On Excess Differencing in Discrete Time ...

On Excess Differencing in Discrete Time Representations of Cointegrated Continuous Time Models with Mixed Sample∗

Marcus J. CHAMBERS University of Essex May 2006; revised June 2006

Abstract: This paper investigates the source of apparent excess differencing that has been observed to occur in discrete time representations corresponding to cointegrated continuous time systems with a mixed sample of stock and flow variables. Two methods of deriving exact discrete time models are considered and excess differencing is found to manifest itself in both, and in each case the source of the differencing is identified. It is also shown that the excess differencing does not arise in models containing only stock variables or only flow variables. Some further analytical and computational results that enable Gaussian estimation to be implemented are also provided.

∗

I would like to thank Zamros Dzulkafli for checking the derivations reported in Section 3, as well as seminar participants at the Workshop in memory of Rex Bergstrom held at the University of Westminster Business School, 2nd December, 2005 for comments on some preliminary results. This version was prepared for the A.R. Bergstrom Memorial Conference held at the University of Essex, 24th–25th May, 2006, the participants of which I also thank for their comments, particularly Peter Phillips. Address for correspondence: Professor Marcus J. Chambers, Department of Economics, University of Essex, Wivenhoe Park, Colchester, Essex CO4 3SQ, England. Tel: +44 1206 872756; fax: +44 1206 872724; e-mail: [email protected].

1. INTRODUCTION To his students and colleagues Rex Bergstrom’s passion for economics and econometrics was clearly evident. Nowhere was this more apparent than when he was discussing continuous time methods, an area in which he worked extensively for much of his career and for which he is perhaps best known. Indeed, he continued working on his continuous time macroeconomic model of the UK and the related econometric techniques beyond his retirement in 1992, and it is an ensuing article, Bergstrom (1997), that provides the motivation for the present paper. The model considered in Bergstrom (1997) is a system of mixed first- and second-order stochastic differential equations with unobservable stochastic trends and mixed stock and flow variables, and it corresponds to the form of the macroeconometric model of the UK developed in Bergstrom and Nowman (2006). The model allows for cointegration among the variables in the system and its discrete time representation is notable for the fact that it is written in terms of the first-differences of the observations. At first sight the discrete time model therefore appears to correspond to a system in which all variables are integrated of order one and not cointegrated, but Bergstrom (2006) provides an example which demonstrates that, in a model with unobservable stochastic trends, the value of the Gaussian likelihood function is unaffected whether the entire system is differenced or whether it is differenced only in directions orthogonal to the cointegration vectors. Since Gaussian estimation was the motivation for the derivation of the exact discrete time model such excess differencing was therefore shown to be unimportant, but Bergstrom (1997, p.485) did acknowledge that a study of the asymptotic sampling properties of the Gaussian estimator would be aided by the derivation of a discrete time model that more closely resembles cointegrated systems formulated directly in discrete time. This paper picks up the issue of excess differencing in cointegrated continuous time models and attempts to identify its source. The model considered is a system of first-order stochastic differential equations with mixed stock and flow variables in which the system matrix displays the reduced rank implied by cointegration. Two solution methods are analysed and each one is found to suffer from excess differencing. The first method derives the discrete time representation for the vector of observed variables directly, while the second treats the stationary linear combinations of the variables (the cointegrating vectors) separately from the orthogonal combinations. The source of the excess differencing is identified in each case, and it is shown that this phenomenon arises only in systems containing mixed stock and flow variables. Indeed, when only stock variables or only flow variables are present, the discrete time representation takes the form of a discrete time error correction model, but the presence of the mixed sample results in an autoregressive representation in first differences. Also derived are the autocovariance matrices of the discrete time distur-

1

bance vectors that are utilised in specifiying the time domain Gaussian likelihood function, and some computational issues are briefly discussed as an aid to the implementation of the formulae derived. It is perhaps also worth mentioning that alternative approaches exist for handling cointegrated continuous time models that do not rely on the exact discrete time model of the type derived in Bergstrom (1997) and that do not display the excess differencing. Harvey and Stock (1988) propose Kalman filter techniques for handling systems with common stochastic trends which avoid the need to derive an exact discrete model for the observations and which can be used to obtain Gaussian estimates. Exact discrete time models in the form of first-order error correction models have been derived by Phillips (1991), Chambers (2003) and Chambers and McCrorie (2007), where in each of these cases the dynamics, potentially generated by higher-order systems, are assigned to the disturbance vector. Phillips suggests frequency domain regression techniques for the estimation of the cointegrating vectors; Chambers uses the representation in a theoretical analysis of the asymptotic efficiency of optimal estimators; and Chambers and McCrorie use frequency domain techniques to approximate the likelihood function and also derive the asymptotic properties of the resulting estimators for the parameters of the cointegrating vectors as well as those governing the dynamics. These methods should be viewed as complementary to those outlined here.

2. THE MODEL To examine the issues raised above we shall consider the following first-order model dy(t) = [a + bt + Ay(t)] dt + ζ(dt), t > 0

(1)

where y(t) is an n × 1 vector of continuous time random variables, ζ(dt) is an n × 1 vector of random measures satisfying E[ζ(dt)] = 0, E[ζ(dt)ζ(dt)0 ] = Σdt and E[ζ(∆1 )ζ(∆2 )0 ] = 0 for disjoint intervals ∆1 and ∆2 , a and b are n × 1 vectors of constants, and A = αβ 0 where α and β are n × r matrices of rank 0 < r < n. The r cointegrating relationships are therefore depicted by the stationary r × 1 vector β 0 y(t). In circumstances in which the cointegrating relationships themselves contain intercepts and deterministic trend terms, the vectors a and b can be restricted appropriately; details of how this is achieved in discrete time cointegrated systems can be found in Pesaran, Shin and Smith (2000) and carry over straightforwardly to the continuous time model. The vector y(t) shall be assumed to comprise both stock and flow variables and, without loss of generality, we shall write y(t) = [y s (t)0 , y f (t)0 ]0 , where y s (t) is an ns × 1 vector of stock variables, y f (t) is an nf × 1 vector of flow variables, and ns + nf = n. Note that the first ns elements of the initial state vector y(0) are observed and equal to y s (0), while y f (0) is unknown.

2

The observations on the stock variables are of the form yts = y s (t) (t = 0, 1, . . . , T ) while those on the flow variables are of the form ytf =

Rt

t−1 y

f (r)dr

(t = 1, . . . , T ), T denoting

sample size. The objective is to relate the unknown parameters of the model (1) to the discrete time observations, which shall be arranged in the n × 1 vector yt = [yts0 , ytf 0 ]0 . Note that wtf = y f (t) is unobservable, as is wts =

Rt

t−1 y

s (r)dr,

these components in the unobservable n × 1 vector wt =

and it is convenient to arrange

[wts0 , wtf 0 ]0 .

