21. Inferring a Continuous-Time System from Discrete-Time Data: An Appreciation of A. W. Phillips (1959)

21. Inferring a Continuous-Time System from Discrete-Time Data: An Appreciation of A. W. Phillips (1959)#

Note

This chapter presents A. W. Phillips’s 1959 Biometrika paper, “The Estimation of Parameters in Systems of Stochastic Differential Equations.” Rather than reproduce the paper, we paraphrase Phillips’s argument step by step and read it as an early application of machinery built up in earlier chapters: covariance-stationary processes and their spectral densities, white noise and physical realizability, rational transfer functions, the sum-of-exponentials structure of rational autocovariances, and the folding formula for sampled processes. The framing and the closing survey draw on an appreciation of Phillips written by Lars Peter Hansen and Thomas J. Sargent.[1]

Why Phillips’s 1959 paper matters#

One way to honour A. W. Phillips is to trace the continuing influence of one of his most enduring contributions to economic dynamics: his 1959 Biometrika paper about how discrete-time observations can be used to restrict a continuous-time linear model. That paper described, with great precision, what later came to be called the problem of aggregation over time, set out a framework for studying it, and obtained useful characterizations of it. Phillips’s paper shared something of the destiny of John F. Muth’s 1960 and 1961 papers on rational expectations: it took the profession years to recognize how much could be done with the ideas. In 1960, both Phillips and Muth were well ahead of most economists in their command of the technicalities of time series analysis and in their appreciation of its potential for economic dynamics. Economists did not really take up the inquiry where Phillips left it until the early 1970s, when A. R. Bergstrom, Christopher Sims, P. C. B. Phillips, and others returned to aggregation over time.

Phillips’s setting is one we have been preparing for throughout these notes. He supposes that a vector of data is generated by a continuous-time stochastic process with a rational spectral density matrix[2], the class of processes that Chapter 8 singled out through the spectral factorization theorem, and that Chapter 11 generated by driving constant-coefficient linear differential equations with white noise. As Phillips recognized, processes with rational spectral densities are the natural environment for studying time aggregation, for a chain of interrelated reasons. A rational continuous-time spectral density implies all of the following:

(a) the discrete-time process obtained by skip-sampling (point-in-time sampling) the continuous process also has a rational spectral density;

(b) the continuous-time autocovariance function is a positive semidefinite function on the real line that is a weighted sum of exponentials \(e^{\lambda_i \tau}\), with \(\operatorname{Re}\lambda_i < 0\), the structure that Chapter 11 derived for solutions of linear stochastic differential equations;

(c) the discrete-time autocovariance sequence is a positive semidefinite sequence on the integers that is a weighted sum of geometric sequences \(\mu_i^{|\tau|}\), with decay factors \(\mu_i = e^{\lambda_i}\), with the same weights and the same exponents as in (b), mapped through the exponential;

(d) the continuous-time data solve a system of linear differential equations driven by a continuous-time vector white noise, itself physically unrealizable in the sense of Chapter 4 but well defined once convolved with an \(L^2\) kernel;

(e) the discrete-time data solve a system of linear difference equations driven by a discrete-time white noise;

(f) the continuous differential equation can be solved to express the data as a convolution of an \(L^2\) kernel with the white noise, the continuous-time Wold representation of Chapter 8;

(g) the discrete difference equation can be solved to express the sampled data as a convolution of an \(\ell^2\) sequence with the discrete white noise.

The bridge between (b) and (c) carries the same weights and relates the exponents by \(\mu_i = e^{\lambda_i}\). It is the foundation of Phillips’s approach and the source of the aliasing problem. Because the complex logarithm is multivalued, for any integer \(k\) the decay factor \(\mu_i = \exp(\lambda_i \pm 2\pi i k)\) is consistent with the same observed discrete data; many continuous-time models therefore share one discrete-time autocovariance sequence. P. C. B. Phillips (1972, 1973, 1974) and Hansen and Sargent (1983, 1991) pursued this thread and, in a sense, completed it; the precise count of observationally equivalent models is the subject of 22. The Dimensionality of the Aliasing Problem in Models with Rational Spectral Densities. What follows reconstructs the steps by which Phillips reached the representations (a)–(g) and turned them into a procedure for inference.

