Random Coefficient Autoregressive Research Articles

Estimation of Random Coefficient Autoregressive Model with Error in Covariates

Measurement error is common in many statistical problems and has received considerable attention in various regression contexts. In this study, we consider the random coefficient autoregressive model with measurement error possibly present in covariates. The least squares and weighted least squares methods are used to estimate the model parameters, and the consistency and asymptotic normality of the two kinds of estimators are proved. Furthermore, we propose an empirical likelihood method based on weighted score equations to construct confidence regions for the parameters. The simulation results show that the weighted least squares estimators are superior to the least squares estimators and that the confidence regions have good finite-sample behavior. At last, the model is applied to a real data example.

Asymptotic Inference in the Random Coefficient Autoregressive Model with Time-functional Variance Noises

Asymptotic Inference in the Random Coefficient Autoregressive Model with Time-functional Variance Noises

Formula omitted]-functionals for change point detection in random coefficient autoregressive models

We study a family of Lp-functionals of the weighted CUSUM test statistic to detect the presence of changepoints in the deterministic part of the autoregressive parameter in a Random Coefficient AutoRegressive sequence of order 1 (RCA(1)). Our statistics are robust against the presence of heteroskedasticity of unknown form, and can be applied irrespective of whether the sequence is stationary or not, with no prior knowledge of stationarity or lack thereof required.

A new RCAR(1) model based on explanatory variables and observations

The random coefficient autoregressive (RCAR) processes are very useful to model time series in applications. It is commonly observed that the random autoregressive coefficient is assumed to be an independent identically distributed (i.i.d.) random variable sequence. To make the RCAR model more practical, this paper considers a new RCAR(1) model driven by explanatory variable and observations. We use the conditional least squares, the quantile regression and the conditional maximum likelihood methods to estimate the model parameters. The consistency and asymptotic normality of the proposed estimates are established. Simulation studies are conducted for the evaluation of the developed approaches and two applications to real-data examples are provided. The results show that the proposed procedures perform well for the simulations and application.

Changepoint Detection in Heteroscedastic Random Coefficient Autoregressive Models

We propose a family of CUSUM-based statistics to detect the presence of changepoints in the deterministic part of the autoregressive parameter in a Random Coefficient Autoregressive (RCA) sequence. Our tests can be applied irrespective of whether the sequence is stationary or not, and no prior knowledge of stationarity or lack thereof is required. Similarly, our tests can be applied even when the error term and the stochastic part of the autoregressive coefficient are non iid, covering the cases of conditional volatility and shifts in the variance, again without requiring any prior knowledge as to the presence or type thereof. In order to ensure the ability to detect breaks at sample endpoints, we propose weighted CUSUM statistics, deriving the asymptotics for virtually all possible weighing schemes, including the standardized CUSUM process (for which we derive a Darling-Erdős theorem) and even heavier weights (so-called Rényi statistics). Simulations show that our procedures work very well in finite samples. We complement our theory with an application to several financial time series.

A test for strict stationarity in a random coefficient autoregressive model of order 1

We propose a test for the null of strict stationarity in a Random Coefficient AutoRegression (RCAR) of order 1. The test can also be used in the case of a standard AR(1) model, and it can be applied under minimal requirements on the existence of moments — in both cases without requiring any modifications or prior knowledge.

Asymptotic behavior for Markovian iterated function systems

Let (U,d) be a complete separable metric space and (Fn)n≥0 a sequence of random functions from U to U. Motivated by studying the stability property for Markovian dynamic models, in this paper, we assume that the random function (Fn)n≥0 is driven by a Markov chain X={Xn,n≥0}. Under some regularity conditions on the driving Markov chain and the mean contraction assumption, we show that the forward iterations Mnu=Fn∘⋯∘F1(u), n≥0, converge weakly to a unique stationary distribution Π for each u∈U, where ∘ denotes composition of two maps. The associated backward iterations M̃nu=F1∘⋯∘Fn(u) are almost surely convergent to a random variable M̃∞ which does not depend on u and has distribution Π. Moreover, under suitable moment conditions, we provide estimates and rate of convergence for d(M̃∞,M̃nu) and d(Mnu,Mnv), u,v∈U. The results are applied to the examples that have been discussed in the literature, including random coefficient autoregression models and recurrent neural network.

