Index

A

absolute continuity of the kernel adaptive expectations in continuous time optimality of adjustment costs and mean square differentiability admissible model aggregation over time effects on a bivariate model aliasing definition dimensionality of ergodicity, [1] in Phillips's system seasonal component see also folding formula aliasing problem dimensionality of analytic decomposition analytic function autocovariance rigidity of annihilation operator and transforms general form

approximation error of adaptive expectations ARMA process as the discrete shadow of a continuous model arrival rate arrival times Arrow-Debreu prices asset prices and local unpredictability autocorrelation function autocorrelation function, normalized see autocovariance function autocovariance smoothness autocovariance function continuity at the origin differentiability of matrix sum of exponentials see also covariogram autoregressive representation of a sampled process averaged process

B

band decomposition used in reverse band-pass filter bandwidth conditions on

Bartlett triangle bivariate prediction of inflation bivariate Wold representation Brownian motion see Wiener process

C

Cagan model aggregation over time Cagan's adaptive expectations continuous time version Cauchy criterion certainty equivalence, [1] Cesaro average characteristic roots closed span of sampled observations coherence and smoothing companion form compensated Poisson process competitive equilibrium and the planning problem consistency of autocovariance estimators of spectral density estimators consumption and income contamination (Geweke), [1] continuity almost sure continuity convention (left versus right)

continuous time econometrics history of continuous time model as the primitive object control algebraic Riccati equation convolution cost-to-go matrix countable infinity of observationally equivalent models counting process covariance ergodicity, [1] and consistent estimation definition see also fourth cumulant covariance stationarity and the aliasing problem definition covariogram see also spectral density Cramér representation statement cross-equation restrictions cross-spectral density definition estimation of cumulant

D

degenerate process detectability differential equation system linear differential operator D diffusion coefficient Dirac delta function, [1] discontinuous kernel discounting in prediction formulas removal of

discrete kernel Marcet's formula for discrete sampling of a continuous time process drift duality filtering and control

E

ensemble average ergodic see ergodicity ergodicity, [1] aliasing, [1] and fourth moments and the theory of estimation covariance covariance, definition criterion for destroyed by sampling estimation, [1] mean square, [1] mean square, definition sample mean spectral criterion sufficient conditions for two conditions

estimation ergodicity, [1] of continuous time systems Euler equations spectral factorization of Euler-Maruyama discretization exponentially weighted forecast

F

feedback and feedforward filter algebraic Riccati equation first order Markov process aliasing in flow variable folding formula applied to a structural system derivation see also Nyquist frequency forecast optimal linear forecast error sampled forecast error covariance forecast error variance over short intervals

Fourier transform of an autocovariance table of tables of fourth cumulant condition on frequency band variance carried by frequency bands sampling frequency domain representation of a process fundamental white noise and the innovations representation definition

G

Gaussian process and covariance ergodicity and strict stationarity generalized Poisson process definition generalized stochastic process definition versus ordinary

geometric distributed lead in decision rules prediction of Granger causality and time aggregation created or destroyed by sampling induced by aggregation spurious

H

Hamilton-Jacobi-Bellman equation Hamiltonian matrix

Hansen, L. P. Hilbert space of a stationary process

I

identification finite versus countable of behavioural equations see also observational equivalence through structural restrictions identity theorem impulse response continuous versus discrete of the innovations representation independent increments of the Wiener process information discarded by aggregation information set of an agent initial value theorem innovation discrete time discrete time versus continuous time recoverability by an econometrician innovation process continuous time

innovations representation instantaneous variance infinite integrated process interarrival times invariant subspace stable inverse optimal predictor problem continuous time univariate isometry between L2 and the process span Isserlis's theorem Ito integral left limit convention solution in the Ito sense Ito isometry Ito's rule Poisson case Wiener case

J

jump at the origin

jump process

K

Kalman filter see Kalman-Bucy filter Kalman gain Kalman-Bucy filter see also innovations representation

kernel vanishing derivatives at the origin Kolmogorov prediction formula see Wiener-Kolmogorov prediction formula

