Stationary AR Research Articles

Moderate deviations for the mildly stationary autoregressive model with dependent errors

ABSTRACT In this paper, we consider the normalized least squares estimator of the parameter in a mildly stationary first-order autoregressive (AR(1)) model with dependent errors which are modelled as a mildly stationary AR(1) process. By martingale methods, we establish the moderate deviations for the least squares estimators of the regressor and error, which can be applied to understand the near-integrated second-order autoregressive processes. As an application, we also obtain the moderate deviations for the Durbin–Watson statistic.

Contrast estimation of time-varying infinite memory processes

This paper extends the study of kernel-based estimation for locally stationary processes proposed in Dahlhaus et al., 2019 to infinite-memory processes models such as locally stationary AR(∞), GARCH(p,q), ARCH(∞) or LARCH(∞) processes. The estimators are computed as localized M-estimators for every contrast satisfying appropriate regularity conditions. We prove the uniform consistency and pointwise asymptotic normality of such kernel-based estimators. We apply our results to common contrasts such as least-square, least-absolute-value, or quasi-maximum likelihood contrast. Numerical experiments demonstrate the efficiency of the estimators on both simulated and real data sets.

Run length distribution for a modified EWMA scheme fitted with a stationary AR(p) model

A modified exponentially weighted moving average (EWMA) scheme is a quality control chart which can quickly detect small shifts in autocorrelated data. In this article, an explicit formula is presented to calculate the average run length (ARL) for a modified EWMA chart with an autoregressive AR(p) model involving exponential white noise. Its performance is compared with a numerical integral equation method via a simulation study to determine whether the explicit formula can help decrease the processing time. Moreover, this scheme is tested against a classical EWMA control chart using the ARL and RMI results. Finally, the explicit formula for ARL is applied to a real numerical example from the health field to demonstrate the efficacy of our method.

The Relationship between Stock Price Index and Industrial Production Index in Construction Industry

This paper investigates the relation between stock price index and industrial production index in construction industry, namely, the relation between stock market and real economy activity. Because stock price is the leading indicator, stock price may be related to industrial production. Previous studies have been studied on the relation between released stock price index and IPI. However, this study is distinguishable in that released stock price index is decomposed into fundamental component and transitory component. The first step in this empirical analysis is to decompose released stock price index. In state-space model, fundamental component is set up as a random walk process and transitory component is modeled as a stationary AR(2) process. In the second step, to check the stationarity, the ADF unit root test is used. In the third step, to elucidate the relationship between stock price index and IPI, Granger causality test is applied. The findings are as follows. The results reveal that released stock price index Granger causes IPI. The fundamental component of stock price index Granger causes IPI, and the transitory component of stock price index Granger causes IPI. The increase in released stock price index and the fundamental component of stock price index causes IPI to increase after one month. This paper indicates that the analysis of stock price index helps to predict IPI in construction industry.

Dealing with serially correlated errors in the context of spurious regression for two independent stationary AR(1) processes

ABSTRACT Serially correlated errors are most likely to appear in regression analysis when time-series data are used either as a true symptom of autocorrelation or as an indication of a false specification among variables. This study examines the problem of serially correlated errors in the context of spurious regression for two independent stationary AR(1) processes, showing evidence of removing the presence of both symptoms using a Monte Carlo analysis.

Stochastic volatility forecasting of the Finnish housing market

ABSTRACT The purpose of the article is to assess the in-sample fit and the out-of-sample forecasting performances of four stochastic volatility (SV) models in the Finnish housing market. The competing models are the vanilla SV, the SV model where the latent volatility follows a stationary AR(2) process, the heavy-tailed SV and the SV with leverage effects. The models are estimated using Bayesian technique, and the results reveal that the SV with leverage effects is the best model for modelling the Finnish house price volatility. The heavy-tailed SV model provides accurate out-of-sample volatility forecasts in most of the studied regions. Additionally, the models’ performances are noted to vary across almost all cities and sub-areas, and by apartment types. Moreover, the AR(2) component substantially improves the in-sample fit of the standard SV, but it is unimportant for the out-of-sample forecasting performance. The study outcomes have crucial implications, such as portfolio management and investment decision-making. To establish suitable time-series volatility forecasting models of this housing market, these study outcomes will be compared to the performances of their GARCH models counterparts.

