Calculation of interim multipliers in a linear dynamic model: a general exposition
Interim multipliers provide a simple yet powerful way to describe the dynamic properties of a dynamic model. We follow Lutkepohl (1991) and derive the dynamic multipliers in a simultaneous system of n linear dynamic equations with n endogenous variables yt and m exogenous variables xt. The structural
form of the model is given by:
r0yt riyt-i + – i+£t – (A.23)
To investigate the dynamic properties of the model it will be more convenient to work with the reduced form of the model:
yt=£ Aiyt-i+£ Bi xt-i+ut (A.24)
defining the n x n matrices Ai = Г-1Гі, i = 1,…,q, and the n x m matrices
Bi = r-1Di, i = 0,…,q. The reduced form residuals are given by ut= r-1et.
It is also useful to define the autoregressive final form of the model as:
yt = A(L)-1B(L)xt + A(L)-1ut (A.25)
= D(L)xt + vt,
where the polynomials are
A(L) = I – A1L—————- AqLq,
B(L) = Bo + B1L + ••• + Bq Lq,
and the final form coefficients are given by the (infinite) rational lag polynomial
D(L) = A(L)-1B(L)
= D0+D1L + ••• + Dj Lj + ••• .
To obtain a simple expression for the interim multipliers it is useful to rewrite the reduced form representation of the model in its companion form as:
Zt = TZt-1 + *xt+Ut (A.26)
forming stacked (n + m)q x 1 vectors with new variables
Zt (yt, . .., yt-q+1, xt, . .., xt-q+1 )
Ut= (ut, 0,…,0)/
and defining a selection matrix
Jnx(n+m)q (In, 0n, . .., 0n n, m, . .., 0n, m).
The matrices T(n+m)qx(n+m)q and ‘i(n+m)qxm are formed by stacking the (reduced form) coefficient matrices Ai, Bi for У і in the following way:
The eigenvalues (characteristic roots) of the system matrix Ф are useful to summarise the characteristics of the dynamic behaviour of the complete system, like whether it will generate ‘oscillations’ as in the case when there is (at least) one pair of complex conjugate roots, or ‘exploding’ behaviour when (at least) one root has modulus greater than 1.
A different way to address the dynamic properties is to calculate the ‘interim multipliers’ of the model, which has the additional advantage that they can be easily graphed.
Successive substitution of Zt in equation (A.26) yields:
Zt = ФZt_l+Фxt+Ut (A.28)
= vz-i+Y, ф я*—+E Ut-j
yt = E J$j *xt-j +X) J$j J’ut-j, (A.29)
is assumed to disappear as і grows sufficiently large. The dynamic multipliers Dj and the interim multipliers Mi can be obtained from (A.29) as the partial derivatives Dj = dyt/dxt-j and their cumulated sums
respectively. We obtain estimates of the multipliers Di and Mi by inserting estimates of the parameters in (A.24) into the companion form matrices Ф and Ф.
D i = JT i’k, і = 0,…
and the interim multipliers are defined in terms of their cumulated sums Mi.
M i = £ Dj (A.31)
= J^^j #
= J(I + Ф + Ф2 + ••• + Ф )’l.
The long-run multipliers are given by
Mж = J2 Dj = J(I – Ф)-1Ф (A.32)
3 = 0