Matei Demetrescu

Combining Significance of Correlated Statistics with Application to Panel Data

The inverse normal method, which is used to combine P‐values from a series of statistical tests, requires independence of single test statistics in order to obtain asymptotic normality of the joint test statistic. The paper discusses the modification by Hartung (1999, Biometrical Journal, Vol. 41, pp. 849–855), which is designed to allow for a certain correlation matrix of the transformed P‐values. First, the modified inverse normal method is shown here to be valid with more general correlation matrices. Secondly, a necessary and sufficient condition for (asymptotic) normality is provided, using the copula approach. Thirdly, applications to panels of cross‐correlated time series, stationary as well as integrated, are considered. The behaviour of the modified inverse normal method is quantified by means of Monte Carlo experiments.

LONG MEMORY TESTING IN THE TIME DOMAIN

An integration test against fractional alternatives is suggested for univariate time series. The new test is a completely regression based, lag augmented version of the LM test by Robinson (1991, Journal of Econometrics 47, 67-84). Our main contributions, however, are the following. First, we let the short memory component follow a general linear process. Second, the innovations driving this process are martingale dierences with eventual conditional heteroskedasticity that is accounted for by means of White’s standard errors. Third, we assume the number of lags to grow with the sample size, thus approximating the general linear process. Under these assumptions limiting normality of the test statistic is retained. The usefulness of the asymptotic results for finite samples is established in Monte Carlo experiments. In particular, we study several strategies of model selection.

Testing for the cointegrating rank of a vector autoregressive process with uncertain deterministic trend term

When applying Johansens procedure for determining the cointegrating rank to systems of variables with linear deterministic trends, there are two possible tests to choose from. One test allows for a trend in the cointegration relations and the other one restricts the trend to be orthogonal to the cointegration relations. The first test is known to have reduced power relative to the second one if there is in fact no trend in the cointegration relations, whereas the second one is based on a misspecified model if the linear trend is not orthogonal to the cointegration relations. Hence, the treatment of the linear trend term is crucial for the outcome of the rank determination procedure. We compare two alternative testing strategies which are applicable if there is uncertainty regarding the proper trend specification. In the first one a specific cointegrating rank is rejected if one of the two tests rejects and in the second one the trend term is decided upon by a pretest. The first strategy is shown to be preferable in applied work.

Panel Unit Root Testing with Nonlinear Instruments for Infinite-Order Autoregressive Processes

The asymptotic null distribution of the nonlinear IV panel unit root test due to Chang (2002, Journal of Econometrics 110, 261-292) is examined under the assumption of an invertible general linear process with a weak summability condition. An autoregressive approximation of order p, with p growing to infinity jointly with the sample size T is employed in the test regression. The conditions under which the analysis is conducted are fairly similar to those usually assumed for the augmented Dickey-Fuller test, with the exception of the conditions imposed on the innovations of the general linear process; these satisfy much stricter conditions needed for the nonlinear IV framework. The asymptotic normality of the nonlinear IV (panel) unit root test is established when p is of a magnitude order lower than the square root of T. Furthermore, the convergence rate of the nonlinear IV estimator of the coefficient associated to the lagged level is found to be lower than square root T, thus leading to inconsistency of the residual variance estimator. Two simple solutions to this problem are suggested.

Joint forecasts of Dow Jones stocks under general multivariate loss function

When forecasts are assessed by a general loss (cost-of-error) function, the optimal point forecast is, in general, not the conditional mean, and depends on the conditional volatility-which, for stock returns, is time-varying. In order to provide forecasts of daily returns of 30 DJIA stocks under a general multivariate loss function, the following issues are addressed. We discuss what conditions define a multivariate loss function, and a simple class of such functions is proposed. Based on suitable combinations of univariate losses, the suggested multivariate functions are convenient for practical applications with many variables. To keep the computational aspect tractable, a flexible multivariate GARCH model is employed in estimating the conditional forecast distributions. The model easily copes with large number of series while allowing for skewness, fat tails, non-ellipticity, and tail dependence. Based on Engles DCC GARCH, it uses multivariate affine generalized hyperbolic distributions as conditional probability law, and the number of parameters to be estimated simultaneously does not depend on the number of series. The model is fitted using daily data from 2002 to 2007 (keeping data from 2008 for out-of-sample forecasts), and a bootstrap procedure is used to derive point forecasts under several multivariate loss functions of the proposed type.

