11.1 Heteroskedasticity: Nature and Detection

11.2 Aims and Learning Objectives By the end of this session students should be able to: Explain the nature of heteroskedasticity Understand the causes and consequences of heteroskedasticity Perform tests to determine whether a regression model has heteroskedastic errors

11.3 Nature of Heteroskedasticity Heteroskedasticity is a systematic pattern in the errors where the variances of the errors are not constant. Ordinary least squares assumes that all observations are equally reliable.

11.4 Y i =   +   X i + U i Regression Model Var(U i ) =   2 Homoskedasticity: Heteroskedasticity: Var(U i ) =  i  2 Or E(U i 2 ) =   2 Or E(U i 2 ) =  i  2

11.5 Homoskedastic pattern of errors XiXi YiYi Income Consumption

XiXi X1X1 X2X2 YiYi f(Y i ) The Homoskedastic Case.. X3X3 X4X4 Income Consumption

11.7 Heteroskedastic pattern of errors XiXi YiYi Income Consumption

11.8. X iX i X1X1 X2X2 YiYi f(Y i ) Consumption X3X3.. The Heteroskedastic Case Income rich people poor people

11.9 Causes of Heteroskedasticity Direct Scale Effects Structural Shift Learning Effects Common Causes Indirect Omitted Variables Outliers Parameter Variation

Ordinary least squares estimators still linear and unbiased. 2. Ordinary least squares estimators not efficient. 3. Usual formulas give incorrect standard errors for least squares. 4. Confidence intervals and hypothesis tests based on usual standard errors are wrong. Consequences of Heteroskedasticity

11.11 Y i =   +   X i + e i heteroskedasticity: Var(e i ) =  i  2 Formula for ordinary least squares variance (homoskedastic disturbances): Formula for ordinary least squares variance (heteroskedastic disturbances): ^ ^ Therefore when errors are heteroskedastic ordinary least squares estimators are inefficient (i.e. not “best”)

11.12 Detecting Heteroskedasticity e i 2 : squared residuals provide proxies for U i 2 Preliminary Analysis Data - Heteroskedasticity often occurs in cross sectional data (exceptions: ARCH, panel data) Graphical examination of residuals - plot e i or e i 2 against each explanatory variable or against predicted Y

11.13 Residual Plots eiei 0 XiXi.. Plot residuals against one variable at a time after sorting the data by that variable to try to find a heteroskedastic pattern in the data

11.14 The Goldfeld-Quandt Test 1. Sort data according to the size of a potential proportionality factor d (largest to smallest) 2. Omit the middle r observations 3. Run separate regressions on first n 1 observations and last n 2 observations 4. If disturbances are homoskedastic then Var(U i ) should be the same for both samples. Formal Tests for Heteroskedasticity

11.15 H o :  1  2 =  2  2 H 1 :  1  2 >  2  2 The Goldfeld-Quandt Test 5. Specify null and alternative hypothesis 6. Test statistic Compare test statistic value with critical value from F-distribution table

11.16 White’s Test 1. Estimate And obtain the residuals 2. Run the following auxiliary regression: 3. Calculate White test statistic from auxiliary regression 4. Obtain critical value from  2 distribution (df = no. of explanatory variables in auxiliary regression) 5. Decision rule: if test statistic > critical  2 value then reject null hypothesis of no heteroskedasticity

11.17 Summary In this lecture we have: 1. Analysed the theoretical causes and consequences of heteroskedasticity 2. Outlined a number of tests which can be used to detect the presence of heteroskedastic errors