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on Econometric Time Series |
| By: | Peter Zadrozny |
| Abstract: | A univariate GARCH(p,q) process is quickly transformed to a univariate autoregressive moving-average process in squares of an underlying variable. For positive integer m, eigenvalue restrictions have been proposed as necessary and sufficient restrictions for existence of a unique mth moment of the output of a univariate GARCH process or, equivalently, the 2mth moment of the underlying variable. However, proofs in the literature that an eigenvalue restriction is necessary and sufficient for existence of unique 4th or higher even moments of the underlying variable, are either incorrect, incomplete, or unecessarily long. Thus, the paper contains a short and general proof that an eigenvalue restriction is necessary and sufficient for existence of a unique 4th moment of the underlying variable of a univariate GARCH process. The paper also derives an expression for computing the 4th moment in terms of the GARCH parameters, which immediately implies a necessary and sufficient inequality restriction for existence of the 4th moment. Because the inequality restriction is easily computed in a finite number of basic arithmetic operations on the GARCH parameters and does not require computing eigenvalues, it provides an easy means for computing "by hand" the 4th moment and for checking its existence for low-dimensional GARCH processes. Finally, the paper illustrates the computations with some GARCH(1,1) processes reported in the literature. |
| Keywords: | state-space form, Lyapunov equations, nonnegative and irreducible matrices |
| JEL: | C32 C63 G12 |
| Date: | 2005 |
| URL: | http://d.repec.org/n?u=RePEc:ces:ceswps:_1505&r=ets |
| By: | Helmut Luetkepohl |
| Abstract: | Vector autoregressive moving-average (VARMA) processes are suitable models for producing linear forecasts of sets of time series variables. They provide parsimonious representations of linear data generation processes (DGPs). The setup for these processes in the presence of cointegrated variables is considered. Moreover, a unique or identified parameterization based on the echelon form is presented. Model specification, estimation, model checking and forecasting are discussed. Special attention is paid to forecasting issues related to contemporaneously and temporally aggregated processes. |
| JEL: | C32 |
| Date: | 2004 |
| URL: | http://d.repec.org/n?u=RePEc:eui:euiwps:eco2004/25&r=ets |
| By: | Paulo M.M. Rodrigues; A.M. Robert Taylor |
| Abstract: | In this paper we derive, under the assumption of Gaussian errors with known error covariance matrix, asymptotic local power bounds for seasonal unit root tests for both known and unknown deterministic scenarios and for an arbitrary seasonal aspect. We demonstrate that the optimal test of a unit root at a given spectral frequency behaves asymptotically independently of whether unit roots exist at other frequencies or not. We also develop modified versions of the optimal tests which attain the asymptotic Gaussian power bounds under much weaker conditions. We further propose nearefficient regression-based seasonal unit root tests using pseudo-GLS de-trending and show that these have limiting null distributions and asymptotic local power functions of a known form. Monte Carlo experiments indicate that the regression-based tests perform well in finite samples. |
| Keywords: | Point optimal invariant (seasonal) unit root tests; asymptotic local power bounds; near seasonal integration |
| JEL: | C22 |
| Date: | 2004 |
| URL: | http://d.repec.org/n?u=RePEc:eui:euiwps:eco2004/29&r=ets |
| By: | Paulo M. M. Rodrigues |
| Abstract: | In this paper, we analyse the properties of recursive trend adjusted unit root tests. We show that OLS based recursive trend adjustment can produce unit root tests which are not invariant when the data is generated from a random walk with drift and investigate whether the power performance previously observed in the literature is maintained under invariant versions of the tests. A finite sample evaluation of the size and power of the invariant procedures is presented. |
| Keywords: | Recursive Trend Adjustment, Unit root tests, Invariance |
| JEL: | C12 C22 |
| Date: | 2004 |
| URL: | http://d.repec.org/n?u=RePEc:eui:euiwps:eco2004/31&r=ets |
| By: | Helmut Luetkepohl |
| Abstract: | Vector autoregressive (VAR) models are capable of capturing the dynamic structure of many time series variables. Impulse response functions are typically used to investigate the relationships between the variables included in such models. In this context the relevant impulses or innovations or shocks to be traced out in an impulse response analysis have to be specified by imposing appropriate identifying restrictions. Taking into account the cointegration structure of the variables offers interesting possibilities for imposing identifying restrictions. Therefore VAR models which explicitly take into account the cointegration structure of the variables, so-called vector error correction models, are considered. Specification, estimation and validation of reduced form vector error correction models is briefly outlined and imposing structural short- and long-run restrictions within these models is discussed. |
| Keywords: | Cointegration, vector autoregressive process, vector error correction model |