The continuous-time system#

Phillips begins with a system of linear stochastic lagged dependences among \(n\) variables \(x_i(t)\) of continuous time, written as a system of convolution integrals,

(205)#\[x_i(t) = \sum_{j \neq i} \int_0^\infty w_{ij}(h)\, x_j(t-h)\, dh + \xi_i(t), \qquad i = 1, \ldots, n.\]

Each weighting function \(w_{ij}(h)\) records how the value of \(x_i\) at time \(t\) depends on the history of \(x_j\), and the \(\xi_i(t)\) are stochastic disturbances. This is the multivariate, two-sided cousin of the one-sided distributed lags that appeared as far back as Chapter 3 (the shot-noise integral \(\int_0^\infty L(\tau) z(t-\tau)\, d\tau\)) and that organized the Wold representation of Chapter 8.

Phillips’s pivotal modelling choice is to require every lagged dependence to be rational: the Laplace transform of each weighting function, which Phillips calls its transfer function, is a ratio of finite-order polynomials,

(206)#\[\int_0^\infty e^{-sh}\, w_{ij}(h)\, dh = \frac{\beta_{ij}(s)}{\alpha_{ij}(s)}.\]

This is exactly the rational-transfer-function language of Chapters 8 and 11. Recalling that the Laplace transform of the derivative operator \(D = d/dt\) is multiplication by \(s\), the identity that ran through the operational calculus of Chapters 8, 11, and 12, Phillips rewrites the convolution system (205) as a system of differential equations and then, collecting terms, as the single operator equation

(207)#\[Q(D)\, x(t) = \xi(t),\]

where \(Q(D)\) is the \(n \times n\) matrix of operators with diagonal entries \(1\) and off-diagonal entries \(-\beta_{ij}(D)/\alpha_{ij}(D)\).

At this point Phillips imposes a restriction whose payoff appears only later, and which we are in a position to read structurally. He insists that each numerator \(\beta_{ij}(s)\) be of strictly lower degree than its denominator \(\alpha_{ij}(s)\). His own justification is that behavioural units cannot respond to change instantaneously: an instantaneous response would require a Dirac \(\delta\)-impulse in the weighting function \(w_{ij}\), and a \(\delta\)-impulse is exactly what a numerator of equal or higher degree would produce in the inverse transform.[3] In the language of Chapter 4, Phillips is asking that the lagged dependences be physically realizable, built from ordinary rather than generalized kernels. By the criteria of Chapters 2 and 9, a strict numerator–denominator degree gap is precisely what buys mean-square continuity, indeed a definite order of mean-square differentiability, for the response. Phillips will later show that this same restriction is what makes the separate behavioural equations identifiable. A modelling assumption motivated by economics turns out to coincide with the smoothness condition the earlier chapters identified, and to carry the weight of identification as well.

The disturbances, white noise, and spectral factorization#

Phillips next specifies the disturbance vector \(\xi(t)\). He takes the system to be stable and the \(\xi_i(t)\) to be stationary with rational spectral densities, and he generates them by passing a more primitive vector \(\zeta(t)\) through a rational filter,

(208)#\[\xi(t) = R(D)\, \zeta(t),\]

with \(R(D)\) again a matrix of proper rational operators. Writing \(\vartheta\), \(\psi\), \(\lambda\) for the (cross-)covariance functions of \(\xi\), of \(\zeta\), and between them, and \(\theta\), \(\Psi\), \(\Lambda\) for their two-sided Laplace transforms, a short convolution argument gives \(\theta(s) = R(s)\,\Psi(s)\,R'(-s)\). It is the continuous-time analogue of the filtering calculation in Chapter 8.

