First-order Autoregressive Process Research Articles

Tail probability bounds for acceptable random variables of partial sums and their application in linear models

Exponential probability inequalities play an important role in many parts of probability and statistics, offering a fundamental tool for tail probability bounding and modeling stochastic processes. The present work proposes a new exponential inequality for acceptable random variables. This class covers dependencies that are not embraced by the standard independence assumptions. Our results extend the theoretical perspective by placing sharper bounds on the above-mentioned variables, which tends to increase the accuracy of probabilistic modeling. Moreover, we study complete convergence in first-order autoregressive processes with acceptable innovations. The complete convergence framework gives further insights into the long-term behavior of the wide class of processes commonly used in time series analysis, econometrics, and other areas of applied research. In turn, our results on the influence of acceptable innovations on the properties of complete convergence establish bridges between theoretical probability inequalities and statistical applications. In addition, these contributions provide new tools in investigating the stability and predictability of the autoregressive models. These considerations consequently set up significant perspectives and enrich a developing body of research related to stochastic processes by reinforcing the theoretical framework as well as the methodological apparatus of probability theory.

Swarming Insects May Have Finely Tuned Characteristic Reynolds Numbers

Over the last few years, there has been much effort put into the development and validation of stochastic models of the trajectories of swarming insects. These models typically assume that the positions and velocities of swarming insects can be represented by continuous jointly Markovian processes. These models are first-order autoregressive processes. In more sophisticated models, second-order autoregressive processes, the positions, velocities, and accelerations of swarming insects are collectively Markovian. Although it is mathematically conceivable that this hierarchy of stochastic models could be extended to higher orders, here I show that such a procedure would not be well-based biologically because some terms in these models represent processes that have the potential to destabilize insect flight dynamics. This prediction is supported by an analysis of pre-existing data for laboratory swarms of the non-biting midge Chironomus riparius. I suggest that the Reynolds number is a finely tuned property of swarming, as swarms may disintegrate at both sufficiently low and sufficiently high Reynolds numbers.

Preliminary Test Estimation for Parallel 2-Sampling in Autoregressive Model

The purpose of this paper is to discuss the problem of estimation and testing the equality of two autoregressive parameters of two first-order autoregressive processes AR(1), where for each process, the observations are made at different time points. The primary interest is to propose the testing procedures for the homogeneity of autocorrelation parameters ρ1 and ρ2. Furthermore, we are interested in estimating ρ1 under uncertain and weak prior information about the possible equality of ρ1 and ρ2, though we may not have full confidence in the tenacity of this information. A large sample test for the homogeneity of the parameters is developed. Pooled “P” (or restricted estimator) and preliminary test “PT” estimators are proposed, and their properties are investigated and compared with the unrestricted estimator “UE” of ρ1.

Statistical inference for the first-order autoregressive process with the fractional Gaussian noise

While the statistical inference of first-order autoregressive processes driven by independent and identically distributed noises has a long history, the statistical analysis for first-order autoregressive processes driven by dependent noises is more recent. This paper considers the problem of estimating all the unknown parameters in first-order autoregressive processes driven by the fractional Gaussian noise, which is a self-similar stochastic process used to model persistent or anti-persistent dependency structures in observed time series. The estimation procedure is built upon the marriage of the log-periodogram regression and the method of moments. The usual asymptotic properties, including consistency and asymptotic normality, are established under some mild conditions. Monte Carlo simulations are performed to demonstrate the feasibility and effectiveness of the proposed method. Finally, the estimation methods are applied to model some realized volatility time series, where we find that the realized volatility is rough. Moreover, the proposed model is compared with some alternative models, including the first-order autoregressive process driven by the standard Gaussian noise, the heterogeneous autoregression model with lag structure (1,5,21) (HAR(1,5,21)), the autoregressive fractionally integrated moving average model (ARFIMA(1,d,0)) and the scaled fractional Brownian motion, in forecasting the logarithmic realized volatility.

