ass="t m0 x4c h8 y70 ff3 fs7 fc0 sc0 ls0 ws0">
diagonal matrix containing the (ordered) eigenvalues of
.
ˆ
Σ
The validity of PC estimation for weakly dependent
processes follows from results in [7] and [8]. In particu-
lar, in [7], under some general conditions, r consis-
tency and asymptotic normality of PC estimation of the
unobserved common factors has been established, at each
point in time, for and ,rT 0

L
ˆˆ
Ξx


ˆ
,
t
f
0Σ
ε
rT, when both
the unobserved factors and the idiosyncratic components
show limited serial correlation, and the latter also display
limited heteroskedasticity in both their time-series and
cross-sectional dimensions (see Theorem 1, p. 145); more-
over, the invariance of the singular value decomposition
to row ordering is discussed in [8] (see the Lemma on the
eigenvalues matrix in [8], p. 175).
PC-VAR estimation of is then be implemented
as follows:
P
..
t
tii
xD
ε
2
12
1) apply PCA to and compute f;
ttt
2) obtain by means of OLS estimation of the
stationary dynamic vector regression model
x

LD

.
L
d
t
(3)
where
p
pLD

ˆˆ
L
DΞ
LLDD L

D

ˆ
PCVAR
LP
has all the roots
outside the unit circle;
3) recover the (implied OLS) estimate of the actual
parameters yield by the unrestricted VAR model in (1)
by solving the linear constraints
. (4)
Note that, by construction, the PC-VAR estimator and
the OLS estimator of the unrestricted VAR model in (1)
are the same estimator, i.e.,

ˆ
PCVAR
LLP
ˆtt
fη

 
ˆˆˆ
L

ΞΞ P
ΞΞ
ˆ
P
OLS
In fact, substituting (2) in (1) yields
.

t
LxP
ηε
ˆ
LLP
ˆ
Ξ, (5)
i.e., the dynamic vector regression in (3), with
and .

ˆ
LΞ
LP

D
I
LDP
ˆ
tt
The implied matrix is then estimated by com-
puting

ˆˆ
Ξ,
as r due to the orthonormality of the eigenvec-
tors. The PC-VAR estimator would therefore show the
same asymptotic properties of the OLS estimator.
ˆ
The case considered is however of no interest for em-
pirical implementations, as it does not allow for any di-
mensionality reduction, relatively to the estimation of the
unrestricted VAR model.
2.1. The Unfeasible Case
Consider the case in which only the first s,
s
r
, prin-
cipal components associated with the s largest ordered
eigenvalues of are considered, with j
ˆ
Σˆ0
, j = s +
1, ···, r. The same results as obtained above (
s
r
, im-
plicitly) would hold.
Rewrite the identity in (2) as
,,*,*,
ˆˆ
ˆˆ
tsstrsrstt tt
 ΞfΞfxxτx
*, ,
ˆ
ˆ
tsst
Ξfx,
ˆ
ˆ
trsrst
(6)
where ,
Ξf0τ,
ˆrst
as
f0
,,
()
(1)( ) ()
(1 )(1)
ˆˆˆˆˆ
ˆ
'',
tstrst srs
rr
rrsrrs
srs



,







fff ΞΞΞ .
Then, substituting (6) in (1) yields

*,*, .
tttttt
LL
xPx τηPxη (7)
PC-VAR would then entail OLS estimation of

,
ˆ
.. .,.
tstt
t
L
iid
f
0Σ
xD ε
ε
(8)
It then follows
 
**
ˆˆˆˆˆˆˆˆ
'LLL L

 DΞPΞPΞΞP,
i.e.,

*
ˆˆˆˆˆ
s
PCVAR
LLL

PDΞDΞ, (9)
2
**1*2 *
p
p
LLL L DDD D
 
*()
()
,1,,,
jj
rrs
rs
LL jp





0DD
, where
2
**1*2 *
p
p
LLL L PPP P

*() ()
() ()
ˆˆ
,1,,,
jjs j
rrs rrs
rs rr
Ljp
 


 


Ξ00ΞPP P
and is the Hadamart product.
2.2. The Feasible Case
s
rConsider the case in which only the first s,
, prin-
cipal components associated with the s largest ordered
eigenvalues of are considered, with
ˆ
Σˆ0
j
, j = s +
1, ···, r.
By rewriting (7) as

 