Phillips now makes the primitive process \(\zeta(t)\) a white noise: he sets \(\Psi(s) = I\). This is the boldest and most modern step in the paper, and it is the white noise of Chapters 3, 4, 6 and 7. Replacing \(s\) by \(i\omega\) turns Laplace transforms into spectral densities, and \(\Psi = I\) says that \(\zeta\) has a flat spectral density at every frequency and zero cross-spectra. Its covariance matrix is \(\delta(\tau) I\), a \(\delta\)-impulse at the origin. As Chapter 4 stressed, such a process has infinite instantaneous variance and is not physically realizable; it exists only as a generalized process, exactly like \(dN/dt\) in Chapter 3 or the formal derivative \(dW/dt\) of the Wiener process in Chapter 6. It is nonetheless a perfectly legitimate input, because convolving it with the proper rational kernel \(R\) returns an ordinary, physically realizable \(\xi(t)\), which is the point of Chapter 4. With \(\Psi = I\),

(209)#\[\theta(s) = R(s)\, R'(-s).\]

Equation (209) is the spectral factorization theorem of Chapter 8, written for a vector disturbance. Phillips has factored the rational spectral density \(\theta\) of the observable disturbance into a stable rational operator \(R\) times its conjugate \(R'(-s)\). That is the factorization \(S(\omega) = \tilde P(i\omega)\tilde P(-i\omega)\) that identified the fundamental white-noise representation in Chapter 8. The condition that the roots of the denominators of \(R\) have negative real parts is the fundamentalness (minimum-phase) condition from that chapter.

Phillips performs the factorization one way; 15. State-Space Models, the Kalman Filter, and Spectral Factorization performs it another. Phillips works in the frequency domain and by hand: he forms the generating function of the sampled autocovariances, factors its numerator polynomial by finding roots, and keeps the factor whose zeros lie outside the unit circle. That is a discrete-time spectral factorization carried out explicitly. The Kalman–Bucy filter reaches the same object by a different route. It solves an algebraic Riccati equation in the time domain, whose stabilizing solution delivers the minimum-phase factor \(T(s) = I + C(sI-A)^{-1}K\) automatically, with no root-finding at all. Phillips in 1959 was doing by hand, and for the sampled process, what the Riccati equation does in one step; the two are the frequency-domain and time-domain faces of the same theorem, and recognizing that is what later made large systems computationally tractable.

The system autocovariance is a matrix sum of exponentials#

Combining (207) and (208), the whole system reads \(Q(D)x(t) = R(D)\zeta(t)\). Clearing denominators by a diagonal operator \(P(D)\) puts it in polynomial form,

(210)#\[F(D)\, x(t) = G(D)\, \zeta(t), \qquad F = PQ,\ G = PR,\]

with \(F(D)\) and \(G(D)\) now matrices of polynomials. Because of the strict numerator–denominator degree gap imposed earlier, the diagonal entry in each row of \(F\) has strictly higher degree than the off-diagonal entries and than the entries of \(G\) in that row; hence \(\det F(D)\) does not vanish identically and the system can be solved, \(x(t) = F^{-1}(D) G(D)\, \zeta(t)\). This is the multivariate version of the operator solution \(x = \tilde P(D) w\) of Chapter 11. Reading \(F^{-1}G\) as a one-sided convolution kernel makes it the continuous-time Wold representation (f) listed above.

Repeating the filtering calculation that produced (209), the spectral density matrix of the observable vector \(x\) is

(211)#\[\Phi(s) = F^{-1}(s)\, G(s)\, G'(-s)\, F'^{-1}(-s).\]

Phillips now extracts the autocovariance. Writing \(p(s) = |F(s)| = \prod_{r=1}^m (s - \lambda_r)\) for the characteristic polynomial and assuming the roots \(\lambda_r\) (the poles of the system) are distinct with negative real parts, he expands (211) in partial fractions and inverts. The unique inverse compatible with stationarity is the one that decays as \(\tau \to \pm\infty\):

(212)#\[\phi(\tau) = \sum_{r=1}^m K_r\, e^{\lambda_r \tau}, \qquad \tau \geq 0, \qquad \phi(-\tau) = \phi(\tau)'.\]

