MSc Time Series Econometrics
Module 2: multivariate time series topics
Lecture 1: VARs, introduction, motivation, estimation,
preliminaries
Tony Yates
Spring 2014, Bristol
Topics we will cover
• Vector autoregressions:
– motivation
– Estimation, MLE, OLS, Bayesian using analytical and
Gibbs Sampling MCMC methods
– Identification [short run restrictions, long run
restrictions, sign restrictions, max share criteria]
– interpretation, use, contribution to macroeconomics
• Factor models in vector autoregressions
• TVP VAR estimation using kernels.
• Bootstrapping
VARs useful sources
• Chris Sims, 'Macroeonomics and reality‘
• Lutz Kilian 'Structural Vector Autoregressions‘
• Fabio Canova: Methods for Applied Business
Cycle research
• James Hamilton 'Time Series Analysis‘
• Helmut Luktepohl 'New introduction to
multiple time series analysis'
Useful sources, ctd
• Stock and Watson: implications of dynamic
factor models for VAR analysis
• Stock and Watson: 'Dynamic factor models'
Matrix/linear algebra pre-requisites
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Scalar, vector, matrix.
Transpose
Inverse (matrix equivalent of dividing).
Diagonal matrix.
Eigenvalues and eigenvectors.
Powers of a matrix.
Matrix series sums. Matrix equivalent of geometric
scalar sums.
• Variance-covariance matrix.
• Cholesky factor of a variance-covariance matrix.
• Givens matrix.
Some applications
• Christiano, Eichenbaum, Evans: ‘Monetary policy
shocks: what have we learned and to what end?’
• Christiano, Eichenbaum and Evans (2005): ‘Nominal
rigidities and the dynamics effects of a monetary policy
shock’
• Mountford, Uhlig (2008): ‘what are the effects of fiscal
policy shocks?’
• Gali (1999): ’Technology, employment and the business
cycle….’
• I’ll remind you of these as we go through.
VAR motivation: Cowles Commission
models
• Dominant paradigm was large scale
macroeconometric models, in policy institutions
especially.
• Many estimated equations.
• Academic origins in foundational work to create
national accounts; Keynesian formulation of
macroeconomics; Haavelmo’s notion of
‘probability model’ applied to this.
• Nice discussion in Sims’ Nobel acceptance
lecture.
Illustrative but silly example of a CC
model
Sims: Equation for C excludes U, TU
Equation for Y excludes C, U
And so on....
C t c 0 c 1 Yt c 2 Yt1 u Ct
Yt y 0 y 1 N t y 2 N t1 y 3 Wt u Yt
Wt w 0 w 1 U t w 2 TU t u Wt
U t u 0 u 1 Yt u 2 Yt1 u Ut
TU t tu 0 tu 1 Yt tu 2 U t u TUt
Those u’s look exogenous
and are meant to be, but
are they really primitive
shocks? Both Sims’ and
Lucas’ critiques would
suggest not
Lucas: equation for C sounds like common
sense, but is it the C that reflects the
solution to a consumers’ problem in a
general equilibrium model? Maybe, or
maybe not
[Nobel prize yielding] Critiques of
Cowles Commission approach
• Lucas (1976):
– Laws of motion have to come from solving problems
of agents in the model
– If not, correlations and therefore coefficients will
change if policy changes. [Lucas ‘Critique’]
• Sims (1981):
– “Macroeconomics and reality”
– ‘Incredible’ identification restrictions
– Spurious: categorisation of some variables as
exogenous & restriction of some dynamic coefficients
to be zero
Response to Sims and Lucas critiques
• Estimate and identify a vector autoregression.
• No incredible restrictions. Everything left in.
• Reduced form shocks span the structural
shocks.
• Structural shocks and their effects sought
through identification, reference to classes of
Lucas-Critique proof models
• ‘Modest policy interventions’ [Sims and Zha].
Cowles Commission variables set out
as a VAR model
Yt 
Ct
A 011 A 012 . . . A 016
Yt
A 021
.
. . .
.
.
.
. . .
.
Ut
.
.
. . .
.
TU t
.
.
. . .
.
Nt
A 061
.
. . . A 066
Wt