Recursive Adjustment for General Deterministic Components and Improved Cointegration Rank Tests

Abstract This paper discusses tests for the cointegration rank of integrated vector autoregressions when the series are recursively adjusted for deterministic components. To this end, the asymptotic properties of recursive, or adaptive, procedures for the removal of general additive deterministic components are analyzed in two different, complementary, situations. When the stochastic component of the examined time series is weakly stationary (as would be the equilibrium errors), the effect of recursive adjustment vanishes with increasing sample size. When the suitably normalized stochastic component converges weakly to some limiting continuous-time process with integrable paths (as would be the case with the common stochastic trends), recursive adjustment has a permanent effect even asymptotically: the normalized recursively adjusted process converges weakly to a recursively adjusted version of the limiting process. The null limiting distributions of the cointegration rank tests can be expressed in terms of recursively adjusted Brownian motions. Moreover, the finite-sample properties of the cointegration rank tests with recursive adjustment are examined in cases of empirical relevance: the considered deterministic components are a constant, and a constant and a linear trend, respectively. Compared to the likelihood ratio tests or the tests with generalized least squares adjustment, improvements in terms of empirical rejection frequencies under the null are found in finite samples; improvements are found under the alternative as well, with the likelihood ratio test performing increasingly better as the magnitude of the initial condition increases. Regarding rank selection, a very simple combination of the three testing procedures with different adjustments performs best.

An extension of the Gauss-Newton algorithm for estimation under asymmetric loss

Estimators obtained by the use of the relevant loss function lead to forecasts with good properties when the same loss function is used to evaluate the forecasts. The provided extension of the Gauss-Newton algorithm is tailored for the associated optimization problem. Due to an approximation of the second derivative of the loss function, it can be viewed as a succession of linear generalized least-squares regressions and is easy to implement. Smoothing loss functions which do not possess derivatives has asymptotic validity. The extension performs well compared to the Newton (with exact Hessian) and BFGS algorithms in a Monte Carlo study employing different loss functions and several autoregressive models.

Spurious Persistence and Unit Roots due to Seasonal Differencing: The Case of Inflation Rates / Künstliche Persistenz und Einheitswurzeln infolge saisonaler Differenzen: Das Beispiel Inflationsraten

Summary Studying annual growth rates (seasonal differences) in case of seasonal data produces much more persistence, autocorrelation and stronger evidence in favour of a unit root than analyzing seasonal growth rates (ordinary differences). First, this statement is quantified theoretically. Second, it is supported experimentally with simulations, and, finally, it is empirically illustrated with quarterly GDP deflators from 7 European economies. Zusammenfassung Wenn man bei saisonalen Daten jährliche Inflationsraten (saisonale Differenzen) untersucht, so fällt die Persistenz viel höher und die Evidenz zugunsten der Einheitswurzelhypothese viel stärker aus als bei saisonalen Inflationsraten (gewöhnliche Differenzen). Unterstellt man ein Zeitreihenmodell, so kann dieser Effekt theoretisch quantifiziert werden. Darüber hinaus wird er anhand von Computerexperimenten demonstriert. Eine Studie von 7 europäischen Inflationsraten zeigt schließlich, dass unsere theoretischen Ergebnisse auch empirisch Bestand haben.

Robust Inference for Near-Unit Root Processes with Time-Varying Error Variances

The autoregressive Cauchy estimator uses the sign of the first lag as instrumental variable (IV); under independent and identically distributed (i.i.d.) errors, the resulting IV t-type statistic is known to have a standard normal limiting distribution in the unit root case. With unconditional heteroskedasticity, the ordinary least squares (OLS) t statistic is affected in the unit root case; but the paper shows that, by using some nonlinear transformation behaving asymptotically like the sign as instrument, limiting normality of the IV t-type statistic is maintained when the series to be tested has no deterministic trends. Neither estimation of the so-called variance profile nor bootstrap procedures are required to this end. The Cauchy unit root test has power in the same 1/T neighborhoods as the usual unit root tests, also for a wide range of magnitudes for the initial value. It is furthermore shown to be competitive with other, bootstrap-based, robust tests. When the series exhibit a linear trend, however, the null distribution of the Cauchy test for a unit root becomes nonstandard, reminiscent of the Dickey-Fuller distribution. In this case, inference robust to nonstationary volatility is obtained via the wild bootstrap.

Testing for stationarity in large panels with cross-dependence, and US evidence on unit labor cost

A new stationarity test for heterogeneous panel data with large cross-sectional dimension is developed and used to examine a panel with growth rates of unit labor cost in the USA. The test allows for strong cross-unit dependence in the form of unbounded long-run correlation matrices, for which a simple parameterization is proposed. A KPSS-type distribution results asymptotically if letting T→∞ be followed by N→∞. Some evidence against stationarity (short memory) is found for the examined series.