1. Autocorrelation

2. The Nature of Autocorrelation
The randomness of the sample implies that the error terms for different observations will be uncorrelated.
When we have time-series data, where the observations follow a natural ordering through time, there is always a possibility that successive errors will be correlated with each other.
In any one period, the current error term contains not only the effects of current shocks but also the carryover from previous shocks. This carryover will be related to, or correlated with, the effects of the earlier shocks. When circumstances such as these lead to error terms that are correlated, we say that autocorrelation exists.
The possibility of autocorrelation should always be entertained when we are dealing with time-series data.

3. For efficiency (accurate estimation / prediction) all systematic information needs to be incorporated into the regression model.
Autocorrelation is a systematic pattern in the errors that can be either attracting (positive) or repelling (negative) autocorrelation.

4. crosses line not enough (attracting)et . .
Postive
Auto. . . . . . . . . . . . . . . . . . . . . t
crosses line randomly et . . . . . . . . . . . . No
Auto. . . . . . . . . . . . . . . . . t
crosses line too much (repelling) . et . . . . . . . .
Negative
Auto. . . . . . . . t . .

5. Regression Model yt = 1 + 2xt + et E(et) = 0 var(et) = 2
zero mean:
E(et) = 0
homoskedasticity:
var(et) = 2
nonautocorrelation:
cov(et, es) = t = s
autocorrelation:
cov(et, es) = t = s

6. Order of Autocorrelation
yt = 1 + 2xt + et
1st Order:
et = et1 + t
et = 1et1 + 2et2 + t
2nd Order:
3rd Order:
et = 1et1 + 2et2 + 3et3 + t
We will assume First Order Autocorrelation:
et = et1 + t
AR(1) :

7. First Order Autocorrelation
yt = 1 + 2xt + et
et = et1 + t
where 1 <  < 1
var(t) = 2
E(t) = 0
cov(t, s) = t = s
These assumptions about t imply the following about et :
E(et) = 0
cov(et, etk) = e2 k for k > 0
2
var(et) = e2 =
corr(et, etk) = k for k > 0
12

8. Autocorrelation creates some Problems for Least Squares:
If we have an equation whose errors exhibit autocorrelation, but we ignore it, or are simply unaware of it, what does it have on the properties of least squares estimates?
1. The least squares estimator is still linear and unbiased but it is not efficient.
2. The formulas normally used to compute the least squares standard errors are no longer correct and confidence intervals and hypothesis tests using them will be wrong.

9. yt = 1 + 2xt + e Autocorrelation: E(et) = 0, var(et) = 2 , cov(et, es) = ,t  s
(Linear)
Where
(Unbiased)

10. xt x  yt = 1 + 2xt + et e2 cov(et, es) = ,t  s
Autocorrelation:
Incorrect formula for least squares variance:
e2
var(b2) =
xt x 
Correct formula for least squares variance:

11. Generalized Least Squares
et = et1 + t
substitute
in for et
yt = 1 + 2xt + et
yt = 1 + 2xt + et1 + t
Now we need to get rid of et1
(continued)

12. (continued) yt = 1 + 2xt + et1 + t yt = 1 + 2xt + et
et = yt  12xt
lag the
errors
once
et1 = yt1  12xt1
yt = 1 + 2xt + yt1  12xt1 + t
(continued)

13. yt = 1 + 2xt + yt1  12xt1 + t
yt  yt1 = 1(1) + 2(xtxt1) + t
yt = 1 xt1*+ 2 xt2* + t , *
t =2, 3, …, T.
yt = yt  yt1 *
xt2 = (xtxt1) *
x1t* = (1)

14. yt = yt  yt1 x1t* = (1) xt2 = xt xt1
yt = 1 xt1*+ 2 xt2* + t , *
Problems estimating this model with least squares:
1. One observation is used up in creating the transformed (lagged) variables leaving only (T1) observations for estimating the model (Cochrane-Orcutt method drops the first observation).
2. The value of  is not known. We must find some way to estimate it.

15. (Option) Recovering the 1st Observation
Dropping the 1st observation and applying least squares
is not the best linear unbiased estimation method.
Efficiency is lost because the variance of the error associated with the 1st observation is not equal to that of the other errors.
This is a special case of the heteroskedasticity
problem except that here all errors are assumed
to have equal variance except the 1st error.