*,
,,
,
ˆˆ
ˆˆ
ˆ
ˆ,
tttt
s
strsrstt
sst t
LL
LL
L




x
ΞfΞf
Ξf
PxPτη
PP η
Pλ
where
tt tt
L ληPτη0
j
, as, for ,
ˆrst
,
f0,
Copyright © 2012 SciRes. OJS
C. MORANA
OJS
253
Copyright © 2012 SciRes.
consistency of the PC-VAR estimator in (9), obtained
from OLS estimation of (8), would require the limiting
uncorrelation condition
ˆ
msT
f0λpli
ˆˆ
to hold, where
,1 ,
ˆ
s
ssp

is the design matrix
containing the temporal information on the lagged prin-
cipal components and is the T vector containing
the temporal information on the error process. The latter
condition would necessarily hold for the case, as

f
λ
ff

Tsp
1
1p

*, 11tt by construction, due to the or-
thogonality of ,
plim xT0τ
ˆ
s
t and , and therefore of and
. The condition
f,
ˆrs
ft*,t
x
t
τ

*,titj, plim xT0

τ,1,,ij p
t
τ
,
, would on the other hand appear to be required for
the case. As under the weak stationarity assump-
tion, for any generic element in the and vectors,
the Wold decomposition would yield
ij
p1
*,t
x
1, ,

*m
mt x*,,
t
x
Lms


*
2
mx
m
, ,
*,..
xt wn

,,
nt

0,

1Ln
nt rs
n
E
2
0,


,..
ntwn

L
,
with and stationary infinite order polyno-

L
mials in the lag operator, provided *,,
,0
mn
xt t



1n
,
vector process
t
x
for the

n.i.d. ,,
tt
tv
L
Φ
0Σ
xv
v
25n
the necessary conditions for consistency would then be
satisfied. Asymptotic normality would also follow under
the same conditions of validity of OLS estimation of un-
restricted VAR models.
3. Monte Carlo Results
Consider the following data generation process (DGP)
where
, 1

p
j
j
j, , is
a polynomial matrix in the lag operator, where the j
coefficient matrices contain randomly extracted values
from the interval (0.4,0.4), constrained to yield a
weakly stationary vector autoregressive process;
LL
ΦIΦ

1,,4p
Φ
1,2 1,
1,2
v
1,
1, 1,
1
1
1
n
nn
nnn


Σ


,
with ,ij
coefficients randomly extracted from the in-
terval (1,1) and constrained to yield an average absolute
off-diagonal element (correlation coefficient)


