Unit Root Model Research Articles

A note on mean squared prediction error under the unit root model with deterministic trend

Assume that observations are generated from the first‐order autoregressive (AR) model with linear time trend and the unknown model coefficients are estimated by least squares. This article develops an asymptotic expression for the mean squared prediction error (MSPE) of the least squares predictor in the presence of a unit root. As a by‐product, we also obtain a connection between the MSPE and the growth rate of the Fisher information. The key technical tool used to derive these results is the negative moment bound for the minimum eigenvalue of the normalized Fisher information matrix.

NULL RECURRENT UNIT ROOT PROCESSES

The classical nonstationary autoregressive models are both linear and Markov. They include unit root and cointegration models. A possible nonlinear extension is to relax the linearity and at the same time keep general properties such as nonstationarity and the Markov property. A null recurrent Markov chain is nonstationary, andβ-null recurrence is of vital importance for statistical inference in nonstationary Markov models, such as, e.g., in nonparametric estimation in nonlinear cointegration within the Markov models. The standard random walk is an example of a null recurrent Markov chain.In this paper we suggest that the concept of null recurrence is an appropriate nonlinear generalization of the linear unit root concept and as such it may be a starting point for a nonlinear cointegration concept within the Markov framework. In fact, we establish the link between null recurrent processes and autoregressive unit root models. It turns out that null recurrence is closely related to the location of the roots of the characteristic polynomial of the state space matrix and the associated eigenvectors. Roughly speaking the process isβ-null recurrent if one root is on the unit circle, null recurrent if two distinct roots are on the unit circle, whereas the others are inside the unit circle. It is transient if there are more than two roots on the unit circle. These results are closely connected to the random walk being null recurrent in one and two dimensions but transient in three dimensions. We also give an example of a process that by appropriate adjustments can be madeβ-null recurrent for anyβ∈ (0, 1) and can also be made null recurrent without beingβ-null recurrent.

On non-stationary threshold autoregressive models

In this paper we study the limiting distributions of the least-squares estimators for the non-stationary first-order threshold autoregressive (TAR(1)) model. It is proved that the limiting behaviors of the TAR(1) process are very different from those of the classical unit root model and the explosive AR(1).

Limit theory for a general class of GARCH models with just barely infinite variance

In this article, limit theory is established for a general class of generalized autoregressive conditional heteroskedasticity models given by ɛt = σtηt and σt = f (σt−1, σt−2,…, σt−p, ɛt−1, ɛt−2,…, ɛt−q), when {ɛt} is a process with just barely infinite variance, that is, {ɛt} is a process with infinite variance but in the domain of normal attraction. In particular, we show that under some regular conditions, converges weakly to a Gaussian process. Applications of the asymptotic results to statistical inference, such as unit root test and sample autocorrelation, are also investigated. The obtained result fills in a gap between the classical infinite variance and finite variance in the literature. Further, when applying our limiting result to Dickey–Fuller (DF) test in a unit root model with integrated generalized autoregressive conditional heteroskedasticity (IGARCH) errors, it just confirms the simulation result of Kourogenis and Pittis (2008) that the DF statistics with IGARCH errors converges in law to the standard DF distribution.

A nonlinear approach to testing the unit root null hypothesis: an application to international health expenditures

In this article, we examine the unit root null hypothesis for per capita total Health Expenditures (HEs), per capita private HEs and per capita public HEs for 29 Organization for Economic Co-operation and Development (OECD) countries. The novelty of our work is that we use a new nonlinear unit root test that allows for one structural break in the data series. We find that for around 45% of the countries, we are able to reject the unit root hypothesis for each of the three HE series. Moreover, using Monte Carlo simulations, we show that our proposed unit root model has better size and power properties than the widely used Augmented Dickey–Fuller (ADF) and Lagrange Multiplier (LM) type tests.

A Bayesian Analysis of Unit Roots and Structural Breaks in the Level, Trend, and Error Variance of Autoregressive Models of Economic Series

In this article, a Bayesian approach is suggested to compare unit root models with stationary autoregressive models when the level, the trend, and the error variance are subject to structural changes (known as breaks) of an unknown date. Ignoring structural breaks in the error variance may be responsible for not rejecting the unit root hypothesis, even if allowance is made in the inferential procedures for breaks in the mean. The article utilizes analytic and Monte Carlo integration techniques for calculating the marginal likelihoods of the models under consideration, in order to compute the posterior model probabilities. The performance of the method is assessed by simulation experiments. Some empirical applications of the method are conducted with the aim to investigate if it can detect structural breaks in financial series, especially with changes in the error variance.

