Indirect inference for locally stationary models

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Abstract

We propose the use of indirect inference estimation to conduct inference in complex locally stationary models. We develop a local indirect inference algorithm and establish the asymptotic properties of the proposed estimator. Due to the nonparametric nature of locally stationary models, the resulting indirect inference estimator exhibits nonparametric rates of convergence. We validate our methodology with simulation studies in the confines of a locally stationary moving average model and a new locally stationary multiplicative stochastic volatility model. Using this indirect inference methodology and the new locally stationary volatility model, we obtain evidence of non-linear, time-varying volatility trends for monthly returns on several Fama–French portfolios.

Introduction

Time-varying economic and financial variables, and relationships thereof, are stable features in applied econometrics. Notable examples include asset pricing models with time-varying features (Ghysels, 1998, Wang, 2003, Koo et al., 2020) and trending macroeconomic models (Stock and Watson, 1998, Phillips, 2001). While classical analyses of time series are built on the assumption of stationarity, data studied in finance and economics often exhibit nonstationary features.

Many different schools of modeling and estimation methods are used to accommodate the nonstationary behavior of observed time series data. In particular, statistical tools developed for locally stationary processes provide a convenient means of conducting analyses of trending economic and financial models. Heuristically, local stationarity implies that a process behaves in a stationary manner (at least) in the vicinity of a given time point but could be nonstationary over the entire time horizon. For certain widely-studied time series models, slowly time-varying parameters ensure local stationarity under some regularity conditions; for instance, see Dahlhaus, 1996, Dahlhaus, 1997 (AR(1)), Dahlhaus and Subba Rao (2006) (ARCH()), Dahlhaus and Polonik (2009) (MA()), Koo and Linton (2012) (diffusion processes) and Koo and Linton (2015) (GARCH(1,1) with a time-varying unconditional variance) among many other classes of locally stationary processes.

While many classes of well-known time series models can be generalized to locally stationary processes, it is worth noting that estimation and inference procedures developed in one class of locally stationary processes often cannot be applied to a different class of locally stationary processes. In particular, many estimation methods for locally stationary processes are composed of estimation approaches that primarily focus on local regression with closed-form estimators, local maximum likelihood estimation (MLE) with a closed-form likelihood function (in the time domain) and spectral density approach (in the frequency domain), all of which could be intractable or difficult to implement for various locally stationary extensions of commonly used structural econometric models; we refer to Vogt (2012), Dahlhaus and Subba Rao (2006) and Dahlhaus and Polonik (2009), for examples. As such, model specifications compatible with the above statistical methods are rather limited and cannot be used for estimation and inference in more complicated locally stationary models, such as, for instance, models with latent variables or unobservable factors.

More importantly, structural models of economic and financial relationships commonly rely on the use of latent variables to represent information that is unavailable to the econometrician. This modeling approach implies, almost by definition, that simple (closed-form) representations for the conditional distributions of the endogenous variables are unavailable, with straightforward estimation methods often infeasible as a consequence. In such cases, if we were to extend common locally-stationary models to include the latent variables that are necessary to structurally model phenomena found in economics and finance, this would render the existing estimation methods used for such models infeasible. For instance, this situation arises in state–space models if either the measurement or state transition densities do not have closed forms, as in the case of stochastic volatility models. A secondary example is the fact that estimation of univariate locally stationary diffusion models cannot be straightforwardly extended to versions of these models with stochastic volatility.

To circumvent the above issue, and to help proliferate the use of locally stationary models and methods in econometrics and finance, we propose a novel nonparametric indirect inference (hereafter, II) method to estimate locally stationary processes. Instead of estimating complex structural locally stationary models directly, we indirectly obtain our estimator by targeting consistent estimators of simpler auxiliary models, and use these consistent estimates to conduct inference on the structural parameters. See, Smith (1993), Gourieroux et al. (1993) and Gourieroux and Monfort (1996) for discussion of indirect inference in parametric models.

To illustrate the main idea behind our nonparametric II approach for locally stationary processes, we consider the following motivating example. Suppose that the true data generating process evolves according to Yt,T=ξ(tT)expht2εt,where ht=ω+δht1+σvt,(εt,vt)N0,1001,where ξ(tT)>0, for all tT. This locally stationary multiplicative stochastic volatility (LS-SV) model decomposes volatility into a short-term, latent volatility process, ht, and a slowly time-varying component, captured by ξ(), and can capture a wide range of volatility behaviors. The above model allows for non-stationary, but slowly changing, volatility dynamics, which may result from the transitory nature of the business cycle.

Suppose that we wish to estimate and conduct inference on the unknown volatility function ξ() in (1). While (G)ARCH-based versions of the locally stationary volatility model have been analyzed by several researchers (see, e.g., Dahlhaus and Subba Rao, 2006, Engle and Rangel, 2008, Fryzlewicz et al., 2008, and Koo and Linton, 2015), since the latent volatility process, ht, pollutes the observed data, Yt,T, it is not entirely clear how to estimate the parameters in (1). Indeed, largely due to this fact, locally stationary volatility models have not been previously explored in the literature, even though their stationary counterparts form the backbone of many empirical studies in finance and financial econometrics.

