Testing exact rational expectations in cointegrated vector autoregressive models

Journal of Econometrics 93 (1999) 73}91

Testing exact rational expectations in cointegrated vector autoregressive models S+ren Johansen , Anders Rygh Swensen
* University of Copenhagen, Denmark and European University Institute, Florence, Italy
Department of Mathematics, University of Oslo, P.O. Box 1053, N-0316 Blindern, Oslo, Norway Received 1 September 1994; accepted 1 September 1998

Abstract This paper considers the testing of restrictions implied by rational expectations hypotheses in a cointegrated vector autoregressive model for I(1) variables. If the rational expectations involve one-step-ahead observations only and the coe$cients are known, an explicit parameterization of the restrictions is found, and the maximum-likelihood estimator is derived by regression and reduced rank regression. An application is given to a present value model.  1999 Elsevier Science S.A. All rights reserved. PACS: C32 Keywords: VAR models; Cointegration; Rational expectations

1. Introduction Expectations play a central role in many economic theories. But the incorporation of this kind of variable into empirical models rises many problems. The variables are in many cases unobserved, either because data on expectations are unavailable, or because there may often be reason to suspect that the available data on expectations are unreliable. There are also problems connected with

* Corresponding author. E-mail address: [email protected] (A.R. Swensen) 0304-4076/99/$ - see front matter  1999 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 7 6 ( 9 9 ) 0 0 0 0 4 - 4

74

S. Johansen, A.R. Swensen / Journal of Econometrics 93 (1999) 73}91

validity. Economic agents may bene"t from not revealing their real expectations. Some sort of proxies must therefore be used. One possibility, when the models contain stochastic elements, is to use conditional expectations in the probabilistic sense given some previous information. When this information is all available information, past and present, contained in the variables of the model, rational expectation is the usual denomination. Another, perhaps more precise, name is model consistent expectations. This emphasizes the aspect that the expectations mean conditional expectations in the model the analysis is based upon, an idea originally introduced by Muth (1960, 1961). However, since rational expectation seems to be the common name for this type of expectations, we shall stick to this usage in the following. It is well known that dynamic models containing rational expectations of future values have a multitude of solutions. In a recent paper Baillie (1989) advocated a procedure for testing the restrictions implied by rational expectations on a set of variables by assuming that the solutions could be described by a vector autoregressive (VAR) model. He then expressed the restrictions implied by the postulated relationships between the expectations as restrictions on the coe$cients of the VAR model. In this paper we shall follow the same approach. Baillie, however, allowed for non-stationary behavior of the variables that could be eliminated by "rst transforming the variables using known cointegrating relationships. It is therefore necessary with some knowledge about how the variables cointegrate. This can represent a problem. If the restrictions completely specify the cointegration vectors, the transformation will be known under the hypothesis, and no problem arises. But it may happen that the cointegration rank in the VAR model is larger than what is speci"ed by the rational expectations hypothesis. The remaining cointegration vectors that are used to transform to stationarity must then be estimated before the transformation is carried out. Since none of the restrictions of the hypothesis are used at this stage, substitution of the unrestricted estimators to transform the system yields a test of the Wald type. This is the case even if at the second stage, when the tests are carried out in the transformed system, the likelihood ratio tests appropriate for stationary VAR models are used. The two-stage tests must therefore be a sort of Wald test. Here we shall develop the likelihood ratio tests when the restrictions implied by the rational expectations hypothesis have a particular form. We therefore pursue another path than that outlined by Baillie to deal with the nonstationary situation. Starting out with the VAR model we only assume that the variables are integrated of order one. It turns out, as one might expect, that the restrictions on the expectations entail restrictions on the cointegrating relationships. In addition some restrictions on the short-run parameters of the model must be satis"ed.

S. Johansen, A.R. Swensen / Journal of Econometrics 93 (1999) 73}91

75

Thus we can de"ne three nested models: The VAR, the cointegrated VAR and the cointegrated VAR with the restrictions implied by rational expectations. In order to apply likelihood methods one will then have to work out the maximum-likelihood estimator in each model and use these to construct the likelihood ratio tests. Our contribution here is to show that the cointegrated VAR model with rational expectations restrictions of a particular form, to be commented on below, admits an explicit solution in terms of reduced rank regression. Thus the procedure we suggest is "rst to "t a VAR model to the data, then check that the cointegration rank is the one predicted by theory and "nally estimate the cointegrated VAR model with rational expectations restrictions. All these tests involve simple explicit calculations and well-known limit distributions. The likelihood ratio test we propose can serve as an alternative to the usual Wald type of test described by Baillie. It is well known from statistical theory that the likelihood ratio test, like the Wald test, is consistent and e$cient under reasonable conditions. Furthermore it is, unlike the Wald test, invariant under reparameterizations. Also, nested hypotheses can be naturally dealt with. The last property may be of value in testing restrictions involving rational expectations, since it makes it possible to decompose the test statistics into a part that can be attributed to the restriction on the cointegration vectors, and a part due to the other parts of the imposed restrictions. We would like to comment on the form of the rational expectations hypotheses we can handle in this framework. They are formulated as linear restrictions involving the conditional expected value of the observations one-step-ahead and the present and lagged observed values. In the case where the restrictions can be expressed as linear combinations with known coe$cients, we can "nd explicit expressions for the maximum-likelihood estimators and the maximal value of the likelihood. However, in a rational expectations context the interesting situations often arise when the coe$cients are not known, but can be expressed as functions of one or more parameters of interest, e.g. discount factors in present value models. The results in the paper are of interest in this case too, since they can be used to evaluate the likelihood at every "xed value of the parameters appearing in the coe$cients. In some cases, a numerical optimization technique can then be used to "nd the maximal value of the likelihood. It may therefore be possible to determine the maximum-likelihood estimators and employ the likelihood ratio test even where the coe$cients depend on unknown parameters. The paper is organized as follows. In the next section we state the type of relationships among expectations we shall consider, and derive the implications for the VAR model when the expectations are considered to be rational in the sense described earlier. In Section 3 we show how a likelihood ratio test can be developed when the restrictions have this form. Assuming that the variables are