Testing for Random Coefficient Autoregressive and Stochastic Unit Root Models

The random coefficient autoregressive model has been utilized for modeling financial time series because it possesses features that are often observed in financial time series. When the mean of the random autoregressive coefficient is one, it is called the stochastic unit root model. This paper proposes two Lagrange multiplier tests for the null hypothesis of random coefficient autoregressive and stochastic unit root models against a more general model. We apply our Lagrange multiplier tests to several stock index data, and find that the stochastic unit root model is rejected, whereas the random coefficient autoregressive model is not. This result indicates that it is important to check the validity of the stochastic unit root model prior to applying it to financial time series data, which may be better modeled by the random coefficient autoregressive model with the mean being not equal to one.

Weighted least squares estimation in a binary random coefficient panel model with infinite variance

This article investigates the asymptotic properties of weighted least squares estimators (WLSE) for a binary random coefficient autoregressive (RCA) panel model with heterogeneous variances of panel variables. It is an extension of Johansen and Lange (2013) to a panel model, which is more practical for macroeconomic time series data. We develop asymptotic properties of the WLSE in cases of finite and infinite variances, respectively, as both sizes of panels and samples tend to infinity. In the latter case with infinite variance, the asymptotic for the WLSE βˆ of the coefficient β is shown to be a curious result βˆ⟶pβ−1. It is proven by using the notion of a tail index and the stable distribution limit. In a Monte Carlo simulation, feasible WLSEs are computed iteratively and some evidences are given to verify our theoretical results.

Testing for strict stationarity in a random coefficient autoregressive model

We propose a procedure to decide between the null hypothesis of (strict) stationarity and the alternative of nonstationarity, in the context of a random coefficient autoregression (RCAR). The procedure is based on randomizing a diagnostic which diverges to positive infinity under the null, and drifts to zero under the alternative. Thence, we propose a randomized test which can be used directly and—building on it—a decision rule to discern between the null and the alternative. The procedure can be applied under very general circumstances: albeit developed for an RCAR model, it can be used in the case of a standard AR(1) model, without requiring any modifications or prior knowledge. Also, the test works (again with no modification or prior knowledge being required) in the presence of infinite variance, and in general requires minimal assumptions on the existence of moments.

Correction to: Random Coefficient Autoregressive Processes: a Markov Chain Analysis of Stationarity and Finiteness of Moments by Paul D. Feigin and Richard L. Tweedie J. Time Series Anal., Vol. 6, No. 1 (1985)

Correction to: <i>Random Coefficient Autoregressive Processes: a Markov Chain Analysis of Stationarity and Finiteness of Moments</i> by Paul D. Feigin and Richard L. Tweedie J. Time Series Anal., Vol. 6, No. 1 (1985)

Comparing Parameter Estimation of Random Coefficient Autoregressive Model by Frequentist Method

This paper compares the frequentist method that consisted of the least-squares method and the maximum likelihood method for estimating an unknown parameter on the Random Coefficient Autoregressive (RCA) model. The frequentist methods depend on the likelihood function that draws a conclusion from observed data by emphasizing the frequency or proportion of the data namely least squares and maximum likelihood methods. The method of least squares is often used to estimate the parameter of the frequentist method. The minimum of the sum of squared residuals is found by setting the gradient to zero. The maximum likelihood method carries out the observed data to estimate the parameter of a probability distribution by maximizing a likelihood function under the statistical model, while this estimator is obtained by a differential parameter of the likelihood function. The efficiency of two methods is considered by average mean square error for simulation data, and mean square error for actual data. For simulation data, the data are generated at only the first-order models of the RCA model. The results have shown that the least-squares method performs better than the maximum likelihood. The average mean square error of the least-squares method shows the minimum values in all cases that indicated their performance. Finally, these methods are applied to the actual data. The series of monthly averages of the Stock Exchange of Thailand (SET) index and daily volume of the exchange rate of Baht/Dollar are considered to estimate and forecast based on the RCA model. The result shows that the least-squares method outperforms the maximum likelihood method.

Parameter Estimation for p-Order Random Coefficient Autoregressive (RCA) Models Based on Kalman Filter

In this paper we elaborate an algorithm to estimate p-order Random Coefficient Autoregressive Model (RCA(p)) parameters. This algorithm combines quasi-maximum likelihood method, the Kalman filter, and the simulated annealing method. In the aim to generalize the results found for RCA(1), we have integrated a subalgorithm which calculate the theoretical autocorrelation. Simulation results demonstrate that the algorithm is viable and promising.