As we shall see below the

key to deriving discrete time representations that can be used for estimation is the ability to solve out the elements of the unobservable vector wt from the difference equations that are implied by the solution to (1), which is given by y(t) = etA y(0) +

Z

t

e(t−s)A (a + bs)ds +

0

where etA =

Z

t

e(t−s)A ζ(ds), t > 0,

(2)

0

P∞

j j=0 (tA) /j!

denotes the matrix exponential. It can be shown (see Appendix

C) that etA can be expressed more simply in terms of the exponential of B = β 0 α, given by 



etA = In + αB −1 etB − Ir β 0 ,

(3)

where In denotes the n × n identity matrix. It is clear from this representation that the matrix etA − In retains the reduced rank of A.

3. METHOD 1 There are three key equations that motivate this method. The first two are implications of the solution of the model, given by (2), from which it follows, defining J = eA , that y(t) = Jy(t − 1) + µ + γt + ηt , t = 1, . . . , T, where µ = Ga − Hb, γ = Gb, G =

R1 0

(4)

esA ds, H =

R1 0

sesA ds, and ηt =

Rt

(t−s)A ζ(ds). t−1 e

Integrating (4) over the interval (t − 1, t] yields Z

t

Z

t−1

y(r)dr = J t−1

y(r)dr + m + γt + vt , t = 1, . . . , T,

(5)

t−2

where m = µ − (γ/2) and vt =

Rr (r−s)A ζ(ds)dr. t−1 r−1 e

Rt

Finally, integrating the model (1)

over (t − 1, t] yields Z

t

y(r)dr + et ,

∆y(t) = g + bt + A

(6)

t−1

where g = a − (b/2) and et =

Rt

t−1 ζ(dr).

These three equations can be used to derive the

following exact discrete time representation, in which vectors and matrices are partitioned conformably with the stock and flow components (so that, for example, the vector αf is nf × 1 and the matrix Jsf is ns × nf ). THEOREM 1. Let y(t) be generated by (1). Then, if ns ≥ r, the observable vector yt

3

satisfies y1 = Φ0 y(0) + φ0 + u1 ,

(7)

∆yt = Φ∆yt−1 + φ1 + φ2 t + ut , t = 2, . . . , T,

(8)

where the parameter matrices and disturbance vectors in (8) and (7) are defined by 

Φ= 

φ0 = 

Jss + Jsf αf Ps 0 αf F Ps µs + γs mf0

0







 , Φ0 =  

 , φ1 = 

Z 0

Jsf

Gf s Gf f

  , Ps = (α0 αs )−1 α0 , F = B −1 (eB − Ir ), s s 

γs − Jsf [bf + αf Ps (gs − bs )]

1Z r

e(r−s)A dsdr, H0 =

0

Z

1Z r

0



 , φ2 = 

mf − αf F Ps (gs − bs )

m0 = G0 a + H0 b, G0 = 

Jss

Jsf (bf − αf Ps bs ) γf − αf F Ps bs

 ,

re(r−s)A dsdr,

0



ust = ∆ηts + Jsf eft−1 − αf Ps est−1 , (t = 2, . . . , T ), us1 = η1s , uft

=

vtf

−

αf F Ps est−1 ,

(t = 2, . . . , T ),

uf1

=

ρf1 ,

Z

1Z r

ρ1 = 0

e(r−s)A ζ(ds)dr.

0

Remark 1. The system of equations in (8) is entirely in terms of the first differences ∆yt . This is somewhat unusual in view of the cointegration in the system, but is entirely in accordance with the results of Bergstrom (1997). Remark 2. Another interesting feature of the system (8) is that it is entirely driven by lagged values of ∆yts but not of ∆ytf . A possible explanation for this is that ∆yts contains, up to a white noise vector, information about the cointegrating relationship β 0

Rt

t−1 y(r)dr.

To see this, the first ns equations of (6) give 



∆yts = gs + bs t + αs βs0 wts + βf0 ytf + est ,

(9)

s and hence the presence of ∆yt−1 as the driving force in the system is acting as a proxy for f s the partially unobservable cointegration term βs0 wt−1 + βf0 yt−1 . Indeed, (6) appears to be

the source of the excess differencing in the discrete time model. Remark 3. The condition ns ≥ r is needed so that the matrix Ps can be computed. It enables (9) to be solved for βs0 wts + βf0 ytf in terms of ∆yts . Remark 4. When the sample consists entirely of stock variables the exact discrete time model is provided directly by (4). Noting, from (3), that J = In + αF β 0 , it is clear that in this case y(t) satisfies the error correction model (ECM) ∆y(t) = αF β 0 y(t − 1) + µ + γt + ηt , where ηt is vector white noise. Similarly, in the case of a sample comprised entirely of flow

4

variables, (5) yields the ECM ∆yt = αF β 0 yt−1 + m + γt + vt , in which vt is a vector MA(1) process. This emphasises the fact that it is essentially the presence of a mixed sample that results in the form of the discrete time model in (8).

4. METHOD 2 An alternative approach is to derive an exact discrete time model for the r stationary linear combinations β 0 yt as well as the n−r linear combinations of the differences ∆yt in orthogonal 0 ∆y , directions to the cointegrating space. These latter combinations will be denoted β⊥ t 0 β = 0. We shall also assume that β and β 0 where β⊥ in an n × (n − r) matrix satisfying β⊥ ⊥ 0 β =I 0 0 are normalised so that β 0 β = Ir , β⊥ n−r and ββ + β⊥ β⊥ = In . The usual approach in ⊥

discrete time is to define the matrix Kd = [β, β⊥ ] which satisfies Kd0 Kd = Kd Kd0 = In and to transform the model by pre-multiplying by Kd0 . When applied to the continuous time system (1) this yields Kd0 dy(t) = Kd0 a + Kd0 bt + Kd0 AKd Kd0 y(t) dt + Kd0 ζ(dt), 



(10)

which can also be written as the pair of equations dβ 0 y(t) =



β 0 a + β 0 bt + β 0 αβ 0 y(t) dt + β 0 ζ(dt),

(11)

0 β⊥ dy(t) =



0 0 0 0 β⊥ a + β⊥ bt + β⊥ αβ 0 y(t) dt + β⊥ ζ(dt).

(12)





The difficulty in working with this representation when there is a mixed sample of stock and flow variables lies in the need to treat the temporal aggregation of the different components 0 y(t) in different ways. In particular, because β 0 y(t) = β 0 y s (t) + β 0 y f (t), its of β 0 y(t) and β⊥ s f

counterpart in terms of observed variables is β 0 yt = βs0 yts + βf0 ytf , which involves elements of β 0 y(t) and β 0

Rt

t−1 y(r)dr

which can’t be extracted separately from each expression.

Our proposed solution to this problem utilises an extended transformation matrix K of dimension n × 2n defined by 

K=

βs

0

βs⊥

0

0

βf

0

βf ⊥

 

and which is normalised so as to satisfy K 0 K = I2n and KK 0 = In . The transformation K 0 y(t) then enables us to pick out terms such as βs0 y s (t) and βf0 y f (t), as well as their orthogonal counterparts, and treat them separately for the purposes of the temporal aggregation.

