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Multivariate Time Series Models II∗
Myles J. Callan
February 21, 2008
Abstract This is the second of a 2 part review of multivariate time series models. It begins by distinguishing between structural and reduced form models in the context of vector autoregressive models. The re- view continues by discussing the how to estimate these models and the restrictions required to identify these models, and discusses inno- vation accounting: i.e. impulse response functions and variance de- composition. The conclusion outlines the various methods employed to indentify VAR models: structural vector autoregression models.
1 Vector Autoregression
In VAR models, we treat the included variables symmetrically (where in- cluded variables are suggested by theory). The structural model is:
yt = −b 12 zt + γ 11 yt− 1 + γ 12 zt− 1 + ηyt zt = −b 21 yt + γ 21 yt− 1 + γ 22 zt− 1 + ηzt
or in matrix notation: [ 1 b 12 b 21 1
] [
yt zt
]
[
γ 11 γ 12 γ 21 γ 22
] [
yt− 1 zt− 1
]
[
ηyt ηzt
]
or: ∗The sources for this material are: Enders, Applied Econometric Time Series; Hendry and Juselius, Explaining Cointegration Analysis: Part II; Cochrane, Time Series for Macroeconomics and Finance (the section on Structural VARs and VECMs); and Juselius, The Cointegrated VAR Model: Econometric Methodology and Macroeconomic Applica- tions.
Bxt = Γ 1 xt− 1 + ηt
Where ηyt and ηzt are white-noise disturbances, and are uncorrelated with each other. That is:
ηt ∼ IID
[
σ^2 y 0 0 σ^2 z
] )
or ηt ∼ IID
0 , Ση
The reduced form of this structural model is an example of a VAR(1):
xt = B−^1 Γ 1 xt− 1 + B−^1 ηt
or:
xt = A 1 xt− 1 + et
Where e 1 t and e 2 t are white-noise disturbances, which, in general, are cor- related with each other. That is:
et ∼ IID
[
σ^21 σ 12 σ 21 σ^22
] )
or et ∼ IID
and σ 12 = σ 21
Where e 1 t =
ηyt − b 12 ηzt 1 − b 12 b 21 and e 2 t =
ηzt − b 21 ηyt 1 − b 12 b 21
2 Innovation Accounting
Impulse Response Functions and Variance Decomposition are components of Innovation Accounting. The starting place in both cases is to identify the structural model. The primitive/structural model cannot be estimated directly, instead the reduced form model is esimated. However, for an eco- nomic interpretation to be possible, it is necessary to identify the structural model—that is, we put restrictions on the reduced form model.
2.1 Impulse Response Functions
To see the nature of the restrictions required, we can compare the vector moving average representations of the structural and reduced form models
xt =
∑^ ∞
i=
Ai 1 et−i
We can decompose this n-step ahead forecast error variance into pro- portions due to “own” shocks and other shocks, which in the bivariate case is:
σ y^2
[
φ 11 (0)^2 + φ 11 (1)^2 +... + φ 11 (n − 1)^2
]
σy(n)^2
and
σ z^2
[
φ 12 (0)^2 + φ 12 (1)^2 +... + φ 12 (n − 1)^2
]
σy(n)^2
3 Structural VAR
Unless the underlying structural model can be identified from the reduced form VAR by using recursive ordering, the results of the Cholesky decom- position does not have any direct economic interpretation. That is, for impulse response analysis and variance decomposition to have an economic interpretation, we need to use the (autonomous) structural shocks (and not the forecast errors/reduced form errors). Structural VAR involves using economic theory, rather than the Cholesky Decomposition, to recover the structural innovations.
To begin with, we need to know how many restrictions we need to impose. In the case of an n-variable VAR model, we have the following (where the number of parameters are displayed beneath each component):
︸︷︷︸^ B
n^2 − n
xt = (^) ︸︷︷︸Γ 1 n^2
xt− 1 + ηt xt = (^) ︸︷︷︸A 1 n^2
xt− 1 + et
Where ηt ∼ IID
0 , Ση ︸︷︷︸ n
and et ∼ IID
n^22 +n
This implies an equivalent way, and more direct way, of explaining the nature of imposing restrictions: As ηt = Bet, if we can get values for the elements of B we can use the residuals ˆet to estimate the structural shocks, ˆηt. We can also use the fact that ηt = Bet, to suggest the various ways in which the restrictions can be imposed. The relationship between the reduced form errors and the structural shocks implies that Ση = E(η t′ηt) = E
Bete′ tB′
BE
ete′ t
B′^ = BΣB′. We can estimate Σ, which gives us an estimate of n^2 +n 2 parameters, whereas Ση^ and^ B^ have^ n
(^2) parameters.
