A characterization of the inverse autocorrelation function

The inverse autocorrelation function of a weakly stationary stochastic process X_t at lag h, γⁱ _h, is shown to equal the negative of the partial correlation between random variables X_t and X_t+h after elimination of the influence of random variables X_k, k≠t5,t+h.     

On the problem of model selection in bilinear time series analyses

One of the problems in bilinear time series (BLTS) analysis is that of identification. Unlike linear models, the identification in BLTS modelling is not always based on the autocorrelation function (or spectrum) since it is sometimes misleading, The authors, therefore., derive in this note the autocorrelation function of a function of a bilinear process which can be used for identification as well as for testing the linearity.     

Derivation of the theoretical autocovariance and autocorrelation function of autoregessive moving average processes

Closed form expressions for the theoretical autocovariance and autocorrelation function of mixed autoregressive moving average processes are presented. The results provide insight into the construction of autocovariances and autocorrelatians and are useful in theoretical analysis, model identification as well as in implementing maximum likelihood estimation algorithms.     

基于自相关视角的弱平稳过程之间的伪回归分析

随机干扰项之间的未知形式自相关是导致相互独立的弱平稳过程之间伪回归的主要原因.通过理论分析和一系列的蒙特卡罗模拟,揭示了数据过程本身的持久性、样本容量T和随机干扰项自相关之间的内在联系.研究发现随机干扰项往往呈现出与数据过程阶数相同的自相关.进一步研究表明,运用广义差分法和Cochrane- Orcutt迭代法虽然能大大减少伪回归概率,但在有些情况下,即使当样本容量较大时,较高阶的Cochrane- Orcutt迭代法仍然无法避免伪回归的发生.     

Time series analysis of categorical data using auto-odds ratio function

In this paper, we consider the auto-odds ratio function (AORF) as a measure of serial association for a stationary time series process of categorical data at two different time points. Numerical measures such as the autocorrelation function (ACF) have no meaningful interpretation, unless the time series data are numerical. Instead, we use the AORF as a measure of association to study the serial dependency of the categorical time series for both ordinal and nominal categories. Biswas and Song [Discrete-valued ARMA processes. Stat Probab Lett. 2009;79(17):1884–1889] provided some results on this measure for Pegram's operator-based AR(1) process with binary responses. Here, we extend this measure to more general set-ups, i.e. for AR(p) and MA(q) processes and for a general number of categories. We discuss how this method can effectively be used in parameter estimation and model selection. Following Weiß [Empirical measures of signed serial dependence in categorical time series. J Stat Comput Simul. 2011;81(4):411–429], we derive the large sample distribution of the estimator of the AORF under independent and identically distributed (iid) set-up. Some simulation results and two categorical data examples (one is ordinal and other nominal) are presented to illustrate the proposed method.     

Wild-bootstrapped variance-ratio test for autocorrelation in the presence of heteroskedasticity

The Breusch–Godfrey LM test is one of the most popular tests for autocorrelation. However, it has been shown that the LM test may be erroneous when there exist heteroskedastic errors in a regression model. Recently, remedies have been proposed by Godfrey and Tremayne [9] and Shim et al. [21]. This paper suggests three wild-bootstrapped variance-ratio (WB-VR) tests for autocorrelation in the presence of heteroskedasticity. We show through a Monte Carlo simulation that our WB-VR tests have better small sample properties and are robust to the structure of heteroskedasticity.     

Testing for Uncorrelated Residuals in Dynamic Count Models With an Application to Corporate Bankruptcy

This article proposes new model checks for dynamic count models. Both portmanteau and omnibus-type tests for lack of residual autocorrelation are considered. The resulting test statistics are asymptotically pivotal when innovations are uncorrelated but possibly exhibit higher order serial dependence. Moreover, the tests are able to detect local alternatives converging to the null at the parametric rate T^{? 1/2}, with T the sample size. The finite sample performance of the test statistics are examined by means of Monte Carlo experiments. Using a dataset on U.S. corporate bankruptcies, the proposed tests are applied to check if different risk models are correctly specified. Supplementary materials for this article are available online.     

存在自相关时检验和估计方法的改进——基于自回归分布滞后模型的自相关检验和参数估计

存在自相关时的自相关检验和参数估计是基础计量经济学的一个重要内容,并且存在自相关时的原模型已转化为自回归分布滞后模型。讨论存在自相关时的自相关检验和参数估计问题,提出了一种基于自回归分布滞后模型的自相关检验法,并同时给出了相应的参数估计。     

Preliminary Estimation in Gaussian Stationary Processes

Preliminary estimation of the kth Lag autocorrelation function in the Gaussian stationary processes is considered. An estimation procedure is derived from the ratio of the sum filter and the difference filter. The performance of this estimator is compared to the sample estimator through a Monte Carlo study.     

