Identi cation Problem in Nonlinear Cointegration
Christian Gourieroux Joann Jasiaky This version: June 14, 2000
CREST and CEPREMAP , e-mail:
[email protected]. y York University, e-mail:
[email protected]
The second author gratefully acknowledges nancial support of the Natural Sciences and Engineering Research Council of Canada
Acknowledgments: We thank Clive Granger and two anonymous referees for helpful comments.
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Introduction
Nonlinear dynamics of macroeconomic or nancial variables is often examined and modelled by using standard tools of time series analysis. However the instruments and concepts such as the autocorrelation and partial autocorrelation functions, spectral densities, unit root tests, cointegrating vectors and E.C.M. representation [Granger (1986), Engle, Granger (1987), ...], as well as long memory [Hosking (1981)] have been conceived for linear dynamics. Despite this incoherency, many authors continue to apply these methods to various nonlinear transformations of time series. A typical example is the autocorrelogram, which occasionally is computed from nonlinearly transformed series [Lepnik (1958), Granger, Newbold (1976), Granger, Hallman (1991), Ding, Granger, Engle (1993)]. Empirical evidence suggests that the pattern of autocorrelations strongly depends on the applied nonlinear transformation. For instance, the autocorrelogram of nancial returns may reveal an absence of (linear) dependence whereas the autocorrelogram of squares or absolute values of the same series may feature long memory [Ding, Granger, Engle (1987)]. As well there exist nonlinear transformations yielding stationary transformed processes whereas some others may induce various nonstationary patterns. This leads to spurious results of standard Dickey-Fuller tests applied to transformed series [Granger, Hallman (1989), Corradi (1995)], or misleading outcomes of a cointegration analysis of nonlinearly transformed series. In this note we focus on identi cation problems which arise in nonlinear cointegration analysis. We rst illustrate in section 2 the poor performance of the traditional, autocorrelation-based measure of persistence and propose to de ne the persistence of nonlinear processes using the concept of persistence by trajectory. We extend it to the notion of persistence space in section 3. This approach allows us to introduce in section 4 the de nitions of nonlinear integration and nonlinear cointegration which are general enough to encompass as special cases various examples from recent literature. We point out the existence of a general identi cation problem in nonlinear cointegration analysis due to multiplicity of cointegration directions. Our results indicate that this diculty can be circumvented by imposing adequate assumptions. From this perspective, some particular assumptions used by researchers in recently published studies on nonlinear cointegrations appear to have ensured the identi cation by eliminating the multiplicity and constraining the set of cointegration directions.
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Persistence by Trajectory
The temporal dependence of a nonlinear transformation g of a stationary process is commonly evaluated by estimating the autocorrelations of the transformed series:
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Corr[g(Xt ); g1 (Xt?h )] h (g) = Max g 1
= Corr[g(Xt ); E [g(Xt )jXt?h ]];
where Xt?h and g1 (Xt?h ) is measurable with respect to this information. This measure of temporal dependence is not adequate in a nonstationary framework, as illustrated by the following example of a gaussian random walk. Let us consider a gaussian random walk: Xt =
t X =1
(2.1)
;
where the noise process ftg consists of i.i.d. standard Normal variables. It has been proved in Ermini, Granger (1993) that the exponential transformations of the random walk are such that: E [exp Xt+h jXt ] = exp
and
h2
2 exp Xt ;
(2.2)
Corr [exp Xt+h; g2 (Xt )] t;h = max g 2
= Corr (exp Xt+h ; exp Xt ) s exp 2 t ? 1 : = exp 2 (t + h) ? 1
(2.3)
The exponential random walk features an explosive behaviour, since the autoregressive coecient 2 exp h2 in the autoregressive representation (2.2) is larger than one. However when we consider the autocorrelation function at large t, we nd: h = tlim !1 t;h = exp ?
2 h
2 h
2 = exp ? 2 : This formula resembles the autocorrelation of a stationary AR(1) process. The ambiguity in determining the range of temporal dependence can be avoided by using a de nition based on asymptotic properties of the prediction. We pursue this approach in the following section.
