Nonparametric cointegration analysis of fractional systems with unknown integration orders

In this paper a nonparametric variance ratio testing approach is proposed for determining the cointegration rank in fractionally integrated systems. The test statistic is easily calculated without prior knowledge of the integration order of the data, the strength of the cointegrating relations, or the cointegration vector(s). The latter property makes it easier to implement than regression-based approaches, especially when examining relationships between several variables with possibly multiple cointegrating vectors. Since the test is nonparametric, it does not require the specification of a particular model and is invariant to short-run dynamics. Nor does it require the choice of any smoothing parameters that change the test statistic without being reflected in the asymptotic distribution. Furthermore, a consistent estimator of the cointegration space can be obtained from the procedure. The asymptotic distribution theory for the proposed test is non-standard but easily tabulated or simulated. Monte Carlo simulations demonstrate excellent finite sample properties, even rivaling those of well-specified parametric tests. The proposed methodology is applied to the term structure of interest rates, where, contrary to both fractional- and integer-based parametric approaches, evidence in favor of the expectations hypothesis is found using the nonparametric approach.     

Studentization and prediction in a multivariate normal setting

In a simple multivariate normal prediction setting, derivation of a predictive distribution can flow from formal Bayes arguments as well as pivoting arguments. We look at two special cases and show that the classical invariant predictive distribution is based on a pivot whose sampling distribution depends on the parameter – that is, the pivot is not an ancillary statistic. In contrast, a predictive distribution derived by a structural argument is based on a pivot with a parameter free distribution (an ancillary statistic). The classical procedure is formal Bayes for the Jeffreys prior. Our results show that this procedure does not have a structural or fiducial interpretation.     

Bayesian analysis of the error correction model

This paper presents a method for estimating the posterior probability density of the cointegrating rank of a multivariate error correction model. A second contribution is the careful elicitation of the prior for the cointegrating vectors derived from a prior on the cointegrating space. This prior obtains naturally from treating the cointegrating space as the parameter of interest in inference and overcomes problems previously encountered in Bayesian cointegration analysis. Using this new prior and Laplace approximation, an estimator for the posterior probability of the rank is given. The approach performs well compared with information criteria in Monte Carlo experiments.     

Distribution approximations for cointegration tests with stationary exogenous regressors

The distribution of a functional of two correlated vector‐Brownian motions is approximated by a Gamma distribution. This functional represents the limiting distribution for cointegration tests with stationary exogenous regressors, but also for cointegration tests based on a non‐Gaussian likelihood. The approximation is accurate, fast and easy to use in comparison with both tabulated critical values and simulated p‐values. The proposed procedure is applied to a UK model investigating purchasing power parity. Copyright © 2005 John Wiley & Sons, Ltd.     

Applied cointegration analysis in the mirror of macroeconomic theory

Cointegration analyses of macroeconomic time series are often not based on fully specified theoretical models. We use a theoretical model to scrutinize common procedures in applied cointegration analysis. Monte Carlo experiments show that (1) some tests of the cointegration vectors do not work well on series generated by an equilibrium business cycle model; (2) cointegration restrictions add little to forecasting; (3) structural VAR models based on weak long-run restrictions seem to work well. The main disadvantages of cointegration analysis without strong links to economic theory are that it makes it hard to estimate and interpret the cointegration vectors.     

Bayesian point estimation of the cointegration space

A neglected aspect of the otherwise fairly well developed Bayesian analysis of cointegration is point estimation of the cointegration space. It is pointed out here that, due to the well known non-identification of the cointegration vectors, the parameter space is not Euclidean and the loss functions underlying the conventional Bayes estimators are therefore questionable. We present a Bayes estimator of the cointegration space which takes the curved geometry of the parameter space into account. This estimate has the interpretation of being the posterior mean cointegration space and is invariant to the order of the time series, a property not shared with many of the Bayes estimators in the cointegration literature. An overall measure of cointegration space uncertainty is also proposed. Australian interest rate data are used for illustration. A small simulation study shows that the new Bayes estimator compares favorably to the maximum likelihood estimator.     

Near unit roots,cointegration, and the term structure of interest rates

The term structure of interest rates is often modelled as a cointegrated system with the yield spreads forming the cointegrating vectors. Testing whether the yield spreads span the cointegration space is problematic because conventional tests on the cointegration vectors tend to overreject when the largest autoregressive roots deviate from unity, as is likely to be the case with interest rates. A new test that is robust w.r.t. deviations from the exact unit root assumption is developed and applied to monthly US interest rate data from 1952:1–1991:2. Taking into account the regime shift in 1979, the hypothesis of the yield spreads being the cointegrating vectors cannot be rejected using the robust test. Copyright © 2000 John Wiley & Sons, Ltd.     

