2. THE DDE SYSTEM AND ITS SPECTRAL PROPERTIES

To deal simultaneously with the general features detailed in the previous section, we shall consider a real-valued stationary continuous time q-dimensional random process {y(t)}_t=−∞^∞ generated according to the system of DDEs defined by

where n ≥ 1 is an integer, D denotes the mean square differential operator, s₀,…,s_n are finite integers, and A_lk(θ) (k = 1,…,s_l; l = 0,…,n) are q × q matrices whose elements are known functions of an unknown p × 1 parameter vector θ, where p ≤ (s₀ + ··· + s_n)q². The lag (or lead) terms ν_lk are finite real numbers satisfying ν_l1 < ν_l2 < ··· < ν_lsl, where ν_lk may equal zero for l ≠ n but ν_nk is restricted to be nonzero for all k. Although we shall refer to the ν_lk as lag parameters we stress that leads can appear in (1) if the relevant ν_lk are negative. The term ζ(dt) is a q × 1 vector of random measures satisfying E [ζ(dt)] = 0, E [ζ(dt)ζ(dt)′] = Ωdt, and E [ζ(Δ₁)ζ(Δ₂)′] = 0 for disjoint sets (intervals) Δ₁ and Δ₂ on the real line and where Ω is an unknown q × q symmetric positive definite covariance matrix. The differential order of the system (the highest order derivative) is equal to n, and the difference order of the system (the largest lag) is equal to max(|ν_lk|).

The system of DDEs in (1) clearly nests the specifications that have arisen recently in economics, as described in the previous section. It also nests the model considered in Chambers (1998), in which n = 1 and all lags are integer valued, in addition to the univariate specification in Chambers and McGarry (2002), although in the latter case the DDE was utlized to capture the cyclical movement of a series in an unobserved components model.

The unknown parameters in (1) are the p elements of θ, the

lag parameters ν_lk (l = 0,…,n; k = 1,…,s_l) contained in the s × 1 vector ν, and the q(q + 1)/2 distinct elements of the symmetric covariance matrix Ω, which are contained in the vector ω. We shall assume that the vectors θ, ν, and ω belong to suitable parameter spaces

, respectively, where

and where

are defined in Assumption 1, which follows. It is convenient to combine the m = p + s + q(q + 1)/2 unknown parameters into the single m × 1 parameter vector β ≡ (θ′,ν′,ω′)′, which belongs to the composite parameter space

.

For reasons that shall become apparent, we propose a frequency domain maximum likelihood estimator of β. Central to formulating the frequency domain likelihood function is the spectral density function of the observed discrete time process generated by (1), which can be derived from the spectral density of the continuous time process y(t). First note that (1) has the representation α(D)y(t) dt = ζ(dt), where

denotes the characteristic function of (1) and whose dependence on θ and ν has been suppressed for notational convenience. The appropriate behavior of α(z) is ensured by restricting the parameter spaces

as follows.

Assumption 1. The parameter spaces

are restricted so that the product space

{θ,ν : the roots of the equation det[α(z)] = 0 have negative real parts and the matrix α(iλ) is nonsingular for all −∞ < λ < ∞}.

The condition governing the roots of the determinantal equation ensures that y(t) is a stationary process, whereas the inversion condition ensures that the spectral density matrix function of the continuous time process y(t) is defined for all frequencies. Proofs of the existence and uniqueness of solutions to systems like (1) are identical to Chambers (1998), and the spectral density matrix of the continuous time process y(t) is given by

which follows because the spectral density matrix of the white noise vector ζ(dt) is (½π)Ω and the frequency response function of the operator D is iλ. We have emphasized the dependence of the spectral density function on the parameter vector β for purposes of clarity.

The underlying time series is allowed to comprise both stock and flow variables, such that y(t) = [y^s(t)′,y^f(t)′]′, where the vector of stock variables (y^s) has dimension q^s × 1 and the vector of flows (y^f) has dimension q^f × 1 and where q^s + q^f = q. The observed discrete time vector is then given by

for t = 1,…,T, where T denotes the sample size. The frequency response function of the time invariant linear filter that transforms y(t) into y_t is given by the matrix

where h(iλ) = (1 − e^−iλ)/iλ is the filter for flows.⁴

where

That f (λ,β) is Hermitian positive definite and integrable⁶

The aliasing phenomenon is, of course, relevant in this setting. It is impossible to detect variation at high frequencies in the continuous time series because it appears as variation below the Nyquist frequency π/Δt in the discrete time series, where Δt is the observation frequency. Using high-frequency data is one solution so that the continuous time spectral density becomes negligible outside [−π/Δt,π/Δt], but such an option is typically not available for the majority of economic time series. For conditions on the identification of the parameters in continuous time models the reader is referred to Hansen and Sargent (1983) and Phillips (1973).

