The relationship between the Beveridge–Nelson decomposition and other permanent–transitory decompositions that are popular in economics

The Beveridge–Nelson (BN) decomposition is a model-based method for decomposing time series into permanent and transitory components. When constructed from an ARIMA model, it is closely related to decompositions based on unobserved components (UC) models with random walk trends and covariance stationary cycles. The decomposition when extended to I(2)$I\left(2\right)$ models can also be related to non-model-based signal extraction filters such as the HP filter. We show that the BN decomposition provides information on the correlation between the permanent and transitory shocks in a certain class of UC models. The correlation between components is known to determine the smoothed estimates of components from UC models. The BN decomposition can also be used to evaluate the efficacy of alternative methods. We also demonstrate, contrary to popular belief, that the BN decomposition can produce smooth cycles if the reduced form forecasting model is appropriately specified.     

Structural information in recursive VAR orderings

This paper investigates structural models that will permit a Cholesky decomposition of the covariance matrix of VAR residuals to identify some structural impulse response functions. Cholesky decompositions are found to be useful identification tools for the set of partially recursive structural models. A partially recursive structure is defined as any block recursive system where the equations in one block can be recursively ordered and where the structural shocks are uncorrelated. Using this class of models, we derive necessary and sufficient conditions for the moving average representation from a Cholesky decomposition to identify structure. The paper concludes by discussing implications of these results for empirical research.     

Permanent-transitory Decomposition in Var Models With Cointegration and Common Cycles

In this paper we derive permanent-transitory decompositions of non-stationary multiple times series generated by (r)nite order Gaussian VAR(p) models with both cointegration and serial correlation common features. We extend existing analyses to the two classes of reduced rank structures discussed in Hecq, Palm and Urbain (1998). Using the corresponding state space representation of cointegrated VAR models in vector error correction form we show how decomposition can be obtained even in the case where the number of common feature and cointegration vectors are not equal to the number of variables. As empirical analysis of US business fluctuations shows the practical relevance of the approach we propose.     

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 times series. In this paper we 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, we 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.     

Recent Developments in Modelling Nonstationary Vector Autoregressions

In this paper we review some recent developments in the modelling of nonstationary vector autoregressions (VARs) which we feel have great potential for furthering applied researchers understanding of the relationships linking the variables making up a VAR. The developments surveyed are the use of model determination criteria in selecting lag length, trend order and cointegrating rank, causality testing in vector error correction models, FM-VAR estimation of levels VARS, common trends and cycles analysis, permanent and transitory decompositions, impulse response asymptotics, and the links between cointegrated VARs and structural models. The techniques are illustrated by applications to the modelling of U.K. equities, dividends and interest rates.     

Isometric operators on Hilbert spaces and Wold decomposition of stationary time series

Wold Theorem plays a fundamental role in the decomposition of weakly stationary time series. It provides a moving average representation of the process under consideration in terms of uncorrelated innovations, whatever the nature of the process is. From an empirical point of view, this result enables to identify orthogonal shocks, for instance in macroeconomic and financial time series. More theoretically, the decomposition of weakly stationary stochastic processes can be seen as a special case of the Abstract Wold Theorem, that allows to decompose Hilbert spaces by using isometric operators. In this work we explain this link in detail, employing the Hilbert space spanned by a weakly stationary time series and the lag operator as isometry. In particular, we characterize the innovation subspace by exploiting the adjoint operator. We also show that the isometry of the lag operator is equivalent to weak stationarity. Our methodology, fully based on operator theory, provides novel tools useful to discover new Wold-type decompositions of stochastic processes, in which the involved isometry is no more the lag operator. In such decompositions the orthogonality of innovations is ensured by construction since they are derived from the Abstract Wold Theorem.     

MULTIPLICATIVE DECOMPOSITION AND INDEX NUMBER THEORY: AN EMPIRICAL APPLICATION OF THE SATO–VARTIA DECOMPOSITION

In de Boer (2008), additive decompositions of aggregate changes in a variable into its factors were considered. We proposed using the ‘ideal’ Montgomery decomposition, developed in index number theory as an alternative to the commonly used methods in structural decomposition analysis, and applied it to the example analyzed by Dietzenbacher and Los (1998) (D&L). In this paper we consider multiplicative decompositions and show that the method proposed by D&L of taking the geometric mean of all elementary decompositions is ‘ideal’. However, it requires the computation of an ever-increasing number of decompositions when the number of factors increases. As an alternative, we propose using the Sato–Vartia decomposition, which is also ‘ideal’, but requires the computation of only one decomposition. Application to the example of D&L reveals that the two methods yield results that are very close to each other.     