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Persistence Spaces
In this section we introduce the persistence by trajectory and propose the concept of persistence space allowing us to de ne nonlinearly integrated (cointegrated) processes.
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We consider a process (Xt ) and a possibly extended universe (Zt ). We assume that the process is homogenous with respect to the universe, i.e. that the conditional distributions of Xt given Zt?1 do not depend on the date t. Consequently, the conditional expectations E [g(Xt+h)jZt ] are also time independent. Let us now introduce a positively valued sequence = ( h ; h 0) converging to zero at in nity.
De nition 1: The by trajectory (b.t.) persistence space of order is de ned by: E b:t: = fg such that there exists a scalar c(g) with: E (g(Xt+h )jZt ) ? c(g) = 0( h ) a.s. g. Therefore we have jE (g(Xt+h )jZt ) ? c(g)j (Zt ) h a.s.. It is easily veri ed that E b:t: is a vector space, and has the monotony property. To separate the stationary and nonstationary parts of the process (Xt ), we consider the vector space: E b:t: (0) = [ E b;t ;
(3.4)
where the union is taken over all possible sequences. A transformation g belongs to the space E b:t: (0) if and only if the prediction E [g(Xt+h )jZt ] becomes independent of Zt when the horizon h tends to in nity. This condition is satis ed by stationary regular processes, traditionally called I(0). On the contrary, it is violated when g(Xt ) features nonstationarities or nonregularities.
De nition 2: If g belongs to E b:t:(0), the transformed process g(Xt) is NLI(0) (nonlinearly
integrated of order 0). If g does not belong to E b:t: (0), the transformed process g(Xt ) is NLI (nonlinearly integrated).
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Nonlinear Cointegration
This section de nes the nonlinear cointegration, relates this concept to de nitions used earlier in the literature and points out the identi cation problem. Let us now consider a bivariate process Xt = (X1t ; X2t )0 homogenous with respect to the universe (Zt ) = (Xt ).
De nition 3: The components X1 and X2 are nonlinearly cointegrated i :
(i) (X1t ) and (X2t ) are NLI with respect to the universe (Zt ) = (Xt ); (ii) There exists a transformation Yt = a(X1t ; X2t ), which depends both on X1 and X2 , satis es a one to one relationship with one of the arguments, and is such that (Yt ) is NLI(0) with respect to the universe (Xt ). Note that two types of restrictions have been imposed on the function a. Firstly a is supposed to depend on both arguments. Indeed in a nonlinear framework there may exist nonstationary processes (X1t ), with transformations Yt = a(X1t ) which are NLI(0). However
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the transformation a(X1t ) does not represent the comovements between X1t ; X2t , since X2t is not involved. Secondly we assume that the function a satis es a one to one relationship with one of the arguments X1 , say. Then we can write: X1t = a?1 1 (Yt ; X2t );
where a?1 1 is the inverse with respect to the rst argument. Therefore it is possible to express X1t as a nonlinear function of X2t and a NLI(0) process. Generally, in nonlinear cointegration analysis an identi cation problem is encountered, due to a multiplicity of cointegration directions. The following example provides an illustration of the problem. Example 6.1: Let us consider three independent gaussian white noises (1;t ); (2;t ); (3;t ), and de ne: Z1;t = 1 Z1;t?1 + 1;t (AR(1) process), j1 j < 1, Z2;t = 2 Z2;t?1 + 2;t (AR(1) process), j2 j < 1, P Z3;t = t =1 3; (random walk), X1;t = sgn(1;t)jZ1;t j jZ3;t j, X2;t = sgn(2;t)jZ2;t j jZ3;t j. The components (X1;t ) and (X2;t ) are NLI due to the presence of the random walk jZ3;t j. Moreover any transformation of X1;t =X2;t, sgn(X1;t), sgn(X2;t ) is NLI(0). For instance the directions X1;t =X2;t + a sgn(X2;t ) are nonlinear cointegration directions for any scalar a.