The difficulty for estimation arises from the computation of the discrete time spectral density function (4). When the lags are integers it is possible to calculate (4) exactly. Chambers (1998) shows how to achieve this by exploiting the periodicity of the exponential terms⁷

Only if ν_lk were integer could the periodicity of the exponential terms be exploited to reduce them to be of the form e^−iλν_lk.

One solution to the problem of computing (4), and the procedure used throughout this paper, is to simply truncate the summation at a finite number M, i.e., to use

Clearly (6) is only an approximation to the true discrete time spectral density function, but if the continuous time spectral density is sufficiently small after some finite point, i.e., if

is negligible, where

, then the approximation should be sufficiently accurate. We do at least know that the continuous time spectral density matrix f (λ,β) behaves like λ⁻²ⁿ uniformly in

as λ → ∞. Although the choice of truncation point M may seem somewhat arbitrary, we provide in the next section conditions on M that ensure the consistency and asymptotic normality of the frequency domain estimator.

The error arising from the use of (6) in place of (4) is given by

A useful property of the error, and one that will be called upon when investigating the asymptotic properties of the estimator, is that R^M(λ,β) = O(M¹⁻²ⁿ) uniformly in (λ,β).⁸

3. FREQUENCY DOMAIN MAXIMUM LIKELIHOOD ESTIMATION

A commonly used frequency domain estimator is the minimizer of the negative of (2/T) times the frequency domain approximation to the Gaussian log-likelihood function, given by

where

is the periodogram,

is the discrete Fourier transform of y_t, and

. However, for the reasons stated earlier, it is not possible to accurately compute the spectral density function F(λ,β). We therefore consider a feasible version of (8), replacing F(λ,β) with F^M(λ,β), resulting in

The likelihood function in (9) contains a bias term that may persist even asymptotically if the truncation is independent of the sample size. We therefore allow M to increase with T and prescribe appropriate rates of increase for the minimizer of (9) to be consistent and asymptotically normally distributed. In fact, we consider the asymptotic properties of two estimators, defined by

where

denotes the parameter space. We shall refer to

as being an infeasible estimator of β because the spectral density function F(λ,β) in (8) cannot be computed accurately. Conversely we shall refer to

as being a feasible estimator of β because the truncated spectral density F^M(λ,β) can be computed exactly for finite M. We shall require the following additional assumptions to establish the consistency of both estimators.

Assumption 2. The elements of the matrices A_lk(θ) (l = 0,…,n; k = 1,…,s_l) are continuous functions of θ ∈ Θ.

Assumption 3. (a)

, where β₀ is the true value of and

is a compact subset of

. (b) For β ≠ β₀, F(λ,β) ≠ F(λ,β₀) on a set of positive Lebesgue measure.

Assumption 4. The random measure ζ(dt) is Gaussian.

Assumption 5. M = O(T^δ) for some δ > 0.

Assumption 2 is a standard continuity assumption that ensures, in particular, that the spectral density function F(λ,β) is continuous in

. Assumption 3(a) is also standard in consistency proofs of optimization estimators, whereas 3(b) is an identification condition. Assumption 4 is equivalent to defining ζ(dt) as the increment of a Brownian motion process and ensures that the processes y(t) in continuous time and y_t in discrete time are also Gaussian. This assumption could undoubtedly be relaxed so that non-Gaussian processes could be considered but at the cost of complicating the proofs. One consequence of doing this would be that the asymptotic covariance matrix in Theorem 2, which follows, would become more complicated and would depend on the higher moments of the process. Finally, Assumption 5 requires the truncation point M to increase at an arbitrary rate with T, so that asymptotically the approximation error is eliminated. The consistency result is as follows.

THEOREM 1. Under Assumptions 1–5,

as T → ∞.

The proof of Theorem 1 first establishes the consistency of the infeasible estimator

by demonstrating that L_T(β) converges uniformly over

to a function L(β) that is uniquely minimized at β = β₀. The consistency of the feasible estimator

is then shown by establishing the uniform convergence of L_T^M(β) to L_T(β). The feasible estimator, based on the truncated spectral density, therefore shares the consistency property of the infeasible estimator, provided that the truncation point increases with sample size.

