Co-integration in Continuous and Discrete Time The State-Space Error-Correction Model

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1 Co-integration in Continuous and Discrete Time The State-Space Error-Correction Model Bernard Hanzon and Thomas Ribarits University College Cork School of Mathematical Sciences Cork, Ireland European Investment Bank Risk Management Directorate Financial Risk Department 100, boulevard Konrad Adenauer, L-2950 Luxembourg Summary In this paper 1 we consider co-integrated I(1) processes in the state-space framework. We introduce the state-space error correction model (SSECM) and provide a complete treatment of how to estimate SSECMs by maximum likelihood methods, including reduced rank regression techniques. In doing so, we follow very closely the Johansen approach for the VAR case; see Johansen (1995). The remaining free parameters will be represented using a novel type of local parametrization. An application to UK money market zero rates shows the usefulness of the new approach. Keywords: Co-integration, state-space models, data-driven local coordinates, maximum likelihood estimation, reduced rank regression. 1. INTRODUCTION In dynamical stochastic modelling, including financial dynamic stochastic modelling, one distinguishes stationary models and non-stationary models. The simplest example of a non-stationary model is the wellknown random walk. Mean reverting models are examples of stationary models. Intuitively, when the processes run over a long enough period of time, a non-stationary process will tend to become large compared to a stationary process. Granger s idea (for which he received the Nobel prize in 2003) was that two variables which are both non-stationary can still allow a linear combination which is stationary: the variables are co-integrated. Such linear combinations are interpreted as equilibrium relations. This idea can be generalized to the case of more than two variables in a straightforward way. An important problem that arises to see if such models are applicable for a certain set of variables is the system identification problem, which asks us to find a stochastic system within an appropriate class of systems allowing for co-integration, that represents available data as good as possible. Here we will use the maximum likelihood principle for this purpose. Within any given class of systems this leads to the maximum likelihood estimator, which is defined as the optimizer of the likelihood function. The optimization problem involved is non-trivial, not in the least due to the co-integration structure in the model. For the class of (vector) autoregressive time series models with a fixed maximal lag, S. Johansen (see e.g. Johansen (1995)) has given an elegant solution method for this optimization problem. The solution method involves an eigenvalue calculation as 1 The present manuscript is an extended summary of the full paper. This paper has been submitted to the 5th World Congress of the Bachelier Finance Society, London, UK, July 2008.

2 2 Bernard Hanzon and Thomas Ribarits its key step. Up till now co-integration has mostly been investigated in the context of discrete-time vector autoregressive (VAR) models. In modern mathematical finance continuous-time stochastic model are the standard. Therefore for application of co-integration models to finance (e.g. to interest rate modelling) a continuous-time formulation will be useful. Here we present a discrete-time and a continuous-time state space model with co-integration and an associated optimization method to arrive at the maximum likelihood estimator. In our approach we endeavor to stay as close as possible to Johansen s celebrated method for the VAR class. One key step is the derivation of the state-space error-correction form of the model. With the goal of arriving at the maximum likelihood estimator, we apply a series of partial optimization steps (called concentration steps in econometrics) on the likelihood function. The crucial step of finding the co-integrating relations, conditional on the choice of the remaining parameters, involves the solution of an eigenvalue problem in complete analogy with Johansen s solution for the class of VAR models. After a number of these partial concentration steps we end up with a concentrated likelihood function in a smaller number of parameters. In order to find an optimum for that function we apply a specific gradient search technique that circumvents the potential problems that are related to the fact that eigenvalues of a matrix are not everywhere differentiable with respect to the entries of that matrix. The method that we apply is a special case of the so-called separable least-squares DDLC method for maximum likelihood estimation of stochastic linear systems, see e.g. Ribarits et al. (2005) and the references given there. The DDLC (data-driven local coordinates) technique applied avoids having to work out a particular canonical form for the state space matrices involved: the method produces its own local coordinates. As we are using a search method to find the maximum likelihood estimator, a starting point needs to be supplied. How we propose to do that is described in a separate section. In a final section an application to UK money market zero rates is described. 