Testing for randomness in a random coefficient autoregression model

We propose a test to discern between an ordinary autoregressive model, and a random coefficient one. To this end, we develop a full-fledged estimation theory for the variances of the idiosyncratic innovation and of the random coefficient, based on a two-stage WLS approach. Our results hold irrespective of whether the series is stationary or nonstationary, and, as an immediate result, they afford the construction of a test for ”relevant” randomness. Further, building on these results, we develop a randomised test statistic for the null that the coefficient is non-random, as opposed to the alternative of a standard RCA(1) model. Monte Carlo evidence shows that the test has the correct size and very good power for all cases considered.

Random coefficient autoregressive processes and the PUCK model with fluctuating potential

The potentials of unbalanced complex kinetics (PUCK) model consists of a random walker subjected to a potential centered at its moving average position. We study the PUCK model with fluctuating quadratic potential, showing that it is a special case of the random coefficient autoregressive (RCAR) process and thus a member of the same class of processes as the autoregressive conditional heteroskedasticity (ARCH) process; both RCAR processes but with different coefficient dependence. For the general Gaussian RCAR process, we use generalized Fibonacci numbers to derive explicit expressions for the mean squared displacement and the autocovariance function. We also obtain the conditions for divergence of variance, which imply tail index of the power-law-tailed cumulative probability distribution. Translating the results to the PUCK model, we analyze US Dollar/Japanese Yen market data and demonstrate that the model simultaneously reproduces empirical facts of price time series: diffusion and autocorrelation of prices and heavy-tailed distribution of price changes.

RCAR and CHARMA Models Versus Stochastic Volatility Models: A Forecasting Analysis Using Macrofinancial Data

Using inflation and return time series, we first evaluate the forecasting performance of two classes of conditional heteroscedastic models: the random coefficient autoregressive (RCAR) models and the conditional heteroscedastic autoregressive moving average (CHARMA) models. Markov Chain Monte Carlo schemes are developed for the estimation of the model parameters. Furthermore, the forecasting ability of these two models is compared against that of several stochastic volatility models that control for time-varying parameters, in mean effects, leverage effects and moving average errors. We found that our proposed models can produce better point and density forecasts than various stochastic volatility specifications, with the CHARMA models outperforming the RCAR models.

Sample path properties of an explosive double autoregressive model

ABSTRACTThis article studies sample path properties of an explosive double autoregressive (DAR) model. After suitable renormalization, it is shown that the sample path converges weakly to a geometric Brownian motion. This further strengthens our understanding of sample paths of nonstationary DAR processes. The obtained results can be extended to nonstationary random coefficient autoregressive (RCA) models. Simulation studies are carried out to support our results.

Statistical inference in a random coefficient panel model

This paper studies the asymptotics of the Weighted Least Squares (WLS) estimator of the autoregressive root in a panel Random Coefficient Autoregression (RCA). We show that, in an RCA context, there is no “unit root problem” : the WLS estimator is always asymptotically normal, irrespective of the average value of the autoregressive root, of whether the autoregressive coefficient is random or not, and of the presence and degree of cross dependence. Our simulations indicate that the estimator has good properties, and that confidence intervals have the correct coverage even for sample sizes as small as (N,T)=(10,25). We illustrate our findings through two applications to macroeconomic and financial variables.

Combined estimating function for random coefficient models with correlated errors

ABSTRACTEstimating functionshave been shown to be convenient to study inference for non linear time series models. Recently, Thavaneswaran et al. (2012) used combined estimating functions to study inference for random coefficient autoregressive (RCA) models with generalized autoregressive heteroscedasticity errors. While most RCA modeling assumes that the random term and the error are independent, Chandra and Taniguchi (2001) studied inference for RCA models with correlated errors using linear estimating functions. In this paper, we derive the quadratic estimating functions for the joint estimation of the conditional mean, variance, and correlation parameters of the RCA models with correlated errors.

A random-coefficient third-order autoregressive process

Many systems and physical processes require non-Gaussian probabilistic models to accurately capture their dynamic behaviour. In this paper, we present a random-coefficient mathematical form that can be used to simulate a third-order Laplace autoregressive (AR) process. The mathematical structure of the random-coefficient AR process has a Markovian property that makes it flexible and simple to implement. A detailed derivation of its parameters as well as its pseudo-code implementation is provided. Moreover, it is shown that the process has an autocorrelation property that satisfies Yule–Walker type of equations. Having such an autocorrelation property makes the developed AR process, particularly, convenient for deriving mathematical models for dynamic systems, as well as signals, whose parameters of interest are Laplace distributed.