TimeSeriesBook.pdf

In matrix notation we can write the observation

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In matrix notation we can write the observation equation for the dynamic factor model as follows: Y t = Λ 0 f t + Λ 1 f t - 1 + . . . + Λ q f t - q + W t where Λ i , i = 0 , 1 , . . . , q , are n × r matrices. The state vector X t equals ( f 0 t , . . . , f 0 t - q ) 0 if we assume that the idiosyncratic component is white noise, i.e. W t = ( W 1 t , . . . , W nt ) 0 WN(0 , R ). The observation equation can then be written compactly as: Y t = GX t + W t where G = (Λ 0 , Λ 1 , . . . , Λ q ). Usually, we assume that R is a diagonal matrix. The correlation between the different time series is captured exclusively by the joint factors. The state equation depends on the assumed dynamics of the factors. One possibility is to model { f t } as a VAR(p) process with Φ(L) f t = e t , e t WN(0 , Σ), and p q + 1, so we can use the state space representation of the VAR(p) process from above. For the case p = 2 and q = 2 we get: X t +1 = f t +1 f t f t - 1 = Φ 1 Φ 2 0 I r 0 0 0 I r 0 f t f t - 1 f t - 2 + e t +1 0 0 = FX t + V t +1 and Q = diag(Σ , 0 , 0). This scheme can be easily generalized to the case p > q + 1 or to allow for autocorrelated idiosyncratic components, assuming for example that they follow autoregressive processes. Real Business Cycle Model (RBC Model) State space models are becoming increasingly popular in macroeconomics, especially in the context of dynamic stochastic general equilibrium (DSGE)
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17.2. FILTERING AND SMOOTHING 349 models. These models can be seen as generalizations of the real business cycle (RBC) models. 7 In these models a representative consumer is supposed to maximize the utility of his consumption stream over his infinite life time. Thereby, the consumer has the choice to consume part of his income or to invest his savings (part of his income which is not consumed) at the market rate of interest. These savings can be used as a mean to finance investment projects which increase the economy wide capital stock. The increased capital stock then allows for increased production in the future. The production process itself is subject to a random shocks called technology shocks. The solution of this optimization problem is a nonlinear dynamic system which determines the capital stock and consumption in every period. Its local behavior can be investigated by linearizing the system around its steady state. This equation can then be interpreted as the state equation of the system. The parameters of this equation F and Q are related, typically in a nonlinear way, to the parameters describing the utility and the production function as well as the process of technology shocks. Thus, the state equation summarizes the behavior of the theoretical model. The parameters of the state equation can then be estimated by relating the state vector, given by the capital stock and the state of the technology, via the observation equation to some observable variables, like real GDP, consumption, investment, or the interest rate. This then completes the state space representation of the model which can be analyzed and estimated using the tools presented in Section 17.3. 8 17.2 Filtering and Smoothing
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