ARMA Processes Research Articles

Consistency of minimum description length model selection for piecewise stationary time series models

This paper establishes the consistency of the minimum description length (MDL) model selection procedure by [10, 11] for a class of non-stationary time series models. We consider a time series model in which the observations are viewed as coming from stationary segments. In other words, the data are assumed to come from a general time series model in which the parameters change at break-points. Each of these segments is modeled by a pre-specified family of parametric stationary time series models. [10, 11] formulated the above problem and used the minimum description length (MDL) principle to estimate the number of break-points, the location of the break-points, the order of the parametric model and the parameter values in each of the segments. The procedure performed well on a variety of examples. In this paper we show consistency of their minimal MDL model selection procedure under general regularity conditions on the likelihood function. Results about the rate of convergence of the break-point-location estimator are also given. Applications are considered for detecting changes in independent random variables, and in ARMA and GARCH processes.

Rate of convergence in the central limit theorem for parameter estimation in a causal, invertible ARMA(p, q) model

In this study we consider the estimators of the parameters of a stable ARMA(p, q) process. The autoregressive parameters are estimated by the instrumental variable technique while the moving average parameters are estimated using a derived autoregressive process. The estimators are shown to be asymptotically normal and their rate of convergence to normality is derived.

Monetary dynamics in post inflation Bolivia

Is inflation always explained as a monetary phenomenon? In this study, the author presents empirical evidence regarding the relation of Money Growth and its effect on Inflation, for the specific case of Bolivia. The text describes the situation of the country analyzed in terms of monetary policy after one of the worst cases of hyperinflation ever; empirical evidence is presented trough two independent estimations, one following the Cochrane-Orcutt procedure, and the other following a Box-Jenkins procedure (ARMA process) that confirms the relation between Money Growth and Inflation in Bolivia using data for the years 1998 to 2010.

Comparing ARMA Processes with Roots of Modulus 1 and Polynomial Regression

Consider a process defined as where 𝔹 is the backward operator, Q a polynomial with all zeros of modulus ≥1, and (Y t ) an ergodic stationary process. We show that , even if some zeros (real or complex) have modulus 1. In particular, if (Y t ) is a white noise, it becomes the innovation of (X t ). It follows that the polynomial regression model where P is a polynomial of degree p and (ϵ t ) is a white noise, is a zero-mean IMA(p + 1, p + 1) and that (ϵ t ) is the innovation of the MA part. A practical consequence of this fact is that the ARIMA model remains competitive even if the model is a genuine polynomial regression. Various numerical simulations illustrate that point.

Modelling volatility and financial market risk of shares on the Johannesburg stock exchange

In this paper, we develop ARMA-GARCH type models for modelling volatility and financial market risk of shares on the Johannesburg Stock Exchange under the assumption of a skewed Student-t distribution. Daily data is used for the period 2002 to 2010. Several GARCH type models are used including threshold GARCH, GARCH-in mean and exponential GARCH. The results suggest that daily returns can be characterized by an ARMA (0, 1) process. This means that shocks to conditional mean dissipate after one period. Empirical results show that ARMA (0,1)-GARCH(1, 1) model achieves the most accurate volatility forecast. These results are useful to financial managers and modellers in both emerging and developed economies.

Statistical analysis of discrete-valued time series using categorical ARMA models

This paper concerns the analysis of discrete-valued time series using a class of categorical ARMA models recently proposed by Biswas and Song (2009). Such ARMA processes are flexible to model discrete-valued time series, allowing a wide range of marginal distributions such as binomial, multinomial, Poisson and nominal/ordinal categorical probability mass functions. To apply these models in the data analysis this paper focuses on the development of a needed statistical toolbox, which includes maximum likelihood estimation and inference, model selection, and goodness-of-fit test. Particularly in AR models a bias-corrected AIC statistic is derived for the order selection, while a randomized conditional moment (RCM) test is furnished to examine the goodness-of-fit. Finite-sample performances of the proposed methods are examined through simulation studies, in which the bias-corrected AIC is shown to outperform the traditional AIC and BIC statistics and the RCM test achieves desirable power. As part of the numeric illustration, a data analysis of categorical time series on infant sleep quality is provided by the application of this new toolbox.

