On trends and constants in periodic autoregressions

Periodic autoregressions are characterised by autoregressive structures that vary with the season. If a time series is periodically integrated, one needs a seasonally varying differencing filter to remove the stochastic trend. When the periodic regression model contains constants and trends with unrestricted parameters, the data can show diverging seasonal deterministic trends. In this paper we derive explicit expressions for parameter restrictions that result in common deterministic trends under periodic trend stationarity and periodic integration.     

Semiparametric Bayesian Inference for Time Series with Mixed Spectra

A Bayesian analysis is presented of a time series which is the sum of a stationary component with a smooth spectral density and a deterministic component consisting of a linear combination of a trend and periodic terms. The periodic terms may have known or unknown frequencies. The advantage of our approach is that different features of the data—such as the regression parameters, the spectral density, unknown frequencies and missing observations—are combined in a hierarchical Bayesian framework and estimated simultaneously. A Bayesian test to detect deterministic components in the data is also constructed. By using an asymptotic approximation to the likelihood, the computation is carried out efficiently using the Markov chain Monte Carlo method in O ( Mn ) operations, where n is the sample size and M is the number of iterations. We show empirically that our approach works well on real and simulated samples.     

Nonparametric Nonlinear Cotrending Analysis,With an Application to Interest and Inflation in the United States

Given the assumption that the components of a vector time series are stationary around nonlinear deterministic time trends, nonlinear cotrending is the phenomenon that one or more linear combinations of the time series are stationary around a linear trend or a constant; hence, the series have common nonlinear deterministic time trends. In this article, I develop nonparametric tests for nonlinear cotrending, and I derive nonparametric estimators of the cotrending vectors. I apply this approach to the federal funds rate and the consumer price index inflation rate in the United States, using monthly data, to analyze the price puzzle.     

A bayesian analysis of trend determination in economic time series

In this paper we provide a comprehensive Bayesian posterior analysis of trend determination in general autoregressive models. Multiple lag autoregressive models with fitted drifts and time trends as well as models that allow for certain types of structural change in the deterministic components are considered. We utilize a modified information matrix-based prior that accommodates stochastic nonstationarity, takes into account the interactions between long-run and short-run dynamics and controls the degree of stochastic nonstationarity permitted. We derive analytic posterior densities for all of the trend determining parameters via the Laplace approximation to multivariate integrals. We also address the sampling properties of our posteriors under alternative data generating processes by simulation methods. We apply our Bayesian techniques to the Nelson-Plosser macroeconomic data and various stock price and dividend data. Contrary to DeJong and Whiteman (1989a,b,c), we do not find that the data overwhelmingly favor the existence of deterministic trends over stochastic trends. In addition, we find evidence supporting Perron's (1989) view that some of the Nelson and Plosser data are best construed as trend stationary with a change in the trend function occurring at 1929.     

A bayesian analysis of trend determination in economic time series

In this paper we provide a comprehensive Bayesian posterior analysis of trend determination in general autoregressive models. Multiple lag autoregressive models with fitted drifts and time trends as well as models that allow for certain types of structural change in the deterministic components are considered. We utilize a modified information matrix-based prior that accommodates stochastic nonstationarity, takes into account the interactions between long-run and short-run dynamics and controls the degree of stochastic nonstationarity permitted. We derive analytic posterior densities for all of the trend determining parameters via the Laplace approximation to multivariate integrals. We also address the sampling properties of our posteriors under alternative data generating processes by simulation methods. We apply our Bayesian techniques to the Nelson-Plosser macroeconomic data and various stock price and dividend data. Contrary to DeJong and Whiteman (1989a,b,c), we do not find that the data overwhelmingly favor the existence of deterministic trends over stochastic trends. In addition, we find evidence supporting Perron's (1989) view that some of the Nelson and Plosser data are best construed as trend stationary with a change in the trend function occurring at 1929.     

Wavelet semi-parametric inference for long memory in volatility in the presence of a trend

Risk of investing in a financial asset is quantified by functionals of squared returns. Discrete time stochastic volatility (SV) models impose a convenient and practically relevant time series dependence structure on the log-squared returns. Different long-term risk characteristics are postulated by short-memory SV and long-memory SV models. It is therefore important to test which of these two alternatives is suitable for a specific asset. Most standard tests are confounded by deterministic trends. This paper introduces a new, wavelet-based, test of the null hypothesis of short versus long memory in volatility which is robust to deterministic trends. In finite samples, the test performs better than currently available tests which are based on the Fourier transform.     