Quantifying U.S. aluminum in-use stocks and their relationship with economic output

A dynamic material flow analysis model is developed to quantify aluminum in-use stocks and old scrap recycling and recovery in the United States for the period of 1900 to 2007. The total in-use aluminum stock in 2007 is estimated as 93 million metric tons, which represents approximately 34% of the cumulative apparent consumption since 1900. Alternately, since 1900 nearly 40% of the cumulative discarded aluminum has not been recycled for domestic use in the U.S. or for export to foreign consumers. Statistical time series analysis is used to explore the relationship between model results of in-use stocks and gross domestic product (GDP). Unlike most previous studies of material consumption and economic activity, which ignore the statistical properties of time series data to the detriment of model estimation and inference, data stationarity is explicitly evaluated through unit root testing and model specification is adjusted accordingly. The annual percentage change in GDP is found to have a large and significant association with the annual percentage change in net additions to in-use stocks. Model sensitivity and uncertainty are quantified through the application of the Fourier Amplitude Sensitivity Test and alternate specifications of product lifetime probability density functions.

Estimation for nearly unit root processes with GARCH errors

In this paper the limiting distribution of the least square estimate for the autoregressive coefficient of a nearly unit root model with GARCH errors is derived. Since the limiting distribution depends on the unknown variance of the errors, an empirical likelihood ratio statistic is proposed from which confidence intervals can be constructed for the nearly unit root model without knowing the variance. To gain an intuitive sense for the empirical likelihood ratio, a small simulation for the asymptotic distribution is given.

M-ESTIMATION FOR A SPATIAL UNILATERAL AUTOREGRESSIVE MODEL WITH INFINITE VARIANCE INNOVATIONS

We study the limiting behavior of theM-estimators of parameters for a spatial unilateral autoregressive model with independent and identically distributed innovations in the domain of attraction of a stable law with indexα∈ (0, 2]. Both stationary and unit root models and some extensions are considered. It is also shown that self-normalizedM-estimators are asymptotically normal. A numerical example and a simulation study are also given.

Bayesian Estimation and Model Selection in the Generalized Stochastic Unit Root Model

We develop Bayesian techniques for estimation and model comparison in a novel Generalized Stochastic Unit Root (GSTUR) model. This allows us to investigate the presence of a deterministic time trend in economic series, while allowing the degree of persistence to change over time. In particular the model allows for shifts from stationarity I(0) to nonstationarity I(1) or vice versa. The empirical analysis demonstrates that the GSTUR model provides new insights on the properties of some macroeconomic time series such as stock market indices, inflation and exchange rates.

Estimating short and long-term residential demand for electricity: New evidence from Sri Lanka

This study investigates the short-run dynamics and long-run equilibrium relationship between residential electricity demand and factors influencing demand – per capita income, price of electricity, price of kerosene oil and price of liquefied petroleum gas – using annual data for Sri Lanka for the period, 1960–2007. The study uses unit root, cointegration and error-correction models. The long-run demand elasticities of income, own price and price of kerosene oil (substitute) were estimated to be 0.78, − 0.62, and 0.14 respectively. The short-run elasticities for the same variables were estimated to be 0.32, − 0.16 and 0.10 respectively. Liquefied petroleum (LP) gas is a substitute for electricity only in the short-run with an elasticity of 0.09. The main findings of the paper support the following (1) increasing the price of electricity is not the most effective tool to reduce electricity consumption (2) existing subsidies on electricity consumption can be removed without reducing government revenue (3) the long-run income elasticity of demand shows that any future increase in household incomes is likely to significantly increase the demand for electricity and (4) any power generation plans which consider only current per capita consumption and population growth should be revised taking into account the potential future income increases in order to avoid power shortages in the country.

A Consistent Model of 'Explosive' Financial Bubbles with Mean-Reversing Residuals

We present a self-consistent model for explosive financial bubbles, which combines a mean-reverting volatility process and a stochastic conditional return which reflects nonlinear positive feedbacks and continuous updates of the investors' beliefs and sentiments. The conditional expected returns exhibit faster-than-exponential acceleration decorated by accelerating oscillations, called log-periodic power law. Tests on residuals show a remarkable low rate (0.2%) of false positives when applied to a GARCH benchmark. When tested on the S&P500 US index from Jan. 3, 1950 to Nov. 21, 2008, the model correctly identifies the bubbles ending in Oct. 1987, in Oct. 1997, in Aug. 1998 and the ITC bubble ending on the first quarter of 2000. Different unit-root testsconfirm the high relevance of the model specification. Our model also provides a diagnostic for the duration of bubbles: applied to the period before Oct. 1987 crash, there is clear evidence that the bubble started at least 4 years earlier. We confirm the validity and universality of the volatility-confined LPPL model on seven other major bubbles that have occurred in the World in the last two decades. Using Bayesian inference, we find a very strong statistical preference for our model compared with a standard benchmark, in contradiction with [Feigenbaum2006] which used a unit-root model for residuals.

