tailieunhanh - Recursive macroeconomic theory, Thomas Sargent 2nd Ed - Chapter 2

Chapter 2 Time series . Two workhorses This chapter describes two tractable models of time series: Markov chains and first-order stochastic linear difference equations. These models are organizing devices that put particular restrictions on a sequence of random vectors. | Chapter 2 Time series . Two workhorses This chapter describes two tractable models of time series Markov chains and first-order stochastic linear difference equations. These models are organizing devices that put particular restrictions on a sequence of random vectors. They are useful because they describe a time series with parsimony. In later chapters we shall make two uses each of Markov chains and stochastic linear difference equations 1 to represent the exogenous information flows impinging on an agent or an economy and 2 to represent an optimum or equilibrium outcome of agents decision making. The Markov chain and the first-order stochastic linear difference both use a sharp notion of a state vector. A state vector summarizes the information about the current position of a system that is relevant for determining its future. The Markov chain and the stochastic linear difference equation will be useful tools for studying dynamic optimization problems. . Markov chains A stochastic process is a sequence of random vectors. For us the sequence will be ordered by a time index taken to be the integers in this book. So we study discrete time models. We study a discrete state stochastic process with the following property MARKOV PROPERTy A stochastic process xt is said to have the Markov property if for all k 1 and all t Prob xt 1 xt xt-i . xt-k Prob xt i xt . We assume the Markov property and characterize the process by a Markov chain. A time-invariant Markov chain is defined by a triple of objects namely - 26 - Markov chains 27 an n-dimensional state space consisting of vectors ei i 1 . n where ei is an n x 1 unit vector whose ith entry is 1 and all other entries are zero an n x n transition matrix P which records the probabilities of moving from one value of the state to another in one period and an n x 1 vector v0 whose i th element is the probability of being in state i at time 0 v0i Prob x0 e . The elements of matrix P are Pij Prob xt i ej xt ei . For these