Testing for Predictability in a Noninvertible ARMA Model

We develop likelihood-based tests for autocorrelation and predictability in a first order non- Gaussian and noninvertible ARMA model. Tests based on a special case of the general model, referred to as an all-pass model, are also obtained. Data generated by an all-pass process are uncorrelated but, in the non-Gaussian case, dependent and nonlinearly predictable. Therefore, in addition to autocorrelation the proposed tests can also be used to test for nonlinear predictability. This makes our tests different from their previous counterparts based on conventional invertible ARMA models. Unlike in the invertible case, our tests can also be derived by standard methods that lead to chi-squared or standard normal limiting distributions. A further convenience of the noninvertible ARMA model is that, to some extent, it can allow for conditional heteroskedasticity in the data which is useful when testing for predictability in economic and financial data. This is also illustrated by our empirical application to U.S. stock returns, where our tests indicate the presence of nonlinear predictability

Residual-based diagnostic tests for noninvertible ARMA models

This paper proposes two residual-based diagnostic tests for noninvertible ARMA models. The tests are analogous to the portmanteau tests developed by Box and Pierce (1970), Ljung and Box (1978) and McLeod and Li (1983) in the conventional invertible case. We derive the asymptotic chi-squared distributions for the tests and study the size and power properties in a Monte Carlo simulation study. An empirical application employing financial time series data points out the usefulness of noninvertible ARMA model in analyzing stock returns and the use of the proposed test statistics

Residual-based diagnostic tests for noninvertible ARMA models

This paper proposes two residual-based diagnostic tests for noninvertible ARMA models. The tests are analogous to the portmanteau tests developed by Box and Pierce (1970), Ljung and Box (1978) and McLeod and Li (1983) in the conventional invertible case. We derive the asymptotic chi-squared distributions for the tests and study the size and power properties in a Monte Carlo simulation study. An empirical application employing financial time series data points out the usefulness of noninvertible ARMA model in analyzing stock returns and the use of the proposed test statistics

Noncausal autoregressions for economic time series

This paper is concerned with univariate noncausal autoregressive models and their potential usefulness in economic applications. In these models, future errors are predictable, indicating that they can be used to empirically approach rational expectations models with nonfundamental solutions. In the previous theoretical literature, nonfundamental solutions have typically been represented by noninvertible moving average models. However, noncausal autoregressive and noninvertible moving average models closely approximate each other, and therefore,the former provide a viable and practically convenient alternative. We show how the parameters of a noncausal autoregressive model can be estimated by the method of maximum likelihood and derive related test procedures. Because noncausal autoregressive models cannot be distinguished from conventional causal autoregressive models by second order properties or Gaussian likelihood, a model selection procedure is proposed. As an empirical application, we consider modeling the U.S. inflation which, according to our results, exhibits purely forward-looking dynamics.Noncausal autoregression; expectations; inflation persistence

Spectral identification and estimation of mixed causal-noncausal invertible-noninvertible models

This paper introduces new techniques for estimating, identifying and simulating mixed causal-noncausal invertible-noninvertible models. We propose a framework that integrates high-order cumulants, merging both the spectrum and bispectrum into a single estimation function. The model that most adequately represents the data under the assumption that the error term is i.i.d. is selected. Our Monte Carlo study reveals unbiased parameter estimates and a high frequency with which correct models are identified. We illustrate our strategy through an empirical analysis of returns from 24 Fama-French emerging market stock portfolios. The findings suggest that each portfolio displays noncausal dynamics, producing white noise residuals devoid of conditional heteroscedastic effects

