I(1) and I(2) Cointegration Analysis Theory and Applications

Transcription

1 I(1) and I(2) Cointegration Analysis Theory and Applications Heino Bohn Nielsen Ph.D. Thesis Institute of Economics University of Copenhagen October 23

2 Contents Introduction and Summary Likelihood Ratio Testing for Cointegration Ranks in I(2) Models Analyzing I(2) Systems by Transformed Vector Autoregressions Cointegration Analysis in the Presence of Outliers An I(2) Cointegration Analysis of Price and Quantity Formation in Danish Manufactured Exports Inflation Adjustment in the Open Economy: An I(2) Analysis of UK Prices Has US Monetary Policy Followed the Taylor Rule? A Cointegration Analysis

3 Introduction and Summary This thesis, entitled I(1) and I(2) Cointegration Analysis: Theory and Applications, consists of six article manuscripts, which broadly reflect the research areas I have been working with during my Ph.D. studies. The manuscripts are self-contained and can be read independently, and the connecting thread through the manuscripts is the use of the cointegrated vector autoregressive model as the statistical framework. The first three manuscripts are theoretical and deal with issues related to the vector autoregressive model with I(1) and I(2) restrictions, namely: likelihood ratio testing for the number of I(), I(1) and I(2) relations in a vector process in Chapter 1; how to make simple inference on the I(2) cointegrating parameters based on a minimal I(2)-to- I(1) transformation in Chapter 2; and how to conduct cointegration inference in the I(1) model in the presence of outliers in Chapter 3. The last three manuscripts are empirical applications analyzing, respectively, price and quantity formation in Danish exports in Chapter 4; the impact of import price shocks on UK inflationinchapter5;andmonetary policy rules for the US in Chapter 6. The remainder of this introduction contains brief summaries of the manuscripts. Chapter 1, Likelihood Ratio Testing for Cointegration Ranks in I(2) Models, is written jointly with Anders Rahbek. A central point in empirical applications of the p dimensional I(2) vector autoregressive model is the determination of the so-called cointegration ranks (r, s), i.e. the number of stationary relations, r, the number of I(1) trends, s, and the number of I(2) trends, p r s. The paper considers the likelihood ratio test for the number of cointegrating and multi-cointegrating relations. We derive the asymptotic distribution of the likelihood ratio test for the (multi-) cointegration ranks, and show that the limiting distribution is identical to the limiting distribution of the much applied test statistic from the Two-Step estimation procedure. However, in finite samples the test statistics differ; and based on two existing empirical examples and corresponding Monte Carlo simulations, we conclude that the likelihood ratio test is clearly preferable to the Two-Step rank test. Chapter 2, Analyzing I(2) Systems by Transformed Vector Autoregressions, is ajointpaperwithhans Christian Kongsted. Most empirical applications considering I(2) behavior transform the original I(2) process to an I(1) process comprising real magnitudes. The transformation requires a homogeneity restriction to apply to the original data, which is equivalent to knowing the loadings to the I(2) trends up to a normalization. This 3

4 Introduction paper assumes that the necessary homogeneity restrictions are valid, and characterizes the relationship between the parameters of the original I(2) vector autoregressive model and the parameters in the I(1) model for the transformed process. It is shown that the parameters of the transformed model are subject to certain restrictions; one set that fits directly into the structure of an I(1) reduced rank regression analysis, and a second set of restrictions that requires a more elaborate estimation algorithm and is commonly ignored in applied work. An empirical example and a simulation study indicate that only a minor loss of efficiency is incurred by ignoring the second set of restrictions. The paper concludes that a properly transformed model provides a practical and effective means for inference on the parameters of the I(2) model. Chapter 3, Cointegration Analysis in the Presence of Outliers, analyzes how inference of the cointegration rank of an I(1) vector autoregressive model can be conducted in the presence of innovational an additive outliers. An innovational outlier is produced by a shock to the innovation term of a data generating process and is propagated through the autoregressive structure. An additive outlier, on the other hand, is superimposed on the levels of the data, and the effects are independently of the autoregressive parameters. Models with innovational dummies can be estimated with reduced rank regression, and the usual practice in applied cointegration analysis is to include innovational dummy variables to account for large residuals. Models with additive dummies, on the other hand, cannot be estimated using reduced rank regression, and the paper proposes a simple algorithm for maximum likelihood estimation of the cointegration model with additive dummies. The paper analyzes how outliers can be modelled with dummy variables. Based on a Monte Carlo simulation, we conclude that additive outliers distort inference on the cointegration rank while innovational outliers are less harmful. Furthermore, unrestricted dummies are misspecified if an outlier is additive, and the potential distortion from misspecified dummies is much larger than the distortions from outliers per se. Thesefindings question the usual practice in applied cointegration analyses. Instead, the paper suggests an outlier detection procedure to determine the types as well as the locations of outliers before testing for the cointegration rank. Alternatively, the focus could be on correcting additive outliers, while the less distorting innovational outliers could be ignored. A maximum likelihood based correction works very well, but requires a non-standard estimation algorithm. An alternative linear interpolation in the levels of the data is almost equally effective. Chapter 4, An I(2) Cointegration Analysis of Price and Quantity Formation in Danish Manufactured Exports, is an application of the I(2) and I(1) cointegration framework to a quarterly data set for the Danish export sector The Danish export price, the competing price on the world market, and the production costs are characterized as integrated of second order, I(2), but a long-run homogeneity restriction seems to cancel the I(2)-trend allowing a transformed process to be analyzed within the cointegrated I(1) framework. 4