Separating states in astronomical sources using hidden Markov models: with a case study of flaring and quiescence on EV Lac

ABSTRACT We present a new method to distinguish between different states (e.g. high and low, quiescent and flaring) in astronomical sources with count data. The method models the underlying physical process as latent variables following a continuous-space Markov chain that determines the expected Poisson counts in observed light curves in multiple passbands. For the underlying state process, we consider several autoregressive processes, yielding continuous-space hidden Markov models of varying complexity. Under these models, we can infer the state that the object is in at any given time. The continuous state predictions from these models are then dichotomized with the help of a finite mixture model to produce state classifications. We apply these techniques to X-ray data from the active dMe flare star EV Lac, splitting the data into quiescent and flaring states. We find that a first-order vector autoregressive process efficiently separates flaring from quiescence: flaring occurs over 30 per cent–40 per cent of the observation durations, a well-defined persistent quiescent state can be identified, and the flaring state is characterized by higher plasma temperatures and emission measures.

Fixed accuracy confidence intervals for variance under first-order stationary autoregressive processes

Fixed accuracy confidence intervals for variance under first-order stationary autoregressive processes

Consistent DCF Methods for Constant-Growth Annuities à la Modigliani & Miller or Miles & Ezzell

Abstract In this paper, we develop two complete discounted-cash flow (DCF) frameworks for the valuation of constant-growth annuities and perpetuities. By ‘complete’ we mean that these frameworks allow the valuation of a firm or project by means of different DCF methods, particularly, the equity method, the free-cash-flow (FCF) method, the adjusted-present-value-method, and the capital-cash-flow method. This also requires the derivation of formulas that allow the translation between different required returns, like the required return on unlevered and levered equity, the discount rate in the FCF method, and the required return on the tax-shield. Our paper departs from the two most advocated and mutually exclusive frameworks when dealing with DCF. The first is based on Modigliani and Miller (M&M), where the FCF at different points in time are independently distributed. The second framework rests on the analysis of Miles and Ezzell (M&E) who presume a first-order autoregressive cash-flow process. Some elements of a ‘complete’ framework exist in the literature, but in our opinion, a complete picture has not been developed yet. The contributions of this paper are the following: (1) We develop (or expand) the set of formulas that are required for the valuation of constant-growth annuities and perpetuities; (2) The formulas we develop in this paper are based on a backward-iteration process, which in itself represents a suitable tool for firm valuation; (3) Using a numerical example, we show that the two mutually exclusive frameworks of M&M or M&E achieve very different valuation results; (4) It turns out that the expected returns and the growth rate of the FCF are partly linked, but this relationship is different in the two frameworks; (5) In our numerical examples, we show how the constant-growth annuity or perpetuity, can be integrated with an explicitly planned FCF.

Bivariate first-order random coefficient integer-valued autoregressive processes based on modified negative binomial operator

In this paper, a new bivariate random coefficient integer-valued autoregressive process based on modified negative binomial operator with dependent innovations is proposed. Basic probabilistic and statistical properties of this model are derived. To estimate unknown parameters, Yule-Walker, conditional least squares and conditional maximum likelihood methods are considered and evaluated by Monte Carlo simulations. Asymptotic properties of the estimators are derived. Moreover, coherent forecasting and possible extension of the proposed model is provided. Finally, the proposed model is applied to the monthly crime datasets and compared with other models.

Almost Sure Central Limit Theorem for Error Variance Estimator in Pth-Order Nonlinear Autoregressive Processes

In this paper, under some suitable assumptions, using the Taylor expansion, Borel–Cantelli lemma and the almost sure central limit theorem for independent random variables, the almost sure central limit theorem for error variance estimator in the pth-order nonlinear autoregressive processes with independent and identical distributed errors was established. Four examples, first-order autoregressive processes, self-exciting threshold autoregressive processes, threshold-exponential AR progresses and multilayer perceptrons progress, are given to verify the results.

