Heteroskedasticity PDF

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HETEROSKEDASTICITY: Introduction: In the classical linear regression model one of the assumptions is that, the error µ has the same variance given any value of the explanatory variable. i.e. 𝑣𝑎𝑟 (𝑢|𝑥) = 𝜎 2 ( Wooldridge, 2015. p. 52). Stated differently, the variance of the unobserved error, u, conditional on the explanatory variables, is constant. This assumption is very central to linear regression models and it suggests homoscedasticity. This assumption fails whenever the variance of the unobserved factors changes across different segments of the population, where the segments are determined by the different values of the explanatory variables. When this assumption fails, the variance of the error term is not constant and the outcome is Heteroskedastic. Heteroskedasticity is thus a violation of the homoscedastic assumption in the classical linear regression model.

The Heteroskedasticity Problem Heteroskedasticity is a systematic pattern in the errors where the variances of the errors are not constant. If heteroskedasticity is present in a regression, for instance (Yi = βi + β1 Xi + ui ) it implies that the variance of the independent variable (Yi) increases as (Xi ) increases. Stated differently the variance of the error is a function of the independent variables [var(ui |xi ) = σ2 h(Xi )]. For example, in a food expenditure equation, heteroskedasticity is present if the variance of the unobserved factors (food expenditure) increases with income. Also in a model where family expenditure depends on the income of the family, Families with low incomes will spend relatively little on vacations, and the 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 be greater variability among such families, resulting in heteroskedasticity.In order to satisfy the regression assumptions and be able to trust the results, the residuals should have a constant variance. Thus, the presence of heteroskedasticity is a problem.

Reasons for Heteroskedasticity: There are several reasons when the variances of error term (𝜇𝑖 ) may be variable, some of which are: 1. The Error-learning model: This model suggests that as people learn, their errors of behaviour becomes more consistent or smaller overtime. It means that over time the standard deviation (𝜎𝑖2 ) decreases. This causes variations in the independent variable and thus heteroskedasticity. 1

2. As incomes grow, people have more discretionary income and hence more scope for choice about the disposition of their income. Thus (𝜎𝑖2 ) is likely to increase with the level of income. This explains why dividend payout I big companies is expected to be larger than in smaller companies 3. Improvement in Data Collection Technique: As data collecting techniques improves, (𝜎𝑖 2 ) is likely to decrease. . This causes variations in the independent variable overtime of collecting data and thus heteroscedasticity. 4. Outliers: This is an observation that is much different (either very small or very large) in relation to the observations in the sample. Inclusion or exclusion of such observations can alter the regression outcome. 5. Misspecification: This stems from omitted significant variables from the model. Omission of such significant variables often result in residuals that do not give accurate conclusions. 6. Skewness: Skewness in the distribution of one or more explanatory variable included in the model can cause heteroskedasticity. Examples is an economic variable such as income and wealth. The distribution of income and wealth in most societies is uneven, with the bulk of the income and wealth being owned by a few at the top. 7. Transformation Error: Heteroskedasticity can occur then models transformation is not done correctly. This can be in the form of data transformation such as ratio or in functional forms such as log-linear. Consequence of Heteroskedasticity 

Heteroskedasticity does not result in biased parameter estimates.

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OLS estimators not efficient

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Confidence intervals and hypothesis testing are for efficient for conclusions

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In addition, the standard errors are biased when heteroscedasticity is present. This in turn leads to bias in test statistics and confidence intervals.

Consequences of Heteroscedasticity for the Least Square Estimator (OLS) The implications for the violation of the classical assumption 𝑣𝑎𝑟 (𝑢|𝑥) = 𝜎 2 is that: 1. The least squares estimator is still a linear unbiased estimator, but it is no longer BEST. i.e the least square fails to be the BLUE. (Hill, Griffiths & Lim, 20018. p. 302) That means that, 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. OLS is not optimal when heteroscedasticity is present because, it gives

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equal weight to all observations when, in fact, observations with larger disturbance variance contain less information than observations with smaller disturbance variance.

2. The standard errors usually computed for the least squares estimator are incorrect. Confidence intervals and hypothesis tests that use these standard errors may be misleading. (Hill, Griffiths & Lim, 20018. p. 302) In the presence of heteroscedasticity, the variances of OLS estimators are not provided by the usual OLS formulas. If standard errors are biased, it affects results for significance test and confidence intervals thus leading to incorrect conclusions. The implication of this is that, if we still use the OLS estimator in the face of heteroscedasticity, our standard errors would be inappropriate and there is a greater probability of making misleading inferences.

Detection of Heteroscedasticity Heteroskedasticity can be detected using two approaches

Formal Methods 

The Park Test

According to the park test, there is heteroskedasticity if the β is statistically significant after running a second stage regression on the estimated residual 𝑙𝑛𝜇 𝑖2 = 𝑙𝑛𝜎 2 + 𝛽𝑙𝑛𝑋𝑖 + 𝑣𝑖 = 𝛼 + 𝛽𝑙𝑛𝑋𝑖 + 𝑣𝑖 

The White Test

Using the White test, if the test statistic is greater than the critical value, we reject the null hypothesis that there is no heteroskedasticity. First, we estimate the regression model and obtain its residuals

𝑌 = 𝛽1 + 𝛽2𝑖 + 𝛽3 𝑋3𝑖 + 𝑢𝑖

2 + 𝛼6 𝑋2𝑖 𝑋3𝑖 + 𝑣𝑖 Run an auxiliary regression: 𝜇 𝑖2 = 𝛼1 + 𝛼2 𝑋2𝑖 + 𝛼3 𝑋3𝑖 + 𝛼4 𝑋22𝑖 + 𝛼5 𝑋3𝑖

Calculate the white test statistic : 𝑛. 𝑅 2 ~𝑋2 𝑑. 𝑓 Obtain critical value for 𝑋 2 distribution to make conslusion

Informal Methods 

Nature of the problem Heteroskedasticity is more likely in cross sectional data than in time series data. More often, in cross-sectional data involving heterogeneous units, heteroscedasticity may be the rule rather than the exception. For example, in a cross-sectional analysis involving

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the investment expenditure in relation to sales, rate of interest, etc., heteroscedasticity is generally expected if small, medium, and large-size firms are sampled together. 

Graphical Method This is done by plotting a graph of the residuals against fitted values. If there is no a prior information about the nature of heteroscedasticity, in practice one can do the regression analysis on the assumption that there is no heteroscedasticity and then do an after examination of the estimated residual squared µ𝑖2 to see if they exhibit any systematic pattern. An examination of the estimated residual squared will show any of the following patterns. Where the graph does not show any systematic pattern between the two variables as in (1). There is no presence of heteroskedasticity.

Source: Gujurati, 2005

Correcting Heteroscedasticity Where the value of the heteroskedastic error term is known the most appropriate method of correcting the heteroskedasticity is by means of the Weighted Least Squares. This gives BLUE estimates and where the heteroskedastic error term is not known, we can make use of the heteroskedastic-consistent estimation methods.

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Bibliography: Greene, W. H. (2000). Econometric analysis (International edition). Gujarati, D., & Porter, D. (2005). Basic econometrics (5th ed.). Boston: McGraw-Hill. Hill, R. C., Griffiths, W. E., Lim, G. C., & Lim, M. A. (2008). Principles of econometrics (Vol. 5). Hoboken, NJ: Wiley. White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica: Journal of the Econometric Society, 817-838. Wooldridge, J. M. (2015). Introductory econometrics: A modern approach. Nelson Education.

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