Publications - Søren Johansen

Weak convergence to derivatives of fractional Brownian motion

Johansen, S., Nielsen, M. Ø. — Sat, 01 Jan 2022 20:29:36 +0100

Adjustment coefficients and exact rational expectations in cointegrated vector autoregressive models

Johansen, S., Swensen, A. R. — Thu, 01 Jul 2021 20:29:36 +0200

In cointegrated vector autoregressive models exact linear rational expectation relations can imply restrictions on the adjustment parameters. We show how such restrictions can be tested, in particular when the restrictions imply weak exogeneity of some variables.

Cointegration and adjustment in the CVAR(∞) representation of some partially observed CVAR(1) models

Johansen, S. — Fri, 01 Mar 2019 20:29:36 +0100

A multivariate CVAR(1) model for some observed variables and some unobserved variables is analysed using its infinite order CVAR representation of the observations. Cointegration and adjustment coefficients in the infinite order CVAR are found as functions of the parameters in the CVAR(1) model. Conditions for weak exogeneity for the cointegrating vectors in the approximating finite order CVAR are derived. The results are illustrated by two simple examples of relevance for modelling causal graphs.

Data revisions and the statistical relation of global mean sea level and surface temperature

Hillebrand, E., Johansen, S., Schmith, T. — Tue, 01 Dec 2020 20:29:36 +0100

We study the stability of estimated linear statistical relations of global mean temperature and global mean sea level with regard to data revisions. Using four different model specifications proposed in the literature, we compare coefficient estimates and long-term sea level projections using two different vintages of each of the annual time series, covering the periods 1880–2001 and 1880–2013. We find that temperature and sea level updates and revisions have a substantial influence both on the magnitude of the estimated coefficients of influence (differences of up to 50%) and therefore on long-term projections of sea level rise following the RCP4.5 and RCP6 scenarios (differences of up to 40 cm by the year 2100). This shows that in order to replicate earlier results that informed the scientific discussion and motivated policy recommendations, it is crucial to have access to and to work with the data vintages used at the time.

Models where the Least Trimmed Squares and Least Median of Squares estimators are maximum likelihood

Berenguer-Rico, V., Johansen, S., Nielsen, B. — Thu, 19 Sep 2019 20:29:36 +0200

The Least Trimmed Squares (LTS) and Least Median of Squares (LMS) estimators are popular robust regression estimators. The idea behind the estimators is to find, for a given h, a sub-sample of h `good' observations among n observations and estimate the regression on that sub-sample. We find models, based on the normal or the uniform distribution respectively, in which these estimators are maximum likelihood. We provide an asymptotic theory for the location-scale case in those models. The LTS estimator is found to be sqrt(h) consistent and asymptotically standard normal. The LMS estimator is found to be h consistent and asymptotically Laplace.

Uniform Consistency of Marked and Weighted Empirical Distributions of Residuals

Berenguer-Rico, V., Johansen, S., Nielsen, B. — Thu, 13 Jun 2019 20:29:36 +0200

A uniform weak consistency theory is presented for the marked and weighted empirical distribution function of residuals. New and weaker sufficient conditions for uniform consistency are derived. The theory allows for a wide variety of regressors and error distributions. We apply the theory to 1-step Huber-skip estimators. These estimators describe the widespread practice of removing outlying observations from an intial estimation of the model of interest and updating the estimation in a second step by applying least squares to the selected observations. Two results are presented. First, we give new and weaker conditions for consistency of the estimators. Second, we analyze the gauge, which is the rate of false detection of outliers, and which can be used to decide the cut-off in the rule for selecting outliers.

Cointegration between trends and their estimators in state space models and CVAR models

Johansen, S., Tabor, M. N. — Sun, 01 Jan 2017 20:29:36 +0100

The cointegrated vector autoregressive model with general deterministic terms

Johansen, S., Nielsen, M. Ø. — Mon, 01 Jan 2018 20:29:36 +0100

In the cointegrated vector autoregression (CVAR) literature, deterministic terms have until now been analyzed on a case-by-case, or as-needed basis. We give a comprehensive unified treatment of deterministic terms in the additive model X-t = gamma Z(t)+Y-t, where Z(t) belongs to a large class of deterministic regressors and Y-t is a zero-mean CVAR. We suggest an extended model that can be estimated by reduced rank regression, and give a condition for when the additive and extended models are asymptotically equivalent, as well as an algorithm for deriving the additive model parameters from the extended model parameters. We derive asymptotic properties of the maximum likelihood estimators and discuss tests for rank and tests on the deterministic terms. In particular, we give conditions under which the estimators are asymptotically (mixed) Gaussian, such that associated tests are x(2)-distributed. (C) 2017 Elsevier B.V. All rights reserved.