76

S. Johansen, A.R. Swensen / Journal of Econometrics 93 (1999) 73}91

integrated of order 1 we discuss in Section 4 the asymptotic distribution of the test. Finally, we demonstrate how the restrictions implied by a present value model can be tested by applying the likelihood ratio test worked out in this paper, and compare it with the usual way of treating these restrictions.

2. The form of the restrictions This section de"nes the statistical model which is assumed to generate the data and formulates the rational expectation hypothesis. We then give some examples from the literature which illustrate the types of model that can be dealt with in the present formulation. Finally we derive the restrictions on the parameters of the statistical model implied by the rational expectation hypothesis. We assume that the p-dimensional vectors of observations are generated according to the vector autoregressive (VAR) model X "A X #2#A X #k#UD #e , t"1,2, ¹, R  R\ I R\I R R

(1)

where X , , X are assumed to be "xed and e ,2, e are independent, \I> 2   2 identically distributed Gaussian vectors, with mean zero and covariance matrix R. The matrices D , t"1,2,¹ consist of deterministic series orthogonal to the R constant term. Model (1) can be reparameterized as DX "PX #P DX #2#P DX #k#UD #e , R R\  R\ I R\I> R R t"1,2,¹,

(2)

where P"A #2#A !I, P "!(A #2#A ), i"2,2, k.  I G G I For +X , to be I(1) we assume that the matrix P has reduced rank R R2 0(r(p and thus may be written P"ab,

(3)

where a and b are p;r matrices of full column rank. This model which we shall use as the starting point, has been treated extensively (see e.g. Johansen, 1991, 1992, 1995; Johansen and Juselius, 1990, 1992). We note that parameters a and b are unidenti"ed because of the multiplicative form in (3). In our treatment of rational expectations we shall, as explained in the introduction, elaborate upon ideas exposited by Baillie (1989). The set of restrictions we consider is of the form E[c X "O ]#c X #c X #2#c X #c"0.  R> R  R \ R\ \I> R\I>

(4)

S. Johansen, A.R. Swensen / Journal of Econometrics 93 (1999) 73}91

77

Here E[ ) "O ] denotes the conditional expectation in the probabilistic sense R taken in model (1), given the variables X ,2, X . The p;q matrices  R c , i"!k#1,2, 1 are known matrices, possibly equal to zero. The q-dimenG sional vector c can contain unknown parameters and is of the form c"Hu where the q;s dimensional matrix H is known, and u is an s-dimensional vector consisting of unknown parameters, 0)s)q. An important special case is where the matrix H is speci"ed so that c is a vector with a set of the elements consisting of zeros and where the rest consists of unrestricted parameters. In addition we will assume that the two matrices c and c #2#c #c are  \I>   of full column rank. Note that we allow lagged values of X to be included in the R restrictions. On letting d "! I\ c , i"0,2, k the restrictions (4) may be \G> HG\ \H reformulated as E[c DX "O ]!d X #d DX #2#d DX #c"0.  R> R  R \ R \I> R\I>

(5)

Thus theoretical restrictions may be expressed in the form (4) or (5) according to what is convenient in the particular application. There are a number of interesting economic hypotheses that are subsumed by formulation (4) or (5). We only mention three, but refer to the paper by Baillie (1989) mentioned above for a more thorough discussion. Example 1. Let X denote the vector (n , n , d , i , i ), where i and R R R R R R R i denote domestic and foreign interest rate, respectively, n and n are the R R R domestic and foreign in#ation rate and d is the depreciation of own currency. R Two hypotheses of interest are the uncovered interest parity hypothesis which can be formulated as i !i "E[d "O ] R R R> R and equality of the expected real interest rates i !E[n "O ]"i !E[n "O ]. R R> R R R> R These hypotheses can be expressed in form (4) where c"c "0, j"2, 3,2, H and where c and c are given by the matrices  





c " 