5 Introduction Based on the I(2)-to-I(1) transformed process, the paper analyzes the long-run and short-run structure of the Danish manufacturing export sector. Two long-run relations are found and identified as a demand-relation for Danish exports and a polynomially cointegrated price relation. The demand relation has a price elasticity of around 3 numerically, and there is a permanent positive effect from the German reunification on the Danish equilibrium market share. In the price formation a large weight to foreign prices and an effect from the rate of inflation to the steady-state markup are found. The latter effect is interpreted as an element of caution in the price setting in an inflationary environment. To characterize the short-run behavior a system of simultaneous equations is developed. Chapter 5, Inflation Adjustment in the Open Economy: An I(2) Analysis of UK Prices, is a joint paper with Christopher Bowdler, analyzing the transmission of import price shocks to UK domestic inflation The paper estimates a cointegrated vector autoregressive model for consumer prices, unit labour costs, import prices and real consumption growth subject to I(1) and I(2) restrictions. The paper argues that a key to understand the inflation adjustment to import price shocks is the accommodating role of real unit labor costs. In the empirical analysis the price variables are found to be integrated of second order, I(2), but a homogeneity restriction allows for a transformation from I(2) to I(1) space. The consumption growth is found to be stationary, and a second long-run relation links UK inflation to real import prices and real unit labor costs. An important finding is that real unit labor costs error correct to this relation, and high real import prices reduce the real wage, such that the pass through to domestic inflation is moderated. To illustrate the inflation dynamics following a foreign price shock the paper applies the structural vector autoregressive methodology to the cointegrated system. The paper performs an impulse response analysis to illustrate that the accommodating real wages limit the impact of import price shocks on domestic inflation. The accommodating effect may explain why the depreciation of sterling in 1992 left inflation unchanged. In contrast, high real import prices in 1974 increased inflation because wage accommodation effects were weaker at that time. Finally, Chapter 6, Has US Monetary Policy Followed the Taylor Rule? A Cointegration Analysis , considers possible monetary policy rules for the US within a cointegrated vector autoregressive model. This paper is written jointly with Anders Møller Christensen. The paper reconsiders the empirical evidence concerning monetary policy rules using the equilibrium correction structure of a cointegrated vector autoregressive model. In particular, the paper argues that a relation involving a policy interest rate can only be interpreted as a monetary policy rule if the policy rate equilibrium corrects to the suggested relation. This is a testable hypothesis in the multivariate cointegration model. Using this idea, it is rejected that US monetary policy can be described by the traditional rule suggested by John B. Taylor in 1993 in a seminal paper in Carnegie- 5

6 Introduction Rochester Conference Series on Public Policy. Instead we find a stable long-term relationship between the Federal funds rate, the unemployment rate, and the long-term interest rate, with deviations from the long-term relation being corrected primarily via changes in the Federal funds rate. This is taken as an indication that the Federal Open Market Committee sets interest rates with a view to activity and to expected inflation and other conditions available in financial markets. 6

7 Introduction Acknowledgements I would like to take the opportunity to thank a number of people for their encouragements and support. First of all I would like to thank my supervisor, Hans Christian Kongsted, for excellent guidance during my three years of Ph.D. studies. His inspiration and many comments have strongly influenced the entire thesis. Secondly, I wish to thank my coauthors, Christopher Bowdler, Anders Møller Christensen, Hans Christian Kongsted, and Anders Rahbek, for very inspiring cooperation; and Katarina Juselius and Dan Knudsen for comments and suggestions. In spring 22 I visited Nuffield College and University of Oxford. The hospitality of these institutions and a financing grant from the Euroclear Bank and the University of Copenhagen are gratefully acknowledged. In particular, I would like to thank Bent Nielsen for help in organizing the visit as well as academic and personal assistance during the stay. Furthermore, I would like to thank David Hendry, Sophocles Mavroeidis, and John Muellbauer for discussions of various subjects. Finally, thanks to Freja for all her love and patience, and to Katinka Coco, born April 23, for making the last six months so much more fun. hbn 7

8 Chapter 1 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models 8

9 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models Heino Bohn Nielsen Institute of Economics, University of Copenhagen Anders Rahbek Dept. of Applied Mathematics and Statistics, University of Copenhagen Abstract This paper considers the likelihood ratio (LR) test for the number of cointegrating and multi-cointegrating relations in the I(2) vector autoregressive (VAR) model. We derive the asymptotic distribution of the LR test for the (multi-) cointegration ranks, and show that this is identical to the asymptotic distribution of the much applied test statistic from the Two-Step procedure, see Johansen (1995), Paruolo (1996), and Rahbek, Kongsted, and Jørgensen (1999). However, conclusions with regards to the cointegration ranks in existing empirical applications change when applying the LR test, and based on Monte Carlo simulations we conclude that the LR test is preferable to the Two-Step based statistic. Keywords: Vector autoregressive model, Cointegration, I(2), Likelihood Ratio, Monte Carlo. JEL Classification: C32. 1 Introduction and Summary For many OECD countries for the post-war period, the first difference of nominal variables, e.g. inflation rates or money growth, seem to behave as unit root processes, implying that the levels of the nominal variables are integrated of second order, I(2). Johansen (1992) shows that a model for I(2) variables can be parameterized as a vector autoregressive (VAR) model with two reduced rank restrictions imposed, see also Johansen (1995), Paruolo (1996), Johansen (1997), Rahbek, Kongsted, and Jørgensen (1999), Paruolo and Rahbek (1999), Paruolo (2) and the survey by Haldrup (1998). Several applications have appeared in the empirical literature on e.g. money demand and open economy Discussions with Søren Johansen and Hans Christian Kongsted are gratefully acknowledged. Ox code for calculating the likelihood ratio test for the (multi-) cointegration ranks in the I(2) model can be obtained from the authors. Maximum likelihood estimation of the I(2) model has also been implemented in the new version of CATS in RATS and in the Matlab program me2 by Pieter Omtzigt. 9

10 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models price determination, see inter alia Juselius (1998), Juselius (1999), Diamandis, Georgoutsos, and Kouretas (2), Banerjee, Cockerell, and Russell (21), Banerjee and Russell (21), Fliess and MacDonald (21), Nielsen (22) and Kongsted (23). A central point in empirical applications of the multivariate p dimensional I(2) model is the determination of the so-called cointegration ranks (r, s), i.e. the number of stationary relations, r, the number of I(1) trends, s, and the number of I(2) trends, p r s. Inthis paper we consider the likelihood ratio (LR) test for the (multi-) cointegration ranks in the I(2) model and derive the asymptotic distribution. Existing literature applies the non-lr test based on the sum of two sets of canonical correlations from a Two-Step estimation procedure, see Johansen (1995), Paruolo (1996), and Rahbek, Kongsted, and Jørgensen (1999). The Two-Step procedure is easy to implement as a sequential application of the reduced rank regression (RRR), known from the estimation of the cointegrated I(1) model, but it does not make use of the rich structure of the I(2) parameterization and ignores a set of restrictions in the first step of the procedure. Maximum likelihood (ML) algorithms for estimation of the I(2) model have been proposed by Johansen (1997) and Boswijk (2) based on different parameterizations. In this paper we derive the asymptotic distribution of the LR test for the cointegration ranks using results from Johansen (1997), and show that the asymptotic distribution of the LR test is identical to the asymptotic distribution of the Two-Step test. This implies that the asymptotic critical values published for the Two-Step test also apply to the LR test. Based on two existing empirical examples and corresponding Monte Carlo simulations we examine some finite sample properties of the LR test and the Two-Step rank test. We find that the size properties of the LR test are excellent. For ranks (r, s), where r> or p r s>, the Two-Step statistic is always larger than the LR statistic, which increases rejection frequencies. In some examples we find that the difference between the LR and the Two-Step statistic is large. Overall we conclude that the LR test seems clearly preferable to the Two-Step rank test. The rest of the paper is organized as follows: Section 2 presents the I(2) model and the used notation. Section 3 presents the LR and the Two-Step test statistics, and the asymptotic distribution of the LR test is then derived in Section 4. Section 5 illustrates some small sample properties of the rank tests based on empirical examples and Monte Carlo simulations. Throughout the paper use is made of the following notation: for any p r matrix α of rank r, r<p,letα indicate a p (p r) matrix whose columns form a basis of the orthogonal complement of span(α). Hence α =ifr = p and α = I p if α =. Define also α = α(α α) 1 and let P α = αα = αα denote the orthogonal projection matrix onto span(α). Finally, the symbols D and P are used to indicate weak convergence and convergence in probability respectively. 1

