Showing papers in "Econometric Theory in 1995"
TL;DR: In this paper, a new parameterization of the multivariate ARCH process is proposed and equivalence relations are discussed for the various ARCH parameterizations, and conditions suffcient to guarantee the positive deffniteness of the covariance matrices are developed.
Abstract: This paper presents theoretical results in the formulation and estimation of multivariate gen- eralized ARCH models within simultaneous equations systems. A new parameterization of the multivariate ARCH process is proposed and equivalence relations are discussed for the various ARCH parameterizations. Constraints suffcient to guarantee the positive deffniteness of the con- ditional covariance matrices are developed, and necessary and suffcient conditions for covariance stationarity are presented. Identifcation and maximum likelihood estimation of the parameters in the simultaneous equations context are also covered.
4,413 citations
TL;DR: This article examined regression tests of whether x forecasts y when the largest autoregressive root of the regressor is unknown and showed that the power loss from using these conservative tests is small.
Abstract: This paper examines regression tests of whether x forecasts y when the largest autoregressive root of the regressor is unknown. It is shown that previously proposed two-step procedures, with first stages that consistently classify x as I(1) or I(0), exhibit large size distortions when regressors have local-to-unit roots, because of asymptotic dependence on a nuisance parameter that cannot be estimated consistently. Several alternative procedures, based on Bonferroni and Scheffe methods, are therefore proposed and investigated. For many parameter values, the power loss from using these conservative tests is small.
379 citations
TL;DR: This paper derived the asymptotic distribution of ordinary least squares estimates of the largest autoregressive root and its t-statistic, a convex combination of the Dickey-Fuller distribution and the standard normal, the mixture depending on the correlation between the equation error and the regression covariates.
Abstract: In the context of testing for a unit root in a univariate time series, the convention is to ignore information in related time series. This paper shows that this convention is quite costly, as large power gains can be achieved by including correlated stationary covariates in the regression equation.The paper derives the asymptotic distribution of ordinary least-squares estimates of the largest autoregressive root and its t-statistic. The asymptotic distribution is not the conventional Dickey-Fuller distribution, but a convex combination of the Dickey-Fuller distribution and the standard normal, the mixture depending on the correlation between the equation error and the regression covariates. The local asymptotic power functions associated with these test statistics suggest enormous gains over the conventional unit root tests. A simulation study and empirical application illustrate the potential of the new approach.
361 citations
TL;DR: In this article, a multivariate test for the existence of I(2) variables in a VAR model is presented, which is illustrated using a data set consisting of U.K. and foreign prices and interest rates as well as the exchange rate.
Abstract: This paper discusses inference for I(2) variables in a VAR model. The estimation procedure suggested consists of two reduced rank regressions. The asymptotic distribution of the proposed estimators of the cointegrating coefficients is mixed Gaussian, which implies that asymptotic inference can be conducted using the χ2 distribution. It is shown to what extent inference on the cointegration ranks can be conducted using the tables already prepared for the analysis of cointegration of I(1) variables. New tables are needed for the test statistics to control the size of the tests. This paper contains a multivariate test for the existence of I(2) variables. This test is illustrated using a data set consisting of U.K. and foreign prices and interest rates as well as the exchange rate.
323 citations
TL;DR: In this article, the authors define the Wiener decomposition of a stationary series as the sum of uncorrelated components, each of which is associated with a single frequency, or a narrow frequency band.
Abstract: The definition of causation, discussed in Granger (1980) and elsewhere, has been widely applied in economics and in other disciplines. For this definition, a series yt is said to cause xt+l if it contains information about the forecastability for xt+l contained nowhere else in some large information set, which includes xt−j, j ≥ 0. However, it would be convenient to think of causality being different in extent or direction at seasonal or low frequencies, say, than at other frequencies. The fact that a stationary series is effectively the (uncountably infinite) sum of uncorrelated components, each of which is associated with a single frequency, or a narrow frequency band, introduces the possibility that the full causal relationship can be decomposed by frequency. This is known as the Wiener decomposition or the spectral decomposition of the series, as discussed by Hannan (1970). For any series generated by , where xt, and are both stationary, with finite variances and a(B) is a backward filterwith B the backward operator, there is a simple, well-known relationship between the spectral decompositions of the two series.