16. Recovering the 1st Observation
The 1st observation should fit the original model as:
y1 = 1 + 2x e1
with error variance: var(e1) = e2 = 2 /(1-2).
We could include this as the 1st observation for our estimation procedure but we must first transform it so that it has the same error variance as the other observations.
Note: The other observations all have error variance 2.

17. y1 = 1 + 2x1 + e1 with error variance: var(e1) = e2 = 2 / (1-2).
The other observations all have error variance 2.
Given any constant c : var(ce1) = c2 var(e1).
If c = 1-2 , then var( 1-2 e1) = (1-2) var(e1).
= (1-2) e2
= (1-2) 2 /(1-2)
= 2
The transformation 1 = 1-2 e1 has variance 2 .

18. y1 = 1 + 2x1 + e1 Multiply through by 1-2 to get:
The transformed error 1 = 2 e1 has variance 2 .
This transformed first observation may now be
added to the other (T-1) observations to obtain
the fully restored set of T observations.

19. We can summarize these results by saying that, providing  is known, we can find the Best Linear Unbiased Estimator for 1 and 2 by applying least squares to the transformed mode
yt = 1 xt1*+ 2 xt2* + t , *
t =1, 2, 3, …, T.
where the transformed variables are defined by
for the first observation, and
for the remaining t = 2, 3, …, T observations.

20. Estimating Unknown  Value
If we had values for the et , we could estimate:
et = et1 + t
First, use least squares to estimate the model:
yt = 1 + 2xt + et
The residuals from this estimation are:
et = yt  b1  b2xt ^

21. et et-1 et-1 ^ et = yt - b1 - b2xt ^ et = et1 + t ^ =
Next, estimate the following by least squares:
et = et1 + t ^
The least squares solution is:
et et-1
et-1 T
t = 2 2 ^
=

22. Durbin-Watson Test
The Durbin-Watson test is by far the most important one for detecting AR(1) errors.
It is assumed that the vt are independent random errors with distribution N(0, v2).
The assumption of normally distributed random errors is needed to derive the probability distribution of the test statistic used in the Durbin-Watson test.

23. et et-1 et ^ d= The Durbin-Watson Test statistic, d, is :2 ^
d=
For a null hypothesis of no autocorrelation, we can use H0:  = 0.
For an alternative hypothesis we could use H1:  > 0 or H1:  < 0 or H1:   0.

24. Testing for Autocorrelation
The test statistic, d , is approximately related to  as: ^
0  d  2(1)  4 ^
When  = 0 , the Durbin-Watson statistic is d  2. ^
When  = 1 , the Durbin-Watson statistic is d  0. ^
When  = 1, the Durbin-Watson statistic is d  4. ^
Tables for critical values for d are not always
readily available so it is easier to use the p-value
that most computer programs provide for d.
Reject H0 if p-value < , the significance level.

25. Test for the first-order autocorrelation
Reject H Inclusive Do not reject H0
H1:  > d < dL dL < d < dU d > dU
H1:  < 0 d > 4  dL 4  dU < d < 4  dL d < 4  dU
d > 4  dL 4  dU < d < 4  dL
H1:   or or dU < d < 4  dU
d < dL dL < d < dU
Note: The lower and upper bounds (dL and dU) depend on sample size n and the number of explanatory variables k (not include intercept).

26.  > 0 dU 2 dL 4  < 0 2 4  dU 4  dL 4  > 0  < 0 2
A. Test for Positive Autocorrelation
 > 0
No evidence of positive autocorrelation
Inclusive dU 2 dL 4
B. Test for Negative Autocorrelation
No evidence of negative autocorrelation
Inclusive
 < 0 2
4  dU
4  dL 4
C. Two-Sided Test for Autocorrelation
No evidence of autocorrelation
 > 0
Inclusive
 < 0
Inclusive 2
4  dL dU
4  dU 4 dL

27. Prediction with AR(1) Errors
When errors are autocorrelated, the previous period
error may help us predict next period error.
The best predictor, yT+1 , for next period is:
yT+1 = 1 + 2xT  eT ^ ~
where 1 and 2 are generalized least squares
estimates and eT is given by: ~ ^ ~ ^ ^
eT = yT  1  2xT

28. For h periods ahead, the best predictor is:
yT+h = 1 + 2xT+h + h eT ^ ~
Assuming |  | < 1, the influence of h eT
diminishes the further we go into the future
(the larger h becomes). ^ ~