| JEL: | C32 |
| Date: | 2005 |
| URL: | http://d.repec.org/n?u=RePEc:eui:euiwps:eco2005/02&r=ets |
| By: | Marcelo Resende |
| Abstract: | This paper further investigated wave behaviours for mergers and acquisitions-M&A in the U.K. during the 1969Q1/2004Q1 period by means of Markov-Switching models. Previous analysis had focused on traditional models that incorporate the potentially limiting assumption of constant transition probabilities across regimes. The consideration of more general models with time-varying transition probabilities across regimes along the lines of Diebold et al (1994) provide a useful route for assessing to which extent M&A waves are driven by economic variables usually considered in the related literature. The empirical implementation considered lagged conditioning variables referring to real output growth, real growth in money supply and real stock market returns. The evidence indicated that one should reject the constant transition probability model in favour of the time-varying transition probability model and therefore the usual aggregate variables considered in the empirical literature on M&A appear indeed to play some role in determining the wave behaviour of M&A in the U.K., though the effects are asymmetric across the different regimes. |
| JEL: | C32 L12 |
| Date: | 2005 |
| URL: | http://d.repec.org/n?u=RePEc:eui:euiwps:eco2005/04&r=ets |
| By: | Jaroslava Hlouskova; Martin Wagner |
| Abstract: | This paper presents results concerning the size and power of first generation panel unit root and stationarity tests obtained from a large scale simulation study, with in total about 290 million test statistics computed. The tests developed in the following papers are included: Levin, Lin and Chu (2002), Harris and Tzavalis (1999), Breitung (2000), Im, Pesaran and Shin (1997 and 2003), Maddala and Wu (1999), Hadri (2000), and Hadri and Larsson (2005). Our simulation set-up is designed to address i.a. the following issues. First, we assess the performance as a function of the time and the cross-section dimension. Second, we analyze the impact of positive MA roots on the test performance. Third, we investigate the power of the panel unit root tests (and the size of the stationarity tests) for a variety of first order autoregressive coefficients. Fourth, we consider both of the two usual specifications of deterministic variables in the unit root literature. |
| Keywords: | Panel Unit Root Test, Panel Stationarity Test, Size, Power, Simulation Study |
| JEL: | C12 C15 C23 |
| Date: | 2005 |
| URL: | http://d.repec.org/n?u=RePEc:eui:euiwps:eco2005/05&r=ets |
| By: | Dietmar Bauer; Martin Wagner |
| Abstract: | We investigate autoregressive approximations of multiple frequency I(1) processes, of which I(1) processes are a special class. The underlying data generating process is assumed to allow for an infinite order autoregressive representation where the coefficients of the Wold representation of the suitably differenced process satisfy mild summability constraints. An important special case of this process class are VARMA processes. The main results link the approximation properties of autoregressions for the nonstationary multiple frequency I(1) process to the corresponding properties of a related stationary process, which are well known (cf. Section 7.4 of Hannan and Deistler, 1988). First, error bounds on the estimators of the autoregressive coefficients are derived that hold uniformly in the lag length. Second, the asymptotic properties of order estimators obtained with information criteria are shown to be closely related to those for the associated stationary process obtained by suitable differencing. For multiple frequency I(1) VARMA processes we establish divergence of order estimators based on the BIC criterion at a rate proportional to the logarithm of the sample size. |
| Keywords: | Unit Roots, Multiple Frequency I(1) Process, Nonrational Transfer Function, Cointegration, VARMA Process, Information Criteria |
| JEL: | C13 C32 |
| Date: | 2005 |
| URL: | http://d.repec.org/n?u=RePEc:eui:euiwps:eco2005/09&r=ets |
| By: | Steve Bond (Institute for Fiscal Studies and Nuffield College, Oxford); Céline Nauges; Frank Windmeijer (Institute for Fiscal Studies) |
| Abstract: | We consider a number of unit root tests for micro panels where the number of individuals is typically large, but the number of time periods is often very small. As we discuss, the presence of a unit root is closely related to the identification of parameters of interest in this context. Calculations of asymptotic local power and Monte Carlo evidence indicate that two simple t-tests based on ordinary least squares estimators perform particularly well. |
| Keywords: | Generalised Method of Moments, identification, unit root tests |
| JEL: | C12 C23 |
| Date: | 2005–07 |
| URL: | http://d.repec.org/n?u=RePEc:ifs:cemmap:07/05&r=ets |
| By: | Gary Koop; Roberto León-González; Rodney W. Strachan |