11 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models 2 Model and Representation In this section we introduce the notation used throughout and present the I(2) model as a submodel of the vector autoregressive model. 2.1 The I(2) Model Consider the p dimensional vector autoregressive model of order k : X t = Π 1 X t Π k X t k + t, t =1, 2,...,T, or in a parameterization convenient for I(2) analysis: Xk 2 2 X t = ΠX t 1 Γ X t 1 + Ψ i 2 X t i + t, (1) where Π = P k i=1 Π i I, Γ = I + P k i=2 (i 1) Π i, and Ψ j = P k i=j+2 (i j 1) Π i. Here t is an iid N(, Ω) sequence with Ω positive definite, and the initial values X k+1,...,x are fixed. The I(2) model, H(r, s), is defined by the two reduced rank restrictions i=1 Π = αβ (2) α Γβ = ξη, (3) where α and β are p r matrices and ξ and η are (p r) s matrices. Note, that (3) can alternatively be written as P α ΓP β = α 1 β 1,whereα 1 = α ξ and β 1 = β η are p s matrices, and by definition span(α 1 ) span(α )andspan(β 1 ) span(β ). We use the notation H(r) =H(r, p r) to denote I(1) models, where α 1 and β 1 are p (p r) matrices, and H(p) denotes the unrestricted VAR. 2.2 Representation Under the additional assumption that the characteristic polynomial, A(z) =I p Π 1 z... Π k z k, has 2(p r) s roots at the point z = 1 and the remaining roots outside the unit circle, the process X t is I(2). Under this hypothesis, denoted H (r, s), X t has the representation: tx sx tx X t = C 2 i + C 1 i + γ 1 + γ 2 t + X t. (4) where s=1 i=1 i=1 C 2 = β 2 (α 2Θβ 2 ) 1 α 2, β C 1 =ᾱ ΓC 2, β 1C 1 =ᾱ 1(I ΘC 2 ), (5) and X t is a stationary I() process. Here Θ = Γβα Γ + I p P k 2 i=1 Ψ i,α 2 =(α, α 1 ), and β 2 =(β,β 1 ). Note that α 2 Θβ 2 has full rank p r s under H (r, s), because the 2(p r) s unit roots correspond exactly to the reduced ranks in (2) and (3). The coefficients γ 1 and γ 2 depend on the initial conditions and satisfy (β,β 1 ) γ 2 =,and 11

12 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models β γ 1 δβ 2γ 2 =,whereδ =ᾱ Γ β 2.Alsonotethatbydefinition (α, α 1,α 2 )and(β,β 1,β 2 ) are square non-singular matrices with orthogonal blocks, and that β 2 is a function of (β,β 1 ). It follows from (4) and (5) that (β,β 1 ) C 2 = so that (β,β 1 ) X t is stationary, whereas the (p r s) linear combinations β 2X t are integrated of order two, I(2). Moreover it turns out that also the r linear combinations β X t δβ 2 X t (6) are stationary. Since the process (6) and (β,β 1 ) X t are stationary, it follows that also β X t ᾱ Γ X t = β X t ᾱ Γ(P β,β1 + P β2 ) X t (7) is a stationary process. 3 Estimations and Rank Test Statistics This section presents the LR test based on the ML algorithm of Johansen (1997) and the usually applied Two-Step rank test based on the estimator of Johansen (1995). 3.1 Likelihood Ratio Test For the likelihood analysis, Johansen (1997) suggests an alternative parameterization based on a multi-cointegration term of the type (7): 2 X t = α[ρ τ X t 1 + ψ X t 1 ]+Ωα (α Ωα ) 1 κ τ X t 1 Xk 2 + Ψ i 2 X t i + t, (8) i=1 where α (p r), ρ ((r + s) r), τ (p (r + s)), ψ (p r), κ ((r + s) (p r)), Ψ i (p p), i =1,...,k 2, and Ω (p p) are all freely varying parameters. The parameters in the previous formulation can be derived from the new parameters from the identities τ =(β,β 1 ), β = τρ, ψ = (α Ω 1 α) 1 α Ω 1 Γ,andκ = α Γ(β,β 1)= (α Γβ,ξ), see Johansen (1997). Here we have used the projection identity α(α Ω 1 α) 1 α Ω 1 + Ωα (α Ωα ) 1 α = I p. The LR test for H(r, s) against the unrestricted alternative, H(p), is given by = 2logQ (H(r, s) H(p)) = T log Ω 1 ˆΩ, (9) S LR r,s where Ω and ˆΩ denote the covariance matrices estimated under H(r, s) andh(p) respectively. No closed form solution for the ML estimator exists, but estimates can be obtained through an iterative algorithm that switches between two steps: For fixed τ, the parameters α and α can be obtained by solving an eigenvalue problem and the remaining 12