Nonstationary Cointegration in the Fractionally Cointegrated VAR Model

Johansen, S., Nielsen, M. Ø. — Mon, 01 Jul 2019 20:29:36 +0200

We consider the fractional cointegrated vector autoregressive (CVAR) model of Johansen and Nielsen (2012a) and make two distinct contributions. First, in their consistency proof, Johansen and Nielsen (2012a) imposed moment conditions on the errors that depend on the parameter space, such that when the parameter space is larger, stronger moment conditions are required. We show that these moment conditions can be relaxed, and for consistency we require just eight moments regardless of the parameter space. Second, Johansen and Nielsen (2012a) assumed that the cointegrating vectors are stationary, and we extend the analysis to include the possibility that the cointegrating vectors are non-stationary. Both contributions require new analysis and results for the asymptotic properties of the likelihood function of the fractional CVAR model, which we provide. Finally, our analysis follows recent research and applies a parameter space large enough that the usual (non-fractional) CVAR model constitutes an interior point and hence can be tested against the fractional model using a Chi-squared-test.

Boundedness of M-estimators for linear regression in time series

Johansen, S., Nielsen, B. — Sat, 01 Jun 2019 20:29:36 +0200

We show boundedness in probability uniformly in sample size of a general M-estimator for multiple linear regression in time series. The positive criterion function for the M-estimator is assumed lower semicontinuous and sufficiently large for large argument. Particular cases are the Huber-skip and quantile regression. Boundedness requires an assumption on the frequency of small regressors. We show that this is satisfied for a variety of deterministic and stochastic regressors, including stationary and random walks regressors. The results are obtained using a detailed analysis of the condition on the regressors combined with some recent martingale results.

Testing the CVAR in the Fractional CVAR Model

Johansen, S., Nielsen, M. Ø. — Thu, 01 Nov 2018 20:29:36 +0100

We consider the fractional cointegrated vector autoregressive (CVAR) model of Johansen and Nielsen (2012a) and show that the likelihood ratio test statistic for the usual CVAR model is asymptotically chi-squared-distributed. Because the usual CVAR model lies on the boundary of the parameter space for the fractional CVAR in Johansen and Nielsen (2012a), the analysis requires the study of the fractional CVAR model on a slightly larger parameter space so that the CVAR model lies in the interior. This in turn implies some further analysis of the asymptotic properties of the fractional CVAR model.

Testing the CVAR in the fractional CVAR model

Johansen, S., Nielsen, M. Ø. — Tue, 24 Oct 2017 20:29:36 +0200

We consider the fractional cointegrated vector autoregressive (CVAR) model of Johansen and Nielsen (2012a) and show that the test statistic for the ususal CVAR model is asymptotically chi-squared distributed. Because the usual CVAR model lies
on the boundary of the parameter space for the fractional CVAR in Johansen and Nielsen (2012a), the analysis requires the study of the fractional CVAR model on a slightly larger parameter space so that the CVAR model lies in the interior. This in
turn implies some further analysis of the asymptotic properties of the fractional CVAR model.

Improved inference on cointegrating vectors in the presence of a near unit root using adjusted quantiles

Franchi, M., Johansen, S. — Sun, 01 Jan 2017 20:29:36 +0100

It is well known that inference on the cointegrating relations in a vector autoregression (CVAR) is difficult in the presence of a near unit root. The test for a given cointegration vector can have rejection probabilities under the null, which vary from the nominal size to more than 90%. This paper formulates a CVAR model allowing for multiple near unit roots and analyses the asymptotic properties of the Gaussian maximum likelihood estimator. Then two critical value adjustments suggested by McCloskey (2017) for the test on the cointegrating relations are implemented for the model with a single near unit root, and it is found by simulation that they eliminate the serious size distortions, with a reasonable power for moderate values of the near unit root parameter. The findings are illustrated with an analysis of a number of different bivariate DGPs.