304 citations
TL;DR: In this paper, the estimation and identification of the functional structures of nonlinear econometric systems of the ARCH type was studied. And the nonparametric kernel estimates for the nonlinear functions characterizing the systems were established strong consistency along with sharp rates of convergence under mild regularity conditions.
Abstract: We consider the estimation and identification of the functional structures of nonlinear econometric systems of the ARCH type. We employ nonparametric kernel estimates for the nonlinear functions characterizing the systems, and we establish strong consistency along with sharp rates of convergence under mild regularity conditions. We also prove the asymptotic normality of the estimates.
285 citations
TL;DR: In this paper, a number of consistency results for nonparametric kernel estimators of density and regression functions and their derivatives are presented, which allow for near-epoch dependent, nonidentically distributed random variables, data-dependent bandwidth sequences, preliminary estimation of parameters, and non-parametric regression on index functions.
Abstract: This paper presents a number of consistency results for nonparametric kernel estimators of density and regression functions and their derivatives. These results are particularly useful in semiparametric estimation and testing problems that rely on preliminary nonparametric estimators, as in Andrews (1994, Econometrica 62, 43–72). The results allow for near-epoch dependent, nonidentically distributed random variables, data-dependent bandwidth sequences, preliminary estimation of parameters (e.g., nonparametric regression based on residuals), and nonparametric regression on index functions.
235 citations
TL;DR: In this article, the authors examined a suitably modified version of the unit root test proposed by Schmidt and Phillips (1992), and showed that a one-time structural break in the intercept does not affect its asymptotic distribution under the null hypothesis, and this is true whether the break is allowed for in the model or not.
Abstract: In this paper, we examine a suitably modified version of the unit root test proposed by Schmidt and Phillips (1992). A one-time structural break in the intercept does not affect its asymptotic distribution under the null hypothesis, and this is true whether the break is allowed for in the model or not. This implies that the asymptotic validity of this test statistic under the null is not affected by the incorrect placement of the structural break, by the allowance for a break when there is no break, or by no allowance for a break when there is a break.
222 citations
TL;DR: In this article, the authors show that for both deterministic and random regressors, the bootstrap distribution converges weakly to the limit distribution of the quantile regression estimator in probability.
Abstract: The asymptotic variance matrix of the quantile regression estimator depends on the density of the error. For both deterministic and random regressors, the bootstrap distribution is shown to converge weakly to the limit distribution of the quantile regression estimator in probability. Thus, the confidence intervals constructed by the bootstrap percentile method have asymptotically correct coverage probabilities.
208 citations
TL;DR: In this article, the authors review a very few results on some basic elements of large sample theory in a restricted structural framework, as described in detail in the recent book by LeCam and Yang (1990, Asymptotics in Statistics: Some Basic Concepts).
Abstract: The primary purpose of this paper is to review a very few results on some basic elements of large sample theory in a restricted structural framework, as described in detail in the recent book by LeCam and Yang (1990, Asymptotics in Statistics: Some Basic Concepts. New York: Springer), and to illustrate how the asymptotic inference problems associated with a wide variety of time series regression models fit into such a structural framework. The models illustrated include many linear time series models, including cointegrated models and autoregressive models with unit roots that are of wide current interest. The general treatment also includes nonlinear models, including what have become known as ARCH models. The possibility of replacing the density of the error variables of such models by an estimate of it (adaptive estimation) based on the observations is also considered.Under the framework in which the asymptotic problems are treated, only the approximating structure of the likelihood ratios of the observations, together with auxiliary estimates of the parameters, will be required. Such approximating structures are available under quite general assumptions, such as that the Fisher information of the common density of the error variables is finite and nonsingular, and the more specific assumptions, such as Gaussianity, are not required. In addition, the construction and the form of inference procedures will not involve any additional complications in the non-Gaussian situations because the approximating quadratic structure actually will reduce the problems to the situations similar to those involved in the Gaussian cases.
206 citations
TL;DR: This paper developed tests for cointegration that can be applied when some of the cointegrating vectors are prespecified under the null or under the alternative hypotheses, which correspond to the standard Wald tests for the inclusion of error correction terms in the VAR.
