Niels Haldrup

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.

The asymptotics of single-equation cointegration regressions with I(1) and I(2) variables

Abstract This paper addresses single-equation regression models containing both I(1) and I(2) variables, possibly with maintained deterministic components. We analyze conditions under which standard Gaussian inference can be validly conducted and the existing literature on spurious regressions for the I(1) case is extended to models with I(2) series. The analysis helps in describing how the residual-based Dickey–Fuller class of tests for noncointegration is affected when both I(1) and I(2) variables may enter the system. New critical values for this case are provided. The paper is completed by an empirical application of money demand in the UK.

Estimation of fractional integration in the presence of data noise

A comparative study is presented regarding the performance of commonly used estimators of the fractional order of integration when data is contaminated by noise. In particular, measurement errors, additive outliers, temporary change outliers, and structural change outliers are addressed. It occurs that when the sample size is not too large, as is frequently the case for macroeconomic data, then non-persistent noise will generally bias the estimators of the memory parameter downwards. On the other hand, relatively more persistent noise like temporary change outliers and structural changes can have the opposite effect and thus bias the fractional parameter upwards. Surprisingly, with respect to the relative performance of the various estimators, the parametric conditional maximum likelihood estimator with modelling of the short run dynamics clearly outperforms the semiparametric estimators in the presence of noise that is not too persistent. However, when a non-zero mean is allowed for, it may reverse the conclusion.

Improving Size and Power in Unit Root Testing

A frequent criticism of unit root tests concerns the poor power and size properties that many of such tests exhibit. However, the past decade or so intensive research has been conducted to alleviate these problems and great advances have been made. The present paper provides a selective survey of recent contributions to improve upon both size and power of unit root tests and in so doing the approach of using rigorous statistical optimality criteria in the development of such tests is stressed. In addition to presenting tests where improved size can be achieved by modifying the standard Dickey-Fuller class of tests, the paper presents theory of optimal testing and the construction of power envelopes for unit root tests under different conditions allowing for serial correlation, deterministic components, assumptions regarding the initial condition, non-Gaussian errors, and the use of covariates.

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 ECMs are estimated. Lee (1992, 1996) also found multicointegration in this data set but without explicitly exploiting the I(2) property.

Separation in cointegrated systems and persistent-transitory decompositions

The notion of separation in cointegrated systems helps identifying possible sub-system structures that may reduce the complexity of larger systems by yielding a more parsimonious representation of the time series. In this paper the authors demonstrate that although the subsystem cointegration analysis in such systems can be conducted in case of both completely and partially separated systems, the dual approach, i.e. calculation of the common stochastic trends, may turn out to yield properties of the trends that differ depending upon the type of separation under consideration. In particular, they demonstrate how persistent-transitory (P-T) decompositions and long- and short-memory factorizations of a multivariate time series will interact across systems when considering the presence (or absence) of different types of separation. Generalizations to non-linear error correction models are briefly discussed. Copyright 1997 by Blackwell Publishing Ltd

Directional Congestion and regime switching in a long memory model for electricity prices

The functioning of electricity markets has experienced increasing complexity as a result of deregulation in recent years. Consequently this affects the multilateral price behaviour across regions with physical exchange of power. It has been documented elsewhere that features such as long memory and regime switching reflecting congestion and non-congestion periods are empirically relevant and hence are features that need to be taken into account when modeling price behavior. In the present paper we further elaborate on the co-existence of long memory and regime switches by focusing on the effect that the direction of possible congestion episodes has on the price dynamics. Under non-congestion prices are identical. The direction of possible congestion is identified by the region with excess demand of power through the sign of price differences and hence three different states can be considered: Non-congestion and congestion periods with excess demand in the one or the other region. Using data from the Nordic power exchange, Nord Pool, we find that the price dynamics and long memory features of the price series generally are rather different across the different states. Also, there is evidence of fractional cointegration at some grid points when conditioning on the states.

Representations of I(2) Cointegrated Systems Using the Smith-Mcmillan Form

Abstract This paper presents a discussion of cointegration amongst I (2) variables and provides a synthesis of various ways I (2) cointegrated systems may be characterized and represented. Following Yoo (1986), Engle and Yoo (1991) and Salmon (1988) we use the Smith-McMillan form of a rational polynomial matrix as a unifying framework to describe the null-space structure of I (2)-cointegrated systems and show how different representations such as the autoregressive and error correction representations, the common stochastic trends representation and various triangular array decompositions, can be derived. Hence we extend the I (1) results of Hylleberg and Mizon (1989) to I (2) systems. The different representations provide different insights into distinct features of multivariate systems that may simultaneously contain several types of equilibrium behaviour that is more complex than that found with I (1) systems. We also discuss how appropriately defined state variables may ease the interpretational difficulties that may arise in polynomially cointegrated systems.

Measurement errors and outliers in seasonal unit root testing

Seasonal and non-seasonal data are frequently observed with noise. For instance, the time series can have irregular abrupt changes and interruptions following as a result of additive or temporary change outliers caused by external circumstances. Equally, the time series can have measurement errors. In this paper we analyse the above types of data irregularities on the behavior of seasonal unit root tests. Outliers and measurement errors can seriously affect seasonal unit root inference and it is shown how the distortion of the tests will depend upon the frequency, magnitude, and persistence of the outliers as well as on the signal to noise ratio associated with measurement errors. Some solutions to the implied inference problems are suggested and shown to work in practice.

Multiple Unit Roots in Periodic Autoregression

In this paper we propose a model selection strategy for a univariate periodic autoregressive time series which involves tests for one or more unit roots and for parameter restrictions corresponding to seasonal unit roots and multiple unit roots at the zero frequency. Examples of models that are considered are variants of the seasonal unit roots model and the periodic integration model. We show that the asymptotic distributions of various test statistics are the same as well-known distributions which are already tabulated. We apply our strategy to three empirical series to illustrate its ease of use. We find that evidence for seasonal unit roots based on nonperiodic models disappears when periodic representations are considered.