Forecasting with Bayesian
Vector Autoregression
Student: Ruja Cătălin
Supervisor: Professor Moisă Altăr
1. Objectives


To apply the BVAR methodology to a group of
Romanian macroeconomic time series.
To show an improvement in forecast
performance compared to the unrestricted
VAR.
2. Improving forecast performance



The BVAR methodology allows a tradeoff
between oversimplification and overfitting.
oversimplification – univariate (ARIMA) models
which cannot capture the important interaction
between variables.
overfitting – unrestricted VAR which tend to
incorporate useless or misleading relationships
(due to small sample size and large number of
coefficients to be estimated).
3. The Bayesian approach



The Bayesian approach allows the use of prior
beliefs, in the form of probabilities, about which
of the possible models will forecast best.
An extensive system of confidence for the priors
is specified.
A statistical procedure is used to revise this
priors in light of the evidence in the data.
4. The BVAR methodology

Consider an unrestricted, time-varying, p
order VAR representation:
(1) X t  At ( L) X t 1  Ct   t ,
 t ~ N (0, )
n-vector of variables
Xt
At (L) (n x n) matrix of polynomials.
5. Specifying the priors

The scalar representation for
i
t
x
the i th component of X :
t
i
i
i
1
i
1
i
1
i
2
i
2
i
2
(2) xt  c  a1,1 xt 1  a1, 2 xt  2   a1, p xt  p  a2,1 xt 1  a2, 2 xt 2   a2, p xt  p 
 ani ,1 xtn1  ani , 2 xtn2   ani , p xtn p   ti

The prior for coefficient vector:
i
i
(3) a1 ~ N (a , A1 )
A1 is generated as a function of the hyperparameters vector π.
6.a The Minnesota priors
Litterman (1985), Kadiyala and Karlsson (1997), Kenny, Meyler and
Quinn (1998), Ritschl and Woiteck (2002)
(4)
a i  (0, 0, 0, 0, 1, 0,, 0)
(i-1)*p+1 elements
(5) var( a i j ,l ) 
 1 2 i
l 2 2 j
2
l
2
for i  j
for i  j
For the constant term a flat (diffuse) prior is specified.
6.b The Minnesota prior

Changes in the hyperparameters which lead to smaller
(larger) variances of coefficients are referred to as
tightening (loosening) the prior. Thus, in the limit, as the
prior is tightened around its mean each equation takes
the form of a random walk whit drift fit to the data:
(6)
xti  xti1  c   ti
7. Specifications
Doan, Litterman and Sims (1984), Racette and Raynauld (1992), Ballabriga
(1997), Ritschl and Woiteck (2002)

The coefficient evolve over time according with:
(7)
a   3a
i
t
i
t 1
 (1   3 )a  vt , vt ~ N (0,V )
i
The matrix V is proportional with A1, with π4 as the
proportionality factor.
8.a The state space representation

The general state space representation of the dynamics of yt, an
(n x 1) vector of variables (the signal vector):
(8)
(9)
 t 1  F ( z t ) t  ut 1
yt  a( zt )  [ H ( zt )]'  t  et
state equation
signal equation
t
(r x 1) state vector
zt (k x 1) vector of exogenous or predetermined variables.
 0 Q( z t )
0 
u t 1



z
,

~
N
,
 e t t 1 

 0   0
R( z t ) 
 t

  
 t 1  ( y 't 1 , y 't  2 , , y '1 , z t' 1 , z t' 2 , , z1' )'
8.b The state space representation
(10)
xti  z t' ati   ti
zt'  (1, xt11 , xt12 ,, xt1 p , xt21 , xt22 ,, xt2 p , , xtn1 , xtn2 , xtn p )
ati  (c i , a1i,1 , a1i, 2 ,  , a1i, p , a2i ,1 , a2i , 2 , , a2i , p , , ani ,1 , ani , 2 , , ani , p )'
i
i
(11)  t  a t  a
(12)  t 1   3  I k  t  ut 1
(13) xti  z t' a i  z t'  t   ti
F ( z t )   3  I k a( z t )  z t' a i H ( z t )  z t
R ( z t )  0 .9   2 i
the state equation
the signal equation
Q( zt )   4  A1
Doan, Litterman and Sims (1984)
9. Forecast performance

The root mean square error
(14)

1
2
RMSE s     X t  s X t  
 T t 1

T
1
2
The log-determinant of the matrices of summed crossproduct of s-step ahead out-of-sample forecast
(15)
s
LDs  log  Es
 t  X t s X t

T
E s    s  t  s  t '
t 1
10. Data

The variables included in the model are:





the industrial output index IPI (index as against previous
month)
ROL/USD exchange rate ER (average monthly nominal
exchange rate)
consumer price index CPI (index as against previous
month)
the net nominal wage and salary earnings NW (monthly
averages)
the monetary aggregate M2 (monthly averages).
11.a Adapted Minnesota priors
4.66
4.65
4.64
(16) a  (0, 0, 0, 0, 0.9, 0,, 0)
i
4.63
4.62
4.61
(i-1)*p+1 elements
4.60
2000
2001
2002
CPI
102.4
102.0
101.6
(17) a i  (0, 0, 0, 0)
101.2
100.8
100.4
100.0
99.6
99.2
99:07
00:01
00:07
01:01
01:07
IPI
02:01
02:07
11.b Adapted Minnesota priors
Doan, Litterman and Sims (1984), Litterman (1985), and Kadiyala and
Karlsson (1997)
(18) var( a i j ,l ) 
 1 2 i
l 2 j wi
for i  j
2
for i  j
l
w
IPI
0.4
ER
1
CPI
0.8
M2
1
NW
1
12.a Estimation results


Hyperparameters values
π1
π2
π3
π4
0.001
0.03
10-6
1
Root mean square error
BVAR model
IPI
CSM
CPI
M2
NW
s=1
0.0047
0.0216
0.0055
0.0201
0.0197
s=2
0.0046
0.0234
0.0057
0.0382
0.0384
s=3
0.0049
0.0244
0.0062
0.0540
0.0575
IPI
CSM
CPI
M2
NW
s=1
0.0070
0.0263
0.0082
0.0147
0.0059
s=2
0.0068
0.0318
0.0088
0.0237
0.0086
s=3
0.0057
0.0358
0.0065
0.0326
0.0136
UVAR model
12.b Forecasting performance

The overall measures of fit
LD1
LD2
LD3
BVAR
-37.84147
-36.65044
-35.53909
UVAR
-37.30522
-34.66569
-34.79627
13. Conclusions
This paper assesses the opportunity of using the
Bayesian Vector Autoregression for forecasting a group
of series for the Romanian economy. Using the
methodology developed by Doan, Litterman and Sims
(1984), Litterman (1984, 1985) it is found that BVAR
performs
better than the unrestricted vector
autoregression. This conclusion is similar to that of
Doan, Litterman and Sims (1984), and more recently
Racette and Raynauld (1992) and Ballabriga, Alvarez
and Jareno (2000).