Stationary ARMA Processes Research Articles

Efficient nonparametric estimation of generalised autocovariances

This paper provides a necessary and sufficient condition for asymptotic efficiency of a nonparametric estimator of the generalised autocovariance function of a stationary random process. The generalised autocovariance function is the inverse Fourier transform of a power transformation of the spectral density and encompasses the traditional and inverse autocovariance functions as particular cases. A nonparametric estimator is based on the inverse discrete Fourier transform of the power transformation of the pooled periodogram. We consider two cases: the fixed bandwidth design and the adaptive bandwidth design. The general result on the asymptotic efficiency, established for linear processes, is then applied to the class of stationary ARMA processes and its implications are discussed. Finally, we illustrate that for a class of contrast functionals and spectral densities, the minimum contrast estimator of the spectral density satisfies a Yule–Walker system of equations in the generalised autocovariance estimator.

A two-step estimation procedure for locally stationary ARMA processes with tempered stable innovations

A two-step estimation procedure for locally stationary ARMA processes with tempered stable innovations

Factorization and discrete-time representation of multivariate CARMA processes

In this paper we show that stationary and non-stationary multivariate continuous-time ARMA (MCARMA) processes have the representation as a sum of multivariate complex-valued Ornstein-Uhlenbeck processes under some mild assumptions. The proof benefits from properties of rational matrix polynomials. A conclusion is an alternative description of the autocovariance function of a stationary MCARMA process. Moreover, that representation is used to show that the discrete-time sampled MCARMA(p,q) process is a weak VARMA(p,p-1) process if second moments exist. That result complements the weak VARMA(p,p-1) representation derived in Chambers and Thornton (2012). In particular, it relates the right solvents of the autoregressive polynomial of the MCARMA process to the right solvents of the autoregressive polynomial of the VARMA process; in the one-dimensional case the right solvents are the zeros of the autoregressive polynomial. Finally, a factorization of the sample autocovariance function of the noise sequence is presented which is useful for statistical inference.

Kullback-Leibler and Rényi divergence rate for Gaussian stationary ARMA processes comparison

In signal processing, ARMA processes are widely used to model short-memory processes. In various applications, comparing or classifying ARMA processes is required. In this paper, our purpose is to provide analytical expressions of the divergence rates of the Kullback-Leibler divergence, the Rényi divergence (RD) of order α and their symmetric versions for two Gaussian ARMA processes, by taking advantage of results such as the Yule-Walker equations and notions such as inverse filtering. The divergence rates can be interpreted as the sum of different quantities: power of one ARMA process filtered by the inverse filter associated with the second ARMA process, cepstrum, etc. Finally, illustrations show that the ranges of values taken by the divergence rates of the RD are sensitive to α, especially when the latter is close to 1.

Efficient Nonparametric Estimation of Generalized Autocovariances

This paper provides a necessary and sufficient condition for asymptotic efficiency of a nonparametric estimator of the generalized autocovariance function of a stationary random process. The generalized autocovariance function is the inverse Fourier transform of a power transformation of the spectral density and encompasses the traditional and inverse autocovariance functions as particular cases. A nonparametric estimator is based on the inverse discrete Fourier transform of the power transformation of the pooled periodogram. The general result on the asymptotic efficiency is then applied to the class of Gaussian stationary ARMA processes and its implications are discussed. Finally, we illustrate that for a class of contrast functionals and spectral densities, the minimum contrast estimator of the spectral density satisfies a Yule-Walker system of equations in the generalized autocovariance estimator.

Indirect inference for locally stationary ARMA processes with stable innovations

The class of locally stationary processes assumes that there is a time-varying spectral representation, that is, the existence of finite second moment. We propose the α-stable locally stationary process by modifying the innovations into stable distributions and the indirect inference to estimate this type of model. Due to the infinite variance, some of interesting properties such as time-varying autocorrelation cannot be defined. However, since the α-stable family of distributions is closed under linear combination which includes the possibility of handling asymmetry and thicker tails, the proposed model has the same tail behaviour throughout the time. In this paper, we propose this new model, present theoretical properties of the process and carry out simulations related to the indirect inference in order to estimate the parametric form of the model. Finally, an empirical application is illustrated.

