TimeSeriesBook.pdf

X y μ γ 1 t 1 1 φ γ 1 t x j 1 u j v j stationary

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= X 0 Y 0 + μ γ 1 t + 1 1 - φ 0 γ 0 1 t X j =1 u j v j + stationary process . (16.1) The Beveridge-Nelson decomposition represents the integrated process { ( X t , Y t ) 0 } as a sum of three components: a linear trend, a multivariate random walk and a stationary process. Multiplying the Beveridge-Nelson decomposition from the left by the cointegration vector β = (1 , - γ ) 0 , we see that both the trend and the random walk component are eliminated and that only the stationary component remains. Because the first column of Ψ(1) consists of zeros, only the second struc- tural shock, namely { v t } , will have a long-run (permanent) effect. The long- run effect is γ/ (1 - φ ) for the first variable, X t , and 1 / (1 - φ ) for the second variable, Y t . The first structural shock (preference shock) { u t } has non long- run effect, its impact is of a transitory nature only. This decomposition into permanent and transitory shocks is not typical for this model, but can be done in general as part of the so-called common trend representation (see Section 16.2.4). Finally, we will simulate the reaction of the system to a unit valued shock in v t . Although this shock only has a temporary influence on ∆ Y t , it will have a permanent effect on the level Y t . Taking φ = 0 . 8, we get long-run effect (persistence) of 1 / (1 - φ ) = 5 as explained in Section 7.1.3. The present discounted value model then implies that this shock will also have a permanent effect on X t too. Setting γ = 1, this long-run effect is given by γ (1 - β ) j =0 β j (1 - φ ) - 1 = γ/ (1 - φ ) = 5. Because this long-run effect
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318 CHAPTER 16. COINTEGRATION 0 5 10 15 20 25 30 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 h Variable Y Variable X Figure 16.1: Impulse response functions of the present discounted value model after a unit shock to Y t ( γ = 1 , β = 0 . 9 , φ = 0 . 8) is anticipated in period t , the period of the occurrence of the shock, X t will increase by more than one. The spread turns, therefore, into positive. The error correction mechanism will then dampen the effect on future changes of X t so that the spread will return steadily to zero. The corresponding impulse responses of both variables are displayed in Figure 16.1. Figure 16.2 displays the trajectories of both variables after a stochastic simulation where both shocks { u t } and { v t } are drawn from a standard nor- mal distribution. One can clearly discern the non-stationary character of both series. However, as it is typically for cointegrated series, they move more or less in parallel to each other. This parallel movement is ensured by the error correction mechanism. The difference between both series which is equal to the spread under this parameter constellation is mean reverting around zero. 16.2 Definition and Representation of Coin- tegrated Processes 16.2.1 Definition We now want to make the concepts introduced earlier more precise and give a general definition of cointegrated processes and derive the different repre-
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16.2. DEFINITION AND REPRESENTATION 319 0 10 20 30 40 50 60 70 80 90 100 240 250 260 270 280 290 300 310 320 330 time Values for X t and Y t Variable X Variable Y Figure 16.2: Stochastic simulation of the present discounted value model
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