y t by left multiplication with 1 (L) as y t = 1 (L) t =ª(L) t 2.5 Variance decomposition and innovation accounting Consider the VAR(p) model where


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1 . Variance decomposition and innovation accounting Consider the VAR(p) model where (L)y t = t, (L) =I m L L p L p is the lag polynomial of order p with m m coe±cient matrices i, i =,...p. Provided that the stationarity condition holds we may obtain a vector MA representation of y t by left multiplication with (L) as where y t = (L) t =ª(L) t (L) =ª(L) =I m +ª L +ª L +. 8
2 The m m coe±cient matrices ª, ª,... may be obtained from the identity (L)ª(L) =(I m as ª j = p i= j i= i L i )(I m + ª j i i i= ª i L i )=I m with ª = I m and j = when i>p, by multiplying out and setting the resulting coe±cient matrix for each power of L equal to zero. For example, start with L = L : L +ª L=(ª )L ª = =ª = Consider next L : i= ª i i ª L ª L L ª =ª + = i= ª i i The result generalizes to any power L j,which yields the transformation formula given above. 9
3 Now, since y t+s =ª(L) t+s = t+s + i= ª i t+s i we have that the e ect of a unit change in t on y t+s is y t+s =ª s. t Now the t 's represent shocks in the system. Therefore the ª i matrices represent the model's response to a unit shock (or innovation) at time point t in each of the variables i periods ahead. Economists call such parameters dynamic multipliers. The response of y i to a unit shock in y j is therefore given by the sequence below, known as the impulse response function, ψ ij,, ψ ij,, ψ ij,,..., where ψ ij,k is the ijth element of the matrix ª k (i, j =,...,m). 8
4 For example if we were told that the rst element in t changes by δ at the same time that the second element changed by δ,..., and the mth element by δ m, then the combined e ect of these changes on the value of the vector y t+s would be given by y t+s = y t+s δ + + y t+s δ m =ª s δ, t mt where δ =(δ,...,δ m ). Generally an impulse response function traces the e ect of a onetime shock to one of the innovations on current and future values of the endogenous variables. 8
5 Example: Exogeneity in MA representation Suppose we have a bivariate VAR system such that x t does not Granger cause y t. Then we can write yt x t = φ() φ () φ () + φ(p) φ (p) φ(p) yt x t yt p x t p + +,t,t Then the coe±cient matrices ª j = j i= ª j i i in the corresponding MA representation are lower triangular as well (exercise): yt x t =,t,t + ψ(i) i= ψ (i),t i ψ(i),t i Hence, we see that variable y does not react to a shock in x.. 8
6 Ambiguity of impulse response functions Consider a bivariate VAR model in vector MA representation, that is, y t =ª(L) t with E( t t)=, where ª(L) gives the response of y t =(y t,y t ) to both elements of t,thatis, t and t. Justaswellwemightbeinterestedinevaluating responses of y t to linear combinations of t and t, for example to unit movements in t and t +. t. This may be done by de ning new shocks ν t = t and ν t = t +. t, or in matrix notation ν t = Q t with Q =.. 8
7 The vector MA representation of our VAR in terms of the new chocks becomes then with y t =ª(L) t =ª(L)Q Q t =: ª (L)ν t ª (L) =ª(L)Q. Note that both representations are observationally equivalent (they produce the same y t ), but yield di erent impulse response functions. In particular, ψ = ψ Q = I Q = Q, which implies that single component shocks may now have contemporaneous e ects on more than one component of y t. Also the covariance matrix of residuals will change, since E(ν t ν t)=e(q t tq ) = unless Q = I. But the fact that both representations are observationally equivalent implies that it is our own choice which linear combination of the ti 'swe ndmostusefultolookatinthe response analysis! 8
8 Orthogonalized impulse response functions Usually the components of t are contemporaneously correlated. For example in our VAR() model of the equitybond data the contemporaneous residual correlations are ================================= FTA DIV R TBILL FTA DIV. R TBILL ================================= If the correlations are high, it doesn't make much sense to ask "what if t has a unit impulse" with no change in t since both come usually at the same time. For impulse response analysis, it is therefore desirable to express the VAR in such a way that the shocks become orthogonal, (thatis,the ti 's are uncorrelated). Additionally it is convenient to rescale the shocks so that they have a unit variance. 8