Jeffrey's divergence between ARFIMA processes

The symmetric Kullback–Leibler divergence known as Jeffrey's divergence (JD) has found applications in signal and image processing, from radar clutter modeling to texture analysis. Recently, several studies were done on the JD between ergodic wide-sense stationary autoregressive (AR) and/or moving average (MA) processes. It was shown that the so-called asymptotic JD increment can be useful to compare ergodic wide-sense stationary ARMA processes. An interpretation of the asymptotic JD increment was also proposed. It consists in calculating the power of the first process filtered by the inverse filter associated with the second process, and conversely. However, in some biomedical applications, econometrics and other areas, long-memory processes have rather to be studied. Therefore, this paper aims at addressing the JD between ergodic wide-sense stationary autoregressive fractionally integrated moving average (ARFIMA) processes. More particularly, we study the influence of the ARFIMA parameters on the value of the asymptotic JD increment. Then, we analyze if the interpretation of the asymptotic JD increment based on inverse filtering is still valid for this type of process. Finally, some simulation results illustrate the theoretical analysis.

A Note on Exponential Inequalities of ψ-Weakly Dependent Sequences

Two exponential inequalities are established for a wide class of general weakly dependent sequences of random variables, called <TEX>${\psi}$</TEX>-weakly dependent process which unify weak dependence conditions such as mixing, association, Gaussian sequences and Bernoulli shifts. The <TEX>${\psi}$</TEX>-weakly dependent process includes, for examples, stationary ARMA processes, bilinear processes, and threshold autoregressive processes, and includes essentially all classes of weakly dependent stationary processes of interest in statistics under natural conditions on the process parameters. The two exponential inequalities are established on more general conditions than some existing ones, and are proven in simpler ways.

Limited information minimal state variable learning in a medium-scale multi-country model

Rational expectations has been the dominant way to model expectations, but the literature has quickly moved to a more realistic assumption of boundedly rational learning where agents are assumed to use only a limited set of information to form their expectations. A standard assumption is that agents form expectations by using the correctly specified reduced form model of the economy, the minimal state variable solution (MSV), but they do not know the parameters. However, with medium-sized and large models the closed-form MSV solutions are difficult to attain given the large number of variables that could be included. Therefore, agents base expectations on a misspecified MSV solution. In contrast, we assume that agents know the deep parameters of their own optimising frameworks. However, they are not assumed to know the structure nor the parameterisation of the rest of the economy, nor do they know the stochastic processes generating shocks hitting the economy. In addition, agents are assumed to know that the changes (or the growth rates) of fundament variables can be modelled as stationary ARMA(p,q) processes, the exact form of which is not, however, known by agents. This approach avoids the complexities of dealing with a potential vast multitude of alternative misspecified MSVs.Using a new multi-country euro area model with boundedly estimated rationality we show that this approach is compatible with the same limited information assumption that was used in deriving and estimating the behavioural equations of different optimising agents. We find that there are strong differences in the adjustment path to the shocks to the economy when agents form expectations using our learning approach compared to expectations formed under the assumption of strong rationality. Furthermore, we find some variation in expansionary fiscal policy in periods of downturns compared to boom periods.

Strictly stationary solutions of ARMA equations with fractional noise

We obtain necessary and sufficient conditions for the existence of strictly stationary solutions of ARMA equations with fractional noise. Here, the underlying noise sequence of the fractional noise is assumed to be i.i.d. but no a priori moment assumptions are made. We also characterize for which i.i.d. driving noise sequences the series defining fractional noise converges almost surely. In the proofs, we use growth estimates for the moments of random walks developed by Manstavičius (1982) and techniques related to those of Brockwell and Lindner (2010) for the existence of strictly stationary ARMA processes with i.i.d. noise.

Effect of increasing the forecast horizon on correlation between forecasted returns and actual returns: an empirical analysis

This study investigates the influence of increasing the forecast horizon on correlation between the forecast returns and the actual returns. ARMA and EGARCH models are used in this paper to capture univariate asset returns. ARMA model is conditional mean model and EGARCH model is conditional variance model and both of them are used for forecasting, parameter estimation and simulation of time series. The persistence in the volatility of the time series is usually exemplified by a highly persistent fitted EGARCH model. Traditional stationary ARMA processes are often used to capture high degree of persistence in financial time series. In this paper, a hybrid of EGARCH and ARMA model is used to forecast stock returns based on their actual returns then we have analysed the correlation trend between actual and forecasted returns by varying the forecast horizon. Results shown in this paper are depicting that correlation between actual and forecast returns undergoes a decaying pattern on increasing forecast horizon. Stock prices of three companies CIPLA, GAIL and LIC have been taken on daily basis from April 1997 to April 2010 and prediction has been done for next 150 days.

