Stationary AR Research Articles

Forecasting the stationary AR(1) with an almost unit root

Although unit root tests have made a great contribution in time series econometrics, their major disadvantage is the low powers they attain on certain occasions, as for the case of the stationary AR(1), when φis close to one. In this study, considering the random walk as the true model, we derive the probability of the prediction interval to include any future value yT + s of AR(1). Using certain estimates from Monte Carlo simulations, we proceed to numerical computations, resulting in the main finding that the values for the specific probability depend upon the location the most recent available observation in the sample possesses in its marginal distribution.

Bias-adjusted estimation in the ARX(1) model

A new point estimator for the AR(1) coefficient in the linear regression model with arbitrary exogenous regressors and stationary AR(1) disturbances is developed. Its construction parallels that of the median-unbiased estimator, but uses the mode as a measure of central tendency. The mean-adjusted estimator is also considered, and saddlepoint approximations are used to lower the computational burden of all the estimators. Large-scale simulation studies for assessing their small-sample properties are conducted. Their relative performance depends almost exclusively on the value of the autoregressive parameter, with the new estimator dominating over a large part of the parameter space.

The fading number of single-input multiple-output fading channels with memory

We derive the fading number of stationary and ergodic (not necessarily Gaussian) single-input multiple-output (SIMO) fading channels with memory. This is the second term, after the double-logarithmic term, of the high signal-to-noise ratio (SNR) expansion of channel capacity. The transmitter and receiver are assumed to be cognizant of the probability law governing the fading but not of its realization. It is demonstrated that the fading number is achieved by independent and identically distributed (i.i.d.) circularly symmetric inputs of squared magnitude whose logarithm is uniformly distributed over an SNR-dependent interval. The upper limit of the interval is the logarithm of the allowed transmit power, and the lower limit tends to infinity sublogarithmically in the SNR. The converse relies inter alia on a new observation regarding input distributions that escape to infinity. Lower and upper bounds on the fading number for Gaussian fading are also presented. These are related to the mean squared-errors of the one-step predictor and the one-gap interpolator of the fading process respectively. The bounds are computed explicitly for stationary mth-order autoregressive AR(m) Gaussian fading processes.

Stationary Autoregressive Models via a Bayesian Nonparametric Approach

Abstract. An approach to constructing strictly stationary AR(1)‐type models with arbitrary stationary distributions and a flexible dependence structure is introduced. Bayesian nonparametric predictive density functions, based on single observations, are used to construct the one‐step ahead predictive density. This is a natural and highly flexible way to model a one‐step predictive/transition density.

Approximate confidence sets for a stationary AR( p) process

Approximate confidence intervals are derived for the autoregressive parameters of a stationary, Gaussian auto-regressive process of arbitrary order and shown to be asymptotically correct to order o ( 1 / n ) , where n is the sample size. Simulation studies are included for small and moderate sample sizes for the case of two auto-regressive parameters, and these indicate excellent approximation for sample sizes as small as n = 10 , 20 . The convergence is in the very weak sense, and the derivation differs from most existing work through its direct focus on Studentized estimation error and its use of Stein's identity.

Asymptotic inference for a nearly unstable sequence of stationary spatial AR models

A nearly unstable sequence of stationary spatial autoregressive processes is investigated, where the autoregressive coefficients are equal, and their sum tends to one. It is shown that the limiting distribution of the least-squares estimator for this coefficient is normal and, in contrast to the doubly geometric process, the typical rate of convergence is n −5/4.

Acoustic Diagnosis on Rolling Bearings Based on Presumption of a Division Term by Locally Stationary AR Model

Acoustic diagnosis system with condenser microphone has some attractive characters what is more cheaper for cost of devices of sensor, is not keeping contact with machinery components and is wide band range on frequency compared with piezoelectric accelerometers. This paper proposes a method to do condition diagnosis about strong non stationary signal generated by operating rotating machineries which are rolling bearings. Especially, a signal processing technique in order to eliminate non stationary noises including operating sound is described by adopting locally stationary AR model a kind of system identification. Further, the optimum order of locally stationary AR model is determined from presumption of division terms. The effectiveness of this method has been identified by experiment based on a rotary simulator device on rolling bearings.

