Polynomial Cointegration Research Articles

Comment on "Polynomial cointegration tests of anthropogenic impact on global warming" by Beenstock et al. (2012) – some hazards in econometric modelling of climate change

Abstract. We outline six important hazards that can be encountered in econometric modelling of time-series data, and apply that analysis to demonstrate errors in the empirical modelling of climate data in Beenstock et al. (2012). We show that the claim made in Beenstock et al. (2012) as to the different degrees of integrability of CO2 and temperature is incorrect. In particular, the level of integration is not constant and not intrinsic to the process. Further, we illustrate that the measure of anthropogenic forcing in Beenstock et al. (2012), a constructed "anthropogenic anomaly", is not appropriate regardless of the time-series properties of the data.

Polynomial cointegration tests of anthropogenic impact on global warming

Abstract. We use statistical methods for nonstationary time series to test the anthropogenic interpretation of global warming (AGW), according to which an increase in atmospheric greenhouse gas concentrations raised global temperature in the 20th century. Specifically, the methodology of polynomial cointegration is used to test AGW since during the observation period (1880–2007) global temperature and solar irradiance are stationary in 1st differences, whereas greenhouse gas and aerosol forcings are stationary in 2nd differences. We show that although these anthropogenic forcings share a common stochastic trend, this trend is empirically independent of the stochastic trend in temperature and solar irradiance. Therefore, greenhouse gas forcing, aerosols, solar irradiance and global temperature are not polynomially cointegrated, and the perceived relationship between these variables is a spurious regression phenomenon. On the other hand, we find that greenhouse gas forcings might have had a temporary effect on global temperature.

A STATE SPACE CANONICAL FORM FOR UNIT ROOT PROCESSES

In this paper we develop a canonical state space representation of autoregressive moving average (ARMA) processes with unit roots with integer integration orders at arbitrary unit root frequencies. The developed representation utilizes a state process with a particularly simple dynamic structure, which in turn renders this representation highly suitable for unit root, cointegration, and polynomial cointegration analysis. We also propose a new definition of polynomial cointegration that overcomes limitations of existing definitions and extends the definition of multicointegration for I(2) processes of Granger and Lee (1989a, Journal of Applied Econometrics4, 145–159). A major purpose of the canonical representation for statistical analysis is the development of parameterizations of the sets of all state space systems of a given system order with specified unit root frequencies and integration orders. This is, e.g., useful for pseudo maximum likelihood estimation. In this respect an advantage of the state space representation, compared to ARMA representations, is that it easily allows one to put in place restrictions on the (co)integration properties. The results of the paper are exemplified for the cases of largest interest in applied work.

Inflation and the Markup in the Euro Area

The paper implements a consistent empirical strategy in order to investigate the behaviour of the markup over the cycle and its contribution to inflation movements.We model the price series as I(2) components and use polynomial cointegration in order to recover a long-run price schedule. We do not reject statistically the reduction of the I(2) framework to an I(1) model as from te mid 1980s. We observe that the markup is fairly counter-cyclical and has a permanent effect on inflation through an error-correcting mechanism. Structural and forecasting equations exhibiting good performance are therefore estimated.

Cointegration analysis with state space models

This paper presents and exemplifies results developed for cointegration analysis with state space models by Bauer and Wagner in a series of papers. Unit root processes, cointegration, and polynomial cointegration are defined. Based upon these definitions, the major part of the paper discusses how state space models, which are equivalent to VARMA models, can be fruitfully employed for cointegration analysis. By detailing the cases most relevant for empirical applications, the I(1), multiple frequency I(1), and I(2) cases, a canonical representation is developed and thereafter some available statistical results are briefly discussed.

Polynomial Cointegration Between Stationary Processes With Long Memory

Abstract. In this article we consider polynomial cointegrating relationships between stationary processes with long range dependence. We express the regression functions in terms of Hermite polynomials and consider a form of spectral regression around frequency zero. For these estimates, we establish consistency by means of a more general result on continuously averaged estimates of the spectral density matrix at frequency zero.

Polynomial Cointegration Between Stationary Processes With Long Memory

Polynomial Cointegration Between Stationary Processes With Long Memory

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.

Polynomial Cointegration Between Stationary Processes with Long Memory

In this paper we consider polynomial cointegrating relationships between stationary processes with long range dependence. We express the regression functions in terms of Hermite polynomials and we consider a form of spectral regression around frequency zero. For these estimates, we establish consistency by means of a more general result on continuously averaged estimates of the spectral density matrix at frequency zero.

A re‐interpretation of the linear quadratic model when inventories and sales are polynomially cointegrated

AbstractEstimation of the linear quadratic model, the workhorse of the inventory literature, traditionally takes inventories and sales to be first‐difference stationary series, and the ratio of the two variables to be stationary. However, these assumptions do not always match the properties of the data for the last two decades in the United States. We propose a model that allows for the non‐stationary characteristics of the data, using polynomial cointegration. We show that the closed‐form solution has other recent models as special cases. The resulting model performs well on aggregate and disaggregated data. Copyright © 2006 John Wiley & Sons, Ltd.

