Christopher Dougherty EC220 - Introduction to econometrics (chapter 13) Slideshow: graphical...

Christopher Dougherty

EC220 - Introduction to econometrics (chapter 13)Slideshow: graphical techniques for detecting nonstationarity

Original citation:

Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 13). [Teaching Resource]

© 2012 The Author

This version available at: http://learningresources.lse.ac.uk/139/

Available in LSE Learning Resources Online: May 2012

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GRAPHICAL TECHNIQUES FOR DETECTING NONSTATIONARITY

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Section 11.7 outlines the time series analysis approach to representing a time series as a univariate ARMA(p, q) process, such as that shown above, for appropriate choice of p and q.


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Much earlier than conventional econometricians, time series analysts recognized the importance of nonstationarity and the need for eliminating it by differencing.


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With the need-for-differencing aspect in mind, the ARMA(p, q) model was generalized to the ARIMA(p, d, q) model where d is the number of times the series has to be differenced to render it stationary.


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The key tool for determining d, and subsequently p and q, was the correlogram.

Autocorrelation function

22 )()(

))((

XktXt

XktXtk

XEXE

XXE

for k = 1, ...


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The autocorrelation function of a series Xt gives the theoretical correlation between the value of a series at time t and its value at time t +k, for values of k from 1 to (typically) about 20, being defined as the series shown above, for k = 1, … The correlogram is its graphical representation.

for k = 1, ...

22 )()(

))((

XktXt

XktXtk

XEXE

XXE


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For example, the autocorrelation function for an AR(1) process Xt = 2Xt–1 + t is k = 2k,

the coefficients decreasing exponentially with the lag provided that 2 < 1 and the process is stationary.



for k = 1, ...

Autocorrelation function of an AR(1) process

ttt XX 12

kk 2

22 )()(

))((

XktXt

XktXtk

XEXE

XXE

0.0

0.2

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0.6

0.8

1.0

1 4 7 10 13 16 19


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The figure shows the correlogram for this process with 2 = 0.8.


for k = 1, ...

kkk 8.02

ttt XX 12

Correlogram of an AR(1) process

22 )()(

))((

XktXt

XktXtk

XEXE

XXE

0.0

0.2

0.4

0.6

0.8

1.0

1 4 7 10 13 16 19


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Higher-order stationary AR(p) processes may exhibit a more complex mixture of damped sine waves and damped exponentials, but they retain the feature that the weights eventually decline to zero.


for k = 1, ...

kkk 8.02

ttt XX 12


22 )()(

))((

XktXt

XktXtk

XEXE

XXE


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By contrast, an MA(q) process has nonzero weights for only the first q lags and zero weights thereafter. In particular, the first autocorrelation coefficient for the MA(1) process Xt = t + 2t–1 is as shown and all subsequent autocorrelation coefficients are zero.



for k = 1, ...

Autocorrelation function of an MA(1) process

12 tttX

22

21 1

22 )()(

))((

XktXt

XktXtk

XEXE

XXE


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In the case of nonstationary processes, the theoretical autocorrelation coefficients are not defined but one may be able to obtain an expression for E(rk), the expected value of the sample autocorrelation coefficients.

0.0

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1 4 7 10 13 16 19

ttt XX 1

Correlogram of a random walk (T = 200)


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For long time series, these coefficients decline slowly. For example, in the case of a random walk, the correlogram for a series with 200 observations is as shown in the figure.

0.0

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ttt XX 1



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Time series analysts exploit this fact in a two-stage procedure for identifying the orders of a series believed to be of the ARIMA(p, d, q) type.



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In the first stage, if the correlogram exhibits slowly declining coefficients, the series is differenced d times until the series exhibits a stationary pattern. Usually one differencing is sufficient, and seldom more than two.



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The second stage is to inspect the correlogram of the differenced series and its partial correlogram, a related tool, to determine appropriate values for p and q.



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This is not an exact science. It requires judgment, a reading of the tea-leaves, and different analysts can come up with different values. However, when that happens, alternative models are likely to imply similar forecasts, and that is what matters.



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Time series analysis is a pragmatic approach to forecasting. As Box, a leading exponent, once said, “All models are wrong, but some are useful.”



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In any case, the complexity of the task is limited by the fact that in practice most series are adequately represented by a process with the sum of p and q no greater than 2.



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There are, however, two problems with using correlograms to identify nonstationarity. One is that a correlogram similar to that for a random walk, shown in the figure, could result from a stationary AR(1) process with a high value of 2.

0.0

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1.0

1 4 7 10 13 16 19

ttt XX 1



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The other is that the coefficients of a nonstationary process may decline quite rapidly if the series is not long. This is illustrated in the figure, which shows the expected values of rk for a random walk when the series has only 50 observations.


ttt XX 1

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0.0

0.2

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1.0

1 4 7 10 13 16 19


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We will now look at an example. The figure showss the data for the logarithm of DPI for 1959–2003.

7.0

7.5

8.0

8.5

9.0

9.5

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

LGDPI


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-0.2

0

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1

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This figure presents the sample correlogram.

Sample correlogram of LGDPI


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At first sight, the falling autocorrelation coefficients suggest a stationary AR(1) process with a high value of 2.



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-0.2

0

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Although the theoretical correlogram for such a process, shown inset, looks a little different in that the coefficients decline exponentially to zero without becoming negative, a sample correlogram would have negative values similar to those in the figure.

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kkk 8.02

ttt XX 12




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-0.2

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1

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However, the correlogram of LGDPI is also very similar to that for the finite nonstationary process shown inset.


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1 4 7 10 13 16 19



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This figure shows the differenced series, which appears to be stationary around a mean annual growth rate of between 2 and 3 percent.

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0.00

0.01

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0.08

1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000

First difference of LGDPI


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Possibly there might be a downward trend, and equally possibly there might be a discontinuity in the series at 1972, with a step down in the mean growth rate after the first oil shock, but these hypotheses will not be investigated here.

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0.00

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1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000

First difference of LGDPI


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This figure shows the corresponding correlogram, whose low, erratic autocorrelation coefficients provide support for the hypothesis that the differenced series is stationary.

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1

1 4 7 10 13 16 19

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 13.3 of C. Dougherty,

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

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

downloaded from the OUP Online Resource Centre

http://www.oup.com/uk/orc/bin/9780199567089/.

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

Christopher Dougherty EC220 - Introduction to econometrics (chapter 13) Slideshow: graphical...

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