TimeSeriesBook.pdf

R t t 1 100 102 200 c t 101 4 see also exercise 172

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R t = 0 , t = 1 , . . . , 100 , 102 , . . . , 200; c > 0 , t = 101. 4 See also exercise 17.2
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17.1. THE STATE SPACE REPRESENTATION 345 This means that W t = 0 and that Y t = X t for all t except for t = 101. For the missing observation, we have G 101 = Y 101 = 0. The variance for this observation is set to R 101 = c > 0. The same idea can be used to obtain quarterly data when only yearly data are available. This problem typically arises in statistical offices which have to produce, for example, quarterly GDP data from yearly observations incor- porating quarterly information from indicator variables (see Section 17.4.1). More detailed analysis for the case of missing data can be found in Harvey and Pierce (1984) and Brockwell and Davis (1991, Chapter 12.3). Structural Time Series Analysis An important application of the state space representation in economics is the decomposition of a given time series into several components: trend, cycle, season and irregular component. This type of analysis is usually coined structural time series analysis (See Harvey, 1989; Mills, 2003). Consider, for example, the additive decomposition of a time series { Y t } into a trend T t , a seasonal component S t , a cyclical component { C t } , and an irregular or cyclical component W t : Y t = T t + S t + C t + W t . The above equation relates the observed time series to its unobserved com- ponents and is called the basic structural model (BSM) (Harvey, 1989). The state space representation is derived in several steps. Consider first the case with no seasonal and no cyclical component. The trend is typically viewed as a random walk with time-varying drift δ t - 1 : T t = δ t - 1 + T t - 1 + ε t , ε t WN(0 , σ 2 ε ) δ t = δ t - 1 + ξ t , ξ t WN(0 , σ 2 ξ ) . The second equation models the drift as a random walk. The two distur- bances { ε t } and { ξ t } are assumed to be uncorrelated with each other and with { W t } . Defining the state vector X ( T ) t as X ( T ) t = ( T t , δ t ) 0 , the state and the observation equations become: X ( T ) t +1 = T t +1 δ t +1 = 1 1 0 1 T t δ t + ε t +1 ξ t +1 = F ( T ) X ( T ) t + V ( T ) t +1 Y t = (1 , 0) X ( T ) t + W t with W t WN(0 , σ 2 W ). This representation is called the local linear trend (LLT) model and implies that { Y t } follows an ARIMA(0,2,2) process (see exercise 17.5.1).
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346 CHAPTER 17. KALMAN FILTER In the special case of a constant drift equal to δ , σ 2 ξ = 0 and we have that ∆ Y t = δ + ε t + W t - W t - 1 . { Y t } therefore follows a MA(1) process with ρ (1) = - σ 2 W / ( σ 2 ε + 2 σ 2 W ) = - (2 + κ ) - 1 where κ = σ 2 ε 2 W is called the signal-to-noise ratio. Note that the first order autocorrelation is necessarily negative. Thus, this model is not suited for time series with positive first order autocorrelation in its first differences. The seasonal component is characterized by S t = S t - d and d t =1 S t = 0 where d denotes the frequency of the data. 5 Given starting values S 1 , S 0 , S - 1 , . . . , S - d +3 , the following values can be computed recursively as: S t +1 = - S t - . . . - S t - d +2 + η t +1 , t = 1 , 2 , . . .
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