| Abstract: | A message coming out of the recent Bayesian literature on cointegration is that it is important to elicit a prior on the space spanned by the cointegrating vectors (as opposed to a particular identified choice for these vectors). In this note, we discuss a sensible way of eliciting such a prior. Furthermore, we develop a collapsed Gibbs sampling algorithm to carry out efficient posterior simulation in cointegration models. The computational advantages of our algorithm are most pronounced with our model, since the form of our prior precludes simple posterior simulation using conventional methods (e.g. a Gibbs sampler involves non-standard posterior conditionals). However, the theory we draw upon implies our algorithm will be more efficient even than the posterior simulation methods which are used with identified versions of cointegration models. |
| Date: | 2005–07 |
| URL: | http://d.repec.org/n?u=RePEc:lec:leecon:05/13&r=ets |
| By: | Rodney W. Strachan |
| Abstract: | This paper generalises 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. |
| Date: | 2005–07 |
| URL: | http://d.repec.org/n?u=RePEc:lec:leecon:05/14&r=ets |
| By: | Zacharias Bragoudakis (Bank of Greece) |
| Abstract: | This paper is an exercise in applied macroeconomic forecasting. We examine the forecasting power of a vector error-correction model (VECM) that is anchored by a long-run equilibrium relationship between Greek national income and productive public expenditure as suggested by the economic theory. We compare the estimated forecasting values of the endogenous variables to the real-historical values using a stochastic simulation analysis. The simulation results provide new evidence supporting the ability of the model to forecast not only one-period ahead but also many periods into the future. |
| Keywords: | Cointegration, Forecasting, Simulation Analysis, Vector error- correction models |
| JEL: | C15 C32 C53 E0 E6 |
| Date: | 2005–07–25 |
| URL: | http://d.repec.org/n?u=RePEc:wpa:wuwpem:0507013&r=ets |
| By: | Catalin Starica (Chalmers & Gothenburg University); Stefano Herzel (University of Perugia); Tomas Nord (Chalmers University of Technology) |
| Abstract: | The paper investigates from an empirical perspective aspects related to the occurrence of the IGARCH effect and to its impact on volatility forecasting. It reports the results of a detailed analysis of twelve samples of returns on financial indexes from major economies (Australia, Austria, Belgium, France, Germany, Japan, Sweden, UK, and US). The study is conducted in a novel, non-stationary modeling framework proposed in Starica and Granger (2005). The analysis shows that samples characterized by more pronounced changes in the unconditional variance display stronger IGARCH effect and pronounced differences between estimated GARCH(1,1) unconditional variance and the sample variance. Moreover, we document particularly poor longer-horizon forecasting performance of the GARCH(1,1) model for samples characterized by strong discrepancy between the two measures of unconditional variance. The periods of poor forecasting behavior can be as long as four years. The forecasting behavior is evaluated through a direct comparison with a naive non-stationary approach and is based on mean square errors (MSE) as well as on an option replicating exercise. |
| Keywords: | stock returns, volatility forecasting, GARCH(1,1), IGARCH effect, hedging, non-stationary, longer horizon forecasting |
| JEL: | C14 C32 |
| Date: | 2005–08–02 |
| URL: | http://d.repec.org/n?u=RePEc:wpa:wuwpem:0508003&r=ets |
| By: | Matthias Mohr (European Central Bank) |
| Abstract: | This paper proposes a new univariate method to decompose a time series into a trend, a cyclical and a seasonal component: the Trend-Cycle filter (TC filter) and its extension, the Trend-Cycle-Season filter (TCS filter). They can be regarded as extensions of the Hodrick-Prescott filter (HP filter). In particular, the stochastic model of the HP filter is extended by explicit models for the cyclical and the seasonal component. The introduction of a stochastic cycle improves the filter in three respects: first, trend and cyclical components are more consistent with the underlying theoretical model of the filter. Second, the end-of- sample reliability of the trend estimates and the cyclical component is improved compared to the HP filter since the pro-cyclical bias in end- of-sample trend estimates is virtually removed. Finally, structural breaks in the original time series can be easily accounted for. |
| Keywords: | economic cycles, time series, filtering, trend-cycle decomposition, seasonality |
| JEL: | C13 C22 E32 |
| Date: | 2005–08–03 |
| URL: | http://d.repec.org/n?u=RePEc:wpa:wuwpem:0508004&r=ets |
| By: | Matthias Mohr (European Central Bank) |
| Abstract: | This zip archive contains implementations of the trend-cycle-season filter in Eviews, Excel, and MatLab. The trend-cycle-season filter is another univariate method to decompose a time series into a trend, a cyclical and a seasonal component: the Trend-Cycle filter (TC filter) and its extension, the Trend-Cycle-Season filter (TCS filter), see paper ewp-em/0508004 at http://econwpa.wustl.edu/eprints/em/pape rs/0508/0508004.abs |
| Keywords: | economic cycles, time series, filtering, trend-cycle decomposition, seasonality |
| JEL: | C13 C22 E32 |
| Date: | 2005–08–03 |
| URL: | http://d.repec.org/n?u=RePEc:wpa:wuwpem:0508005&r=ets |
| By: | Isabel Proenca (ISEG-UTL) |
