On Assessing Prediction Error in Autoregressive Models

Zhang and Shaman (Assessing prediction error in autoregressive models. Trans. Am. Math. Soc. 347, (1995), 627–37) pose the problem of estimating the conditional mean square one-step-ahead prediction error (CMOPE) for a Gaussian first-order autoregressive process. They put forward a certain estimator (with small asymptotic bias) of CMOPE and propose that its effectiveness be judged by its asymptotic correlation with CMOPE. Unfortunately, the derivation of this correlation by Zhang and Shaman (1995) is incomplete. It is very difficult to complete this derivation. For this reason we use Monte Carlo simulation to gain some insight into the correlation of the estimator with CMOPE. The results of this simulation show that the estimator is extremely poor. We then propose an alternative estimator (with small asymptotic bias) of CMOPE which is shown from Monte Carlo simulation results to have higher large-sample correlation with CMOPE than the estimator of CMOPE put forward by Zhang and Shaman (1995).     

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.     

Self-Weighted Lad-Based Inference for Heavy-Tailed Continuous Threshold Autoregressive Models

This note investigates the self-weighted least absolute deviation estimation (SLADE) of a heavy-tailed continuous threshold autoregressive (TAR) model. It is shown that the SLADE is strongly consistent and asymptotically normal. The SLADE is global in the sense that the convergence rate is first obtained before deriving its limiting distribution. Moreover, a test for the continuity of TAR model is considered. A sign-based portmanteau test is developed for diagnostic checking. An empirical example is given to illustrate the usefulness of our method. Combined with the results (Yang and Ling, 2017), a complete asymptotic theory on the SLADE of a heavy-tailed TAR model is established. This enriches asymptotic theory of statistical inference in threshold models.     

Asymptotic mean-squared forecast error when an autoregression with linear trend is fitted to data generated by an I(0) or I(1) process

Abstract.  Assume that a time series is generated by an autoregression which has atmost one unit root. A correctly specified model, including linear time trend, is estimated by ordinary least squares, but no allowance is made for any unit root in the generating process. We investigate the impact of estimation error on the mean-squared error of forecasts calculated from the fitted model.     

Median‐unbiased Estimation and Exact Inference Methods for First‐order Autoregressive Models with Conditional Heteroscedasticity of Unknown Form

Abstract. Consider the first‐order autoregressive model y_t = φy_t?1 + ?_t, t = 1,…, T, with arbitrary initial non‐zero value y₀. Assuming that the error terms ?_t are independently distributed according to median‐zero distributions [ Zieliński (1999) Journal of Time Series Analysis, Vol. 20, p. 477] shows that the estimator conjectured by Hurwicz (1950) Statistical Inference in Dynamic Economic Models. New York, NY: Wiley – the median of the consecutive ratios y_t/y_t?1– is an exactly median‐unbiased estimator of the autoregressive parameter φ. This paper shows that the Hurwicz estimator remains median‐unbiased under more general distributional assumptions, without assuming statistical independence. In particular, no restrictions are placed on the degree of heterogeneity and dependence of the conditional variance process. A computationally efficient method is also proposed to build exact confidence intervals for the autoregressive parameter which are valid in finite samples for any value of φ on the real line.     

On Residual Variance Estimation in Autoregressive Models

In this paper we consider time series models belonging to the autoregressive (AR) family and deal with the estimation of the residual variance. This is important because estimates of the variance are involved in, for example, confidence sets for the parameters of the model, estimation of the spectrum, expressions for the estimated error of prediction and sample quantities used to make inferences about the order of the model. We consider the asymptotic biases for moment and least squares estimators of the residual variance, and compare them with known results when available and with those for maximum likelihood estimators under normality. Simulation results are presented for finite samples     

Computer Algebra Derivation of the Bias of Linear Estimators of Autoregressive Models

Abstract. A symbolic method which can be used to obtain the asymptotic bias and variance coefficients to order O(1/n) for estimators in stationary time series is discussed. Using this method, the large‐sample bias of the Burg estimator in the AR(p) for p = 1, 2, 3 is shown to be equal to that of the least squares estimators in both the known and unknown mean cases. Previous researchers have only been able to obtain simulation results for the Burg estimator's bias because this problem is too intractable without using computer algebra. The asymptotic bias coefficient to O(1/n) of Yule–Walker as well as least squares estimates is also derived in AR(3) models. Our asymptotic results show that for the AR(3), just as in the AR(2), the Yule–Walker estimates have a large bias when the parameters are near the nonstationary boundary. The least squares and Burg estimates are much better in this situation. Simulation results confirm our findings.     