2. THE DISCRETE-TIME STATE-SPACE ERROR CORRECTION MODEL Consider the following linear, time invariant, discrete-time state-space system: x t+1 = Ax t + Bε t, x t0 = x 0 (2.1) y t = Cx t + ε t, t = t 0, t 0 + 1, t 0 + 2,.... (2.2) Here, x t is the n-dimensional state vector, which is, in general, unobserved; A R n n, B R n p and C R p n are parameter matrices; y t is the observed p-dimensional output. In addition, (ε t ) t N is a p- dimensional Gaussian discrete-time white noise process with Eε t = 0 and Eε t ε t = Σ. The state-space system in (2.1, 2.2) is called minimal if rk [ B, AB,..., A n 1 B ] =rk [ C, A C,..., (A ) n 1 C ] = n. In the remainder of the paper we will assume minimality of the state-space systems except if the contrary is explicitly stated. The matrix A is called asymptotically stable if λ max (A) < 1 is satisfied, where λ max (A) denotes an eigenvalue of A of maximum modulus and stable if the sequence of matrices {A k } k=0 is bounded. Here we will consider models for which A is stable and for which each eigenvalue is either 1 or lies within the open unit disc. It can be shown that the requirement of stability of A can be replaced by the equivalent requirement that the geometric and algebraic multiplicities of 1 as an eigenvalue of A coincide. This multiplicity will be denoted by µ = c, where c stands for the number of common trends in the model. It can be shown that µ satisfies 0 µ p. The sub-class of models for which 1 µ p 1 is denoted by I(1). This is the simplest class of co-integrated models. The number r := p µ = p c is called the rank of co-integration. It is also the number of linearly independent co-integrating vectors (this will become clear later). Note that in the model used the process noise (noise in the state equation (2.1)) coincides with the

3 Co-integration in Continuous and Discrete Time The State-Space Error-Correction Model 3 measurement noise (noise in the measurement equation (2.2)). Starting with other noise specifications, e.g. with independent process and measurement noise, one can arrive at a specification with equal process and measurement noise by applying the Kalman filter and using the innovations ε t = y t CE(x t y t 1, y t 2,...) as the noise terms in the model. In general one will get a model with time-varying parameters in this way, however the time-variation tends to be small after a sufficient amount of time, and here we will assume that the parameters are constant in this innovations representation. Note that we can solve for y t by eliminating the state vector sequence from the equations, to obtain: y t = CA t t0 x 0 + Σ t t0 j=1 CAj 1 Bε t j + ε t, t t 0 (2.3) To simplify the exposition let us assume that x 0 = 0. Furthermore let us formally define ε t = 0, y t = 0, for all t with t < t 0. Let L denote the lag operator, so Ly t = y t 1. Then we can write: where k(z) = y t = k(l)ε t, CA j 1 Bz j + I = Cz(I za) 1 B + I, z C, z σ(a) j=1 where σ(a) denotes the spectrum of A. Let k(z) := k(z) 1 = z C(I zā) 1 B + I, with Ā = A BC, B = B, C = C. Then ε t = k(l)y t = CĀj 1 Byt j + y t j=1 It is well-known (and an explicit proof will be given in the full paper) that the geometric multiplicity µ of 1 as an eigenvalue of A can be read of from the rank of k(1), in fact rank( k(1)) = r = p µ An equivalent statement is that there must exist full column rank p r matrices α and β such that C(I Ā) 1 B + I = αβ Defining k(z) := I + ( k(z) k(1)z)/(z 1) we find k(0) = 0 and we can rewrite ε t = k(l)y t in the so-called error correction form y t = αβ y t 1 + k(l) y t + ε t where y t = y t y t 1. If y t 1 is far from the kernel of the matrix β and β y t 1 is large then a large correction can be expected in y t. This motivates the (standard) terminology error-correction form. Note that except for the innovation ε t the terms on the right-hand side of this equation are known at time t 1. Therefore this form can be used as a regression model, in which the coefficients α, β and those determining k have to be determined. The error-correction form can be rewritten as where x t+1 = Ā x t + B y t, (2.4) y t = αβ y t 1 + CĀ(I Ā) 1 x t + ε t, (2.5) αβ = C(I Ā) 1 B I. This is called the state-space error-correction form or the state-space error correction model (SSEMC).