13 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models parameters can be found from regression. For fixed values of these parameters, τ can be estimated by generalized least squares. Convergence to the global maximum of the likelihood function is not guaranteed, but the value of the likelihood function increases in each iteration. In our implementation, we use the Two-Step estimates, presented below, as starting values for the ML iterations. 3.2 Two-Step Rank Test The Two-Step estimator is based on the parameterization in (1). The first step imposes the restriction (2) to obtain Xk 2 2 X t = αβ X t 1 Γ X t 1 + Ψ i 2 X t i + t. (1) Ignoring the second restriction (3), the parameters α and β are estimated using RRR, similar to estimation in I(1) models. The test for reduced rank of Π is the familiar trace test of H(r) inh(p) : px Q r = T log (1 λ i ), i=r+1 where 1 λ 1... λ p are the eigenvalues from the corresponding RRR, see Johansen (1996, Chapter 6). The second step is estimated conditional on r and on the estimated α and β from the first step. The model is decomposed in a marginal model for α 2 X t and a model for α 2 X t conditional on α 2 X t. The restriction (3) concerns alone the marginal equation: α 2 X t = ξη β Xk 2 X t 1 α Γββ X t 1 + α Ψ i 2 X t i + α t, which can be estimated with RRR. Conditional on r, the test for reduced rank, s, of α Γβ, can be written as Q r,s = T p r X i=s+1 i=1 i=1 log (1 ζ i ), where 1 ζ 1... ζ p r are the second step eigenvalues. The Two-Step test for H(r, s) against the unrestricted H(p) isgivenby S 2S r,s = Q r + Q r,s. 3.3 Rank Determination Different values of the cointegration ranks define the sequence of partially nested models illustrated in Table 1 for the trivariate case. In the determination of the cointegration ranks in empirical applications, economic theory could give some guidance. When a particular model is suggested from theory, 13

14 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models the corresponding hypothesis, H(r, s), could be tested against H(p). If little is known a priori, however, an estimate (br, bs) of the ranks can be obtained from the data by a sequential application of the rank test. The idea is to start testing the most restricted model, H(, ), and in case of rejection to proceed left-to-right and top-to-bottom in Table 1. The estimator can be written as (br, bs) = {(r, s) S r,s c r,s ; S r,s >c r,s for the indices (r <r,s p r )and(r = r, s <s)}, where S r,s and c r,s denote the test statistics and critical values respectively. The estimator (br, bs) will select the correct ranks with a limiting probability (1 π), where π is the size of each test in the sequence, see Johansen (1995). [Table 1 around here] 4 Asymptotic Distribution of the Likelihood Ratio Test In this section we state the asymptotic distribution of the likelihood ratio test for the cointegration ranks in the I(2) model. We introduce the following notation: for two stochastic processes X u and Y u on the unit interval u [, 1], define the process X u corrected for Y u by µz 1 X u Y = X u X s Y sds 4.1 No Deterministic Components µz 1 Y s Y sds 1 Y u. Consider the hypothesis H(r, s) against the unrestricted H(p). We state the asymptotic distribution of the LR statistic in (9). Theorem 1 Under H (r, s), thenast, S LR r,s = 2logQ(H(r, s) H(p)) D Q r + Q r,s, where Q r ( Z 1 µz 1 1 Z ) 1 = tr dw u G u G u G udu G u dwu ( Z 1 µz 1 1 Z ) 1 Q r,s = tr dw 2u W2u W 2u W2udu W 2u dw2u. Here W u =(W1u,W 2u ) is a (p r) dimensional standard Brownian motion on the unit interval, u [, 1], withw 1u of dimension s and W 2u of dimension p r s. Furthermore, Ã! W 1u G u = R u W. 2vdv W2 14

15 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models The proof is given in the appendix. Note, that the asymptotic distribution of the LR test is identical to the asymptotic distribution of the Two-Step rank test derived in Johansen (1995, Theorem 7). In particular, the two components, Q r and Q r,s, are the asymptotic distributions of the statistics from the first and second step of the Two-Step rank test, Q r and Q r,s. This implies that the critical values for the Two-Step rank test, see Johansen (1995) for the present case, can also be applied to the LR test. 4.2 Linear Trends In empirical applications it is often important to include deterministic linear trends. Rahbek, Kongsted, and Jørgensen (1999) propose a specification with deterministic linear trends in all directions of the process, including the multi-cointegrating relations (6), at the same time avoiding quadratic trends. An important feature of this model, denoted H (r, s) in the following, is that both the LR and Two-Step tests for cointegration ranks are asymptotically similar with respect to the parameters of the deterministic terms, see also Nielsen and Rahbek (2). In terms of the parameterization in (8), H (r, s) can be represented as: 2 X t = α[ρ τ Xt 1 + ψ Xt 1]+Ωα (α Ωα ) 1 κ τ Xt 1 Xk 2 + Ψ i 2 X t i + t, (11) i=1 where τ =(τ,τ 1 ) ((p +1) (r + s)), ψ = ψ,ψ 1 ((p +1) r), and, finally, X t 1 = (Xt 1,t) is p + 1 dimensional. Imposing the assumptions in section 2.2, the model is referred to as H (r, s). Under H (r, s) the process X t in (11) has a representation similar to (4) with γ 1 and γ 2 being functions of initial values in addition to τ 1 and ψ 1, see Rahbek, Kongsted, and Jørgensen (1999). As a result, X t is as noted an I(2) process with linear trends in all linear combinations of the process, including the multi-cointegrating ones. The result in Theorem 1 is extended to the model H (r, s) in Theorem 2. As the proof mimics the proof of Theorem 1, we state the result without proof, simply noting that the asymptotic distributions of the ML estimators are identical to the asymptotic distributions of the Two-Step estimators in Rahbek, Kongsted, and Jørgensen (1999). Note that H (r, s) in terms of the parametrization in (1), suitable for the Two-Step analysis, corresponds to restricting a constant and linear regressor, µ 1 + µ 2 t,asfollows: µ 2 = αb 1 and α µ 1 = α Γβb 1 ξη 1, (12) where b 1 is 1 r and η 1 is 1 s, see Rahbek, Kongsted, and Jørgensen (1999) for further details. Theorem 2 Under H (r, s), thenast, 2logQ(H (r, s) H (p)) D Q r + Q r,s, 15

