TimeSeriesBook.pdf

Applications of this framework to the problem of

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ally suited to deal with missing observations. Applications of this framework to the problem of disaggregation were provided by Bernanke et al. (1997:1) and Cuche and Hess (2000), among others. We will illustrate this approach below. Starting point of the analysis are the yearly growth rates of GDP and indicator variables which are recorded at the quarterly frequency and which are correlated with GDP growth. In our application, we will consider the growth of industrial production ( IP ) and the index on consumer sentiment ( C ) as indicators. Both variables are available on a quarterly basis from 1990 onward. For simplicity, we assume that the annualized quarterly growth rate of GDP, { Q t } , follows an AR(1) process with mean μ : Q t - μ = φ ( Q t - 1 - μ ) + w t , w t WN(0 , σ 2 w ) In addition, we assume that GDP is related to industrial production and consumer sentiment by the following two equations: IP t = α IP + β IP Q t + v IP,t C t = α C + β C Q t + v C,t where the residuals v IP,t and v C,t are uncorrelated. Finally, we define the relation between quarterly and yearly GDP growth as: J t = 1 4 Q t + 1 4 Q t - 1 + 1 4 Q t - 2 + 1 4 Q t - 3 , t = 4 , 8 , 12 . . . 12 Similarly, one may envisage the disaggregation of yearly data into monthly ones or other forms of disaggregation.
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17.4. EXAMPLES 361 We can now bring these equations into state space form. Thereby the observation equation is given by Y t = A t + G t X t + W t with observation and state vectors Y t = J t IP t C t , t = 4 , 8 , 12 , . . . ; 0 IP t C t , t 6 = 4 , 8 , 12 , . . . X t = Q t - μ Q t - 1 - μ Q t - 2 - μ Q t - 3 - μ
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362 CHAPTER 17. KALMAN FILTER and time-varying coefficient matrices A t = μ α IP α C , t = 4 , 8 , 12 , . . . ; 0 α IP α C , t 6 = 4 , 8 , 12 , . . . G t = 1 4 1 4 1 4 1 4 β IP 0 0 0 β C 0 0 0 , t = 4 , 8 , 12 , . . . ; 0 0 0 0 β IP 0 0 0 β C 0 0 0 , t 6 = 4 , 8 , 12 , . . . R t = 0 0 0 0 σ 2 IP 0 0 0 σ 2 C , t = 4 , 8 , 12 , . . . ; 1 0 0 0 σ 2 IP 0 0 0 σ 2 C , t 6 = 4 , 8 , 12 , . . . The state equation becomes: X t +1 = FX t + V t +1 where F = φ 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 Q = σ 2 w 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 On my homepage http://www.neusser.ch/ you will find a MATLAB code which maximizes the corresponding likelihood function numerically. Figure 17.2 plots the different estimates of GDP growth and compares them with the data released by SECO.
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17.4. EXAMPLES 363 Q1-91 Q4-92 Q4-94 Q4-96 Q4-98 Q4-00 Q4-02 Q4-04 Q4-06 -3 -2 -1 0 1 2 3 4 percent yearly GDP filterted quarterly GDP smoothed quarterly GDP quarterly estimates of GDP published by SECO Figure 17.2: Estimates of quarterly GDP growth rates for Switzerland 17.4.2 Structural Time Series Analysis A customary practice in business cycle analysis is to decompose a time series into several components. As an example, we estimate a structural time series model which decomposes a times series additively into a local linear trend, a business cycle component, a seasonal component, and an irregular compo- nent. This is the specification studied as the basic structural model (BSM) in
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