0

0

0

!1

0

0

0

1

0

0

1

1

and c " !1  0

!1

!1

0

0 . 0 0

78

S. Johansen, A.R. Swensen / Journal of Econometrics 93 (1999) 73}91

An important point in this example is that q"2, which means that a test for (4) is a simultaneous test for the uncovered parity hypotheses and the equality of the expected real interest rates. Restriction (4) is more general than it may appear. Consider the expression  I\ dH\E[ f  X "O ]#f  X # f  X #f"0,  R>H R  R \H R\H H H

(6)

where d is a known constant and f , , f , f are known matrices, with the \I> 2  exception that f may have the same form as c. Using iterated conditional expectations in a similar expression at time t#1, multiplying by d and subtracting from (6) yields ( f !df )E[X "O ]#( f !df )X   R> R  \ R I\ # ( f !df )X #f  X #(1!d) f"0, \H \H> R\H \I> R\I> H which shows that restrictions of form (6) can be written in form (4). Formulation (6) covers the two following examples: Example 2. Campbell and Shiller (1987) studied a present value model for two variables > and y having the form R R  > "c(1!d) dHE[y "O ]#c, R R>H R H where c is a coe$cient of proportionality, d a discount factor and c a constant that may be unknown. This relation is of form (6), which can be seen by taking f "(1,!c(1!d)) and f "(0,!cd(1!d)).   In a related paper Campbell (1987) treated a system with X "(y , y , co ), R R R R where y and y are capital and labor income, respectively and co is consumpR R R tion. The permanent income hypothesis he investigated is of form  co "c[y #(1!d) dHE[y "O ]]. R IR JR>H R H Thus in the case where c and d are known, under the hypothesis, these are examples of the hypotheses that can be cast in form (4). Example 3. In a study of money demand Cuthbertson and Taylor (1990) considered restrictions of the form  (m!p) "j(m!p) #(1!j)(1!jD) (jD)HE[cz "O ], R R\ R>H R H

S. Johansen, A.R. Swensen / Journal of Econometrics 93 (1999) 73}91

79

where m!p is real money balances, and cz are the determinants of the long-run real money demand. The restrictions are deduced from a model where agents minimize the expected discounted present value of an in"nite-period cost function measuring both the cost of being away from the long-run equilibrium and the cost of adjustment, conditional on information at time t. The constant j, which satis"es 0(j(1, depends on the relative importance of the two cost factors. Taking X "(m !p , z ) and f "(!j, 0,2, 0), f "(1,!(1!j) R R R R \  (1!jD)c), f "(0,!jD(1!j)(1!jD)c) we see that this is a situation  covered by the formulation in (6) if j, D and c are known. A recent application of a similar model to the demand for labor can be found in Engsted and Haldrup (1994). This example illustrates that a formulation of restrictions (4}5) which allows restrictions on lagged values is desirable. An important feature of (4) or (5), which should be noticed, is that the restriction is exact in the sense that the expectations are formed on the basis of the variables of the model. There are no expectations involving stochastic processes unobservable to the econometrician. It is precisely this property that allows us to derive testable restrictions on the coe$cients of the VAR model based on the economic restrictions (4). This distinction is emphasized by Hansen and Sargent (1981, 1991). We conclude this section by reformulating restrictions (4) and (5) as restrictions on the coe$cients of the statistical model (2). Taking the conditional expectation of DX given X ,2,X , we get, by using (2) and multiplying by c R>  R  c E[DX "O ]"c PX #c P DX #2#c P DX  R> R  R   R  I R\I> #c k#c UD .   R>

(7)

Inserting this expression into (5) implies that the following conditions must be satis"ed: c P"d , c P "!d , i"2,2, k,    G \G> c k"!c"!Hu and 

c U"0. 

This can be summarized as follows. Proposition 1. Restrictions (4) or (5) take the following form in terms of model (2): (i) bac "! I\ c "d .  H\ \H  (ii) c P " I\ c "!d , i"2,2, k.  G HG\ \H \G> c k"!Hu and c U"0.  

(8)

80

S. Johansen, A.R. Swensen / Journal of Econometrics 93 (1999) 73}91

Note that the conditions of the "rst part of the proposition may be formulated as d "! I\ c 3sp(b), i.e. the vector d must belong to the space spanned  H\ \H  by the columns of b. Also, multiplying both sides by the matrix (bb)\b, one has the following restrictions on the adjustment parameters a: ac "  (bb)\bd .  Thus the restrictions implied by the rational expectation hypothesis are simultaneous restrictions on all parameters. The restrictions on P ,2, P , k, U  I are linear and the restrictions on a and b are non-linear due to the multiplicative structure P"ab.