Tests for Parameter Instability in Regressions with 1(1) Processes

This article derives the large-sample distributions of Lagrange multiplier (LM) tests for parameter instability against several alternatives of interest in the context of cointegrated regression models. The fully modified estimator of Phillips and Hansen is extended to cover general models with stochastic and deterministic trends. The test statistics considered include the SupF test of Quandt, as well as the LM tests of Nyblom and of Nabeya and Tanaka. It is found that the asymptotic distributions depend on the nature of the regressor processes—that is, if the regressors are stochastic or deterministic trends. The distributions are noticeably different from the distributions when the data are weakly dependent. It is also found that the lack of cointegration is a special case of the alternative hypothesis considered (an unstable intercept), so the tests proposed here may also be viewed as a test of the null of cointegration against the alternative of no cointegration. The tests are applied to three data sets—an aggregate consumption function, a present value model of stock prices and dividends, and the term structure of interest rates.     

The role of the drift in I(2) systems

Summary This paper discussed the role of the drift in vector autoregressive processes allowing for integrated components up to order 2. It is shown how the drift can generate linear and quadratic deterministic trends. A two-stage statistical analysis of the system in the presence of quadratic trends is proposed. The analysis of the system allows to define a consistent sequence of tests on the numbers of common components integrated of a given order, called the integration indices of the system. The relevant asymptotic distributions are non-standard, belong to the Limiting Gaussian Functional family and are tabulated by simulation. The proposed procedure can also be consistently combined with other procedures proposed by the author for the cases of a linear trend and of no deterministic trends in the system. Invited paper at the Conference held in Bologna, Italy, 27–28 May 1993, on “Statistical Tests: Methodology and Econometric Applications”. Partial financial support is acknowledged from Italian MURST grants 40% and 60%.     

Asymptotic Properties of OLS Estimates in Autoregressions with Bounded or Slowly Growing Deterministic Trends

ABSTRACT

We propose a general method of modeling deterministic trends for autoregressions. The method relies on the notion of L ₂-approximable regressors previously developed by the author. Some facts from the theory of functions play an important role in the proof. In its present form, the method encompasses slowly growing regressors, such as logarithmic trends, and leaves open the case of polynomial trends.     

The role of information in nonstationary regression

The role of standard likelihood-based measures of information and efficiency is unclear when regressions involve nonstationary data. Typically the standardized score is not asymptotically Gaussian and the standardized Hessian has a stochastic, rather than deterministic limit. Here we consider a time series regression involving a deterministic covariate which can be evaporating, slowly evolving or nonstationary. It is shown that conditional information, or equivalently, profile Kullback–Leibler and Fisher information remain informative about both the accuracy, i.e. asymptotic variance, of profile maximum likelihood estimators, and the power of point optimal invariant tests for a unit root. Specifically, these information measures indicate fractional, rather than linear trends that may minimize inferential accuracy. Such is confirmed in a numerical experiment.     

Doubly periodic non-homogeneous Poisson models for hurricane data

Non-homogeneous Poisson processes with periodic claim intensity rate have been proposed as claim counts in risk theory. Here a doubly periodic Poisson model with short- and long-term trends is studied. Beta-type intensity functions are presented as illustrations. The likelihood function and the maximum likelihood estimates of the model parameters are derived.Doubly periodic Poisson models are appropriate when the seasonality does not repeat exactly the same short-term pattern every year, but has a peak intensity that varies over a longer period. This reflects periodic environments like those forming hurricanes, in alternating El Niño/La Niña years. An application of the model to the data set of Atlantic hurricanes affecting the United States (1899–2000) is discussed in detail.     

SEMIFAR forecasts, with applications to foreign exchange rates

SEMIFAR models introduced in Beran (1997. Estimating trends, long-range dependence and nonstationarity, preprint) provide a semiparametric modelling framework that enables the data analyst to separate deterministic and stochastic trends as well as short- and long-memory components in an observed time series. A correct distinction between these components, and in particular, the decision which of the components may be present in the data have an important impact on forecasts. In this paper, forecasts and forecast intervals for SEMIFAR models are obtained. The forecasts are based on an extrapolation of the nonparametric trend function and optimal forecasts of the stochastic component. In the data analytical part of the paper, the proposed method is applied to foreign exchange rates from Europe and Asia.     

Generalized fixed‐T panel unit root tests

Panel data unit root tests, which can be applied to data that do not have many time series observations, are based on very restrictive error and deterministic component specification assumptions. In this paper, we develop a new, doubly modified estimator, based on which we propose a panel unit root test that allows for multiple structural breaks, linear and nonlinear trends, heteroscedasticity, serial correlation, and error cross‐section heterogeneity, when the number of time series observations is finite. The test has the additional perk that it is invariant to the initial condition.     