Potentially, everything is a
function of everything else
lagged
Yt1 A 1 Yt2 
. . . Zt
Simultaneity encoded
in the reduced form
errors. To be
disentangled into
structural shocks
through identification.
VAR topics and contributions to macro
(1)
• Estimating parameters in a DSGE model:
– Rotemburg and Woodford (1998)
– Christiano, Eichenbaum and Evans (2005)
• Identify monetary policy ‘shock’
• Choose parameters of the model so that
impulse response to this shock in the DSGE
model as close as possible to corresponding
IRF in the VAR.
VAR topics and contributions to macro
(2)
• Evaluating the RBC claim that technology
shocks cause business cycles
– Gali (1999)
• Identified technology shocks as the only thing
that could change N/Y in long run
• Showed that these caused hours work to fall,
not rise
• Inconsistent with RBC model (‘make hay while
the sun shines’)
VAR topics and contributions to macro
(3)
• Cogley and Sargent (2005)
– VAR with time-varying parameters
– Multivariate counterpart to persistence,
‘predictability’
– Bayesian estimation
• By how much did inflation predictability change
in the post-war period?
• If it changed a lot, what does that suggest about
its causes?
Misc technical preliminaries to help
you read the papers
• The lag polynomial operator ‘L’
y t 1 y t1 2 y t2 
...
p y tp
0 L 0 y t1 1 L 1 y t1 
...
p1 L p1 y t1


L
y t1
Ly t y t1 , L 1 y t y t1
Lag / lead operator denoted by positive,
negative powers of L
Misc technical preliminaries...
• Cholesky decomposition
11

.
.
21 22
.

31 32 33
a 0 0

. b 0
a 0 0
a . .
. b 0
0 b .
. . c
0 0 c
Sigma is a v-cov matrix,
with elements symmetric
about the diagonal;
Cholesky factor on the
RHS
a . .
PP 
. . c
0 b .
0 0 c
1 0 0
PP I 
0 1 0
0 0 1
a, b, c 0
Here we decompose further
using an orthonormal
matrix P
Diagonals of vcov matrix are positive beause
these correspond to variances…
Misc technical preliminaries
• Givens matrix is an example of an orthonormal
matrix
1 0 0 0
P
0 c s 0
0 s c 0
, c cos
, s sin
0 0 0 1
Also known as a Givens ‘rotation’
Useful theorem: any orthonormal matrix can be shown to be a product of
Givens matrices with different thetas, the number depending on the
dimension of the orthonormal matrix concerned.
Products of orthonormal matrices

Pa P
a I, P b P b I

Pa Pb 

P a P b I
If two matrices are orthonormal, then so is the product of those matrices.
The VAR impulse response function
y t  y t1 e t
y irf 
y1
e
y2
e
y3

2 e
...
...
yn
n1 e
Impulse response function in a univariate time series model
Take some unit value for e1, then substitute into eq for y repeatedly
AR(1) impulse response function
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
60
70
80
90
100
Y_t=0.85*y_t-1+e_t; e(1)=1
IRF shrinks back to zero as we multiply 1 by successively higher
powers of 0.85
Matlab code to plot IRF in an AR(1)
Matlab code to plot an AR(1) impulse
response
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%compute impulse response in an AR(1).
%illustration for MSc Time Series Econometrics module.
%y_t=rho*y_t-1+e_t.
%calibrate parameters
rho=0.85;
%persistence parameter in the ar(1)
samp=100;
%length of time we will compute y for.
y=zeros(samp,1);
%create a vector to store our ys.
%semi colons suppress output of every command
%to screen.
e=1;
%value of the shock in period 1.
y(1)=e;
%first period, y=shock.
for t=2:samp
y(t)=rho*y(t-1);
end
%loop to project forwards effect of the unit shock.
%now plot the impulse response
time=[1:samp];
%create a vector of numbers to record the progression of time
plot(time,y)
VAR impulse response
Yt 
Yirf,1 
y1
y2
AYt1 e t 
t
a 11 a 12
y1
a 21 a 22
y2