5

It is convenient to define the random vector z(t) = K 0 y(t) = [z1 (t)0 , z2 (t)0 ]0 , where 

z1 (t) = 



βs0 y s (t)



 (2r × 1), z2 (t) = 

βf0 y f (t)

0 y s (t) βs⊥

βf0 ⊥ y f (t)

  (2(n − r) × 1),

as well as the matrix C = K 0 αβ 0 K and vectors a = K 0 a and b = K 0 b which are of the form  

C=

βs0 αs

  0  β α = f f  0  βs⊥ αs 0  

C1 0 C2



βs0 αs

0 0

βf0 αf

0 0 

0 α βs⊥ s

      b1 a1 , a =  . , b =    b2 0 0  a2

βf0 ⊥ αs βf0 ⊥ αs 0 0

Then, defining ζ(dt) = K 0 ζ(dt), the transformed system can be written i

h

dz(t) = a + bt + Cz(t) dt + ζ(dt),

(13)

whose solution is of the same form as that given in (2) for y(t): t

Z

tC

z(t) = e z(0) +

(t−s)C

e



Z



t

a + bs ds +

e(t−s)C ζ(ds), t > 0.

(14)

0

0

The solution vector z(t) therefore also satisfies the stochastic difference equation z(t) = eC z(t − 1) + ct + t ,

(15)

where 

ct = 



c1t

Z =

c2t



t

(t−s)C

e t−1





a + bs ds and t = 

1t 2t

 Z =

t

e(t−s)C ζ(ds).

t−1

From the form of the matrix C we find that 

Cj = 

C1j

0

C2 C1j−1

0

where Ψ1 (s) = 

eC = 

Θ





 and so esC = 

j j−1 j=1 s C1 /j

P∞

esC1



0

C2 Ψ1 (s) I2(n−r)

,

!. It is also convenient to define



0

Γ I2(n−r)

,

where Θ = eC1 and Γ = C2 Ψ1 (1). Furthermore, from the form of the submatrix C1 , it can be shown that 

Θ=

Θss

Θsf

Θf s Θf f





=

Θss

Θss − Ir

Θf f − Ir

Θf f





=

Θsf + Ir

Θsf

Θf s

Θf s + Ir

 .

This is an important result in what follows and can be traced as the source of the apparent excess differencing in the discrete time model below. Note, too, that the partitions of the matrix Θ are all r × r, corresponding now to the r × 1 components βs0 yts and βf0 ytf , rather than the stock and flow vectors themselves, as in section 3. Also, in Theorem 2 below, the subscripts 1 and 2 denote partitions of vectors conformably with z1 and z2 and, hence, are 2r × 1 and 2(n − r) × 1, respectively. 6

THEOREM 2. Let y(t) be generated by (1). Then, assuming that ns ≥ r, nf ≥ r, and that Θsf and Θf s are nonsingular, the observable vector yt satisfies f 0 f s 0 s β 0 y1 = K10 βs y (0) + K10 βf y (0) + τ 11 + ξ 11 ,

(16)

f 0 f 0 s 0 s [y1 − y(0)] = N10 βs y (0) + N10 βf y (0) + τ 21 + ξ 21 , β⊥

(17)

β 0 y2 = Θss βs0 [y1s − y s (0)] + Θf f βf0 [y1f − y f (0)] + K 20 β 0 y(0) + τ 12 + ξ 12 , 0 β⊥ ∆y2 = Γss βs0 [y1s − y s (0)] + Γf f βf0 [y1f − y f (0)] + N 20 β 0 y(0) + τ 22 + ξ 22 ,

(18) (19)

f s β 0 ∆yt = K2s βs0 ∆yt−1 + K2f βf0 ∆yt−1 + τ 1t + ξ 1t , t = 3, . . . , T,

(20)

f 0 s β⊥ ∆yt = N1s βs0 ∆yt−1 + N1f βf0 ∆yt−1 + τ 2t + ξ 2t , t = 3, . . . , T,

(21)

where f s s e f s , N f = Γsf + Γ ef f , K10 = Θss + Ψf0 s , K10 = Θsf + Ψf0 f , N10 = Γss + Γ 10

K 20 = Θss Θf f + Θf f Θss − Θss − Θf f + Ir , N 20 = Γss Θf f + Γf f Θss , f −1 K2s = Θsf Θf f Θ−1 sf Θss − Θsf Θf s , K2 = Θf s Θss Θf s Θf f − Θf s Θsf , f −1 N1s = Γss + Γsf Θf f Θ−1 sf , N1 = Γf f + Γf s Θss Θf s ,

Z

Ψ0 =

1

rC1

e

Z e = C2 Ψ2 (1), Ψ2 (r) = dr, Γ

0

r

Ψ1 (s)ds, 0

f f s τ 11 = cs11 + ψe11 , τ 12 = cs12 + Θsf cf11 + ψ12 + Θf s ψ11 , f f s τ 21 = cs21 + ψe21 , τ 22 = cs22 + Γsf cf11 + ψ22 + Γf s ψ11 , f f −1 f s s s τ 1t = τ1t + τ1t , τ 2t = Γsf Θ−1 sf τ1t + Γf s Θf s τ1t + τ2t + τ2t , t = 3, . . . , T, f −1 s s s s s τ1t = cs1t − Θsf Θf f Θ−1 sf c1,t−1 + Θsf c1,t−1 , τ2t = c2t − Γsf Θsf c1t , t = 3, . . . , T, f f f f f −1 f s τ1t = ψ1t − Θf s Θss Θ−1 f s ψ1,t−1 + Θf s ψ1,t−1 , τ2t = ψ2t − Γf s Θf s ψ1t , t = 3, . . . , T,



ψe1 =  

ψt = 

ψe11 ψe21 ψ1t ψ2t



1Z r

Z =  Z =

0

0

t

Z





e(r−s)C a + bs dsdr,

r





e(r−s)C a + bs dsdr, t = 2, . . . , T,

t−1 r−1

f f s ξ 11 = s11 + νe11 , ξ 12 = s12 + Θsf f11 + ν12 + Θf s ν11 , f f s ξ 21 = s21 + νe21 , ξ 22 = s22 + Γsf f11 + ν22 + Γf s ν11 , f f −1 f s s s ξ 1t = ξ1t + ξ1t , ξ 2t = Γsf Θ−1 sf ξ1t + Γf s Θf s ξ1t + ξ2t + ξ2t , t = 3, . . . , T, f −1 s s s s s ξ1t = s1t − Θsf Θf f Θ−1 sf 1,t−1 + Θsf 1,t−1 , ξ2t = 2t − Γsf Θsf 1t , t = 3, . . . , T,

7

f f f f f −1 f s ξ1t = ν1t − Θf s Θss Θ−1 f s ν1,t−1 + Θf s ν1,t−1 , ξ2t = ν2t − Γf s Θf s ν1t , t = 3, . . . , T,



νe1 =  

νt = 

νe11 νe21 ν1t ν2t



1Z r

Z =  Z =

0

0

t

Z

e(r−s)C ζ(ds)dr,

r

e(r−s)C ζ(ds)dr, t = 2, . . . , T.

t−1 r−1

Remark 1. The representation in (20) appears to be over-differenced in that β 0 ∆yt , rather than β 0 yt , is on the left-hand-side. Once again this would seem to be due to the simultaneous presence of both stock and flow variables. If the sample comprised only stock variables, so that β 0 yt = βs0 yts , then (34) and (35) in Appendix A provide the exact discrete time model for t = 1, . . . , T : 0 β 0 yt = Θβ 0 yt−1 + c1t + 1t , β⊥ ∆yt = Γβ 0 yt−1 + c2t + 2t .