3. Derivation of the maximum-likelihood estimators and the likelihood ratio test We now present an algorithm for calculating the maximum-likelihood estimator under the restrictions implied by the rational expectations hypothesis, and also derive a formula for the likelihood ratio test. We conclude with a summary of the method. In order to derive the maximum-likelihood estimator under the restrictions imposed by the rational expectations model we need to maximize the likelihood function under the constraints given by Proposition 1. Rather than maximizing under constraints it is easier to maximize with respect to freely varying parameters, and we therefore derive in Proposition 2 an expression for the matrix P in terms of freely varying parameters. This allows us to derive regression equations and hence to "nd the maximum likelihood estimator by both regular and reduced rank regressions. It is convenient for the formulation to introduce some notation. If a is any p;q matrix of full column rank q(p, then a is a p;(p!q) matrix with , columns which are orthogonal to the columns of a, so that (a, a ) is nonsingular. , If we can write a"(a , a ), where a is q;q of full rank, then we can take    a "(!a a\, I). We de"ne a "a(aa)\, such that aa "I. ,   The expression we shall use in deriving the maximum-likelihood estimators is then given by: Proposition 2. The p;p matrix P has reduced rank r and satisxes Pb"d,

(9)

where b and d are p;q matrices of full column rank if and only if there exist (p!q);(r!q) matrices g and m of full rank and a (p!q);q matrix H such that P has the representation P"bd#b gmd #b Hd. , , ,

(10)

S. Johansen, A.R. Swensen / Journal of Econometrics 93 (1999) 73}91

81

Proof. Assuming that (9) is true we consider



bPd (b, b )P(d, d )" , , b Pd ,

bPd

 

dd , " b Pd b Pd , , ,

0



b Pd , ,

.

If P has rank r, then r*q, which is the rank of dd. Then b Pd must have , , rank r!q, and can be written b Pd "gm for matrices of rank r!q. If we , , de"ne the (p!q);q matrix H as H"b Pd, we get the representation ,



dd P"(b, b ) , H

0



gm

(d, d )"bd#b Hd#b gmd , , , , ,

which proves one part of the proposition. Next, assume that P can be represented as in (10). Then bP"d. That the rank is reduced can be seen from



dd (b, b )P(d, d )" , , H

0



gm

,

which has rank equal to rank(dd)#rank(gm)"q#(r!q)"r.

䊐

Next, we want to derive the consequences of the restrictions derived in Proposition 1 on the model equations. If we insert the explicit parameterization for the matrix P we "nd that (2) with b"c and d"d becomes   DX "bdX #b gmd X #b HdX R R\ , , R\ , R\ #P DX #2#P DX #k#UD #e .  R\ I R\I> R R

(11)

By multiplying (11) with a"b and b we get, after taking into account the , restrictions in Proposition 1 aDX "gmd X #HdX R\ R , R\ # aP DX #2#aP DX  R\ I R\I> # ak#aUD #ae , R R bDX "dX !d DX !2!d DX !Hu#be . R R\ \ R\ \I> R\I> R

(12) (13)

We have now decomposed the model equations (2) into two sets of regression equations. The "rst set of q equations is a non-linear regression because of the reduced rank matrix gm, and the second set of p!q equations is a simple linear

82

S. Johansen, A.R. Swensen / Journal of Econometrics 93 (1999) 73}91

regression. The error terms ae and be are correlated, hence we derive the R R conditional model for aDX given bDX and the past using o"aRb(bRb)\. R R This gives aDX "gmd X #o(bDX #d DX #d DX ) R , R\ R \ R\ \I> R\I> # (H(dd)\!o)dX R\ # aP DX #2#aP DX  R\ I R\I> # (oHu#ak)#aUD #u . R R

(14)

where the errors are u "(a!ob)e . Note that they are independent of the R R errors be of (13). R We intend to "nd the maximum-likelihood estimators and the maximal value of the likelihood by considering separately the marginal model given by (13), and the conditional model (14) described above. This can be done since the likelihood can be written as a product of the likelihoods of the conditional and the marginal models. But in addition we must verify that there are no constraints between the parameters of the two models. If they cannot vary independently of each other, it is not possible to express the maximum value of the product as the product of the maximal values of the two factors. The parameters of the marginal model are u and bRb"R . The parameters  of the conditional model are g, m, o, c"(H(dd)\!o), t "aP , i"2,2, k, G G U "aU, "(oHu#ak) and R "aRa!aRb(bRb)\bRa. We have to    show that the range of variation of the regression coe$cients in the conditional model is the same for any value of the parameters in the marginal model u and R . It is a well-known property of the multivariate Gaussian distribution that  the possible values of o and R are the same for any value of R .   The parameter u, however, enters both the conditional and the marginal model and could therefore generate restrictions across the equations. We can show that the possible values of the coe$cient are any p!q vector no matter which value of u is taken. For this it is enough to see that ak can take on any value in RN\O. We decompose k as k"a k #bM k "a k !bM Hu    and see that k "ak is unrestricted by the choice of u, and hence that is  unrestricted by the choice of u. Now consider the estimation of the conditional system. We must consider two cases. If q"r all cointegrating relations are known, b"d, and the matrices m and g are zero. In this case equations (12) and (13) contain only stationary processes, and inference can proceed with regression of the variable aDX on R bDX #d DX #2#d DX , dX , DX ,2, DX ,D R \ R\ \I> R\I> R\ R\ R\I> R

S. Johansen, A.R. Swensen / Journal of Econometrics 93 (1999) 73}91

83

and 1, t"1,2, ¹. Let R be the residuals and compute S " 2 R R /¹. R  R R R Thus in this case the method suggested shows which regressions to perform as a consequence of the error structure found to describe the process. If q"1,2, r!1, we regress the variables aDX and d X on R , R\ bDX #d DX #2#d DX , dX , DX ,2, DX ,D R \ R\ \I> R\I> R\ R\ R\I> R and 1, t"1,2, ¹. De"ning the residuals as R and R Eq. (14) takes the form R R R "gmR #error. R R De"ne the (p!q);(p!q) matrices S , i, j"1, 2 by GH 1 2 S " R R , GH ¹ GR HR R

i, j"1, 2.

