Absorption of shocks in nonlinear autoregressive models

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Abstract

It generally is difficult, if not impossible, to fully understand and interpret nonlinear time series models by considering the estimated values of the model parameters only. To shed light on the characteristics and implications of a nonlinear model it can then be useful to consider the effects of shocks on the future patterns of the time series variable. Most interest in such impulse response analysis has concentrated on measuring the persistence of shocks, or the magnitude of their (ultimate) effect. A framework is developed and implemented that is useful for measuring the rate at which this final effect is attained, or the rate of absorption of shocks. It is shown that the absorption rate can be used to examine whether the propagation of different types of shocks, such as positive and negative shocks or large and small shocks follows different patterns. The nonlinear floor-and-ceiling model for US output growth is used to illustrate the various concepts. The presence of substantial asymmetries in both persistence and absorption of shocks is documented, with interesting differences arising across magnitudes of shocks and across regimes in the model. Furthermore, it appears that asymmetry became much less pronounced due to a large decline in output volatility in the 1980s.

Introduction

Nonlinear time series models are becoming increasingly popular in empirical macroeconomics and empirical finance for describing and forecasting variables such as output, (un)employment, stock returns and interest rates, see Granger and Teräsvirta (1993) and Franses and van Dijk (2000) for reviews. Examples of often considered models are the threshold autoregressive (TAR) model, see Tong (1990), the smooth transition autoregressive (STAR) model, see Chan and Tong, 1986, Teräsvirta, 1994 and van Dijk et al. (2002), the Markov-switching model put forward in Hamilton (1989), and the artificial neural network (ANN) model advocated by Kuan and White (1994), among others. A key feature of these (and other) nonlinear time series models is that they assume that the model structure (lag length, parameters, variance) experiences occasional changes or, put differently, they assume the presence of different regimes. Hence, these models can, for example, describe asymmetric business cycle behavior as observed in output and unemployment, or different behavior in different states of the financial market (for example, in bull and bear markets) as observed in stock returns and interest rates.

A common property of many of these (univariate) nonlinear models (and this holds true even more so for their multivariate counterparts) is that only considering (estimates of) the model parameters generally is not sufficient to completely grasp the implied properties of time series generated by the model. Put differently, it is difficult to interpret a specific nonlinear model and to understand why it is useful in a particular application. Therefore, to shed light on the characteristics of a nonlinear model it is often useful to consider the effects of shocks on the future patterns of the time series variable. Impulse response functions provide a convenient tool for measuring such effects. Recent applications of impulse response analysis in nonlinear models in empirical macroeconomics and finance can be found in Weise (1999), Balke (2000), Dufour and Engle (2000), Taylor and Peel (2000), Altissimo and Violante (2001), Taylor et al. (2001), Balke et al. (2002), Skalin and Teräsvirta (2002), Atanasova (2003), Chen et al. (2004), Dufrenot et al. (2004), Grier et al. (2004) and Camacho (2005), among others.

Most applications of impulse response analysis concentrate on measuring the persistence of shocks, indicated by the magnitude of their (ultimate) effect on the time series variable. Interestingly, far less attention typically is given to measuring the rate at which this final effect is attained, that is, how fast shocks are ‘absorbed’ by a time series. Due to the properties of impulse responses in linear models, they can be used straightforwardly to gain insight in this rate of absorption of shocks as well, see, for example, Lütkepohl (2005) for a discussion of impulse response functions in linear models. However, impulse response analysis in nonlinear models is more complicated, as discussed at length in Gallant et al. (1993), Koop et al. (1996), and Potter (2000). The complications arise because in nonlinear models (1) the effect of a shock depends on the history of the time series up to the point where the shock occurs, (2) the effect of a shock need not be proportional to its size and (3) the effect of a shock depends on shocks occurring in periods between the moment at which the impulse occurs and the moment at which the response is measured. Because of these properties of impulse responses, assessing the absorption time of shocks in nonlinear models also is more involved, as will become clear below. In this paper we develop and implement a framework that can be used for measuring and analyzing absorption of shocks in nonlinear models. Among others, we demonstrate that our absorption measure can be used to address relevant questions such as

  • (1)

    Are positive and negative shocks absorbed at the same rate? Are shocks of different magnitudes absorbed at the same rate?

  • (2)

    Does the rate of absorption of shocks depend on the initial values or the history of the time series?

  • (3)

    Are shocks absorbed at the same rate by the different components of a multivariate time series?