A Note on Diagnosing Multivariate Conditional Heteroscedasticity Models

In this paper we consider several tests for model misspecification after a multivariate conditional heteroscedasticity model has been fitted. We examine the performance of the recent test due to Ling and Li ( J. Time Ser. Anal. 18 (1997), 447–64), the Box–Pierce test and the residual-based F test using Monte Carlo methods. We find that there are situations in which the Ling–Li test has very weak power. The residual-based diagnostics demonstrate significant under-rejection under the null. In contrast, the Box–Pierce test based on the cross-products of the standardized residuals often provides a useful diagnostic that has reliable empirical size as well as good power against the alternatives considered.     

基于SVR误差补偿技术的神经网络城市污水厂水质预测

城市污水处理是一个复杂的生化处理过程,现在的神经网络技术无法做到对此过程精准建模.为了解决该问题,提出了一种基于SVR误差补偿技术的神经网络水质预测算法.该算法先利用BP神经网络对水质处理过程进行映射,再利用SVR误差补偿模型获得BP网络的预测补偿,进行预测数据校正.为了验证补偿模型的性能,还组织了马尔科夫补偿模型的对...     

Estimation in Random Coefficient Autoregressive Models

Abstract. We propose the quasi‐maximum likelihood method to estimate the parameters of an RCA(1) process, i.e. a random coefficient autoregressive time series of order 1. The strong consistency and the asymptotic normality of the estimators are derived under optimal conditions.     

Robust Estimation in Vector Autoregressive Moving-Average Models

Bustos and Yohai proposed a class of robust estimates for autoregressive moving-average (ARMA) models based on residual autocovariances (RA estimates). In this paper an affine equivariant generalization of the RA estimates for vector ARMA processes is given. These estimates are asymptotically normal and, when the innovations have an elliptical distribution, their asymptotic covariance matrix differs only by a scalar factor from the covariance matrix corresponding to the maximum likelihood estimate. A Monte Carlo study confirms that the RA estimates are efficient under normal errors and robust when the sample contains outliers. A robust multivariate goodness-of-fit test based on the RA estimates is also obtained.     

Sequential Adaptive Estimators in Nonparametric Autoregressive Models

Abstract

We construct a sequential adaptive procedure for estimating the autoregressive function at a given point in nonparametric autoregression models with Gaussian noise. We make use of the sequential kernel estimators. The optimal adaptive convergence rate is given as well as the upper bound for the minimax risk.     

A Note on the Information Matrix for Multiplicative Seasonal Autoregressive Moving‐Average Models

Abstract. It is shown that there is an invariance property for each of the elements of the information matrix of a multiplicative seasonal autoregressive moving‐average time‐series model, which enables the integral specification of Whittle (1953a,b) to be solved in a straightforward way. The resulting non‐iterative closed procedure shares the property possessed by the piecemeal approach of Godolphin and Bane (2006) of being independent of the seasonal period, but our procedure is preferable if one or more orders of the seasonal components of the model are greater than unity. The procedure is therefore simpler, in general, than the iterative method of Klein and Mélard (1990) that depends necessarily on the seasonal period. In the strictly non‐seasonal case this invariance property prescribes a non‐iterative closed procedure for evaluating the information matrix which improves on the methods of Godolphin and Unwin (1983) , Friedlander (1984) , McLeod (1984) and Klein and Spreij (2003) . Three illustrations of the approach are given.     

Optimal Detection of Exponential Component in Autoregressive Models

Abstract. In this article, the problem of detecting the eventual existence of an exponential component in an AR(1) model, that is, the problem of testing ordinary AR(1) dependence against the alternative of an exponential autoregression [EXPAR(1)] model, was considered. A local asymptotic normality property was established for EXPAR(1) models in the vicinity of AR(1) ones. Two problems arose in this context, which were quite typical in the study of nonlinear time‐series models. The first was a problem of parameter identification in the EXPAR(1) model. A special parameterization was developed so as to overcome this technical problem. The second problem was related to the fact that the underlying innovation density had to be treated as a nuisance. The problem at hand, indeed, appeared to be nonadaptive. These problems were solved using semi‐parametrically efficient pseudo‐Gaussian methods (which did not require Gaussian observations).     