(15)

By now well-known arguments the maximum-likelihood estimators of m are given by mI "(v ,2, v ) where v ,2, v are eigenvectors in the eigenvalue  P\O  N\O problem "jS !S S\S ""0,    

(16)

which has solutions jI '2'jI . Here the normalization mI S mI "I is  N\O  P\O used. The estimator of g is given by g"S mI .  The part of the maximized-likelihood function stemming from the conditional model is therefore



"S "“P\O (1!jI )/"aa" G ¸\2 "  G    "S "/"aa" 

when q"1,2, r!1, when q"r.

The part stemming from the marginal model (13) follows from results for standard multivariate Gaussian models, and equals ¸\2 ""RI "/"bb",    where 1 2 RI " (bDX !dX #d DX #2#d DX  ¹ R R\ \ R\ \I> R\I> R #Hu)(bDX !dX #d DX #2#d DX R R\ \ R\ \I> R\I> #Hu) (17)

84

S. Johansen, A.R. Swensen / Journal of Econometrics 93 (1999) 73}91

and u is the maximum-likelihood estimator for u. Hence the maximum value of the likelihood function under the hypothesis is given by P\O ¸\2 ""RI ""S " “ (1!jI )/"bb""aa", &    G G

(18)

where the product is to be taken as 1 when q"r. In Johansen and Juselius (1990) it is shown that the maximum value of the likelihood in the reduced rank model de"ned by (2) and (3) is given by ¸\2""S "“P (1!jK ), where S , jK , i"1,2, r arise from maximizing the

  G G  G likelihood in a manner similar to the one described above. In this case only restriction (3) is taken into account. Collecting the results above we therefore have the following result for the likelihood ratio test of restrictions (4) in the VAR model (2) with reduced rank condition (3) imposed. Proposition 3. Consider the rational expectation restrictions of form (4) with c , , c , c known. Assume that b"c , a"c and d"!(c #c # \I> 2    ,   2#c ) have full column rank. The likelihood ratio statistic of a test for the \I> restrictions (4) in the reduced rank VAR model satisfying (3) against a VAR model satisfying only reduced rank condition (3), is P !2 ln Q"¹ ln "S "!¹ ln (1!jK )#¹ ln "RI "  G  G P\O !¹ ln "S "#¹ ln(1!jI )!¹ ln("bb""aa"),  G G where RI , S and jI , i"1,2, r!q are given by (15)}(17),   G S , jK , i"1,2, r are estimates from VAR model (2) satisfying (3).  G

and

Remark 1. The normalization mI S mI "I of the estimated cointegration  P\O vectors, which are not speci"ed by theory, is not the only one possible. In fact, the same considerations as for the general cointegrated VAR model satisfying (3), apply in the case where (4) is imposed in addition. What is really estimated is the space spanned by columns of b when the restrictions from the rational expectations hypothesis are taken into account. Thus choosing a normalization is the same as choosing a coordinate system for this space. For a fuller discussion on this point we refer to Johansen (1995, pp. 71}72). Remark 2. In the case where q"r, the restrictions from the rational expectations hypothesis (4) completely specify the cointegration vector b, and there is no

S. Johansen, A.R. Swensen / Journal of Econometrics 93 (1999) 73}91

85

need to use a reduced rank regression. This is necessary in the case q"1,2, r!1, where b is only partially speci"ed by restrictions (4). Remark 3. Although it is an essential part of the restrictions (4) that the matrices c , , c , c are completely known we can, as pointed out in the introduc\I> 2   tion, combine the results above with a grid search or a numerical optimization procedure to obtain the maximum-likelihood estimators in the cases where the matrices are speci"ed as functions of some unknown parameters. Section 5 contains an example. We conclude this section with a summary of the steps that are necessary to compute the maximal value of the likelihood under the rational expectations hypothesis we consider, when the rank of the VAR model is r. Summary of the computation of ¸\2 under the hypothesis given by (4) &  or (5): (i) Set b"c and d "! I\ c , i"0,2, k. Let d"d . Compute  \G> HG\ \H  a"b . , (ii) First consider the marginal model (13) bDX "dX !d DX !2!d DX !Hu#g , R R\ \ R\ \I> R\I> R

(iii) (iii)

(iii)

(iv)

where g ,2, g are independent N(0, R ) distributed q;1 vectors and u is  2  a s;1 vector of parameters. Let u and RI be the maximum-likelihood  estimators in this model. Next consider the conditional model (14) (a) If q"r, regress aDX on bDX #d DX #2#d DX , R R \ R\ \I> R\I> dX , DX ,2, DX , D and 1. Let R be the residuals. ComR\ R\ R>I\ R R pute S " 2 R R .  2 R R R (b) If q"1,2, r!1, regress aDX and dM  X on bDX #d D R , R\ R \ X #2#d DX , dX , DX ,2, DX , D and 1. R\ \I> R\I> R\ R\ R>I\ R Let R and R be the residuals. Compute S " 2 R R , i, j"1, 2. R R GH 2 R GR HR Let jI '2'jI be the ordered solutions of the eigenvalue problem  N\O "jS !S S\S ""0.     Compute ¸\2 from (18), where S is taken from step (iii.a) if q"r and &   from step (iii.b) if q"1,2, r!1.