9 So we want to pick a Q such that E(ν t ν t )=I. This may be accomplished by coosing Q such that Q Q =,sincethen E(ν t ν t)=e(q t tq )=E(Q Q )=I. Unfortunately there are many di erent Q's, whose inverse S = Q act as "square roots" for,thatis,ss =. This may be seen as follows. Choose any orthogonal matrix R (that is, RR = I) and set S = SR. We have then S S = SRR S = SS =. Which of the many possible S's, respectively Q's, should we choose? 8
10 Before turning to a clever choice of Q (resp. S), let us brie y restate our results obtained so far in terms of S = Q. If we nd a matrix S such that SS =,and transform our VAR residuals such that ν t = S t, then we obtain an observationally equivalent VAR where the shocks are orthogonal (i.e. uncorrelated with a unit variance), that is, E(ν t ν t)=s E( t t)s = S S = I. The new vector MA representation becomes y t =ª (L)ν t = i= ψ i ν t i, where ψi = ψ i S (m m matrices) so that ψ = S = I m. The impulse response function of y i to a unit shock in y j is then given by the othogonalised impulse response function ψ ij,, ψ ij,, ψ ij,,.... 8
11 Choleski Decomposition and Ordering of Variables Note that every orthogonalization of correlated shocks in the original VAR leads to contemporaneous e ects of single component shocks ν ti to more than one component of y t,sinceψ = S will not be diagonal unless was diagonal already. One generally used method in choosing S is to use Cholesky decomposition which results in a lower triangular matrix with positive main diagonal elements for ª,e.g. y,t y,t = ψ () ψ () ψ () ν,t ν,t +ª ()ν t +... Hence Cholesky decomposition of implies that the second shock ν,t does not a ect the rst variable y,t contemporaneously, but both shocks can have a contemporaneous effect on y,t (and all following variables, if we had choosen an example with more than two components). Hence the ordering of variablesisimportant! 88
12 It is recommended to try various orderings to see whether the resulting interpretations are consistent. The principle is that the rst variable should be selected such that it is the only one with potential immediate impact on all other variables. The second variable may have an immediate impact on the last m components of y t, but not on y t, the rst component, and so on. Of course this is usually a di±cult task in practice. Variance decomposition The uncorrelatedness of the ν t 's allows to decompose the error variance of the s stepahead forecast of y it into components accounted for by these shocks, or innovations (this is why this technique is usually called innovation accounting). Because the innovations have unit variances (besides the uncorrelatedness), the components of this error variance accounted for by innovations to y j is given by s l= ψ (l) ij,asweshallseebelow. 89
13 Consider an orthogonalized VAR with m components in vector MA representation, y t = l= ψ (l)ν t l. The s stepahead forecast for y t is then E t (y t+s )= l=s ψ (l)ν t+s l. De ning the s stepahead forecast error as we get e t+s = y t+s E t (y t+s ) e t+s = s l= ψ (l)ν t+s l. It's i'th component is given by e i,t+s = s m l= j= ψ (l) ij ν j,t+s l = m s j= l= ψ (l) ij ν j,t+s l. 9
14 Now, because the shocks are both serially and contemporaneously uncorrelated, we get fortheerrorvariance V(e i,t+s )= = m s j= l= m s j= l= V(ψ (l) ij ν j,t+s l ) ψ (l) ij V(νj,t+s l ). Now, recalling that all shock components have unit variance, this implies that V(e i,t+s )= m j= s l= ψ (l) ij, where s l= ψ (l) ij accounts for the error variance generatd by innovations to y j,asclaimed. Comparing this to the sum of innovation responses we get a relative measure how important variable js innovations are in the explaining the variation in variable i at di erent stepahead forecasts, i.e., R ij,l = s l= ψ (l) ij mk= s l= ψ (l). ik 9
