Abstract
This chapter provides a survey of methods of continuous time modelling based on an exact discrete time representation. It begins by highlighting the techniques involved with the derivation of an exact discrete time representation of an underlying continuous time model, providing specific details for a second-order linear system of stochastic differential equations. Issues of parameter identification, Granger causality, nonstationarity and mixed frequency data are addressed, all being important considerations in applications in economics and other disciplines. Although the focus is on Gaussian estimation of the exact discrete time model, alternative time domain (state space) and frequency domain approaches are also discussed. Computational issues are explored, and two new empirical applications are included along with a discussion of applications in the field of macroeconometric modelling.
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Notes
- 1.
McCrorie (2009) lists a number of contributions that use an exact discrete time model.
- 2.
Rex Bergstrom spent over 20 years of his academic career at the University of Essex and had both direct and indirect influences on the current authors. He taught both Marcus Chambers and Roderick McCrorie at the Masters level and supervised the PhD thesis of Chambers (1990). Chambers, in turn, was the PhD supervisor of McCrorie (1996) and Thornton (2009).
- 3.
This paper was based on Phillips’s M.A. dissertation supervised by Bergstrom at the University of Auckland in 1969. It represented the first of many contributions by Phillips on continuous time econometrics; Yu (2014) provides a survey of this work.
- 4.
Most non-linear models are not directly amenable to the derivation of exact discrete time representations and typically result in transition densities that have no closed-form solution. See, however, Phillips and Yu (2009), Fergusson and Platen (2015) and Thornton and Chambers (2016), for examples where a closed-form density is apposite.
- 5.
The initial conditions are usually assumed to be fixed which imparts a type of nonstationarity on an otherwise stable system. This is a different type of nonstationarity to that which has dominated the econometrics literature in recent years and which we discuss in Sect. 14.2.4.
- 6.
The use of a vector of random measures to specify the disturbance vector in a continuous time model in the econometrics literature is due to Bergstrom (1983) who built on the work of Rozanov (1967). A common alternative is to replace ζ(dt) with Σ 1∕2dW(t) where dW(t) denotes the increment in a vector of Wiener processes and Σ 1∕2(Σ 1∕2)′ = Σ. Note, though, that the latter specification imposes Gaussianity on the system, whereas the distribution of ζ(dt) is unspecified beyond its first two moments.
- 7.
Such dependencies are, however, emphasised in Sect. 14.2.2 where we discuss issues of identification.
- 8.
In fact, Σ ϕ is an np × np matrix of zeros except for the n × n bottom right-hand corner block which is equal to Σ.
- 9.
- 10.
Note that this inconsistency arises owing to no new information on Dx(0) becoming available as T →∞.
- 11.
Bergstrom (1986) also includes results for a system that contains exogenous stock and flow variables.
- 12.
The coefficient matrix multiplying u(t) is set to an identity in order to identify the parameters of the model in view of u(t) having covariance matrix Σ.
- 13.
Hence the presence of the phrase ‘an exact discrete time representation’ rather than ‘the exact discrete time representation’ in the title of this chapter.
- 14.
The state space form in (14.20) is augmented by an additional n f elements in a vector y 0(t) that corresponds to the aggregated or observed flow variables.
- 15.
This identification problem is therefore different in nature and on top of the classical identification problem which seeks to avoid observational equivalence through model and estimator choice; see, for example, Chambers and McCrorie (2006). In open systems, namely, systems involving exogenous variables, the solution of the stochastic differential equation depends on a continuous time record of the exogenous variables and so some sort of approximation of the time paths is necessary to achieve identification; see, in particular, Bergstrom (1986), Hamerle et al. (1991, 1993) and McCrorie (2001) for explicit discussion of this issue.
- 16.
McCrorie and Chambers (2006, Section 3.1) outline and discuss the concept of Granger causality in the context of continuous and discrete time models.
- 17.
Thornton and Chambers (2013) provide a recent discussion of temporal aggregation in macroeconomics with continuous time models in view.
- 18.
Stock (1987) had earlier provided an example that cointegration as a property was invariant to temporal aggregation.
- 19.
- 20.
The model considered by Chambers (2016) also includes a vector of intercepts and deterministic trends.
- 21.
Such factors include, but are not restricted to, the order of the continuous time system, the dimension of the vector x(t), the sample size, the way in which the likelihoods are programmed and the optimisation algorithm used.
- 22.
This paper was reprinted 25 years later with an update as Moler and Van Loan (2003).
- 23.
- 24.
See Gandolfo (1981) for a textbook treatment.
- 25.
At the time of writing (July 2017), Pier Carlo Padoan is Italy’s Minister of Economy and Finance, a position he has held since February 2014.
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Acknowledgements
We thank an editor and two anonymous referees for helpful comments that have led to improvements in this paper and the Scottish Institute for Research in Economics for arranging facilities in the School of Economics, University of Edinburgh, for the authors to meet to work on this chapter. The first author also thanks the Economic and Social Research Council for financial support under grant number ES/M01147X/1.
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14.1 Electronic Supplementary Material
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Appendix
Appendix
The Gauss code below was used in the simulation exercise. Note that n is used in the code as the data span and t is the sample size, whereas, in the text in Sect. 14.3, it is T and N, respectively, that are used for these quantities.
/∗ Simulation of continuous time AR(1) process at different frequencies ∗/ new; a=-1.0; /∗ Continuous time AR parameter ∗/ hv=1|1/2|1/3|1/4|1/6|1/12;/∗ Discrete time sampling intervals∗/ n=100; /∗ Data span ∗/ x0=0; /∗ Initial value ∗/ nreps=100000; /∗ Number of replications ∗/ s2=1; /∗Continuous time innovation variance∗/ rndseed 6665; /∗ seed for random numbers ∗/ rhv=rows(hv); tv=n./hv; maxt=maxc(tv); hmin=minc(hv); hrel=hv/hmin; eahm=exp(a∗hmin); e2ahm=exp(2∗a∗hmin); eahv=exp(a∗hv); s2m=s2∗(e2ahm-1)/(2∗a); sm=sqrt(s2m); cta=zeros(nreps,rhv); /∗ nreps times number of h values ∗/ dta=cta; eta=cta; nogood=0; for i (1,nreps,1); u=sm∗rndn(maxt,1); xm=datagen(u); /∗ maxt times 1 ∗/ for hi (1,rhv,1); h=hv[hi,1]; t=tv[hi,1]; xh=reshape(xm,tv[hi,1],hrel[hi,1]); x=xh[.,hrel[hi,1]]; bhat=x[2:t,1]/x[1:t-1,1]; if bhat le 0; ahat=0; nogood=nogood+1; else; ahat=ln(bhat)/h; endif; ehat=(x[2:t,1]-x[1:t-1,1])/(h∗x[1:t-1,1]); cta[i,hi]=ahat; dta[i,hi]=bhat; eta[i,hi]=ehat; endfor; endfor; stop; proc datagen(e); local x; x = recserar(e, x0, eahm); retp( x ); endp;
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Chambers, M.J., McCrorie, J.R., Thornton, M.A. (2018). Continuous Time Modelling Based on an Exact Discrete Time Representation. In: van Montfort, K., Oud, J.H.L., Voelkle, M.C. (eds) Continuous Time Modeling in the Behavioral and Related Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-77219-6_14
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