Common cycles in seasonal non-stationary time series

This paper extends the notion of common cycles to quarterly time series having unit roots both at the zero and seasonal frequencies. It is shown that common cycles are present in the Hylleberg–Engle–Granger–Yoo decomposition of these series when there exists a linear combination of their seasonal differences which follows an MA process of order, at most, three. The pitfalls of seasonal adjustment for common cycles analysis are also documented. Inference on common cycles in seasonally cointegrated series is derived from existing statistical methods for codependence. Concepts and methods are illustrated with an empirical analysis of the comovements between consumption and output using Italian data. Copyright © 1999 John Wiley & Sons, Ltd.     

Common trends and cycles in I(2) VAR systems

This paper analyzes common cycles in I(2) vector autoregressive (VAR) systems. We consider different choices of stationary variables extracted from a VAR, including deviations from equilibria. This extension is based on the equilibrium dynamics representation of the system, introduced in this paper. Inference on the number of common features is addressed via reduced rank regression, as well as estimation of the cofeature relations and testing. An application to Australian prices is presented; it is found that the deviation from one equilibrium relation is an innovation process, whereas no common cycles can be obtained for the acceleration rates.     

Structural Decomposition Techniques: Sense and Sensitivity

Structural decomposition techniques are widely used to break down the growth in some variable into the changes in its determinants. In this paper, we discuss the problems caused by the existence of a multitude of equivalent decomposition forms which are used to measure the contribution of a specific determinant. Although it is well known that structural decompositions are not unique, the extent of the problem and its consequences seem to have been largely neglected. In an empirical analysis for The Netherlands between 1986 and 1992, results are calculated for 24 equivalent decomposition forms. The outcomes exhibit a large degree of variability across the different forms. We also examine the two approaches that have been used predominantly in the literature. The average of the two so-called polar decompositions appears to be remarkably close to the average of the full set of 24 decompositions. The approximate decomposition with mid-point weights appears to be almost exact. Although this last alternative might seem a solution to the problem of the marked sensitivity, in fact, it only conceals the problem.     

A multivariate unobserved component analysis of US housing market

This study explores price dynamics and price relationships in the US housing market with a focus on four regions: Northeast, Midwest, South, and West. It applies a multivariate state-space model to identify the common trends and common cycles in US regional markets. The study finds that the principal source of secular price variability in the Northeast and West markets is due to two common stochastic trends, while a large share of transitional price variability in the Northeast, West and Midwest originates from three common stochastic cycles. The study estimates the relationships between the common unobserved components and economic variables and finds that unemployment, federal funds rate, corporate default risk, economic expansion, unanticipated inflation in the construction market are significant underlying economic phenomena that impact the evolution of the common movements in both the short run and the long run housing dynamics. Authorship is equally shared between the authors.     

Cyclical capital structure decisions

Past literature has documented clear trends in the leverage ratio and other capital structure choices made by US firms. We expand this line of research by showing that aggregate capital structure ratios of US firms, during the last decades, are characterised not only by time trends but also by clear cycles. We set the start and end dates of these cycles using a ‘classical approach’. The cycles relating to the ratio of new shares versus debt are shorter and are more intense than the cycles regarding the term of the new debt obligations. The cycles that describe the ratio of retained earnings versus new equity issues are wider in relative terms and with similar duration to the cycles of decisions on external versus internal financing. This means that the decision to substitute debt for shares (or vice versa) is much more common, frequent and significant, than the decision term debt.     

COMMON SEASONAL FEATURES: GLOBAL UNEMPLOYMENT

Seasonal patterns in economic time series are generally examined from a univariate point of view. Using extensions of the unit root literature, important classes of seasonal processes are deterministic, stationary stochastic or mean reverting, and unit root stochastic. Time series tests have been developed for each of these. This paper examines seasonality in a multivariate context. Systems of economic variables can have trends, cycles and unit roots as well as the various types of seasonality. Restrictions such as cointegration and common cycles are here applied also to examine multivariate seasonal behaviour of economic variables. If each of a collection of series has a certain type of seasonality but a linear combination of these series can be found without seasonality, then the seasonal is said to be ‘common’. New tests are developed to determine if seasonal characteristics are common to a set of time series. These tests can be employed in the presence of various other time series structures. The analysis is applied to OECD data on unemployment for the period 1975.1 to 1993.4, and it is found that four diverse countries (Australia, Canada, Japan and USA) not only have common trends in their unemployment, but also have common deterministic seasonal features and a common cycle/stochastic seasonal feature. Such a collection of characteristics were not found in other groups of OECD countries.     