Evaluation of metrics and thresholds for use in national-scale river harmful algal bloom assessments

The spatiotemporal distribution of harmful algal blooms (HABs) in rivers remains poorly understood, and there is an urgent need to develop a consistent set of metrics to better document HAB occurrences and forecast future events. Using data from seven sites in the Illinois River Basin, we computed metrics focused on HAB conditions related to excess algal growth and hypoxia. Daily mean chlorophyll and dissolved oxygen (DO) concentrations, gross primary productivity (GPP), and net ecosystem productivity (NEP) rates, focused on water quality status, identifying the timing of the transition from a clear-water to an algal dominated state. Early warning indicators (EWIs), the first-order autoregressive process (Ar1) and standard deviation (SD) of chlorophyll concentrations, focused on future events, forecasting blooms. Metrics were compared to either literature-derived or statistical-based thresholds and were normalized by total number of daily samples for an exceedance rate. Exceedances of a daily mean chlorophyll concentration averaged 50 % across all sites using a 10 µg L−1 threshold but increasing the threshold to 50 μg L−1 reduced the average exceedance rate to 5 %. The average exceedance rate for GPP (∼8 g O2 m2d−1 threshold) was 15 %, similar to the daily amplitude DO concentration (∼3 mg L−1 threshold), but the average for NEP (0 g O2 m2 d−1 threshold) was higher, at 28 %. The number of days with at least 1 continuous DO concentration below the threshold of 5, 3, or 2 mg L−1, had basin wide exceedance rates of 9 %, 3 %, and 2 %, respectively. Thresholds for EWIs, Ar1 and SD, were exceeded at 5 of the 7 sites with high chlorophyll concentrations and GPP rates. The correlation between proxies for algal biomass (chlorophyll concentration) and productivity (GPP) was strongest for sites in the middle region of the basin, with R2 values between 0.54 and 0.74. Although, cyanotoxin concentrations are the most commonly used metrics by states to define an inland water HAB, there is a paucity of publicly available data. The wider availability of chlorophyll and oxygen data combined with the results from this study suggest that biomass and productivity state and event-based metrics may be a promising way to assess and predict the vulnerability of rivers to some of the deleterious effects of HABs at broad spatial scales.

The implications of economic freedom and gender ideologies on women's opportunity-to-necessity entrepreneurship

PurposeThis study explores the relationship between economic freedom and gender ideologies on the allocation of women’s opportunity-to-necessity entrepreneurship across countries. Opportunity entrepreneurship is typically understood as one’s best option for work, whereas necessity entrepreneurship describes the choice as driven by no better option for work. Specifically, we examine how economic freedom (i.e. each country’s policies that facilitate voluntary exchange) and gender ideologies (i.e. each country’s propensity for gendered separate spheres) affect the distribution of women’s opportunity-to-necessity entrepreneurship across countries.Design/methodology/approachWe construct our sample by matching data from the following country-level sources: the Global Entrepreneurship Monitor’s Adult Population Survey (APS), the Fraser Institute’s Economic Freedom Index (EFI), the European/World Value Survey’s Integrated Values Survey (IVS) gender equality index, and other covariates from the IVS, Varieties of Democracy (V-dem) World Bank (WB) databases. Our final sample consists of 729 observations from 109 countries between 2006 and 2018. Entrepreneurial activity motivations are measured by the ratio of the percentage of women’s opportunity-driven total nascent and early-stage entrepreneurship to the percentage of female necessity-driven total nascent and early-stage entrepreneurship at the country level. Due to a first-order autoregressive process and heteroskedastic cross-sectional dependence in our panel, we estimate a fixed-effect regression with robust standard errors clustered by country.FindingsAfter controlling for multiple macro-level factors, we find two interesting findings. First, economic freedom positively affects the ratio of women’s opportunity-to-necessity entrepreneurship. We find that the size of government, sound money, and business and credit regulations play the most important role in shaping the distribution of contextual motivations over time and between countries. However, this effect appears to benefit efficiency and innovation economies more than factor economies in our sub-sample analysis. Second, gender ideologies of political equality positively affect the ratio of women’s opportunity-to-necessity entrepreneurship, and this effect is most pronounced for efficiency economies.Originality/valueThis study offers one critical contribution to the entrepreneurship literature by demonstrating how economic freedom and gender ideologies shape the distribution of contextual motivation for women’s entrepreneurship cross-culturally. We answer calls to better understand the variation within women’s entrepreneurship instead of comparing women’s and men’s entrepreneurial activity. As a result, our study sheds light on how structural aspects of societies shape the allocation of women’s entrepreneurial motivations through their institutional arrangements.