Each coefficient \(K_r\) is a rank-one matrix, \(K_r = k_r l_r\), built from the left and right null vectors of \(F(\lambda_r)\). It is the residue of the pole \(\lambda_r\). Equation (212) is the centrepiece, and it is precisely the result (b) promised at the outset: a rational spectral density produces an autocovariance that is a weighted sum of exponentials, with decay rates equal to the poles. In the scalar case it is exactly the \(p(\tau) = \sum_j g_j e^{\lambda_j \tau}\) of Chapter 11 and the sum-of-exponentials entries in the table of Chapter 8. Phillips’s contribution is the matrix bookkeeping, the rank-one residues \(K_r\), that carries the scalar fact into a multivariate system.

From continuous time to discrete time: sampling and aliasing#

Here Phillips builds the bridge to the data. Sample the continuous autocovariance at unit intervals, \(\phi_\tau \equiv \phi(\tau)\) for integer \(\tau\), and form the one-sided generating function \(M(z) = \sum_{\tau \ge 0} \phi_\tau z^\tau\). Substituting (212) and summing the geometric series gives

(213)#\[M(z) = \sum_{r=1}^m \frac{K_r}{1 - \mu_r z}, \qquad \mu_r = e^{\lambda_r}.\]

This is result (c), and it is the exact statement that sampling preserves the sum-of-exponentials structure: the same residues \(K_r\) reappear, and each continuous decay rate \(\lambda_r\) is mapped to a discrete decay factor \(\mu_r = e^{\lambda_r}\). The discrete-time autocovariance sequence is a sum of geometric sequences \(K_r \mu_r^{|\tau|}\), the autocovariances of a vector ARMA process. Sampling a rational continuous-time model lands us inside the rational discrete-time models studied by Bartlett, Quenouille, Whittle, and Wold, which is exactly why Phillips can hope to estimate the continuous system from discrete data.

The map from the discrete data back to the continuous parameters runs through

(214)#\[\lambda_r = \log \mu_r.\]

Equation (214) is where the aliasing problem is born, and it is the same phenomenon as the folding formula of Chapter 17. The complex logarithm is multivalued: if \(\mu_r\) is complex, then \(\lambda_r\), \(\lambda_r + 2\pi i\), \(\lambda_r - 2\pi i\), \(\ldots\) all exponentiate to the same \(\mu_r\), so infinitely many continuous-time decay rates produce the same sampled autocovariance. Sampling at unit intervals can resolve frequencies only up to the Nyquist frequency \(\pi\); higher continuous-time frequencies are folded (aliased) back into the observable band, exactly as Chapter 17’s \(S^d(\omega) = \sum_n S(\omega - 2\pi n)\) folds the continuous spectrum. The data always recover the decay rate \(\operatorname{Re}\lambda_r\), since \(\operatorname{Re}\lambda_r = \log|\mu_r|\). The data leave the oscillation frequency \(\operatorname{Im}\lambda_r\) undetermined. Counting precisely how many continuous models survive is the project of 22. The Dimensionality of the Aliasing Problem in Models with Rational Spectral Densities, where the requirement that the implied continuous spectral density be positive semidefinite is shown to cut the countable infinity down to a finite set.

Phillips also gives the constructive direction. From a finite stretch of sampled autocovariances \(\phi_\tau\) one can fit a matrix generating function \(M(z) = U^{-1}(z) V(z)\) with polynomial \(U, V\); the coefficients of \(U\) satisfy a linear difference equation in the \(\phi_\tau\), after which a partial-fraction expansion returns the residues \(K_r\) and decay factors \(\mu_r\), and (214) returns the continuous poles \(\lambda_r\). The whole continuous spectral density \(\Phi(s)\) can then be reassembled from the \(K_r\) and \(\lambda_r\) via (212).