15 Thus, while impulse response functions traces the e ects of a shock to one endogenous variable on to the other variables in the VAR, variance decomposition separates the variation in an endogenous variable into the component shocks to the VAR. Example. Let us choose in our example two orderings. One according to the feedback analysis [(I: FTA, DIV, R, TBILL)], andanorderingbasedupontherelativetiming of the trading hours of the markets [(II: TBILL, R, DIV, FTA)]. In EViews the order is simply de ned in the Cholesky ordering option. Below are results in graphs with I: FTA, DIV, R, TBILL; II: R, TBILL DIV, FTA, and III: General impulse response function. 9
16 Impulse responses: Order {FTA, DIV, R, TBILL} Response to Cholesky One S.D. Innovations ± S.E. Response of DFTA to DFTA Response of DFTA to DDIV Response of DFTA to DR Response of DFTA to DTBILL Response of DDIV to DFTA Response of DDIV to DDIV Response of DDIV to DR Response of DDIV to DTBILL Response of DR to DFTA Response of DR to DDIV Response of DR to DR Response of DR to DTBILL Response of DTBILL to DFTA Response of DTBILL to DDIV Response of DTBILL to DR Response of DTBILL to DTBILL Order {TBILL, R, DIV, FTA} Response to Cholesky One S.D. Innovations ± S.E. Response of DFTA to DFTA Response of DFTA to DDIV Response of DFTA to DR Response of DFTA to DTBILL Response of DDIV to DFTA Response of DDIV to DDIV Response of DDIV to DR Response of DDIV to DTBILL Response of DR to DFTA Response of DR to DDIV Response of DR to DR Response of DR to DTBILL Response of DTBILL to DFTA Response of DTBILL to DDIV Response of DTBILL to DR Response of DTBILL to DTBILL
17 Impulse responses continue: Response to Generalized One S.D. Innovations ± S.E. Response of DFTA to DFTA Response of DFTA to DDIV Response of DFTA to DR Response of DFTA to DTBILL Response of DDIV to DFTA Response of DDIV to DDIV Response of DDIV to DR Response of DDIV to DTBILL Response of DR to DFTA Response of DR to DDIV Response of DR to DR Response of DR to DTBILL Response of DTBILL to DFTA Response of DTBILL to DDIV Response of DTBILL to DR Response of DTBILL to DTBILL The general impulse response function are de ned as GI(j, δ i, F t )=E[y t+j it = δ i, F t ] E[y t+j F t ]. That is di erence of conditional expectation given an one time shock occurs in series j. These coincide with the orthogonalized impulse responses if the residual covariance matrix is diagonal. Pesaran, M. Hashem and Yongcheol Shin (998). Impulse Response Analysis in Linear Multivariate Models, Economics Letters, 8, 9. 9
18 Variance decomposition graphs of the equitybond data Variance Decomposition ± S.E. Percent DFTA variance due to DFTA Percent DFTA variance due to DDIV Percent DFTA variance due to DR Percent DFTA variance due to DTBILL Percent DDIV variance due to DFTA Percent DDIV variance due to DDIV Percent DDIV variance due to DR Percent DDIV variance due to DTBILL Percent DR variance due to DFTA Percent DR variance due to DDIV Percent DR variance due to DR Percent DR variance due to DTBILL Percent DTBILL variance due to DFTA Percent DTBILL variance due to DDIV Percent DTBILL variance due to DR Percent DTBILL variance due to DTBILL
19 On estimation of the impulse response coe±cients Consider the VAR(p) model (L)y t = t, with (L) =I m L L p L p. Then under stationarity the vector MA representation is y = +ª t +ª t + When we have estimates of the ARmatrices i denoted by ^ i, i =,...,p; the next problem is to construct estimates ^ª j for the MA matrices ª j. Recall that ª j = j i= ª j i i with ª = I m,and j = when i>p. The estimates ^ª j can be obtained by replacing the i 's by their corresponding estimates ^ i. 9
20 Next we have to obtain the orthogonalized impulse response coe±cients. This can be done easily, for letting S bethecholeskydecomposition of such that we can write = SS, y t = i= ª i t i = i= ª i SS t i where = i= ª i ν t i, and ν t = S t. Then ª i =ª is Cov(ν t )=S S = I. The estimates for ª i are obtained by replacing ª t with their estimates ^ª t and using Cholesky decomposition of ^. 9
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