CHAPTER 1
Why Should We Care About Nonlinearity?

Linear processes and linear models dominate research and applications of time series analysis. They are often adequate in making statistical inference in practice. Why should we care about nonlinearity then? This is the first question that came to our minds when we thought about writing this book. After all, linear models are easier to use and can provide good approximations in many applications. Empirical time series, on the other hand, are likely to be nonlinear. As such, nonlinear models can certainly make significant contributions, at least in some applications. The goal of this book is to introduce some nonlinear time series models, to discuss situations under which nonlinear models can make contributions, to demonstrate the value and power of nonlinear time series analysis, and to explore the nonlinear world. In many applications, the observed time series are indirect (possibly multidimensional) observations of an unobservable underlying dynamic process that is nonlinear. In this book we also discuss approaches of using nonlinear and non-Gaussian state space models for analyzing such data.

To achieve our objectives, we focus on certain classes of nonlinear time series models that, in our view, are widely applicable and easy to understand. It is not our intention to cover all nonlinear models available in the literature. Readers are referred to Tong (1990), Fan and Yao (2003), Douc et al. (2014), and De Gooijer (2017) for other nonlinear time series models. The book, thus, shows our preference in exploring the nonlinear world. Efforts are made throughout the book to keep applications in mind so that real examples are used whenever possible. We also provide the theory and justifications for the methods and models considered in the book so that readers can have a comprehensive treatment of nonlinear time series analysis. As always, we start with simple models and gradually move toward more complicated ones.

1.1 Some Basic Concepts

A scalar process xt is a discrete-time time series if xt is a random variable and the time index t is countable. Typically, we assume the time index t is equally spaced and denote the series by {xt}. In applications, we consider mainly the case of xt with t = 1. An observed series (also denoted by xt for simplicity) is a realization of the underlying stochastic process.

A time series xt is strictly stationary if its distribution is time invariant. Mathematically speaking, xt is strictly stationary if for any arbitrary time indices {t1, ., tm}, where m > 0, and any fixed integer k such that the joint distribution function of is the same as that of . In other words, the shift of k time units does not affect the joint distribution of the series. A time series xt is weakly stationary if the first two moments of xt exist and are time invariant. In statistical terms, this means E(xt) = µ and Cov(xt, xt + l) = ?l, where E is the expectation, Cov denotes covariance, µ is a constant, and ?l is a function of l. Here both µ and ?l are independent of the time index t, and ?l is called the lag-l autocovariance function of xt. A sequence of independent and identically distributed (iid) random variates is strictly stationary. A martingale difference sequence xt satisfying E(xt│xt - 1, xt - 2, .) = 0 and Var(xt│xt - 1, xt - 2, .) = s2 > 0 is weakly stationary. A weakly stationary sequence is also referred to as a covariance-stationary time series. An iid sequence of Cauchy random variables is strictly stationary, but not weakly stationary, because there exist no moments. Let xt = stet, where et ~ iidN(0, 1) and s2t = 0.1 + 0.2xt - 12. Then xt is weakly stationary, but not strictly stationary.

Time series analysis is used to explore the dynamic dependence of the series. For a weakly stationary series xt, a widely used measure of serial dependence between xt and xt - l is the lag-l autocorrelation function (ACF) defined by

where l is an integer. It is easily seen that ?0 = 1 and ?l = ?- l so that we focus on ?l for l > 0. The ACF defined in Equation (1.1) is based on the Pearson's correlation coefficient. In some applications we may employ the autocorrelation function using the concept of Spearman's rank correlation coefficient.

1.2 Linear Time Series

A scalar process xt is a linear time series if it can be written as

where µ and ?i are real numbers with ?0 = 1, S8i = -8|?i| < 8, and {at} is a sequence of iid random variables with mean zero and a well-defined density function. In practice, we focus on the one-sided linear time series

where ?0 = 1 and S8i = 0|?i| < 8. The linear time series in Equation (1.3) is called a causal time series. In Equation (1.2), if ?j ? 0 for some j < 0, then xt becomes a non-causal time series. The linear time series in Equation (1.3) is weakly stationary if we further assume that Var(at) = s2a < 8. In this case, we have E(xt) = µ, Var(xt) = s2aSi = 08?2i, and ?l = s2aS8i = 0?i?i + l.

The well-known autoregressive moving-average (ARMA) models of Box and Jenkins (see Box et al., 2015) are (causal) linear time series. Any deviation from the linear process in Equation (1.3) results in a nonlinear time series. Therefore, the nonlinear world is huge and certain restrictions are needed in our exploration. Imposing different restrictions leads to different approaches in tackling the nonlinear world which, in turn, results in emphasizing different classes of nonlinear models. This book is no exception. We start with some real examples that exhibit clearly some nonlinear characteristics and employ simple nonlinear models to illustrate the advantages of studying nonlinearity.

1.3 Examples of Nonlinear Time Series

To motivate, we analyze some real-world time series for which nonlinear models can make a contribution.

Example 1.1 Consider the US quarterly civilian unemployment rates from 1948.I to 2015.II for 270 observations. The quarterly rate is obtained by averaging the monthly rates, which were obtained from the Federal Reserve Economic Data (FRED) of the Federal Reserve Bank of St. Louis and were seasonally adjusted. Figure 1.1 shows the time plot of the quarterly unemployment rates. From the plot, it is seen that (a) the unemployment rate seems to be increasing over time, (b) the unemployment rate exhibits a cyclical pattern reflecting the business cycles of the US economy, and (c) more importantly, the rate rises quickly and decays slowly over a business cycle. As usual in time series analysis, the increasing trend can be handled by differencing. Let rt be the quarterly unemployment rate and xt = rt - rt - 1 be the change series of rt. Figure 1.2 shows the time plot of xt. As expected, the mean of xt appears to be stable over time. However, the asymmetric pattern in rise and decay of the unemployment rates in a business cycle shows that the rate is not time-reversible, which in turn suggests that the unemployment rates are nonlinear. Indeed, several nonlinear tests discussed later confirm that xt is indeed nonlinear.

Figure 1.1 Time plot of US quarterly civilian unemployment rates, seasonally adjusted, from 1948.I to 2015.II.

Figure 1.2 Time plot of the changes in US quarterly civilian unemployment rates, seasonally adjusted, from 1948.I to 2015.II.

If a linear autoregressive (AR) model is used, the Akaike information criterion (AIC) of Akaike (1974) selects an AR(12) model for xt. Several coefficients of the fitted AR(12) model are not statistically significant so that the model is refined. This leads to a simplified AR(12) model as

where the variance of at is s2a = 0.073 and all coefficient estimates are statistically significant at the usual 5% level. Figure 1.3 shows the results of model checking, which consists of the time plot of standardized residuals, sample autocorrelation function (ACF), and the p values of the Ljung-Box statistics Q(m) of the residuals. These p values do not adjust the degrees of freedom for the...