Recovering and identifying the behavioural equations#

Knowing \(\Phi(s)\) is not the same as knowing the structural operators \(Q\) and \(\theta\), the behavioural equations and the disturbance covariances one wants. Phillips shows how to recover them, and exactly here the degree restriction from §”The continuous-time system” earns its keep.

When every equation of the system has the same order \(u\), the strict degree gap forces the diagonal of \(F(s)\) to be a degree-\(u\) polynomial while the off-diagonal entries have degree at most \(u-1\). Counting coefficients against the residue conditions \(F(\lambda_r) K_r = 0\), Phillips finds exactly enough independent equations to pin down every coefficient of \(F\); the structural \(Q(s) = P^{-1}(s) F(s)\) and the disturbance spectrum \(\theta(s)\) then follow. The behavioural equations are identified. When the equations are of different orders, identification can fail, but Phillips shows it is restored if one knows that the structural disturbances are mutually uncorrelated: requiring the off-diagonal entries of \(G(s)G'(-s)\) to vanish forces the admissible change-of-basis matrix to be diagonal, so the system is again identified. This is the same kind of side information, exclusion and orthogonality restrictions, that 22. The Dimensionality of the Aliasing Problem in Models with Rational Spectral Densities and the rational-expectations literature later used to overcome aliasing.

The discrete-time system implied by sampling#

Phillips closes by writing the sampled data’s own dynamics. Forming the two-sided generating function and factoring it, he shows that the sampled vector \(x_t\) obeys a stochastic difference equation

(215)#\[U(E^{-1})\, x_t = W(E^{-1})\, \epsilon_t,\]

driven by a discrete white noise \(\epsilon_t\), where \(E^{-1}\) is the lag operator \(E^{-h} y_t = y_{t-h}\). This is representation (e)–(g): the continuous system \(F(D)x = G(D)\zeta\) has a discrete-time ARMA shadow, the discrete Wold representation of the sampled data. Phillips’s point is that because the sampled continuous process and this discrete ARMA process have identical second-moment properties, the sampling behaviour of estimators of the continuous system can be studied through the well-developed theory of estimators for discrete ARMA models.

Every step of Phillips’s procedure runs on sample autocovariances \(\hat\phi_\tau\) computed from a single realization, and the argument presupposes that these converge to the population \(\phi_\tau\). That is the property called covariance ergodicity, and it is not implied by covariance stationarity: it is a restriction on fourth moments, not second. For the processes Phillips works with it does hold. A Gaussian process with a rational spectral density has an exponentially decaying autocovariance, which suffices. A1. Ergodicity and the Consistent Estimation of Second Moments gives the conditions, and notes the one circumstance, exhibited in 17. Discrete Sampling: The Folding Formula, in which sampling itself destroys the property. It is the same identification of continuous-time content with a discrete-time representation that organizes the companion chapters on time aggregation (18. Time Aggregation, 20. Aggregation Over Time and the Inverse Optimal Predictor Problem for Adaptive Expectations in Continuous Time).

Phillips’s legacy#

Phillips set out the framework; the profession spent decades unpacking it. Two threads run directly out of the 1959 paper. The first is the aliasing / identification thread of (214): P. C. B. Phillips (1972, 1973, 1974) and Hansen and Sargent (1983, 1991) characterized exactly when, and how nearly, a continuous-time model can be recovered from skip-sampled data. 22. The Dimensionality of the Aliasing Problem in Models with Rational Spectral Densities gives that analysis. The second, pursued by Christopher Sims (1971) and John Geweke (1978), asks how closely discrete-time distributed lags resemble continuous-time ones. 18. Time Aggregation takes up that question, asking when the discrete moving-average coefficients \(C_k\) resemble a sampled continuous kernel \(p(\tau)\); Sims’s formula expresses the discrete lag as a convolution of the continuous lag with a function of the regressors’ autocovariance, a time-series version of the omitted-variable theorem in which the “omitted variables” are the continuum of unobserved sampling instants.