AR(1)모형에서 자기회귀계수의 다중검정을 위한 베이지안방법

본 논문은 베이즈인자(Bayes factor)를 이용하여 정상(stationary) AR(1)모형의 자기회귀계수에 대해 다중검정하는 방법을 제시한다. 모수들에 대한 사전분포로는 무정보 사전분포(noninformative prior distribution)를 가정한다. 이러한 경우에 통상적으로 사용되는 베이즈인자를 근사없이 정확히 계산하여 각 모형에 대한 사후확률(posterior probability)을 얻는다. 최종적으로 모의실험 자료 및 실제 자료에 적용하여 이론의 결과가 잘 부합되는지를 검토한다. This paper presents the multiple testing method of an autoregressive parameter in stationary AR(1) model using the usual Bayes factor. As prior distributions of parameters in each model, uniform prior and noninformative improper priors are assumed. Posterior probabilities through the usual Bayes factors are used for the model selection. Finally, to check whether these theoretical results are correct, simulated data and real data are analyzed.

Efficiency and Validity Analyses of Two-Stage Estimation Procedures and Derived Testing Procedures in Quantitative Linear Models with AR(1) Errors

In a quantitative linear model with errors following a stationary Gaussian, first-order autoregressive or AR(1) process, Generalized Least Squares (GLS) on raw data and Ordinary Least Squares (OLS) on prewhitened data are efficient methods of estimation of the slope parameters when the autocorrelation parameter of the error AR(1) process, ρ, is known. In practice, ρ is generally unknown. In the so-called two-stage estimation procedures, ρ is then estimated first before using the estimate of ρ to transform the data and estimate the slope parameters by OLS on the transformed data. Different estimators of ρ have been considered in previous studies. In this article, we study nine two-stage estimation procedures for their efficiency in estimating the slope parameters. Six of them (i.e., three noniterative, three iterative) are based on three estimators of ρ that have been considered previously. Two more (i.e., one noniterative, one iterative) are based on a new estimator of ρ that we propose: it is provided by the sample autocorrelation coefficient of the OLS residuals at lag 1, denoted r(1). Lastly, REstricted Maximum Likelihood (REML) represents a different type of two-stage estimation procedure whose efficiency has not been compared to the others yet. We also study the validity of the testing procedures derived from GLS and the nine two-stage estimation procedures. Efficiency and validity are analyzed in a Monte Carlo study. Three types of explanatory variable x in a simple quantitative linear model with AR(1) errors are considered in the time domain: Case 1, x is fixed; Case 2, x is purely random; and Case 3, x follows an AR(1) process with the same autocorrelation parameter value as the error AR(1) process. In a preliminary step, the number of inadmissible estimates and the efficiency of the different estimators of ρ are compared empirically, whereas their approximate expected value in finite samples and their asymptotic variance are derived theoretically. Thereafter, the efficiency of the estimation procedures and the validity of the derived testing procedures are discussed in terms of the sample size and the magnitude and sign of ρ. The noniterative two-stage estimation procedure based on the new estimator of ρ is shown to be more efficient for moderate values of ρ at small sample sizes. With the exception of small sample sizes, REML and its derived F-test perform the best overall. The asymptotic equivalence of two-stage estimation procedures, besides REML, is observed empirically. Differences related to the nature, fixed or random (uncorrelated or autocorrelated), of the explanatory variable are also discussed.

103 AR モデルによる転がり軸受音響診断へのサンプリング周波数および AR フィルタ長の影響

Acoustic diagnosis system with condenser microphone has some attractive characters what is more cheap for cost of devices of sensor, is not keeping contact with machinery components and is wide band range on frequency compare with piezoelectric accelerometers. This paper proposes a method to do condition diagnosis about non stationary signal generated by operating rotary machineries which are a rolling bearing. Especially, a signal processing technique in order to eliminate non stationary noises including operating sound is described by adopting locally stationary AR model a kind of system identification. Further, the optimum order of locally stationary AR model is determined from presumption of division terms. The effectiveness of this method has been identified by experimentation based on a rotary simulator device on a rolling bearing.

