Christopher Dougherty

EC220 - Introduction to econometrics (chapter 12)Slideshow: autocorrelation

 

 

 

 

Original citation:

Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 12). [Teaching Resource]

© 2012 The Author

This version available at: http://learningresources.lse.ac.uk/138/

Available in LSE Learning Resources Online: May 2012

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AUTOCORRELATION

1

Assumption C.5 states that the values of the disturbance term in the observations in the sample are generated independently of each other.

1

Y = 1 + 2

X

Y

X

AUTOCORRELATION

2

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.

1

Y

X

Y = 1 + 2

X

AUTOCORRELATION

3

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.

Y

1

X

Y = 1 + 2

X

First-order autoregressive autocorrelation: AR(1)

AUTOCORRELATION

ttt uu 1

8

ttt uXY 21

A particularly common type of autocorrelation, at least as an approximation, is first-order autoregressive autocorrelation, usually denoted AR(1) autocorrelation.

First-order autoregressive autocorrelation: AR(1)

AUTOCORRELATION

ttt uu 1

8

ttt uXY 21

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.

First-order autoregressive autocorrelation: AR(1)

Fifth-order autoregressive autocorrelation: AR(5)

AUTOCORRELATION

ttt uu 1

ttttttt uuuuuu 5544332211

8

ttt uXY 21

Here is a more complex example of autoregressive autocorrelation. It is described as fifth-order, and so denoted AR(5), because it depends on lagged values of ut up to the fifth lag.

First-order autoregressive autocorrelation: AR(1)

Fifth-order autoregressive autocorrelation: AR(5)

Third-order moving average autocorrelation: MA(3)

AUTOCORRELATION

ttt uu 1

ttttttt uuuuuu 5544332211

3322110 tttttu

8

ttt uXY 21

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.

First-order autoregressive autocorrelation: AR(1)

Fifth-order autoregressive autocorrelation: AR(5)

Third-order moving average autocorrelation: MA(3)

AUTOCORRELATION

ttt uu 1

ttttttt uuuuuu 5544332211

3322110 tttttu

8

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.

ttt uXY 21

AUTOCORRELATION

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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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ttt uu 1

AUTOCORRELATION

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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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ttt uu 1

AUTOCORRELATION

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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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ttt uu 10.0

AUTOCORRELATION

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ttt uu 11.0

AUTOCORRELATION

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ttt uu 12.0

AUTOCORRELATION

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With equal to 0.3, a pattern of positive autocorrelation is beginning to be apparent.

ttt uu 13.0

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AUTOCORRELATION

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ttt uu 14.0

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AUTOCORRELATION

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ttt uu 15.0

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AUTOCORRELATION

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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.

ttt uu 16.0

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AUTOCORRELATION

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ttt uu 17.0

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AUTOCORRELATION

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ttt uu 18.0

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AUTOCORRELATION

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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.

ttt uu 19.0

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AUTOCORRELATION

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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.

ttt uu 195.0

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AUTOCORRELATION

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Next we will look at negative autocorrelation, starting with the same set of 50 independently

distributed values of t.

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ttt uu 10.0

AUTOCORRELATION

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We will take larger steps this time.

ttt uu 13.0

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AUTOCORRELATION

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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.

ttt uu 16.0

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AUTOCORRELATION

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Now the pattern of negative autocorrelation is very obvious.

ttt uu 19.0

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1

============================================================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 6.359334Adjusted R-squared 0.998515 S.D. dependent var 0.437527S.E. of regression 0.016859 Akaike info criter-5.263574Sum squared resid 0.011937 Schwarz criterion -5.143130Log likelihood 121.4304 F-statistic 14797.05Durbin-Watson stat 0.633113 Prob(F-statistic) 0.000000============================================================

AUTOCORRELATION

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Next, we will look at a plot of the residuals of the logarithmic regression of expenditure on housing services on income and relative price.

AUTOCORRELATION

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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.

-0.04

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0.01

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0.04

1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003

AUTOCORRELATION

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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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-0.01

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0.01

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1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003

Copyright Christopher Dougherty 2011.

These slideshows may be downloaded by anyone, anywhere for personal use.

Subject to respect for copyright and, where appropriate, attribution, they may be

used as a resource for teaching an econometrics course. There is no need to

refer to the author.

The content of this slideshow comes from Section 12.1 of C. Dougherty,

Introduction to Econometrics, fourth edition 2011, Oxford University Press.

Additional (free) resources for both students and instructors may be

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Individuals studying econometrics on their own and who feel that they might

benefit from participation in a formal course should consider the London School

of Economics summer school course

EC212 Introduction to Econometrics

http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx

or the University of London International Programmes distance learning course

20 Elements of Econometrics

www.londoninternational.ac.uk/lse.

11.07.25