Transcript Chapter 6 - Facultypages.morris.umn.edu

CHAPTER 6
REGRESSION DIAGNOSTIC III:
AUTOCORRELATION
Damodar Gujarati
Econometrics by Example
AUTOCORRELATION
 One of the assumptions of the classical linear
regression (CLRM) is that the covariance between
ui, the error term for observation i, and uj, the error
term for observation j, is zero.
 Reasons for autocorrelation include:
The possible strong correlation between the shock in
time t with the shock in time t+1
More common in time series data
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CONSEQUENCES
 If autocorrelation exists, several consequences ensue:
 The OLS estimators are still unbiased and consistent.
 They are still normally distributed in large samples.
 They are no longer efficient, meaning that they are no longer
BLUE.
 In most cases standard errors are underestimated.
 Thus, the hypothesis-testing procedure becomes suspect, since
the estimated standard errors may not be reliable, even
asymptotically (i.e., in large samples).
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DETECTION OF AUTOCORRELATION
 Graphical method
 Plot the values of the residuals, et, chronologically
 If discernible pattern exists, autocorrelation likely a problem
 Durbin-Watson test
 Breusch-Godfrey (BG) test
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DURBIN-WATSON (d) TEST
 The Durbin-Watson d statistic is defined as:
t n
d
 (e  e
t 2
t n
e
t 1
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Econometrics by Example
t 1
t
2
t
)
2
DURBIN-WATSON (d) TEST ASSUMPTIONS




Assumptions are:
1. The regression model includes an intercept term.
2. The regressors are fixed in repeated sampling.
3. The error term follows the first-order autoregressive (AR1)
scheme:
ut  ut 1  vt
where ρ (rho) is the coefficient of autocorrelation, a value between -1
and 1.
 4. The error term is normally distributed.
 5. The regressors do not include the lagged value(s) of the
dependent variable, Yt.
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DURBIN-WATSON (d) TEST (CONT.)
 Two critical values of the d statistic, dL and dU, called the lower and upper
limits, are established
 The decision rules are as follows:




1. If d < dL, there probably is evidence of positive autocorrelation.
2. If d > dU, there probably is no evidence of positive autocorrelation.
3. If dL < d < dU, no definite conclusion about positive autocorrelation.
4. If dU < d < 4 - dU, probably there is no evidence of positive or negative
autocorrelation.
 5. If 4 - dU < d < 4 - dL, no definite conclusion about negative autocorrelation.
 6. If 4 - dL < d < 4, there probably is evidence of negative autocorrelation.
 d value always lies between 0 and 4
 The closer it is to zero, the greater is the evidence of positive
autocorrelation, and the closer it is to 4, the greater is the evidence of
negative autocorrelation. If d is about 2, there is no evidence of positive or
negative (first) order autocorrelation.
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BREUSCH-GODFREY (BG) TEST
 This test allows for:
 (1) Lagged values of the dependent variables to be included as
regressors
 (2) Higher-order autoregressive schemes, such as AR(2), AR(3), etc.
 (3) Moving average terms of the error term, such as ut-1, ut-2, etc.
 The error term in the main equation follows the following AR(p)
autoregressive structure:
ut  1ut 1  2ut 2  ...   put  p  vt
 The null hypothesis of no serial correlation is:
1  2  ...   p  0
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BREUSCH-GODFREY (BG) TEST (CONT.)
 The BG test involves the following steps:
 Regress et, the residuals from our main regression, on the regressors in
the model and the p autoregressive terms given in the equation on the
previous slide, and obtain R2 from this auxiliary regression.
 If the sample size is large, BG have shown that: (n – p)R2 ~ X2p
 That is, in large samples, (n – p) times R2 follows the chi-square distribution with
p degrees of freedom.
 Rejection of the null hypothesis implies evidence of autocorrelation.
 As an alternative, we can use the F value obtained from the auxiliary
regression.
 This F value has (p , n-k-p) degrees of freedom in the numerator and
denominator, respectively, where k represents the number of parameters in the
auxiliary regression (including the intercept term).
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REMEDIAL MEASURES
 First-Difference Transformation
 If autocorrelation is of AR(1) type, we have: ut  ut 1  vt
 Assume ρ=1 and run first-difference model (taking first difference
of dependent variable and all regressors)
 Generalized Transformation
 Estimate value of ρ through regression of residual on lagged
residual and use value to run transformed regression
 Newey-West Method
Generates HAC (heteroscedasticity and autocorrelation
consistent) standard errors
 Model Evaluation
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Econometrics by Example