A related strand compares continuous and discrete impulse responses, Phillips’s representations (d)–(e) and (f)–(g). Hansen and Sargent (1991), staying within Phillips’s rational world, traced when discrete impulse responses resemble the underlying continuous ones to the mean-square continuity and differentiability of the process. Chapters 2 and 9 showed that the numerator and denominator degrees of the rational spectral density fix those properties. That is the degree gap Phillips insisted on. Albert Marcet (1991) broke with the rational assumption, keeping the kernel in \(L^2\) but allowing discontinuities, and showed how much harder aggregation over time becomes when the smoothness Phillips assumed is absent (23. Temporal Aggregation of Economic Time Series).

Finally, rational expectations threw new light on the aliasing problem. The cross-equation restrictions that rational expectations imposes often supply just enough identifying information to pin down a unique continuous-time model, sometimes without using the rational spectral-density restriction. Christiano (1984), Christiano and Eichenbaum (1987), Heaton (1993), and Hansen and Sargent (1991) exploited this route. Christiano offered an image of how it works.[4] More recently, Hansen and Scheinkman (1995) extended the questions Phillips raised under (b) and (c) to nonlinear continuous-time Markov processes, using moment conditions for test functions to identify them from discrete data; the operator counterparts of Phillips’s \(\lambda_r \mapsto \mu_r = e^{\lambda_r}\) reappear there as the relation between the generator of the continuous process and the transition operator of its sampled version.

In ten pages Phillips assembles rational transfer functions, the spectral factorization theorem, white noise as a generalized input, the sum-of-exponentials autocovariance, the sampling map \(\mu = e^{\lambda}\), and the discrete ARMA shadow of a continuous system. He turns them into a program for inference. Almost every structure that this book builds appears in his paper.

Exercises#

These exercises make Phillips’s continuous-to-discrete bridge concrete. They use only numpy.

import numpy as np

Exercise 25

Aliasing and the multivalued logarithm. Phillips’s map (214), \(\lambda_r = \log \mu_r\), is the root of the aliasing problem because the complex logarithm is multivalued. Take a complex continuous-time pole \(\lambda = -0.3 + i\).

(a) Show that \(\lambda\) and the aliases \(\lambda_k = -0.3 + i(1 + 2\pi k)\), for \(k = -3, \ldots, 3\), all map to the same discrete decay factor \(\mu = e^{\lambda}\).

(b) Confirm that the decay rate \(\operatorname{Re}\lambda = \log|\mu|\) is recovered from \(\mu\), whereas the oscillation frequency \(\operatorname{Im}\lambda\) is determined only up to a multiple of \(2\pi\), that is, only up to the Nyquist frequency \(\pi\) of unit-interval sampling (Chapter 17).

Exercise 26

Phillips’s procedure end-to-end on the Ornstein–Uhlenbeck process. Let \(x\) be the scalar Ornstein–Uhlenbeck process of Chapters 7–8, \(dx = -a\, x\, dt + b\, dW\), with \(a = 0.7\), \(b = 1\). Its continuous-time autocovariance is \(\phi(\tau) = \tfrac{b^2}{2a} e^{-a|\tau|}\) (a one-term version of (212) with pole \(\lambda = -a\)).

(a) Sample \(\phi\) at the integers and form Phillips’s generating function \(M(z) = \sum_{\tau \ge 0} \phi_\tau z^\tau\). Verify it equals \(\phi_0/(1 - \mu z)\) with \(\mu = e^{-a}\), as (213) predicts.

(b) Recover the continuous-time parameters from the sampled autocovariances alone: \(\mu = \phi_1/\phi_0\), then \(a = -\log\mu\) from (214), and \(b^2 = 2a\,\phi_0\).

(c) Exhibit the discrete-time shadow (215): the sampled process is the AR(1) \(x_t = \mu\, x_{t-1} + \epsilon_t\). Report the innovation variance \(\sigma_\epsilon^2 = \phi_0(1 - \mu^2)\) and check that the AR(1) autocovariances \(\phi_0\mu^{|\tau|}\) reproduce the sampled \(\phi_\tau\).