,
1
1
12
nn
ij
iji k
nn

 ,
0,0.05,0.10,0.15,0.20,0.25,0.30k
2, 4,, 24
1,,4p
, covering the
cases of main interest.
The estimated models are the PC-VAR(p,r) models,
considering r principal components, rand
rn
lags, and the unrestricted VAR(p) model,
equivalent to the PC-VAR(p,r) model with
(25).
The temporal (usable) sample size is T and the
number of replications is 10,000.
100
System level simulation results, i.e., the mean absolute
bias and root mean square error, across parameters and
equations, are reported in Tables 1 and 2.
As shown in Tables 1 and 2, PC-VAR estimation im-
proves upon unrestricted OLS VAR estimation in terms
Table 1. Monte Carlo results.a
# PC (explained total variance)
ρ = 0.00
2 (0.17) 4 (0.31) 6 (0.43) 8 (0.53) 10 (0.62) 12 (0.70) 14 (0.77) 16 (0.83)18 (0.88) 20 (0.93) 22 (0.96) 24 (0.99) 25 (1.0)
p = 1
Bias 0.025 0.023 0.021 0.019 0.017 0.016 0.014 0.013 0.012 0.012 0.012 0.012 0.012
RMSE 0.054 0.060 0.064 0.067 0.071 0.074 0.078 0.083 0.088 0.095 0.102 0.112 0.118
p = 2
Bias 0.024 0.022 0.020 0.019 0.017 0.015 0.014 0.013 0.012 0.012 0.012 0.013 0.014
RMSE 0.051 0.057 0.062 0.067 0.072 0.077 0.083 0.090 0.098 0.108 0.119 0.134 0.142
p = 3
Bias 0.023 0.022 0.020 0.018 0.017 0.016 0.014 0.014 0.013 0.014 0.015 0.017 0.019
RMSE 0.049 0.056 0.062 0.068 0.074 0.082 0.090 0.101 0.114 0.130 0.150 0.178 0.197
p = 4
Bias 0.022 0.021 0.019 0.018 0.017 0.016 0.015 0.015 0.015 0.017 0.021 0.030 0.044
RMSE 0.047 0.054 0.061 0.069 0.077 0.087 0.100 0.116 0.138 0.169 0.221 0.326 0.499
C. MORANA
254
Continued
# PC (explained total variance)
ρ = 0.05
2 (0.19) 4 (0.33) 6 (0.45) 8 (0.55) 10 (0.64)(0.89) 20 (0.93) 22 (0.96) 24 (0.99) 25 (1.0)12 (0.72) 14 (0.78) 16 (0.84) 18
p = 1
Bias 0.028 0.026 0.024 0.022 0.020 0.017 0.014 0.013 0.012 0.012 0.012 0.013
Bias 0.026 0.024 0.022 0.020 0.019 0.017 0.014 0.013 0.012 0.012 0.013 0.014
Bias 0.025 0.023 0.022 0.020 0.018 0.017 0.014 0.014 0.014 0.015 0.017 0.019
Bias 0.024 0.022 0.021 0.019 0.018 0.017 0.015 0.016 0.017 0.021 0.030 0.044
0.016
RMSE 0.053 0.060 0.064 0.067 0.071 0.075 0.079 0.084 0.090 0.096 0.105 0.115 0.121
p = 2
0.015
RMSE 0.050 0.056 0.062 0.067 0.072 0.078 0.084 0.091 0.100 0.110 0.122 0.137 0.146
p = 3
0.016
RMSE 0.048 0.055 0.061 0.067 0.074 0.082 0.091 0.102 0.115 0.132 0.153 0.182 0.202
p = 4
0.016
RMSE 0.046 0.053 0.061 0.068 0.077 0.087 0.100 0.117 0.140 0.173 0.226 0.334 0.511
# P) C (explained total variance
ρ = 0.10
2 (0.23) 4 (0.39) 6 (0.51) 8 (0.60) 10 (0.68)(0.91) 20 (0.94) 22 (0.97) 24 (0.99) 25 (1.0)12 (0.75) 14 (0.81) 16 (0.86) 18
p = 1
Bias 0.030 0.029 0.027 0.025 0.022 0.020 0.015 0.014 0.013 0.012 0.013 0.013
Bias 0.027 0.027 0.025 0.023 0.021 0.019 0.015 0.014 0.013 0.013 0.014 0.015
Bias 0.026 0.026 0.024 0.022 0.020 0.018 0.016 0.015 0.015 0.016 0.018 0.020
Bias 0.025 0.024 0.023 0.021 0.019 0.018 0.016 0.016 0.018 0.022 0.032 0.047
0.017
RMSE 0.052 0.058 0.063 0.068 0.072 0.076 0.081 0.087 0.094 0.101 0.111 0.122 0.129
p = 2
0.017
RMSE 0.048 0.055 0.061 0.067 0.073 0.079 0.086 0.095 0.104 0.115 0.129 0.145 0.155
p = 3
0.017
RMSE 0.046 0.053 0.060 0.067 0.075 0.084 0.094 0.106 0.120 0.139 0.162 0.193 0.214
p = 4
0.017
RMSE 0.044 0.052 0.059 0.068 0.077 0.089 0.103 0.121 0.146 0.181 0.238 0.355 0.542
PC-VAR Estimation of Vector Autoregressive Models
Open Journal of Statistics, 2012, 2, 251-259
http://dx.doi.org/10.4236/ojs.2012.23030 Published Online July 2012 (http://www.SciRP.org/journal/ojs)
PC-VAR Estimation of Vector Autoregressive Models*
Claudio Morana1,2,3,4
1Dipartimento di Economia Politica, Università di Milano Bicocca, Milano, Italy