Testing for cointegration using partially linear models

A partially linear model of cointegration is developed where stationary covariates enter nonparametrically. We propose tests for cointegration using singular values of the estimated autoregressive matrix. The tests are based on eigenvalues of standardized matrices and are relatively simple to compute. Asymptotic theory of the proposed test is developed. It is shown that the limiting distribution of the proposed test is similar to that of several tests in the recent literature. A Gamma approximation of the distribution is discussed to facilitate inference. Finite sample properties of the proposed procedure are illustrated in some limited Monte Carlo experiments. An empirical application to US macroeconomic time series is conducted to highlight the approach.     

A Wald test for the cointegration rank in nonstationary fractional systems

This paper develops new methods for determining the cointegration rank in a nonstationary fractionally integrated system, extending univariate optimal methods for testing the degree of integration. We propose a simple Wald test based on the singular value decomposition of the unrestricted estimate of the long run multiplier matrix. When the “strength” of the cointegrating relationship is less than 1/2, the test statistic has a standard asymptotic distribution, like Lagrange Multiplier tests exploiting local properties. We consider the behavior of our test under estimation of short run parameters and local alternatives. We compare our procedure with other cointegration tests based on different principles and find that the new method has better properties in a range of situations by using information on the alternative obtained through a preliminary estimate of the cointegration strength.     

Likelihood based testing for no fractional cointegration

We consider two likelihood ratio tests, the so-called maximum eigenvalue and trace tests, for the null of no cointegration when fractional cointegration is allowed under the alternative, which is a first step to generalize the so-called Johansen’s procedure to the fractional cointegration case. The standard cointegration analysis only considers the assumption that deviations from equilibrium can be integrated of order zero, which is very restrictive in many cases and may imply an important loss of power in the fractional case. We consider the alternative hypotheses with equilibrium deviations that can be mean reverting with order of integration possibly greater than zero. Moreover, the degree of fractional cointegration is not assumed to be known, and the asymptotic null distribution of both tests is found when considering an interval of possible values. The power of the proposed tests under fractional alternatives and size accuracy provided by the asymptotic distribution in finite samples are investigated.     

The price-wage inflationary spiral in Poland

In this paper the problem of price-wage relationship modelling in the case of a mixed economy is addressed. The empirical investigation was based on Polish annual data for the period of a centrally planned system (1964–1989) and on quarterly data for the period of transition towards a market economy (1990.1–1990.3). The traditional approach proved to be inappropriate because of the variables' nonstationarity. Identification of long-run behaviour was attempted by applying the two-step Engle-Granger's, or alternatively, Johansen's maximum likelihood (ML) procedures. The ML estimator provided better estimates of cointegration vectors and, even more important, allowed as many as three to be found.The main conclusion which can be drawn from the empirical findings is that three variables: price index, average wages and labour productivity, form a multi-dimensional equilibrium space. This property of the described phenomena needs to be taken into serious account when building macroeconometric models explaining the behaviour of the Polish economy.The existence of these three cointegration vectors is troublesome because of unusual problems of interpretation. However, if it is not as a result of misspecification and/or small sample bias, it proves that much remains to be learned about the price-wage mechanisms functioning in economies having a mixed character.     

Analysis of cointegrated VARMA processes

VARMA (vector autoregressive moving average) processes are proposed for modelling cointegrated variables. For this purpose the echelon form is combined with the error correction form. Procedures for estimating the Kronecker indices which characterize the echelon form and for specifying the cointegration rank are discussed. The asymptotic distribution of the coefficient estimators is given. An example based o n US macroeconomic data illustrates the procedure and demonstrates its feasibility in practice.     

Bootstrapping cointegrating regressions

In this paper, we consider bootstrapping cointegrating regressions. It is shown that the method of bootstrap, if properly implemented, generally yields consistent estimators and test statistics for cointegrating regressions. For the cointegrating regression models driven by general linear processes, we employ the sieve bootstrap based on the approximated finite-order vector autoregressions for the regression errors and the first differences of the regressors. In particular, we establish the bootstrap consistency for OLS method. The bootstrap method can thus be used to correct for the finite sample bias of the OLS estimator and to approximate the asymptotic critical values of the OLS-based test statistics in general cointegrating regressions. The bootstrap OLS procedure, however, is not efficient. For the efficient estimation and hypothesis testing, we consider the procedure proposed by Saikkonen [1991. Asymptotically efficient estimation of cointegration regressions. Econometric Theory 7, 1–21] and Stock and Watson [1993. A simple estimator of cointegrating vectors in higher order integrating systems. Econometrica 61, 783–820] relying on the regression augmented with the leads and lags of differenced regressors. The bootstrap versions of their procedures are shown to be consistent, and can be used to do asymptotically valid inferences. A Monte Carlo study is conducted to investigate the finite sample performances of the proposed bootstrap methods.     