The Qualitative Expectations Hypothesis

Frydman, R., Johansen, S., Rahbek, A., Tabor, M. N. — Mon, 26 Jun 2017 20:29:36 +0200

We introduce the Qualitative Expectations Hypothesis (QEH) as a new approach to modeling macroeconomic and financial outcomes. Building on John Muth's seminal insight underpinning the Rational Expectations Hypothesis (REH), QEH represents the market's forecasts to be consistent with the predictions of an economist's model. However, by assuming that outcomes lie within stochastic intervals, QEH, unlike REH, recognizes the ambiguity faced by an economist and market participants alike. Moreover, QEH leaves the model open to ambiguity by not specifying a mechanism determining specific values that outcomes take within these intervals. In order to examine a QEH model's empirical relevance, we formulate and estimate its statistical analog based on simulated data. We show that the proposed statistical model adequately represents an illustrative sample from the QEH model. We also illustrate how estimates of the statistical model's parameters can be used to assess the QEH model's qualitative implications.

Improved inference on cointegrating vectors in the presence of a near unit root using adjusted quantiles

Franchi, M., Johansen, S. — Thu, 27 Apr 2017 20:29:36 +0200

It is well known that inference on the cointegrating relations in a vector autoregression (CVAR) is difficult in the presence of a near unit root. The test for a given cointegration vector can have rejection probabilities under the null, which vary from the nominal size to more than 90%. This paper formulates a CVAR model allowing for many near unit roots and analyses the asymptotic properties of the Gaussian maximum likelihood estimator. Then a critical value adjustment suggested by McCloskey for the test on the cointegrating relations is implemented, and it is found by simulation that it eliminates size distortions and has reasonable power for moderate values of the near unit root parameter. The findings are illustrated with an analysis of a number of different bivariate DGPs.

Model discovery and Trygve Haavelmo's legacy

Hendry, D. F., Johansen, S. — Thu, 01 Jan 2015 20:29:36 +0100

The role of cointegration for optimal hedging with heteroscedastic error term

Gatarek, L., Johansen, S. — Wed, 08 Mar 2017 20:29:36 +0100

The role of cointegration is analysed for optimal hedging of an h-period portfolio. Prices are assumed to be generated by a cointegrated vector autoregressive model allowing for stationary martingale errors, satisfying a mixing condition and hence some heteroscedasticity. The risk of a portfolio is measured by the conditional variance of the h-period return given information at time t. If the price of an asset is nonstationary, the risk of keeping the asset for h periods diverges for large h. The h-period minimum variance hedging portfolio is derived, and it is shown that it approaches a cointegrating vector for large h, thereby giving a bounded risk. Taking the expected return into account, the portfolio that maximizes the Sharpe ratio is found, and it is shown that it also approaches a cointegration portfolio. For constant conditional volatility, the conditional variance can be estimated, using regression methods or the reduced rank regression method of cointegration. In case of conditional heteroscedasticity, however, only the expected conditional variance can be estimated without modelling the heteroscedasticity. The fi…ndings are illustrated with a data set of prices of two year forward contracts for electricity, which are hedged by forward contracts for fuel prices. The main conclusion of the paper is that for optimal hedging, one should exploit the cointegrating properties for long horizons, but for short horizons more weight should be put on the remaining dynamics.

Cointegration between trends and their estimators in state space models and CVAR models

Johansen, S., Tabor, M. N. — Sun, 01 Jan 2017 20:29:36 +0100

In a linear state space model, y_{t+1}=BT_{t}+eps_{t+1}, we investigate if the unobserved trend, T_{t}, cointegrates with the extracted trend E_{t}T_{t}, and with the estimated trend E^_{t}T_{t}, in the sense that the spreads T_{t}-E_{t}T_{t} and E_{t}T_{t}-E^_{t}T_{t} are stationary. We find that this result holds for BT_{t}-BE_{t}T_{t} and BE_{t}T_{t}-B^E^_{t}T_{t}. For the trends T_{t} and E^_{t}T_{t}, however, this type cointegration depends on the identification of B and T_{t}. The same results are found, if the observations, y_{t}, from the state space model are analysed using a cointegrated vector autoregressive model, where the trend is defined as the common trend. Finally we investigate cointegration between trends and their estimators based on the two models, and find the same results. We illustrate with two examples and confirm the results by a small simulation study.