2Center for Research on Pensions and Welfare Policies, Collegio Carlo Alberto, Moncalieri, Italy
3Fondazione Eni Enrico Mattei, Milano, Italy
4International Centre for Economic Research, ICER, Torino, Italy
Email: claudio.morana@unimib.it
Received May 7, 2012; revised June 8, 2012; accepted June 22, 2012
ABSTRACT
In this paper PC-VAR estimation of vector autoregressive models (VAR) is proposed. The estimation strategy success-
fully lessens the curse of dimensionality affecting VAR models, when estimated using sample sizes typically available
in quarterly studies. The procedure involves a dynamic regression using a subset of principal components extracted
from a vector time series, and the recovery of the implied unrestricted VAR parameter estimates by solving a set of lin-
ear constraints. PC-VAR and OLS estimation of unrestricted VAR models show the same asymptotic properties. Monte
Carlo results strongly support PC-VAR estimation, yielding gains, in terms of both lower bias and higher efficiency,
relatively to OLS estimation of high dimensional unrestricted VAR models in small samples. Guidance for the selection
of the number of components to be used in empirical studies is provided.
Keywords: Vector Autoregressive Model; Principal Components Analysis; Statistical Reduction Techniques
1. Introduction
In this paper principal components vector autoregressive
estimation (PC-VAR) for large scale dynamic economet-
ric models is proposed. Vector autoregressive models
(VAR), when estimated using economic time series of
sample sizes typically available in empirical quarterly
studies, are subject to the curse of dimensionality. Recent
contributions, dealing with this issue in the framework of
factor augmented VAR (FVAR) models ([1-4]; see also
[5] for a survey and new results), have mostly relied on
principal components (PC) estimation (in the frequency
or time domain) of the underlying unobserved common
factor structure. Results of [1,6,7] have in fact proved
consistency and asymptotic normality of PC estimation
under various scenarios, including the exact and approxi-
mate factor model case, weakly stationary (short memory)
and I(1) integrated processes, also showing conditional
heteroskedasticity; see also [8] for additional implica-
tions of temporal dependence for PC estimation.
The proposed approach is different from FVAR mod-
eling, as no reference to an underlying factor structure is
made, and PC estimation is performed to yield accurate
estimation of the parameters of an unrestricted VAR
model, rather than of a dynamic factor model. The pro-
cedure involves a dynamic regression using a subset of
principal components extracted from a vector time series,
and the recovery of the implied unrestricted VAR pa-
rameter estimates by solving a set of linear constraints.
PC-VAR and OLS estimation of unrestricted VAR mod-
els show the same asymptotic properties.
Monte Carlo results strongly support PC-VAR estima-
tion, yielding gains, in terms of both lower bias and
higher efficiency, relatively to OLS estimation of high
dimensional unrestricted VAR models in small samples.
Guidance for the selection of the number of components
to be used in empirical studies is provided.
After this introduction, the paper is organized as fol-
lows. In section two PC-VAR estimation is presented,
while in section three Monte Carlo analysis is performed;
see [9] for an empirical application of the procedure.
2. PC-VAR Estimation of VAR Models
Consider the vector autoregressive (VAR) model

