The effect of non-normal disturbances and conditional heteroskedasticity on multiple cointegration tests

!t is well-known that Johansen's multiple cointegration tests' results and those of Johansen and Juselius' tests for restricrions on cointegrating vectors and their weights have far-reaching implications for economic modelling and analysis. Therefore, it is important to ensure that the tests have desirable finite sample properties. Although the statistics are derived under Gaussian distribution,the asympotic results are derived under a much wider class of distributions. Using simulation, this paper investigates the effect of non-normal disturbances on these tests in finite samples. Further, ARCH/GARCH type conditional heteroskedasticity is present in many economic and financial time series. This paper examines the finite properties of the tests when the error term follows ARCH/GARCH type processes. From the evidence, it appears that researchers should not be overly concerned by the possibility of small departures from non-normality when using Johansen's suggested techniques even in finite samples. ARCH and GARCH effects may be more problematic, however. In particular it becomes more important ro test whether the restriction implicit in the integrated (or near-integrated) ARCH-type Drocess actually holds in time series for the application of the cointegraiion rank tests and the test for restrictions on cointegrating weights. The tests for restrictions on cointegrating vectors apper to be robust for non-normal errors and for all ARCH and GARCH type processes considered.     

协整平滑转移回归中的线性检验 ——基于完全修正最小二乘法的扩展

 在解释变量内生条件下,Choi,Saikkonen(2004)使用动态最小二乘法估计协整平滑转移回归模型,并基于动态最小二乘的估计结果构造 统计量检验协整向量的非线性。本文系统解析了 的构造并指出其不足,针对这一不足,本文将动态最小二乘法扩展为完全修正的最小二乘法,并进而基于完全修正的最小二乘法估计结果构造 统计量检验协整向量的非线性。本文的仿真试验表明,在有限样本下, 与 的检验势没有显著差异,但 的水平扭曲小于 。     

DYNAMIC FACTOR MODELS

This paper introduces nonlinear dynamic factor models for various applications related to risk analysis. Traditional factor models represent the dynamics of processes driven by movements of latent variables, called the factors. Our approach extends this setup by introducing factors defined as random dynamic parameters and stochastic autocorrelated simulators. This class of factor models can represent processes with time varying conditional mean, variance, skewness and excess kurtosis. Applications discussed in the paper include dynamic risk analysis, such as risk in price variations (models with stochastic mean and volatility), extreme risks (models with stochastic tails), risk on asset liquidity (stochastic volatility duration models), and moral hazard in insurance analysis.

We propose estimation procedures for models with the marginal density of the series and factor dynamics parameterized by distinct subsets of parameters. Such a partitioning of the parameter vector found in many applications allows to simplify considerably statistical inference. We develop a two- stage Maximum Likelihood method, called the Finite Memory Maximum Likelihood, which is easy to implement in the presence of multiple factors. We also discuss simulation based estimation, testing, prediction and filtering.     

Curved Chaotic Map Time Series Models and Their Stochastic Reversals

This paper considers two types of chaotic map time series models, including the well-known tent, logistic and binary-shift maps as special cases; these are called curved tent and curved binary families. Deterministic behaviour is investigated by invariant distributions, Lyapunov exponents, and by serial dependency. Stochastic time reversal of the families is shown to produce models which have a broader range of stochastic and chaotic properties than their deterministic counterparts. The marginal distributions may have concentrations and restricted supports and are shown to be a non-standard class of invariant distribution. Dependenc y is generally weaker with the reversed stochastic models. The work gives a broad statistical account of deterministic and stochastically reversed map models, such as are emerging in random number generation, communica tion systems and cryptography     

Bayesian Inference in Cointegrated I (2) Systems: A Generalization of the Triangular Model

This paper generalizes the cointegrating model of Phillips (1991) to allow for I (0), I (1) and I (2) processes. The model has a simple form that permits a wider range of I (2) processes than are usually considered, including a more flexible form of polynomial cointegration. Further, the specification relaxes restrictions identified by Phillips (1991) on the I (1) and I (2) cointegrating vectors and restrictions on how the stochastic trends enter the system. To date there has been little work on Bayesian I (2) analysis and so this paper attempts to address this gap in the literature. A method of Bayesian inference in potentially I (2) processes is presented with application to Australian money demand using a Jeffreys prior and a shrinkage prior.     

A Comparative Study of Observation- and Parameter-driven Zero-inflated Poisson Models for Longitudinal Count Data

Longitudinal count data with excessive zeros frequently occur in social, biological, medical, and health research. To model such data, zero-inflated Poisson (ZIP) models are commonly used, after separating zero and positive responses. As longitudinal count responses are likely to be serially correlated, such separation may destroy the underlying serial correlation structure. To overcome this problem recently observation- and parameter-driven modelling approaches have been proposed. In the observation-driven model, the response at a specific time point is modelled through the responses at previous time points after incorporating serial correlation. One limitation of the observation-driven model is that it fails to accommodate the presence of any possible over-dispersion, which frequently occurs in the count responses. This limitation is overcome in a parameter-driven model, where the serial correlation is captured through the latent process using random effects. We compare the results obtained by the two models. A quasi-likelihood approach has been developed to estimate the model parameters. The methodology is illustrated with analysis of two real life datasets. To examine model performance the models are also compared through a simulation study.     