Non identification of structural change in non stationary AR(1) models

This article studies the identification problem in AR(1) models with a change in the AR parameter at an unknown time k 0. Consider the model where denotes the indicator function and is a sequence of i.i.d. random variables which are in the domain of attraction of the normal law with zero means and possibly infinite variances. Two cases were investigated: case (I) and case (II) where c is a fixed constant. We derived the limiting distributions of the least squares estimator of the break fraction τ 0() for the cases above under some mild conditions. The results showed that the change point (or k 0) is non identifiable for the aforementioned two cases by the least squares method. Monte Carlo experiments were conducted to examine the identification of τ 0 under finite sample situations. Our theoretical results are supported by the Monte Carlo experiments.

Local Power of Kolmogorov’s and Omega-Squared Type Criteria in Autoregression

A stationary AR(p) model is considered. The autoregression parameters are unknown as well as the distribution of innovations. Based on the residuals from the parametric estimates, an analog of the empirical distribution function is defined and tests of Kolmogorov’s and ω2 type are constructed for testing hypotheses on the distribution of innovations. The asymptotic power of these tests under local alternatives is obtained.

Unified Inference for an AR Process Regardless of Finite or Infinite Variance GARCH Errors*

Statistical inference in finance often depends on certain moment conditions such as finite or infinite variance, yet it is practically challenging to disentangle these conditions. This article develops a class of unified unit root tests for AR(1) models and a weighted least squares estimator along with robust inference for a stationary AR(r) model regardless of finite or infinite variance GARCH errors. The inferential framework applies the empirical likelihood method to some weighted score equations without estimating the GARCH errors. In contrast to extant unit root tests relying on bootstrap or subsampling methods to approximate critical values, the proposed unit root tests can be easily implemented with critical values obtained directly from a chi-squared distribution using the Wilks theorem. Extensive simulation studies confirm the good finite sample performance of the proposed methods before we illustrate them empirically with financial ratios for stock return predictability and HKD/USD exchange rate returns.

BIAS-CORRECTED INFERENCE FOR A MODIFIED LEE–CARTER MORTALITY MODEL

AbstractAs a benchmark mortality model in forecasting future mortality rates and hedging longevity risk, the widely employed Lee–Carter model (Lee, R.D. and Carter, L.R. (1992) Modeling and forecasting U.S. mortality. Journal of the American Statistical Association, 87, 659–671.) suffers from a restrictive constraint on the unobserved mortality index for ensuring model’s identification and a possible inconsistent inference. Recently, a modified Lee–Carter model (Liu, Q., Ling, C. and Peng, L. (2018) Statistical inference for Lee–Carter mortality model and corresponding forecasts. North American Actuarial Journal, to appear.) removes this constraint and a simple least squares estimation is consistent with a normal limit when the mortality index follows from a unit root or near unit root AR(1) model with a nonzero intercept. This paper proposes a bias-corrected estimator for this modified Lee–Carter model, which is consistent and has a normal limit regardless of the mortality index being a stationary or near unit root or unit root AR(1) process with a nonzero intercept. Applications to the US mortality rates and a simulation study are provided as well.

On the Asymptotic Power of Tests of Fit under Local Alternatives in Autoregression

We consider a stationary AR(p) model. The autoregression parameters are unknown as well as the distribution of innovations. Based on the residuals from the parameter estimates, an analog of empirical distribution function is defined and the tests of Kolmogorov’s and ω2 type are constructed for testing hypotheses on the distribution of innovations. We obtain the asymptotic power of these tests under local alternatives.

On the Power of Pearson’s Test under Local Alternatives in Autoregression with Outliers

We consider a stationary linear AR(p) model with contamination (gross errors in the observations). The autoregression parameters are unknown, as well as the distribution of innovations. Based on the residuals from the parameter estimates, an analog of the empirical distribution function is defined and a test of Pearson’s chi-square type is constructed for testing hypotheses on the distribution of innovations. We obtain the asymptotic power of this test under local alternatives and establish its qualitative robustness under the hypothesis and alternatives.

On the Empirical Distribution Function of Residuals in Autoregression with Outliers and Pearson’s Chi-Square Type Tests

We consider a stationary linear AR(p) model with observations subject to gross errors (outliers). The distribution of outliers is unknown and arbitrary, their intensity is γn−1/2 with an unknown γ, n is the sample size. The autoregression parameters are unknown, they are estimated by any estimator which is n1/2-consistent uniformly in γ ≤ Γ < ∞. Using the residuals from the estimated autoregression, we construct a kind of empirical distribution function (e.d.f.), which is a counterpart of the (inaccessible) e.d.f. of the autoregression innovations. We obtain a stochastic expansion of this e.d.f., which enables us to construct a test of Pearson’s chi-square type for testing hypotheses about the distribution of innovations. We establish qualitative robustness of this test in terms of uniform equicontinuity of the limiting level with respect to γ in a neighborhood of γ = 0.

