Transcript Autocorrelation

Type
author name/s here
Dougherty
Introduction to Econometrics,
5th edition
Chapter heading
Chapter 12: Autocorrelation
© Christopher Dougherty, 2016. All rights reserved.
AUTOCORRELATION
Y
1
X
Assumption C.5 states that the values of the disturbance term in the observations in the
sample are generated independently of each other.
1
AUTOCORRELATION
Y
1
X
In the graph above, it is clear that this assumption is not valid. Positive values tend to be
followed by positive ones, and negative values by negative ones. Successive values tend
to have the same sign. This is described as positive autocorrelation.
2
AUTOCORRELATION
Y
1
X
In this graph, positive values tend to be followed by negative ones, and negative values by
positive ones. This is an example of negative autocorrelation.
3
AUTOCORRELATION
Yt   1   2 X t  ut
First-order autoregressive autocorrelation: AR(1)
ut  ut 1   t
A particularly common type of autocorrelation, at least as an approximation, is first-order
autoregressive autocorrelation, usually denoted AR(1) autocorrelation.
8
AUTOCORRELATION
Yt   1   2 X t  ut
First-order autoregressive autocorrelation: AR(1)
ut  ut 1   t
It is autoregressive, because ut depends on lagged values of itself, and first-order, because
it depends only on its previous value. ut also depends on t, an injection of fresh
randomness at time t, often described as the innovation at time t.
8
AUTOCORRELATION
Yt   1   2 X t  ut
First-order autoregressive autocorrelation: AR(1)
ut  ut 1   t
Fifth-order autoregressive autocorrelation: AR(5)
ut  1ut 1   2 ut 2   3 ut 3   4 ut 4   5 ut 5   t
Here is a more complex example of autoregressive autocorrelation. It is described as fifthorder, and so denoted AR(5), because it depends on lagged values of ut up to the fifth lag.
8
AUTOCORRELATION
Yt   1   2 X t  ut
First-order autoregressive autocorrelation: AR(1)
ut  ut 1   t
Fifth-order autoregressive autocorrelation: AR(5)
ut  1ut 1   2 ut 2   3 ut 3   4 ut 4   5 ut 5   t
Third-order moving average autocorrelation: MA(3)
ut  0 t  1 t 1  2 t 2  3 t 3
The other main type of autocorrelation is moving average autocorrelation, where the
disturbance term is a linear combination of the current innovation and a finite number of
previous ones.
8
AUTOCORRELATION
Yt   1   2 X t  ut
First-order autoregressive autocorrelation: AR(1)
ut  ut 1   t
Fifth-order autoregressive autocorrelation: AR(5)
ut  1ut 1   2 ut 2   3 ut 3   4 ut 4   5 ut 5   t
Third-order moving average autocorrelation: MA(3)
ut  0 t  1 t 1  2 t 2  3 t 3
This example is described as third-order moving average autocorrelation, denoted MA(3),
because it depends on the three previous innovations as well as the current one.
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AUTOCORRELATION
ut  ut 1   t
3
2
1
0
1
-1
-2
-3
We will now look at examples of the patterns that are generated when the disturbance term
is subject to AR(1) autocorrelation. The object is to provide some bench-mark images to
help you assess plots of residuals in time series regressions.
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AUTOCORRELATION
ut  ut 1   t
3
2
1
0
1
-1
-2
-3
We will use 50 independent values of , taken from a normal distribution with 0 mean, and
generate series for u using different values of .
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AUTOCORRELATION
ut  0.0ut 1   t
3
2
1
0
1
-1
-2
-3
We have started with  equal to 0, so there is no autocorrelation. We will increase 
progressively in steps of 0.1.
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AUTOCORRELATION
ut  0.1ut 1   t
3
2
1
0
1
-1
-2
-3
( = 0.1)
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AUTOCORRELATION
ut  0.2ut 1   t
3
2
1
0
1
-1
-2
-3
( = 0.2)
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AUTOCORRELATION
ut  0.3ut 1   t
3
2
1
0
1
-1
-2
-3
With  equal to 0.3, a pattern of positive autocorrelation is beginning to be apparent.