The role of initial values in conditional sum-of-squares estimation of nonstationary fractional time series models

Johansen, S., Nielsen, M. Ø. — Fri, 01 Jan 2016 20:29:36 +0100

The cointegrated vector autoregressive model with general deterministic terms

Johansen, S., Nielsen, M. Ø. — Tue, 09 Aug 2016 20:29:36 +0200

In the cointegrated vector autoregression (CVAR) literature, deterministic terms have until now been analyzed on a case-by-case, or as-needed basis. We give a comprehensive unified treatment of deterministic terms in the additive model X(t)=
Z(t) + Y(t), where Z(t) belongs to a large class of deterministic regressors and Y(t) is a zero-mean CVAR. We suggest an extended model that can be estimated by reduced rank regression and give a condition for when the additive and extended models are asymptotically equivalent, as well as an algorithm for deriving the additive model parameters from the extended model parameters. We derive asymptotic properties of the maximum likelihood estimators and discuss tests for rank and tests on the deterministic terms. In particular, we give conditions under which the estimators are asymptotically (mixed) Gaussian, such that associated tests are khi squared distributed.

Rejoinder: Asymptotic Theory of Outlier Detection Algorithms for Linear Time Series Regression Models

Johansen, S., Nielsen, B. — Wed, 01 Jun 2016 20:29:36 +0200

Asymptotic Theory of Outlier Detection Algorithms for Linear Time Series Regression Models

Johansen, S., Nielsen, B. — Wed, 01 Jun 2016 20:29:36 +0200

Outlier detection algorithms are intimately connected with robust statistics that down-weight some observations to zero. We define a number of outlier detection algorithms related to the Huber-skip and least trimmed squares estimators, including the one-step Huber-skip estimator and the forward search. Next, we review a recently developed asymptotic theory of these. Finally, we analyse the gauge, the fraction of wrongly detected outliers, for a number of outlier detection algorithms and establish an asymptotic normal and a Poisson theory for the gauge.

Tightness of M-estimators for multiple linear regression in time series

Johansen, S., Nielsen, B. — Mon, 30 May 2016 20:29:36 +0200

We show tightness of a general M-estimator for multiple linear regression in time series. The positive criterion function for the M-estimator is assumed lower semi-continuous and sufficiently large for large argument: Particular cases are the Huber-skip and quantile regression. Tightness requires an assumption on the frequency of small regressors. We show that this is satisfied for a variety of deterministic and stochastic regressors, including stationary an random walks regressors. The results are obtained using a detailed analysis of the condition on the regressors combined with some recent martingale results.

Analysis of the Forward Search using some new results for martingales and empirical processes

Johansen, S., Nielsen, B. — Sun, 01 May 2016 20:29:36 +0200

The Forward Search is an iterative algorithm for avoiding outliers in a regression analysis suggested by Hadi and Simonoff (J. Amer. Statist. Assoc. 88 (1993) 1264-1272), see also Atkinson and Riani (Robust Diagnostic Regression Analysis (2000) Springer). The algorithm constructs subsets of "good" observations so that the size of the subsets increases as the algorithm progresses. It results in a sequence of regression estimators and forward residuals. Outliers are detected by monitoring the sequence of forward residuals. We show that the sequences of regression estimators and forward residuals converge to Gaussian processes. The proof involves a new iterated martingale inequality, a theory for a new class of weighted and marked empirical processes, the corresponding quantile process theory, and a fixed point argument to describe the iterative aspect of the procedure.

An asymptotic invariance property of the common trends under linear transformations of the data

Johansen, S., Juselius, K. — Wed, 01 Jan 2014 20:29:36 +0100

It is well known that if X_t is a nonstationary process and Y_t is a linear function of X_t, then cointegration of Y_t implies cointegration of X_t. We want to find an analogous result for common trends if X_t is generated by a finite order VAR with i.i.d. (0,Ω_x) errors ε_xt. We first show that Y_t has an infinite order VAR representation in terms of its white noise prediction errors, ε_yt, which are a linear process in ε_xt, the prediction error for X_t. We then apply this result to show that the limit of the common trends for Y_t generated by ε_yt, are linear functions of the common trends for X_t, generated by ε_xt. We illustrate the findings with a small analysis of the term structure of interest rates.