On the other hand, if the sample consisted entirely of flow variables, so that β 0 yt = βf0 ytf , then (43) and (44) in Appendix A describe the exact discrete time model for t = 2, . . . , T : 0 β 0 yt = Θβ 0 yt−1 + ψ1t + ν1t , β⊥ ∆yt = Γβ 0 yt−1 + ψ2t + ν2t .

Neither of these representations requires the stationary linear combination β 0 yt to be differenced, emphasising that it is the necessity of treating the stock and flow components differently in a mixed sample that results in the differencing of this variable. Remark 2. The differences of the lagged stock and flow variables on the right-handsides of (20) and (21) appear separately rather than as the single term β 0 ∆yt−1 . This is yet another manifestation of the fact that the variables are, by necessity, treated differently in the derivation of the discrete time model. In the proof of Theorem 2 in Appendix A it can be seen in (48) and (53) that two seemingly more natural representations occur, which are f f s s + τ 1t + ξ 1t , − K2s βs0 yt−2 − K2f βf0 yt−2 β 0 yt = K1s βs0 yt−1 + K1f βf0 yt−1 f f 0 s s β⊥ ∆yt = N1s βs0 yt−1 + N1f βf0 yt−1 + N2s βs0 yt−2 + N2f βf0 yt−2 + τ 2t + ξ 2t .

The lagged terms on the right-hand-sides of these equations, however, are not necessarily stationary, because the stationarity of β 0 yt does not imply the stationarity of βs0 yts and βf0 ytf themselves. The differences of these components that appear in Theorem 2 arise because of the relationships between the coefficient matrices depicted in Proposition 1 in Appendix C and ensure the stationarity of the individual terms in the resulting discrete time representation. Remark 3. The source of the apparent excess differencing in the discrete time representation can be traced to the relationship between the submatrices of the matrix Θ = eC1 .

8

These have an impact most notably in transforming the seemingly natural representation for β 0 yt in (48), reproduced above, into the first-differenced representation (20). The form of the matrix eC1 itself arises due to the transformation from continuous time to discrete time and the necessity to treat stock variables and flow variables differently; the apparent excess differencing does not arise with stocks or flows alone (see Remark 1). Remark 4. The conditions ns ≥ r and nf ≥ r arise so that an appropriately normalised transformation matrix K can be defined. In the usual discrete time case it is straightforward to take Kd = [β, β⊥ ] where, if β ∗ denotes the underlying matrix of cointegrating parameters of interest, it is possible to take β = β ∗ (β ∗0 β ∗ )−1/2 (because n > r) which satisfies β 0 β = Ir . In the situation here it is necessary to treat βs and βf separately, and such a normalisation is not feasible if ns < r or nf < r, both of which are possible while still retaining n > r. Remark 5. The increments y1s − y s (0) and y1f − y f (0) that appear in (17), (18) and (19) are stationary (net of trend). This can be seen by noting that Θss = Θsf + Ir and Ψf0 f = Ψf0 s + Ir and substituting into (54) and (57), respectively, in Appendix A to give i

h

f . βs0 [y1s − y s (0)] = Θsf β 0 y(0) + cs11 + s11 , βf0 y1f − y f (0) = Ψf0 s β 0 y(0) + ψe1f + νe11

5. GAUSSIAN ESTIMATION AND AUTOCOVARIANCES Estimates of the unknown parameters of the model (1) can be obtained by maximising the Gaussian likelihood function. In the case of Method 1 it is convenient to define the nT × 1 vector u = (u01 , . . . , u0T )0 and its nT × nT block-diagonal Toeplitz covariance matrix Ω = E(uu0 ). Assuming that u is multivariate Gaussian (which is equivalent to assuming that ζ(dt) is Gaussian or, equivalently, the increment of a Brownian motion process), minus twice the negative of the logarithm of the Gaussian likelihood function can be written (ignoring the constant term) ln L1 = ln |Ω| + u0 Ω−1 u. In order for this approach to be followed it is therefore necessary to derive expressions for the autocovariances of the discrete time disturbance vectors that define the matrix Ω. This is facilitated by first reducing the expressions involving double integrals of the random measure ζ(dt) to single integrals using the approach of Bergstrom (1997) and McCrorie (2000); details can be found in Appendix B. Theorem 3 provides the autocovariances of the discrete time disturbance vector ut in Method 1. THEOREM 3. Let ut be defined as in Theorem 1. Then E(u1 u01 )

Z

= Ω11 =

1

J0 (s)ΣJ0 (s)0 ds,

0

9

E(ut u0t ) = Ω0 =

Z

1

0

E(ut u0t−1 ) = Ω1 =

1

Z

J0 (s)ΣJ0 (s)0 ds +

J1 (s)ΣJ1 (s)0 ds, t = 2, . . . , T,

0 1

Z

J1 (s)ΣJ0 (s)0 ds, t = 2, . . . , T,

0

where J0 (s) = S1 F0 (s) + S2 F1 (s), J1 (s) = S2 F2 (s) − S1 F0 (s) + S3 , 

S1 = 

Ins

0

0

0





0

 , S2 = 

F0 (s) = esA , F1 (s) =

s

Z



0



 , S3 = 

0 Inf

erA dr, F2 (s) =

0

1

Z

−Jsf αf Ps Jsf −αf F Ps

0

 ,

erA dr.

s

The disturbance vector ut is therefore a vector MA(1) process implying that ∆yt in (8) is a vector ARMA(1,1) process. The autocovariances for the discrete time disturbances in Method 2 are provided in Theorem 4. 0

0

THEOREM 4. Let ξ t = (ξ 1t , ξ 2t )0 , t = 1, . . . , T . Then 0 E(ξ 1 ξ 1 )

=

0

Z

0

Z

1

1

W0 (s)ΣW0 (s)0 ds +

E(ξ 3 ξ 1 ) =

W3 (s)ΣW3 (s)0 ds,

0 1

W2 (s)ΣW0 (s)0 ds,

0 1

Z

=

0

1

Z

W1 (s)ΣW0 (s) ds + 0 0

=

1

Z

0

W0 (s)ΣW0 (s) ds + 0

Z

W1 (s)ΣW1 (s) ds + 0

Z

0

W2 (s)ΣW3 (s)0 ds,

0

1

Z

E(ξ t ξ t−1 ) =

1

W1 (s)ΣW0 (s)0 ds +

E(ξ t ξ t−2 ) =

1

W2 (s)ΣW2 (s)0 ds, t = 3, . . . , T,

0

Z

0

Z

0

1

Z

0

Z

0

W3 (s)ΣW0 (s)0 ds,

0

E(ξ 2 ξ 2 ) =

0 E(ξ t ξ t )

W0 (s)ΣW0 (s)0 ds,

0

E(ξ 2 ξ 1 ) =

0 E(ξ 3 ξ 2 )

1

Z

1

W2 (s)ΣW1 (s)0 ds, t = 4, . . . , T,

0 1

W2 (s)ΣW0 (s)0 ds, t = 4, . . . , T,

0

where W0 (s) = S4 R0 (s)+S6 R1 (s), W1 (s) = S5 R0 (s)+S6 R2 (s)+S7 R1 (s), W2 (s) = S7 R2 (s), W3 (s) = S8 R0 (s) + S6 R2 (s) + S9 R1 (s), 

