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RH demonstrate all LIMDEP cmds in the empirical

examples during the session (by actually

executing them live during the class)

RH demonstrate copying LIMDEP cmds in the Word

file into the text editor so can subsequently

execute those cmds for each of the procedures

- RH
- insert parts of this into 7424 notes to use as

part of current heterosked. PowerPoint slides - add note to first of current 7424 heterosked. PPT

file to see this file for parts to insert into it

for more about heteroskedas. - also use empirical examples as exercises for 7424

Heteroskedasticity LIMDEP 7.0 for Windows

Heteroskedasticity - Overview

- Basics
- definition
- consequences
- when it can occur
- two types
- Detection
- Whites
- Goldfeld-Quandt
- Breusch-Pagan

Overview (cont.)

- Correcting
- Whites HCCM method
- ARCH Models
- brief overview only

Basics - Definition

- HETEROSKEDASTICITY
- ?2 varies over observations
- cross section usually
- (sometimes) scale of Y varies within sample
- HOMOSKEDASTICITY
- ?2 constant over observations

Definition (cont.)

- ?12 0 . . . . . . . 0
- ?2 ? 0 ?22 0 . . . 0 ? ?2 I
- 0 0 0 . . . ?n2
- What do you notice about this covariance matrix?

Y

NO HETEROSKEDASTICITY

X

Want dispersion of DV around regression line

(?2) roughly constant across sample

Regression line

Y

HETEROSKEDASTICITY

X

Common case dispersion of DV around regression

line (?2) not constant

Consequences for OLS

- ? E? X 0
- E? ?X ?2 ? ? ?2 I
- OLS b
- ? b (XX)-1Xy
- ? ? (XX)-1X?
- ? unbiased
- ? consistent
- ? asymptotically normal
- no longer efficient
- usual inference procedures based on F and t

distributions no longer valid

Basics (cont.)

- When can heteroskedasticity occur?
- Examples
- Household income and consumption
- Lower income households
- Little flexibility in spending

When can it occur? (cont.)

- Most of income spent on necessities
- Food
- Shelter
- Clothing
- Transportation
- Cable TV

When can it occur? (cont.)

- Little change in dispersion of consumption around

mean consumption - Small dispersion ?2

When can it occur? (cont.)

- Higher income households
- Much more flexibility in spending
- Once necessities purchased. . .
- much remains to spend in different ways
- Some might be large consumers
- Some might be large savers investors

When can it occur? (cont.)

- Higher income households (cont.)
- Large dispersion of consumption around mean

consumption

When can it occur? (cont.)

- Point
- Lower income households, small dispersion ?2
- Higher income households, larger dispersion ?2

?2

?2

When can it occur? (cont.)

- Large firms vs. small firms in same industry
- same principle for many cases whose units vary

widely in size - States (population)
- Counties (population)
- Colleges (enrollment)
- Etc.

Two Types of Heteroskedasticity

- PURE
- Occurs in the error term of a correctly specified

regression equation. - Usually due to the nature of data.
- IMPURE
- Occurs in the error term of an incorrectly

specified regression equation. - For example, omitting an important variable can

sometimes cause heteroskedasticity

Two Types (cont.)

- Each type fixed differently
- IMPURE
- correct the specification error
- PURE
- use one of the remedies we will cover later

Two Types (cont.)

- GUIDANCE
- Test for heteroskedasticity
- Fix it (if your model has it)
- Test for it again
- Model still have it?
- Specification error causing it
- IMPURE HETEROSKEDASTICITY

Detection of Heteroskedasticity

- Three tests
- Whites
- Goldfeld-Quandt
- Breusch-Pagan/Godfrey

Detection (cont.)

- Vary according to generality and power
- recall power is prob. of rejecting false H0
- generality
- how specific must be about form of

heteroskedasticity - always for all tests
- H0 ?2i ?2 for all i
- HA not H0

Detection (cont.)

- Whites Test
- most general
- does not require prior knowledge of form of

heteroskedasticity - least powerful

Whites Test (cont.)

- Models Used - Preview
- first
- (A) Yt ?1 ?2Xt2 ?3Xt3 ut
- second
- (B) û2t ?0 ?1Xt2 ?2Xt3 ?3X2t2 ?4X2t3

?5Xt2Xt3 ?t

Whites Test (cont.)

- Steps
- Step 1 Estimate (A) by OLS
- Step 2 Compute the residual
- ût Yt - ?1-hat - ?2-hat Xt2 - ?3-hat Xt3,
- and square it.
- LIMDEP automatically calculates residual
- you calculate residual-squared

Whites Test (cont.)

- Step 3 Regress the squared residual û2t against

a constant, Xt2, Xt3, X2t2, X2t3, and Xt2Xt3.