Semiparametric estimation for partially linear models with [formula omitted]-weak dependent errors

Semiparametric estimators are developed for a partially linear regression model with ψ -weakly dependent errors. The ψ -weak dependence condition, introduced by Doukhan and Louhich [Doukhan, P., and Louhich, S. (1999). A new weak dependence condition and applications to moment inequalities. Stochastic Processes and their Applications, 84, 313–342], unifies weak dependence conditions such as mixing, association, Gaussian sequences and Bernoulli shifts. The class of ψ -weak dependent processes includes many important nonlinear processes such as stationary threshold autoregressive processes and bilinear processes as well as stationary ARMA processes. Asymptotic normalities are established for semiparametric generalized least squares estimators of the parametric component and for estimators of the nonparametric function. Expansions are obtained for the biases and variances of the estimators. Real data set and simulated data set analyses are provided for a model with a threshold autoregressive error process.

The ECB's New Multi-Country Model for the Euro Area: NMCM - With Boundedly Rational Learning Expectations

Rational expectations has been the dominant way to model expectations, but the literature has quickly moved to a more realistic assumption of boundedly rational learning where agents are assumed to use only a limited set of information to form their expectations. A standard assumption is that agents form expectations by using the correctly specified reduced form model of the economy, the minimal state variable solution (MSV), but they do not know the parameters. However, with medium-sized and large models the closed-form MSV solutions are difficult to attain given the large number of variables that could be included. Therefore, agents base expectations on a misspecified MSV solution. In contrast, we assume agents know the deep parameters of their own optimising frameworks. However, they are not assumed to know the structure nor the parameterisation of the rest of the economy, nor do they know the stochastic processes generating shocks hitting the economy. In addition, agents are assumed to know that the changes (or the growth rates) of fundament variables can be modelled as stationary ARMA(p,q) processes, the exact form of which is not, however, known by agents. This approach avoids the complexities of dealing with a potential vast multitude of alternative mis-specified MSVs. Using a new Multi-country Euro area Model with Boundedly Estimated Rationality we show this approach is compatible with the same limited information assumption that was used in deriving and estimating the behavioral equations of different optimizing agents. We find that there are strong differences in the adjustment path to the shocks to the economy when agent form expectations using our learning approach compared to expectations formed under the assumption of strong rationality. Furthermore, we find that some variation in expansionary fiscal policy in periods of downturns compared to boom periods.

Robust estimation for order of hidden Markov models based on density power divergences

In this paper, we study the robust estimation for the order of hidden Markov model (HMM) based on a penalized minimum density power divergence estimator, which is obtained by utilizing the finite mixture marginal distribution of HMM. For this task, we adopt the locally conic parametrization method used in [D. Dacunha-Castelle and E. Gassiate, Testing in locally conic models and application to mixture models. ESAIM Probab. Stat. (1997), pp. 285–317; D. Dacunha-Castelle and E. Gassiate, Testing the order of a model using locally conic parametrization: population mixtures and stationary arma processes, Ann. Statist. 27 (1999), pp. 1178–1209; T. Lee and S. Lee, Robust and consistent estimation of the order of finite mixture models based on the minimizing a density power divergence estimator, Metrika 68 (2008), pp. 365–390] to avoid the difficulties that arise in handling mixture marginal models, such as the non-identifiability of the parameter space and the singularity problem with the asymptotic variance. We verify that the estimated order is consistent and simulation results are provided for illustration.

Discrete-valued ARMA processes

This paper presents a unified framework of stationary ARMA processes for discrete-valued time series based on Pegram’s [Pegram, G.G.S., 1980. An autoregressive model for multilag markov chains. J. Appl. Probab. 17, 350–362] mixing operator. Such a stochastic operator appears to be more flexible than the currently popular thinning operator to construct Box and Jenkins’ type stationary ARMA processes with arbitrary discrete marginal distributions. This flexibility allows us to yield an ARMA model for time series of binomial or categorical observations as a special case, which was unavailable with the extended thinning operator [Joe, H., 1996. Time series models with univariate margins in the convolution-closed infinitely divisible class. J. Appl. Probab. 33, 664–677] because the binomial/categorical distribution is not infinitely divisible. We also study parameter estimation and comparison with the thinning operator based method, whenever applicable. Real data examples are used to examine and illustrate the proposed method.