Maximum Likelihood Estimation of Dynamic Linear Panel Data Models with Fixed Effects

In this paper we consider inference procedures for two types of dynamic linear panel data models with fixed effects. First, we show that the closure of the stationary ARMA panel model with fixed effects can be consistently estimated by the First Difference Maximum Likelihood Estimator and we derive its large N, fixed T, as well as its large T, arbitrary N asymptotic distributions. These results allow us to formulate two simple likelihood based unit root tests which use the table of the standard normal distribution. We also establish the asymptotic properties of the Modified ML Estimator for the conditional AR(1) panel model with fixed effects under various asymptotic plans. Then we show that when N tends to infinity but T is fixed, the FDMLE does not attain the (generalised) Cramer-Rao lowerbound for the stationary AR(1) panel model with fixed effects and is asymptotically less efficient than the Random Effects MLE. However, when T tends to infinity, the FDMLE, the Fixed Effects MLE, the REMLE are asymptotically equivalent to the Modified MLE and the Within ML (or LSDV) Estimator, i.e. under normality some of the moment conditions implicitly exploited by the first three estimators become asymptotically redundant.

GENERALIZED SIGNED-RANK ESTIMATORS FOR AUTOREGRESSION PARAMETERS

ABSTRACT A new class of weighted signed-rank-based estimates for estimating the parameter vector of an autoregressive time series is considered. The Wilcoxon signed-rank estimate and the GR-estimates of Terpstra et al. (GR-Estimates for an Autoregressive Time Series. Statistics and Probability Letters 2001, 51, 165–172; Generalized Rank Estimates for an Autoregressive Time Series: A U-Statistic Approach. Statistical Inference for Stochastic Processes 2001, 4, 155–179) can be viewed as special cases of the so-called GSR-estimates. Asymptotic linearity properties are derived for the GSR-estimates. Based on these properties, and a symmetry assumption, the GSR-estimates are shown to be asymptotically normal at rate n 1/2. The theory of U-Statistics along with a characterization of weak dependence that is inherent in stationary AR(p) models are the primary tools used to obtain the results. Tests of hypotheses as well as standard errors for confidence interval procedures can be based on such results. An efficiency study indicates that, for an appropriately chosen set of weights, the GSR-estimate is more efficient than the GR-estimate. Furthermore, the GSR-estimate has an added advantage in that an intercept term can be estimated simultaneously; unlike the GR-estimate. Two examples and a small simulation study are used to illustrate the computational and robust aspects of the GSR-estimates.

DIAGNOSIS FOR ROTARY MACHINERY BASED ON LOCALLY STATIONARY AR MODEL

This paper proposes a method to do condition diagnosis about non stationary signal generated by operating rotary machineries which are bearing, gear, rotor and so on. Especially, a signal processing technique in order to eliminate slight non stationary noises including operating sound is described by adopting locally stationary AR model [1] a kind of system identification. The effectiveness of this modeling has been identified by numerical simulation based on a rotary simulator device on a failure rolling bearing.

A Note on Mean‐squared Prediction Errors of the Least Squares Predictors in Random Walk Models

An asymptotic expression for the mean‐squared prediction error (MSPE) of the least squares predictor is obtained in the random walk model. It is shown that the term of order 1/n in this error, where n is the sample size, is twice as large as the one obtained from the first‐order autoregressive (AR(1)) model satisfying the stationary assumption. Moreover, while the correlation between the squares of the (normalized) regressor variable and normalized least squares estimator is asymptotically negligible in the stationary AR(1) model, we have found that the correlation has significantly negative value in the random walk model. To obtain these results, a new methodology, which is found to be useful in dealing with the moment properties of a strongly dependent process, is introduced.