Fama-Macbeth Test Under Firm Dynamics

We analyze the bias in the standard error caused by the correlated variance-covariance error matrix from using the Fama-McBeth method with panel data. We propose a three step Generalized Fama-McBeth method to model the correlated errors. In the first step we estimate the betas using cross-sectional regressions. Next, the betas are fitted with a a stationary ARMA process. We then test the drift parameter of the ARMA process of the betas. We show that this is equivalent to testing for the beta being equal to zero. We also model a stochastic entry/exit process for firms in a dataset which helps estimate and test the betas with this method for a more general class of error process. This approach is much simpler and easier to implement than the methods commonly used to correct for the OLS standard errors for large data sets.

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.

Annual stream flow simulation by ARMA processes and prediction by Kalman filter

Algeria is a semi-arid country where water resources are not sufficient to cope with the important socio-economic development demands. Any sustainable development strategy is basically dependent on a rigorous management of water resources potential, which presents a true challenge to be tackled for such countries. Classic mathematical models based on time invariability and ignorance of the stochastic and non-linear nature of hydrological variables are not sufficient for simulation and prediction studies. The present study subscribes to the stochastic hydrological processes modeling and prediction in case of time-varying linear systems through adaptive Kalman filter (KF) methodology. It focuses upon stream flows as a water resources component, which is directly related to the socio-economic development meanwhile it opts for Kalman filter as principal tool of work. The main objective is the application of (KF) technique to model and predict annual streamflow volumes in Northern Algeria and the obtained results are satisfactory.

Multivariate CARMA processes, continuous-time state space models and complete regularity of the innovations of the sampled processes

The class of multivariate L\'{e}vy-driven autoregressive moving average (MCARMA) processes, the continuous-time analogs of the classical vector ARMA processes, is shown to be equivalent to the class of continuous-time state space models. The linear innovations of the weak ARMA process arising from sampling an MCARMA process at an equidistant grid are proved to be exponentially completely regular ($\beta$-mixing) under a mild continuity assumption on the driving L\'{e}vy process. It is verified that this continuity assumption is satisfied in most practically relevant situations, including the case where the driving L\'{e}vy process has a non-singular Gaussian component, is compound Poisson with an absolutely continuous jump size distribution or has an infinite L\'{e}vy measure admitting a density around zero.

Modeling dependence dynamics through copulas with regime switching

Measuring dynamic dependence between international financial markets has recently attracted great interest in financial econometrics because the observed correlations rose dramatically during the 2008–09 global financial crisis. Here, we propose a novel approach for measuring dependence dynamics. We include a hidden Markov chain (MC) in the equation describing dependence dynamics, allowing the unobserved time-varying dependence parameter to vary according to both a restricted ARMA process and an unobserved two-state MC. Estimation is carried out via the inference for the margins in conjunction with filtering/smoothing algorithms. We use block bootstrapping to estimate the covariance matrix of our estimators. Monte Carlo simulations compare the performance of regime switching and no switching models, supporting the regime-switching specification. Finally the proposed approach is applied to empirical data, through the study of the S&P500 (USA), FTSE100 (UK) and BOVESPA (Brazil) stock market indexes.

Money growth and inflation: evidence from post-inflation Bolivia

In this short study, the author presents empirical evidence regarding the relation of money growth and its effect on inflation, for the case of Bolivia. The text describes the situation of the country analysed in terms of monetary policy after one of the worst cases of hyperinflation ever; empirical evidence is presented through two independent estimations, one following the Cochrane-Orcutt procedure, and the other following a Box-Jenkins procedure (ARMA process) that confirms the relation between money growth and inflation in Bolivia using data for the years 1998 to 2010. Evidence confirms the impact of money growth on inflation; the evidence suggests that the level of current inflation in Bolivia has a strong relation with the level of money quantity of two past periods on average. Moreover, the results show that monetary expansion affects the level of inflation in even when controlling for time impacts. The author also tries to assess the impact that a period of high inflation has in the development of growth policies for the case of Bolivia.

