Order Autoregressive Process Research Articles

The joint distribution of the maximum and minimum of an AR(1) process

Consider a sequence of n observations from an autoregressive process of order 1 with maximum Mn and minimum mn. We give their joint cumulative distribution function first in terms of n repeated integrals and then, for the case, where the marginal distribution of the observations is absolutely continuous, as a weighted sum of nth powers of eigenvalues of a certain Fredholm kernel. This enables good approximations for the joint distribution when n repeated integrals is not a practical solution.

Temporal trends in δ18O composition of precipitation in Germany: insights from time series modelling and trend analysis

AbstractTemporal and spatial variations of stable oxygen (18O) and hydrogen (2H) isotope measurements in precipitation act as important proxies for changing hydro‐meteorological and regional and global climate patterns. Temporal trends in time series of the stable isotope composition in precipitation were rarely observed, and they are poorly understood. These might be a result of a lack of proper trend detection tools and effort for exploring trend processes. Here, we investigate temporal trends of δ18O in precipitation at 17 observation stations in Germany between 1978 and 2009. We test if significant trends in the isotope time series from different models can be observed. Mann–Kendall trend tests are applied on the isotope series, using general multiplicative seasonal autoregressive integrate moving average (ARIMA) models, which account for first and higher order serial correlations. Effects of temperature, precipitation, and geographic parameters on isotope trends are also investigated in the proposed models. To benchmark our proposed approach, the ARIMA results are compared with a trend‐free pre‐whitening procedure, the state of the art method for removing the first order autocorrelation in environmental trend studies. Moreover, we further explore whether higher order serial correlations in isotope series affects our trend results. Overall, three out of the 17 stations show significant changes when higher order autocorrelation are adjusted, and four show a significant trend when temperature and precipitation effects are considered. The significant trends in the isotope time series generally occur only at low elevation stations. Higher order autoregressive processes are shown to be important in the isotope time series analysis. Results suggest that the widely used trend analysis with only the first order autocorrelation adjustment may not adequately take account of the high order autocorrelated processes in the stable isotope series. The investigated time series analysis method including higher autocorrelation and external climate variable adjustments is shown to be a better alternative. Copyright © 2014 John Wiley & Sons, Ltd.

Asymptotic normality of Huber-Dutter estimators in a linear EV model with AR(1) processes

The paper studies a linear errors-in-variables model with first order autoregressive processes. The Huber-Dutter (HD) estimators of unknown parameters are given, and the asymptotic normality of the HD estimators is investigated. Finally, a simple example is given to illustrate our estimation method. MSC:60F05, 60G10, 62F35, 62M10, 60G42.

Multi-Curve HJM Modelling for Risk Management

We present a HJM approach to the projection of multiple yield curves developed to capture the volatility content of historical term structures for risk management purposes. Since we observe the empirical data at daily frequency and only for a finite number of time-to-maturity buckets, we propose a modelling framework which is inherently discrete. In particular, we show how to approximate the HJM continuous time description of the multi-curve dynamics by a Vector Autoregressive process of order one. The resulting dynamics lends itself to a feasible estimation of the model volatility-correlation structure and market risk-premia. Then, resorting to the Principal Component Analysis we further simplify the dynamics reducing the number of covariance components. Applying the constant volatility version of our model on a sample of curves from the Euro area, we demonstrate its forecasting ability through an out-of-sample test.

WHAT WORKS BEST AND WHEN: ACCOUNTING FOR MULTIPLE SOURCES OF PURESELECTION BIAS IN PROGRAM EVALUATIONS

AbstractMost evaluations are still quasi‐experimental and most recent quasi‐experimental methodological research has focused on various types of propensity score matching to minimize conventional selection bias on observables. Although these methods create better‐matched treatment and comparison groups on observables, the issue of selection on unobservables still looms large. Thus, in the absence of being able to run randomized controlled trials (RCTs) or natural experiments, it is important to understand how well different regression‐based estimators perform in terms of minimizing pure selection bias, that is, selection on unobservables. We examine the relative magnitudes of three sources of pure selection bias: heterogeneous response bias, time‐invariant individual heterogeneity (fixed effects [FEs]), and intertemporal dependence (autoregressive process of order one [AR(1)]). Because the relative magnitude of each source of pure selection bias may vary in different policy contexts, it is important to understand how well different regression‐based estimators handle each source of selection bias. Expanding simulations that have their origins in the work of Heckman, LaLonde, and Smith (), we find that difference‐in‐differences (DID) using equidistant pre‐ and postperiods and FEs estimators are less biased and have smaller standard errors in estimating the Treatment on the Treated (TT) than other regression‐based estimators. Our data analysis using the Job Training Partnership Act (JTPA) program replicates our simulation findings in estimating the TT.