e, Ae, A 2 e, . . . 
,e 

t1
e1
e2
e1
e2
A multivariate example.
For IRF to e1, choose e=[1,0] for first period, then project forwards....
Matlab code to plot VAR(1) impulse
response function
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%compute impulse response in an VAR(1).
%illustration for MSc Time Series Econometrics module.
%y_t=A*y_t-1+e_t.
clear all;
%bit of housekeeping to clear all variables so each time you run program as you are debugging
%you know you are not adding onto previous
%values
%calibrate parameters
A=[0.6 0.2;
0.2 0.6];
samp=100;
y=zeros(samp,2);
e=[1;0];
y(1,:)=e;
%length of time we will compute y for.
%create a matrix this time to store our 2 by samp bivariate time series y={y1,y2}.
%we will simulate a shock to the first equation. Note shock in first period is now a 2 by 1 vector.
%first period, y=shock. the colon ':' means 'corresponding values in this dimension'
for t=2:samp
y(t,:)=A*y(t-1,:)';
%loop to project forwards effect of the unit shock.
end
%' is transpose
%now plot the impulse response
time=[1:samp];
%create a vector of numbers to record the progression of time
subplot(2,1,1)
plot(time,y(:,1))
subplot(2,1,2)
plot(time,y(:,2))
VAR(1) impulse response
1
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
0.25
0.2
0.15
0.1
0.05
0
Y_t=A*Y_t-1’+e_t, e_t=[1 0], A=[0.6 0.2;0.2 0.6]
Note that the eigenvalues of A are 0.4 and 0.8
Using a VAR to forecast
f
Yth | e s0,st
f
Yth
 h
 e tA
Forecast at future horizon h, conditional
on starting from steady state= IRF to the
latest estimated shock.
 h  h1  h2
 hn
 e t A e t1 A e t2 A 
...
e tn A
Forecast conditional on all the shocks estimated to have occurred:
Sum of IRF to that shock at increasing horizons.
Terms further to the right get smaller as higher powers of h [=smaller for
stable VAR]
Reflects response to shocks that hit further and further back in time.
Forecasts further and further out shrink back to steady state for the same
reason. Higher powers of h , A has eigenvalues <1 in absolute value.
VAR(1) representation of a VAR(p)
Yt A 1 Yt1 A 2 Yt2 
. . . A p Ytp e t
Recall that you
learned how to write
the VAR
representation of an
AR(p).
Yt
Yt 
Yt1
AYt1 e t
...
Ytp1
This is analogous.
A1 A2 . . . Ap
A
Ik
0 ...
0
Ik
0 ...
0
0
Ik
0
0
et
,et 
0
...
0
Moving average representation of a
VAR(p)

Yt  Ai e ti
i0
We have used the VAR(1) form of the VAR(p).
In words, it means Y is the sum of shocks, where each shock taken to higher
and higher powers as we go back in time.
Notice how this is related to the formula for the VAR impulse response we
computed before.
Persistence, memory, predictability,
stability
• When a shock hits, how long does it takes for its effects to
die out?
• Applications:
– Business cycle theory concerned with mechanisms for
propagation. Consumption and output not white noise: why?
– Bad monetary and fiscal policy could be part of the story about
why shocks take time to die out
• Time series notions of persistence etc are one way to
characterise propagation and bad policy.
• Why ‘bad’, because more persistence means larger
variance, and larger variance for most utility functions is
bad
Persistence and variance
y t  y t1 e t
var
y t  2 var
y t1 var
e t 2cov
y t1 , e t 
 2 var
y t1 var
et 
NB : var
y t var
y t1 
var
et
var
y t 
1 2
Persistence, higher rho, means lower denominator, means higher
unconditional variance of y
Most economic models asssume agents don’t like variance
So persistence is interesting economically, since it usually indicates
something ‘bad’ is happening
Persistence and predictability
j1