Maximum likelihood estimation of change point from stationary to nonstationary in autoregressive models using dynamic linear model

AbstractChange point estimation is a useful concept in time series models that could be applied in several fields such as financing, quality control. It helps to decrease costs of decision making and production by monitoring stock market and production lines, respectively. In this paper, the maximum likelihood technique is developed to estimate change point at which the stationary AR(1) model changes to a nonstationary process. Filtering and smoothing of dynamic linear model are used to estimate unknown parameters after change point. We also assume that correlation exists between samples' statistics. Simulation results show the effectiveness of the proposed estimators to estimate the change point of stationary. In addition based on Shewhart control chart, filtering has a better accuracy in comparison to smoothing. A real example is provided to illustrate the application.

STATIONARY INTEGRATED ARCH(∞) AND AR(∞) PROCESSES WITH FINITE VARIANCE

We prove the long standing conjecture of Ding and Granger (1996) about the existence of a stationary Long Memory ARCH model with finite fourth moment. This result follows from the necessary and sufficient conditions for the existence of covariance stationary integrated AR(∞), ARCH(∞), and FIGARCH models obtained in the present article. We also prove that such processes always have long memory.

LASSO order selection for sparse autoregression: a bootstrap approach

ABSTRACTAutoregressive models are widely employed for predictions and other inferences in many scientific fields. While the determination of their order is in general a difficult and critical step, this task becomes more complicated and crucial when the time series under investigation is realization of a stochastic process characterized by sparsity. In this paper we present a method for order determination of a stationary AR model with a sparse structure, given a set of observations, based upon a bootstrapped version of MAICE procedure [Akaike H. Prediction and entropy. Springer; 1998], in conjunction with a LASSO-type constraining procedure for lag suppression of insignificant lags. Empirical results will be obtained via Monte Carlo simulations. The quality of our method is assessed by comparison with the commonly adopted cross-validation approach and the non bootstrap counterpart of the presented procedure.

Stationary Gaussian Markov processes as limits of stationary autoregressive time series

We consider the class, C p , of all zero mean stationary Gaussian processes, { Y t : t ∈ ( − ∞ , ∞ ) } with p derivatives, for which the vector valued process { ( Y t ( 0 ) , … , Y t ( p ) ) : t ≥ 0 } is a p + 1 -vector Markov process, where Y t ( 0 ) = Y ( t ) . We provide a rigorous description and treatment of these stationary Gaussian processes as limits of stationary AR( p ) time series.

Peculiarities of finding characteristic functions of the generating process in the model of stationary linear AR(2) process with negative binomial distribution

The linear random AR(2) autoregressive process having the negative binomial distribution has been considered. It has the form ξt + a1ξt-1 + a2ξt-2 = ςt, t ∈ Z, where {a1, a2 ≠ 0} are the autoregressive parameters; Z = {...,-1,0,1,...} is the sequence of integers; {ξt,t ∈ Z} is the random process with discrete time and independent values having the infinitely divisible distribution law that is called generating process. The method of finding the characteristic function of the generating process for linear autoregressive process having negative binomial distribution is presented. This inverse problem is solved by using properties of the characteristic function of stationary linear autoregressive process that can be presented in the Kolmogorov canonical form and as a linear stationary autoregressive process. An example of finding the Poisson spectrum of jumps and the characteristic function for the linear second order autoregressive process (AR(2)) with negative binomial distribution has been also presented.

Asymptotics of self-weighted M-estimators for autoregressive models

In this paper, we consider a stationary autoregressive AR(p) time series \(y_t=\phi _0+\phi _1y_{t-1}+\cdots +\phi _{p}y_{t-p}+u_t\). A self-weighted M-estimator for the AR(p) model is proposed. The asymptotic normality of this estimator is established, which includes the asymptotic properties under the innovations with finite or infinite variance. The result generalizes and improves the known one in the literature.

Finite Sample Bias and MSE of Fully Aggregated Estimator For AR(1) Model with a General Error Distribution

In this paper, we derive the finite sample bias and MSE of the fully aggregated estimator (FAE) for the stationary AR(1) model with intercept, which is proposed by Han, Phillips, and Sul (2011). Our analytical results show why the FAE is less biased than the ordinary least square estimator in finite sample case and is not biased by non-normality of error distribution and by intercept term at least O(1/T ), where T is sample size. We also propose a second order unbiased FAE using the analytical result. Finally, we examine the Monte Carlo simulation and show that it is consistent with the theoretical results.