Multivariate normal-Laplace distribution and processes

The normal-Laplace distribution is considered and its properties are discussed. A multivariate normal-Laplace distribution is introduced and its properties are studied. First order autoregressive processes with these stationary marginal distributions are developed and studied. A generalized multivariate normal-Laplace distribution is introduced. Multivariate geometric normal-Laplace distribution and multivariate geometric generalized normal-Laplace distributions are introduced and their properties are studied. Estimation of parameters and some applications are also discussed.

Asymptotic Behavior of Conditional Least Squares Estimators for Unstable Integer‐valued Autoregressive Models of Order 2

ABSTRACTIn this paper, the asymptotic behavior of the conditional least squares estimators of the autoregressive parameters, of the mean of the innovations, and of the stability parameter for unstable integer‐valued autoregressive processes of order 2 is described. The limit distributions and the scaling factors are different according to the following three cases: (i) decomposable, (ii) indecomposable but not positively regular, and (iii) positively regular models.

Extracting the Influential Commodities in Stochastic Model of Simple Laspeyre Price Index Numbers with AR(2) Errors

This paper, on the first hand, deals with the problem of estimation of Laspeyre price index number when the errors are assumed to be generated from AR(2) process. The general expression of hat matrix and DFBETA measure to find the influential consumer commodities in stochastic Laspeyre price model with AR(2) errors are developed on the other. The hat values show the noteworthy findings that the corresponding weights of consumer items have large influence on the parameter estimates for simple Laspeyre price index number and are not affected by the parameter of autoregressive process of order two. While, DFBETA measures are the functions of both weights and autocorrelation parameters. Lastly, an example is presented with reference to price data of Pakistan, and shows its practical importance in financial time series.

Technical Analysis: Does Recent Market Data Substantiate the Efficient Market Hypothesis?

Currently, there is a controversy regarding fundamental predictability of the market, which has created a gap in understanding the fundamental principles of market performance. The problem this study addressed was the unknown predictability of the market that stems from under-investigated trading rules with current market data. The purpose of this quantitative study was to investigate the discrepant results of predictability from various types of technical analysis tools utilizing recent stock market data from the Dow Jones Industrial Average (DJIA) 1987 to 2009 to verify that these tools still produce positive trading results. The hypotheses explored whether market predictability is validated when those analytic techniques that were successfully applied in previous studies are applied to the original study using current stock market data. The sample used in this study was an ex-post facto research design that investigated simple analytic trading tools. The theoretical support for this study was from a model-based bootstrap study that found consistent positive returns by using simple technical analysis trading tools. A bootstrap methodology was used to construct tests on 5,800 trading days of the DJIA collected from the Center for Research in Security Prices. Replication of the study was analyzed via t test, random walk, autoregressive process of order one (AR (1)), a generalized autoregressive conditional heteroskedasticity in-mean model (GARCH-M), exponential GARCH (EGARCH) of the null model, descriptive analysis, autocorrelation, skewness, and kurtosis using a 5% significance level. The possible social significance of this study is the future benefit investors may realize by utilizing technical analysis trading strategies and protective buy (sell) stops to obtain positive returns in their long-term investment strategies.

An Improved Mean Shift Algorithm for Vehicle Tracking

Classical mean shift tracking algorithm doesnt show good performance when the tracked objects move fast, change in size or pose. This paper proposes an improved mean shift method used for vehicle tracking. Firstly, a position prediction model based on second order auto-regression process is used to find the initial position of mean shift iteration, reduce times of iteration and enhance the tracking accuracy. Secondly, we employ a position search method based on the weight image to improve the tracking result when the result of basic mean shift tracking is not good. The proposed algorithm is tested in a real traffic video to track a vehicle changing in size and pose with more accurate result than basic mean shift tracking algorithm.

Modeling sea surface temperatures of the North Atlantic Ocean

An assessment of the trends and patterns of sea surface temperatures in the North Atlantic Ocean was carried out using statistical methods. The data were obtained from the climate research unit, comprising twenty-five 5 by 10° regions, covering latitudes 40 to 65°N and longitudes 10 to 60°W between 1973 and 2008. Linear regression was initially used to model the seasonally adjusted sea surface temperatures. The data was filtered with a second order autoregressive process to remove autocorrelations between temperature lags. A multivariate linear regression model was then fit to the data to account for the spatial correlation, giving the covariance matrix of the estimated slopes. A three-factor model was fit to the covariance matrix to form three larger regions. A spline linear regression model with one knot at 1990 was then fit to these three regions. The temperatures in these three regions were found to increase by approximately 0.13 °C per decade during 1973–1989. During 1990–2008, the increase in temperatures in the two of three larger regions identified by the factors analysis was at least 0.40 °C per decade.

