TimeSeriesBook.pdf

If investors expect positive negative change in the

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model of Campbell (1987). If investors expect positive (negative) change in the dividends tomorrow, they want to buy (sell) the share thereby increasing (decreasing) its price already today. In the context of the permanent income hypothesis expected positive income changes lead to a reduction in today’s saving rate. If, on the contrary, households expect negative income changes to occur in the future, they will save already today (“saving for the rainy days”). Inserting for P t Y t + j , j = 0 , 1 , . . . , the corresponding forecast equation P t Y t + h = μ (1 - φ h ) + φ h Y t , we get: S t = βγμ (1 - φ ) (1 - β )(1 - βφ ) + βγφ 1 - βφ Y t + u t . The remarkable feature about this relation is that { S t } is a stationary process because both { Y t } and { u t } are stationary, despite the fact that { Y t } and { X t } are both an integrated processes of order one. The mean of S t is: E S t = βγμ 1 - β . From the relation between S t and ∆ Y t and the AR(1) representation of { Y t }
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314 CHAPTER 16. COINTEGRATION we can deduce a VAR representation of the joint process { ( S t , Y t ) 0 } : S t Y t = μ (1 - φ ) βγ (1 - β )(1 - βφ ) + βγφ 1 - βφ 1 + 0 βγφ 2 1 - βφ 0 φ ! S t - 1 Y t - 1 + u t + βγφ 1 - βφ v t v t . Further algebraic transformation lead to a VAR representation of order two for the level variables { ( X t , Y t ) 0 } : X t Y t = c + Φ 1 X t - 1 Y t - 1 + Φ 2 X t - 2 Y t - 2 + Z t = μ (1 - φ ) γ (1 - β )(1 - βφ ) 1 + 0 γ + γφ 1 - βφ 0 1 + φ X t - 1 Y t - 1 + 0 - γφ 1 - βφ 0 - φ X t - 2 Y t - 2 + 1 γ 1 - βφ 0 1 u t v t Next we want to check whether this stochastic difference equation possesses a stationary solution. For this purpose, we must locate the roots of the equation det Φ( z ) = det( I 2 - Φ 1 z - Φ 2 z 2 ) = 0 (see Theorem 12.1). As det Φ( z ) = det 1 - γ - γφ 1 - βφ z + γφ 1 - βφ z 2 0 1 - (1 + φ ) z + φz 2 ! = 1 - (1 + φ ) z + φz 2 , the roots are z 1 = 1 and z 2 = 1. Thus, only the root z 1 lies outside the unit circle whereas the root z 2 lies on the unit circle. The existence of a unit root precludes the existence of a stationary solution. Note that we have just one unit root, although each of the two processes taken by themselves are integrated of order one. The above VAR representation can be further transformed to yield a representation of process in first differences { (∆ X t , Y t ) 0 } : X t Y t = μ (1 - φ ) γ (1 - β )(1 - βφ ) 1 - 1 - γ 0 0 X t - 1 Y t - 1 + 0 γφ 1 - βφ 0 φ X t - 1 Y t - 1 + 1 γ 1 - βφ 0 1 u t v t . This representation can be considered as a generalization of the Dickey-Fuller regression in first difference form (see equation (7.1)). In the multivariate
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16.1. A THEORETICAL EXAMPLE 315 case, it is known as the vector error correction model (VECM) or vector error correction representation. In this representation the matrix Π = - Φ(1) = - 1 γ 0 0 is of special importance. This matrix is singular and of rank one. This is not an implication which is special to this specification of the present discounted value model, but arises generally as shown in Campbell (1987) and Campbell and Shiller (1987). In the VECM representation all vari- ables except ( X t - 1 , Y t - 1 ) 0 are stationary by construction. This implies that - Π( X t - 1 , Y t - 1 ) 0 must be stationary too, despite the fact that { ( X t , Y t ) 0 } is not stationary as shown above.
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