16 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models where Q r Q r,s ( Z 1 µz 1 = tr dw u G 1u ( Z 1 µz 1 = tr dw 2u G 2u 1 Z ) 1 G 1u G 1udu G 1u dwu 1 Z 1 G 2u G 2udu G 2u dw2u ). Here W u =(W1u,W 2u ) is a (p r) dimensional standard Brownian motion on the unit interval, u [, 1], withw 1u of dimension s and W 2u of dimension p r s. Furthermore, W 1u Ã! R G 1u = u W 2vdv and G W 2u 2u =. 1 u (1,W 2 ) Critical values are given in inter alia Rahbek, Kongsted, and Jørgensen (1999) and in Doornik (1998). 5 Finite Sample Properties The asymptotic equivalence of the LR test and the Two-Step rank test does not hold for finite samples in general. The relation between the test statistics is stated in the following proposition. Proposition 1 Consider the hypothesis H(r, s) in H(p). In finite samples, Equality holds if (p r s) r =. S LR r,s S 2S r,s. Proof: Under the alternative, H(p), estimation is identical for the ML and the Two- Step analysis, but under the null, H(r, s), the Two-Step procedure does not maximize the likelihood function. To see this compare (1) and (8) to obtain Γ = αψ Ωα α 1 Ωα κ τ = α α Ω 1 α 1 α Ω 1 Γ + Ωα α 1 Ωα α Γββ + Ωα α 1 Ωα ξη β. The first step of the Two-Step procedure ignores the restriction imposed on the last term, and instead of the 2s (p r) s 2 free parameters in ξη the Two-Step procedure allows for (p r) 2 parameters. In two special cases, the Two-Step procedure maximizes the likelihood function under H(r, s). First in I(1) models, where s = p r and α Γβ is non-singular. And secondly if r = where the second step of the Two-Step procedure is conditional on α = β =. 16

17 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models Note, that the magnitude of the difference depends on the number of redundant parameters as well as the sample correlation between the terms in (1) and the redundant terms. The number of additional parameters is largest for models on the diagonal of Table 1, where s =, while the LR and the Two-Step statistics coincide for model located in the first row and last column of Table 1. Also note that in the model H (r, s), the second restriction in (12) is also ignored in the first step of the Two-Step procedure, which introduces an additional distortion. To illustrate the difference we consider two empirical examples, namely the analysis of Australian inflation and the markup of prices and cost taken from Banerjee, Cockerell, and Russell (21); and the Danish import price determination from Kongsted (23). In both cases we revisit the rank determination based on the original data and use the estimated model as a data generating process (DGP) in a small Monte Carlo simulation. 5.1 Australian InflationandtheMarkuponCosts The first empirical example is the analysis of Banerjee, Cockerell, and Russell (21), who consider the relation between inflation and the markup of prices on costs based on a set of quarterly Australian data covering the effective sample 1972 : : 2. The three dimensional process consists of the logs of consumer prices, import prices and unit labor costs. Furthermore, four (assumed) stationary and weakly exogenous variables are included. Banerjee, Cockerell, and Russell (21) analyze a second order VAR based on the deterministic specification of Paruolo (1996) and use the Two-Step estimator. To simplify the analysis we exclude in this paper the conditioning variables and apply the deterministic specification of Rahbek, Kongsted, and Jørgensen (1999). The left hand side of Table 2 reports the Two-Step rank tests. Starting from the most restricted model H (, ) it is possible to reject all models with r =ata5%level. In the second row, H (1, ) is easily rejected whereas the model H (1, 1) has a p value of 72%. This is also the preferred model in the original study. The LR tests are reported in the right hand side of the Table 2. As noted, all test statistics in first row and last column are identical to the Two-Step results, and the remaining test statistics are all lower. Using the LR test there are less evidence for rejecting H (1, ). The LR test statistic is 48, corresponding to a p value of 6%, as compared to a clearly significant Two-Step statistic of 78. For the theoretically preferred H (1, 1) the difference is smaller. [Table 2 around here] Simulations Based on the Estimated Model. To analyze the difference, we set up a small Monte Carlo simulation. As DGP we use the model H (1, 1) with parameters set to the ML estimates, and generate time series, X 11,...,X 1,...X T,forT ranging between 5 and 1, by replacing t with random draws from an iid normal with the estimated 17

18 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models covariance matrix. The initial values X 11 and X 1 are taken from the actual data and the first 1 observations are discarded to eliminate the importance of this choice. The rejection frequencies at a nominal 5% level are reported in Table 3 for the two tests. The rejection frequencies of H (r, s) against H (p) are identical for the Two-Step rank test and the LR test for models with r =, while there are big differences when testing H (1, ) and the true model H (1, 1). For the LR test the rejection frequencies for the true model H (1, 1) are very close to the nominal 5% for all sample lengths. The rejection frequency of H (1, ) against H (3) are relatively low in small samples. The Two-Step test is sized distorted, with rejection frequencies for the true model H (1, 1) around 2% in small samples. At the same time the rejection frequencies for H (1, ) are higher than the LR test. For a small sample, T = 5, the rejection frequency for H (1, ) is 94% compared to a rejection frequency of 32% for the LR test. The distributions of the test statistics are reported in Figure 1 for the case T = 75. The graphs are organized according the nesting structure in Table 1. The lower right graph reports the results for tests of the true model, H (1, 1). The distribution of the LR statistics almost coincides with the asymptotic distribution, while the distribution of the Two-Step statistics is more dispersed and displaced to the right. The differences in rejection frequencies translate into differences in the distributions of the ranks estimated from the sequential test procedure, (br, bs), cf. Table 4. Note, that in small samples the correct ranks (r, s) =(1, 1) are rarely chosen, primarily due to high Type II error probabilities. For this particular DGP, the proportion of correctly selected models is higher using the Two-Step rank test than the LR test in small samples. For longer samples the ordering changes. [Table 3 around here] [Table 4 around here] [Figure 1 around here] 5.2 Danish Import Price Determination In this section we consider the Danish import price determination from Kongsted (23). Data consists of import, competing, and domestic prices, as well as an interest rate. Kongsted (23) estimates a VAR(2) for the effective sample 1975 : : 4. The deterministic specification includes a linear trend in all directions and an unrestricted impulse dummy for 1992 : 2, and we adopt the same specification. The theoretically preferred setup according to Kongsted (23) is H (2, 1) with two stationary relations and one I(2) trend. The Two-Step rank tests, also reported in Kongsted (23), are given in the left hand sideoftable5. Itispossibletorejectallmodelswithr =,whileh (1, 2) has a p value of 9%. The first model not rejected at a 1% level is the preferred H (2, 1). 18

