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Cointegrating VAR Models
and Probability Forecasting:
Applied to a Small Open Economy
Gustavo Sánchez
April 2009
Summary

VEC and Cointegrating VAR Models


Estimate Parameters
Probability Forecasting


Simulate Forecasts
Summary Statistics to estimate
probabilities of events
Point Forecast and Confidence Interval
Forecast for lgdp
15.8
16
16.2 16.4 16.6
16.3 16.4 16.5 16.6 16.7
Forecast for lm1
Forecast for loilp
6.8
3
7
3.5
7.2
4
7.4
4.5
7.6
Forecast for lcpi
2008q4
2009q1
2009q2
2009q3
2009q4 2008q4
95% CI
2009q1
forecast
2009q2
2009q3
2009q4
Probability of Inflation Greater than 45
Proportion estimation
Number of obs
=
225
Proportion
Std. Err.
[95% Conf. Interval]
0
0.2888889
0.0302838
0.2292113
0.348566
1
0.7111111
0.0302838
0.6514336
0.770789
inf_45
0
.02
Density
.04
.06
Density Inflation
0
20
kernel = epanechnikov, bandwidth = 1.9987
40
inf
60
80
Cointegrating VAR models




Based on the vector error correction (VEC) model
specification.
The specification assumes that the economic theory
characterizes the long-run equilibrium behavior
The short-run fluctuations represent deviations
from that equilibrium.
The short-run and long-run (economic) concepts
are linked to the statistical concept of stationarity.
Cointegrating VAR models
Reduced form for a VEC model
p 1
zt  a  bt  zt 1   i zt i  t
i 1
Where:
zt :
 :
I(1) Endogenous variables
Matrices containing the long-run adjustment coefficients and
coefficients for the cointegrating relationships
i : Matrix with coefficients associated to short-run dynamic effects
a,b :
t :
Vectors with coeficients associated to the intercepts and trends
Vector with innovations
Cointegrating VAR models
Reduced form for a VEC model
p 1
zt  a  bt  zt 1   i zt i  t
i 1


Identifying α and β requires r2 restrictions
(r: number of cointegrating vectors).
Johansen FIML estimation identifies α and β by
imposing r2 atheoretical restrictions.
Cointegrating VAR models

Garrat et al. (2006) describe the Cointegrating VAR
approach:



Use economic theory to impose restrictions to identify αβ.
Exact identification is not necessarily achieved by the
theoretical restrictions.
Test whether the overidentifying restrictions are valid.
** Restrictions on VEC system **
*** Restrictions on Beta lm1 ***
constraint 1 [_ce1]lm1=1
.
.
.
constraint 6 [_ce1]ltipp906bn=0
*** Restrictions on Beta lmt ***
constraint 8 [_ce2]lmt=1
.
.
.
constraint 11 [_ce2]ltipp906bn=0
*** Restrictions on alpha ***
constraint 12 [D_loilp]l._ce1=0
constraint 13 [D_loilp]l._ce2=0
** VEC specification **
vec
lm1 lmt lcpi loilp ltcpn lxt ltipp906bn lgdp ///
if tin(1991q1,2008Q4), lags(2) rank(2)
///
bconstraints(1/11) aconstraints(12/13)
///
noetable
Vector error-correction model
Sample: 1991q1 - 2008q4
No. of obs
AIC
HQIC
SBIC
Log likelihood = 659.9591
Det(Sigma_ml) = 1.51e-18
Cointegrating equations
Equation
Parms
chi2
P>chi2
------------------------------------------_ce1
2
50.19532
0.0000
_ce2
3
1639.412
0.0000
------------------------------------------Identification: beta is overidentified
Identifying constraints:
( 1) [_ce1]lm1 = 1
( 2) [_ce1]lmt = 0
( 3) [_ce1]lxt = 0
( 4) [_ce1]loilp = 0
( 5) [_ce1]lcpi = 0
( 6) [_ce1]ltipp906bn = 0
( 7) [_ce2]lm1 = 0
( 8) [_ce2]lmt = 1
( 9) [_ce2]lxt = 0
(10) [_ce2]ltcpn = 0
(11) [_ce2]ltipp906bn = 0
=
72
= -15.80442
= -14.6589
= -12.92697
-----------------------------------------------------------------------------beta |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------_ce1
|
lm1 |
1
.
.
.
.
.
lmt | (dropped)
lcpi | (dropped)
loilp | (dropped)
ltcpn |
.215578
.0697673
3.09
0.002
.0788365
.3523194
lxt | (dropped)
ltipp906bn | (dropped)
lgdp | -4.554976
.6489147
-7.02
0.000
-5.826825
-3.283127
_cons |
57.02687
.
.
.
.
.
-------------+---------------------------------------------------------------_ce2
|
lm1 | (dropped)
lmt |
1
.
.
.
.
.
lcpi | -.0317544
.0087879
-3.61
0.000
-.0489784
-.0145304
loilp | -.0780758
.0255611
-3.05
0.002
-.1281746
-.027977
ltcpn | (dropped)
lxt | (dropped)
ltipp906bn | (dropped)
lgdp | -2.519458
.1105036
-22.80
0.000
-2.736041
-2.302875
_cons |
26.26122
.
.
.
.
.
------------------------------------------------------------------------------
*** Point Forecast ***
fcast compute y_, step(4)
keep
y_lm1 y_lmt y_lcpi
///
y_loilp y_ltcpn y_lxt
///
y_ltipp906bn y_lgdp quarter
keep
if tin(2009q1,2009q4)
save
"filename"
** Residuals from the VEC equations **
foreach x of varlist lm1 lmt lxt loilp ///
ltcpn lcpi
///
ltipp906bn lgdp {
predict res_`x' if e(sample),
///
residuals
///
equation(D_`x')
}
Probability Forecasting



