1. Autocorrelation II
Lecture 21
Lecture 21

2. Today’s plan Durbin’s h-statistic Finite Distributed Lags
Koyck Transformations and Adaptive Expectations
Seasonality
Testing in the presence of higher order serially correlated forms.
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3. Returning to the Durbin-Watson
Last time we talked about how to test for autocorrelation using the Durbin-Watson test
We found autocorrelation in the data in L_20.xls
We used this figure: 2 4 H1 dL dU
H0:  =0
Reject null
Accept null
4-dU
4-dL
d = 0.331
1.475
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4. Generalized least squares (3)
Need an estimate of  : we can transform the variables such that:
where:
Known as Cochrane-Orcutt transformation.
Estimating equation (3) allows us to estimate in the presence of first-order autocorrelation
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5. Problems 1) The model presented by may still have some autocorrelation
the D-W test doesn’t tell us anything about this
we have to retest the model
2) We may lose information when we lag our variables
to get around this information loss, we can use the Prais-Winsten formula to transform the model:
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6. Problems (2)
3) We might want to include a lagged endogenous variable in the model
including the lagged endogenous variable Yt-1 biases the Durbin-Watson test towards 2
this means it’s biased towards the null of no autocorrelation
in this instance, we’ll use Durbin’s h-statistic (1970):
v = square of the standard error on the coefficient (g) of the lagged endogenous variable
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7. Durbin’s h-statistic
Durbin’s h-statistic is normally distributed and is approximated by the z-statistic (standard normal)
null hypothesis: H0:  = 0
the null can be rejected at the 5% level of significance
L21.xls has example.
Problems with the h-statistic
the product nv must be less than one (where n = # of observations)
if nv  1, the h-statistic is undefined
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8. A note on consistency
Model with lagged endogenous variable and first-order serially correlated error may be mis-specified.
Yt = b0 + b1Yt-1 + ut
and
ut = rut-1 + et
If so, presence of first-order serial correlation may induce omitted variable bias.
Need to include additional lagged endogenous variable term:
Yt = a0 + a1Yt-1 + a2Yt-2 + et
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9. Yt = a + b0Xt + b1Xt-1 +…+bkXt-k + et
Why lags?
This mainly relates to macroeconomic models
economic events such as consumer expenditure, production, or investment
for instance: consumer expenditure this year may be related to consumer expenditure last year
In a general distributed lag model:
Yt = a + b0Xt + b1Xt-1 +…+bkXt-k + et
where k = any large number less than t-2
can eliminate coefficients b1 to bk by using a t-test
number of lags included is ad-hoc
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10. Problems for OLS
Lags lead to severe problems for ordinary least squares
loss of information (degrees of freedom)
independent variables (X) are highly correlated [multi-collinearity problem]
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11. Why lags are useful Psychological reasons: behavior is habit-forming
so things like labor market behavior and patterns of money holding can be captured using lags
Technological reasons: a firm’s production pattern
Institutional: unions
Multipliers: short run and long run multipliers (how to read finite distributed lags in a model).
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12. Ad-hoc nature of lags What can we do? Two approaches
Koyck transformation
Adaptive expectations
Different implications on the assumptions about economic processes
will end up with the same estimating equation
looking only at the end product, we won’t be able to tell the Koyck transformation from adaptive expectations
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13. Koyck transformation Model: Yt = a + b0Xt + b1Xt-1 +…+bkXt-k + et
The Koyck transformation suggests that the further back in time we go, the less important is that factor
for instance, information from 10 years ago vs. information from last year
The transformation suggests:
Where 0 <  < 1
j = 1,…k
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14. Koyck transformation (2)
So,
Can use the expression for bj to rewrite the model
Yt = a + b0 (Xt + Xt-1 + 2Xt-2 + ….+ kXt-k) + et (4)
this imposes the assumption that earlier information is relatively less important
Lagging the equation and multiplying it by , we get:
Yt-1 = a + b0 (Xt-1 + 2Xt-2 + ….+ kXt-k) +  et-1 (5)
Subtracting (5) from (4), we get
Yt = a(1- ) + b0Xt + Yt-1 + vt where vt = et - et-1
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15. Koyck transformation (3)
Why is this transformation useful?
Allows us to take the ad-hoc lag series and condense it into a lagged endogenous variable
now we only lose one observation due to the lagged endogenous variable
the  given by the estimation gives the coefficient of autocorrelation
Problem: by construction, we have first-order autocorrelation
use Durbin h-statistic
but estimating equation might be mis-specified!
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16. Adaptive expectations
Another way to approach the problem of the ad-hoc nature of lags
Can use the example of trying to measure the natural rate of unemployment
In 1968, Friedman estimated the equation:
Yt = a + bXt* + ut
where Xt* = natural rate of unemployment
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17. Adaptive expectations (2)
Using adaptive expectations we have that
Xt* - Xt-1* =  (Xt - Xt-1*)
Can rewrite the equation: Xt* - (1 -  )Xt-1* =  Xt
Using a lag operator where:
LXt = Xt-1
L2Xt = Xt-2
where
Xt* = expectation
Xt = observed
0 <  < 1
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18. Adaptive expectations (3)
We can then rewrite :  Xt = (1 - L )Xt*
where  = (1-  )
This can be rewritten as:
now we have the natural rate of unemployment in terms of the observed rate of unemployment
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19. Adaptive expectations (3)
Substituting into the model we get:
Upon further multiplication and substitution we arrive at:
this looks very similar to that for the Koyck transformation
where
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20. Problems with the approaches
For the lagged endogenous variables in the ad-hoc lag structure, we are uncertain as to which economic model of agent behavior underlies the estimating equation
We have 1st-order autocorrelation by the construction of the model
use the Durbin h-statistic
Yt-1 and et-1 (ut-1) are sure to be correlated [ E(X,e)  0]
this leads to biased estimates
we’ll deal with this using instrumental variables and simultaneous equations
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21. Other topics
Seasonality and the use of dummy variables in time series models.
Trends and their use in time series models
Testing and correcting in the presence of higher orders of serial correlation.
Lecture 21