Detecting and Resolving Heteroscedasticity in time series model

  Detecting and Resolving Heteroscedasticity A good start should be made by first defining the words homoscedasticity and heterosc...


 Detecting and Resolving Heteroscedasticity



A good start should be made by first defining the words homoscedasticity and heteroscedasticity. Both words can be split into two parts, having as a first part, the Greek words homo(which means same or equal) or hetero( which means different or unequal), and the second part, the Greek word skedastic(which means spread or scatter) . So, homoscedasticity means equal spread, and heteroscedasticity on the other hand means unequal spread. In econometrics, the measure we usually use for spread is  the variance, and therefore heteroscedasticity deals with unequal variance.

One of the assumption of OLS is that the error should be homoscedastic. The violation of this assumption is called heteroscedasticity.


Var(ut)=ơ2   for all observations. That is, the variance of the error term is constant (Homoscedasticity) over the sample period. If the error terms do not have constant variance, they are said to be heteroscedastic.


Heteroscedasticity can be given in terms of income and expenditure patterns. People with low levels of income do not have much flexibility in spending their money. A large proportion of their income will be spent on buying food, clothing and transportation; at low levels of income, consumption patterns will not differ much and the spread will be more or less low. On the other hand, rich people have a much wider choice and flexibility in spending. Some might consume a lot, some might be large savers or investors in the stock market. So the spread for high incomes will be definitely higher than that of lower incomes




Errors may increase as the value of an independent variable increases. Annual family expenditures for education differ among rich and poor families or educated and uneducated families. A research may include two income group of families may face heteroscedasticity problem.


A research may include two income group of families may face heteroscedasticity problem.


Heteroscedasticity is occurred, If;


- There are subpopulation differences or other interaction effects. For instance, the effect of education in employment differs for villagers and urban parts.


- There are model misspecifications. For instance, instead of using Y, using the log of Y or instead of using X, using X2.


- There are omitted variables. Omitting the important variables from the model may cause bias. In the correctly specified model, the patterns of heteroscedasticity are expected disappear.




If the plot of residuals shows some uneven envelope of residuals, so that the width of the envelope is considerably larger for some values of than for others, a more formal test for heteroscedasticity should be conducted.




H0 : the variance of the error term is constant (homoscedastic)


H1 : the variance of the error term is heteroscedastic

The aim of this section is to examine the consequences of heteroscedasticity, to present tests for detecting heteroscedasticity in econometric model as well as to show the way of resolving heteroscedasticity.

Consequences


A general approach
Consider a classical linear regression model:

Y= A0 + A1X1 + A2X2 + A3X3 +…+ AKXK  +  U  [1]
If the error term U in the equation above is known to be heteroscedastic, then the consequences on the OLS estimator can be summarized as follows:

·         Heteroscedasticity does not result in unbiased parameter estimates. This is because none of the explanatory variable is correlated with the error term. A correctly specified equation that suffers from heteroscedasticity still gives parameters which are relatively good.


·          OLS estimates are no longer BLUE. That is, among all of 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. The standard errors are biased when heteroscedasticity is present. This in turn leads to bias in test statistics and confidence interval. Therefore the test statistics are no more reliable.

Detection


In order to choose which efficient method to use, we have to test for the presence of heteroscedasticity. Let’s examine some of the informal and formal methods of detecting heteroscedasticity. Most of the method used follows the examination of the OLS residual Ûi since they are what we observe, and not the disturbance Ui. The hope is that they are good estimate of Ui and may be fulfilled in sample size that is fairly large.




THE INFORMAL WAY


It can be inspected through visual inspection. Do a visual inspection of residuals plotted against fitted values; or, plot the independent variables suspected to be correlated with the variance of the error term. It is often helpful to plot the Ûi2 against the fitted (Ŷ) as well as scatter of these variables against the independent variables Xi.