To derive the asymptotic distributions of the two estimators, two additional assumptions are required.

Assumption 5′. M = O(T^δ) for some δ > 1/(4n − 2).

Assumption 6. The elements of the matrices A_lk(θ) (l = 0,…,n; k = 1,…,s_l) are twice continuously differentiable functions of θ ∈ Θ.

Assumption 5′ represents a strengthening of Assumption 5 and is required for the asymptotic distribution theory. Note that δ is a decreasing function of the differential order n, so that in models with higher order derivatives, a slower rate of increase of M with T will suffice. The intuition behind this is fairly simple because we know that the error of truncation (R^M) is inversely related to the order of derivative (n) and therefore as the order of derivative increases, the same magnitude for R^M can be obtained with a smaller truncation point. In models containing just a first derivative the condition reduces to δ > 0.5 as in Chambers and McGarry (2002). Assumption 6 is common in the asymptotic analysis of optimization estimators and ensures that the objective functions (8) and (9) can be differentiated to second order.

THEOREM 2. Under Assumptions 1–4, 5′, and 6,

as T → ∞, where the (i,j) element of S(β₀) is given by

for i,j = 1,…,m, and ∂F(λ,β₀)/∂β_i denotes the matrix ∂F(λ,β)/∂β_i evaluated at β = β₀.

Both estimators share the same normal limiting distribution, provided the truncation point for the infeasible estimator increases at the appropriate rate with sample size, as required in Assumption 5′. The proof of Theorem 2 relies on standard arguments for the infeasible estimator

, based on a mean value expansion of the score vector, whereas the result for

is obtained by establishing that

as T → ∞. It is worth noting that the elements of the matrix S(β₀) can be consistently estimated using

, where the (i,j) element of the latter is defined by the computationally more convenient expression

where

denotes the matrix ∂F(λ,β)/∂β_i evaluated at

. The matrix

is obtained in the same way with

replaced by

.

The estimators

are both asymptotically efficient under the stated assumptions. Departures from these assumptions, however, such as model misspecification or non-Gaussianity, can be expected to lead to inefficiencies, even asymptotically. Appropriately designed sampling experiments could be utilized to assess the extent of such inefficiencies in finite samples, but such an investigation remains beyond the scope of the present paper.

The asymptotic normality of the feasible estimator and the likelihood-based approach to estimation both mean that tests of hypotheses concerning β can be conducted using the standard trinity of Wald, likelihood ratio, and Lagrange multiplier test statistics. Under the appropriate null hypothesis standard techniques can be used to show that these statistics will be asymptotically distributed as chi-squared with degrees of freedom equal to the number of restrictions being tested. For example, tests of the nonlinear restrictions h(β) = 0, where

is twice continuously differentiable and the matrix ∂h/∂β′ evaluated at β = β₀ has full rank, have asymptotic χ_c² distributions under the null, but other hypotheses are also of interest. Prominent among these is a test of equality of lag parameters across the elements of y(t). In a simple setting, suppose

in which case the null of interest would be H₀ : ν₁ = ν₂ = ··· = ν_q. This might correspond to a test of whether the time-to-build factors in a group of categories of investment expenditures were equal. Within the same setting another hypothesis might be that the ν_i were all equal to a particular integer value, e.g., unity. If this null hypothesis were found to be consistent with the data then its imposition would enable the spectral density function to be computed exactly using the techniques of Chambers (1998).

4. CONCLUDING COMMENTS

In this paper we have considered the problem of estimating the unknown parameters, including lag and lead parameters, in a system of DDEs of sufficient generality that nests the important recent DDE specifications that have appeared in economics, thereby expanding the scope for estimation of DDEs beyond those that have appeared in the econometrics literature to date. The main technical results of this paper concern the derivation of the asymptotic properties of the (infeasible) frequency domain maximum likelihood estimator and its feasible counterpart based on an approximation to the discrete time spectral density function. The feasible frequency domain maximum likelihood estimator, based on the truncated spectral density, is shown to share the same desirable asymptotic properties as the infeasible estimator, based on the exact spectral density, provided the truncation parameter increases at an appropriate rate with the sample size (which is given by Assumption 5 for consistency and by Assumption 5′ for asymptotic normality). It is therefore possible to obtain consistent and asymptotically normal estimates of the lag (or lead) parameters themselves in addition to the other parameters in the DDE system and to perform asymptotic chi-squared inference on hypotheses concerning those parameters.