S4 =  

S6 =  

S8 = 

Ir 0 0

0

0 In−r 0

0 Ir 0 0

0

0

0 In−r 0 0

0 Γsf

0 0



 , S5 =  

0

0 Θsf





 , S7 = 





 , S9 = 

−Θsf Θf f Θ−1 Θsf sf

0 0

−Γsf Θf f Θ−1 sf

0 0

Γsf

Θf s −Θf s Θss Θ−1 0 0 fs Γf s

−Γf s Θss Θ−1 fs

Θf s 0 0 0 Γf s

0 0 0 10

 ,

0 0

 ,  ,



R0 (s) =  

R2 (s) = 

esC1



0



 K 0 , R1 (s) = 

C2 Ψ1 (s) In−r Ψ3 (1) − Ψ3 (s)

0

−C2 Ψ2 (s)

s−1

Ψ3 (s)

0

C2 Ψ2 (s) s

  K 0 , Ψ3 (s) =

Z

s

  K 0,

erC1 dr.

0 0

0

Theorem 4 shows that the vector ξ t is an MA(2) process. Defining ξ = (ξ 1 , . . . , ξ T )0 and 0

Ω = E(ξξ ) the objective function for Gaussian estimation can be written 0

ln L2 = ln |Ω| + ξ Ω

−1

ξ.

Although we have derived expressions that determine the components needed for Gaussian estimation using either ln L1 or ln L2 , some further investigation of these expressions is necessary from a practical computational point of view.

6. COMPUTATIONAL ISSUES Inspection of the formulae in Theorems 1 and 2 reveals that a number of the matrices and vectors appear as integrals involving the matrix exponential and other functions. The autocovariances used in computing the Gaussian likelihood functions ln L1 and ln L2 also involve integrals of matrix exponentials. Theorem 5 contains computable representations for the relevant expressions in Method 1. THEOREM 5. The matrices G, H, G0 and H0 , and functions J0 (s) and J1 (s), in Method 1 have the following representations: 1

Z

G=

esA ds = Π1 + Π2 F β 0 , G0 =

0

Z

H= 0

Z

1Z r

0 1

1 se ds = Π1 + Π2 Π3 β 0 , H0 = 2 sA

0

Z

1 e(r−s)A dsdr = Π1 + Π2 B −1 (F − Ir ) β 0 , 2

1Z r

(r−s)A

re 0

0

1 1 dsdr = Π1 + Π2 B −1 Π3 − Ir β 0 , 3 2 



Ji (s) = Ji0 + Ji1 s + Ji2 esB β 0 , i = 0, 1, where Π1 = In − αB −1 β 0 , Π2 = αB −1 , and Π3 = B −1 (eB − F ), J00 = S1 Π1 − S2 Π2 B −1 β 0 , J01 = S2 Π1 , J02 = S1 Π2 + S2 Π2 B −1 , J10 = S2 (Π1 + Π2 B −1 eB β 0 ) − S1 Π1 + S3 , J11 = −S2 Π1 , and J12 = −S1 Π2 − S2 Π2 B −1 . The integrals determining the autocovariance matrices Ω11 , Ω0 and Ω1 then have the representation Z 0

1

Ji (s)ΣJk (s)0 ds =

Z

1h

0

i

h

i0

Ji0 + Ji1 s + Ji2 esB β 0 Σ Jk0 + Jk1 s + Jk2 esB β 0 ds

1 1 1 0 0 0 0 0 + Ji0 ΣβF 0 Jk2 + Ji1 ΣJk0 + Ji1 ΣJk1 = Ji0 ΣJk0 + Ji0 ΣJk1 2 2 3 0 0 0 0 +Ji1 ΣβΠ03 Jk2 + Ji2 F β 0 ΣJk0 + Ji2 Π3 β 0 ΣJk1 + Ji2 Π4 Jk2 , where Π4 =

R1 0

0

esB β 0 ΣβesB ds. 11

(22)

The only remaining integral and matrix exponential that require computation are therefore Π4 and eB . In fact, both can be obtained from the computation of a single matrix exponential, this being 

M = exp 

−B β 0 Σβ 0

B0





=

M11 M12 0

M22

 .

0 M Jewitt and McCrorie (2005), using results of van Loan (1978), show that Π4 = M22 12 0 . Hence the problem is reduced to the computation of eM , and a number of and eB = M22

methods exist for this purpose. The most staightforward of these is truncation of the infinite series representation at some sufficiently large integer n, resulting in the approximation eM ≈ (eM )n = I2r +

n X Mj j=1

j!

.

The integer n can be chosen such that the elements of the difference (eM )n − (eM )n−1 are sufficiently small in absolute value. An investigation of alternative methods of computing matrix exponentials in the context of continuous time macroeconometric models can be found in Jewitt and McCrorie (2005), and although they recommend a method based on a trigonometric representation, they conclude that the problem does not appear to be ill-conditioned for the types of models typically considered in the literature so that the truncation method should be adequate. Method 2 requires similar, but somewhat distinct, considerations, and is slightly more complex than Method 1 from a computational point of view. As no particularly new insights are obtained by presenting the relevant expressions here it will suffice to highlight some differences that arise compared to Method 1. The first of these is that, for some expressions, it appears necessary to truncate infinite series. A leading example is in the computation of the submatrix Γ of eC which is defined as Γ = C2 Ψ1 (1) where Ψ1 (1) =

j−1 j=1 C1 /j!

P∞

is closely

related to the exponential of C1 by noting that C1 Ψ1 (1) = eC1 − I2r . The singularity of C1 rules out simple inversion and premultiplication, so truncation of the infinite series would ape = C2 Ψ2 (1) where Ψ2 (1) = pear necessary. This also applies to Γ

R1 0

Ψ1 (s)ds can be obtained

by integrating the expansion of Ψ1 (s) term by term to give Ψ2 (1) =

j−1 j=1 C1 /j!(j

P∞

+ 1);

again this would appear to require truncation. Secondly, the deterministic components in Method 2 are derived from subvectors of ct and ψt , and these can be shown to be in the form of linear trends by working through the algebra.

