VAR PROCESSES
1 Introduction to Vector Processes
Reading on matrices:
Chris Orme, Lecture Notes in Linear Algebra, cost
£1.00, from Room N.4.3, Dover St. building.
Suppose we want to model n related time series
|1w> |2w> ===> |nw. Define
5 the 6vector
|1w
9 |2w :
:
(n × 1)
yw = 9
7 .. 8 >
|nw
For example, to analyse monetary policy, might use
rate of interest (uw), the output growth (jw) and the rate
of inflation ({sw):
6
5
uw
yw = 7 jw 8 =
{sw
Develop a multivariate time series model for yw.
1.1 Vector white noise
The n × 1 vector white noise process %w satisfies
H(%w) = 0
H(%w%0w) = P
H(%w%0v) = 0> v 6= w
Thus:
• each element has mean zero;
• variance-covariance matrix is constant over time;
• elements have zero autocorrelations and zero
cross-correlations over time. For example, n = 2 &
v = w 1, we require
H(%w%0w1) =
·
H(%1w%1>w1) H(%1w%2>w1)
H(%2w%1>w1) H(%2w%2>w1)
¸
= 0=
Vector white noise has not only H(%lw%l>wm ) = 0> m =
1> 2> ===,
but also cross-correlations over time,
eg. H(%1w%2>wm ) = H(%2w%1>wm ) = 0> m = 1> 2> ===.
Implication:
All past elements of %wm are uncorrelated with
current %w.
1.2 VAR(S ) processes
Vector autoregressive process of order S , or VAR(S ),
is
yw = + x1yw1 + x2yw2 + === + xS ywS + %w
where %w is vector white noise with H(%w%0w) = P.
Analogous to AR(s).
Note:
Any interrelations at w between |lw captured in P;
All interrelations over time between |lw & |n>wm
captured by VAR coefficients.
In lag operator notation:
x(O)yw = + %w
where x(O) is the n × n matrix polynomial
x(O) = In x1O x2O2 === xS OS =
Example: VAR(1) with = 0 is
yw = + xyw1 + %w=
For n = 2:
·
|1w
|2w
¸
·
with
·
|1>w1
|2>w1
¸
·
¸
%1w
%2w
·
¸
!11|1>w1 + !12|2>w1 + %1w
=
!21|1>w1 + !22|2>w1 + %2w
=
!11 !12
!21 !22
¸·
H(%21w)
+
H(%1w%2w)
H(%2w%1w) H(%22w)
· 2
¸
1 12
= P=
12 22
H(%w%0w) =
and
x(O) = I2 xO =
·
¸
1 !11O !12O
!21O 1 !22O
¸
Points to note:
• For VAR with S A 1, an additional subscript (or
superscript) is needed for elements of xm to indicate
lag m , in addition to the two subscripts used in a
VAR(1) for the equation and the variable.
• The VAR(S ) treats each of the n variables in
the same way: there is no distinction between
endogenous and exogenous variables.
• The VAR focuses on dynamic interrelationships
between the variables.
2 Properties of VAR Processes
2.1 Stationarity
A VAR process is (second-order) stationary if:
H(yw) = µ for all w
H(yw µ)(yw µ)0 = K0 for all w
H(yw µ)(ywm µ)0 = Km for all w and any m
Km = H(yw µ)(ywm µ)0 is the autocovariance
matrix at lag m .
Example: VAR between interest rates & inflation, at
lag m = 1 :
K1 = H(yw µ)(yw1 µ)0
·
¸
¤
£
uw u
uw1 u {sw1 {s
= H
{sw {s
·
H(uw u )(uw1 u )
=
H({sw {s)(uw1 u )
¸
H(uw u )({sw1 {s)
H({sw {s)({sw1 {s)
Diagonal elements of Km are lag m autocovariances;
Off-diagonal elements of Km are lag m crosscovariances.
Assume = 0, and recall that, for nonsingular
matrix A,
1
A1 =
dgm[A]
|A|
In VAR x(O)yw = %w , yw = x1(O)%w.
Premultiply by |x(O)| :
,
|x(O)| yw = dgm[x(O)]%w
where dgm[x(O)] is adjoint matrix of x(O).
A determinant is scalar, so same AR |x(O)|
applies to each variable.
,
Each variable in yw is stationary if all roots of
¯
¯
2
S¯
¯
|x(})| = In x1} x2} === xS } = 0
lie outside the unit circle.
This is also the stationarity condition for the VAR.
