Download PDF by Hansen L., Sargent T.J.: Recursive Models of Dynamic Linear Economies

By Hansen L., Sargent T.J.

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5) Although preferences for consumption goods are state and date-separable, they are not separable across components. Following Frisch (1932), Heckman and MaCurdy (1980), and Browning, Deaton, and Irish (1985), it is convenient to exploit the separability across time and states and to analyze the implied consumption demands in each state of the world and time period separately. For a given state of the world and time period t , the contribution of ct to the utility function is 1 − β t (Πct − bt )′ (Πct − bt ).

1) where Ub and Ud are matrices used to select entries of zt . The law of motion for {zt : t = 0, 1, . } is zt+1 = A22 zt + C2 wt+1 for t = 0, 1, . . 2) where z0 is a given initial condition. We make the following technical assumption: Assumption 2: The eigenvalues of the matrix A22 have absolute values that are less than or equal to one. 2) can accommodate a rich variety of time series processes. The matrices Ub and Ud can be chosen to pick off appropriate components of zt in such a way as to make bt or dt follow any of those stochastic processes.

20 Stochastic Linear Difference Equations the mean of x10 is (I − A11 )−1 A12 , then the covariance and mean of x1t remain constant over time. In this case, the process is said to be covariance stationary. If the eigenvalues of A11 are all less than unity in modulus, then Vt → V as t → ∞ , starting from any initial value V0 . 16), notice that if all of the eigenvalues of A11 are less than unity in modulus, then limj→∞ vj = V . That is, the covariance matrix of j -step ahead forecast errors converges to the unconditional covariance matrix of x as the horizon j goes to infinity.

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Recursive Models of Dynamic Linear Economies by Hansen L., Sargent T.J.


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