In this paper, we generalize the II approach of Gourieroux et al. (1993) to present a convenient estimator for unknown functions in locally stationary models, such as the LS-SV model. This approach to II estimation relies on a locally stationary auxiliary model that can be easily estimated using the observed data and that captures the underlying features of interest in the structural model. For example, in the context of the LS-SV model, a reasonable auxiliary model would be the locally stationary GARCH model: Yt,T=ρ(tT)σtzt, where σt+12=α0+α1zt2+βσt2,where ρ(tT)>0 for all tT, and where zt is an error process.

The remainder of this paper further develops the ideas behind this estimation method in the context of a general locally stationary model and establishes the asymptotic properties of the proposed estimation procedure under regularity conditions. To establish the asymptotic properties of these II estimators, we must first develop conditions that guarantee locally stationary models admit consistent estimators of their corresponding limit values. This is itself a novel result since the vast majority of research into locally stationary models has focused on estimators defined by relatively simple criterion functions, and all under the auspices of correct model specification. Indeed, Kristensen and Lee (2019) is the only other study of which the authors are aware that treats genuinely misspecified locally stationary models. These new results for locally stationary estimators of the auxiliary model enable us to deduce the asymptotic properties of our proposed II estimator for the structural model parameters.

The estimation procedure proposed herein is demonstrated through two Monte Carlo examples, and an empirical application. The empirical application applies the LS-SV model to examine the volatility structure of several commonly analyzed Fama–French portfolios. We find that most of these portfolios display time-varying volatility patterns that broadly track the underlying (low-frequency) expansion and contractions of the United States economy.

The remainder of the paper is organized as follows. Section 2 introduces the general model and the related framework. In Section 2.3, we present our general approach and define the corresponding local II (L-II) estimators for a general locally stationary model. Section 3 develops asymptotic results that demonstrate the properties of this estimation procedure. Simulation results for a simple example of a locally stationary moving average model of order one are discussed in Section 4. In Section 5 we analyze the locally stationary stochastic volatility model. We consider a small Monte Carlo to demonstrate our estimation method, then apply this method to analyze the volatility behavior of Fama–French portfolio returns, where we find ample evidence for smoothly time-varying nonlinear volatility dynamics over the sample period. All proofs are relegated to Appendix A. The proof of Corollary 2 and additional details for the LS-SV model are provided in the supplementary appendix.

Throughout this paper, the following notations are used. The symbol R denotes the real numbers, while N denotes the natural numbers. For g:RdR denoting a given function, we let gsupxRd|g(x)| denote the sup-norm, and some arbitrary norm. However, for xRd, we abuse notation and let x denote the Euclidean norm. Let || denote the absolute value function, and for Ω a d×d positive-definite matrix, we let xΩ2xΩx denote the weighted norm of x. For an unknown parameter θ, the subscript 0 denotes the true value of θ. The quantities Op() and op() denote the usual big O and little o in probability. C denotes a generic constant that can take different values in different places.

Section snippets

Structural models

We assume the researcher is interested in conducting inference on a model in the class of locally stationary processes.

Definition 1

Let {Yt,T}t=1,,T;T=1,2, denote a triangular array of observations. The process {Yt,T} is locally stationary if there exists a stationary process {ytT,t} for each re-scaled time point tT[0,1], such that for all T, Pmax1tT|Yt,TytT,t|CTT1=1,where {CT} is a measurable process satisfying, for some η>0, supTE|CT|η<.

The magnitude of η captures the degree of approximation

Asymptotic behavior of L-II

This section establishes the asymptotic properties of the L-II estimator. We establish the convergence (in probability) of θˆ() to θ0() and provide the asymptotic distribution of θˆ() under a fairly general setup.

Before presenting the details, we introduce the limit quantities that will be needed for our results. Consider the limit objective function and its minimizer corresponding to sample quantities, i.e. (7), (8), such that, for uU=[δ,1δ] and a small, positive δ=o(1), ρ0(u;θ0(u))arg

Simple example

In this section, we consider a simple generalization of the time-varying moving average model that allows the roots of the moving average lag polynomial to be time-varying. After presenting the model, we demonstrate how our L-II approach can be applied to estimate the model and present simulation results on the effectiveness of this strategy.

Time-varying multiplicative stochastic volatility model

The use of stochastic volatility to capture the conditional heteroskedastic movements of asset returns is now commonplace in economics and finance. Recently, however, several authors have suggested that volatility should be decomposed into short and long-run components (see, e.g., Engle and Rangel, 2008 and Engle et al., 2013). Such a decomposition has given rise to the class of multiplicative time-varying GARCH models, e.g. Koo and Linton (2015). Such models decompose volatility into a

Discussion

We propose a novel II estimator for locally stationary processes and thereby extend, for the first time, the use of II estimation to general classes of semiparametric models with slowly time-varying parameters. As part of this study, we also propose a novel local stationary multiplicative stochastic volatility (LS-SV) model. We leave two important topics for future research: the efficiency of the L-II estimator, and the ensuing semiparametric efficiency bound for the class of locally stationary

Acknowledgments

We are grateful to Jianqing Fan, the associated editor and three anonymous referees whose comments improved the paper significantly. We would like to thank seminar and conference participants for helpful comments. Frazier acknowledges financial support from the Australian Research Council Grant No. DE200101070. Koo acknowledges financial support from the Australian Research Council Grant No. DE170100713.

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