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Statistical Analysis of Cointegrated Vectors
 Journal of Economic Dynamics and Control
, 1988
"... We consider a nonstationary vector autoregressive process which is integrated of order 1, and generated by i.i.d. Gaussian errors. We then derive the maximum likelihood estimator of the space of cointegration vectors and the likelihood ratio test of the hypothesis that it has a given number of dimen ..."
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Cited by 2576 (12 self)
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We consider a nonstationary vector autoregressive process which is integrated of order 1, and generated by i.i.d. Gaussian errors. We then derive the maximum likelihood estimator of the space of cointegration vectors and the likelihood ratio test of the hypothesis that it has a given number of dimensions. Further we test linear hypotheses about the cointegration vectors. The asymptotic distribution of these test statistics are found and the first is described by a natural multivariate version of the usual test for unit root in an autoregressive process, and the other is a x2 test. 1.
A new approach to the economic analysis of nonstationary time series and the business cycle
 ECONOMETRICA
, 1989
"... ..."
A Simple Estimator of Cointegrating Vectors in Higher Order Cointegrated Systems
 ECONOMETRICA
, 1993
"... Efficient estimators of cointegrating vectors are presented for systems involving deterministic components and variables of differing, higher orders of integration. The estimators are computed using GLS or OLS, and Wald Statistics constructed from these estimators have asymptotic x2 distributions. T ..."
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Cited by 507 (3 self)
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Efficient estimators of cointegrating vectors are presented for systems involving deterministic components and variables of differing, higher orders of integration. The estimators are computed using GLS or OLS, and Wald Statistics constructed from these estimators have asymptotic x2 distributions. These and previously proposed estimators of cointegrating vectors are used to study longrun U.S. money (Ml) demand. Ml demand is found to be stable over 19001989; the 95 % confidence intervals for the income elasticity and interest rate semielasticity are (.88,1.06) and (.13,.08), respectively. Estimates based on the postwar data alone, however, are unstable, with variances which indicate substantial sampling uncertainty.
Critical values for cointegration tests
 Eds.), LongRun Economic Relationship: Readings in Cointegration
, 1991
"... This paper provides tables of critical values for some popular tests of cointegration and unit roots. Although these tables are necessarily based on computer simulations, they are much more accurate than those previously available. The results of the simulation experiments are summarized by means of ..."
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Cited by 475 (3 self)
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This paper provides tables of critical values for some popular tests of cointegration and unit roots. Although these tables are necessarily based on computer simulations, they are much more accurate than those previously available. The results of the simulation experiments are summarized by means of response surface regressions in which critical values depend on the sample size. From these regressions, asymptotic critical values can be read off directly, and critical values for any finite sample size can easily be computed with a hand calculator. Added in 2010 version: A new appendix contains additional results that are more accurate and cover more cases than the ones in the original paper.
Testing for Common Trends
 Journal of the American Statistical Association
, 1988
"... Cointegrated multiple time series share at least one common trend. Two tests are developed for the number of common stochastic trends (i.e., for the order of cointegration) in a multiple time series with and without drift. Both tests involve the roots of the ordinary least squares coefficient matrix ..."
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Cited by 455 (7 self)
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Cointegrated multiple time series share at least one common trend. Two tests are developed for the number of common stochastic trends (i.e., for the order of cointegration) in a multiple time series with and without drift. Both tests involve the roots of the ordinary least squares coefficient matrix obtained by regressing the series onto its first lag. Critical values for the tests are tabulated, and their power is examined in a Monte Carlo study. Economic time series are often modeled as having a unit root in their autoregressive representation, or (equivalently) as containing a stochastic trend. But both casual observation and economic theory suggesthat many series might contain the same stochastic trendso that they are cointegrated. If each of n series is integrated of order 1 but can be jointly characterized by k < n stochastic trends, then the vecto representation of these series has k unit roots and n k distinct stationary linear combinations. Our proposed tests can be viewed alternatively as tests of the number of common trends, linearly independent cointegrating vectors, or autoregressive unit roots of the vector process. Both of the proposed tests are asymptotically similar. The firstest (qf) is developed under the assumption that certain components of the process have a finiteorder vector autoregressive (VAR) representation, and the nuisance parameters are handled by estimating this VAR. The second test (q,) entails computing the eigenvalues of a corrected sample firstorder autocorrelation matrix, where the correction is essentially a sum of the autocovariance matrices. Previous researchers have found that U.S. postwar interest rates, taken individually, appear to be integrated of order 1. In addition, the theory of the term structure implies that yields on similar assets of different maturities will be cointegrated. Applying these tests to postwar U.S. data on the federal funds rate and the three and twelvemonth treasury bill rates providesupport for this prediction: The three interest rates appear to be cointegrated.