Abstract: Many economic models imply that ratios, simple differences, or “spreads” of variables are I(0). In these models, cointegrating vectors are composed of 1's, 0's, and —1's and contain no unknown parameters. In this paper, we develop tests for cointegration that can be applied when some of the cointegrating vectors are prespecified under the null or under the alternative hypotheses. These tests are constructed in a vector error correction model and are motivated as Wald tests from a Gaussian version of the model. When all of the cointegrating vectors are prespecified under the alternative, the tests correspond to the standard Wald tests for the inclusion of error correction terms in the VAR. Modifications of this basic test are developed when a subset of the cointegrating vectors contain unknown parameters. The asymptotic null distributions of the statistics are derived, critical values are determined, and the local power properties of the test are studied. Finally, the test is applied to data on foreign exchange future and spot prices to test the stability of the forward–spot premium.
TL;DR: In this paper, the authors investigate the finite sample properties of likelihood ratio tests for cointegrating ranks and conclude that 100 observations are not sufficient to ensure reasonably good performance uniformly over the values of the nuisance parameters that affect the distributions of the test statistics.
Abstract: This paper investigates through Monte Carlo simulation the finite sample properties of likelihood ratio tests for cointegrating ranks that were proposed by Johansen (1991, Econometrica 59, 1551-1580). We transform the model into a canonical form so that the experiment is well controlled without loss of generality and then conduct a comprehensive simulation study. As expected, the test performance is very sensitive to the value of the stationary root(s) of the process. We also find that the test performance depends crucially on the correlation between the innovations that drive the stationary and the nonstationary components of the process. We conclude that 100 ob.servations are not sufficient to ensure reasonably good performance uniformly over the values of the nuisance parameters that affect the distributions of the test statistics.
TL;DR: In this article, the authors considered unit root tests based on M estimators and developed an asymptotic theory for these tests, which is shown how the distributions of the tests depend on nuisance parameters and how tests can be constructed that are invariant to these parameters.
Abstract: This paper considers unit root tests based on M estimators. The asymptotic theory for these tests is developed. It is shown how the asymptotic distributions of the tests depend on nuisance parameters and how tests can be constructed that are invariant to these parameters. It is also shown that a particular linear combination of a unit root test based on the ordinary least-squares (OLS) estimator and on an M estimator converges to a normal random variate. The interpretation of this result is discussed. A simulation experiment is described, illustrating the level and power of different unit root tests for several sample sizes and data generating processes. The tests based on M estimators turn out to be more powerful than the OLS-based tests if the innovations are fat-tailed.
TL;DR: In this article, the authors investigate a bias in an asymptotic expansion of the simulated maximum likelihood estimator introduced by Lerman and Manski (pp. 305-319 in C. Manski and D. McFadden (eds.), Structural Analysis of Discrete Data with Econometric Applications, Cambridge: MIT Press, 1981) for the estimation of discrete choice models.
Abstract: In this article, we investigate a bias in an asymptotic expansion of the simulated maximum likelihood estimator introduced by Lerman and Manski (pp. 305–319 in C. Manski and D. McFadden (eds.), Structural Analysis of Discrete Data with Econometric Applications, Cambridge: MIT Press, 1981) for the estimation of discrete choice models. This bias occurs due to the nonlinearity of the derivatives of the log likelihood function and the statistically independent simulation errors of the choice probabilities across observations. This bias can be the dominating bias in an asymptotic expansion of the simulated maximum likelihood estimator when the number of simulated random variables per observation does not increase at least as fast as the sample size. The properly normalized simulated maximum likelihood estimator even has an asymptotic bias in its limiting distribution if the number of simulated random variables increases only as fast as the square root of the sample size. A bias-adjustment is introduced that can reduce the bias. Some Monte Carlo experiments have demonstrated the usefulness of the bias-adjustment procedure.
TL;DR: In this paper, the authors consider the effect of model selection on prediction regions and show that the use of asymptotic results for the construction of prediction regions requires the same sort of care as use of such results for constructing confidence regions for the parameters of interest, and that a great deal of care must be exercised in any attempt at such an application.