(all combinations of Xs) - This is the auxiliary regression corresponding to

model (B) above - (B) û2t ?0 ?1Xt2 ?2Xt3 ?3X2t2 ?4X2t3

?5Xt2Xt3 ?t

Whites Test (cont.)

- Step 4 Compute the statistic TR2,
- (multiply T times R2)
- where T is the size of the sample
- and R2 is the unadjusted R-squared from the

auxiliary regression in Step 3 (model B)

Whites Test (cont.)

- Reject the null hypothesis that
- H0 ?1 ?2 ?3 ?4 ?5 0
- (H0 is no heteroskedasticity)
- if TR2 gt the upper 5 percent point on the

chi-square distribution with 5 d.f. - (if TR2 gt critical value, then have

heteroskedasticity in model) - d.f. always number of slope ?i in model B

Whites Test (cont.)

- is nonconstructive
- if reject H0 of homoskedasticity
- test gives no guidance about form of

heteroskedasticity

Empirical Example-Whites Test

- See hard copy

LIMDEP Commands

- See hard copy

Detection (cont.)

- Goldfeld-Quandt test
- less general than Whites
- more powerful than Whites

Goldfeld-Quandt (cont.)

- Goal see if ?2 constant across sample
- Premise of test
- Compare estimated ?2 in two parts of sample
- No heteroskedasticity if ?2 equal in both parts
- Heteroskedasticity if ?2 unequal in both parts

Goldfeld-Quandt

- NOTE
- Must have idea of what variable ("Z") associated

with heteroskedasticity - Must have data on that Z

Goldfeld-Quandt (cont.)

- Steps
- Step 1 Order the observations by the size of Z,

the variable thought to be related to the size of

the error term's variance. (example family

income in the earlier example)

Goldfeld-Quandt (cont.)

- Step 2 Omit the middle d observations, where d

may be one-fifth, one-third, or similar. - Max. of one-third
- This must leave TWO EQUAL-SIZED SUBSAMPLES.

Goldfeld-Quandt (cont.)

- d must be small enough to allow sufficient

observations (? 30) in each subsample - If this is impossible, use
- White's test (covered earlier) or
- Breusch-Pagan/Godfrey test (covered next)

Goldfeld-Quandt (cont.)

- Step 3 Estimate one regression model on the

subsample with values of Z associated with a

small variance of the error term. - Call the error sum of squares from this

regression ESS1.

Goldfeld-Quandt (cont.)

- Step 4 Estimate the same regression model on

the subsample with values of Z associated with a

large variance of the error term. - Call the error sum of squares from this

regression ESS2.

Goldfeld-Quandt (cont.)

- Step 5 Form the test statistic ESS2/ESS1.
- This is distributed as an F statistic with (n -

k) degrees of freedom in both the numerator and

the denominator where - n is the size of each subsample
- k is no. of coefficients, including
- intercept

Goldfeld-Quandt (cont.)

- Note
- always form this test statistic with the larger

ESS in the numerator

Empirical Example - Goldfeld-Quandt

- See hard copy
- See end of this presentation for a different

example

LIMDEP Commands

- See hard copy

Detection (cont.)

- Breusch-Pagan/Godfrey test
- Compared to Whites Goldfeld-Quandt
- least general
- most powerful

Breusch-Pagan (cont.)

- Hypothesizes that ?i2 ?2f(?0 ?zi)
- where zi is vector of independent variables
- model homoskedastic if ? 0
- possible forms of f(?0 ?zi)
- (?0 ?zi)
- (?0 ?zi)2
- exp(?zi)

Breusch-Pagan (cont.)

- Step 1
- estimate regression model
- (A) Yt ?1 ?2Xt2 . . . ?KXtK ?t
- Step 2
- form ui ei2 / (ee/N)
- Step 3
- regress ui on zi
- call this model (B)

Breusch-Pagan (cont.)

- Step 4
- form LM statistic as 1/2 of explained sum of

squares from model (B) - LM statistic
- asymptotically distributed as chi-squared
- d.f. number of variables in zi

Empirical Example - Breusch-Pagan

- See hard copy

LIMDEP Commands

- See hard copy

Robust Estimation of Asymptotic Covariance

Matrices

- Recall
- E? ?X ?2 ? ? ?2 I
- b no longer efficient
- source of inefficiency is not knowing ?
- values of elements ?i2
- very common case

Robust Estimation (cont.)

- best hope
- estimate elements of ?
- Whites heteroskedasticity consistent estimator

of ? (HCCM) - gives asymptotic cov. matrix of b

White Heteroskedasticity Consistent Estimator

- Extremely useful
- Can estimate asymptotic cov matrix of b
- (values of elements of ?)
- without knowing nature of heteroskedasticity

Estimation - HCCM (cont.)