4. The asymptotic distribution of the test statistics First we consider the case where the coe$cients c , , c , c appearing in \I> 2   rational expectations hypotheses (4) are assumed to be known. At the end of the

86

S. Johansen, A.R. Swensen / Journal of Econometrics 93 (1999) 73}91

section we will brie#y comment on situations where these matrices contain unknown parameters that must be estimated. In treating the distributional properties of the estimators and tests we will throughout the section assume that the following conditions are satis"ed. Let P(z) denote the characteristic polynomial of the VAR model (2), i.e. P(z)" (1!z)I!zP!(1!z)zP !2!(1!z)zI\P , and let !W equal the de I rivative of P(z) evaluated at z"1. Under the conditions that (i) "P(z)""0 implies that "z"'1 or z"1, (ii) restriction (3) and (iii) a Wb has rank p!r, , , Johansen (1991) derived an explicit representation of X in terms of the errors. R In particular, the vector DX and the components of bX are stationary. R R Therefore, the columns of b are the cointegration vectors in the sense of Engle and Granger (1987). Using these results one can "nd the asymptotic distribution of the estimators of a, b and the other unknown parameters (see Johansen, 1991; or Ahn and Reinsel, 1990). Normalized by ¹, the deviation bK !b converges towards a mixed Gaussian distribution. The other estimators converge at the rate ¹\. The asymptotic distribution of these estimators is a multivariate Gaussian distribution, except for the distribution of the estimator for the constant term, which is more complicated. The asymptotic covariance matrix of the estimators of b and the other parameters is block diagonal. This has the consequence that a test on the b parameters may be carried out separately from the rest. The conditions derived in Proposition 1 are conditions on all of the parameters. It seems natural, however, to test "rst the restrictions on the cointegrating relations b ignoring the other restrictions, and then test the restrictions on the other parameters. The test of the restrictions on b can be done by the maximum-likelihood procedure developed by Johansen and Juselius, (1990), and amounts to carrying out a s test. If this hypothesis is not rejected, one can proceed to test the restrictions on the other parameters implied by Proposition 1, treating b as known. This means that the processes involved can be transformed to stationary processes. Hence this part of the testing can be carried out following well known procedures developed for inference in stationary time series. This is in essence the traditional Wald-type test for models of the kind we treat in this paper. It has the advantage of applying to more complicated situations, e.g. those which arise if there is more than one lead involved in the rational expectations hypotheses. As shown in the previous section it is possible to carry out the tests as likelihood ratio tests in situations where c , , c , c in restrictions (4) are \I> 2   known. This has the advantage that if the hypothesis of rational expectations is accepted then the calculations give the estimated parameter values. We shall indicate the asymptotic distribution in the case covered by Proposition 3. By the results referred to above the asymptotic distribution is s, and the number of degrees of freedom is the di!erence between the number of

S. Johansen, A.R. Swensen / Journal of Econometrics 93 (1999) 73}91

87

free parameters in the general case and the number of parameters under the hypothesis. Since there are pr#(p!r)r#(k!1)p#p#mp#p(p#1)/2 parameters in model (2) satisfying (3) when there are m deterministic series, and formulation (13)}(12) has (p!q)r#(p!r)(r!q)#(k!1)p(p!q)#s# (p!q)#m(p!q)#p(p#1)/2 parameters, the number of degrees of freedom is rq#(p!r)q#(k!1)pq!s#(m#1)q"kpq!s#(m#1)q. The more general case where the matrices c , , c , c contain unknown \I> 2   parameters, d, is more complicated. However, when the cointegrating relations can be expressed as smooth functions b(d), the results of Johansen (1991) apply. The likelihood ratio test can be carried out as usual where the degrees of freedom of the s are modi"ed to take account of the additional parameters. In special cases it may also be possible to separate the parameters d so that b can be expressed as a smooth function of one part. Then one can also use the s approximation to the distribution of the likelihood ratio test. Example 3 of Section 2 is an instance, where the parameters c only appear in the cointegration vector, since b"(!jD, 0,2, 0) and d"(1!j)(1!jD) (!1, c).