4 4 Bernard Hanzon and Thomas Ribarits 3. THE CONTINUOUS-TIME STATE-SPACE ERROR CORRECTION MODEL Consider the following continuous-time stochastic linear state-space model (with formats of vectors and matrices analogous to the discrete time case) dx t = AX t + BdW t, x t0 = x 0 (3.6) dy t = CX t dt + dw t, t [t 0, ) (3.7) Here we assume A to be continuous-time stable, i.e. exp(at), t [0, ) is bounded, and each element in the spectrum of A is zero or lies in the open left half plane. Then the geometric and algebraic multiplicities of 0 as an eigenvalue of A coincide and will be denoted by µ. If 1 µ p 1 then this is a co-integration model of class I(1), but now for continuous time. The model is in innovations form. Here W t denotes a Wiener process with covariance matrix Σ so that W t W s has a normal distribution with mean zero and covariance matrix (t s)σ if s < t. We can now write dw t = dy t CX t dt = dy t + t s=t 0 C exp( Ā(t s) B)dY s dt In integral form this gives (using the simplifying assumption that W t0 = 0 and Y t0 = 0) W t = Y t + t s=t 0 ( CĀ 1 exp(ā(t s)) B CĀ 1 B)dYs Note that the integral is measurable with respect to the strict past of the process Y t at time t, because the integrand is zero at s = t. We cannot split the integral without losing this essential property. To obtain the desired regression equation we therefore choose an arbitrary small number t > 0 and derive the following approximate equation: Here the co-integration rank constraint is Y t Y t t (I CĀ 1 B)Yt t CĀ 1 X t t + W t. rank(i CĀ 1 B) = r. The representation obtained is the continuous-time state-space error correction model in integral form. 4. MAXIMIZATION OF THE LIKELIHOOD FUNCTION The proposed method for the maximization of the likelihood function is outlined here for the discrete-time state-space error correction model. A similar approach can be applied to the continuous time model with discrete-time (but not necessarily with equidistant time points) observations. Let us assume n > p for simplicity of exposition and let us define the following quantities: Z 0t := y t, Z 1t := y t 1, Z 2t := x t, M i,j := 1 T T Z it Z jt, i, j {0, 1, 2} Note that Z 2t depends on Ā, B via x t = j=1 (Ā)j 1 B yt j. The maximum likelihood estimator is the value of (α, β, Ā, B, C, Σ) that minimizes L subject to C(I Ā) 1 B = (Ip + αβ ) where L is given by: L = log det(σ) + 1 T ΣT t=1(z 0t αβ Z 1t CĀ(I Ā) 1 Z 2t ) Σ 1 (Z 0t αβ Z 1t CĀ(I Ā) 1 Z 2t ), Note that L is minus the logarithm of the likelihood function, but will, with some abuse of language, be called the log-likelihood function here. t=1

5 Co-integration in Continuous and Discrete Time The State-Space Error-Correction Model 5 As the criterion is quadratic in the combined elements of C and α and the constraint is linear in those combined elements, we can solve for C and α as a function of the other parameters. Substituting the result into the log-likelihood function we obtain a so-called concentrated log-likelihood function L 2c that depends on (β, Ā, B, Σ) and is given by where L 2c = log det Σ + 1 T T (R 0t S 01 β(β S 11 β) 1 β R 1t ) Σ 1 (R 0t S 01 β(β S 11 β) 1 β R 1t ) t=1 and where R 0t and R 1t are given by S ij = 1 T T R it R jt, i, j {0, 1} t=1 R 0t = Z 0t ([M 02 (I Ā)Ā]H 11 H 21 )Ā(I Ā) 1 Z 2t, and H = R 1t = Z 1t ([M 12 (I Ā)Ā]H 11 H 21 )Ā(I Ā) 1 Z 2t ( ) ( ) H11 H 12 Ā(I Ā) = 1 1 M 22 (I Ā) 1 Ā (I Ā) 1 B H 21 H 22 ((I Ā) 1 B) 0 For given β, Ā, B the optimal choice ˆΣ of matrix Σ is ˆΣ = S 00 S 01 β(β S 11 β) 1 β S 10 and substituting this in the concentrated log-likelihood function L 2c gives, after some manipulations, ( L 3c (β, Ā, B) det(β (S 11 S 10 S00 1 = log det(s 00 ) + log S ) 01)β) det(β + p S 11 β) Note that the quotient in the second term of this expression can be viewed as a generalized Rayleigh quotient. The problem of optimizing this with respect to β can be solved by considering the associated generalized eigenvalue problem (λs 11 S 01 S00 1 S 10)v = 0, λ R, v R p, v 0. Let λ i, i = 1, 2,..., r denote the largest r