A Invertibility, Fundamentalness and Blaschke matrices

In document Monetary policy shocks - a nonfundamental look at the data (Page 23-38)

Consider the Wold representation for an n-dimensional time series process, Xt,

Xt= C(L)t (18)

where t denotes fundamental innovations. They are defined given the econo-metrician’s information set Ft as

t= Xt− E[Xt|Ft−1], (19) where Ft−1={Xs}s<t.

VAR modeling is based on inverting (19) and truncating the resulting au-toregressive polynomial. Impulse-responses can then be simulated by inverting the estimated VAR for an estimated analog to (18).

Now suppose that the true data generating process was given by

Xt=C(L) t (20)

If any roots of the polynomialC(L) are inside the unit circle, then (20) fails to be invertible. The structural shock process,t, is then called nonfundamental and fails to coincide with t.

The possible non-invertibility of the underlying economic structure can be motivated along two different paths. To show this, consider a bivariate economy with an exogenous scalar x1,t and an endogenous scalar x2,t.

First, it could of course be that the driving process, x1,t, has a noninvertible moving average part. This property would then carry over to the structural moving average representation of the variable of interest. For example, suppose the shock process is given by

x1,t= (1− cL)t, (21)

where|c| > 1, and model its impact on the variable of interest as

x2,t= (1− dL)x1,t= (1− dL)(1 − cL)t (22) The structural moving-average representation for x2,t, (22), would then triv-ially be non-invertible as well.

Apart from theoretical time series considerations about the potential non-invertible nature of the moving-average part of the data-generating process, there may be particular reasons to assume that certain underlying economic structures may lead to a noninvertible representations. Examples are given in Hansen and Sargent (1991) and Lippi and Reichlin (1993, 1994). These papers show that noninvertibility is likely to be an issue of particular importance in

forward-looking rational expectations models when the econometrician’s infor-mation set, Ftis a strict subset of the relevant information set, Ft, that defines the structural innovations:

span(Ft)⊂ span(Ft) (23)

In this case noninvertibility can be an issue even if the data-generating pro-cess for the exogenous propro-cess, (21), is invertible. To see this suppose that x1,t is observable to the agents of the economy who determine x2,tas a function of their forecasts on x1,t, but that x1,t is not observable to the econometrician.

For example, suppose that

(1− α1L)(1− α2L)x2,t= x1,t+ βE[x1,t+1] (24) Then, combining (21) and (24) I obtain

(1− α1L)(1− α2L)x2,t = x1,t+1+ βE[x1,t+1] (25)

= (1− cL − βc)t,

which is noninvertible as long as the |1−β∗cc | < 1, which is the case, for example, for c = β = 0.8.

To illustrate the difference between the true impulse-response given the data-generating process (25) and a misspecified autoregression approximation take the following example. For the numerical example I assume that c = β = 0.8; α1 = 0.45 and α2 = 0.9. Obviously, for this parameterization, the process (25) is not invertible. I then simulate this process and subsequently estimate a autoregression on it. Figure 6 plots the impulse response function of for both the structural (nonfundamental) representation and the misspecified autoregression based on the fundamental representation.

What is special about this structure is that the misspecified autoregression is observationally equivalent to the nonfundamental data-generating process. It is not only that there exists a fundamental representation - the Wold representa-tion - that can be inverted and truncated to yield an approximate autoregression.

The misspecified autoregression is actually observationally equivalent in that its autocovariance-generating function is exactly that of the nonfundamental data-generating process. The reason for this is that the nonfundamental and the fundamental representation are linked by a Blaschke matrix, i.e. a matrix that reweighs the structural shocks across time to define the fundamental innovation (see section 3). In order to see where the Blaschke matrix enters rewrite (25) as follows

(1− α1L)(1− α2L)x2,t= (1 + θL)t, (26) where θ = 1−βcc t and σ2 = (1− βc)2σ2. The autocovariance-generating function for x2is then

gx2(z) = (1 + θz)∗ (1 + θz−1)

(1− α1z)(1− α2z)∗ (1 − α1z−1)(1− α2z−1)∗ σ2. (27)

On the other hand the autocovariance-generating function for x2 given the econometrician’s model, a first-order autoregression of x2,t, is

gxeco2 (z) = 1

(1− α2z)(1− α2z−1)σu2, (28) where σ2u = θ2σ2. The ratio between these two autocovariance-generating functions is given by (1−α(θ+z)(θ+z−1)

1z)(1−α1z−1) ∗ θ−2. This ratio describes the transfer function that allows to move from one representation to another. With θ =−α11, as is the case in this numerical example, this transfer function is a Blaschke ma-trix as discussed in Section 3 of the paper. The reason for the nonidentifiability is thus the fact that the Blaschke matrix is a filter that has a gain at frequency zero of exactly unity.

Finally, note that if x1 where observable to the econometrician, then com-bining (21) and (24) I can solve for x2,t

(1− cL)(1 − α1L)(1− α2L)x2,t = (1− βc − cL)x1,t (29)

= cx1,t−1

+(1− βc)(1 − cL)t,

which is invertible. Thus the question of invertibility in forward-looking rational expectations models is clearly dependent on the econometrician’s infor-mation set.

B References References

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Working Paper 5451

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o u t p u t r e s p o n s e s t o a 1 p e r c e n t c o n t r a c t i o n a r y m o n e t a r y s h o c k

- 1 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 0 . 2

1 4 7 1 0 1 3 1 6 1 9 2 2 2 5 2 8 3 1

t i m e

percentage response

V A R

v e i l

m o n e y m a t t e r s

Figure 1:

c u m m u l a t i v e e f f e c t o f a 1 p e r c e n t c o n t r a c t i o n a r y m o n e t a r y p o l i c y

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t i m e

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V A R v e i l

m o n e y m a t t e r s

Figure 2:

c u m m u l a t i v e e f f e c t o f a 1 p e r c e n t c o n t r a c t i o n a r y m o n e t a r y s h o c k o n

o u t p t

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1

a l p h a

percentage response

o u t p u t r e s p o n s e

Figure 3:

price response to a 1 percent contractionary monetary response

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4

1 3 5 7 9 11 13 1 5 1 7 1 9 21 23 25 27 29 31

t i m e

percentage response

V A R veil

Figure 4:

federal funds response to a 1 percent contractionary monetary policy shock

-1 -0.5 0 0.5 1 1.5

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

time

percentage response

VAR veil

Figure 5:

Impulse-Responses to the Data-Generating Process and the Autoregressive Fundamental

Approximation

-1 -0.5 0 0.5 1 1.5

1 3 5 7 9 1 1 13 15

Nonfundamental Data-Generating Process Fundamental Approximation

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In document Monetary policy shocks - a nonfundamental look at the data (Page 23-38)