5. Application to a present value model In order to illustrate the results of the previous sections we consider the present value model treated by Campbell and Shiller (1987) and discussed in Example 2 in Section 2. We follow their approach analyzing X "(> , y ) R R R\ where > is the value of the stock at the beginning of period t and y is the R R dividend paid during period t. Since, as Campbell and Shiller point out, the coe$cient of proportionality c equals d/(1!d), the present value model is > "  dGE[y "O ],where the right-hand side is the sum of the discounted R G R\>G R value of expected future dividends. The apriori model is a two-dimensional reduced rank VAR model, and this is estimated by methods described in Johansen and Juselius (1990). Expressing the present value model as restrictions of form (6) implies that q"1 and that the non-zero vectors are f "(1, 0), f "!d(0, 1). Using the   same reformulation as described after Example 1, the restriction may be expressed as


 
 

(1, 0)

> > R !E (d, d) R> O "0. R y y R\ R

(19)

We can now apply the machinery from Section 3 to "nd the maximumlikelihood estimate of the parameters of the reduced rank VAR model when the restriction from the present value model is imposed, and get an expression for

88

S. Johansen, A.R. Swensen / Journal of Econometrics 93 (1999) 73}91

the maximal value of the likelihood. In the notation from that section, this means that




b"!d

1 1

,







1 d!1 a"b " , d" , , !1 d




d d " . , 1!d

Eqs. (12) and (13) become, with g"m"0 and k"2




D> R\ D(> !y )"hI ((d!1)> #dy )#n R R\ R\ R\ Dy R\



#k !k #(e !e ),   R R !dD(> #y )"(d!1)> #dy !d(e #e ). R R\ R\ R\ R R Here n"aP , 

H . hI " d#(1!d)

Notice how all the parameters except for d are isolated in the "rst equation. The fact that d appears in both equations means that a full system estimation is necessary in order to be able to draw inferences on d. The part of the maximized-likelihood stemming from the marginal model, ¸\2 , is the mean sum of squares of bDX !dX "> !d(> #y ),   R R\ R\ R R\ divided by "bb""2d. This is step (ii) of the summary in Section 3. Since in this case r"q"1, reduced rank regression is not necessary when estimating the conditional system, i.e step (iii) (a) is the appropriate one. This amounts to regressing aDX "(D> !Dy ) on bDX "!d(D> #Dy ), R R R\ R R R\ dX "(d!1)> #dy , D> , Dy and 1. The part of the maxiR\ R\ R\ R\ R\ mized-likelihood stemming from the conditional model, ¸\2 , is just the   mean residual sum of squares from this regression, divided by "aa""2. Thus for d known, "nding the value of the maximized likelihood when the present value restriction is imposed on a two-dimensional reduced rank VAR model essentially means running a regression and computing two sums of squares. When d is unknown, one can in this particular example repeat the procedure for each d, where d is ranging over the possible values of the parameter. One then gets the maximized likelihood as a function of the unknown d, and the value corresponding to the maximum of this function will be the maximum-likelihood estimator of d. It may be worth commenting on how these problems are usually handled. A survey is given in the paper by Baillie (1989). However, a key reference for

S. Johansen, A.R. Swensen / Journal of Econometrics 93 (1999) 73}91

89

present value models is Campbell and Shiller (1987). They build on ideas introduced by Hansen and Sargent (1981), see also Hansen and Sargent (1991) for a more accessible reference. For comparison we discuss brie#y how example (19) is analyzed by Campbell and Shiller (1987). The cointegrating relation is S "> !dy /(1!d) and the R R R\ number of cointegrating relations is completely speci"ed by r"q"1. Campbell and Shiller "rst determine the order of integration of the variables, and if these are I(1) they build a VAR model for S and Dy . Then the restrictions R R\ implied by the rational expectations hypothesis are derived for the coe$cients of the VAR. When d is known the VAR is estimated under the restrictions and a test of the restrictions is performed. The approach suggested here is to build a VAR for the variables > and R y in levels allowing for cointegration. This allows one to test that the rank is R\ correctly speci"ed taking into account the dynamics of the model. Next one can investigate the stationarity of the individual variables and the speci"cation of the cointegrating relation by asymptotic s tests. Finally, we derive the restrictions implied by the economic hypothesis and estimate the cointegrated VAR under these restrictions. Thus for known d and r"q"1, there is only a slight di!erence between the methods. In general, if the coe$cients c are known and r"q our method consists of G transforming the process to stationarity and testing the implied restrictions in a cointegrated VAR with given cointegrating relations. In this context we can decompose the test into a test for the cointegrating rank and a test for the restrictions on the other parameters. The main di!erence is in the case where r'q, where the analysis proposed in the present paper shows how a combination of reduced rank regression and regression gives the maximum-likelihood estimator and hence the likelihood ratio test of the restrictions on the VAR. If d is unknown the coe$cients c depend on unknown parameters, which have G to be estimated. In this case the algorithm for "nding the maximum-likelihood estimator requires iteration. Once that is done, however, one obtains an e$cient estimate of all parameters in the model under investigation, including d. The way this problem is handled by Campbell and Shiller is to suggest various estimators for d and then proceed with an analysis for known d as above. Thus the analysis will depend on which value is taken as an estimator for d. An additional advantage of the procedure we suggest is that it is easy to construct asymptotic con"dence intervals for d. Let ¸(d) denote the value of the logarithm of the likelihood as a function of d, i.e. the pro"le or concentrated likelihood for d. If dK is the maximum likelihood estimator, a 100(1!a) con"dence region is then given by +d:!2 log(¸(d))#2 log(¸(dK )))c , where c is the 1!a \? \? percentile in the s distribution with 1 degree of freedom. To illustrate the computation of the likelihood ratio tests, we have considered the data for real annual prices on stocks and dividends for the period 1871 to