Note on the bias in the estimation of the serial correlation coefficient of AR(1) processes

We derive approximating formulas for the mean and the variance of an autocorrelation estimator which are of practical use over the entire range of the autocorrelation coefficient ρ. The least-squares estimator 1 $$\Sigma _{i = 1}^{n - 1} \in _i \in _{i = 1} /\Sigma _{i = 1}^{n - 1} \in _i^2 $$ is studied for a stationary AR(1) process with known mean. We use the second order Taylor expansion of a ratio, and employ the arithmetic—geometric series instead of replacing partial Cesaro sums. In case of the mean we derive Marriott and Pope’s (1954) formula, with (n - 1)-1 instead of (n)-1, and an additional term ∝ (n - 1)-2. This new formula produces the expected decline to zero negative bias as ρ approaches unity. In case of the variance Bartlett’s (1946) formula results, with (n — 1)-1 instead of (n)-1. The theoretical expressions are corroborated with a simulation experiment. A comparison shows that our formula for the mean is more accurate than the higher-order approximation of White (1961), for ¦ρ¦> 0.88 and n ≥ 20. In principal, the presented method can be used to derive approximating formulas for other estimators and processes.

A NONPARAMETRIC TEST OF CHANGING CONDITIONAL VARIANCES IN AUTOREGRESSIVE TIME SERIES

A nonparametric test for detecting changing conditional variances in stationary AR(p) time series is proposed in this paper. For AR(1) models, the test statistic is a Kolmogorov-Smirnov type statistic and the asymptotic theory is developed under both the null and the alternative hypotheses. For AR(p) models (p ≥ 2), an approximate test procedure is proposed. The empirical upper percentage points for our test are tabulated for both p = 1 and p = 2 cases and a bootstrap procedure is suggested for the p ≥ 3 case. Monte Carlo simulations demonstrate that the test has very good powers for finite samples under both normal and non-normal errors.

Generalized Rank Estimates For An Autoregressive Time Series: A U-Statistic Approach

A class of weighted rank-based estimates for estimating the parameter vector of an autoregressive time series is considered. This class of estimates is similar to, and contains, the class proposed by Terpstra et al. [54]. Asymptotic linearity properties are derived for the so called GR-estimates. Based on these properties, the GR-estimates are shown to be asymptotically normal at rate n 1/2. The theory of U-statistics along with a characterization of weak dependence that is inherent in stationary AR(p) models are the primary tools used to obtain the results. The so called pair-wise slopes estimator, which is a special case of this class of estimates, is discussed in an AR(1) context.

Bounds for the least squares extrapolation in non-linear AR(1) processes

It is proved that formula for least squares extrapolation in stationary non-linear AR(1) process is valid also for non-stationary non-linear AR(1) processes. This formula depends on the distribution of the corresponding white noise. If the non-linear function used in the model is non-decreasing and concave, upper and lower bounds are derived for least squares extrapolation such that the bounds depend only on the expectation of the white noise. It is shown in an example that the derived bounds in some cases give a good approximation to the least squares extrapolation. Copyright © 2001 by John Wiley & Sons, Ltd.

Properties of Predictors in Overdifferenced Nearly Nonstationary Autoregression

We analyze the effect of overdifferencing a stationary AR(p+1) process whose largest root is near unity. It is found that, if the process is nearly nonstationary, the estimators of the overdifferenced model ARIMA(p,1,0) are root‐T consistent. It is also found that this misspecified ARIMA(p,1,0) has lower predictive mean squared error, to terms of small order, than the properly specified AR(p+1) model due to its parsimony. The advantage of the overdifferenced predictor depends on the remaining roots, the prediction horizon and the mean of the process.

The Dynamics of DAX Implied Volatilities

On the basis of transaction data, this paper analyzes the strike pro…le of implied volatilities of German DAX options for a time to expiration of 45 days. Using WLS spline regressions over the sample period from 1995 to 1999, we estimate a time series of smile characteristics, which we then try to attribute to economic fundamentals. Their choice is motivated by common theoretical explanations of the smile. The strike pattern almost exclusively appears as a “skew” rather than a “smile”. We …nd that the dynamics of the smile pro…le can be accurately modelled by a stationary AR(1) process. Market uncertainty, measured by volatility of