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Chapter 8 Heteroskedasticity

- 1. Introduction

Introduction

- A CLRM assumption that the disturbances ui are

homoscedastic - They all have the same variance.
- So var(ui) ?2 a constant
- Suppose the variance varies across observations
- Then we have heteroskedasticity.
- So var(ui) (?i)2 which varies with each

observation

Example

- Average wages rise with the size of firm. Suppose

wages look like this

Example

- Can we expect the variance of wages to be

constant? - The variance increases as firm size increases.
- So larger firms pay more on average, but there is

more variability in their wages.

Savings Example

- Savings increases with income, so does the

variability of savings or spending - As incomes grow, people have more discretionary

income, so more scope for choice about how to

dispose of it.

Overall

- Heteroskedasticity is more likely in cross

sectional than time-series data.

- 2. Consequences of
- Heteroskedasticity

Consequences

- If we have heteroskedasticity , what happens to

our estimator? - Still linear
- Still unbiased
- Not the most efficient - it does not have minimum

variance. - So it is not BLUE.

Consequences

- If we use usual variance formulas, they will be

biased - This is because the estimator is not an unbiased

estimator of ?2 - So F tests and t tests are unreliable

- 3. Detecting Heteroskedasticity

Detecting Heteroskedasticity

- Past research indicates it
- Know that scale effects will exist
- Ex spending patterns in relation to income
- Firm profitability or investment spending in

relation to the size of the firm.

Detecting Heteroskedasticity

- Examine residuals
- Assume no heteroskedasticity and run OLS and then

look at estimated residuals - In 2-variable model
- Plot squared residuals against the independent

variable. - Plain residual has no correlation with X or Y

Detecting Heteroskedasticity

- In multivariate model, do against different Xs,

or against the predicted value of Y. - Predicted Y is a linear combination of the Xs.
- Graph could show linear or quadratic relationship
- plots provide clues as to the nature of the

heteroskedasticity and how we might transform

variables.

- 4. Park Test

Park Test

- If we find some evidence of heteroskedasticity by

looking at the residuals we can do an explicit

test. - Regress the variance on the X variables.
- Ln(?i)2b1 b2lnXi vi

Park Test

- Dont know the variance
- Use squared residuals as a proxy
- Run ln(ei)2b1 b2lnXi vi
- For a multivariate model, run the squared

residuals against each X variable, or against the

predicted Y. - If b2 is significantly different from 0, then we

have heteroskedasticity.

- 4. Glejser Test

Glejser test

- Similar test to the Park test.
- Regress the absolute values of ei on X.
- The form of the regression may vary
- Can run on square root of X or 1/X etc.
- If significant ts, then heteroskedasticity

- 5. Goldfeld-Quandt Test

Goldfeld-Quandt test

- Order the observations according to the magnitude

of the X thought to be related. - Divide observations into two groups, one with low

values of X and one with high, omitting some

central observations.

Goldfeld-Quandt test

- Run two separate regressions
- Calculate F test
- FESSlarge X/df/ESSsmall X/df
- Should be unity for homoskedasticity.

- 6. Remedial Measures When ? is Known

Remedies

- OLS estimators are not efficient under

heteroskedasticity, though they are unbiased and

consistent. - We can transform the model to get rid of the

heteroskedasticity. - If we know ?2 then use the method of weighted

least squares.

Weighted Least Squares

- Suppose have heteroskedasticity as in the firm

data example - Wages increase with size of firm, but also the

variance increases. - To correct for heteroskedasticity Give less

weight to data points from populations with

greater variability and more weight to those that

have smaller variability.

Weighted Least Squares

- OLS gives equal weight to all observations.
- Weighting observations is called weighted least

squares - A subset of generalized least squares
- Using this method leads to BLUE estimators.

Weighted Least Squares

- Start with basic model
- Yi b1 b2Xi ui
- Y wages and X firm size
- Assume the true error variance (?i)2 is known for

each observation. - Divide through by the standard deviation
- Y/ ?i b1(1/ ?i) b2Xi/ ?i ui/?i

Weighted Least Squares

- Look at the error term.
- Let vi ui/?i
- If is vi homoskedastic then OLS on this model

will give us BLUE estimators - Square vi
- (vi)2 (ui)2/(?i)2
- E(vi)2 E(ui)2/(?i)2

Weighted Least Squares

- Since (?i)2 is known, E(vi)2 becomes 1/(?i)2

E(ui)2 - But E(ui)2 (?i)2
- So this becomes 1/(?i)2 (?i)2 1
- So the transformed error terms is heteroskedastic
- Estimate this model by OLS to get BLUE

estimators. - So WLS is OLS on the transformed variables.

Weighted Least Squares

- In OLS, minimize
- (ei)2 (Y- b1 - b2X2)2
- In GLS, minimize
- w(Y- b1 - b2X2)2
- where w 1/?i

Weighted Least Squares

- In GLS we minimize a weighted sum of squares
- The weights we are using are inversely

proportional to ?i - observations from a population with large

variance will get smaller weights and vice versa.

Weighted Least Squares

- OLS minimizes the sum of squared residuals
- But these residuals are very large for the

observations with a large variance, so it is

giving these more weight. - GLS corrects for this

- 7. Remedial Measures When ? is not Known

Remedies

- We have to make assumptions about (?i)2 if we

dont know it. - Plot residuals against X and find a cone shape
- This indicates that the error variance is

linearly related to X. - Now transform the model by dividing Y, the

intercept, X and the error term by square root of

X.

Remedies

It can be proved that the error variance in this

model is homoskedastic and we can estimate by OLS

(actually its a form of GLS).

Remedies

- Plot residuals against X and find a trumpet shape
- This indicates that the error variance increases

proportional to the square of X. - Now transform the model by dividing Y, the

intercept, X and the error term by X.

Remedies

Slope has become intercept and intercept becomes

slope. But this changes back when we multiply

out by X

Remedies

- Log transformation of both sides
- Log transformation compresses the scales in which

the variables are measured - This reduces the differences.
- Cannot do this if some Y and X values are zero or

negative.