chap12_2

Econometrics

Chapter 12 – Autocorrelation

Autocorrelation In the classical regression model, it

is assumed that E(utus) = 0 if t is not equal to s.

What happens when this assumption is violated?

First-order autocorrelation1 1 2 2

where:t o t t k kt tY X X X u

1

2 2

( ) 0( ) 0 for

( )

1

t t t

t

t s

t

u uEE t s

E

Positive first-order autocorrelation

Negative first-order autocorrelation

Incorrect model specification and apparent autocorrelation

Violation of assumption of classical regression model

1 1

21 1

2

( ) [( )( )]

( )

0

t s t t t

t t t

u

E u u E u u

E u u

1corr( )t tu u

Consequences of first-order autocorrelation OLS estimators are unbiased and

consistent OLS estimators are not BLUE Estimated variances of residuals is

biased Biased estimator of standard errors of

residuals (usually a downward bias) Biased t-ratios (usually an upward

bias)

Detection Durbin-Watson statistic

21

2

2

1

( )N

t tt

N

tt

u ud

u

2(1 )d

Acceptance and rejection regions for DW statistic

o

1

H : 0H : 0

AR(1) correction: known 1 1 2 2

1where: t o t t k kt t

t t t

Y X X X uu u

Lagging this relationship 1 period:1 1 1 1 2 2 1 1 1t o t t k kt tY X X X u

Multiplying this by -1 1 1 1 2 2 1 1 1t o t t k kt tY X X X u

With a little bit of algebra: 1 1 1 1 1 2 2 2 1 1 11t t o t t t t k kt kt t tY Y X X X X X X u u

1 1 1 1 1 2 2 2 1 11t t o t t t t k kt kt tY Y X X X X X X

AR(1) correction: known Solution? quasi-difference each variable:

1

1 1 1 1

2 2 2 1

1

t t t

t t t

t t t

kt kt kt

Y Y Y

X X X

X X X

X X X

Regress:1 1 2 2t o t t k kt tY X X X

AR(1) correction: known This procedure provides unbiased and

consistent estimates of all model parameters and standard errors.

If a unit root is said to exist. In this case, quasi-differencing is equivalent to differencing:

1 1 2 2t o t t k kt tY X X X

Generalized least squares This approach is referred to as:

Generalized Least Squares (GLS) GLS estimation strategy:

If one of the assumptions of the classical regression model is violated, transform the model so that the transformed model satisfies these assumptions.

Estimate the transformed model

AR(1) correction: unknown Cochrane-Orcutt procedure:

1. Estimate the original model using OLS. Save the error terms

2. Regress saved error term on lagged error term (without a constant) to estimate

3. Estimate a quasi-differenced version of original model. Use the estimated parameters to generate new estimate of error term.

4. Go to step 2. Repeat this process until change in parameter estimates become less than selected threshold value.

This results in unbiased and consistent estimates of all model parameters and standard errors.

Prais-Winsten estimator Cochrane-Orcutt method involves the

loss of 1 observation. Prais-Winsten estimator is similar to

Cochrane-Orcutt method, but applies a different transformation to the first observation (see text, p. 444).

Monte Carlo studies indicate substantial efficiency gain from the use of the Prais-Winsten estimator (relative to the Cochrane-Orcutt method)

Hildreth-Lu estimator A grid search algorithm – helps

ensure that the estimator reaches a global minimum sum of squared error terms rather than a local minimum sum of squared error terms

Maximum likelihood estimator selects parameter values that

maximize the computed probability of observing the realized outcomes for the dependent and independent variables

an asymptotically efficient estimator

Higher-order autocorrelation AR(p):

1 1 2 2t t t p t pu u u u

Detection of AR(p) error process Breusch-Godfrey test:

1. Estimate the parameters of original model and save error term

2. Regress estimated error term on all independent variables in the original model and the first p lagged error terms

3. Compute the Breusch-Godfrey Lagrange Multiplier test statistic: NR2

4. An AR(p) process is found to exist if the LM statistic exceeds the critical value for a variate with p degrees of freedom.

Use Box-Pierce or Ljung-Box statistic (see p. 451)

Lagged dependent variable as regressor Durbin-Watson statistic is biased

downward when a lagged dependent variable is used as a regressor.

Use Durbin’s h test or Lagrange Multiplier test (test statistic = (N-1)R2 in this case).

Correction: Hatanaka’s estimator (on pp. 458-9 of the text)

Correction of AR(p) process Use Prais-Winsten (modified for an

AR(p) process) or maximum likelihood estimator