19 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models The LR statistics are reported in the right hand side of the table. The tests for r = are identical, but now it is marginally harder to reject H (1, 2). If this model is nevertheless rejected, the next potential model is H (2, ) with two I(2) trends. The LR test statistic of 45 is markedly lower than the corresponding Two-Step test statistic of 75. [Table 5 around here] Simulations Based on the Estimated Model I. Again we set up a small Monte Carlo simulation. We use as a DGP the estimated H (2, 1), and exclude the dummy variable both in the DGP and in the estimation model. The rejection frequencies are reported in Table 6 for different sample lengths. Now the small sample distributions differ for several model: H (1, ),H (1, 1),H (1, 2),H (2, ) and H (2, 1), and there are large differences in the rejection frequencies. The Two-Step rank test is heavily over-sized with small-sample rejection frequencies of the true null H (2, 1) above 3%. The actual size converges very slowly to the nominal 5%, and for T = 5 the rejection frequency is still twice the nominal size. The LR test has excellent size properties, but the rejection frequencies for more restricted models indicate a low power. The distributions of the test statistics are reported in Figure 2. Again the distributions of the Two-Step rank tests are more dispersed and displaced to the right. The differences are largest where s =, but there are substantial differences for several models. [Table 6 around here] [Figure 2 around here] Simulations Based on the Estimated Model II. FinallywetakethemodelH (2, ) as the DGP to illustrate the properties if s = in the DGP, such that a large distortion appears in the Two-Step estimation of the model with correct ranks. The rejection frequencies are reported in Table 7. Note, that the true model H (2, ) is almost always rejected using the Two-Step rank test and this is case for all relevant sample lengths. For T = 2, corresponding to 5 years of quarterly observations, the actual size is 66%. The LR tests, on the other hand, have good size properties, with rejection frequencies close to the nominal size. [Table 7 around here] 19

20 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models A Proof of Theorem 1 In this appendix we derive the asymptotic distribution of the LR test for cointegration ranks in the I(2) model. In order to motivate the notation and ease the presentation of the I(2) test we start by considering the well-known likelihood ratio test for cointegration rank in the I(1) model. Throughout we use the notation θ and ˆθ to denote the ML estimates of a parameter θ under the null hypothesis and under the alternative respectively. A.1 Asymptotics for the I(1) LR Test Consider the well-known I(1) model and the rank test in this case. The present rederivation of the LR test is not based on the usual representation in terms of eigenvalues and canonical correlations, see e.g. Johansen (1996, Chapter 11), but is instead based on a linear regression type formulation. It is assumed here that the reader is familiar with the well-established literature on I(1) VAR models. For simplicity and without loss of generality consider the simplest case of the p dimensional I(1) VAR(1) model as given by X t = ΠX t 1 + t, (13) and the hypothesis H(r) parameterized as Π = αβ. Wewanttoderivetheasymptotic distribution of the likelihood ratio 2logQ(H(r) H(p)) = T log Ω 1 ˆΩ, where Ω, ˆΩ are the ML estimates of the covariance matrix Ω under the hypothesis H(r) and the alternative respectively. Denote by H (r) the model with exactly p r unit roots in the characteristic polynomial, A(z), with the remaining roots outside the unit circle. Lemma 1 Under H (r), then as T, the LR statistic 2logQ(H(r) H(p)) converges in distribution to ( µz 1 µz 1 1 Z ) 1 tr W u dwu W u Wudu W u dwu, (14) where W u is (p r)-dimensional standard Brownian motion on the unit interval, u [, 1]. Proof: Recall that under H(r) the parameters α and β are non-identified. Identification is obtained by normalization on the known p r matrix c such that β c = β(c β) 1 and α c = αβ c, and hence Π = αβ = α c β c. Denote by α,β and Ω the true parameters corresponding to the null, H (r). In order to derive the asymptotic distribution of 2logQ( H(r) H(p)) introduce the simple auxiliary null hypothesis H aux that β is known, β = β, as given by the equation, X t = αβ X t 1 + t. (15) 2

21 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models Using that by definition H aux H(r) H(p), and therefore in particular Q(H(r) H(p)) = Q(H(r) H aux ) Q(H aux H(p)), this implies that 2logQ(H(r) H(p)) = 2logQ(H aux H(p)) [ 2logQ(H aux H(r))]. Turn first to 2logQ(H aux H(r)) and introduce the (p r) r dimensional parameter B = β (β β ), (16) where β is normalized by c = β. NotethatthehypothesisH aux is given by B =. The asymptotic distribution of the ML estimates of α and β under H(r) and normalized by c = β,ˆα and ˆβ respectively, is given in Johansen (1996) and Johansen (1997). It follows that µz 1 1 Z 1 T ˆB = T β ³ˆβ β D B = F u Fudu F u dvuω 1 α (α Ω 1 α ) 1, (17) with F u = β MV u,wherev u is a Brownian motion on u [, 1] with covariance Ω,and M = β (α β ) 1 α. Under H(r), with α and β normalized on c = β, X t = αβ X t 1 + t = αβ β β + β β Xt 1 + t = α B Z 1t + Z t + t. (18) Here Z 1t = β X t 1 and Z t = β X t 1 are I(1) and I() processes respectively. Define the estimated residuals ³ ˆ t = X t ˆα ˆB Z 1t + Z t, t = X t ᾰz t and ˆ t = X t ˆαZ t. Then Ω = 1 T P T t=1 t t,and where à X T =ˆα ˆB 1 T ˆΩ = 1 T TX ˆ tˆ t = 1 T t=1 TX ˆ tˆ t Y T + YT X T, t=1! TX Z 1t Z1t ˆBˆα and Y T = t=1 à 1 T! TX ( X t ˆαZ t ) Z1t ˆBˆα. Using (17), the consistency of ˆα and the continuous mapping theorem, it follows that, µz 1 D TX T α B F u Fudu B α = α (α Ω 1 α ) 1 α Ω 1 Z 1 t=1 µz 1 1 Z 1 dv u Fu F u Fudu F u dv uω 1 α (α Ω 1 α ) 1 α. Next, by convergence to stochastic integrals, Ã! 1 TX TY T = t Z1t (ˆα α ) 1 TX t Zt T T T ˆBˆα t=1 t=1 Z 1 µz 1 1 Z 1 D dv u Fu F u Fudu F u dvuω 1 α (α Ω 1 α ) 1 α 21