It is basically an estimation of the probability
that a single or joint event occurs.
We could define the event in terms of the levels of
one or more variables, for one or more future time
periods.
It is associated to the uncertainty inherent to
the predictions produced by regression models.
Probability Forecasting


This methodology can be applied to a wide
diversity of models. Our focus here is on the
predictions from a cointegrating VAR model.
In general, forecasting based on econometric
models are subject to:
Future uncertainty
 Parameters uncertainty
 Model uncertainty
 Measurement and policy uncertainty

Probability Forecasting

Future and parameter uncertainty

Let’s consider the standard linear regression model:
yt  xt   ut
Where
u ~ N (0,  2 )
Probability Forecasting

Future and parameter uncertainty

For example, for
σ2
( j ,s )
y
known we could simulate T 1
yT( j ,1s )  xT' ˆ ( j )  uT( s)1
Where:
ˆ ( j )
j-th random draw from
uT( s)1
s-th random draw from
;
j=1,2,…,J ; s=1,2,…,S

N ˆT ,  2 ( X ' X ) 1

N 0,  2


( j)
which is independent from the random draw for ˆ
Probability Forecasting

Computations for VAR cointegrating models

Let’s consider the VEC model
p 1
zt   zt 1   i zt i  a0  a1t  H t
'
i 1
Non-Parametric Approach
1. Simulated errors are drawn from in sample residuals
2. The Choleski decomposition for the estimated Var-Cov
matrix of the error term is used in a two-stage
procedure combined with the simulated errors in (1).
** Matrix for Simulation (First Stage, Pag.167) **
matrix sigma=e(omega) /* V-C Matrix of the residuals */
matrix P=cholesky(sigma)
mkmat res_lm1 res_lmt res_lxt res_loilp ///
res_ltcpn res_lcpi
///
res_lgdp res_ltipp906bn ///
if tin(1991q1,2008q4),
///
matrix(res)
matrix invP_res=inv(P)*res'
matrix invP_rs1=invP_res‘
svmat invP_rs1,names(col)
** Program for Residual Resampling **
program mysim_np, rclass
preserve
bsample 4 if tin(1991q1,2008q4) /* 4 frcst. per. */
mkmat IP_R_D_lm1 IP_R_D_lm IP_R_D_lcpi ///
IP_R_D_loilp IP_R_D_ltcpn IP_R_D_lxt ///
IP_R_D_ltipp906bn IP_R_D_lgdp,
///
matrix(IP_R)
matrix PE_tr=P*IP_R'
matrix PE=PE_tr'
svmat PE,names(col)
●
●
●
●
●
●
●
●
●
****** Simulation ******
simulate “varlist", rep(###)
saving("filename",replace):
mysim_np
///
///
command: mysim_np
s_lm1_1: r(res_lm1_1)
s_lm1_2: r(res_lm1_2)
●
●
●
●
●
●
●
●
●
s_lgdp_3: r(res_lgdp_3)
s_lgdp_4: r(res_lgdp_4)
Simulations (###)
─┼─ 1 ─┼─ 2 ─┼─ 3 ─┼─ 4 ─┼─ 5
....................................................
●
●
●
●
●
●
●
●
●
50
**** Probability Forecasting ****
generate dgdp=gdp/gdp2008*100-100
if year==2009 &
replication>0
generate inf=cpi/cpi2008*100-100 ///
if year==2009 &
replication>0
///
///
///
generate gdp_n__inf45=cond(dgdp<0 & inf>45,1,0)
proportion gdp_n__inf35
Probability of Negative GDP and Inflation>45
Proportion estimation
Number of obs
=
225
Proportion
Std. Err.
[95% Conf. Interval]
0
.68
.0311677
.6185805
.7414195
1
.32
.0311677
.2585805
.3814195
gdp_1__inf45
Density GDP
.1
0
0
.05
.02
Density
.04
.15
.06
.2
Density Inflation
0
20
kernel = epanechnikov, bandwidth = 1.9987
40
inf
60
80
-10
-5
0
dgdp
kernel = epanechnikov, bandwidth = 0.6461
5
Cointegrating VAR Models
and Probability Forecasting:
Applied to a Small Open Economy
Gustavo Sánchez
April 2009