eigenvalues for this problem and let v 1, v 2,..., v r denote corresponding (linearly independent) eigenvectors. Then ˆβ = [v 1, v 2,..., v r ]T is an optimal choice for β for any arbitrary non-singular r r matrix T. Substituting this into L 3c gives L 4c (Ā, B) r = log det S 00 + log(1 λ i ) + p. Note that the resulting function L 4c depends in a rather complicated way on the matrices Ā, B. Here we would like to apply a gradient-search algorithm starting from a good initial point. At first finding the gradient of L 4c might look extremely complicated because of the dependence on the eigenvalues λ 1, λ 2,..., λ r which may even be non-differentiable at some points. Our solution is surprisingly simple: for any given Ā, B we calculate the vector of partial derivatives of the original log-likelihood function L with respect to the entries of Ā, B and evaluate the result in i=1 β = ˆβ, Σ = ˆΣ, α = ˆα, C = ˆ C and call the resulting vector of partial derivatives. We have the following result:

6 6 Bernard Hanzon and Thomas Ribarits (i) If L 4c (Ā, B) is differentiable at (Ā, B) then coincides with the vector of partial derivatives of L 4c with respect to the entries of Ā, B at Ā, B. (ii) If is non-zero at (Ā, B) the value of L 4c can be improved by making a sufficiently small step in the direction of. (iii) If = 0 we have arrived at a critical point of the log-likelihood function L. By considering the state-space error correction model again, it can be seen that if we replace (Ā, B, C) by (T 1 ĀT1 1 1, T 1 B, CT 1 ), and (α, β) by(αt2 1, T 2 β) where T 1 and T 2 are non-singular matrices of appropriate sizes, then the criterion value remains the same. This implies for L 4c that for fixed Ā, B all pairs (T 1 ĀT1 1, T 1 B), T1 nonsingular, give the same function value. The set of all such pairs forms an equivalence class; the DDLC method constructs a system of np local coordinates that is transversal to the tangent space of this equivalence class (in fact orthogonal with respect to some suitable metric) and calculates the gradient with respect to these local coordinates. 5. MODEL SELECTION AND INITIAL PARAMETER ESTIMATES FOR SSECMS Note that up to now we have assumed to know both the state dimension n, i.e. the size of the Ā matrix in (2.4) and the number of common trends c. In practice, these integer-valued parameters have to be estimated from data. Similarly, it is desirable to start the likelihood optimization procedure with some educated guess for (Ā, B). In principle, one can think of many ways of how to find estimates for n, c and, thereafter, initial estimates for (Ā, B). In the application introduced in the following section, we will proceed along the lines of the classical Johansen approach to co-integration: First, we estimated a conventional VAR model (in levels). Let ˆk HQ denote the estimated lag order of such VAR model of the form y t = k j=1 A(j)y t j + ε t, using the Hannan-Quinn criterion function. Given the lag order ˆk HQ, we then performed the classical trace test in Johansen s VAR error correction model to determine r and thus c = p r. The resulting VAR error correction model is readily obtained as well, and one can then transform it back to a conventional VAR model which can, in turn, be straightforwardly written in state-space form, (A, B, C ) say. Note that the state-space representation of a VAR model is widely known as the companion form of the VAR model. By construction, the A matrix has one as c-fold eigenvalue. Applying a suitable state transformation of the form (Ã, B, C ) = (T A T 1, T B, C T 1 ) yields a state-space system with block diagonal à matrix, i.e. ( ) xt+1,1 = x t+1,2 à { ( }} ) { ( Ic 0 xt,1 0 à st ( ) st y t = C1, C }{{} C ( xt,1 x t,2 ) + B {( }} ){ B1 B st ε t (5.8) x t,2 ) + ε t (5.9) where (Ãst st st, B, C ) represents a stable system, i.e. λ max(a st ) < 1. In order to determine n, one can now apply a so-called balance and truncate approach to the stable subsystem. Balanced model truncation is a simple and, nevertheless, efficient model reduction technique for state-space systems; see e.g. Scherrer (2002) and the references provided there for the concept of stochastic