22 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models Therefore by joint convergence, and as 1 P T T t=1 ˆ tˆ P t Ω, Ω P Ω, 2logQ(H aux H(r)) = T log Ω 1 ˆΩ = Ttr Ω 1 YT + YT ª X T + op (1) ( Z 1 µz 1 1 Z ) 1 = tr Ω 1 α (α Ω 1 α ) 1 α Ω 1 dv u Fu F u Fudu F u dvu + o P (1). This implies directly, ( 2logQ(H aux H(p)) = tr Ω 1 Z 1 µz 1 1 Z ) 1 dv u Fu F u Fudu F u dvu + o P (1). Hence by the joint convergence of 2logQ(H aux H(p)) and 2logQ(H aux H(r)) under H (r), and using the skew projection, I p = α (α Ω 1 α ) 1 α Ω 1 +Ω α (α Ω α ) 1 α, 2logQ(H(r) H(p)) = 2logQ(H aux H(p)) [ 2logQ(H aux H(r))] ( Z 1 µz 1 1 Z ) 1 = tr α α 1 Ω α α dv u Fu F u Fudu F u dvu + o P (1) ( µz 1 µz 1 1 Z ) 1 = tr W u dwu W u Wudu W u dwu + o P (1), with W u =(α Ω α ) 1/2 V u. This completes the proof of Lemma 1. A.2 Asymptotics for the I(2) LR Test The proof in the I(2) case is completely analogous to the proof in the I(1) case with the main difference being the more sophisticated parameterization. Again we consider, without loss of generality, the simplest case of the I(2) model, the VAR(2) model as given by 2 X t = ΠX t 1 Γ X t 1 + t. The hypothesis of interest is H(r, s) against the alternative H(p). Under H(r, s) weuse the parameterization in (8), 2 X t = α[ρ τ X t 1 + ψ X t 1 ]+Ωα (α Ωα ) 1 κ τ X t 1 + t. (19) We want to derive the asymptotic distribution of the likelihood ratio test, 2logQ(H(r, s) H(p)) = T log Ω 1 ˆΩ. Proof of Theorem 1: As before, with θ a parameter, the corresponding true parameter is denoted θ. Henceforth, the parameters β and τ under H(r, s) are normalized on c = β and c = τ respectively such that β β = I r and τ τ = I r+s. Furthermore, set 22

23 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models α =(I p β (α β ) 1 α ) β such that all other parameters are identified, see Johansen (1997). Note in particular that ρ = τ β which is (r + s) r. Introduce the parameters defined in Johansen (1997): B = β 2 (ψ ψ ), B 1 = β 1 (β β ), B 2 = β 2 (β β ), C = β 2 (τ τ ) ρ, ³ where ρ = I ρ (ρ ρ ) 1 ρ ρ. Note that ρ = τ β = ρ + τ β 1B 1 = ρ(b 1 )and define similarly ρ (B 1 ). By Johansen (1997, Lemma 1) it follows that under H (r, s) for the ML estimates ˆB, ˆB 1, ˆB 2 and Ĉ derived under H(r, s), where Here T ˆB D B, T ˆB 1 D B 1, T 2 ˆB2 D B 2 and T Ĉ D C, (2) B = B,B 1,B2 µz 1 1 Z 1 C = H u Hudu H u = H u H 1u H 2u µz 1 1 Z 1 = H u Hudu = H u dv 1u (21) H u dv 2u. (22) β 2C 2 V u β 1C 1 V u β 2C 2 R u V sds with V u a Brownian motion on u [, 1] with covariance Ω.Furthermore, V 1u = α Ω 1 α 1 α Ω 1 V u (23) ³ V 2u = ρ κ α 1 Ω α κ ρ 1 ρ κ α 1 Ω α α V u (24) ³ = ξ α 1 Ω α ξ 1 ξ α 1 Ω α α V u. Now under H(r, s) the model in (19) can be written as, 2 X t = A Z t + A 1 Z 1t + A 2 Z 2t + t (25) where Z t,z 1t and Z 2t are I(), I(1) and I(2) regressors respectively, defined by Ã! β Z t = X t 1 + ψ X t 1 τ X t 1 Ã! β Z 1t = 2 X t 1 β 1X t 1 Z 2t = β 2X t 1. Finally, A = ³α, α (ψ ψ ) τ + Ωα α 1 Ωα κ ³ A 1 = αb + Ωα α 1 Ωα κ ρ (B 1 ) C + ρ (B 1 ) B2,αB 1 (26) (27) A 2 = αb 2. (28) 23

24 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models Introduce the auxiliary hypothesis, H aux where ψ (p r), β(p r) andτ (p (r + s)) are fixed at their true values, ψ,β and τ, corresponding to B,B 1,B 2 and C all identically zero. Note that under H aux, the model equation in (25) reduces to 2 X t = A Z t + t, and furthermore that H aux H(r, s) H(p) such that 2logQ(H(r, s) H(p)) = 2logQ(H aux H(p)) [ 2logQ(H aux H(r, s))]. Turn first to 2logQ(H aux H(r, s)) and define the corresponding estimated residuals ˆ t = 2 X t ÂZ t Â1Z 1t Â2Z 2t, t = 2 X t ĂZ t and ˆ t = 2 X t ÂZ t. Then Ω = 1 T P T t=1 t t,and where X T = Y T = ˆΩ = 1 T TX ˆ tˆ t = 1 T t=1 à 1 ³Â1, Â2 T à 1 TX T t=1 TX t=1 TX ˆ tˆ t + X T Y T YT, t=1 Z 1t,Z2t Z 1t,Z2t! ³Â1, Â2 ³ X t ÂZ t Z 1t,Z 2t! ³Â1, Â2. With D T =blockdiag³ 1 1 T I p r, I T p r s it follows that 3/2 à DT 1 TX Z T 1t,Z2t Z 1t,Z2t! Z 1 D D T t=1 H u H udu. (29) Next, using (2) together with the definitions of A i in (26)-(28) and consistency of the remaining parameters, it follows that TD 1 T ³Â1, D Â2 α B,B1,B2 ³ Ω α α 1 Ω α ξ C,,. (3) Combining (29) and (3) and using the definitions (21)-(22) it follows that h D TX T X = α B,B1,B2 ³ i Ω α α 1 Ω α ξ C,, Z 1 h H u Hudu α B,B1,B2 ³ i Ω α α 1 Ω α ξ C,, Note that, as α α =bydefinition, tr ( Z 1 µz 1 1 Z ) 1 Ω 1 X ª = tr α dv 1u Hu H u Hudu H u dv1uα ( Z α + tr Ω 1 µz 1 1 Z 1 1 α ξ dv 2u Hu H u Hudu H u dv 2uξ ), 24