Abstract: Potscher (1991, Econometric Theory7, 163–181) has recently considered the question of how the use of a model selection procedure affects the asymptotic distribution of parameter estimators and related statistics. An important potential application of such results is to the generation of confidence regions for the parameters of interest. It is demonstrated that a great deal of care must be exercised in any attempt at such an application. We also consider the effect of model selection on prediction regions. It is demonstrated that the use of asymptotic results for the construction of prediction regions requires the same sort of care as the use of such results for the construction of confidence regions.
TL;DR: In this article, the asymptotic theory for least absolute deviation estimation of a shift in linear regressions is developed and rates of convergence for the estimated regression parameters and the estimated shift point are derived.
Abstract: This paper develops the asymptotic theory for least absolute deviation estimation of a shift in linear regressions. Rates of convergence and asymptotic distributions for the estimated regression parameters and the estimated shift point are derived. The asymptotic theory is developed both for fixed magnitude of shift and for shift with magnitude converging to zero as the sample size increases. Asymptotic distributions are also obtained for trending regressors and for dependent disturbances. The analysis is carried out in the framework of partial structural change, allowing some parameters not to be influenced by the shift. Efficiency relative to least-squares estimation is also discussed. Monte Carlo analysis is performed to assess how informative the asymptotic distributions are.
TL;DR: In this paper, a new class of tests for parameter stability, the moving-estimates (ME) test, is proposed, and it is shown that in the standard situation the ME test asymptotically equivalent to the maximal likelihood ratio test under the alternative of a temporary parameter shift.
Abstract: In this paper a new class of tests for parameter stability, the moving-estimates (ME) test, is proposed. It is shown that in the standard situation the ME test asymptotically equivalent to the maximal likelihood ratio test under the alternative of a temporary parameter shift. It is also shown that the asymptotic null distribution of the ME test is determined by the increments of a vector Brownian bridge and that under a broad class of alternatives the ME test is consistent and has nontrivial local power in general. Our simulations also demonstrate that the proposed test has power superior to other competing tests when parameters are temporarily instable.
TL;DR: In this article, the authors developed a complete analytical study of the Studentized t ratio and derived closed forms (which are free of integrals) and a simple summation-free integral are derived for the exact limiting density and distribution functions of the t ratio.
Abstract: An encompassing formula to calculate density and distribution functions for unit root statistics was given in Abadir (1992, Oxford Bulletin of Economics and Statistics 54, 305-323) and was applied there to computing the exact limiting density and distribution of the Studentized t ratio. That formula was not a closed form, and it included a double sum and an integral. The purpose of the present paper is to develop a complete analytical study of the t ratio. To do so, closed forms (which are free of integrals) and a simple summation-free integral are derived for the exact limiting density and distribution functions of the t ratio. The density of t being asymmetric, different forms emerge for different signs of t. The forms have in common the use of confluent hypergeometric functions such as the incomplete gamma and the parabolic cylinder functions, thus departing from the simple standard Normal distribution that arises in the cases of a stable or an explosive root. It is also shown that, depending on the sign and value of t; the density and distribution may be well approximated by different proportions of the standard Normal. The shifted Normal distribution is also considered as an approximation. Numerical results available from previous studies are extended and refined using the new formulae, whose relative merits are then analyzed. These merits include analytical features of the distribution that other methods could not uncover. The paper also uses these features for an analytical comparison of the densities of the t ratio and the normalized autocorrelation coefficient.
TL;DR: In this article, a robust statistical approach to nonstationary time series regression and inference is presented, which allows for endogeneities in non-stationary regressors and serial dependence in the shocks that drive the regressors, and the errors that appear in the equation being estimated.
Abstract: This paper provides a robust statistical approach to nonstationary time series regression and inference. Fully modified extensions of traditional robust statistical procedures are developed that allow for endogeneities in the nonstationary regressors and serial dependence in the shocks that drive the regressors and the errors that appear in the equation being estimated. The suggested estimators involve semiparametric corrections to accommodate these possibilities, and they belong to the same family as the fully modified least-squares (FM-OLS) estimator of Phillips and Hansen (1990, Review of Economic Studies 57, 99-125). Specific attention is given to fully modified least absolute deviation (FM-LAD) estimation and fully modified M (FM-M) estimation. The criterion function for LAD and some M-estimators is not always smooth, and this paper develops generalized function methods to cope with this difficulty in the asymptotics. The results given here include a strong law of large numbers and some weak convergence theory for partial sums of generalized functions of random variables. The limit distribution theory for FM-LAD and FM-M estimators that is developed includes the case of finite variance errors and the case of heavytailed (infinite variance) errors. Some simulations and a brief empirical illustration are reported.