Testing for Equal Predictability of Stationary ARMA Processes

In this work we use a measure of predictability of a time series following a stationary ARMA process to develop a test of equal predictability of two or more time series. The test is derived by a set of propositions which links the structure of the AR and MA coefficients to the predictability measure. A particular case of this general approach is constituted by time series having a Wold decomposition with weights having the same sign; in this framework the equal predictability is equivalent to parallelism among ARMA models and the null hypothesis of equal predictability is simply a set of linear restrictions. The ARMA representation of the GARCH models presents non-negative weights, so that this test can be extended to verify the equal predictability of squared time series following GARCH structures.

Test for Parameter Change in Linear Processes Based on Whittle's Estimator

In this article, we develop a cusum test for testing for parameter changes in linear processes based on Whittle's estimator. It is shown that under regularity conditions, the test statistic converges to the sup of a Brownian bridge. The result is particularly useful in handling the change point test in stationary ARMA processes. A simulation result is provided for illustration.

Financial applications of ARMA models with GARCH errors

Purpose – Financial returns are often modeled as stationary time series with innovations having heteroscedastic conditional variances. This paper seeks to derive the kurtosis of stationary processes with GARCH errors. The problem of hypothesis testing for stationary ARMA(p, q) processes with GARCH errors is studied. Forecasting of ARMA(p, q) processes with GARCH errors is also discussed in some detail.Design/methodology/approach – Estimating‐function methodology was the principal method used for the research. The results were also illustrated using examples and simulation studies. Volatility modeling is the subject of the paper.Findings – The kurtosis of stationary processes with GARCH errors is derived in terms of the model parameters (ψ), Ψ‐weights, and the kurtosis of the innovation process. Hypothesis testing for stationary ARMA(p, q) processes with GARCH errors based on the estimating‐function approach is shown to be superior to the least‐squares approach. The fourth moment of the l‐steps‐ahead forec...

Asymptotics for estimation and testing procedures under loss of identifiability

Statistical analyses commonly make use of models that suffer from loss of identifiability. In this paper, we address important issues related to the parameter estimation and hypothesis testing in models with loss of identifiability. That is, there are multiple parameter points corresponding to the same true model. We refer the set of these parameter points to as the set of true parameter values. We consider the case where the set of true parameter values is allowed to be very large or even infinite, some parameter values may lie on the boundary of the parameter space, and the data are not necessarily independently and identically distributed. Our results are applicable to a large class of estimators and their related testing statistics derived from optimizing an objective function such as a likelihood. We examine three specific examples: (i) a finite mixture logistic regression model; (ii) stationary ARMA processes; (iii) general quadratic approximation using Hellinger distance. The applications to these examples demonstrate the applicability of our results in a broad range of difficult statistical problems.

Whittle estimation in a heavy-tailed GARCH(1,1) model

The squares of a GARCH( p, q) process satisfy an ARMA equation with white noise innovations and parameters which are derived from the GARCH model. Moreover, the noise sequence of this ARMA process constitutes a strongly mixing stationary process with geometric rate. These properties suggest to apply classical estimation theory for stationary ARMA processes. We focus on the Whittle estimator for the parameters of the resulting ARMA model. Giraitis and Robinson (2000) show in this context that the Whittle estimator is strongly consistent and asymptotically normal provided the process has finite 8th moment marginal distribution. We focus on the GARCH(1,1) case when the 8th moment is infinite. This case corresponds to various real-life log-return series of financial data. We show that the Whittle estimator is consistent as long as the 4th moment is finite and inconsistent when the 4th moment is infinite. Moreover, in the finite 4th moment case rates of convergence of the Whittle estimator to the true parameter are the slower, the fatter the tail of the distribution. These findings are in contrast to ARMA processes with iid innovations. Indeed, in the latter case it was shown by Mikosch et al. (1995) that the rate of convergence of the Whittle estimator to the true parameter is the faster, the fatter the tails of the innovations distribution. Thus the analogy between a squared GARCH process and an ARMA process is misleading insofar that one of the classical estimation techniques, Whittle estimation, does not yield the expected analogy of the asymptotic behavior of the estimators.