Essays on noninvertible ARMA models

The theory for conventional Gaussian, causal and invertible autoregressive moving average (ARMA) models has developed into a what can be considered as a basis of modern time series analysis. The concept of noninvertibility is plausible only under non-Gaussian processes because non-Gaussianity is a necessary condition for the statistical identification of the noninvertible ARMA model. Therefore, if the Gaussianity assumption is relaxed, we can study a richer class of models which are, unlike their invertible counterparts, capable of capturing nonlinear patterns in the data. The aim of this thesis is to consider some of the novel results in ARMA modeling of stationary time series data, and to expand these results to a particular case of noninvertibility and non-Gaussianity of the model. It also aims at providing insights on applicability of noninvertible ARMA models in financial time series analysis. The first essay proposes two residual-based diagnostic tests for noninvertible ARMA models. The tests are analogous to the portmanteau tests developed by Box and Pierce (1970) and McLeod and Li (1983) in the conventional invertible case. We derive the asymptotic chi-squared distribution for the tests under the null of correctly specified model, and study the size and power properties in a Monte Carlo simulation study. An empirical application employing financial time series data points out the usefulness of noninvertible ARMA model in analyzing stock returns and the use of the proposed test statistics. The second essay studies properties of the maximum likelihood estimator of a noninvertible ARMA model with errors that follow an alpha-stable distribution and have infinite variance. To ensure the identification of the noninvertible ARMA model considered, we restrict the analysis to non-Gaussian distributions. Estimators of the autoregressive and moving average parameters are shown to be square root of n consistent and to converge to a non-standard limiting distribution that is obtained as a maximizer of a certain random function. Estimators of the parameters in the alpha-stable distribution have the conventional square root of n rate of convergence. Finite sample properties of the estimators are studied in a simulation experiment, and an application to financial trading volume data illustrates the applicability of the model. The third and last essay looks for nonlinear predictability in stock returns. For many theoretical asset pricing models, predictability follows as an implication of risk aversion of agents. A closed form solutions for the next periods asset return depends on how the agents form their expectations about the future state of the world. By no means should this predictability be linear. First, we provide evidence of predictability of returns of U.S. stock portfolios and individual financial sector stocks using noninvertible ARMA(1,1) model and two-stage predictability testing procedure. Second, we provide a straightforward extensions to this procedure and allow for a larger model than the non-invertible ARMA(1,1).Väitöskirja koostuu kolmesta itsenäisestä esseestä, jotka käsittelevät ei-kääntyvää autoregressiivistä liukuvan keskiarvon (ARMA) mallia. Tavanomaiset ARMA-mallit on havaittu erittäin hyödyllisiksi aikasarja-aineistojen analysoinnissa, ja niille on kehitetty hyvin laaja teoreettinen pohja muun muassa mallin parametrien estimointiin ja mallin sopivuuden tarkasteluun. Useissa tapauksissa teoreettiset tulokset on johdetut olettaen, että mallien virhetermi on normaalijakautunut. Tämän oletuksen vallitessa ei-kääntyvän mallin tilastollinen identifioiminen on kuitenkin mahdotonta. Tässä väitöstutkimuksessa luovutaan normaalisuusoletuksesta, jonka johdosta voimme tarkastella ei-kääntyviä malleja. Nämä mallit ovat hyödyllisiä muun muassa siksi, että niiden avulla voidaan mallintaa aineistoissa havaittavia epälineaarisuuksia. Väitöstutkimuksessa johdetaan tavanomaisten ARMA-mallien keskeisiä teoreettisia tuloksia ei-normaalisille ja ei-kääntyville ARMA-malleille. Ensimmäinen essee käsittelee estimoidun ei-kääntyvän ARMA-mallin sopivuuden testaamista. Siinä johdettavien asymptoottiseen teoriaan nojaavien testisuureiden avulla on mahdollista tarkastella, toteuttaako estimoidun mallin virhetermit niiltä vaadittua riippumattomuusoletusta. Jos hypoteesi riippumattomuudesta jää voimaan, voidaan estimoitua mallia pitää hyvänä aineiston analysoinnissa. Toinen essee käsittelee ei-kääntyvän ARMA-mallin parametrien estimointia tilanteessa, jossa aineistossa on havaittavissa hyvin suuria poikkeamia. Tätä prosessia voidaan mallintaa olettamalla virhetermeille alpha-stabiili jakauma, jonka varianssi ja mahdollisesti myös ensimmäinen momentti ovat äärettömän suuria. Mallin parametrien estimaattoreiden jakauma lähestyy jotain tiettyä mutta ei kuitenkaan tunnettua jakaumaa. Virhetermin jakauman parametrien estimaattorit sen sijaan ovat asymptoottisesti normaalijakautuneita. Kolmannessa esseessä tutkitaan osakeportfolioiden tuottojen ennustettavuutta käyttäen ei-kääntyvää ARMA-mallia. Tämä malli soveltuu erityisen hyvin ennustettavuuden tarkasteluun silloin, kun havainnot ovat lineaarisesti riippumattomia, mutta aikasarjoissa on havaittavissa epälineaarisia piirteitä. Esseessä analysoidaan useiden osakeportfolioiden tuottosarjoja hyödyntäen mallille kehitettyä estimointiteoriaa, mallinvalintamenetelmiä, sekä kaksivaiheista ennustettavuustestiä