7. CONCLUDING COMMENTS This paper has considered two approaches to deriving discrete time representations corresponding to a cointegrated system of first-order stochastic differential equations with mixed

12

stock and flow variables. Both methods are shown to exhibit the excess differencing that was apparent in the exact discrete time model of Bergstrom (1997) corresponding to a system of mixed first- and second-order differential equations. In Method 1, which solves the system in terms of the observable vector, the source of the differencing is identified to be an equation linking the differences of the stock variables to the partially unobservable cointegrating relationships which have to be solved out of the system, while in Method 2, which solves the system in terms of the cointegrating relationships and their orthogonal counterparts, the source of the differencing is embedded within a particular matrix exponential. Furthermore, the excess differencing is shown to be due to the simultaneous presence of stock variables and flow variables, and arises because of the necessity to treat the variables differently in order to relate them to the discrete time observations. The excess differencing does not arise in systems containing only stock variables or only flow variables, in which cases the discrete time models are shown to be in the form of error correction models.

REFERENCES Bergstrom, A.R. (1997) Gaussian estimation of mixed-order continuous-time dynamic models with unobservable stochastic trends from mixed stock and flow data. Econometric Theory 13, 467–505. Bergstrom, A.R. (2006) The effects of differencing on the Gaussian likelihood of models with unobservable stochastic trends: a simple example. Unpublished. Bergstrom, A.R. & K.B. Nowman (2006) A Continuous Time Econometric Model of the United Kingdom with Stochastic Trends. Cambridge: Cambridge University Press. Chambers, M.J. (2003) The asymptotic efficiency of cointegration estimators under temporal aggregation. Econometric Theory 19, 49–77. Chambers, M.J. & J.R. McCrorie (2007) Frequency domain estimation of temporally aggregated Gaussian cointegrated systems. Journal of Econometrics, forthcoming. Harvey, A.C. &J.H. Stock (1988) Continuous time autoregressive models with common stochastic trends. Journal of Economic Dynamics and Control 12, 365–384. Jewitt, G. & J.R. McCrorie (2005) Computing estimates of continuous time macroeconometric models on the basis of discrete data. Computational Statistics and Data Analysis 49, 397–416. McCrorie, J.R. (2000) Deriving the exact discrete analog of a continuous time system. Econometric Theory 16, 998–1015.

13

Pesaran, H.M., Y. Shin & R.J. Smith (2000) Structural analysis of vector error correction models with exogenous I(1) variables. Journal of Econometrics 97, 293–343. Phillips, P.C.B. (1991) Error correction and long run equilibrium in continuous time. Econometrica 59, 967–980. van Loan, C.F. (1978) Computing integrals involving the matrix exponential. IEEE Transactions on Automatic Control AC–23, 395–404.

14

APPENDIX A: PROOFS OF THEOREMS 1 AND 2 Proof of Theorem 1. We shall first establish (8). The last nf equations of (5) are f s ytf = Jf s wt−1 + Jf f yt−1 + mf + γf t + vtf .

(23)

From (3) it follows that Jf s = αf F βs0 and Jf f = Inf + αf F βf0 , hence (23) can be written 



f s ∆ytf = αf F βs0 wt−1 + βf0 yt−1 + mf + γf t + vtf .

(24)

Multiplying the first ns equations of (6) by αs0 and rearranging we obtain 



(αs0 αs ) βs0 wts + βf0 ytf = αs0 (∆yts − gs − bs t − est ) . Solving and lagging by one period yields f s s βs0 wt−1 + βf0 yt−1 = Ps ∆yt−1 − gs − bs (t − 1) − est−1 .




(25)

Substituting (25) into (24) results in the equation for the flow variables: s ∆ytf = αf F Ps ∆yt−1 + φf1 + φf2 t + uft .

(26)

Next, differencing the first ns equations of (4) yields f s ∆yts = Jss ∆yt−1 + Jsf ∆wt−1 + γs + ∆ηts ,

(27)

while first-differencing the last nf equations of (6) yields 



∆wtf = gf + bf t + αf βs0 wts + βf0 ytf + eft .

(28)

Lagging by one period and using (25) gives f s ∆wt−1 = gf + bf (t − 1) + αf Ps ∆yt−1 − gs − bs (t − 1) − est−1 + eft−1 ,




(29)

which can be substituted into (27) to yield s ∆yts = (Jss + Jsf αf Ps )∆yt−1 + φs1 + φs2 t + ust .

(30)

Stacking (30) above (26) gives (8) as required. To derive (7), first set t = 1 in (4) and take the first ns equations: y1s = Jss y s (0) + Jsf y f (0) + φs0 + us1 .

(31)

Next, integrating (2) over the interval (0, 1] gives Z

1

y(r)dr = Gy(0) + G0 a + H0 b + ρ1 ,

(32)

0

the last nf equations of which imply that y1f = Gf s y s (0) + Gf f y f (0) + mf0 + ρf1 .

(33)

Stacking (31) above (33) yields (7) as required.

15

2

Proof of Theorem 2. We shall establish (20) and (21) first. From (15) and the form of

eC

we find that

z1 (t) = Θz1 (t − 1) + c1t + 1t ,

(34)

∆z2 (t) = Γz1 (t − 1) + c2t + 2t .

(35)

From (34) we then obtain f s + cs1t + s1t , βs0 yts = Θss βs0 yt−1 + Θsf βf0 wt−1

βf0 wtf

(36)

f s = Θf s βs0 yt−1 + Θf f βf0 wt−1 + cf1t + f1t .

(37)

f Solving (36) for βf0 wt−1 and lagging by a further period results in the pair of equations f 0 s 0 s s s βf0 wt−1 = Θ−1 sf βs yt − Θss βs yt−1 − c1t − 1t ,






(38) 

f 0 s 0 s s s βf0 wt−2 = Θ−1 sf βs yt−1 − Θss βs yt−2 − c1,t−1 − 1,t−1 ,

(39)

while lagging (37) by one period gives f f s + cf1,t−1 + f1,t−1 . = Θf s βs0 yt−2 + Θf f βf0 wt−2 βf0 wt−1

(40)

Substituting the right hand sides of (38) and (39) into (40) and solving for βs0 yts yields s s s s βs0 yts = K1s βs0 yt−1 − K2s βs0 yt−1 + τ1t + ξ1t ,

(41)

where K1s = Θss + Θsf Θf f Θ−1 sf . For the flows, integrating (15) over (t − 1, t] yields Z

t

z(r)dr = eC

t−1

Z

t−1

z(r)dr + ψt + νt .

(42)

t−2

From the form of eC we have the pair of equations Z

t

Z

t−1

z1 (r)dr = Θ t−1 Z t

∆

z2 (r)dr = Γ t−1

z1 (r)dr + ψ1t + ν1t ,

(43)

z1 (r)dr + ψ2t + ν2t .

(44)

t−2 Z t−1 t−2

Partitioning (43) we obtain f s s s βs0 wts = Θss βs0 wt−1 + Θsf βf0 yt−1 + ψ1t + ν1t ,

(45)

βf0 ytf

(46)

f f f s = Θf s βs0 wt−1 + Θf f βf0 yt−1 + ψ1t + ν1t ,

which can be solved in the same way as (36) and (37) to yield f f f f − K2f βf0 yt−1 + τ1t + ξ1t , βf0 ytf = K1f βf0 yt−1

(47)

where K1f = Θf f + Θf s Θss Θ−1 f s . Combining (41) and (47) we obtain f f s s β 0 yt = K1s βs0 yt−1 + K1f βf0 yt−1 − K2s βs0 yt−2 − K2f βf0 yt−2 + τ 1t + ξ 1t .