Interpolation,outliers mid inverse autocorrelations

The paper addresses the problem of estimating missing observations in an infinite realization of a linear, possibly nonstationary, stochastic processes when the model is known. The general case of any possible distribution of missing observations in the time series is considered, and analytical expressions for the optimal estimators and their associated mean squared errors are obtained. These expressions involve solely the elements of the inverse or dual autocorrelation function of the series.

This optimal estimator -the conditional expectation of the missing observations given the available ones- is equal to the estimator that results from filling the missing values in the series with arbitrary numbers, treating these numbers as additive outliers, and removing with intervention analysis the outlier effects from the invented numbers.     

Estimating Spatial Autocorrelation With Sampled Network Data

Spatial autocorrelation is a parameter of importance for network data analysis. To estimate spatial autocorrelation, maximum likelihood has been popularly used. However, its rigorous implementation requires the whole network to be observed. This is practically infeasible if network size is huge (e.g., Facebook, Twitter, Weibo, WeChat, etc.). In that case, one has to rely on sampled network data to infer about spatial autocorrelation. By doing so, network relationships (i.e., edges) involving unsampled nodes are overlooked. This leads to distorted network structure and underestimated spatial autocorrelation. To solve the problem, we propose here a novel solution. By temporarily assuming that the spatial autocorrelation is small, we are able to approximate the likelihood function by its first-order Taylor’s expansion. This leads to the method of approximate maximum likelihood estimator (AMLE), which further inspires the development of paired maximum likelihood estimator (PMLE). Compared with AMLE, PMLE is computationally superior and thus is particularly useful for large-scale network data analysis. Under appropriate regularity conditions (without assuming a small spatial autocorrelation), we show theoretically that PMLE is consistent and asymptotically normal. Numerical studies based on both simulated and real datasets are presented for illustration purpose.     

Moving average models—time series in m-dimensions

Stochastic models for discrete time series in the time domain are well known but such models lack consideration of spatial dependency I We expand on their work by constructing spatially dependent moving average models. Definitions of order, stationarity, invertibility, autocorrelation function, and spectrum are made as natural extensions of those in zero dimensions and are implemented in the one and two-space dimensional models.     

Inference for Local Autocorrelations in Locally Stationary Models

For nonstationary processes, the time-varying correlation structure provides useful insights into the underlying model dynamics. We study estimation and inferences for local autocorrelation process in locally stationary time series. Our constructed simultaneous confidence band can be used to address important hypothesis testing problems, such as whether the local autocorrelation process is indeed time-varying and whether the local autocorrelation is zero. In particular, our result provides an important generalization of the R function acf() to locally stationary Gaussian processes. Simulation studies and two empirical applications are developed. For the global temperature series, we find that the local autocorrelations are time-varying and have a “V” shape during 1910–1960. For the S&P 500 index, we conclude that the returns satisfy the efficient-market hypothesis whereas the magnitudes of returns show significant local autocorrelations.     

The twelfth order analogue of the durbin-watson test

This paper provides five percent significance bounds on critical values of the twelfth or-der analogue of the Durbin-Watsontest. These tables are useful for testing for twelfth order autocorrelation in regression models with monthly data which include either an intercept or a full set of monthly seasonal dummies.     

On the bias of sampled autocorrelations

The author believes that, once again, the time has come to study how sample autocorrelation behaviour departs from asymptotic theory. Amongst other things, the classical formulae for bias need revising in the light of current definitions, and to make them immediately applicable to practical situations. Considerable extensions are also necessary to cover more general ARIMA models and, further, to obtain results valid for short series. Motivation is provided by demonstrating the varied implications of bias for Box-Jenkins identifications estimation, verification and forecasting.     

局部空间自相关指标对比研究

本文研究了最常用的局部空间自相关指标;局部Moran指数和局部G系数,基于模拟的空间区域;设计了一些有代表性的空间聚集方案进行计算,比较了两种指标的探测结果,得出局部G系数要优于局部Moran指数的初步结论。     

空间误差自相关随机前沿模型及其估计

将空间计量经济学的思想引入随机前沿分析，构建了基于横截面数据的空间误差自相关随机前沿模型，推导出模型的似然函数以求得参数估计，并给出了各生产单元技术效率的估计。     

The moving-range chart and autocorrelated processes

This paper investigates the behavior of the moving-range statistic and the performance of the standard moving-range chart in the presence of autocorrelation. The distributional properties and time-series properties of the moving-range statistic are presented. The average run length (ARL) properties of the standard moving-range chart in the presence of autocorrelation and a new proposed chart are presented.