Examples: VAR(1) with n = 2
1. !11 = ·1> !12¸= =6>
· !21 =¸=5>
· !22 =¸=7·. Then
¸
|1w
1 =6
|1>w1
%
=
+ 1w
|2w
|2>w1
%2w
=5 =7
and
·
¸
1 O =6O
x(O) = I2 xO =
=5O 1 + =7O
,
¯
¯
¯ 1 } =6} ¯
¯
|x(})| = ¯¯
=5} 1 + =7} ¯
= (1 })(1 + =7}) + =3} 2
= 1 =3} =7} 2 + =3} 2
= 1 =3} =4} 2 = (1 =8})(1 + =5})
Both roots are outside the unit circle and the VAR
is stationary, despite !11 = 1.
2. !11 = =9> !
¯
¯ 12 = =1> !21 = =2> !22¯ = ¯=8>
¯ 1 !11} !12} ¯ ¯ 1 =9} =1} ¯
¯
¯=¯
|x(})| = ¯¯
¯
¯
=2} 1 =8} ¯
!21} 1 !22}
= (1 =9})(1 =8}) =02} 2
= 1 1=7} + =7} 2 = (1 })(1 =7})=
System is nonstationary. Due to factor (1 }), both
|1w> |2w L(1).
2.2 Mean and Autocovariances
For stationary VAR(S ), take expectations in
yw = + x1yw1 + x2yw2 + === + xS ywS + %w
,
& hence
µ = (I
µ = + x1µ + x2µ + === + xS µ
n
x1 x2 === xS )1 = x1(1)=
This is a generalisation of the AR(s) mean result.
Stationarity ensures |x(1)| =
6 0>
since |x(})| does not have a unit root=
Can also obtain Km as function of stationary VAR
coefficient matrices x1> ===> xS .
2.3 Moving Average Representation
Every stationary VAR process has an infinite order
vector moving average (VMA) representation:
yw = x1(O)[ + %w]
= µ + x1(O)%w
4
X
= µ+
[c%wc=
c=0
as µ = x1(1) ; note that [0 = In =
For VAR(1), x1(O) = (In + xO+x2O2 + ===)> so
that
yw = µ+(In + xO+x2O2 + ===)%w
4
X
= µ+
xl%wl=
l=0
and [c = xc.
For general VAR(S ), [c is a function of the VAR
coefficient matrices x1> ===> xS .
3 Interpreting VARs
There are many coefficients in a VAR. Say S = 4
with n = 3; each VAR equation involves 3 × 4 = 12
coefficients plus intercept
, 3 × 13 = 39 coefficients in the system.
With, n = 6 and S = 4, 6 × (6 × 4 + 1) = 150
coefficients!
Dynamic interrelationships in the VAR can be
complex.
Say VAR(1) in jw> {sw and uw:
{sw = 1 + !11{sw1 + !12jw1 + !13uw1 + %sw
jw = 2 + !21{sw1 + !22jw1 + !23uw1 + %jw
uw = 3 + !31{sw1 + !32jw1 + !33uw1 + %uw
What is the effect of u on future {s?
• Rate of interest directly affects future inflation
through !23uw1.
• Also indirect effect through jw (uw1 influences jw and
jw1 appears in {sw equation).
• Both jw1 & {sw1 affect uw, leading to a feedback
effect through uw.
Impulse response functions used for VAR interpretation.
3.1 Impulse Response Functions
Impulse response function is the dynamic effect of a
disturbance %mw.
For monetary policy VAR, an interest rate
disturbance %uw affects each of j> {s & u.
For n variables there are n 2 impulse response
functions, one for each of n disturbances on n
variables.
VMA representation implies
Cyw+c
= [c
C%0w
so that
C|l>w+c
= #clm
C%mw
where #clm is (l> m)wk element of [c.
, impulse response function for |m on |l is
#clm >
As [0 = In
C|lw
=
C%mw
c = 0> 1> 2> ====
½
1
0
if l = m
=
if l 6= m
This is response to %mw = 1=
Can set %mw = dm and show impulse responses as
dm #clm (c = 0> 1> 2> ===).
Shock dm often taken as estimated standard
deviation of %m .
3.2 Orthogonalised Impulse Response Functions
Impulse response functions above do not take
account of disturbance covariance matrix P.
Effectively take %mw = dm > %lw = 0> l 6= m
despite H(%lw%mw) = lm 6= 0=
Therefore, transform %mw to xmw that are mutually
orthogonal;
that is, H(xlwxmw) = 0> l 6= m .