The idea is to find out whether the Ŷ or X is systematically related to the squared residual. If there is no systematic pattern among these variables, one can suggest that there is no presence of heteroscedasticity and the model is healthy. However, there is evidence of heteroscedasticity in the model if these variables exhibit a systematic pattern (e.g linear or quadratic relationships). The knowledge of the relationship between these variables is useful because it enables us to transform the data such that the transformed data eliminate the heteroscedatic disturbance.




THE FORMAL WAY


There are various tests for heteroscedasticity but in this section, the test are limited the ones mostly use by expert. We will apply each test using EVIEWS software


The Breusch-Pagan Test

Proposed by Breusch and Pagan(1980). It is a Lagrange multiplier test for heteroscedasticity and designed to detect any linear form of heteroscedasticity. It is sensitive to the normality assumption of OLS.


It tests the null hypothesis that the error variances are all equal versus the alternative that the error variances are a multiplicative function of one or more variables. Consider the following model:






Yi= A1 + A2X2 i+ A3X3 i+…+ AKXKi  +  Ui     [2]


Where var(Ui)= ơi2. The test procedure is as follows:


Step 1: Run a regression using OLS and obtain the residual Ûi of the equation.




Step 2: Estimate the auxiliary regression: Ûi on the explanatory variables.




Ûi2 = B1 + B2Z2 i+ B3Z3i+…+ BPZPi  +  ei         [3]


ZPi is a set of variables that we think determine the variance of the


error term. Note Zpi=Xpi.




Step 3: Formulate the null and the alternative hypotheses. The null hypothesis of


homoscedasticity is that:


H0: B1= B2= B3=…=BP= 0       


While the alternative hypotheses is that at least one of the B’s is different from zero and


that at least one of the Z’s affects the variance of the residuals which will make it heteroscedastic.




Step 4: Compute the LM = nR2 statistic, where n is the number of observations used in


order to estimate the auxiliary regression in step 2, and R2 is the coefficient of


determination of this regression. The LM statistic follows the x2 distribution


with p - 1 degrees of freedom.




Step 5: Reject the null and conclude that there is significant evidence of heteroscedasticity


when LM -statistical is bigger than the critical value (LM -stat >


X2p-1,ɑ). Alternatively, compute the p value and reject the null if the p value is


less than the level of significance ɑ (usually ɑ = 0.05).

Application in EVIEWS

Let’s consider series Y with a monthly data specification, range from 2001m01 to 2009m04 having 100 observation. We will take a sample from 2001m01 to 2007m12 (84 0bs).


Using  AR(1) process where Y depend on the lag of  itself by one period.



At the menu we click QUICK>>ESTIMATE EQUATION>> then type in the series y c y(-1)>>OK






At the regression output we click VIEW>>RESIDUAL DIAGNOSTICS>> HETEROSKEDASTICITY TEST


Using BREUSCH-PAGAN-GODFREY TEST>> OK


H0: There is no heteroscedasticity


H1: There is heteroscedasticity


You can simply put, if the prob. chi square close to the obs*R-squared is >0.05, we do not reject the H0 and conclude that the model is homoscedastic (constant variance). If the prob. chi square close to the obs*R-squared is < 0.05, we reject the H0 and conclude that there is an evidence of heteroscedasticity.


INTERPRETATION


From the result, we will use the prob. Chi square value since the Chi square critical value is not given. The test statistic is Obs*R-squared =14.04403 and the value of the prob. Chi square of 0.0002 is <0.05 or 5% level of significance. We will reject the H0 of “no heteroscedasticity or homoscedasticity”. Therefore the model is heteroscedastic.


Test for ARCH Effect

The ARCH test belongs to the family of the LM test. It is common for financial variables. Engel (1982) detected that large and small forecast errors tend to occur in clusters so that the conditional variance of error term is the autoregressive function of the past errors. Ignoring ARCH effect can result in inefficiency of the estimation. ARCH (q) effect can be written as follows:




e2t= A0 + A1 e2t-1 + A2e2t-2 + A3 e2t-1 +…+ AKe2t-K  +  Ut  [4]


This is a test for ARCH (q) versus ARCH (0).