Forecasting the Stock Exchange of Thailand uses Day of the Week Effect and Markov Regime Switching GARCH

Problem statement: We forecast return and volatility of the Stock Exchange of Thailand (SET) Index. Approach: In this study, we modeled the SET Index returns using mean equation with day of the week effect and autoregressive moving-average. Next we forecast the volatility of the SET Index by using the GARCH-type model and the Markov Regime Switching GARCH (MRS-GARCH) model. Results: When we model the SET Index by the ARMA (3, 3) process, we find that Friday is the day of the effect of the SET Index. The empirical analysis demonstrates that the MRS-GARCH models outperform all GARCH-type models in forecasting volatility at long term horizons (two weeks and a month). Conclusion: The ARMA (3, 3) and the Friday is the day of the effect of the SET Index return. The MRS-GARCH models outperform at long term horizons.

Asymptotic distribution of the estimated parameters of an ARMA(p, q) process in the presence of explosive roots

Asymptotic distribution of the estimated parameters of an ARMA(p, q) process in the presence of explosive roots

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.

A Persistence Based Decomposition of Macroeconomic and Financial Time Series

The Classical Wold Decomposition Theorem allows to split a weakly stationary time series x into a non-deterministic component, driven by uncorrelated innovations, and a deterministic term. This decomposition is a special case of the Abstract Wold Theorem, which deals with isometric operators defined on Hilbert spaces. As the lag operator is isometric on the Hilbert space H_t(x) spanned by the sequence {x_{t-k}_k}, the Classical Wold Decomposition for time series obtains. Moreover, the \emph{scaling operator} is isometric on the Hilbert space H_t(e), spanned by the classical Wold innovations of x, and it provides an Extended Wold Decomposition. Thus, the process x may be seen as a sum, across scales, of uncorrelated components that explain different layers of persistence, from temporary fluctuations to low-frequency shocks. Multiscale impulse response functions are, then, defined. Conversely, the sum of suitable uncorrelated components delivers a weakly stationary process. This decomposition fruitfully applies to ARMA and fractional ARIMA processes.

Identifying Distributional Characteristics in Random Coefficients Panel Data Models

We study the identification of panel models with linear individual-specific coefficients, when T is fixed. We show identification of the variance of the effects under conditional uncorrelatedness. Identification requires restricted dependence of errors, reflecting a trade-off between heterogeneity and error dynamics. We show identification of the density of individual effects when errors follow an ARMA process under conditional independence. We discuss GMM estimation of moments of effects and errors, and introduce a simple density estimator of a slope effect in a special case. As an application we estimate the effect that a mother smokes during pregnancy on child�s birth weight.

An interpolation algorithm for multivariate ARMA processes

Soltani and Mohammadpour (2009) presented an algorithm for the best linear interpolator of unrecorded innovations in discrete time multivariate second order stationary processes. In this paper, we develop an interpolation procedure for multivariate ARMA processes, using the interpolation of the underlying innovations. In this case, the coefficients of the model and the past and future multivariate data are the inputs of the algorithm. We also obtain a closed form expression for the best linear interpolator of single value for MA(1) and AR(1) time series models.

A single series representation of multiple independent ARMA processes

This article shows that multiple independent time series from the same ARMA process can be represented by a single univariate ARMA time series through an interleaving of the original series. Using this result, existing univariate modelling software can be used to fit a single ARMA time series model simultaneously to multiple independent realizations of the same ARMA process. The interleaving approach and its properties will be presented and compared with alternative estimation options. It will be applied to the modelling of 66 years of daily maximum temperatures for Perth, Western Australia and to other time series models.

GMM Estimation of the Covariance Structure of Longitudinal Data on Earnings

In this article, we discuss generalized method of moments estimation of the covariance structure of longitudinal data on earnings, and we introduce and illustrate a Stata program that facilitates the implementation of the generalized method of moments approach in this context. The program, gmmcovearn, estimates a variety of models that encompass those most commonly used by labor economists. These include models where the permanent component of earnings follows a random growth or random walk process and where the transitory component can follow either an AR(1) or an ARMA(1,1) process. In addition, time-factor loadings and cohort-factor loadings may be incorporated in the transitory and permanent components.