Define uw by
uw = C%w
where C is lower triangular and nonsingular (n × n)>
with diagonal elements of unity and
H(uwu0w) = D> D diagonal.
This is the Cholesky decomposition.
Due to orthogonality, no disturbance xmw provides
information about another xlv(l 6= m);
therefore xmw affects no |lw(l 6= m).
As %w = C1uw, VMA is
yw = µ+
,
4
X
[cC1uwc
c=0
Cyw+c
1
=
[
C
=
c
Cu0w
The orthogonalised impulse response function is for
|m on |l is:
C|l>w+c
c = 0> 1> 2> ===
Cxmw
which is the (l> m ) element of [cC1.
In practice, computed with xmw = dm
& all other xlw = 0.
dm typically equal to estimated standard deviation
of xmw.
It is important to understand that lower triangular
C implies a causal order:
• %1w $ %2w;
• %1w> %2w $ %3w;
• %1w> %2w> ===> %n1>w $ %nw=
That is, within w, variables ordered first may cause
later ones, but not vice versa.
Ordering of variables can have a substantial impact on orthogonalised impulse response functions.
Some econometricians and statisticans are
sceptical about them.
VAR itself gives no information about causal
ordering and hence it depends on a priori economic
beliefs.
Example: Consider again the monetary policy VAR,
with variables are ordered
5 as: 6
{sw
yw = 7 jw 8 =
uw
Orthogonalised impluse response functions here
implicitly assume:
• current inflation may influence current growth;
• current inflation & growth may influence current
interest rates;
• current interest rates & growth do not affect inflation
in period w;
• interest rates at w do not affect current growth.
Estimated VAR(4) model for UK inflation, GDP
growth and interest rates.
Variable order 1: Inflation, growth, interest rates
Variable order 2: Growth, inflation, interest rates
¾Same VAR coefficients
Orthogonalised Impulse Responses to one SE
shock to GROWTH
Effect on INFLATION
********************************************
Horizon
Variable
Variable
Order 1
Order 2
0
0.00
-.15842
1
.09203
.01082
2
-.00937
-.07926
3
.09854
.05633
4
.16212
.09323
5
.12177
.06113
6
.09203
.03498
7
.07004
.02097
8
.06767
.01070
9
.06448
.01914
10
.05878
.01655
11
.05008
.01104
12
.04518
.00893
*******************************************
4 Modelling Issues
4.1 Estimation and Inference
Estimation of a VAR(S ) process is by OLS, separately
for each equation.
Although H(%w%0w) = P (not necessarily diagonal),
OLS is optimal.
Comment: Assumes unrestricted VAR (ie, same
explanatory variables in each equation and no
restrictions between coefficients).
Hypothesis tests for individual coefficients use
usual w-ratios.
A test of, say xS = 0> requires a system test, as
coefficients in all n equations are involved.
These are based on
W
1 X 0
b
PS =
ewew=
W S w=1
with residuals ew (w = S + 1> ===> W ) from VAR(S ).
Note: OLS ¯equation
by equation can be shown to
¯
¯b ¯
minimise ln ¯P
S¯ =
Usual system test is a likelihood ratio test,
¯i
¯
¯
h ¯
¯
¯b
¯
¯b
OU = W ln ¯P(U)¯ ln ¯P(X)¯
b
b
where P(U)>
are the restricted and unrestricted
P(X)
estimated disturbance variance matrices.
When the restrictions are valid, asymptotically
OU "2gi with gi = number of restrictions.
For testing
K0 : xS = 0
unrestricted model is VAR(S );
restricted model is VAR(S 1).
d
Under this K0, OU "2n2 . [There are n zero
restrictions in each of n equations.]
4.2 VAR Order Specification
As for AR(s), two approaches for specification of S :
1. "Testing down": Start with maximum order S . Test
K0 : xS = 0=
If K0 not rejected, move to order S 1 and test
K0 : xS 1 = 0=
Continue until K0 is rejected.
2. Use a model specification criterion to select
between orders 0> 1> ===> S . Most common are
and
¯ ¯ 2S n 2
¯b ¯
DLF(S ) = ln ¯P
S¯ +
W S
¯ ¯ S n2 ln(W s)
¯b ¯
VLF(S ) = ln ¯P
=
S¯ +
W S
Select S that minimises criterion.
Note: These are generalisations of criteria for
univariate AR(S ), where n = 1.