(48)

But, from Proposition 1 in Appendix C, K1s = K2s + Ir and K1f = K2f + Ir ; making these 16

substitutions yields (20) as required. In orthogonal directions to β, we obtain from (35) f 0 s βs⊥ ∆yts = Γss βs0 yt−1 + Γsf βf0 wt−1 + cs2t + s2t .

(49)

f Substituting (38) for βf0 wt−1 and rearranging yields 0 s s s + τ2t + ξ2t , βs⊥ ∆yts = M0s βs0 yts + M1s βs0 yt−1

(50)

−1 s where M0s = Γsf Θ−1 sf and M1 = Γss − Γsf Θsf Θss . Similarly, from (44) we obtain f f f s . + ν2t + ψ2t βf0 ⊥ ∆ytf = Γf s βs0 wt−1 + Γf f βf0 yt−1

(51)

s Substituting for βs0 wt−1 using an equation analogous to (38) and rearranging yields f f f βf0 ⊥ ∆ytf = M0f βf0 ytf + M1f βf0 yt−1 + τ2t + ξ2t ,

(52)

f −1 where M0f = Γf s Θ−1 f s and M1 = Γf f − Γf s Θf s Θf f . Combining (50) and (52), and using

(41) and (47) to substitute for βs0 yts and βf0 ytf respectively, yields, after some manipulation and simplification of the matrices, f f s 0 s + τ 2t + ξ 2t , + N2s βs0 yt−2 + N2f βf0 yt−2 β⊥ ∆yt = N1s βs0 yt−1 + N1f βf0 yt−1

(53)

f −1 where N2s = Γsf (Θf s − Θf f Θ−1 sf Θss ) and N2 = Γf s (Θsf − Θss Θf s Θf f ). Using Proposition

1 in Appendix C, which shows that N2s = −N1s and N2f = −N1f , results in (21). Turning to (16), setting t = 1 in (36) gives the stock component βs0 y1s = Θss βs0 y s (0) + Θsf βf0 y f (0) + cs11 + s11 .

(54)

For the flow component, integrate (14) over (0, 1]: Z

1

Z

1

z(r)dr =

rC

e

0

Z

drz(0) +

0

Z

= 0

1Z r

e 0

1

(r−s)C



0



Z

a + bs dsdr + 0

erC drz(0) + ψe1 + νe1 ,

1Z r

e(r−s)C ζ(ds)dr

0

(55)

Because Z

1

0



erC dr = 

Ψ0 0 e Γ

I

 ,

the equations involving z1 are Z 0

1

z1 (r)dr = Ψ0 z1 (0) + ψe11 + νe11 .

(56)

Picking out the flow components: f f βf0 y1y = Ψf0 s y s (0) + Ψf0 f y f (0) + ψe11 + νe11 .

(57)

Combining (54) and (57) yields (16). In orthogonal directions, setting t = 1 in (49) yields 0 βs⊥ ∆y1s = Γss βs0 y s (0) + Γsf βf0 y f (0) + cs21 + s21 .

17

(58)

From (55), the equations for z2 are Z 0

1

e 1 (0) + z2 (0) + ψe21 + νe21 , z2 (r)dr = Γz

(59)

the flow component of which is e f s β 0 y s (0) + Γ e f f β 0 y f (0) + β 0 y f (0) + ψef + νef . βf0 ⊥ y1f = Γ s f f⊥ 21 21

(60)

Combining (58) and (60) yields (17) as required. Turning to (18), setting t = 2 in (36) yields βs0 y2s = Θss βs0 y1s + Θsf βf0 w1f + cs12 + s12 ,

(61)

while putting t = 1 in (37) gives βf0 w1f = Θf s βs0 y s (0) + Θf f βf0 y f (0) + cf11 + f11 .

(62)

Substituting (62) into (61) yields the stock component of (18): βs0 y2s = Θss βs0 y1s + Θsf Θf s βs0 y s (0) + Θsf Θf f βf0 y f (0) + cs12 + Θsf cf11 + s12 + Θsf f11 .

(63)

For the flows, putting t = 2 in (46) yields f f , + ν12 βf0 y2f = Θf s βs0 w1s + Θf f βf0 y1f + ψ12

(64)

while setting t = 1 in (45) yields s s βs0 w1s = Θss βs0 y s (0) + Θsf βf0 y f (0) + ψ11 + ν11 .

(65)

Substituting (65) into (64) gives f f s s + Θf s ν11 . + Θf s ψ11 + ν12 βf0 y2f = Θf f βf0 y1f + Θf s Θss βs0 y s (0) + Θf s Θsf βf0 y f (0) + ψ12

(66)

Combining (63) and (66) yields f 0 f s 0 s βf y (0) + τ 12 + ξ 12 , β 0 y2 = Θss βs0 y1s + Θf f βf0 y1f + K20 βs y (0) + K20 f s = Θ Θ where K20 sf f s + Θf s Θss and K20 = Θsf Θf f + Θf s Θsf . But, using the fact that

Θf s = Θf f − Ir and Θsf = Θss − Ir and making these substitutions, we can show that s = K −Θ and K f = K −Θ , yielding (18) as required. Finally, for the orthogonal K20 20 ss 20 ff 20

components, putting t = 2 in (49) gives 0 βs⊥ ∆y2s = Γss βs0 y1s + Γsf βf0 w1f + cs22 + s22 .

(67)

Substituting (62) into (67) gives 0 βs⊥ ∆y2s = Γss βs0 y1s + Γsf Θf s βs0 y s (0) + Γsf Θf f βf0 y f (0) + cs22 + Γsf cf11 + s22 + Γsf f11 .

(68)

Putting t = 2 in (51): f f βf0 ⊥ ∆y2f = Γf s βs0 w1s + Γf f βf0 y1f + ψ22 + ν22 .

(69)

18

Substituting (65) into (69) yields f f s s βf0 ⊥ ∆y2f = Γf f βf0 y1f + Γf s Θss βs0 y s (0) + Γf s Θsf βf0 y f (0) + ψ22 + Γf s ψ11 + ν22 + Γf s ν11

(70)

which, combined with (68), yields f 0 f 0 s 0 s β⊥ ∆y2 = Γss βs0 y1s + Γf f βf0 y1f + N20 βs y (0) + N20 βf y (0) + τ 22 + ξ 22 , f s = Γ Θ where N20 sf f s + Γf s Θss and N20 = Γsf Θf f + Γf s Θsf . But, from Proposition 1 in s = N − Γ and N f = N − Γ , which Appendix C, Γsf = Γss and Γf s = Γf f , so that N20 20 ss 20 f 20

2

results in (19) as required.