Gatekeeping procedures with clinical trial applications

The objective of this paper is to give an overview of a relatively new area of multiplicity research that deals with the analysis of hierarchically ordered multiple objectives. Testing procedures for this problem are known as gatekeeping procedures and have found a variety of applications in clinical trials. This paper reviews main classes of these procedures, including serial and parallel gatekeeping procedures, and tree gatekeeping procedures that account for logical restrictions among multiple objectives. We focus on procedures based on marginal p-values; extensions to procedures that exploit the joint distribution of the p-values are also noted. Clinical trial examples are used to illustrate the procedures and their important properties.     

Structural Instability and the Production-Smoothing Model of Inventories

The production-smoothing model of inventories implies that inventories, labor inputs, sales, and factor input prices are cointegrated if sales and factor prices are I(1) with one cointegrating vector for each state variable held. These propositions are tested in six nondurable-goods industries. All industries provide evidence of cointegration. Fewer quasi-fixed factors are found than previous research often assumed. Estimates of cointegrating vectors provide implausible parameter estimates. Rank stability tests, with fixed or seqentially chosen breakpoints, indicate that the cointegrating matrix has unstable rank. Parameter estimates of cointegrating vectors do not provide much support for the production-smoothing model of inventories.     

The selection of zero-non-zero patterned cointegrating vectors in error-correction modelling

An effective and efficient search algorithm has been developed to select from an 1(1) system zero-non-zero patterned cointegrating and loading vectors in a subset VECM, B^q(l)y(t-1) + B^q-1 (L)Ay(t) = ε( t ) , where the long term impact matrix Bq(l) contains zero entries. The algorithm can be applied to higher order integrated systems. The Finnish money-output model presented by Johansen and Juselius (1990) and the United States balanced growth model presented by King, Plosser, Stock and Watson (1991) are used to demonstrate the usefulness of this algorithm in examining the cointegrating relationships in vector time series.     

Tests for serial correlation in mean and variance of a sequence of time series objects

In this era of Big Data, large-scale data storage provides the motivation for statisticians to analyse new types of data. The proposed work concerns testing serial correlation in a sequence of sets of time series, here referred to as time series objects. An example is serial correlation of monthly stock returns when daily stock returns are observed. One could consider a representative or summarized value of each object to measure the serial correlation, but this approach would ignore information about the variation in the observed data. We develop Kolmogorov–Smirnov-type tests with the standard bootstrap and wild bootstrap Ljung–Box test statistics for serial correlation in mean and variance of time series objects, which take the variation within a time series object into account. We study the asymptotic property of the proposed tests and present their finite sample performance using simulated and real examples.     

Analysis of cointegrated models with measurement errors

We study the asymptotic properties of the reduced-rank estimator of error correction models of vector processes observed with measurement errors. Although it is well known that there is no asymptotic measurement error bias when predictor variables are integrated processes in regression models [Phillips BCB, Durlauf SN. Multiple time series regression with integrated processes. Rev Econom Stud. 1986;53:473–495], we systematically investigate the effects of the measurement errors (in the dependent variables as well as in the predictor variables) on the estimation of not only cointegrating vectors but also the speed of the adjustment matrix. Furthermore, we present the asymptotic properties of the estimators. We also obtain the asymptotic distribution of the likelihood ratio test for the cointegrating ranks. We investigate the effects of the measurement errors on estimation and test through a Monte Carlo simulation study.     

The selection of zero-non-zero patterned cointegrating vectors in error-correction modelling

An effective and efficient search algorithm has been developed to select from an 1(1) system zero-non-zero patterned cointegrating and loading vectors in a subset VECM, B ^q (l)y(t-1) + B ^q-1 (L)Ay(t) = ?( t ) , where the long term impact matrix Bq(l) contains zero entries. The algorithm can be applied to higher order integrated systems. The Finnish money-output model presented by Johansen and Juselius (1990) and the United States balanced growth model presented by King, Plosser, Stock and Watson (1991) are used to demonstrate the usefulness of this algorithm in examining the cointegrating relationships in vector time series.     

Rank correlation plots for use with correlated input variables

A method for inducing a desired rank correlation matrix on multivariate input vectors for simulation studies has recently been developed by Iman and Conover (1982). The primary intention of this procedure is to produce correlated input variables for use with computer models. Since this procedure is distribution free and allows the exact marginal distributions to remain intact it can be used with any marginal distributions for which it is reasonable to think in terms of correlation. In this paper we present a series of rank correlation plots based on this procedure when the marginal distributions are normal, lognormal, uniform and loguniform. These plots provide a convenient tool both for aiding the modeler in determining the degree of dependence among input variables (rather than guessing) and for communicating with the modeler the effect of different correlation assumptions. In addition this procedure can be used with sample multivariate data by sampling directly from the respective marginal empirical distribution functions.     

