasedness and consistency properties of the OLS estimators, but they are no longer efficient, not even asymptotically (i.e., large sample size). This lack of efficiency makes the usual hypothesis-testing procedure of dubious value. Documenting the consequences of heteroscedasticity is easier than detecting it. There are several diagnostic tests available, but one cannot tell for sure which will work in a given situation. Even if heteroscedasticity is suspected and detected, it is not easy to correct the problem. If the sample is large, one can obtain White s heteroskedasticity corrected standard errors of OLS estimators and conduct statistical inference based on these standard errors. Otherwise, on the basis of OLS residuals, one can make educated guesses of the likely pattern of heteroscedasticity and transform the original data in such a way that in the transformed data there is no heteroskedasticity. 6
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