- NOTE estimates of elements of ?, ?i2 , used in

estimates of - asymptotic cov matrix of b
- standard errors of b (from cov matrix estimates)
- Hence, can use usual tests based upon OLS results

Estimation - HCCM (cont.)

- (repeating) Can use usual tests based upon OLS

results - t-tests
- ?-hats unbiased consistent even with

heteroskedasticity - Estimated ?-hat variances consistent
- (Square root is denominator of t-test)
- So, t-test OK in large samples

?-hat t ------- s?-hat

Estimation - HCCM (cont.)

- (repeating) Can use usual tests based upon OLS

results - can use F statistic
- although test statistic approximately ? F
- instead of exactly ? F
- good enough

Estimation - HCCM (cont.)

- 3. How do this?
- LIMDEP does it for you!

LIMDEP Commands

- REGRESS lhs name of DV
- rhs list of IV names incl. ONE
- Hetero
- Hetero tells LIMDEP to estimate the revised,

robust cov. matrix - NOTE
- does not change ? estimates

Empirical Example - HCCM

- See hard copy

LIMDEP Commands

- See hard copy
- See previous slide

ARCH Model

- recent analyses of
- exchange rates
- market returns
- inflation
- other financial data
- evidence of clustering of large and small ?t
- variance of forecast error depends on size of

preceding ?t

ARCH Model (cont.)

- AutoRegressive Conditional Heteroskedastic (ARCH)

model - Engle (1982)
- Simple version
- yt xt? ?t
- ?t ut sq. rt.(?0 ?1 ?2t-1)
- u ? N(0,1)
- no autocorrelation

ARCH Model (cont.)

- Var?t ?t-1 ?0 ?1 ?2t-1
- Conditional on ?t-1, ?t heteroskedastic
- Can you explain why? (notes)
- See Econometric Analysis,
- W.H. Greene, (4th ed.), section 18.5 for more

12.6 General Conclusions

- The literature is drifting towards using

Whites HCCM estimator of the elements of

asymptotic cov matrix of b - Advantages
- simple to do
- robust to unknown heteroskedasticity
- usual case is unknown heteroskedasticity

12.6 General Conclusions (cont.)

- Summary
- Testing for heteroskedasticity
- Whites
- Goldfeld-Quandt
- Breusch-Pagan/Godfrey
- Correcting heteroskedasticity
- Whites HCCM estimator

Empirical Example - Goldfeld-Quandt

- Example
- Data
- 1987 state population (millions)
- State expenditures on travel (billions)
- State personal income (billions)
- Data arranged in ascending order of population

POP

TRAVEL

INC

Goldfeld-Quandt Example (cont.)

- NOTES
- Range of travel expends in group 1 1.39
- Suggests possibly smaller variance
- Will drop middle group (2)
- Range of travel expends in group 3 31.41
- Suggests possibly larger variance

- State POP TRAVEL INC
- WY .490 .760 6.3
- AK .524 .857 9.7
- (skip) (skip) (skip) (skip)
- UT 1.680 2.148 19.4
- WV 1.898 1.432 20.9
- (skip) (skip) (skip) (skip)
- WA 4.542 4.469 71.0
- WI 4.807 4.386 70.5
- (skip) (skip) (skip) (skip)
- NY 17.835 17.193 320.0
- CA 27.653 35.797 491.4

17 states

Drop middle group Here drop middle 1/3

17 states

Goldfeld-Quandt Example (cont.)

- Hypotheses (always)
- H0 no heteroskedasticity
- HA heteroskedasticity exists
- Model A TRAVEL ?0 ?1INC ?
- 1 Data arranged according to POP (Z)

Goldfeld-Quandt Example (cont.)

- 2 ignore middle 17 states for remaining steps
- 3 estimate Model A on states Wyoming through

Utah (subsample 1) - ESS1 3.025
- 4 estimate Model A on states Wisconsin through

California (subsample 2) - (1) ESS2 12.657

Goldfeld-Quandt Example (cont.)

- 5 form test statistic
- (FC ESS2 / ESS1
- 12.657 / 3.025 4.18
- So, FC 4.18

Goldfeld-Quandt Example (cont.)

- determine F
- F12,15,.05 2.48 F20,15,.05 2.33
- So (interpolating), F (15,15,.05) 2.42
- Recall that 15 found by n-k where
- n is each subsample's size (17)
- k 2 here

Goldfeld-Quandt Example (cont.)

- 4.18 gt 2.42
- FC gt F, so reject H0
- Heteroskedasticity exists in this sample.