The null hypothesis of homoscedasticity


H0: A1=A2=…=AK=0


While the alternative hypotheses is that at least one of the A’s is different from zero, which makes it heteroscedastic.


Application in EVIEWS


Using an AR(1) process where Y depend on the lag of  itself by one period.






At the menu we click QUICK>>ESTIMATE EQUATION>> then type in the series y c y(-1)>>OK




At the regression output we click VIEW>>RESIDUAL DIAGNOSTICS>> HETEROSKEDASTICITY TEST



Using ARCH TEST >> we put 12 as the number of lags  specify because we are using monthly data>> OK



H0: There is no heteroscedasticity


H1: There is heteroscedasticity


You can simply put, if the prob. chi square close to the obs*R-squared is >0.05, we do not reject the H0 and conclude that the model is homoscedastic (constant variance). If the prob. chi square close to the obs*R-squared is < 0.05, we reject the H0 and conclude that there is an evidence of heteroscedasticity.


INTERPRETATION


From the result, we will use the prob. Chi square value since the Chi square critical value is not given. The test statistic is Obs*R-squared =25.63375 and the value of the prob. Chi square of 0.0121 is <0.05 or 5% level of significance. We will reject the H0 of “no heteroscedasticity or homoscedasticity”. Therefore the model is heteroscedastic.





White Test for Heteroscedasticity (White, 1980)


White (1980) developed a more general test for heteroscedasticity that eliminates the problems that appeared in the previous tests. White's test is also an LM test, but it has


the advantages that it does not assume any prior knowledge of heteroscedasticity, it does not depend on the normality assumption as the Breusch-Pagan test.


It is actually a special case of Breusch-Pagan test. It involves an auxiliary regression of squared residuals, but excludes any higher order terms. It is very general. If the number of the observation is small, power of the white test becomes weak. It can be performed by obtaining least squares residuals and modeling the square residuals as a multiple regression which includes independent variables and their squares and second degree products (interaction term). Consider the following three variables.


Yi= A1 + A2X2 i+ A3X3 i+ Ui    [5]


Step 1: Given the data, run a regression model of the above equation and obtain the residuals Ûi..




Step 2: estimate the following auxiliary regression:


Ûi2.= B1 + B2X2 i+ B3X3i + B4X2i2 + B5X3i2 + B6X2X3 + ei  [6]


In essence, regress the squared residual from the original regression on constant, all the original explanatory variables X, the squared explanatory variables, and the cross product(s) of the explanatory variables.




Step 3: Formulate the null and the alternative hypotheses. The null hypothesis of homoscedasticity is that:


H0:  B2=B3= B4= B5=B6=0




Step 4: Compute the LM = nR2 statistic, where n  is the number of observations used in order  to estimate the auxiliary regression in step 2, and R2 is the coefficient of determination of this regression. The LM statistic follows the chi-square distribution with 6- 1 degrees of freedom( df equal to the number of regressors excluding the constant term in the auxiliary regression).




step 5:  Reject the null and conclude that there is significant evidence of heteroscedasticity


when LM -statistical is bigger than the critical value (LM -stat >


X2p-1,ɑ). Alternatively, compute the p value and reject the null if the p value is


less than the level of significance ɑ (usually ɑ = 0.05).


White's test is more general, and because of its merits, it is recommended over all the previous tests, although one practical problem is that  If a model has several explanatory variables, then introducing all the explanatory variables, their squared (or higher powered) terms, and their cross products can quickly consume degrees of freedom. If cross product terms are absent in the White test, then it is a test of pure heteroscedasticity. Introduction of cross product term lead to a test for heteroscedasticity and misspecification.