L

Laplace transform, [1] one-sided linear quadratic economy and local unpredictability linear regulator continuous time Hamiltonian of see also duality linear stochastic differential equation definition linearly deterministic process definition

linearly indeterministic process definition local unpredictability and aggregation bias kernel criterion see also mean square differentiability locally unpredictable process and vector autoregressions definition long-run variance, [1]

M

Marcet, A. Markov process first order vector martingale locally matrix covariogram matrix sign algorithm matrix sign function mean function mean square continuity criterion for definition kernel criterion mean square convergence mean square derivative definition mean square differentiability criterion in terms of the autocovariance definition higher order kernel criterion kernel criterion for see also local unpredictability the count n-1-m

mean square ergodicity, [1], [2] definition see also covariance ergodicity spectral criterion measurement noise minimum phase money creation and inflation moving average representation one-sided sampled Muth, J. F., [1]

N

non-fundamentalness and the innovations representation definition see also spectral factorization theorem nonstationarity in prediction formulas

nonstationary process examples of Nyquist frequency and folding

O

observation equation observational equivalence class of models in Phillips's system see also aliasing problem

one-sided kernel ordinary stochastic process Ornstein-Uhlenbeck process and aliasing definition orthogonal increments

P

partial fraction expansion periodogram definition inconsistency of permanent income and local unpredictability permanent income theory Phillips, A. W. physical realizability definition point-in-time sampling Poisson counting process definition moments of poles and residues positive semidefinite function positive semidefiniteness as an identifying restriction power spectrum see also spectral density

predictability smoothness, [1] prediction as projection see also annihilation operator Wiener-Kolmogorov formula prediction error prediction formulas, continuous time present value of a forecast probability space process noise product process projection linear least squares onto a closed span onto integer shifts of a kernel projection theorem

Q

quadratic convergence

quadratic variation

R

random constant, [1] and ergodicity created by sampling random measure orthogonal random spectral measure definition random telegraph wave construction of random walk as a local martingale as a nonstationary process Poisson driven

rational autocovariance rational expectations model continuous time rational spectral density and linear SDEs factorization of rational spectral density matrix and identification Riccati equation filtering numerical solution of Riccati equation, algebraic see control algebraic Riccati equation right half plane zeros

S

sample mean ergodicity sample path continuity of nondifferentiability of realization sampled data see discrete sampling sampled innovation sampled process moving average representation of sampled spectral density sampling ever more finely frequency bands sampling interval shrinking sampling rate span Sargent, T. J. seasonal component aliasing folded to zero frequency second-order stationarity see covariance stationarity serial correlation see autocovariance function shot noise and the Poisson process Sims, C. A., [1] skip sampling and identification Slutsky's condition smoothness autocovariance predictability, [1] social planning problem fictitious span sampling rate

spectral density definition see also spectral factorization theorem spectral distribution atom at zero frequency spectral distribution function spectral factorization time domain construction spectral factorization theorem in Phillips's argument statement spectral folding illustration of spectral smoothing and consistency bandwidth spectral window square integrable kernel stabilizability state-space model continuous time stationarity covariance (wide sense) second-order strict wide sense stationary solution of a linear SDE stationary variance stochastic differential equation drift and diffusion Ito sense solution Poisson driven Wiener driven stochastic process stock variable Stratonovich integral, [1] strict stationarity structural parameters recovery of sum of exponentials kernel

T

Taylor series of an autocovariance temporal aggregation and adaptive expectations general theory see also time aggregation time aggregation and vector autoregressions see also aggregation over time

time average as a filter variance of total power preserved by folding transfer function and prediction definition transform tables transition equation

U

undiscounted transformation unit averaging, [1] versus point sampling

univariate prediction of inflation unrealizable disturbance

V

value function gradient of

vector autoregression and time aggregation vector information structure

W

white noise and physical realizability as a generalized process as a limit of shot noise see also fundamental white noise wide-sense stationarity see covariance stationarity Wiener process as a limit of Poisson processes definition

Wiener-Hopf technique see annihilation operator Wiener-Kolmogorov prediction formula Wold decomposition continuous time see also moving average representation Wold kernel behaviour at the origin Wold representation discrete time Wold's theorem