  • (4)

    Are shocks absorbed at the same rate by linear combinations of the components in a multivariate time series and by the individual components themselves?

It should be stressed at the outset that our absorption measure should not be considered as a substitute for conventional impulse response analysis and persistence measures but rather as a complement. Our absorption measure allows one to obtain a more complete picture of the propagation mechanism of a nonlinear model, as it can highlight interesting asymmetry properties of shocks to economic time series different from the ones that are revealed by impulse response functions as such.

Finally, an alternative approach to absorption is developed by Lee and Pesaran (1993) and Pesaran and Shin (1996). They examine the time profile of the effect of shocks by means of so-called ‘persistence profiles’, defined as the difference between the conditional variances of n-step and (n-1)-step ahead forecasts, viewed as a function of n. Also see Galbraith (2003) on the related concept of ‘content horizons’, defined as the ratio of the conditional variance of n-step ahead forecasts and the unconditional variance of the time series of interest.

Our paper proceeds as follows. In Section 2, we briefly review the main aspects of impulse response analysis in nonlinear time series models and the generalized impulse (GI) response functions introduced by Koop et al. (1996). In Section 3, we develop our measure of absorption of shocks. To facilitate the understanding of the concept of absorption, we concentrate on univariate models first. In this section we also demonstrate how to address the question of asymmetric absorption, that is the question whether positive and negative shocks are absorbed differently. In Section 4, we generalize our absorption measure to multivariate models. Particular attention is given to the question whether shocks are absorbed at the same rate by the different components of a multivariate time series. We also outline how to measure absorption for a linear combination of the components of a multivariate time series. In Section 5, we discuss an empirical application to quarterly US GDP growth rates using the floor-and-ceiling model of Pesaran and Potter (1997). We document the presence of substantial asymmetries in both persistence and absorption of shocks, with interesting differences arising across magnitudes of shocks and across regimes in the model. Furthermore, it appears that asymmetry became much less pronounced due to a large decline in output volatility in the 1980s. Finally, Section 6 contains some concluding remarks.

Section snippets

Preliminaries

Consider the multivariate nonlinear autoregressive time series modelYt=FYt-1,,Yt-p;θ+Vt,where Yt=Y1t,,Ykt is a (k×1) random vector, F(·) is a known function that depends on the (q×1) parameter vector θ, Vt=V1t,,Vkt is a (k×1) vector of random disturbances with EVt|Ωt-1=0 and EVtVt|Ωt-1=HYt-1,,Yt-r;ξ, where the (k×k) conditional covariance matrix HYt-1,,Yt-r;ξHt=Ht,ij,i,j=1,,k depends on the (s×1) parameter vector ξ.

Throughout, we use upper-case letters to denote random variables and

Absorption of shocks in univariate models

Irrespective of whether shocks are persistent or not, it should be of interest to assess how fast innovations are absorbed, that is, the rate at which the GI approaches the final response GIYvt,ωt-1. In this section we discuss how absorption can be measured.

Absorption of shocks in multivariate models

The concept of absorption can also be used to investigate the properties of multivariate nonlinear models. In this section, we first define the multivariate extension of the univariate absorption measure used so far. Next, we discuss how to measure whether shocks are absorbed at the same rate by the different components of a multivariate time series, which we call common absorption.

Impulse response and absorption in the floor-and-ceiling model

US output growth has been by far the most popular macroeconomic application of nonlinear regime-switching time series models. In particular, following Hamilton (1989) many attempts have been made to describe the apparent asymmetric behavior over the business cycle in US output by means of Markov-switching models, see Clements and Krolzig (2002) and Kim et al. (2005) for recent contributions. Alternatively, threshold and smooth transition models have also been used for this purpose, see Potter

Concluding remarks

In this paper we proposed a new tool that can be used to examine the properties of univariate and multivariate nonlinear time series models. This tool, which we called the absorption rate, can be viewed as complementary to the familiar impulse response function, as both consider different aspects of the propagation of shocks. The absorption rate can be used to examine whether the propagation of different types of shocks, such as large and small shocks, positive and negative shocks, and shocks

Acknowledgements

Helpful discussions with Clive Granger and Simon Potter are gratefully acknowledged, as well as useful comments and suggestions from an associate editor and an anonymous referee. The first draft of this paper (which circulated under the title ‘Common persistence in nonlinear autoregressive models’) was written while the third author was enjoying the hospitality of the Department of Economics of the University of California, San Diego.

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