Ajuste dos modelos Gompertz e Logístico aos dados de crescimento de frutos de coqueiro anão verde

O objetivo do trabalho foi estudar o ajuste dos modelos não-lineares, gompertz e logístico, na descrição do desenvolvimento de frutos de coqueiro da variedade anão verde, com base nos dados de diâmetros externos, longitudinal e transversal. Os resultados indicaram que tanto para o diâmetro longitudinal externo (DEL) como para o diâmetro transversal externo (DET), o modelo logístico foi o mais adequado na descrição do crescimento do fruto. O ajuste dos modelos, para ambas as variáveis, apresentou autocorrelação residual positiva. Para o DEL o resíduo foi modelado segundo um processo autoregressivo de primeira ordem (AR1) e para a variável DET o processo autoregressivo considerado foi de segunda ordem (AR2). A consideração do processo autoregressivo (AR1 ou AR2) resultou em estimativas dos parâmetros mais precisas. De acordo com o modelo logístico, o fruto do coqueiro anão verde, totalmente desenvolvido, tem diâmetros externos longitudinal e transversal, respectivamente, de 21,56cm e 15,35cm.

JIT delivery with stochastic lead time

We examine the effect of stochastic lead times on Just-in-Time (JIT) delivery. We find that with stochastic lead times there is a possibility of order crossover, and what order crossover does is to transform the original lead times into effective lead times, which is an AR(1) process that is an autoregressive process of Order 1. The mean of this process is the same as the mean of the original lead time, but its variance could be much smaller. The implication is that when we consider order crossover in the analysis, the cost could be less than otherwise (but never less than that with deterministic lead times). The literature on JIT with stochastic lead times has never considered order crossover, which produces the effective delivery times (EDT). Here, we demonstrate some important properties of the EDT: that it is a Cauchy sequence, and hence it converges; that it is an AR(1) process; and that it stochastically dominates the parent lead time distribution.

On composite likelihood estimation of a multivariate INAR(1) model

In several circumstances the collected data are counts observed in different time points, while the counts at each time point are correlated. Current models are able to account for serial correlation but usually fail to account for cross‐correlation. Motivated by the lack of appropriate tools for handling such type of data, we define a multivariate integer‐valued autoregressive process of order 1 (MINAR(1)) and examine its basic statistical properties. Apart from the general specification of the MINAR(1) process, we also study two specific parametric cases that arise under the assumptions of a multivariate Poisson and a multivariate negative binomial distribution for the innovations of the process. To overcome the computational difficulties of the maximum likelihood approach we suggest the method of composite likelihood. The performance of the two methods of estimation, that is, maximum likelihood and composite likelihood, is compared through a small simulation experiment. Extensions of the time‐invariant model to a regression model are also discussed. The proposed model is applied to a trivariate data series related to daily traffic accidents in three areas in the Netherlands.

Unified asymptotic theory for nearly unstable AR([formula omitted]) processes

A unified asymptotic theory for nearly unstable higher order autoregressive processes and their least squares estimates is established. A novel version of Jordan’s canonical decomposition with perturbations together with a suitable plug-in principle is proposed to develop the underlying theories. Assumptions are stated in terms of the domain of attraction of partial Fourier transforms. The machinery is applied to recapture some of the classical results with the driving noise being martingale differences. Further, we show how to extend the results to higher order fractional ARIMA models in nearly unstable settings, thereby offering a comprehensive theory to analyse nearly unstable time series.

On the time series measure of conservatism: a threshold autoregressive model

In this note we propose an alternative test specification for Basu’s (1997) time series measure of conservatism that is related to the threshold unit root test of Enders and Granger (1998). We argue that a regression of changes in earnings on the lagged levels—rather than lagged changes—, including an interaction term for negative values, has three conceptual advantages compared to the conventional setup: (1) a smooth, non-oscillating impulse-response pattern to an unexpected shock in earnings (2) a more efficient estimate of persistence in the long run and (3) it can be extended to higher order autoregressive processes. We apply both approaches to a common dataset of firms from the S&P500 index. We confirm the conventional finding that negative shocks are transitory and display stronger mean reversion than positive shocks. However, while most of the literature reports mixed evidence on positive shocks, we find clear evidence that positive shocks are transitory as well. In a Monte Carlo simulation we explain this finding by documenting that larger standard errors in the Basu specification can lead to incorrect inference when the decision between persistent and transitory shocks is close.

