Introduction to Econometrics, 5th edition

Introduction to Econometrics, 5th edition
Type author name/s here Dougherty Introduction to Econometrics, 5th edition Chapter heading Chapter 13: Introduction to Nonstationary Time Series © Christopher Dougherty, All rights reserved.

GRAPHICAL TECHNIQUES FOR DETECTING NONSTATIONARITY
ARIMA(p, q) process 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. 6

ARIMA(p, q) process Much earlier than conventional econometricians, time series analysts recognized the importance of nonstationarity and the need for eliminating it by differencing. 6

ARIMA(p, q) process 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. 6

ARIMA(p, q) process Autocorrelation function for k = 1, ... The key tool for determining d, and subsequently p and q, was the correlogram. 6

ARIMA(p, q) process Autocorrelation function for k = 1, ... 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, … 6

ARIMA(p, q) process Autocorrelation function for k = 1, ... Autocorrelation function of an AR(1) process For example, the autocorrelation function for an AR(1) process Xt = b2Xt–1 + et is rk = b2k, the coefficients decreasing exponentially with the lag provided that b2 < 1 and the process is stationary. 6

Autocorrelation function for k = 1, ... Correlogram of an AR(1) process The correlogram is the graphical representation of the autocorrelation function. The figure shows the correlogram for an AR(1) process with b2 = 0.8. 7

Autocorrelation function for k = 1, ... Correlogram of an AR(1) process 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. 8

ARIMA(p, q) process Autocorrelation function for k = 1, ... Autocorrelation function of an MA(1) process 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 = et + a2et–1 is as shown and all subsequent autocorrelation coefficients are zero. 9

Correlogram of a random walk (T = 200) 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. 10

Correlogram of a random walk (T = 200) 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. 11

ARIMA(p, q) process 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. 12

ARIMA(p, q) process 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. 13

ARIMA(p, q) process 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. 14

ARIMA(p, q) process 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. 15

ARIMA(p, q) process Time series analysis is a pragmatic approach to forecasting. As Box, a leading exponent, once said, “All models are wrong, but some are useful.” 16

ARIMA(p, q) process 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. 17

Correlogram of a random walk (T = 200) 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 b2. 18

Correlogram of a random walk (T = 50) 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. 19

LGDPI We will now look at an example. The figure showss the data for the logarithm of DPI for 1959–2003. 20

Sample correlogram of LGDPI This figure presents the sample correlogram. 21

Sample correlogram of LGDPI At first sight, the falling autocorrelation coefficients suggest a stationary AR(1) process with a high value of b2. 22

Correlogram of an AR(1) process Sample correlogram of LGDPI 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. 23

Correlogram of a random walk (T = 50) Sample correlogram of LGDPI However, the correlogram of LGDPI is also very similar to that for the finite nonstationary process shown inset. 24

First difference of LGDPI This figure shows the differenced series, which appears to be stationary around a mean annual growth rate of between 2 and 3 percent. 25

First difference of LGDPI 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. 26

Sample correlogram of first difference of LGDPI This figure shows the corresponding correlogram, whose low, erratic autocorrelation coefficients provide support for the hypothesis that the differenced series is stationary. 27

Copyright Christopher Dougherty 2016.
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, fifth edition 2016, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own 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 or the University of London International Programmes distance learning course 20 Elements of Econometrics

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