Infant mortality rates: time trends and fractional integration

This paper examines the existence of time trends in the infant mortality rates in a number of countries in the twentieth century. We test for the presence of deterministic trends by adopting a linear model for the log-transformed data. Instead of assuming that the error term is a stationary I(0), or alternatively, a non-stationary I(1) process, we allow for the possibility of fractional integration and hence for a much greater degree of flexibility in the dynamic specification of the series. Indeed, once the linear trend is removed, all series appear to be I(d) with 0<d<1, implying long-range dependence. As expected, the time trend coefficients are significantly negative, although of a different magnitude from those obtained assuming integer orders of differentiation.     

Extended complex error correction models for seasonal cointegration

In this paper, we extend the complex error correction model (ECM) of [Cubadda, G. (2001). Complex reduced rank models for seasonally cointegrated time series. Oxford Bulletin of Economics and Statistics, 63, 497–511] to models with two types of deterministic terms: (i) restricted seasonal dummies and constant; (ii) restricted seasonal dummies and unrestricted constant. These types of deterministic terms are most frequently adopted in the analysis of seasonal cointegration by many practitioners and researchers, because the other type–where all seasonal dummies and constant terms are unrestricted–may yield oscillating trends. We obtain the limiting distribution of the likelihood ratio (LR) test for the seasonal cointegrating (CI) rank in the extended models. We also provide asymptotic and finite critical values for the test.     

Quasi-random resampling for the bootstrap

Quasi-random sequences are known to give efficient numerical integration rules in many Bayesian statistical problems where the posterior distribution can be transformed into periodic functions on then-dimensional hypercube. From this idea we develop a quasi-random approach to the generation of resamples used for Monte Carlo approximations to bootstrap estimates of bias, variance and distribution functions. We demonstrate a major difference between quasi-random bootstrap resamples, which are generated by deterministic algorithms and have no true randomness, and the usual pseudo-random bootstrap resamples generated by the classical bootstrap approach. Various quasi-random approaches are considered and are shown via a simulation study to result in approximants that are competitive in terms of efficiency when compared with other bootstrap Monte Carlo procedures such as balanced and antithetic resampling.     

Functional-coefficient cointegration models in the presence of deterministic trends

In this article, we extend the functional-coefficient cointegration model (FCCM) to the cases in which nonstationary regressors contain both stochastic and deterministic trends. A nondegenerate distributional theory on the local linear (LL) regression smoother of the FCCM is explored. It is demonstrated that even when integrated regressors are endogenous, the limiting distribution is the same as if they were exogenous. Finite-sample performance of the LL estimator is investigated via Monte Carlo simulations in comparison with an alternative estimation method. As an application of the FCCM, electricity demand analysis in Illinois is considered.     

价格持续期的SEMIFAR-ACD模型

本文提出了一类同时包含确定性,差分平稳以及平稳长相关趋势的描述交易价格持续期新模型---SEMIFAR-ACD模型。研究了模型的估计方法和各估计量的渐近性质,构建了相应的估计算法,并应用实际数据,将SEMIFAR-ACD模型与普通ACD模型模拟效果进行比较,论证了SEMIFAR-ACD模型更好的描述数据的性能。     

Mining periodic patterns and cascading bursts phenomenon in individual e-mail communication

ABSTRACT

Quantitative understanding of human activity is very important as many social and economic trends are driven by human actions. We propose a novel stochastic process, the Multi-state Markov Cascading Non-homogeneous Poisson Process (M2CNPP), to analyze human e-mail communication involving both periodic patterns and bursts phenomenon. The model parameters are estimated using the Generalized Expectation Maximization (GEM) algorithm while the hidden states are treated as missing values. The empirical results demonstrate that the proposed model adequately captures the major temporal cascading features as well as the periodic patterns in e-mail communication.     

Modelling trends in groundwater levels by segmented regression with constraints

This paper provides a statistically unified method for modelling trends in groundwater levels for a national project that aims to predict areas at risk from salinity in 2020. It was necessary to characterize the trends in groundwater levels in thousands of boreholes that have been monitored by Agriculture Western Australia throughout the south-west of Western Australia over the last 10 years. The approach investigated in the present paper uses segmented regression with constraints when the number of change points is unknown. For each segment defined by change points, the trend can be described by a linear trend possibly superimposed on a periodic response. Four different types of change point are defined by constraints on the model parameters to cope with different patterns of change in groundwater levels. For a set of candidate change points provided by the user, a modified Akaike information criterion is used for model selection. Model parameters can be estimated by multiple linear regression. Some typical examples are presented to demonstrate the performance of the approach.