Inflation and the Divergence of Relative Prices: Evidence from a Cointegration Analysis

The relation between inflation and RPV plays a prominent role in explaining the costs of inflation. This study investigates whether the CPI subcategories drift apart more over a period of high inflation rates than during one of low inflation. The wider dispersion of the subcategories is reflected in an increasing number of common stochastic trends in the system of sub price indices. The results for US data as well as for cross-country comparisons indicate that the influence of inflation on the dispersion of relative prices cannot be revealed by counting cointegrating relations. Thus, the number of stochastic trends or cointegrating relations is not a reliable indicator for the distorting effect of inflation on the dispersion of relative prices.     

Stochastic Volatility Models Based on OU-Gamma Time Change: Theory and Estimation

We consider stochastic volatility models that are defined by an Ornstein–Uhlenbeck (OU)-Gamma time change. These models are most suitable for modeling financial time series and follow the general framework of the popular non-Gaussian OU models of Barndorff-Nielsen and Shephard. One current problem of these otherwise attractive nontrivial models is, in general, the unavailability of a tractable likelihood-based statistical analysis for the returns of financial assets, which requires the ability to sample from a nontrivial joint distribution. We show that an OU process driven by an infinite activity Gamma process, which is an OU-Gamma process, exhibits unique features, which allows one to explicitly describe and exactly sample from relevant joint distributions. This is a consequence of the OU structure and the calculus of Gamma and Dirichlet processes. We develop a particle marginal Metropolis–Hastings algorithm for this type of continuous-time stochastic volatility models and check its performance using simulated data. For illustration we finally fit the model to S&P500 index data.     

Testing for spurious and cointegrated regressions: A wavelet approach

This paper proposes a wavelet-based approach to analyze spurious and cointegrated regressions in time series. The approach is based on the properties of the wavelet covariance and correlation in Monte Carlo studies of spurious and cointegrated regression. In the case of the spurious regression, the null hypotheses of zero wavelet covariance and correlation for these series across the scales fail to be rejected. Conversely, these null hypotheses across the scales are rejected for the cointegrated bivariate time series. These nonresidual-based tests are then applied to analyze if any relationship exists between the extraterrestrial phenomenon of sunspots and the earthly economic time series of oil prices. Conventional residual-based tests appear sensitive to the specification in both the cointegrating regression and the lag order in the augmented Dickey–Fuller tests on the residuals. In contrast, the wavelet tests, with their bootstrap t-statistics and confidence intervals, detect the spuriousness of this relationship.     

基于多元经验模式分解的股票收益与宏观经济关系分析

提出一种基于多元经验模式分解的股票市场收益与宏观经济活动关系的分析方法。通过月度道琼斯指数和美国工业生产指数的联合多元经验模式分解,得到多元金融时间序列的多尺度分量。采用希尔伯特—黄变换和边际谱确定每个尺度的主周期,进而在不同尺度下对多元时间序列进行相关性分析及Granger因果检验。结果表明:股票指数在中、长周期的某些尺度上是工业生产指数的Granger原因,序列之间具有明显的相关性,股票指数领先工业生产指数16个月到32个月不等。     

Standard errors for the Laspeyres index number with autocorrelated error models

This paper deals with the stochastic approach to Laspeyres price index number with the assumption of serial correlation of orders 1 and 2. The first round of estimation provides the estimates of Laspeyres index numbers in the presence of serial correlation assuming that variance is independent of time. In the second round of estimation, we use the weighted least square approach to derive the standard errors of Laspeyres index number assuming variance is dependent on time. These standard errors are linked to the variability of relative prices and are simple to evaluate. It shows that the larger index numbers are expected to estimate with less degree of precision. The results are illustrated with price data of Pakistan.     

A Reexamination of Friedman's Consumption Puzzle

The different average and marginal consumption propensities estimated from time series data constitute a classic puzzle of the theory of consumption. This article argues that if consumption and income possess a common stochastic trend (and thus are cointegrated), both the average propensity to consume (APC) and the marginal propensity to consume (MPC) will be consistent but biased in small samples. Upon correcting for this small sample bias, the puzzling discrepancies between the APC and the MPC estimated using annual data for the United States from 1897 to 1949 become substantially smaller. This supports an alternative resolution of the puzzle based on the theory of cointegration.     

BAYESIAN ANALYSIS OF VECTOR ARFIMA PROCESSES

A general framework is presented for Bayesian inference of multivariate time series exhibiting long-range dependence. The series are modelled using a vector autoregressive fractionally integrated moving-average (VARFIMA) process, which can capture both short-term correlation structure and long-range dependence characteristics of the individual series, as well as interdependence and feedback relationships between the series. To facilitate a sampling-based Bayesian approach, the exact joint posterior density is derived for the parameters, in a form that is computationally simpler than direct evaluation of the likelihood, and a modified Gibbs sampling algorithm is used to generate samples from the complete conditional distribution associated with each parameter. The paper also shows how an approximate form of the joint posterior density may be used for long time series. The procedure is illustrated using sea surface temperatures measured at three locations along the central California coast. These series are believed to be interdependent due to similarities in local atmospheric conditions at the different locations, and previous studies have found that they exhibit ‘long memory’ when studied individually. The approach adopted here permits investigation of the effects on model estimation of the interdependence and feedback relationships between the series.