Lecture 11 - Vector Autorgressions (VARs)
(Reference – Section 6.3, Hayashi)
The AR(p) model for the univariate time series yt
provides a simple and useful model for forecasting
the future values of y from its own past history.
However, it seems overly constraining to only use its
own past history. Wouldn’t we generate better
forecasts if we could also use the past history of other
variables that help determine yt?
The VAR(p) model is a natural generalization of the
AR(p) model that provides an equally simple
forecasting model that uses the past history of a
vector of time series (including y itself) to forecast
the future values of y.
Let Yt = [y1t …ynt]’, t = …,-1,0,1,… be an ndimensional vector time series. Assume that it has the
following p-th order vector autoregressive
(VAR(p)) representation Yt = A0 + A1Yt-1 + … + ApYt-p + t
where A0 is an nx1 constant vector; A1,…,Ap are nxn
constant matrices; and t is an nx1 vector white noise
process (i.e., E(t) = 0, E(tt’) = Σ, and E(ts’) = 0 if
t ≠ s).
So each component of Yt , say yit, depends on p
lagged values of itself and the other n-1 variables in
the system.
The stationarity condition for the VAR(p) model ,
which assures the existence of a well-defined onesided VMA(∞) representation:
The roots of the determinantal equation
det( In – A1z - …-Apzp) = 0
all lie outside the unit circle or, equivalently, the
roots of the determinantal equation
det( Inzp – A1zp-1 - …-Ap) = 0
all lie inside the unit circle.
The vector version of the Wold Decomposition
Theorem says that if Yt is a zero mean (linearly
indeterministic) covariance stationary process, then it
has a one-sided VMA(∞) representation whose
coefficients are absolutely summable. (See Hayashi,
p. 387, for the definition of the absolute summability
condition in the vector case.) If, in addition, Yt has a
VAR(p) representation then the VAR(p)’s
coefficients will satisfy the stationarity condition.
Note that
i)
a covariance stationary process will have a
VMA form, but may or may not have a
VAR form. (If the VMA can be inverted to
imply VAR whose coefficients meet the
stationarity condition, then the VMA is said
to be
invertible.)
ii) a stochastic process can have a VAR form
without being stationary. (E.g.,Yt =Yt-1 + t )
Suppose that Yt has a VAR(p) form (which may or
may not meet the stationarity condition) with known
coefficients.
Consider the problem of forecasting yi,t+s given Yt,Yt1,…
For now, let’s assume p = 1 (i.e., a first-order VAR) –
Yt+1 = A0 + A1Yt + t+1
E(Yt+1│Yt,Yt-1,…) = A0 + A1Yt
Yt+2 = A0 + A1Yt+1 + t+2
E(Yt+2 │Yt,Yt-1,…) = A0 + A1 E(Yt+1│Yt,Yt-1,…)
= A0 + A1(A0 + A1Yt)
= (I + A1)A0 + A12Yt
…
E(Yt+s│Yt,Yt-1,…) = (I + A1+…+A1s-1)A0 + A1sYt
and
E(yi,t+s│Yt,Yt-1,…) = [(I + A1+…+A1s-1)A0 + A1sYt]i ,
For p > 1:
Yt+1 = A0 + A1Yt + …+ ApYt-p + t+1
Yˆt 1 t  E(Yt+1│Yt,Yt-1,…) = A0 + A1Yt +…+ApYt-p
Yˆt  2 t  E (Yt  2 Yt , Yt 1 ,...)  A0  A1Yˆt 1 t  A2Yt  ...  A p Yt  p 1
…
Yˆt  s t  E (Yt  s Yt , Yt 1 ,...)  A0  A1Yˆt  s 1 t  A2Yˆt  s  2 t  ...  A p Yˆt  s  p t
where
Then,
Yˆt  s  j t  Yt  s  j
if t+s-j < t.
yˆ i ,t  s t  [Yˆt  s t ]i
Practical considerations include –
- selection of variables to include in the VAR
- selection of the lag length, p
- estimation of the VAR (Note that OLS
equation by equation = SURE)
The VAR’s value in macroeconomics and financial
economics –
- summarizes dynamic interrelationships in
the data that can guide the development of
theoretical models. (What are the facts that
theory needs to explain?)
- basis for structural economic analysis, i.e.,
“story-telling”. (The “unrestricted” VAR is a
reduced-form model. The trick is coming up
with a way to “identify” the underlying
structural relationships.)
In constrast to traditional simultaneous
equation modeling this approach does not
rely on exogeneity restrictions or exclusion
restrictions for identification. That is, it uses
a different (and more plausible?) style of
identfication. (More on this in Econ 674).