The permanent income hypothesis in five major industrial countries: a multivariate trend-cycle decomposition test

This paper presents a new test of the permanent income hypothesis in five major industrial countries. The test first decomposes consumption and income into their long run trend (permanent) and short run cyclical (transitory) components, using the recently developed multivariate stochastic detrending approach developed by Vahid and Engle (1997), among others. This approach exploits the presence of possible common stochastic trends and cycles among the variables in the system to arrive at a more efficient decomposition of these variables. Using the decomposition results, and in contrast to many articles in the literature, the paper finds support for the permanent income hypothesis. Specifically, the paper finds that, while permanent consumption is related to permanent income, transitory consumption is related to neither permanent nor transitory income.

Multivariate Wold decompositions: a Hilbert A-module approach

Decisions in Economics and Finance - Orthogonal decompositions are essential tools for the study of weakly stationary time series. Some examples are given by the classical Wold decomposition of...     

On the Decomposition by Subgroups of the Gini Index and Zenga's Uniformity and Inequality Indexes

Besides a brief historical outline of the Gini index decomposition proposals, we compare, in a subgroups framework, the decompositions of the Gini index and of the uniformity and inequality indexes recently proposed by Zenga. The two decompositions follow a similar scheme: in both cases the overall index can be at first expressed as a weighted average of convenient "cross" measures, with their own interpretation, and afterwards it is decomposed into a within and a between term by merely distinguishing measures evaluated within the same subgroup from the ones regarding different subgroups. This procedure does not depend on an a priori definition of the within or the between term and allows their contribution to be naturally evaluated preserving the structure of the index itself. In the last section the two decompositions are applied to the Italian net household wealth provided by the 2006 central Bank of Italy sample survey on household budgets.     

Bayesian non-parametric signal extraction for Gaussian time series

We consider the problem of unobserved components in time series from a Bayesian non-parametric perspective. The identification conditions are treated as unknown and analyzed in a probabilistic framework. In particular, informative prior distributions force the spectral decomposition to be in an identifiable region. Then, the likelihood function adapts the prior decompositions to the data.     

Decomposing profit change: Konüs,Bennet and Luenberger indicators

We introduce complementary decompositions of profit change that, relying on the duality between the profit function and the directional distance function, shed light on the different sources of profit growth including measures of technical efficiency, allocative efficiency and technological change. Our decompositions extend the literature on Konüs and Bennet quantity and price indicators to profit change. The first decomposition is ‘exact’ in the sense of Diewert, by completely exhausting the sources of profit change into profit inefficiency change (including technical and allocative inefficiency change), technological change, and output and input price change. The second decomposition equates the Bennet quantity indicator to a productivity measure represented by the Luenberger indicator plus allocative inefficiency change. We deem it ‘complete’ because in contrast to the existing literature, it retains the information on allocative inefficiency change while preventing the existence of residual terms capturing price variations, whose meaningful interpretation has not been addressed until now. Our proposed solution takes advantage of the flexibility of the directional distance function when choosing a suitable directional vector. All decompositions have the same structural form and therefore their components can be compared to each other vis-à-vis, providing alternative measures of equivalent sources of profit growth.     

Trends, lead times and forecasting

The innovations representation for a local linear trend can adapt to long run secular and short term transitory effects in the data. This is illustrated by the theoretical power spectrum for the model which may possess considerable power at frequencies that might be associated with cycles of several years' duration. Whilst advantageous for short term forecasting, the model may be of less use when interest is in the underlying long run trend in the data. In this paper we propose a generalisation of the innovations representation for a local linear trend that is appropriate for representing short, medium and long run trends in the data.     

Does It Make Sense to Apply the Static Open Input–Output Model for Imputation and Structural Decomposition?

This paper aims to stimulate discussion on the question concerning whether input–output (10) analysts should go on applying the static open 10 model in times when data supply and computer capacity allow us to switch to semi-closed 10 models, which are rated as comparative improvements. It is argued that the most frequent of the open 10 model's applications, i.e. imputations and structural decompositions, with a switch to a semi-closed model, lose the properties which make them so attractive: imputations no longer are straightforward assignments of production and primary inputs to the components of total final demand; structural decomposition no longer is an additive assignment of changes of production and primary inputs to sources of structural change. In the author's opinion, imputations and structural decompositions on the basis of the open 10 model should be abandoned.