Randomness Test of Thinning Parameters for the NBRCINAR(1) Process

Non-negative integer-valued time series are usually encountered in practice, and a variety of integer-valued autoregressive processes based on various thinning operators are commonly used to model these count data with temporal dependence. In this paper, we consider a first-order integer-valued autoregressive process constructed by the negative binomial thinning operator with random coefficients, to address the problem of constant thinning parameters which might not always accurately represent real-world settings because of numerous external and internal causes. We estimate the model parameters of interest by the two-step conditional least squares method, obtain the asymptotic behaviors of the estimators, and furthermore devise a technique to test the constancy of the thinning parameters, which is essential for determining whether or not the proposed model should consider the parameters’ randomness. The effectiveness and dependability of the suggested approach are illustrated by a series of thorough simulation studies. Finally, two real-world data analysis examples reveal that the suggested approach is very useful and flexible for applications.

INAR(1) process with weighted negative binomial Lindley distributed innovations and applications to criminal and COVID-19 data

In this study, we introduce a pliant stationary first-order integer-valued autoregressive (INAR) process with weighted negative binomial Lindley innovations. The main properties of the model are derived. The methods of conditional maximum likelihood, conditional least square and Yule-Walker are used for estimating the process parameters, while the efficiency of these three methods is evaluated through a simulation study. Finally, the practical aspect of the proposed INAR(1) process is discussed on two time series of the monthly number of criminal mischief reports in Pittsburgh and the daily COVID-19 death cases in Paraguay.

On Sample Size Needed for Block Bootstrap Confidence Intervals to Have Desired Coverage Rates

Block bootstrap is widely used in constructing confidence intervals for parameters estimated from stationary time series. Theoretically, the method should provide valid confidence intervals as the length of the time series goes to infinity. In practice, however, it is necessary to know how large of a finite sample is required for block bootstrap confidence intervals to work well. This study aims to answer this question in a simple simulation setting where the data are generated from a first-order autoregressive process. The empirical coverage rates of several commonly used bootstrap confidence intervals for the mean, standard deviation, and the lag-1 autocorrelation coefficient are compared. A quite large sample is found necessary for the intervals to have the right coverage rates even when estimating a simple parameter like the mean. Some block bootstrap methods could fail when estimating the lag-1 autocorrelation. It is surprising that the coverage property even deteriorates as the sample size increases with some commonly used block bootstrap confidence intervals including the percentile intervals and bias-corrected intervals. KEYWORDS: Autocorrelation; Bias-Correction; Centering; Dependent Data; Percentile; Resampling; Simulation; Time Series

A Time-Varying Mixture Integer-Valued Threshold Autoregressive Process Driven by Explanatory Variables.

In this paper, a time-varying first-order mixture integer-valued threshold autoregressive process driven by explanatory variables is introduced. The basic probabilistic and statistical properties of this model are studied in depth. We proceed to derive estimators using the conditional least squares (CLS) and conditional maximum likelihood (CML) methods, while also establishing the asymptotic properties of the CLS estimator. Furthermore, we employed the CLS and CML score functions to infer the threshold parameter. Additionally, three test statistics to detect the existence of the piecewise structure and explanatory variables were utilized. To support our findings, we conducted simulation studies and applied our model to two applications concerning the daily stock trading volumes of VOW.