TL;DR: An asymptotic theory of the tests of non-nested hypotheses in the stationary dynamic case is got and various test procedures of the encompassing hypothesis are proposed.
Abstract: We define, in a dynamic framework, the notions of binding functions, images, reflecting sets, indirect identification, indirect information, and encompassing. We study the properties of the notion of encompassing when the true distribution does not necessarily belong to one of the two competing models of interest. In this context we propose various test procedures of the encompassing hypothesis. Some of these procedures are based on simulations, and some of them are linked with the notion of indirect estimation (in particular, the GET and simulated GET procedures). As a by-product, we get an asymptotic theory of the tests of non-nested hypotheses in the stationary dynamic case.
TL;DR: In this article, a simple regression model with integrated and stationary regressors and nonlinearities in parameters is considered, and the consistency and order of consistency of the long-run parameter estimator are obtained by employing extensions of well-known sufficient conditions for consistency.
Abstract: Problems with the asymptotic theory of nonlinear maximum likelihood estimation in integrated and cointegrated systems are discussed in this paper. One problem is that standard proofs of consistency generally do not apply; another one is that, even if the consistency has been established, it can be difficult to deduce the limiting distribution of a maximum likelihood estimator from a conventional Taylor series expansion of the score vector. It is argued in this paper that the latter difficulty can generally be resolved if, in addition to consistency, an appropriate result of the order of consistency of the long-run parameter estimator of the model is available and the standardized sample information matrix satisfies a suitable extension of previous stochastic equicontinuity conditions. To make this idea applicable in particular cases, extensions of the author's recent stochastic equicontinuity results, relevant for many integrated and cointegrated systems with nonlinearities in parameters, are provided. As an illustration, a simple regression model with integrated and stationary regressors and nonlinearities in parameters is considered. In this model, the consistency and order of consistency of the long-run parameter estimator are obtained by employing extensions of well-known sufficient conditions for consistency. These conditions are applicable quite generally, and their verification in the special case of this paper suggests how to proceed in more complex models.
TL;DR: In this paper, Choi and Ahn's (1993, Testing the Null of Stationarity for Multiple Time Series, working paper, The Ohio State University) multivariate tests for the null of stationarity and use Park's (1992, Econometrica 60, 119-143) canonical cointegrating regression residuals to make the tests free of nuisance parameters in the limit.
Abstract: This paper introduces various consistent tests for the null of cointegration against the alternative of noncointegration that can be applied to a system of equations as well as to a single equation. The tests are analogs of Choi and Ahn's (1993, Testing the Null of Stationarity for Multiple Time Series, working paper, The Ohio State University) multivariate tests for the null of stationarity and use Park's (1992, Econometrica 60, 119–143) canonical cointegrating regression (CCR) residuals to make the tests free of nuisance parameters in the limit. The asymptotic distributions of the tests are complex but expressed in unified manner by using standard vector Brownian motion. These distributions are tabulated by simulation for some practical cases. Furthermore, the rates of divergence of the tests are reported. Because there are methods for estimating cointegrating matrices other than CCR, it is illustrated for a model without time trends that the tests we introduce work exactly the same way in the limit when Phillips and Hansen's (1990, Review of Economic Studies 57, 99–125) fully modified ordinary least-squares (OLS) procedure is used. Also, is shown that difficulties arise when OLS residuals are used to formulate the tests. Small-scale simulation results are reported to examine the finite sample performance of the tests. The tests are shown to work reasonably wellin finite samples. In particular, it is illustrated that using the multivariate tests introduced in this paper can be a better testing strategy in terms of the finite sample size and power than applying univariate tests several times to each equation in a system of equations.