7 Co-integration in Continuous and Discrete Time The State-Space Error-Correction Model 7 balancing, which relies upon the determination of the singular values corresponding to the system. For given (Ãst st st, B, C ) the computation of these singular values boils down to the solution of two matrix equations. Using an information-type criterion such as SV C(n), we consider the number of singular values that differ significantly from zero as the estimated order of the stable sub-system. By adding c, we then obtain the estimated system order ˆn. Also, by simply truncating (Ã, B, C ) in (5.8,5.9) to size ˆn, we obtain an initial state-space model which can be written in SSECM form, yielding initial estimates for (Ā, B). 6. AN APPLICATION TO UK MONEY MARKET RATES We consider a four dimensional vector (p = 4) of UK money market (zero) rates r 3, r 6, r 9 and r 12 which have been directly obtained from the 4 LIBOR rates corresponding to maturities 3 months, 6 months, 9 months and 1 year. The data set is comprised of 913 daily observations covering the 2.5 year period from 1 January 2004 to 30 June 2006; see Figure 1. For estimation purposes, we used the first 2 years of observations only, and we evaluated the performance of the SSECM (in comparison with Johansen s VAR error correction model) on data corresponding to the first 6 months in For a start, we performed a VAR analysis on the estimation data set: Using the Hannan-Quinn criterion, the lag order k of the AR polynomial has been determined as ˆk HQ = 2, leading to an (unrestricted) estimated model of the form y t = A(1)y t 1 + A(2)y t 2 + ε t. Note that the BIC criterion also led to ˆk BIC = 2, whereas the AIC criterion yielded ˆk AIC = 4. Based on the estimation of this VAR(2) model, the classical Johansen trace test (without intercept) yielded r = 3. Hence, we found only one common trend (c = 1) driving the four interest rates 2. In the next step, following Section 5, we wrote the VAR(2) error correction model in state-space form and applied a suitable state transformation to obtain a system of the form (5.8,5.9). Note that this system had state dimension 8, and c = 1 (as r = 3), i.e. the first state is I(1). In order to reduce the state dimension, we employed a balance and truncate approach to the 7-dimensional stable subsystem: The singular values corresponding to the stable subsystem, sometimes also referred to as canonical correlations, are used for defining the information-type criterion SV C(n) in Figure 1. As can be seen there, SV C(n) is minimized by choosing 3 canonical correlations. Hence, the stable sub-system was truncated taking into account only 3 states and an SSECM of state dimension n = 4 was then estimated: such SSECM has 31 free parameters. Note that the estimated VAR(2) error correction model also had 31 free parameters (although it represents, of couse, a different model class). In order not to depend on only one choice of the initial pair (Ā, B) for the subsequent iterative optimization algorithm, we did not only take the SSECM model corresponding to Johansen s VAR(2) estimate as starting point for the estimation of the SSECM. Instead, we randomly generated 99 other initial points ( centered at the one obtained directly from the VAR(2) model). Most likely, one would normally only retain the best fitting model out of the 100 estimated ones esimation data corresponding to the best model are indicated by a square in Figure 2 but we decided to present the top 20 below and to compare their performance to the Johansen VAR estimate. Referring to Figure 2, the following observations can be made: 2 Such observation is in line with the so-called expectations hypothesis of the term structure of interest rates. It should be noted, however, that more common trends are often found if one were to include e.g. also swap rates with maturities of 2,3,4, etc. years in the analysis.