Testing the Null of Cointegration with Structural Breaks*

We propose a Lagrange Multiplier‐type statistic to test the null hypothesis of cointegration allowing for the possibility of a structural break, in both the deterministic and the cointegration vectors. Our proposal focuses on the presence of endogenous regressors. The test complements the usual non‐cointegration tests so as to obtain stronger evidence of cointegration. We consider the cases of known and unknown dates of the break. In the latter case, we show that minimizing the Sum of Squared Residuals results in a super‐consistent estimator of the break fraction. Finally, the behaviour of the tests is studied through Monte Carlo experiments.     

Singular value decomposition in cointegration analysis: a note regarding the difference stationary series

Many methods and techniques have been developed gradually to compute cointegration vectors. We present here a comparatively simple method for computing the matrix of cointegrating vectors, by applying singular value decomposition. With this method, one can easily accommodate in the cointegrating vectors any deterministic factors, such as a dummy, apart from the constant term and the trend. Besides the errors corresponding to the finally selected cointegrating vector have the property of minimum variance. It is noted that this procedure is not mentioned in the relevant literature. Additionally, a sequential technique is introduced, for determining the order of integration for a given series. With this procedure one can directly detect whether the differencing process produces a stationary series or not, since it seems to be a common belief that differencing a variable (one or more times) we will always get a stationary series [Harris, R.: Using Cointegration Analysis in Econometric Modelling. Prentice Hall, London (1995)]. It will be seen that this is not necessarily the case.     

A semi-parametric Bayesian approach to the instrumental variable problem

We develop a Bayesian semi-parametric approach to the instrumental variable problem. We assume linear structural and reduced form equations, but model the error distributions non-parametrically. A Dirichlet process prior is used for the joint distribution of structural and instrumental variable equations errors. Our implementation of the Dirichlet process prior uses a normal distribution as a base model. It can therefore be interpreted as modeling the unknown joint distribution with a mixture of normal distributions with a variable number of mixture components. We demonstrate that this procedure is both feasible and sensible using actual and simulated data. Sampling experiments compare inferences from the non-parametric Bayesian procedure with those based on procedures from the recent literature on weak instrument asymptotics. When errors are non-normal, our procedure is more efficient than standard Bayesian or classical methods.     

Bootstrap testing for the null of no cointegration in a threshold vector error correction model

We develop a test for the linear no cointegration null hypothesis in a threshold vector error correction model. We adopt a sup-Wald type test and derive its null asymptotic distribution. A residual-based bootstrap is proposed, and the first-order consistency of the bootstrap is established. A set of Monte Carlo simulations shows that the bootstrap corrects size distortion of asymptotic distribution in finite samples, and that its power against the threshold cointegration alternative is significantly greater than that of conventional cointegration tests. Our method is illustrated with used car price indexes.     

Implications of ERM2 for Poland's monetary policy

We propose an extension to the inflation targeting regime currently pursued by Poland. It incorporates the exchange rate stability constraints as imposed by the obligatory participation in the ERM2 that Poland needs to satisfy prior to adopting the euro. The modified policy is based on the forward-looking inflation targeting supplemented with the exchange rate stability objective. Its effective implementation depends on the determined long-term equilibrium exchange rate and the observed degree of exchange rate volatility. Both are empirically estimated by employing the Johansen cointegration tests and the threshold generalized autoregressive heteroscedasticity model with the in-mean extension and generalized error distribution (TGARCH-M-GED).     

A nonparametric approach to assess undergraduate performance

Nonparametric methodologies are proposed to assess college students' performance. Emphasis is given to gender and sector of high school. The application concerns the University of Campinas, a research university in Southeast Brazil. In Brazil college studies are based on a somewhat rigid set of subjects for each major. For this reason a simple GPA comparison may hide true performance. Therefore, we define individual vectors of course grades. These vectors are used in pairwise comparisons of common subject grades for individuals who entered college in the same year. The relative college performances of any two students are compared with their relative performances on the entrance exam score. A procedure based on generalized U-statistics is developed to test if there is selection bias in the entrance exam by some predefined groups, which is equipped with asymptotically normal distribution under both null and alternative hypotheses. Maximum power is attained by employing the union intersection principle, and resampling techniques such as nonparametric bootstrap are employed to generate the empirical distribution of the test statistics and get p-values.     

Model-based asymptotic inference on the effect of infrequent large shocks on cointegrated variables

Quasi-maximum-likelihood (QML) estimation of a model combining cointegration in the conditional mean and rare large shocks (outliers) with a factor structure in the innovations is studied. The goal is not only to robustify inference on the conditional-mean parameters, but also to find regularities and conduct inference on the instantaneous and long-run effect of the large shocks. Given the cointegration rank and the factor order, χ²${\chi }^2$ asymptotic inference is obtained for the cointegration vectors, the short-run parameters, and the direction of each column of both the factor loading matrix and the matrix of long-run impacts of the large shocks. Large shocks, whose location is assumed unknown a priori, can be detected and classified consistently into the factor components.