APPENDIX B: PROOFS OF THEOREMS 3, 4, AND 5 Proof of Theorem 3. First note that ηt = t

Z

Z

r

t−1 r−1 Z tZ t

e(r−s)A ζ(ds)dr +

s

t−1 t

Z

F1 (t − s)ζ(ds) +

=

t−1 F0 (t

− s)ζ(ds). We can also write

e(r−s)A ζ(ds)dr

vt = =

Rt

t−1 t−1

Z

t−1

s+1 Z t−1

Z

e(r−s)A ζ(ds)dr

t−2

F2 (t − 1 − s)ζ(ds),

(71)

t−2

where F1 (t − s) =

Rt s

e(r−s)A dr and F2 (t − 1 − s) =

R s+1

e(r−s)A dr. The expressions for F1 (s)

t−1

and F2 (s) are obtained by making the appropriate substitutions. In a similar way we obtain 1Z r

Z

ρ1 =

e(r−s)A ζ(ds)dr =

0

0

1Z 1

Z s

e(r−s)A ζ(ds)dr =

Z

0

1

F1 (1 − s)ζ(ds).

(72)

0

From the definitions we can write ut = S1 ηt − S1 ηt−1 + S2 vt + S3 et−1 and u1 = S1 η1 + S2 ρ1 . Substituting for ηt , vt and et we obtain Z

u1 =

1

J0 (1 − s)ζ(ds), ut =

Z

0

t

J0 (t − s)ζ(ds) +

Z

t−1

J1 (t − 1 − s)ζ(ds), t = 2, . . . , T.

t−2

t−1

2

The autocovariances are derived from these latter expressions. Proof of Theorem 4. First note that t =

Rt

t−1 R0 (t

− s)ζ(ds) while, proceeding as in

the proof of Theorem 3, Z

t

Z

r

e(r−s)C ζ(ds)dr

νt = t−1 r−1 Z tZ t

e(r−s)C ζ(ds)dr +

= s

Z

t−1 t

= t−1

R1 (t − s)ζ(ds) +

Z

Z

s+1 Z t−1

t−1 t−1

e(r−s)C ζ(ds)dr

t−2

R2 (t − 1 − s)ζ(ds),

(73)

t−2

recalling that ζ(dt) = K 0 ζ(dt). Similarly it can be shown that ν1 =

R1 0

R1 (1 − s)ζ(ds). From

the definitions of ξ 1t and ξ 2t it follows that ξ t has the representation ξ t = S4 t + S5 t−1 + S6 νt + S7 νt−1 while ξ 1 = S4 1 + S6 ν1 and ξ 2 = S4 2 + S8 1 + S6 ν2 + S9 ν1 . Substituting 19

the expressions for t and νt in terms of the integrals with respect to ζ(dt) we obtain the representations 1

Z

ξ1 =

W0 (1 − s)ζ(ds), ξ 2 =

0 t

Z

ξt =

W0 (t − s)ζ(ds) +

t−1

Z

2

Z

W0 (2 − s)ζ(ds) +

1

Z

1

W3 (1 − s)ζ(ds),

0

t−1

W1 (t − 1 − s)ζ(ds) +

Z

t−2

W2 (t − 2 − s)ζ(ds)

t−3

t−2

2

for t = 3, . . . , T , from which the autocovariances are straightforwardly derived. Proof of Theorem 5. From (3) we can write esA = Π1 + Π2 esB β 0 , from which 1

Z

G=

(Π1 + Π2 esB β 0 )ds = Π1 + Π2

0

1

esB dsβ 0 .

0

Noting that

R1

sB −1 B 0 e ds = B (e − Ir ) = F yields the required esA and so, using the representation above,

also involves 1

Z

H = Π1

Z

sds + Π2 0

Now

Z

R1 0

1

expression. The matrix H

sesB dsβ 0 .

0

sesB ds = B −1 [sesB ]10 − B −2 [esB ]10 = B −1 eB − B −2 (eB − Ir ) = Π3 , resulting in the

required expression. Turning to G0 , making the change of varible to q = r − s we obtain G0 =

R1 Rr 0

[

0

eqA dq]dr. Now, using previous arguments,

Rr 0

eqA dq = Π1 r + Π2 B −1 (erB − Ir )β 0 ,

and integrating again with respect to r yields the stated expression. For H0 , we find that Z

1

H0 =

Z

r 0

r



qA

Z

e dq dr = Π1 0

1

2

r dr + Π2 B

−1

0

Z

1

rB

re

0

drβ − Π2 B

−1 0

Z

1

rdr

β

0

0

which results in the required expression. The expansions for J0 (s) and J1 (s) arise from noting that F0 (s) = Π1 + Π2 esB β 0 , F1 (s) = Π1 s + Π2 B −1 (esB − Ir )β 0 and F2 (s) = Π1 (1 − s) + Π2 B −1 (eB − esB )β 0 , the latter two of which are obtained by carrying out the integration as above. Substituting these into the expressions for J0 (s) and J1 (s) yields the relevant matrices Jik (i = 0, 1; k = 0, 1, 2) which can then be plugged into the autocovariances and 2

integrated term by term.

APPENDIX C: SUPPLEMENTARY RESULTS Proof of (3). From the definition of A and etA we obtain etA = In +

P∞

j=1 t

j (αβ 0 )j /j!.

But (αβ 0 )j = α(β 0 α)j−1 β 0 and so, defining B = β 0 α, etA = In + α

∞ j X t

j! j=1

B j−1 β 0 = In + αB −1

∞ j X t

j! j=1

Bj β0, 2

thereby yielding (3).

PROPOSITION 1. Let Θ and Γ be defined as in section 4, K2s , let K2f , N1s and N1f be defined as in Theorem 2, and let K1s , K1f , N2s and N2f be defined as in the Proof of Theorem 2. Then: 20

(i) K1s = K2s + Ir and K1f = K2f + Ir ; (ii) Γsf = Γss and Γf s = Γf f ; (iii) N2s = −N1s and N2f = −N1f . Proof. (i) Because Θss = Θsf + Ir we find that K2s = Θsf Θf f + Θsf Θf f Θ−1 sf − Θsf Θf s . But Θf f = Θf s + Ir implying Θsf Θf f − Θsf Θf s = Θsf . Hence −1 s K2s = Θsf + Θsf Θf f Θ−1 sf = Θss − Ir + Θsf Θf f Θsf = K1 − Ir ,

as required. The proof of K1f = K2f + Ir follows in the same way with appropriate modifications. (ii) Recall that Γ = C2 Q, where Q =

j−1 j=1 C1 /j!.

P∞

From the form of C1 and C2 , and

partitioning Q conformably, we obtain 

Γ=

C21 C21 C22 C22

 

Qss

Qss − Ir

Qf f − Ir

Qf f





=

C21 Q C21 Q C22 Q C22 Q

 ,

where Q = Qss + Qf f − Ir , thereby establishing the claim. (iii) We begin with the substitutions Θss = Θsf + Ir and Θf f = Θf s + Ir : N2s = Γsf Θf s − Γsf (Θf s + Ir )Θ−1 sf (Θsf + Ir ) = Γsf Θf s − Γsf (Θf s + Ir ) − Γsf (Θf s + Ir )Θ−1 sf = −Γsf − Γsf Θf f Θ−1 sf , and hence N2s = −N1s using part (ii) for Γsf . That N2f = −N1f is proved in an identical way with the appropriate modifications.

21

2