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AUTOCORRELATION
ut  0.4ut 1   t
3
2
1
0
1
-1
-2
-3
( = 0.4)
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AUTOCORRELATION
ut  0.5ut 1   t
3
2
1
0
1
-1
-2
-3
( = 0.5)
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AUTOCORRELATION
ut  0.6ut 1   t
3
2
1
0
1
-1
-2
-3
With  equal to 0.6, it is obvious that u is subject to positive autocorrelation. Positive
values tend to be followed by positive ones and negative values by negative ones.
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AUTOCORRELATION
ut  0.7ut 1   t
3
2
1
0
1
-1
-2
-3
( = 0.7)
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AUTOCORRELATION
ut  0.8ut 1   t
3
2
1
0
1
-1
-2
-3
( = 0.8)
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AUTOCORRELATION
ut  0.9ut 1   t
3
2
1
0
1
-1
-2
-3
With  equal to 0.9, the sequences of values with the same sign have become long and the
tendency to return to 0 has become weak.
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AUTOCORRELATION
ut  0.95ut 1   t
3
2
1
0
1
-1
-2
-3
The process is now approaching what is known as a random walk, where  is equal to 1 and
the process becomes nonstationary. The terms ‘random walk’ and ‘nonstationary’ will be
defined in the next chapter. For the time being we will assume |  | < 1.
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AUTOCORRELATION
ut  0.0ut 1   t
3
2
1
0
1
-1
-2
-3
Next we will look at negative autocorrelation, starting with the same set of 50 independently
distributed values of t.
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AUTOCORRELATION
ut  0.3ut 1   t
3
2
1
0
1
-1
-2
-3
We will take larger steps this time.
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AUTOCORRELATION
ut  0.6ut 1   t
3
2
1
0
1
-1
-2
-3
With  equal to –0.6, you can see that positive values tend to be followed by negative ones,
and vice versa, more frequently than you would expect as a matter of chance.
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AUTOCORRELATION
ut  0.9ut 1   t
3
2
1
0
1
-1
-2
-3
Now the pattern of negative autocorrelation is very obvious.
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AUTOCORRELATION
============================================================
Dependent Variable: LGHOUS
Method: Least Squares
Sample: 1959 2003
Included observations: 45
============================================================
Variable
Coefficient Std. Error t-Statistic Prob.
============================================================
C
0.005625
0.167903
0.033501
0.9734
LGDPI
1.031918
0.006649
155.1976
0.0000
LGPRHOUS
-0.483421
0.041780 -11.57056
0.0000
============================================================
R-squared
0.998583
Mean dependent var 6.359334
Adjusted R-squared
0.998515
S.D. dependent var 0.437527
S.E. of regression
0.016859
Akaike info criter-5.263574
Sum squared resid
0.011937
Schwarz criterion -5.143130
Log likelihood
121.4304
F-statistic
14797.05
Durbin-Watson stat
0.633113
Prob(F-statistic) 0.000000
============================================================
Next, we will look at a plot of the residuals of the logarithmic regression of expenditure on
housing services on income and relative price.
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AUTOCORRELATION
0.04
0.03
0.02
0.01
0
1959
1963
1967
1971
1975
1979
1983
1987
1991
1995
1999
2003
-0.01
-0.02
-0.03
-0.04
This is the plot of the residuals of course, not the disturbance term. But if the disturbance
term is subject to autocorrelation, then the residuals will be subject to a similar pattern of
autocorrelation.
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AUTOCORRELATION
0.04
0.03
0.02
0.01
0
1959
1963
1967
1971
1975
1979
1983
1987
1991
1995
1999
2003
-0.01
-0.02
-0.03
-0.04
You can see that there is strong evidence of positive autocorrelation. Comparing the graph
with the randomly generated patterns, one would say that  is about 0.7 or 0.8.
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Copyright Christopher Dougherty 2016.
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Introduction to Econometrics, fifth edition 2016, Oxford University Press.
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2016.05.22