j
0
P t 1   h

ei 
h0
e
i
A ht t A ht

A ht t A ht

ei
ei
Multivariate predictability: if set horizon=2, dimension of
Y_t=1, then this formula delivers rho^2, ie persistence squared.
AR ‘stability’
y t 
y t1 e t , || 1
Univariate model.
e, 
e, 2 e, . . . n e
limt e 0
t
Effects of shock eventually die
out. So series converges to
something.
Stability=stationarity=series have convergent sums=first and second moments
independent of t, and computable

y t e t 
e t1 2 e t2 
. . . n e tn  s e ts
s0
lims s e ts 0
The contribution of a shock very far
back goes to zero
Here we take the perspective that today’s data is the sum of the
effects of shocks going back into the infinite past.
Since today’s data is finite and well-defined, then it must be
that shocks infinitely far back have no effect. Otherwise today’s
data would be infinitely large.
VAR(1) stability
Yt AYt1 e t
x eig
A
;|x | 1, x
det
I K Ax0  |x | 1
Stability condition echoes the AR case,
but where does the dependence on
the eigenvalues come from?
Yt A 0 e t A 1 e t1 A 2 e t2 
...
A n e tn
A 0 L 0 e A Le t A 2 L 2 e t 
...
Ln A n et
Here we write out a vector Y as the sum of contributions from shocks
going back further and further into time.
Explaining VAR(1) eigenvalue stability
condition
A n PDn P 1
Dn 
n1 0
0
0 n2
0
0
P
0 nd
v1 v2 vd
The crucial thing is to make this A^n go to zero
as n goes to infinity.
Importance of eigenvalues in this happening
comes from the fact that we can write a
square matrix using the eigenvalueeigenvector decomposition.
And then compute the power of A using the
powers of the eigenvales of A.
So to make A^n go to zero, we have to make
all the diagonal elements of D go to zero, and
by analogy with the AR(1) case, this means the
eigenvalues have to be < 1 in absolute value.
VAR(p) stability
Yt
Yt 
Yt1
AYt1 e t
...
Ytp1
A
A1 A2 . . . Ap
et
Ik
0 ...
0
0
Ik
0 ...
0
0
Ik
0
0
x eig
A
; |x | 0, x
,et 
...
0
VAR estimation
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Linear, multivariate model
Suggests estimation by...
OLS!
Or MLE, which in these circumstances [linear;
Gaussian errors] is equivalent.
• MLE cumbersome because you may have
many parameters over which to optimise
• Why is this cumbersome? Well, you tell me.
OLS estimation of VAR parameters
y t 
y t1 e t

y 
y2 , . . . yT
, x 
y 2 , . . . y T1 

x x 1 x y

A XX
1
XY
Univariate case
Multivariate case for
VAR(1) representation of
VAR(p)
OLS estimate of reduced form residual
variance covariance matrix
y t 
y t1 e t
 


1/Te e , e y 
x
y y 2 . . . y T , x y 1 . . . y T1 


1/T
e e , e Y AX
Univariate case.
Take data, subtract prediction, multiply
residual vector by transpose....
Analogously for the multivariate
case
The likelihood function for a VAR
• Why bother if we can use OLS? Given the
drawbacks of MLE?
• Log posterior is sum of log likelihood and log
prior: so we need it for Bayesian estimation
• Key to understanding many concepts:
– Origin and derivation of standard errors from slope of
LF
– Estimation when we can’t evaluate the LF but have to
approximate it by simulation
– Pseudo-ML when the data are non-Gaussian
Likelihood fn for an AR(1)
y t  y t1 e t , e t  N
0, 2e 
E
y 1 0
2e
var
y 1 E
y 1 0 
1 
2
y 1 , y 01
f y1 
y 01 ; , 2e 
y 02
1
1
exp
2
2e /
1 2 
2 2e 
1 2 
y 2  y 1 e 2

y 2 y 1 y 01  N
y 01 , 2e 
f y 2 y 1 
y 02
y 01 , , 2e 

y 02 y 01 2
1
exp
2
2
22e /
1 2 
2 e 
1 
LF for an AR(1)/ctd...
f yy 1 ,y 2 ... f y 1 f y 2 y 1 f y 3 y 2 . . . f y Ty T1
y 02
logf yy 1 ,y 2 ... 0. 5 log
2
0. 5 log
1  2 1
2/
1 2 
2
2


