Granger Representation Theorem Research Articles

A note on Johansen's rank conditions and the Jordan form of a matrix

This note presents insights on the Jordan structure of a matrix which are derived from an extension of the and conditions in Johansen (1996). It is first observed that these conditions not only characterize, as it is well known, the size (1 or 2) of the largest Jordan block in the Jordan form of the companion matrix but more generally the geometric multiplicities, the algebraic multiplicities and the whole Jordan structure for eigenvalues of index 1 or 2. In the context of the Granger representation theorem, this means that the Johansen rank conditions do more than determine the order of integration of the process. It is then shown that an extension of these conditions leads to the characterization of the Jordan structure of any matrix.

Cashing In or Paying Out? Analysis of the Relation Between Corporate Earnings and Cash

This paper examines the relation between corporate earnings and cash in the US on annual basis in the period 2002 to 2022. We document that both earnings and cash holdings are non-stationary and therefore we use the Granger representation theorem and the methods of cointegration analysis and make an attempt to model the relation between these company variables. We document a negative sign of the cointegration coefficient estimate that is statistically significant. The statistical significance confirms that cash and earnings are cointegrated, and depend on each other. However, the significance of the models with earnings as the dependent variable is larger, and the coefficient estimates are more reliably negative. These results indicate that earnings depend on cash to a larger degree than cash depends on earnings.

Cointegration and Error Correction Mechanisms for Singular Stochastic Vectors

Large-dimensional dynamic factor models and dynamic stochastic general equilibrium models, both widely used in empirical macroeconomics, deal with singular stochastic vectors, i.e., vectors of dimension r which are driven by a q-dimensional white noise, with q < r . The present paper studies cointegration and error correction representations for an I ( 1 ) singular stochastic vector y t . It is easily seen that y t is necessarily cointegrated with cointegrating rank c ≥ r − q . Our contributions are: (i) we generalize Johansen’s proof of the Granger representation theorem to I ( 1 ) singular vectors under the assumption that y t has rational spectral density; (ii) using recent results on singular vectors by Anderson and Deistler, we prove that for generic values of the parameters the autoregressive representation of y t has a finite-degree polynomial. The relationship between the cointegration of the factors and the cointegration of the observable variables in a large-dimensional factor model is also discussed.

Analysis of the Relation between Corporate Sales and Earnings

This paper examines the relation between corporate sales and earnings in the US on annual basis in the period 1950 to 2016. We document that both total revenues and earnings are non-stationary and therefore we use the Granger representation theorem and the methods of cointegration analysis and make a first attempt at representing the relation between these very important company characteristics with a statistical model. We document that such a relation does not exist between aggregate revenues and earnings. However, industry analysis provides long-run revenue-earnings relations for the different S&P industry sectors. Knowing and understanding better the top-line and bottom-line relation could further help external analysts in their quest for better valuations and forecasts. Many studies have focused on revenues alone and the factors affecting revenues, and earnings alone and the factors affecting earnings. In contrast to these studies this paper examines the relation between corporate sales and earnings, i.e. between the company‘s top-line and bottom-line. To the best of our knowledge this has not been done in the literature so far.

Dynamic Factor Models, Cointegration, and Error Correction Mechanisms

The paper studies Non-Stationary Dynamic Factor Models such that: (1) the factors Ft are I(1) and singular, i.e. Ft has dimension r and is driven by a q-dimensional white noise, the common shocks, with q < r, and (2) the idiosyncratic components are I(1). We show that Ft is driven by r -c permanent shocks, where c is the cointegration rank of Ft, and q - (r - c) < c transitory shocks, thus the same result as in the non-singular case for the permanent shocks but not for the transitory shocks. Our main result is obtained by combining the classic Granger Representation Theorem with recent results by Anderson and Deistler on singular stochastic vectors: if (1 - L)Ft is singular and has rational spectral density then, for generic values of the parameters, Ft has an autoregressive representation with a finite-degree matrix polynomial fulfilling the restrictions of a Vector Error Correction Mechanism with c error terms. This result is the basis for consistent estimation of Non-Stationary Dynamic Factor Models. The relationship between cointegration of the factors and cointegration of the observable variables is also discussed.

Dynamic Factor Models, Cointegration, and Error Correction Mechanisms

In this paper we study Dynamic Factor Models when the factors Ft are I(1) and singular, i.e. rank(Ft) < dim(Ft). By combining the classic Granger Representation Theorem with recent results by Anderson and Deistler on singular stochastic vectors, we prove that, for generic values of the parameters, Ft has an Error Correction representation with two unusual features: (i) the autoregressive matrix polynomial is finite, (ii) the number of error-terms is equal to the number of transitory shocks plus the dierence between the dimension and the rank of Ft. This result is the basis for the correct specification of an autoregressive model for Ft. Estimation of impulse-response functions is also discussed. Results of an empirical analysis on a US quarterly database support the use of our model.