In both cases, the difference between the number of parameters in the struc- tural form and that of the reduced form is n (^2) −n 2.^ This is the number of restrictions that we have to impose on the system for the structural model (and therefore errors) to be identified. The Cholesky decomposition imposes exactly this number of restrictions.^1 Implicitly, it restricts the B matrix (the matrix of contemporaneous interactions) as follows:
b 12 = b 13 =... = b 1 n = 0 =⇒ e 1 t = η 1 t b 23 =... = b 2 n = 0 =⇒ e 2 t = c 21 η 1 t+ η 2 t
... = b 3 n = 0 =⇒ e 3 t = c 31 η 1 t+ c 22 η 2 t+ η 3 t ... ... bn− 1 n = 0 =⇒ ent = cn 1 η 1 t+ cn− 12 η 2 t+... + ηnt Where cij refers to the ijth^ element of C = B−^1. The important point to note is that the order in which you list your variables in the VAR command in Eviews matters. The reason being that the Cholesky decomposition, used by Eviews when computing the Impulse Response functions and the Vari- ance decomposition for an unrestricted VAR, imposes this recursive ordering on the system, implying an economic meaning—that the earlier variables are causally prior to the latter ones—that you may not intend.
The other methods of imposing restriction are:
- Coefficient restrictions (ex. b12 = 1).
- Variance restrictions (ex. σy = σz ).
- Symmetry restrictions (ex. b12 = b21).
- Blanchard-Quah restrictions
(^1) Later, in the section on the Blanchard-Quah restrictions, it will be shown that you can normalize the variance of the structural shocks to equal 1 without changing the nature of the (restrictions) problem. If we work with the normalized structural shocks, t, the Cholesky decomposition is easier to explain. The variance/covariance matrix of t is Σ = I. The Cholesky decomposition is Q−^1 Q−^1 ′^ = Σ (where Σ = E(ete′ t), and Q−^1 is equivalent to (BD)−^1 below if a recursive ordering is suggested by economic theory (i.e. if restrictions of the type discussed in this section are imposed on B, the contemporaneous relationship between the variables)).
With this Q, E(′) = E(Qete′ tQ′) = QΣQ′^ = I
From this we get 3 equations with 4 unknowns. The long run restriction, that the demand shock must have a cumulative effect of zero, supplies the 4 th^ equation, as follows:
xt = A(L)xt− 1 + et = A(L)Lxt + et so that
[I − A(L)L]xt = et and xt = [I − A(L)L]−^1 et
Expanding this Vector Moving Average representation yields:^4 [ ∆yt zt
]
Det
[
1 − A 22 (L)L A 21 (L)L
A 12 (L)L 1 − A 11 (L)L
] [
e 1 t e 2 t
]
Det
[
k=0 a^22 (k)L k+1 ∑∞ k=0 a^21 (k)L k+ ∑∞ k=0 a^12 (k)L k+1 (^1) − ∑∞ k=0 a^11 (k)L k+
] [
e 1 t e 2 t
]
Which implies:
∆yt =
Det
∑^ ∞
k=
a 22 (k)Lk+
e 1 t +
∑^ ∞
k=
a 21 (k)Lk+1e 2 t
Det
∑^ ∞
k=
a 22 (k)Lk+
(c 11 1 t + c 12 2 t) +
∑^ ∞
k=
a 21 (k)Lk+1(c 21 1 t + c 22 2 t)
Given that 1 t has a cumulative effect that sums to zero, this implies:
( 1 −
∑^ ∞
k=
a 22 (k)Lk+
c 11 1 t +
∑^ ∞
k=
a 21 (k)Lk+1c 21 1 t = 0
We now have 4 equations and 4 unknowns and the parameters are identified.
(^4) Where Det is the determinant of [I − A(L)L].