25 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models that is, the cross product terms vanish. Next, by convergence to stochastic integrals, à 1 TY T = T T à X T = Z 1 t=1 Z 1 = Z 1 dv u H u TX t=1! ³ Z X t t ÂZ 1t,Z2t DT TDT 1 ³Â1, Â2! 1 t Z TD 1t,Z2t 1 DT ³Â1, Â2 + op (1) D T dv u H u h α B,B µz 1 1,B2 1 Z 1 H u Hudu µz 1 1 Z 1 dv u Hu H u Hudu T ³ i Ω α α 1 Ω α ξ C,, H u dv 1uα H u dv 2uξ α Ω α 1 α Ω. Therefore by joint convergence, and as T 1 P T t=1 ˆ tˆ P t Ω, Ω P Ω, and furthermore using the definition of V 1 and V 2 in (23)-(24), 2logQ(H aux H(r, s)) = T log Ω 1 ˆΩ = Ttr Ω 1 X Y Y ª + o P (1) ( Z 1 µz 1 1 Z ) 1 = tr Ω 1 α (α Ω 1 α ) 1 α Ω 1 dv u Hu H u Hudu H u dvu ³ 1 + tr ½ α α Ω α ξ ξ α 1 Ω α ξ 1 ξ α 1 Ω α α Z 1 µz 1 1 Z ) 1 dv u Hu H u Hudu H u dvu + o P (1). This implies directly that also ( 2logQ(H aux H(p)) = tr Ω 1 Z 1 µz 1 1 Z ) 1 dv u Hu H u Hudu H u dvu + o P (1). Using the projection I p = α (α Ω 1 α ) 1 α Ω 1 + Ω α (α Ω α ) 1 α and collecting terms, it follows that 2logQ(H(r, s) H(p)) = 2logQ(H aux H(p)) [ 2logQ(H aux H(r, s))] ( Z 1 µz 1 1 Z ) 1 = tr α α 1 Ω α α dv u Hu H u Hudu H u dvu (31) ³ 1 tr ½ α α Ω α ξ ξ α 1 Ω α ξ 1 ξ α 1 Ω α α (32) Z 1 µz 1 1 Z ) 1 dv u Hu H u Hudu H u dvu + o P (1). (33) 25

26 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models The term in (31) can be rewritten as ( Z 1 µz 1 1 Z ) 1 tr α α 1 Ω α α dv u Hu H u Hudu H u dvu + ( Z 1 µz 1 1 Z ) 1 tr α α 1 Ω α α dv u G u G u G udu G u dvu where G u = Ã H 1u H 2u H! = H 1u R 1 H 1uH u du h R 1 H uh u du i 1 H u H 2u R 1 H 2uH u du h R 1 H uh u du i 1 H u. Note that that the term appearing in (32) can be written as Hence, α α Ω α 1 ξ ³ξ α Ω α 1 ξ 1 ξ α Ω α 1 α = α α Ω α 1 α α ξ ξ α Ω α ξ 1 ξ α. 2logQ(H(r, s) H(p)) ( Z 1 µz 1 = tr α α 1 Ω α α dv u G u ( Z 1 µz 1 tr α 2 α 1 2 Ω α 2 α 2 dv u Hu H u Hudu 1 Z 1 ) G u G udu G u dvu + 1 Z 1 ) H u dv u and the result in Theorem 1 follows by defining the standard Brownian motion W u = α Ω α 1/2 α V u., 26

27 Likelihood Ratio Testing for Cointegration Ranks in I(2) Models References Banerjee, A., L. Cockerell, and B. Russell (21): An I(2) Analysis of Inflation and the Markup, Journal of Applied Econometrics, 16, Banerjee, A., and B. Russell (21): The Relationship Between the Markup and Inflation in the G7 Economies and Australia, The Review of Economics and Statistics, 83(2), Boswijk, H. P. (2): Mixed Normality and Ancillarity in I(2) Systems, Econometric Theory, 16, Diamandis, P., D. Georgoutsos, and G. Kouretas (2): The Monetary Model in the Presence of I(2) Components: Long- Run Relationships, Short-Run Dynamics and Forecasting of the Greek Drachma, Journal of International Money and Finance, 19, Doornik, J. A. (1998): Approximations to the Asymptotic Distribution of Cointegration Tests, Journal of Economic Surveys, 12(5), Fliess, N., and R. MacDonald (21): The Instability of the Money Demand Function: and I(2) Interpretation, Oxford Bulletin of Economics and Statistics, 63(4), Haldrup, N. (1998): A Review of the Econometric Analysis of I(2) Variables, Journal of Economic Surveys, 12, Johansen, S. (1992): A Representation of Vector Autoregressive Processes Integrated of Order 2, Econometric Theory, 8, (1995): A Statistical Analysis of Cointegration for I(2) Variables, Econometric Theory, 11, (1996): Likelihood-Based Inference in Cointegrated Autoregressive Models.OxfordUniversity Press, Oxford, 2nd edn. (1997): Likelihood Analysis of the I(2) Model, Scandinavian Journal of Statistics, 24, Juselius, K. (1998): A Structured VAR for Denmark under Changing Monetary Regimes, Journal of Business and Economic Statistics, 16(4), (1999): Price Convergence in the Medium and Long Run: An I(2) Analysis of Six Price Indices, in Cointegration, Causality, and Forecasting - A Festschrift in Honour of Clive W.J. Granger, ed. byr. F. Engle, and H. White. Oxford University Press, Oxford. Kongsted, H. C. (23): An I(2) Cointegration Analysis of Small-Country Import Price Determination, Econometrics Journal, 6, Nielsen, B., and A. Rahbek (2): Similarity Issues in Cointegration Analysis, Oxford Bulletin of Economics and Statistics, 62(1), Nielsen, H. B. (22): An I(2) Cointegration Analysis of Price and Quantity Formation in Danish Manufactured Exports, Oxford Bulletin of Economics and Statistics, 64(5), , Chapter 4 in this Thesis. Paruolo, P. (1996): On the Determination of Integration Indices in I(2) Systems, Journal of Econometrics, 72, (2): Asymptotic Efficiency of the Two Stage Estimator in I(2) Systems, Econometric Theory, 16, Paruolo, P., and A. Rahbek (1999): Weak Exogeneity in I(2) VAR Systems, Journal of Econometrics, 93, Rahbek, A., H. C. Kongsted, and C. Jørgensen (1999): Trend-Stationarity in the I(2) Cointegration Model, Journal of Econometrics, 9,