Energy use, exports, imports and GDP: New evidence from the OECD countries

This paper aims to investigate the relationship between energy use-GDP, energy use-exports-trade and energy use-imports on aggregate level in the OECD countries. While the energy use-growth nexus is widely examined in the literature, studies on the energy use-trade nexus are very scarce. Thus, this study also serves to fill this gap by presenting a new research on the OECD countries. We employ the panel cointegration technique and use the Granger representation theorem to examine the presence of long-run relationship and the causality between pairs of variables. To investigate the presence of causality, several tests developed by Canning and Pedroni (2008) are used. We find that the pairs of energy use-GDP, energy use-exports and energy use-imports are cointegrated and there is two-way Granger causality between each pairs. We also estimate the long-run elasticities by employing panel dynamic ordinary least squares. The results suggest that the sign of long-run elasticity is positive for all pairs. Particularly, 1% increase in GDP, export and import causes almost 0.32%, 0.21% and 0.16% increase in energy use respectively.

Assessing Tourism Development from Sen’s Capability Approach

The purpose of this study is to assess tourism development in the context of Sen’s capability approach. The study developed a model to investigate the relationship between tourism development and human development while focusing on two countries, Nicaragua and Costa Rica. The study applied a cointegration technique based on the Granger representation theorem. Overall, tourism development and human development reveals a tenuous relationship in both cases, reflecting some threshold effect. The importance of tourism growth is merited in the distribution of its benefits and the extent that tourism receipts are allocated to support human development (public health, education, safety, etc.). Rising incomes will not necessarily translate into human development performance, thereby rendering support to Sen’s contention that well-being should not be measured by its instrumental antecedents (such as income) alone. Private incomes through tourism expansion seem to matter most at lower levels of human development.

ANALISIS TINGKAT PARTISIPASI WANITA DALAM ANGKATAN KERJA DI JAWA TENGAH PERIODE TAHUN 1982-2000

At industrialization phase, women will be shifted by men because the increasing of unemployment. When economic development reaches a certain stage, the trend of development turned with increasing women employment. Purpose of the research that will be reached is to estimate influence of women unemployment, women resident that managing household, and women residents that still school on the level of women participation in the labor force in Central Java (1982-2000). This study clarifies the factors that influence the level of participation of women in the labor force and improve the existence theory of the labor force participation rate. One way to analyze the effect of short-run and long run is to use a dynamic model. In this research model used is the Engle Granger Error Correction Models (EG-ECM) which based on "granger representation theorem". The result knows the variable having influence which significance in short-run are women resident that managing household and women resident that still school.

In memory of Clive Granger: an advisory board member of the journal

In memory of Clive Granger: an advisory board member of the journal

Granger's representation theorem: A closed‐form expression for I(1) processes

Summary The Granger representation theorem states that a cointegrated vector autoregressive process can be decomposed into four components: a random walk, a stationary process, a deterministic part, and a term that depends on the initial values. In this paper, we present a new proof of the theorem. This proof enables us to derive closed-form expressions of all terms of the representation and allows a unified treatment of models with different deterministic specifications. The applicability of our results is illustrated by examples. For example, the closed-form expressions are useful for impulse response analyses and facilitate the analysis of cointegration models with structural changes.

The Dynamics of Consumers' Expenditure: The UK Consumption ECM Redux

Simple intertemporal consumption theory implies that non-durable consumption is a random walk, but that consumption cointegrates with income and wealth. By the Granger representation theorem, there must be a (vector) error correction mechanism ((V)ECM) representation of the data; but from the theory, the equilibrating ECM cannot be in consumption. Instead, even with generalisations such as habit persistence, this equilibration should take place via income or wealth. Furthermore, unless the relative price of durables and non-durables is constant, the relative price needs to be taken into account in modelling. In this paper, the short-run dynamics and long-run relationship between non-durable consumption, non-asset income, wealth and the relative price of durable goods are examined. A cointegrating relationship is found to exist. Estimating VECMs, it is found that the adjustment towards the long-run common trend does indeed occur partly via changes in wealth, consistent with forward-looking behaviour on the part of agents. The result implies that consumption will predict asset returns, and this is confirmed by a regression of excess equity returns on the lagged disequilibrium term. A decomposition of shocks hitting the system reveals that between 30% and 90% of fluctuations in non-human wealth are transitory. Even if the lower figure applies, this means a substantial part of short-term fluctuations in wealth is decoupled from permanent consumption.

Stability results for nonlinear error correction models

This paper studies analogs of Granger's representation theorem in the context of a general nonlinear vector autoregressive error correction model. The model allows for nonlinear autoregressive conditional heteroskedasticity and the conditional distribution involved can be a mixture distribution of a rather general type. Mixture models of this kind can be thought of as generalizations of threshold models and they have attracted attention in the recent time series and econometrics literature. The paper develops a useful transformation which shows how the nonlinear error correction model can be transformed to a nonlinear vector autoregressive model so that available results on the stationarity or nonstationarity of the latter can be used for the former. The most satisfactory results are obtained in a model in which a specific structural relation between the nonlinearity and equilibrium correction prevails. Without this structural relation only a lower bound for the number of long-run equilibrium relations can explicitly be determined because the exact number depends on properties of the first and second moments of a nonlinear stationary component of the process.