There exist various ways to handle the multiplicity problem. (i) We can try to determine the set of all nonlinear cointegration directions using nonparametric methods. Most likely this approach is unfeasible except for cases where either the associated cointegration space, or its supplement are included in a vector space of a small dimension. (ii) Alternatively we can impose identifying constraints. Especially, we can restrict the admissible distributions of the process (Xt ) and/or the admissible forms of cointegration directions. This approach is implicitely followed in the literature as illustrated by the examples below.
Restriction on the admissible cointegration directions
In case of a bivariate process we know that linear cointegration directions are included in a space of dimension equal or less than one. Therefore the linearity of the cointegration direction is an identifying constraint. However there may exist strictly nonlinear cointegration directions in the absence of linear cointegration directions, and some nonlinear directions may coexist with the linear
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ones.
Constraint on the error term
In a series of papers Park, Phillips (1998;a,b), Karlsen, Myklebust, Tjostheim (1999) considered the estimation of nonlinear relations of the type X1;t = f (X2;t ) + u1;t , where X2;t features nonstationarities. Thus the cointegration directions are already constrained to satisfy an additive decomposition. However this restriction is not sucient to identify the function f and has to be completed by conditions on the error term. The authors cited above have introduced various assumptions on the error term u1;t. In Park, Phillips (1998;a), Karlsen, Myklebust, Tjostheim (1999) the assumption imposes the independence between the processes X2 and u1 . It is easy to see that this is an identi cation condition of the regression function f . Indeed, if we consider two admissible decompositions X1;t = f (X2;t ) + u1;t = f~(X2;t ) + u~1;t , we deduce u1;t = f~(X2;t ) ? f (X2;t ) + u~1;t , and (u1;t ) is independent of (X2;t ) if and only if f = f~. In Park, Phillips (1998;b), the error term is assumed to be a martingale di erence sequence with respect to the information (Xt ). If we consider two admissible decompositions X1;t = f (X2;t ) + u1;t = f~(X2;t ) + u~1;t, we get u1;t ? u~1;t = f~(X2;t ) ? f (X2;t). We nd that f is identi able if and only if the unique martingale di erence sequence satisfying the deterministic function of (X2;t ) is zero. This explains the multiplicity encountered in example 6.1, where sgn(X2;t) is a nonzero martingale di erence sequence.
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References [1] Corradi, V. (1995): "Nonlinear Transformations of Integrated Time Series: A Reconsideration", Journal of Time Series Analysis, 16, 6, 537-549. [2] Ding, Z., Granger, C. and R. Engle (1993): "A Long Memory Property of Stock Market Returns and a New Model", Journal of Empirical Finance, 1, 83-106. [3] Engle, R. and C. Granger (1987): "Cointegration and Error Correction Representation, Estimation and Testing", Econometrica, 55, 251-276. [4] Ermini, L. and C. Granger (1993): "Some Generalizations on the Algebra of I(1) Processes, Journal of Econometrics, 58, 369-384. [5] Gourieroux, C. and J. Jasiak (1998): "Nonlinear Autocorrelograms: An Application to Inter Trade Durations", CREST D.P. 9841. [6] Granger, C. (1986): "Developments in the Study of Cointegrated Economic Variables", Oxford Bulletin of Economics and Statistics", 48, 213-228. [7] Granger, C. and Hallman (1991): "Nonlinear Transformations of Integrated Time Series", Journal of Time Series Analysis", 12, 207-234. [8] Granger, C. and P. Newbold (1976): "Forecasting Transformed Series", Journal of the Royal Statistical Society, B, 38, 189-203. [9] Hosking, J. (1981): "Fractional Di erencing", Biometrika, 68, 165-176. [10] Karlsen, H., Myklebust C., and D. Tjostheim (1999): "Nonparametric Estimates in a Nonlinear Cointegration Type Model", University of Bergen, D.P. [11] Lepnik, R. (1958): "The E ect of Instantaneous Nonlinear Devices on Cross-Correlation", I.R.E. Trans. Information Theory, IT-4, 73-76. [12] Park, J., and P.C.B. Phillips (1998a): "Asymptotics for Nonlinear Transformations of Integrated Time Series", Cowles Foundation D.P. [13] Park, J., and P.C.B. Phillips (1998b): "Nonlinear Regression with Integrated Time Series", Cowles Foundation D.P.