Estimating the effect of latent time-varying count exposures using multiple lists.

A major challenge in longitudinal built-environment health studies is the accuracy of commercial business databases that are used to characterize dynamic food environments. Different databases often provide conflicting exposure measures on the same subject due to different source credibilities. As on-site verification is not feasible for historical data, we suggest combining multiple databases to correct the bias in health effect estimates due to measurement error in any 1 datasource. We propose a joint model for the time-varying health outcomes, observed count exposures, and latent true count exposures. Our model estimates the time-specific quality of sources and incorporates time dependence of true count exposure by Poisson integer-valued first-order autoregressive process. We take a Bayesian nonparametric approach to flexibly account for location-specific exposures. By resolving the discordance between different databases, our method reduces the bias in the longitudinal health effect of the true exposures. Our method is demonstrated with childhood obesity data in California public schools with respect to convenience store exposures in school neighborhoods from 2001 to 2008.

Bivariate Poisson Generalized Lindley Distributions and the Associated BINAR(1) Processes

This paper proposes new bivariate distributions based on the Poisson generalized Lindley distribution as marginal. These models include the basic bivariate Poisson generalized Lindley (BPGL) and the Sarmanov-based bivariate Poisson generalized Lindley (SPGL) distributions. Subsequently, we introduce the BPGL and SPGL distributions as joint innovation distributions in a novel bivariate first-order integer-valued autoregressive process (BINAR(1)) based on binomial thinning. The model parameters in the BPGL and SPGL distributions are estimated using the method of maximum likelihood (ML) while we apply the conditional maximum likelihood (CML) for the BINAR(1) process. We conduct some simulation experiments to assess the small and large sample performances. Further, we implement the new BINAR(1)s to the Pittsburgh crime series data and they show better fitting criteria than other competing BINAR(1) models in the literature.

Bayesian Analysis of Threshold Autoregressive Model with First Order Autoregressive Innovations

Financial assets exhibit dramatic changes in behaviour. This work examined a two-regime Threshold autoregressive (TAR) models when the innovations follow a first-autoregressive order process. The Bayesian method is proposed to build in the linear first-order autoregressive process with identical distributed innovations. The practical usefulness of this method is demonstrated with simulated and real-life data using U.S.A quarterly real GDP as an example. In simulation experiments and real life example, an increase in first order process parameter, ρ value leads to better estimates in the proposed model. Also, the proposed model was compared with TAR model where the disturbance term does not exhibit regime switching. The proposed model performed well than the traditional TAR model using the simulated and real life data. An increase in first order process parameter, ρ will lead to better estimates and forecast. Hence, the proposed model performed well.

A new First-Order mixture integer-valued threshold autoregressive process based on binomial thinning and negative binomial thinning

In this paper, we introduce a new first-order mixture integer-valued threshold autoregressive process, based on the binomial and negative binomial thinning operators. Basic probabilistic and statistical properties of this model are discussed. Conditional least squares (CLS) and conditional maximum likelihood (CML) estimators are derived and the asymptotic properties of the estimators are established. The inference for the threshold parameter is obtained based on the CLS and CML score functions. Moreover, the Wald test is applied to detect the existence of the piecewise structure. Simulation studies are considered, along with an application: the number of criminal mischief incidents in the Pittsburgh dataset

Multivariate threshold integer-valued autoregressive processes with explanatory variables

To capture the multivariate count time series showing piecewise phenomena, we introduce a class of first-order multivariate threshold integer-valued autoregressive process. The component of autoregressive coefficient is driven by explanatory variables via a logistic link function. Basic probabilistic and statistical properties of the model are discussed. The model parameters are estimated by means of conditional least squares and conditional maximum likelihood methods. A new algorithm to estimate the threshold parameter of the model is also provided. Moreover, the nonlinearity test of the model and existence test of explanatory variables are well addressed using a Wald-type test. Forecasting problem and model extension problem are also considered. Finally, some numerical results of the estimates and a real data example are presented.