TL;DR: In this article, weak and strong laws of large numbers for weakly dependent heterogeneous random variables are presented. But the main feature of the strong law of large number for mixingale sequences is the less strict decay rate that is imposed on the mixingale numbers as compared to previous results.
Abstract: This paper provides weak and strong laws of large numbers for weakly dependent heterogeneous random variables. The weak laws of large numbers presented extend known results to the case of trended random variables. The main feature of our strong law of large numbers for mixingale sequences is the less strict decay rate that is imposed on the mixingale numbers as compared to previous results.
TL;DR: In this article, the properties of systems likelihood procedures for cointegrated systems when the I(2) variables are present are considered, and two alternative methods are proposed: one based on the full system likelihood, whereas another is based on subsystem likelihood.
Abstract: This paper considers the properties of systems likelihood procedures for cointegrated systems when the I(2) variables are present. Two alternative methods are proposed: one is based on the full system likelihood, whereas another is based on the subsystem likelihood. By eliminating all unit roots in the system by the use of prior information concerning the presence of unit roots, these procedures yield estimates whose asymptotic distributions are mixed normal, free from nuisance parameters, and median-unbiased. Both methods are extensions of a full system maximum likelihood procedure by Phillips (1991a) to I(2) models. Three cases of cointegration with I(2) variables are considered in order to cover a wide variety of cointegration relationships. A triangular ECM representation and the two ML estimates are derived for each case, and the asymptotics are discussed as well. The asymptotic efficiency concerning the two estimates are considered.
TL;DR: In this paper, Econometric Theory is used as an alternative approach for economic forecasting in the context of economic forecasting, and the authors present a method based on the SFI-PB-ARTICLE-1995-003.
Abstract: Keywords: Econometric Theory Note: Times Cited: 7 Reference SFI-PB-ARTICLE-1995-003 URL: ://A1995QL93000007 Record created on 2008-03-12, modified on 2017-05-12
TL;DR: In this article, the authors studied the path solutions of a multivariate rational expectations model and derived the dimension of the set of solutions in terms of martingale differences and the dimensions of linear stationary solutions when restricting themselves to the linear case.
Abstract: The aim of this paper is the study of the path solutions of a multivariate rational expectations model. We describe several procedures for solving such dynamic systems based on either the adjoint operator method or the Smith form. As a by-product, we derive the dimension of the set of solutions in terms of martingale differences and the dimension of the set of linear stationary solutions when we restrict ourselves to the linear case. These dimensions are functions of the number of equations in the system, of the maximum lead, and of the orders of some eigenvalues of the characteristic equation associated with the system.
TL;DR: In this paper, asymptotic distributions of some test statistics in near-integrated AR processes are studied and exact formulas for the distribution functions are given as well as approximative results obtained by saddlepoint approximation techniques.
Abstract: Asymptotic distributions of some test statistics in near-integrated AR processes are studied. Some exact formulas for the distribution functions are given as well as approximative results obtained by saddlepoint approximation techniques.
TL;DR: In this article, the unknown structural parameters of a continuous/discrete state space model are estimated by maximum likelihood in the presence of irregular sampling, missing values, and cross-sections of time series (panel data).
Abstract: The unknown structural parameters of a continuous/discrete state space model are estimated by maximum likelihood in the presence of irregular sampling, missing values, and cross-sections of time series (panel data). Exogenous (control) variables are included, and the sampling scheme and missing data pattern can be different for each variable and system. Furthermore, the derived non-linear optimization algorithm with analytical score function can be used for the discrete time case as well.
TL;DR: In this article, a nonparametric conditional moment test of stability of an econometric model against the alternative of instability is proposed, which allows for more than one structural change, although in this case it has to be fairly smooth.
Abstract: This paper considers a nonparametric conditional moment test of stability of an econometric model against the alternative of instability. The alternative hypothesis allows for more than one structural change, although in this case it has to be fairly smooth. This complements existing results for stability in a parametric setting. Also, it is shown that the test is always consistent, unlike the available “parametric” tests, which normally rely on the assumption of a correct specification of the model, at least under the null hypothesis of no structural instability. Moreover, we show that the test has local power comparable to the parametric ones; that is, its asymptotic efficiency is greater than zero. A Monte Carlo experiment about the performance of our test is described.