90

S. Johansen, A.R. Swensen / Journal of Econometrics 93 (1999) 73}91

1986 used by Campbell and Shiller (1987). As they indicate in their Table 3B, a VAR model of order 2 is appropriate, although there are some signs of conditional heteroscedascity in the residuals. Using the maximal eigenvalue and trace statistics of Johansen and Juselius (1990) a reduced rank model of order r"1 is seen to "t well. The normalized eigenvector is (1,!33.55) corresponding to a discount rate of 1/33.55"2.98% and a discount factor d"33.55/34.55"0.971. One can now "t a present value model corresponding to a particular discount factor d by  running the regression described earlier. Notice that the three models, the reduced rank VAR model, the reduced rank VAR model with normalized cointegration vector (1,!d /(1!d )), and "nally   the reduced rank VAR present value model with an imposed discount factor equal to d , are nested. Hence since we use likelihood ratio tests, we can  decompose the s into two parts. The "rst term is the test statistic for testing whether the normalized cointegration vector equals (1,!d /(1!d )). The last   term corresponds to the imposition of the present value model on the short-run dynamics of the process. It should be compared to a s distribution with 4 degrees of freedom. Finally, we estimated the discount factor d as described above by iterating the regressions for various values of d . The maximum-likelihood estimator  equals 0.972. Constructing a 90% con"dence interval as explained above, yields the interval (0.965, 0.0978). The pro"le likelihood is rather steep, so that the 99% con"dence interval is not much wider; (0.957, 0.983). When we test the present value model within the reduced rank model, the appropriate s value is 7.45 with 4 degrees of freedom with P-value slightly above 0.1. Thus we "nd that although the present value model cannot be rejected at a 10% level using the likelihood ratio test, for reasons explained by Campbell and Shiller (1987), the estimated discount factor does not correspond to a reasonable value. To carry out the computations for estimating d a program in GAUSS 2.1 was written. It consists of less than 40 statements. A run on a 486 personal computer, doing 101 regressions, i.e. evaluating the likelihood for d ranging from 0.9 to 1.0 with a step length of 0.001, takes less than "ve seconds.

Acknowledgements We wish to thank Niels Haldrup for drawing our attention to restrictions involving lagged variables and M. Hashem Pesaran for suggesting a simpli"cation of the treatment of present value models. Also the referees, an the associated editor and the coeditor C. Hsiao are thanked for some valuable suggestions and careful reading of the manuscript.

S. Johansen, A.R. Swensen / Journal of Econometrics 93 (1999) 73}91

91

References Ahn, S.K., Reinsel, G.C., 1990. Estimation for partially non-stationary multivariate autoregressive models. Journal of the American Statistical Association 85, 813}823. Baillie, R.T., 1989. Econometric tests of rationality and market e$ciency. Econometric Reviews 8, 151}186. Campbell, J.Y., 1987. Does saving anticipate declining labor income? An alternative test of the permanent income hypotheses. Econometrica 55, 1249}1273. Campbell, J.Y., Shiller, R.J., 1987. Cointegration tests and present value models. Journal of Political Economy 95, 1062}1088. Cuthbertson, K., Taylor, M.P., 1990. Money demand, expectations, and the forward-looking model. Journal of Policy Modeling 12, 289}315. Engle, R.F., Granger, C.W.J., 1987. Co-integration and error correction: Representation, estimation and testing. Econometrica 55, 251}276. Engsted, T., Haldrup, N., 1994. The linear quadratic adjustment cost model and the demand for labour. Journal of Applied Econometrics 9 (Suppl.), S145}S159. Hansen, L.P., Sargent, T.J., 1981. Exact linear rational expectations models: Speci"cation and estimation. Research Department Sta! Report 71, Federal Reserve Bank of Minneapolis. Hansen, L.P., Sargent, T.J., 1991. Exact linear rational expectations models: Speci"cation and estimation. In: Hansen, L.P., Sargent, T.J. (Eds.), Rational Expectations Econometrics. Westview Press, Boulder. Johansen, S., 1991. Estimation and hypothesis testing of cointegration in Gaussian vector autoregressive models. Econometrica 59, 1551}1580. Johansen, S., 1992. Cointegration in partial systems and the e$ciency of single-equation analysis. Journal of Econometrics 52, 389}402. Johansen, S., 1995. Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press, Oxford. Johansen, S., Juselius, K., 1990. Maximum likelihood estimation and inference on cointegration } with application to the demand for money. Oxford Bulletin of Economics and Statistics 52, 169}210. Johansen, S., Juselius, K., 1992. Testing structural hypotheses in a multivariate cointegration analysis of the PPP and the UIP for UK. Journal of Econometrics 53, 211}244. Muth, J.F., 1960. Optimal properties of exponentially weighted forecasts. Journal of the American Statistical Association 55, 299}306. Muth, J.F., 1961. Rational expectations and the theory of price movements. Econometrica 29, 315}335.