1. Autocorrelation
Lecture 18
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2. Today’s plan Definition and implications
How to test for first-order autocorrelation
Durbin-Watson test as an introduction to the presence of first-order serial correlation in the model
How to to estimate allowing for autocorrelation: GLS - generalized least squares as a means of allowing for non-constant variance.
Phillips curve as an example: Note, the original Phillips curve has no lagged variables included in it. Consideration of the model: understand wage inflation and unemployment; or just prediction of inflation?
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3. Definitions and implications
Autocorrelation is a time-series phenomenon
1st-order autocorrelation implies that neighboring observations are correlated
the observations aren’t independent draws from the sample
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4. Definitions and implications (2)
In terms of the Gauss-Markov (or BLUE) theorem:
the model is Yt = a + bXt + et
The model is still linear and unbiased if autocorrelation exists:
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5. Definitions and implications (3)
Autocorrelation will affect the variance:
if s = t, then we would have:
but if s  t, and Y observations are not independent, we have nonzero covariance terms:
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6. Definitions and implications (4)
Think of a numerical example to demonstrate this
assuming t = 3: if we wanted to estimate :
We want to consider the efficiency, or the variance of
If BLUE: all covariance terms are zero.
If covariance terms are nonzero: we no longer have minimum variance
minimum variance is defined as
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7. Summary of implications
1) Estimates are linear and unbiased
2) Estimates are not efficient. We no longer have minimum variance
3) Estimated variances are biased either positively or negatively
4) Unreliable t and F test results
5) Computed variances and standard errors for predictions are biased
Main idea: autocorrelation affects the efficiency of estimators
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8. How does autocorrelation occur?
Autocorrelation occurs through one of the following avenues:
1) Intertia in economic series through construction
With regards to unemployment this was called hysteresis: this means that certain sections of society who are prone to unemployment
2) Incorrect model specification
There might be missing variables or we might have transformed the model to create correlation across variables (more on this later)
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9. How does autocorrelation occur?
3) Cobweb phenomenon
agents respond to information with lags
4) Data manipulation
example: constructing annual information based on quarterly data
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10. Graphical results
With no autocorrelation in the error term: we would expect all errors to be randomly dispersed around zero within reasonable boundaries
Simply graphing the estimated errors against time indicates the possibility of autocorrelation:
we look for patterns of errors over time
patterns can be positive, negative, or zero
Graph error vs. time, we have positive correlation in the error term
errors from one time period to the next tend to move in 1 direction, with a positive slope
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11. Phillips Curve L_18.xls : Phillips Curve data
Can calculate predicted wage inflation using the observed unemployment rate and the estimated regression coefficients
Can then calculate the estimated error of the regression equation
Can also calculate the error lagged one time period
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12. Durbin-Watson statistic
We will use the Durbin-Watson statistic to test for autocorrelation
This is computed by looking over T-1 observations where t = 1, …T
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13. Durbin-Watson statistic (2)
The assumptions behind the Durbin-Watson statistic are:
1) You must include an intercept in the regression
2) Values of X are independent and fixed
3) Disturbances, or errors, are generated by:
this says that errors in this time period are correlated with errors in the last time period and some random error vt
 is the coefficient of autocorrelation and is bounded -1    1
 can be calculated as
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14. How to estimate 
This estimation matters because it will be used in the model correction
 can be estimated by this equation:
Once we have the Durbin-Watson statistic d, you can obtain an estimate for 
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15. How to estimate  (2) How the test works:
The values for d range between 0 and 4 with 2 as the midpoint
using :
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16. How to estimate  (3) We can represent this in the following figure: 24 H1 dL dU
H0:  =0
Reject null
Cannot reject null
4-dU
4-dL
indeterminate
dL represents the D-W upper bound
dU represents the D-W lower bound
The mirror image of dL and dU are 4- dL and 4-dU
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17. Procedure
Table on the second handout for today is the Durbin-Watson statistical table and an additional table for this analysis
1) Run model:
2) Compute:
3) Compute d statistic
4) Find dL and dU from the tables K’ is the number of parameters minus the constant and T is the number of observations
5) Test to see if autocorrelation is present
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18. Example (2) Returning to L18.xls 2 4 H1 dL dU H0:  =0 Reject nullH1 dL dU
H0:  =0
Reject null
Accept null
4-dU
4-dL
0.331
1.475
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19. Generalized least squares
What can we do about autocorrelation?
Recall that our model is: Yt = a + bXt + et (1)
We also know: et = et-1 + vt
We will have to estimate the model using generalized least squares (GLS)
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20. Generalized least squares (2)
Let’s take our model and lag it by one time period:
Yt-1 = a + bXt-1 + et-1
Multiplying by :
Yt-1 = a + bXt-1 + et (2)
Subtracting our (2) from (1), we get
Yt - Yt-1 = a(1-) + b(Xt-1 -  Xt-1) + vt
where
vt = et - et-1
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21. Generalized least squares (3)
Now we need an estimate of  : we can transform the variables such that:
where:
Estimating equation (3) allows us to estimate without first-order autocorrelation
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22. Estimating  There are several approaches
One way is by using a short cut:
thinking back to the Durbin-Watson statistic,
we can rewrite the expression for d as:
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23. Estimating  (2) Collecting like terms, we have:
Solving for , we can get an estimate in terms of d:
Since earlier we defined  as:
we can use this to get a more precise estimate
There are three or four other methods in the text
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