Exercise 27

Sampling preserves the sum-of-exponentials structure. Consider the second-order scalar system \((D + a_1)(D + a_2)\, x = w\) driven by unit white noise, with \(a_1 = 0.5\), \(a_2 = 1.3\). Its spectral density is \(S(\omega) = 1/[(a_1^2 + \omega^2)(a_2^2 + \omega^2)]\), and (Chapter 8, table; Chapter 11) its autocovariance is the two-term sum of exponentials \(\phi(\tau) = K_1 e^{-a_1|\tau|} + K_2 e^{-a_2|\tau|}\) with \(K_1 = \tfrac{1}{2a_1(a_2^2 - a_1^2)}\), \(K_2 = -\tfrac{1}{2a_2(a_2^2 - a_1^2)}\).

(a) Sample at the integers and verify Phillips’s identity (213): the generating function \(\sum_{\tau \ge 0}\phi_\tau z^\tau\) equals \(\dfrac{K_1}{1 - \mu_1 z} + \dfrac{K_2}{1 - \mu_2 z}\) with \(\mu_r = e^{-a_r}\), the same residues \(K_r\), and exponents mapped by \(\mu = e^{\lambda}\).

(b) Recover the continuous poles from the discrete decay factors via (214), \(\lambda_r = \log\mu_r\), and confirm \(\lambda_r = -a_r\).

References#

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Bartlett, M. S. (1946). On the theoretical specification and sampling properties of autocorrelated time series. Journal of the Royal Statistical Society B, 8, 27–41.

Bergstrom, A. R. (1983). Gaussian Estimation of Structural Parameters in Higher Order Continuous Time Dynamic Models. Econometrica, 51, 117–152.

Christiano, L. J. (1984). The Effects of Aggregation over Time on Tests of the Representative Agent Model of Consumption. Mimeo, Federal Reserve Bank of Minneapolis.

Christiano, L. J., and M. Eichenbaum (1987). Temporal Aggregation and Structural Inference in Macroeconomics. Carnegie–Rochester Conference Series on Public Policy, 26, 63–130.

Frazer, R. A., Duncan, W. J., and Collar, A. R. (1950). Elementary Matrices. Cambridge University Press.

Geweke, J. (1978). Temporal Aggregation in the Multiple Regression Model. Econometrica, 46, 643–661.

Hansen, L. P., and T. J. Sargent (1983). The Dimensionality of the Aliasing Problem in Models with Rational Spectral Densities. Econometrica, 51, 377–387.

Hansen, L. P., and T. J. Sargent (1991). Identification of Continuous Time Rational Expectations Models from Discrete Time Data. In L. P. Hansen and T. J. Sargent, Rational Expectations Econometrics. Boulder: Westview Press.

Hansen, L. P., and J. A. Scheinkman (1995). Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes. Econometrica, 63, 767–804.

Heaton, J. (1993). The Interaction Between Time-Nonseparable Preferences and Time Aggregation. Econometrica, 61, 353–385.

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Marcet, A. (1991). Temporal Aggregation of Economic Time Series. In L. P. Hansen and T. J. Sargent, Rational Expectations Econometrics. Boulder: Westview Press.

Muth, J. F. (1960). Optimal Properties of Exponentially Weighted Forecasts. Journal of the American Statistical Association, 55, 299–306.

Muth, J. F. (1961). Rational Expectations and the Theory of Price Movements. Econometrica, 29, 315–335.

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Phillips, P. C. B. (1974). The Estimation of Some Continuous Time Models. Econometrica, 42, 803–823.

Quenouille, M. H. (1957). The Analysis of Multiple Time Series. London: Charles Griffin.

Sims, C. A. (1971). Discrete Approximations to Continuous Time Distributed Lags in Econometrics. Econometrica, 39, 545–563.

Whittle, P. (1953). The Analysis of Multiple Stationary Time Series. Journal of the Royal Statistical Society B, 15, 125–139.

Wold, H. (1949). A Large-Sample Test for Moving Averages. Journal of the Royal Statistical Society B, 11, 297–305.

Notes#