A Multivariate I(2)cointegration analysis of German hyperinflation

This paper re-examines the Cagan model of German hyperinflation during the 1920s under the twin hypotheses that the system contains variables that are I(2) and that a linear trend is required in the cointegrating relations. Using the recently developed I(2) cointegration analysis developed by Johansen (1992, 1995, 1997) extended by Paruolo (1996) and Rahbek et al. (1999) we find that the linear trend hypothesis is rejected for the sample. However, we provide conclusive evidence that money supply and the price level have a common I(2) component. Then, the validity of Cagan's model is tested via a transformation of the I(2) to an I(1) model between real money balances and money growth or inflation. This transformation is not imposed on the data but it is shown to satisfy the statistical property of polynomial cointegration. Evidence is obtained in favour of cointegration between the two sets of variables which is however weakened by the sample dependence of the trace test that the application of the recursi...

Multicointegration in Stock-Flow Models

Multicointegration, in the sense of Granger and Lee (1990), frequently occurs in models of stock-flow adjustment and implies cointegration amongst I(2) variables and their differences (polynomial cointegration). The purpose of this article is two-fold. First, we demonstrate that based on a multicointegrated vector autoregression (VAR) two equivalent error correction model (ECM) representations can be derived; the first is expressed in terms of adjustments in the flows of the variables (the standard I(2) ECM), and the second is expressed in terms of adjustments in both the stocks and the flows. Secondly, we apply I(2) estimation and testing procedures for multicointegrated time series to analyze data for US housing construction. We find that stocks of housing units started and completed exhibit polynomial cointegration (and hence the flows are multicointegrated) and the associated ECM's are estimated. Lee (1992, 1996) also found multicointegration in this data set but without explicitly exploiting the I(2) property.

MIXED NORMALITY AND ANCILLARITY IN I(2) SYSTEMS

This paper studies asymptotic likelihood inference on cointegration parameters in systems integrated of order two. We start with so-called triangular systems and then extend the analysis to vector autoregressions. We show that even when all unit root restrictions have been imposed, the asymptotic observed information is not (locally) ancillary, which implies that the log-likelihood ratio is not locally asymptotically mixed normal. The results are applied to inference on polynomial cointegration. Some similarities and differences with I(1) systems are also discussed.

Multicointegration in Stock‐Flow Models

Multicointegration, in the sense of Granger and Lee (1990), frequently occurs in models of stock‐flow adjustment and implies cointegration amongst I(2) variables and their differences (polynomial cointegration). The purpose of this article is two‐fold. First, we demonstrate that based on a multicointegrated vector autoregression (VAR) two equivalent error correction model (ECM) representations can be derived; the first is expressed in terms of adjustments in the flows of the variables (the standard I(2) ECM), and the second is expressed in terms of adjustments in both the stocks and the flows. Secondly, we apply I(2) estimation and testing procedures for multicointegrated time series to analyze data for US housing construction. We find that stocks of housing units started and completed exhibit poly‐ nomial cointegration (and hence the flows are multicointegrated) and the associated ECM’s are estimated. Lee (1992, 1996) also found multicointegration in this data set but without explicitly exploiting the I(2) property.

An Econometric Analysis of I(2) Variables

This paper provides a selective survey of the recent literature dealing with I(2) variables in economic time series, that is, processes that require to be differenced twice in order to become stationary. With reference to particular economic models intuition is provided of why I(2)‐and polynomial cointegration are features likely to occur in economics. The properties of I(2) series are discussed and I review topics such as: Testing for double unit roots, representations of I(2) cointegrated systems, and hypothesis testing in single equations as well as in systems of equations. Different data sets are used to illustrate the various econometric and statistical techniques.

A PARAMETRIC CHARACTERIZATION OF INTEGRATED VECTOR AUTOREGRESSIVE (VAR) PROCESSES

This paper provides a polynomial factorization theorem that is then used to extend the characterization parts of the parametric representation theorems of Johansen (1992, Econometric Theory 8, 188–202) for vector autoregressive processes integrated of up to order 2. A characterization theorem is provided in the general case of an I(d) process. For the discussion of the complicated polynomial cointegration properties of such processes, the case of an I(3) process is considered as an example.

Polynomial cointegration estimation and test

This paper develops statistical tools to analyze the multivariate time series which can be represented with a polynomial error correction model as introduced by Gregoir and Laroque (1993). We propose an identification criterion for the error correction terms, which fits with the estimation procedure. The estimation proceeds in a number of steps, through repeated applications of principal component analysis: test for the overall cointegration dimension and its decomposition into the dimensions of the error correction terms of various degrees, estimation of the error correction terms themselves, and finally estimation of the full model, given the previous results, by ordinary least squares and overall specification test. The asymptotic distribution of the test statistics at each of these various steps is nonstandard, and we provide statistical tables of its main percentiles, for the two cases where there is (or not) a constant term on the right-hand side of the model and the degree of the polynomial error correction terms is at most two.

Specification of dynamic wage-price relations in Poland (1980?1991)

This paper focuses on the analysis of wage-price relationships during the period 1980–1991 in Poland. The dynamics of wage and price series under radical structural changes are examined, as well as the impact of earlier sharp price increases. The process of wages-to-prices adjustment both in the long- and short-term is analysed. The hypothesis that the introduction of economic reform (at the beginning of 1990) would result in a tendency towards the stabilization of real wages is investigated. This is done through integration and cointegration analysis of wage and price series with special attention being paid to the problems arising from theI(2) character of the variables investigated. The concept of polynomial cointegration is applied to formulate error correction terms for the short-run model of wages. The computations have been made using quarterly data. The results reflect an inhomogeneity of the period investigated, especially the effects of the introduction of economic reform at the beginning of 1990. The nonstationarity of real wages is confirmed, but not their tendency towards stabilization.