The forward discount anomaly and the risk premium: A survey of recent evidence
 JOURNAL OF EMPIRICAL FINANCE
, 1996
"... Forward exchange rate unbiasedness is rejected in tests from the current floating exchange rate era. This paper surveys advances in this area since the publication of Hodrick's (1987) survey. It documents that the change in the future exchange rate is generally negatively related to the forward ..."
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Cited by 394 (11 self)
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Forward exchange rate unbiasedness is rejected in tests from the current floating exchange rate era. This paper surveys advances in this area since the publication of Hodrick's (1987) survey. It documents that the change in the future exchange rate is generally negatively related to the forward discount. Properties of the expected forward forecast error are reviewed. Issues such as the relation of uncovered interest parity to real interest parity, and the implications of uncovered interest parity for cointegration of various quantities are discussed. The modeling and testing for risk premiums is surveyed. Included in this area are tests of the consumption CAPM, tests of the latent variable model, and portfoliobalance models of risk premiums. General equilibrium models of the risk premium are examined and their empirical implications explored. The survey does not cover the important areas of learning and peso problems, tests of rational expectations based on survey data, or the models of irrational expectations and speculative bubbles.
Inference in Linear Time Series Models with Some Unit Roots," Econometrica
, 1990
"... This paper considers estimation and hypothesis testing in linear time series models when some or all of the variables have unit roots. Our motivating example is a vector autoregression with some unit roots in the companion matrix, which might include polynomials in time as regressors. In the general ..."
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Cited by 376 (12 self)
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This paper considers estimation and hypothesis testing in linear time series models when some or all of the variables have unit roots. Our motivating example is a vector autoregression with some unit roots in the companion matrix, which might include polynomials in time as regressors. In the general formulation, the variable might be integrated or cointegrated of arbitrary orders, and might have drifts as well. We show that parameters that can be written as coefficients on mean zero, nonintegrated regressors have jointly normal asymptotic distributions, converging at the rate T'/2. In general, the other coefficients (including the coefficients on polynomials in time) will have nonnormal asymptotic distributions. The results provide a formal characterization of which t or F testssuch as Granger causality testswill be asymptotically valid, and which will have nonstandard limiting distributions.
An autoregressive distributed lag modelling approach to cointegration analysis
 Cambridge University
, 1999
"... This paper examines the use of autoregressive distributed lag (ARDL) models for the analysis of longrun relations when the underlying variables are I(1). It shows that after appropriate augmentation of the order of the ARDL model, the OLS estimators of the shortrun parameters are p Tconsistent wi ..."
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Cited by 355 (5 self)
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This paper examines the use of autoregressive distributed lag (ARDL) models for the analysis of longrun relations when the underlying variables are I(1). It shows that after appropriate augmentation of the order of the ARDL model, the OLS estimators of the shortrun parameters are p Tconsistent with the asymptotically singular covariance matrix, and the ARDLbased estimators of the longrun coe¢cients are superconsistent, and valid inferences on the longrun parameters can be made using standard normal asymptotic theory. The paper also examines the relationship between the ARDL procedure and the fully modi…ed OLS approach of Phillips and Hansen to estimation of cointegrating relations, and compares the small sample performance of these two approaches via Monte Carlo experiments. These results provide strong evidence in favour of a rehabilitation of the traditional ARDL approach to time series econometric modelling. The ARDL approach has the additional advantage of yielding consistent estimates of the longrun coe¢cients that are asymptotically normal irrespective of whether the underlying regressors are I(1) or I(0).