Measuring Conditional Persistence in Nonlinear Time Series*
| Date | 01 June 2007 |
| DOI | http://doi.org/10.1111/j.1468-0084.2006.00437.x |
| Published date | 01 June 2007 |
Measuring Conditional Persistence in Nonlinear
Time Series*
George Kapetanios
Department of Economics, Queen Mary, University of London, London, UK
(e-mail: g.kapetanios@qmul.ac.uk)
Abstract
The persistence properties of economic time series have been a primary object of
investigation in a variety of guises since the early days of econometrics. Recently,
work on nonlinear modelling for time series has introduced the idea that
persistence of a shock at a point in time may vary depending on the state of the
process at that point in time. This article suggests investigating the persistence
of processes conditioning on their history as a tool that may aid parametric non-
linear modelling. In particular, we suggest that examining the nonparametrically
estimated derivatives of the conditional expectation of a variable with respect to its
lag(s) may be a useful indicator of the variation in persistence with respect to its
past history. We discuss in detail the implementation of the measure and present
a Monte Carlo investigation. We further apply the persistence analysis to real
exchange rates.
I. Introduction
The persistence properties of economic time series have been a primary object of
investigation in a variety of guises since the early days of econometrics. The majority
of studies have concentrated on linear models and their persistence properties.
Traditionally, stationary processes have been investigated but following the advent of
unit root econometrics, and the implication of the existence of permanent shocks to
economic variables, non-stationary processes have been investigated as well.
*The author would like to thank the Associate Editor and two anonymous referees for very helpful
comments that greatly improved the paper.
JEL Classification numbers: C22, C14, F31.
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 69, 3 (2007) 0305-9049
doi: 10.1111/j.1468-0084.2006.00437.x
363
Blackwell Publishing Ltd and the Department of Economics, University of Oxford, 2006. Published by Blackwell Publishing Ltd,
9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.
The persistence properties of nonlinear processes have received increased
attention in recent years. Important milestones in this literature include articles
by Gallant, Rossi and Tauchen (1993) and Koop, Pesaran and Potter (1996)
on generalized impulse response analysis. The use of simulation techniques has
enabled the investigation of the impulse responses and persistence properties of
nonlinear processes without the need for analytical expressions for the evaluation of
the relevant expectations.
The evaluation of persistence in nonlinear econometric models is of paramount
importance for a number of economic problems. These include the investigation of the
purchasing power parity (PPP) hypothesis and the implications of the Fisher equation
for the stationarity of real interest rates. Using a linear model to investigate such
problems has led to the rejection of the economic hypotheses in question since data
appear non-stationary according to standard unit root tests. Nevertheless, recent work
(see e.g. Kapetanios, Shin and Snell, 2003) indicates that the use of tests and models
designed for nonlinear processes may uncover evidence supporting economic theory.
Taking the analysis one step further involves looking at the effect of shocks in
different parts of the state space and investigating whether shocks have different
effects for different process histories. It would be of great help to the nonlinear
modelling of such processes if initial diagnostic methods, that took into account
variability in persistence over histories of the process, were available.
Generalized impulse responses may be useful in this context. However, these have
usually been considered in the parametric context of particular nonlinear models.
Crucially, they focus on the evolution of the shock response over the future rather than
the shape of the response as a function of the process history at a given horizon. This
latter shape would be of use to choosing specific nonlinear models such as threshold
autoregressive (TAR), exponential smooth transition autoregressive (ESTAR) or
logistic smooth transition autoregressive (LSTAR) models for further modelling.
1
For
example, some stationary processes may appear to have a hump-shaped response
depending on the state of the process when the shock impacts, suggesting a three-
regime TAR or ESTAR model. Another process may have a sigmoid (s-shaped)
response suggesting the use of a two-regime TAR or LSTAR model.
This article suggests that investigating the persistence of processes conditioning
on their history as a possible tool for further parametric nonlinear modelling. In
particular, we suggest that examining the derivatives of the conditional expectation
of a variable with respect to its lags may be a useful indicator of the variation in
persistence with respect to its past history. This measure is related to the generalized
impulse responses proposed by Koop et al. (1996). But there are important
differences. On the one hand, the new measure provides information that may help
nonlinear time-series modelling as discussed in the previous paragraph. This type of
information is not easy to extract from impulse responses. On the other hand, we
1
For an introduction and details on these popular nonlinear time-series models, see Granger and Tera¨svirta
(1993).
364 Bulletin
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