| Abstract: | Deconvolving kernel estimators when noise is Gaussian entail heavy calculations. In order to obtain the density estimates numerical evaluation of a specific integral is needed. This work proposes an approximation to the deconvolving kernel which simplifies considerably calculations by avoiding the typical numerical integration. Simulations included indicate that the lost in performance relatively to the true deconvolving kernel, is almost negligible in finite samples. |
| Keywords: | deconvolution, density estimation, errors-in-variables, kernel, simulations |
| JEL: | C1 C2 C3 C4 C5 C8 |
| Date: | 2005–08–05 |
| URL: | http://d.repec.org/n?u=RePEc:wpa:wuwpem:0508006&r=ets |
| By: | Jonathan B. Hill (Department of Economics, Florida International University) |
| Abstract: | We develop tests of linearity that are consistent against a class of Compound Smooth Transition Autoregressive (CoSTAR) models of the conditional mean. Our method is an extension of the sup-test developed by Bierens (1990) and Bierens and Ploberger (1997), provides maximal power against popular STAR alternatives and is consistent against any deviation from the null hypothesis. Moreover, the test method can be extended to consistent tests of number of threshold regimes, flexible parametric forms, conditional homoscedasticity against linear or smooth transition GARCH, and nonlinear causality tests. Of particular note, we improve on Bierens's (1990) test theory by considering a vector conditional moment that leads to a sup-test statistic that is never degenerate under the alternative of functional mis-specification. Such non-degeneracy will even help improve on the optimal tests of Andrews and Ploberger (1994). Moreover, our test is a true test against smooth transition alternatives, whereas the universally employed polynomial regression test of Luukkonen et al (1988) and Teräsvirta (1994) requires the assumption that the true data generating mechanism is STAR. A simulation study demonstrates that the suggested STAR sup-statistic renders a test with superlative empirical size and power attributes, in particular in comparison to the Bierens (1990) test, the neural test by Lee, White and Granger (1993), and specifically the polynomial regression test employed throughout the STAR literature. Finally, we apply the new tests to various macroeconomic processes. |
| Keywords: | conditional moment tests, vector weighted tests, consistent tests of functional form, smooth transition autoregression, non-degenerate test |
| JEL: | C12 C22 C45 C52 |
| Date: | 2004–04 |
| URL: | http://d.repec.org/n?u=RePEc:fiu:wpaper:0406&r=ets |
| By: | Jonathan B. Hill (Department of Economics, Florida International University) |
| Abstract: | The universal method for testing linearity against smooth transition autoregressive (STAR) alternatives is the linearization of the STAR model around the null nuisance parameter value, and performing F-tests on polynomial regressions in the spirit of the RESET test. Polynomial regressors, however, are poor proxies for the nonlinearity associated with STAR processes, and are not consistent (asymptotic power of one) against STAR alternatives, let alone general deviations from the null. Moreover, the most popularly used STAR forms of nonlinearity, exponential and logistic, are known to be exploitable for consistent conditional moment tests of functional form, cf. Bierens and Ploberger (1997). In this paper, pushing asymptotic theory aside, we compare the small sample performance of the standard polynomial test with an essentially ignored consistent conditional moment test of linear autoregression against smooth transition alternatives. In particular, we compute an LM sup-statistic and characterize the asymptotic p-value by Hansen's (1996) bootstrap method. In our simulations, we randomly select all STAR parameters in order not to bias experimental results based on the use of "safe", "interior" parameter values that exaggerate the smooth transition nonlinearity. Contrary to past studies, we find that the traditional polynomial regression method performs only moderately well, and that the LM sup-test out-performs the traditional test method, in particular for small samples and for LSTAR processes. |
| Keywords: | Smooth transition AR, consistent conditional moment test, Lagrange Multiplier, bootstrap |
| JEL: | C1 C2 C4 C5 C8 |
| Date: | 2004–07 |
| URL: | http://d.repec.org/n?u=RePEc:fiu:wpaper:0412&r=ets |
| By: | Jonathan B. Hill (Department of Economics, Florida International University) |
| Abstract: | This paper develops a simple sequential multiple horizon non-causation test strategy for trivariate VAR models (with one auxiliary variable). We apply the test strategy to a rolling window study of money supply and real income, with the price of oil, the unemployment rate and the spread between the Treasury bill and commercial paper rates as auxiliary processes. Ours is the first study to control simultaneously for common stochastic trends, sensitivity of causality tests to chosen sample period, null hypothesis over-rejection, sequential test size bounds, and the possibility of causal delays. Evidence suggests highly significant direct or indirect causality from M1 to real income, in particular through the unemployment rate and M2 once we control for cointegration. |
| Keywords: | multiple horizon causality, Wald tests, parametric bootstrap, money-income causality, rolling windows, cointegration |
| JEL: | C12 C32 C53 E47 |
| Date: | 2004–07 |
| URL: | http://d.repec.org/n?u=RePEc:fiu:wpaper:0413&r=ets |