Hetroscedasticity

1
Hetroscedasticity

• Lecture week 5
• Prepared by
• Dr. Zerihun Gudeta

2
OLS and the violation of CLRM assumptions

• The disturbance term has constant variance. It is
  homoscedastic i.e. E(ui2) ?2 where ?2 is a
  constant.
• The Gauss-Markov theorem OLS estimators are
  BLUE.
• Consequence a straightforward OLS procedure
  becomes inappropriate.

3
Causes and nature of heteroscedasticity

• When the variance (or standard deviation) of a
  variable gets related to magnitude.
• The standard deviation will tend to be related
  proportionately to the mean, rather than being
  independent of it.
• A means must be devised to adjust for a changing
  standard deviation or variance.
• For time series data you work with stationary
  series.
• For cross-section data variables exhibit
  different magnitudes and hence some sort of
  adjustment such as converting the data to natural
  logarithms is needed. Variation in such data can
  be measured by the coefficient of variation (CV)
  SD/Mean

4
Implications of heteroscedasticity

• Unbiased and consistent (b/c E(u)0 ).
• Inefficient in the sense that we will be able to
  obtain other estimators with lower variance.
• May underestimates the variances and result in
  larger t-statistics. Thus one may make false
  inferences about the significance of the
  parameters or one might think some variables in
  the equation are significant (important) when
  they are in fact not.

5
Detection of hetroscedasticity

• Examine a variety of graphs to determine whether
  the variance of the residuals is roughly constant
  across the sample and whether the residual
  variance is related to an independent variable
• A scatterplot of the dependent variable against
  each independent variable
• A bar graph of the residuals and the squared
  residuals (a proxy for the disturbance variance)
• A scatterplot of the residuals and/or squared
  residuals against the fitted values of Y
• A scatterplot of the residuals and/or squared
  residuals against the independent variable (s)
• Statistical tests the Park, Glejser and White
  tests.

6
Solutions to hetroscedasticity problem

• If the structure of variance is known,
  Generalized Least Squares (GLS) can be used to
  derive BLUE estimates.
• What GLS does reweighs the observations such
  that those with higher variance are assigned less
  importance when computing the coefficient
  estimates.
• What if the true structure of the variance is
  not known which is true in reality?
• Make some reasoned assumption about the pattern
  of heteroscedasticity usually with respect to its
  relationship to an independent variable i.e. use
  Weighted Least Square (WLS) technique.

7
Example

• Relationship between expenditure on research
  develompent (R_D) and sales.
• Steps
• 1. R_D a bSales

8
Step 1
9
Step 2
10
Step 3