Optimal Forecasting of Noncausal Autoregressive Time Series

In this paper, we propose a simulation-based method for computing point and density forecasts for univariate noncausal and non-Gaussian autoregressive processes. Numerical methods are needed to forecast such time series because the prediction problem is generally nonlinear and no analytic solution is therefore available. According to a limited simulation experiment, the use of a correct noncausal model can lead to substantial gains in forecast accuracy over the corresponding causal model. An empirical application to U.S. inflation demonstrates the importance of allowing for noncausality in improving point and density forecasts.Noncausal autoregression; density forecast; inflation

Essays on noncausal and noninvertible time series

Over the last two decades, there has been growing interest among economists in nonfundamental univariate processes, generally represented by noncausal and non-invertible time series. These processes have become increasingly popular due to their ability to capture nonlinear dynamics such as volatility clustering, asymmetric cycles, and local explosiveness - all of which are commonly observed in Macroeconomics and Finance. In particular, the incorporation of both past and future components into noncausal and noninvertible processes makes them attractive options for modeling forward-looking behavior in economic activities. However, the classical techniques used for analyzing time series models are largely limited to causal and invertible counterparts. This dissertation seeks to contribute to the field by providing theoretical tools robust to noncausal and noninvertible time series in testing and estimation. In the first chapter, "Quantile Autoregression-Based Non-causality Testing", we investigate the statistical properties of empirical conditional quantiles of non-causal processes. Specifically, we show that the quantile autoregression (QAR) estimates for non-causal processes do not remain constant across different quantiles in contrast to their causal counterparts. Furthermore, we demonstrate that non-causal autoregressive processes admit nonlinear representations for conditional quantiles given past observations. Exploiting these properties, we propose three novel testing strategies of non-causality for non-Gaussian processes within the QAR framework. The tests are constructed either by verifying the constancy of the slope coefficients or by applying a misspecification test of the linear QAR model over different quantiles of the process. Some numerical experiments are included to examine the finite sample performance of the testing strategies, where we compare different specification tests for dynamic quantiles with the Kolmogorov-Smirnov constancy test. The new methodology is applied to some time series from financial markets to investigate the presence of speculative bubbles. The extension of the approach based on the specification tests to AR processes driven by innovations with heteroskedasticity is studied through simulations. The performance of QAR estimates of non-causal processes at extreme quantiles is also explored. In the second chapter, "Estimation of Time Series Models Using the Empirical Distribution of Residuals", we introduce a novel estimation technique for general linear time series models, potentially noninvertible and noncausal, by utilizing the empirical cumulative distribution function of residuals. The proposed method relies on the generalized spectral cumulative function to characterize the pairwise dependence of residuals at all lags. Model identification can be achieved by exploiting the information in the joint distribution of residuals under the iid assumption. This method yields consistent estimates of the model parameters without imposing stringent conditions on the higher-order moments or any distributional assumptions on the innovations beyond non-Gaussianity. We investigate the asymptotic distribution of the estimates by employing a smoothed cumulative distribution function to approximate the indicator function, considering the non-differentiability of the original loss function. Efficiency improvements can be achieved by properly choosing the scaling parameter for residuals. Finite sample properties are explored through Monte Carlo simulations. An empirical application to illustrate this methodology is provided by fitting the daily trading volume of Microsoft stock by autoregressive models with noncausal representation. The flexibility of the cumulative distribution function permits the proposed method to be extended to more general dependence structures where innovations are only conditional mean or quantile independent. In the third chapter, "Directional Predictability Tests", joint with Carlos Velasco, we propose new tests of predictability for non-Gaussian sequences that may display general nonlinear dependence in higher-order properties. We test the null of martingale difference against parametric alternatives which can introduce linear or nonlinear dependence as generated by ARMA and all-pass restricted ARMA models, respectively. We also develop tests to check for linear predictability under the white noise null hypothesis parameterized by an all-pass model driven by martingale difference innovations and tests of non-linear predictability on ARMA residuals. Our Lagrange Multiplier tests are developed from a loss function based on pairwise dependence measures that identify the predictability of levels. We provide asymptotic and finite sample analysis of the properties of the new tests and investigate the predictability of different series of financial returns.This thesis has been possible thanks to the financial support from the grant BES-2017-082695 from the Ministerio de Economía Industria y Competitividad.Programa de Doctorado en Economía por la Universidad Carlos III de MadridPresidente: Miguel ángel Delgado González.- Secretario: Manuel Domínguez Toribio.- Vocal: Majid M. Al Sadoo