COMPARISON OF TIME AND CROSS-SECTIONAL AGGREGATION UNDER A TIME SERIES RANDOM COMPONENT MODEL

Abstract. Small-area estimation under a stationary time series random component model is considered. Cross-sectional aggregation and varying degrees of time aggregation are treated as competing prediction methods. An estimated mean-squared prediction error criterion is used to compare these methods. Some exact and asymptotic properties of this criterion are developed, a consistent estimator of the associated asymptotic variance is presented and simultaneous approximate confidence intervals for the mean-squared prediction errors are discussed. Time aggregation of a single series is considered as a special case. In addition, an extension to the assessment of mean-squared prediction errors of synthetic small-area predictors is outlined.     

基于改进误差修正方法的灰色动态模型群水质预测

介绍了一种新的水质预测方法一灰色动态模型群法的基本原理和计算过程。将该方法应用于浑河沈阳段某监测断面的水质预测中,预测所得数值经过一种新的误差修正方法后和实际监测水质数据同时进行水质标识指数法评价,以便得出一个简单直观的结果对其精度进行检验。结论得出,应用新的误差修正方法的灰色动态模型群的预测结果与实际监测的水质结果一致,表明该方法能克服由数据序列随机波动性大而引起的GM（1,1）动态模型群预测精度不高的问题,可以作为水质预测中的一种有效的误善修正方法。     

Conditional variance estimation in regression models with long memory

In this article we study asymptotic properties of a non‐parametric kernel estimator of the conditional variance in a random design model with parametric mean and heteroscedastic errors, for a class of long‐memory errors and predictors. We establish small and large bandwidths asymptotics, which show a different behaviour compared with that of kernel estimators of the conditional mean. We distinguish between an oracle case (i.e. where the errors are directly observed) and a non‐oracle case (where the errors are replaced with residuals) and show non‐equivalence between the oracle and non‐oracle case. We also discuss a practical problem of bandwidth choice. Theoretical results are justified by simulation studies. We apply our theory to DJA and FTSE indices.     

混凝土碳化深度预测模型分析

碳化深度预测模型计算值与实测碳化深度值的对比分析表明,基于水灰比W/C的经验模型、基于水灰比W/C和水泥用量C的经验模型、单纯基于抗压强度的Smolczyk模型的碳化深度预测值与实测值误差较大,应根据实际情况选择使用。牛荻涛模型和张海燕模型作为以抗压强度为主要参数的经验模型,较为可靠。张誉模型基于扩散理论及碳化机理建立,具有充分的理论依据和较高的预测精度。     

Testing for a Unit Root in Noncausal Autoregressive Models

This work develops maximum likelihood‐based unit root tests in the noncausal autoregressive (NCAR) model with a non‐Gaussian error term formulated by Lanne and Saikkonen (2011, Journal of Time Series Econometrics 3, Issue 3, Article 2). Finite‐sample properties of the tests are examined via Monte Carlo simulations. The results show that the size properties of the tests are satisfactory and that clear power gains against stationary NCAR alternatives can be achieved in comparison with available alternative tests. In an empirical application to a Finnish interest rate series, evidence in favour of an NCAR model with leptokurtic errors is found.     

Bayesian Prediction Mean Squared Error for State Space Models with Estimated Parameters

Hamilton (A standard error for the estimated state vector of a state-space model. J. Economet. 33 (1986), 387–97) and Ansley and Kohn (Prediction mean squared error for state space models with estimated parameters. Biometrika 73 (1986), 467–73) have both proposed corrections to the naive approximation (obtained via substitution of the maximum likelihood estimates for the unknown parameters) of the Bayesian prediction mean squared error (MSE) for state space models, when the model's parameters are estimated from the data. Our work extends theirs in that we propose enhancements by identifying missing terms of the same order as that in their corrections. Because the approximations to the MSE are often subject to a frequentist interpretation, we compare our proposed enhancements with their original versions and with the naive approximation through a simulation study. For simplicity, we use the random walk plus noise model to develop the theory and to get our empirical results in the main body of the text. We also illustrate the differences between the various approximations with the Purse Snatching in Chicago series. Our empirical results show that (i) as expected, the underestimation in the naive approximation decreases as the sample size increases; (ii) the improved Ansley–Kohn approximation is the best compromise considering theoretical exactness, bias, precision and computational requirements, though the original Ansley–Kohn method performs quite well; finally, (iii) both the original and the improved Hamilton methods marginally improve the naive approximation. These conclusions also hold true with the Purse Snatching series.