Transcript
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