Application in EVIEWS

Using an AR(1) process where Y depend on the lag of  itself by one period



At the menu we click QUICK>>ESTIMATE EQUATION>> then type in the series y c y(-1)>>OK










At the regression output we click VIEW>>RESIDUAL DIAGNOSTICS>> HETEROSKEDASTICITY TEST










Using WHITE TEST>> excluding white cross term>>OK





H0: There is no heteroscedasticity


H1: There is heteroscedasticity


You can simply put, if the prob. chi square close to the obs*R-squared is >0.05, we do not reject the H0 and conclude that the model is homoscedastic (constant variance). If the prob. chi square close to the obs*R-squared is < 0.05, we reject the H0 and conclude that there is an evidence of heteroscedasticity.


INTERPRETATION


From the result, we will use the prob. Chi square value since the Chi square critical value is not given. The test statistic is Obs*R-squared =16.39008 and the value of the prob. Chi square of 0.0001 is <0.05 or 5% level of significance. We will reject the H0 of “no heteroscedasticity or homoscedasticity”. Therefore the model is heteroscedastic.



RESOLVING HETEROSCEDASTICITY



If we find heteroscedasticity present, there are two ways to remediation: (i) The weighted least squares (WLS). (ii) White’s heteroscedasticity-consistent variances and standard errors. Weighted least squares is a difficult option but more superior when one makes it work (see Basic econometrics by Gujarati or Applied econometrics: A modern approach by Dimitrios Asterious and Stephen G.Hall).




A common way to eliminate the problem of heteroscedasticity is to transform the variables i.e using logarithms of all variables. Variables that have negative or zeroes values should not be log.




Application in EVIEWS


Using an AR(1) process where Y depend on the lag of  itself by one period.




Under the command interface create a new series lnY (transforming Y into logY). Type series lny=log(y) to generate a new series.






At the menu we click QUICK>>ESTIMATE EQUATION>> then type in the series lny c lny(-1)>>OK.









At the regression output we click VIEW>>RESIDUAL DIAGNOSTICS>> HETEROSKEDASTICITY TEST









Using WHITE TEST>> excluding white cross term>>OK. White test is because it’s a general test and it has advantage over others








H0: There is no heteroscedasticity


H1: There is heteroscedasticity


You can simply put, if the prob. chi square close to the obs*R-squared is >0.05, we do not reject the H0 and conclude that the model is homoscedastic (constant variance). If the prob. chi square close to the obs*R-squared is < 0.05, we reject the H0 and conclude that there is an evidence of heteroscedasticity.


INTERPRETATION


From the result, we will use the prob. Chi square value since the Chi square critical value is not given. The test statistic is Obs*R-squared =1.331963 and the value of the prob. Chi square of 0.2485 is >0.05 or 5% level of significance. We do not reject the H0 of “no heteroscedasticity or homoscedasticity”. Therefore the model is homoscedastic.






CONCLUSION



The assumption of a classical linear regression model is that the errors have equal or constant variance. If the assumption is not satisfied, heteroscedasticity exist in the model. Heteroscedasticity does not affect the unbiasedness and consistency properties of the OLS estimators but these estimators are no longer efficient.

REFERENCES


Breusch,T.S & Pagan,A.R.(1980). The Lagrange Multiplier Test and Its Applications to Model Specification in Econometrics. Review of Economic studies 47(1). p.239-253

Dimitrios Asterious & Stephen G.Hal (2007). Applied econometrics: A modern approach. Palgrave Macmillan.

Engle, R.F.(1982). Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation, Econometrics. 50(4).p.987-1007.

 Gujarati D.N(2004). Basic Econometrics, 4th Edition. The McGraw−Hill Companies. P 387-420.


Uğur ERGÜN & Ali Göksu (2013). Applied Econometrics With Eviews Applications. Sarajevoe International Burch University. P 90-150.

White,H.(1980). A Heteroscedasticity- Consistent Covariance Matrix Estimator and a Direct Test for Heteroscedasticity. Econometrica. 48.p.817-838.

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