A statistical approach to construct composite climate records of dated archives

Proxy records of dated environmental archives like stalagmites are used for reconstruction of past climate and therefore are of fundamental interest for the paleoclimate community. However, dating conditions are often not perfect to obtain precise ages with small uncertainties. On this matter, the use of statistical approaches applied to reproduced climate signals of several nearby situated specimens, like stalagmites from one cave, can help to reduce age uncertainties. A new method implemented in MATLAB uses a Monte Carlo approach on absolute age determinations to find the best correlation between climate proxies of several signal reproducing adjacent archives. Therefore, the program is able to combine climatic proxies to construct a composite record. This “intra-site correlation age modelling” (iscam) approach offers great advantages. The age uncertainty can be significantly reduced within the overlapping time intervals and it can be tested if the signal of interest is indeed similar in both records. Additionally, iscam allows to enlarge the time span of a single record while at the same time the signal to noise ratio of the combined record improves in periods where replicates exist. Significance levels of the correlation can be calculated against the red-noise background from a first order autoregressive process (AR1), which allows to determine adequate age uncertainties. The method was designed to synchronize time series of nearby locations, where changes in the climate signal occur simultaneously. Applying this method to geographically dispersed locations might not be appropriate due to unknown leads and lags in the climate system.

Simultaneous confidence bands for Yule–Walker estimators and order selection

Let $\{X_k,k\in{\mathbb{Z}}\}$ be an autoregressive process of order $q$. Various estimators for the order $q$ and the parameters ${\bolds \Theta}_q=(\theta_1,...,\theta_q)^T$ are known; the order is usually determined with Akaike's criterion or related modifications, whereas Yule-Walker, Burger or maximum likelihood estimators are used for the parameters ${\bolds\Theta}_q$. In this paper, we establish simultaneous confidence bands for the Yule--Walker estimators $\hat{\theta}_i$; more precisely, it is shown that the limiting distribution of ${\max_{1\leq i\leq d_n}}|\hat{\theta}_i-\theta_i|$ is the Gumbel-type distribution $e^{-e^{-z}}$, where $q\in\{0,...,d_n\}$ and $d_n=\mathcal {O}(n^{\delta})$, $\delta >0$. This allows to modify some of the currently used criteria (AIC, BIC, HQC, SIC), but also yields a new class of consistent estimators for the order $q$. These estimators seem to have some potential, since they outperform most of the previously mentioned criteria in a small simulation study. In particular, if some of the parameters $\{\theta_i\}_{1\leq i\leq d_n}$ are zero or close to zero, a significant improvement can be observed. As a byproduct, it is shown that BIC, HQC and SIC are consistent for $q\in\{0,...,d_n\}$ where $d_n=\mathcal {O}(n^{\delta})$.

Pricing catastrophe swaps: A contingent claims approach

In this paper, we comprehensively analyze the catastrophe (cat) swap, a financial instrument which has attracted little scholarly attention to date. We begin with a discussion of the typical contract design, the current state of the market, as well as major areas of application. Subsequently, a two-stage contingent claims pricing approach is proposed, which distinguishes between the main risk drivers ex-ante as well as during the loss reestimation phase and additionally incorporates counterparty default risk. Catastrophe occurrence is modeled as a doubly stochastic Poisson process (Cox process) with mean-reverting Ornstein–Uhlenbeck intensity. In addition, we fit various parametric distributions to normalized historical loss data for hurricanes and earthquakes in the US and find the heavy-tailed Burr distribution to be the most adequate representation for loss severities. Applying our pricing model to market quotes for hurricane and earthquake contracts, we derive implied Poisson intensities which are subsequently condensed into a common factor for each peril by means of exploratory factor analysis. Further examining the resulting factor scores, we show that a first order autoregressive process provides a good fit. Hence, its continuous-time limit, the Ornstein–Uhlenbeck process should be well suited to represent the dynamics of the Poisson intensity in a cat swap pricing model.

Bayesian hierarchical Poisson models with a hidden Markov structure for the detection of influenza epidemic outbreaks

Considerable effort has been devoted to the development of statistical algorithms for the automated monitoring of influenza surveillance data. In this article, we introduce a framework of models for the early detection of the onset of an influenza epidemic which is applicable to different kinds of surveillance data. In particular, the process of the observed cases is modelled via a Bayesian Hierarchical Poisson model in which the intensity parameter is a function of the incidence rate. The key point is to consider this incidence rate as a normal distribution in which both parameters (mean and variance) are modelled differently, depending on whether the system is in an epidemic or non-epidemic phase. To do so, we propose a hidden Markov model in which the transition between both phases is modelled as a function of the epidemic state of the previous week. Different options for modelling the rates are described, including the option of modelling the mean at each phase as autoregressive processes of order 0, 1 or 2. Bayesian inference is carried out to provide the probability of being in an epidemic state at any given moment. The methodology is applied to various influenza data sets. The results indicate that our methods outperform previous approaches in terms of sensitivity, specificity and timeliness.