Optimal hedging with the cointegrated vector autoregressive model

Gatarek, L., Johansen, S. — Mon, 27 Oct 2014 20:29:36 +0100

We derive the optimal hedging ratios for a portfolio of assets driven by a Coin-
tegrated Vector Autoregressive model (CVAR) with general cointegration rank. Our hedge is optimal in the sense of minimum variance portfolio.
We consider a model that allows for the hedges to be cointegrated with the hedged asset and among themselves. We nd that the minimum variance hedge for assets driven by the CVAR, depends strongly on the portfolio holding period. The hedge is dened as a function of correlation and cointegration parameters. For short holding periods the correlation impact is predominant. For long horizons, the hedge ratio should overweight the cointegration parameters rather then short-run correlation information. In the innite horizon, the hedge ratios shall be equal to the cointegrating vector. The hedge ratios for any intermediate portfolio holding period should be based on the weighted average of correlation and cointegration parameters.
The results are general and can be applied for any portfolio of assets that can be
modeled by the CVAR of any rank and order.

Outlier detection algorithms for least squares time series regression

Johansen, S., Nielsen, B. — Mon, 27 Oct 2014 20:29:36 +0100

We review recent asymptotic results on some robust methods for multiple regression. The regressors include stationary and non-stationary time series as well as polynomial terms. The methods include the Huber-skip M-estimator, 1-step Huber-skip M-estimators, in particular the Impulse Indicator Saturation, iterated 1-step Huber-skip M-estimators and the Forward Search. These methods classify observations as outliers or not. From the asymptotic results we establish a new asymptotic theory for the gauge of these methods, which
is the expected frequency of falsely detected outliers. The asymptotic theory involves normal distribution results and Poisson distribution results. The theory is applied to a time series data set.

Times Series

Johansen, S. — Mon, 27 Oct 2014 20:29:36 +0100

An overvies of results for the cointegrated VAR model for nonstationary I (1) variables is given. The emphasis is on the analysis of the model and the tools for asymptotic inference. These include: formulation of criteria on the parameters, for the process to be nonstationary and I (1), formulation of hypotheses of interest on the rank, the cointegrating relations and the adjustment coefficients. A discussion of the asymptotic distribution results that are used for inference. The results are illustrated by a few examples. A number of extensions of the theory are pointed out.

Outlier Detection in Regression Using an Iterated One-Step Approximation to the Huber-Skip Estimator

Johansen, S., Nielsen, B. — Tue, 01 Jan 2013 20:29:36 +0100

Least squares estimation in a simple random coefficient autoregressive model

Johansen, S., Lange, T. — Sun, 01 Dec 2013 20:29:36 +0100

The question we discuss is whether a simple random coefficient autoregressive model with infinite variance can create the long swings, or persistence, which are observed in many macroeconomic variables. The model is defined by =ρyt-+,t= 1,.;n, where is an i.i.d. binary variable with p=P( =1), independent of i.i.d. with mean zero and finite variance. We say that the process is persistent if the autoregressive coefficient ρ̂ of on yt- is close to one. We take p

Statistical analysis of global surface temperature and sea level using cointegration methods

Schmith, T., Johansen, S., Thejll, P. — Sun, 01 Jan 2012 20:29:36 +0100

A necessary moment condition for the fractional functional central limit theorem

Johansen, S., Nielsen, M. Ø. — Fri, 01 Jun 2012 20:29:36 +0200

Asymptotic analysis of the Forward Search

Johansen, S., Nielsen, B. — Thu, 28 Feb 2013 20:29:36 +0100

The Forward Search is an iterative algorithm concerned with detection of outliers and other unsuspected structures in data. This approach has been suggested, analysed and applied for regression models in the monograph Atkinson and Riani (2000). An asymptotic analysis of the Forward Search is made. The argument involves theory for a new class of weighted and marked empirical processes, quantile process theory, and a fixed point argument to describe the iterative element of the procedure.

Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model

Johansen, S., Nielsen, M. Ø. — Thu, 01 Nov 2012 20:29:36 +0100

The Analysis of Nonstationary Time Series Using Regression, Correlation and Cointegration

Johansen, S. — Sun, 01 Jan 2012 20:29:36 +0100

The Selection of ARIMA Models with or without Regressors

Johansen, S., Riani, M., Atkinson, A. C. — Tue, 13 Nov 2012 20:29:36 +0100

We develop a $C_{p}$ statistic for the selection of regression models with
stationary and nonstationary ARIMA error term. We derive the asymptotic
theory of the maximum likelihood estimators and show they are consistent and
asymptotically Gaussian. We also prove that the distribution of the sum of
squares of one step ahead standardized prediction errors, when the
parameters are estimated, differs from the chi-squared distribution by a
term which tends to infinity at a lower rate than $\chi _{n}^{2}$. We
further prove that, in the prediction error decomposition, the term
involving the sum of the variance of one step ahead standardized prediction
errors is convergent. Finally, we provide a small simulation study.
Empirical comparisons of a consistent version of our $C_{p}$ statistic with
BIC and a generalized RIC show that our statistic has superior performance,
particularly for small signal to noise ratios. A new plot of our time series
$C_{p}$ statistic is highly informative about the choice of model.

Kointegration og fælles stokastiske trende

Johansen, S. — Thu, 01 Jan 2004 20:29:36 +0100

Exact rational expectations, cointegration, and reduced rank regression

Johansen, S., Swensen, A. R. — Tue, 01 Jan 2008 20:29:36 +0100

We interpret the linear relations from exact rational expectations models as restrictions on the parameters of the statistical model called the cointegrated vector autoregressive model for non-stationary variables. We then show how reduced rank regression, Anderson (1951), plays an important role in the calculation of maximum likelihood estimation of the restricted parameters.

A Bartlett correction factor for tests on the cointegrating relations

Johansen, S. — Sat, 01 Jan 2000 20:29:36 +0100

Likelihood ratio tests for restrictions on cointegrating vectors are asymptotically [chi]² distributed. For some values of the parameters this asymptotic distribution does not give a good approximation to the finite sample distribution. In this paper we derive the Bartlett correction factor for the likelihood ratio test and show by some simulation experiments that it can be a useful tool for making inference.

Cointegration; An Overview

Johansen, S. — Thu, 01 Jan 2004 20:29:36 +0100

Cointegration. Overview and Development

Johansen, S. — Sun, 01 Jan 2006 20:29:36 +0100

A Note on testing restrictions for the cointegration parameters of a VAR with I(2) variables

Johansen, S., Lütkepohl, H. — Sat, 01 Jan 2005 20:29:36 +0100

The interpretation of cointegrating coefficients in the cointegrated vector autoregressive model

Johansen, S. — Sat, 01 Jan 2005 20:29:36 +0100

Testing Weak Exogeneity and the Order of Cointegration in UK Money Demand Data

Johansen, S. — Wed, 01 Jan 1992 20:29:36 +0100

Matematisk Statistik

More on testing exact rational expectations in vector autoregressive models: Restricted drift term

Johansen, S., Swensen, A.R. — Wed, 01 Jan 2003 20:29:36 +0100

A Statistical Analysis of Cointegration for I(2) Variables

Johansen, S. — Sun, 01 Jan 1995 20:29:36 +0100

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 ¿ sup2/sup 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.

A representation theory for a class of vector autoregressive models for fractional processes

Johansen, S. — Tue, 01 Jan 2008 20:29:36 +0100

Based on an idea of Granger (1986), we analyze a new vector autoregressive model defined from the fractional lag operator 1-(1-L)^{d}. We first derive conditions in terms of the coefficients for the model to generate processes which are fractional of order zero. We then show that if there is a unit root, the model generates a fractional process X(t) of order d, d>0, for which there are vectors ß so that ß'X(t) is fractional of order d-b, 0

A Representation of Vector Autoregressive Processes Integrated of Order 2.

Johansen, S. — Wed, 01 Jan 1992 20:29:36 +0100

Matematisk statistik

Determination of Cointegration Rank in the Presense of a Linear Trend

Johansen, S. — Wed, 01 Jan 1992 20:29:36 +0100

Matematisk Statistik

Statistical Analysis of some Non-Stationary Time series

Johansen, S. — Fri, 01 Jan 1999 20:29:36 +0100