8 8 Bernard Hanzon and Thomas Ribarits 5.4 Information Type Criterion using Canonical Correlations Interest Rate SVC Days as from 1 January System order Figure 1. Left: UK money market (zero) rates from 1 January 2004 to 30 June 2006 for maturities of 3 months, 6 months, 9 months and 1 year, respectively. Right: Estimation of the state dimension of the stable part of the SSECM. The best 20 SSECMs with respect to in-sample likelihood fit were all better than the estimated VAR(2) model; see the bottom left part of Figure 2. The out-of-sample fit, as measured by log det ˆΣ+p, where ˆΣ denotes the empirical variance-covariance matrix of the prediction errors obtained on the validation sample, was better for all the 20 SSECMs, too. However, the differences were rather small. The estimated co-integrating spaces for VAR and SSECMs almost coincided. This is evidenced by looking at the top right graph of Figure 2 showing the (negligible) Hausdorff distances between the Student Version of MATLAB estimated co-integrating spaces. Note that the Hausdorff distance between two sub-spaces (of the same dimension) of R n may take values between 0 (the sub-spaces coincide) and 1 (the sub-spaces are orthogonal to each other). As co-integrating sub-spaces can be estimated super-consistently, this observation does not come as a big surprise. In all cases considered, the search algorithm used for the estimation of the SSECMs terminated after reasonably many iterations; see the bottom right graph of Figure 2. Student Version of MATLAB As a concluding remark we want to emphasize that the Hausdorff distance between the estimated cointegrating spaces and the column span of was in the order of magnitude of This shows that the interest rate spreads r 3 r 6, r 6 r 9 and r 9 r 12

9 Co-integration in Continuous and Discrete Time The State-Space Error-Correction Model 9 can be considered to be stationary. The development of a formal test procedure in the framework of SSECMs is a topic of future research. Out of Sample Likelihood Comparison of Predictive Power Initial System Gap Contegrating Spaces SSECM < > AR 7.5 x 10 4 Comparison of Cointegrating Spaces Initial System 26.3 In Sample Likelihood Fit 50 Number of Iterations in the Search Algorithm Likelihood Value Number of Iterations Initial System Initial System Figure 2. Best 20 SSECMs with respect to in-sample likelihood fit (bottom left). The other graphs show the out-of-sample fit (top left), the Hausdorff distance between the estimated co-integrating space for VAR and SSECMs (top right) and the number of iterations the estimation algorithm took until it found a local optimum (bottom right). Student Version of MATLAB

10 10 Bernard Hanzon and Thomas Ribarits REFERENCES Johansen, S. (1995). Likelihood-Based Inference in Cointegrated Vector Auto-Regressive Models. Oxford University Press. Ribarits, T., M. Deistler, and B. Hanzon (2005). An analysis of separable least squares data driven local coordinates for maximum likelihood estimation of linear systems. Automatica, Special Issue on Data- Based Modeling and System Identification 41 (3), Scherrer, W. (2002). Local optimality of minimum phase balanced truncation. In Proceedings of the 15th IFAC World Congress, Barcelona, Spain.