Nonlinear error correction models

The relationship between cointegration and error correction (EC) models is well characterized in a linear context, but the extension to the nonlinear context is still a challenge. Few extensions of the linear framework have been done in the context of nonlinear error correction (NEC) or asymmetric and time varying error correction models. In this paper, we propose a theoretical framework based on the concept of near epoch dependence (NED) that allows us to formally address these issues. In particular, we partially extend the Granger Representation Theorem to the nonlinear case.

Foreign exchange market efficiency and cointegration

The analysis of market efficiency in the foreign exchange market adopted a new approach after Granger (Oxford Bulletin of Economics and Statistics, 48(3), 1986) stated that assets in an efficient market could not be cointegrated. If they were, there would be a market inefficiency since there would be Granger causality running at least in one direction and thus one price could be used to forecast the other. The interpretation that the literature has given to the relationship between cointegration and market efficiency has been that noncointegration is a necessary and sufficient condition for market efficiency. In the authors' opinion, the fact that two spot exchange rates are cointegrated does not necessarily imply that inefficiency exists. In this article, it is argued that when the economy is composed of N exchange rates and the closed system is analysed without dynamics, as it is the case when considering the no-arbitrage condition, the Granger Representation Theorem (GRT) does not tell one anything abo...

Defence spending and economic growth: A causal analysis for Greece and Turkey*

There are a number of studies which consider the relation between military spending and economic growth using Granger causality techniques rather than a well‐defined economic model. Some have used samples of groups of countries, finding no consistent results. Others have focused on case studies of individual countries, which has the advantage of the researchers bringing to bear much more data than the cross country samples and a greater knowledge of the structure of the economy and the budget. This paper adds to the literature by providing an analysis of two countries, Greece and Turkey, which are particularly interesting case studies given their high military burdens, the poor relations between the two and the resulting arms race in the area. In addition to analysing the data using standard “pre‐cointegration” Granger causality techniques, this paper employs modern vector autoregressive (VAR) methodology that utilises cointegration via Granger's representation theorem. The standard Granger causality tests suggest a positive effect of changing military burden on growth for Greece, but this is not sustained when the cointegration between output and military burden is taken into account. The only evidence of significant Granger causality is a negative impact of military burden on growth in Turkey.

Lietuvos eksporto matematinis modeliavimas ir prognozavimas

Economic activities and indicators of a small open country are crucially influenced by the dynamics of export volume. This paper examines the Lithuanian export trends by different countries using a modem non-stationary time series and econometric theory.To avoid spurious regression, time series are modelled as an integrated process. The stationarity of Lithuanian export and the main explored exogenous variables (GDP, import values, exchange rates and CPI) are analysed using the augmented Dicky–Fuller test. Most of the indicators are integrated of order one. Further investigated modelling alternatives are the vector autoregression (VAR) and the error correction model (ECM). Although according to the Granger representation theorem each finite order VAR has a respective ECM, it is known that, in general, co-integrated (in level) process does not admit a pure VAR representation in first differences. Hence VAR is used, when there is no statistical co-integration evidence. In eleven of fourteen cases ECM proved to be appropriate for the Lithuanian export modelling. VAR turned out to be needed only in three cases.The forecasted value of total Lithuanian exports in year 2000 is LTL 14293 mn. Export to the EU, the CIS, and the CEFTA groups are forecasted to be LTL 7006 mn, LTL 2420 mn, and LTL 1043 mn, respectively. The forecasts of Lithuanian exports to the three different country groups are formed assuming constant shares between the analysed and the remaining countries in a certain group. Conditional export forecasts are based on the exogenous variables forecasts. These are mo­delled using the SAS/ETS software as minimum RMSE trend, exponential smoothing or ARIMA model.

Modelling of cointegration in the vector autoregressive model

A survey is given of some results obtained for the cointegrated VAR. The Granger representation theorem is discussed and the notions of cointegration and common trends are defined. The statistical model for cointegrated I(1) variables is defined, and it is shown how hypotheses on the cointegrating relations can be estimated under suitable identification conditions. The asymptotic theory is briefly mentioned and a few economic applications of the cointegration model are indicated.

Granger's Representation Theorem: A Closed-Form Expression for I(1) Processes

The Granger representation theorem states that a cointegrated vector autoregressive process can be decomposed into four components: a random walk, a stationary process, a deterministic part, and a term that depends on the initial values. In this paper, we present a new proof of the theorem. This proof enables us to derive closed-form expressions of all terms of the representation and allows a unified treatment of models with different deterministic specifications. The applicability of our results is illustrated by examples. For example, the closed-form expressions are useful for impulse response analyses and facilitate the analysis of cointegration models with structural changes.

Dynamic Nonmetropolitan Export-Base Modeling

This paper investigated a time series export-base model of nonmetro-politan Nevada counties. Confirmatory factor analysis was used to bifurcate county employment, and county basic and nonbasic employment were tested for cointegration. Employing the Granger Representation Theorem, an error correction mechanism model was estimated. Error correction mechanism model forecasts were performed and long-run forecasts were evaluated. Given current microcomputer hardware and software, this paper provides a procedure to conduct dynamic rural impact analysis by employing a dynamic export-base model.