T 1
/2log
2


T 1
/2log
2 
T

y 0t 
y 0t1 2


2
2

t2
Logging turns complicated product into long sum
We can maximise this and ignore constants.
Likelihood for a VAR(p)
Yt A 1 Yt1 A 2 Yt2 . . . A p Ytp , 
Yt Yt1 , Yt2 , . . . Ytp1  N

Yt1 , Yt2 , . . . Ytp 
Yt 

A1 , A2 . . . . Ap 
A 
AYt , 
Yt Yt1 , Yt2 , . . . Ytp1  N
n/2 |1 |
 2
Y0t , Y0t1 , Y0t2 . . . . ;A, 
f Y tY t1 ,Y t2 ,...Y tp1 
0.5

Y0t AY0t 
Y0t AY0t 1 

0. 5

exp
T


f Y tY t1 ,... 
. . .   log
f Y 2 ... 
f Y 1 ... log
f YY 1 ,Y 2 ...Y Tlog
log
t1
T/2log|1 |

2
Tn/2log

T

Y0t AY0t 
Y0t AY0t 1 

0. 5 
t1
Recap
• Remember likelihood assumes Gaussian errors
• In some circumstances you can get consistent
estimates of parameters (but not standard
errors) even if this is violated.
Distributions for a VARs impulse
response functions
y t 
y t1 e t
IRF is easy in an AR1.
e, 
e, 2 e, . . . n e
irf
Yth
A e, h 0, 1, 2. . .
h
It involves powers of
only 1 coefficient. So
distributions of rho can
be used to compute
distributions of the
elements of the IRFs
vector. Work it out.
Things harder with a VAR as involves
many, jointly distributed coefficients.
Bootstrapping algorithm for VAR IRFs

Yt AYt1 e t
Suppose have estimated a VAR(p)

 
1. Draw, with replacement, a time series of shocks, e i e i1 , e i2 , , , 

2. Create a new time series of observeables using Yt AYt1 e it
3. Re-estimate the VAR to produce A i

4. Compute IRFs using a unit shock e 0 
1, 0, 0. . . 
and powers of A i
5. set i i 1, return to step 1. if i iter
Set iter=200 or so.
Algorithm will generate iter vectors h long, h=max chosen horizon of
the IRF.
Q: how to do step 1 with computer random number generator?
Why estimate VARs
• Estimate RBC/DSGE model by choosing the
parameters to match the VARs IRF
• Estimate using indirect inference.
• Test implications of model by identifying a
shock.
• Accounting for the business cycle by
identifying a shock.
• Forecasting.
Indirect inference with VARs
1. Estimate a VAR on the data
2. Generate data from DSGE model for candidate parameter values 
i
3. Estimate a VAR on the generated data
4. Compute score S i distance of 1 from 2
5. Let i i 1;Go back to 2. i iterm
6.   i such that S
 i min
S
Key reference is Gourieroux et al
Measure of distance, eg, euclidian norm, perhaps weighted in some way.
VAR impulse response matching [Rotemburg and Woodford] is akin to
indirect inference.
In some cases, DSGE models don’t have VAR representations, though
they have VAR approximations
When they do, more properly called a partial information method, rather
than indirect inference.
IRFs one of the infinite moments summarised by the likelihood
Next topic, in fact most of rest of
course, will be on identification in
VARs
• Apologies – we are going to do some
economics.
• These are going to be ‘credible’ identification
restrictions, contrast with Cowles Commission.
• But still contestable.
• They will be based on classes of business cycle
models.
• Enables a test of a theory without making too
many auxiliary assumptions that could fail.
From VARs to SVARs
• Short-run, ‘zero’ restrictions [Sims, CEE]
• Long run restrictions. [Blanchard-Quah, Gali]
• Sign restrictions. [Faust, Uhlig, Mountford and
Uhlig, Canova and de Nicolo]
• Heteroskedasticity-based identification
[Rigobon]