A Bayesian Analysis of Unit Roots in Panel Data Models with Cross-Sectional Dependence

In this paper a Bayesian approach to unit root testing for panel data models is proposed based on the comparison of stationary autoregressive models with and without individual determinist trends, with their counterpart models with unit autoregressive roots. This is done under cross-sectional dependence among the units of the panel. Simulation experiments are conducted with the aim to assess the performance of the suggested inferential procedure, as well as to investigate if the Bayesian model comparison approach can distinguish unit root models from stationary autoregressive models under or without cross-section dependence. The approach is applied to real GDP data for a panel of G7.

A Consistent Model of 'Explosive' Financial Bubbles with Mean-Reversing Residuals

We present a self-consistent model for explosive financial bubbles, which combines a mean-reverting volatility process and a stochastic conditional return which reflects nonlinear positive feedbacks and continuous updates of the investors' beliefs and sentiments. The conditional expected returns exhibit faster-than-exponential acceleration decorated by accelerating oscillations, called "log-periodic power law." Tests on residuals show a remarkable low rate (0.2%) of false positives when applied to a GARCH benchmark. When tested on the S&P500 US index from Jan. 3, 1950 to Nov. 21, 2008, the model correctly identifies the bubbles ending in Oct. 1987, in Oct. 1997, in Aug. 1998 and the ITC bubble ending on the first quarter of 2000. Different unit-root tests confirm the high relevance of the model specification. Our model also provides a diagnostic for the duration of bubbles: applied to the period before Oct. 1987 crash, there is clear evidence that the bubble started at least 4 years earlier. We confirm the validity and universality of the volatility-confined LPPL model on seven other major bubbles that have occurred in the World in the last two decades. Using Bayesian inference, we find a very strong statistical preference for our model compared with a standard benchmark, in contradiction with Chang and Feigenbaum (2006) which used a unit-root model for residuals.

Maximum likelihood estimation and inference methods for the covariance stationary panel AR(1)/unit root model

This paper considers Maximum Likelihood (ML) based estimation and inference procedures for linear dynamic panel data models with fixed effects.

REGIME-SWITCHING AUTOREGRESSIVE COEFFICIENTS AND THE ASYMPTOTICS FOR UNIT ROOT TESTS

Most of the asymptotic results for Markov regime-switching models with possible unit roots are based on specifications implying that the number of regime switches grows to infinity as the sample size increases. Conversely, in this note we derive some new asymptotic results for the case of Markov regime switches that are infrequent in the sense that their number is bounded in probability, even asymptotically. This is achieved by (inversely) relating the probability of regime switching to the sample size. The proposed asymptotic theory is applied to a well-known stochastic unit root model, where the dynamics of the observed variable switches between a unit root regime and a stationary regime.

Nonlinear regression for unit root models with autoregressive errors

This paper shows that the nonlinear least squares estimator for unit root models has the limiting distribution free of nuisance parameters and is more efficient than the augmented Dickey–Fuller estimator when the sum of coefficients for lagged variables is negative.

Unit Root Model Selection

Some limit properties for information based model selection criteria are given in the context of unit root evaluation and various assumptions about initial conditions. Allowing for a nonparametric short memory component, standard information criteria are shown to be weakly consistent for a unit root provided the penalty coefficient Cn →∞ and Cn/n → 0a sn →∞ . Strong consistency holds when Cn/(log log n) 3 → ∞ under conventional assumptions on initial conditions and under a slightly stronger condition when initial conditions are infinitely distant in the unit root model. The limit distribution of the AIC criterion is obtained.

Comparison of Some Tests on Non-Stationary Gaussian AR (1) (A Simulation Study)

A condense study was done on non-stationary Gaussian AR(1), concerning testing the stationarity, the unit root and deterministic trend model . A proposed test was given and a comparison was done using where some new results are obtained.

The stochastic unit root model and fractional integration: An extension to the seasonal case

AbstractIn a recent paper, Yoon (Working Paper, Department of Economics and Related Studies, University of York, 2003. Presented at the ESF‐EMM Second Annual Meeting, Rome, Italy, 2003) asserts that the stochastic unit root (STUR) model is closely related to long memory processes, and, in particular, that it is a special case of an I(d) process, with d = 1.5. In this paper we question this claim by using parametric and semiparametric techniques for modelling long memory. We extend the analysis by considering both non‐normality and seasonality, and shed light, theoretically and by means of Monte Carlo methods, on the relationship between the seasonal STUR and the seasonal I(d) models. The results show that methods that are specifically designed for testing I(d) statistical models are not appropriate for testing the STUR model. Moreover, they have in some cases very low power against STUR alternatives. Copyright © 2007 John Wiley & Sons, Ltd.