TAILIEUCHUNG - Handbook of Econometrics Vols1-5 _ Chapter 20

Chapter 10 CONTINUOUS TIME STOCHASTIC MODELS AND ISSUES OF AGGREGATION OVER TIME Since the publication of the influential articles of Haavelmo (1943) and Mann and Wald (1943) and the subsequent work of the Cowles Commission [see, especially, Koopmans (1950a)], most econometric models of complete economies have been formulated as systems of simultaneous stochastic difference | Chapter 20 CONTINUOUS TIME STOCHASTIC MODELS AND ISSUES OF AGGREGATION OVER TIME A. R. BERGSTROM University of Essex Contents 1. Introduction 1146 2. Closed first-order systems of differential and integral equations 1150 . Stochastic limit operations and stochastic differential equations 1150 . Random measures and systems with white noise disturbances 1157 . Estimation 1171 3. Higher order systems 1193 4. The treatment of exogenous variables and more general models 1202 5. Conclusion 1209 References 1210 Handbook of Econometrics Volume II Edited by Z. Griliches and . Intriligatoi Elsevier Science Publishers BV 1984 1146 A. R. Bergstrom 1. Introduction Since the publication of the influential articles of Haavelmo 1943 and Mann and Wald 1943 and the subsequent work of the Cowles Commission see especially Koopmans 1950a most econometric models of complete economies have been formulated as systems of simultaneous stochastic difference equations and fitted to either quarterly or annual data. Models of this sort which are discussed in Chapter 7 of this Handbook can be written in either the structural form k ry Box Bryt r u 1 r l or the reduced form k yt nox X nry _r v 2 r 1 where y is an n X1 vector of observable random variables endogenous variables x is an m X1 vector of observable non-random variables exogenous variables ut is a vector of unobservable random variables disturbances F is an n X n matrix of parameters Bo is an n X m matrix of parameters BY . Bk are n X n matrices of parameters IIr - F lBr r 0 . k and v F 1ul. It is usually assumed that E m 0 E mjmz 0 s t and E u u l S implying that E i z 0 E vsv 0 s t and E rzi z Î2 . The variables xil . xml ylt . ynl will usually include aggregations over a quarter or year of flow variables such as output consumption exports imports and investment as well as levels at the beginning or end of the quarter or year of stock variables representing inventories fixed capital and financial assets. They will also .

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