.. .,
tt
tiid
-
0, Σ
IPL xη
η
t
x1r
(1)
zero mean I(0) vector process, where is an
2
12
p
1, ,tT, and
p
LLL L PPP P has all
the roots outside the unit circle, with
j
P1, ,jp,
ˆ
ˆ
tt
Ξfx
ˆˆ
tt
,
being a square matrix of coefficients of order r.
PC-VAR estimation relies on the following identity
, (2)
fΞx1r
is the
vector of estimated princi- where
C
opyright © 2012 SciRes. OJS
C. MORANA
252
pal components of t, is the matrix of or-
thogonal eigenvectors associated with the r (ordered)
eigenvalues of (
xˆ
Ξ
rr
ˆ
Σ
tt
Σ). This follows from the
eigenvalue-eigenvector decomposition of, i.e.,
, where
E
xx
Σ
ˆˆ
1
ΞΣ
ˆˆ
Ξ=Γ
ˆ
, r1
ˆdi ˆ
,ag
Γrr is the
# P) C (explained total variance
ρ = 0.15
2 (0.29) 4 (0.46) 6 (0.58) 8 (0.67) 10 (0.74)(0.92) 20 (0.95) 22 (0.98) 24 (0.99) 25 (1.0)12(0.80) 14 (0.85) 16 (0.89) 18
p = 1
Bias 0.031 0.031 0.030 0.027 0.025 0.022 0.017 0.015 0.014 0.013 0.014 0.014
Bias 0.028 0.028 0.027 0.025 0.022 0.020 0.016 0.015 0.014 0.014 0.015 0.016
Bias 0.027 0.027 0.025 0.024 0.022 0.020 0.017 0.016 0.016 0.017 0.020 0.022
Bias 0.025 0.025 0.024 0.022 0.021 0.019 0.017 0.018 0.020 0.024 0.034 0.051
0.019
RMSE 0.051 0.057 0.063 0.068 0.073 0.079 0.085 0.092 0.101 0.110 0.121 0.135 0.143
p = 2
0.018
RMSE 0.047 0.053 0.060 0.067 0.074 0.082 0.090 0.100 0.112 0.125 0.140 0.159 0.171
p = 3
0.018
RMSE 0.045 0.052 0.059 0.067 0.076 0.086 0.098 0.112 0.129 0.150 0.177 0.212 0.236
p = 4
0.018
RMSE 0.043 0.050 0.058 0.068 0.079 0.092 0.109 0.130 0.158 0.197 0.260 0.390 0.600
Copyright © 2012 SciRes. OJS
C. MORANA 255
Continued
# PC (explained total variance)
ρ = 0.20 2( 0.35) 4 (0.54) 6 (0.66) 8 (0.74) 10 (0.80)(0.94) 20 (0.97) 22 (0.98) 24 (1.0) 25 (1.0)12 (0.85) 14 (0.89) 16 (0.92) 18
p = 1
Bias 0.031 0.032 0.031 0.029 0.027 0.024 0.019 0.017 0.015 0.015 0.016 0.016
Bias 0.028 0.028 0.028 0.026 0.024 0.022 0.018 0.017 0.016 0.016 0.017 0.018
Bias 0.027 0.027 0.026 0.025 0.023 0.021 0.018 0.018 0.018 0.019 0.022 0.025
Bias 0.025 0.026 0.025 0.024 0.022 0.021 0.019 0.019 0.022 0.027 0.039 0.053
0.021
RMSE 0.050 0.056 0.062 0.068 0.075 0.083 0.092 0.101 0.112 0.124 0.139 0.156 0.166
p = 2
0.020
RMSE 0.046 0.052 0.059 0.067 0.076 0.086 0.097 0.110 0.124 0.141 0.160 0.183 0.198
p = 3
0.020
RMSE 0.044 0.050 0.058 0.067 0.078 0.091 0.106 0.124 0.144 0.170 0.202 0.244 0.273
p = 4
0.019
RMSE 0.042 0.049 0.057 0.067 0.081 0.097 0.117 0.142 0.175 0.222 0.296 0.447 0.692
# P) C (explained total variance
ρ = 0.25 2 (0.41) 4 (0.63) 6 (0.75) 8 (0.83) 10 (0.88)(0.97)20 (0.98) 22 (0.99) 24 (1.0) 25 (1.0)12 (0.91) 14 (0.94) 16 (0.96) 18
p = 1
Bias 0.031 0.032 0.032 0.031 0.030 0.028 0.023 0.021 0.020 0.019 0.021 0.022
RMSE
Bias 0.028 0.029 0.029 0.028 0.026 0.025 0.022 0.021 0.020 0.021 0.023 0.025
RMSE
Bias 0.027 0.027 0.027 0.026 0.025 0.024 0.022 0.022 0.023 0.025 0.030 0.033
RMSE
Bias 0.026 0.026 0.025 0.025 0.024 0.023 0.022 0.024 0.028 0.036 0.053 0.081
RMSE
0.025
0.049 0.055 0.061 0.068 0.078 0.091 0.106 0.124 0.144 0.167 0.193 0.222 0.240
p = 2
0.023
0.045 0.051 0.058 0.067 0.078 0.093 0.111 0.133 0.157 0.185 0.217 0.255 0.278
p = 3
0.023
0.043 0.049 0.057 0.067 0.081 0.100 0.123 0.151 0.184 0.225 0.274 0.339 0.383
p = 4
0.022
0.042 0.048 0.055 0.067 0.084 0.107 0.138 0.176 0.227 0.297 0.406 0.623 0.977
# P) C (explained total variance
ρ = 0.30 2 (0.44) 4 (0.67) 6 (0.79) 8 (0.86) 10 (0.88)(0.97) 20 (0.98) 22 (0.99) 24 (1.0) 25 (1.0)12 (0.91) 14 (0.94) 16 (0.96) 18
p = 1
Bias 0.031 0.032 0.032 0.031 0.030 0.029 0.026 0.025 0.025 0.026 0.029 0.031
Bias 0.028 0.029 0.029 0.028 0.027 0.026 0.024 0.024 0.025 0.027 0.031 0.034
Bias 0.027 0.028 0.027 0.027 0.026 0.025 0.025 0.026 0.028 0.033 0.040 0.045
Bias 0.026 0.026 0.026 0.025 0.024 0.024 0.025 0.029 0.035 0.047 0.071 0.111
0.028
RMSE 0.049 0.055 0.061 0.069 0.079 0.095 0.115 0.141 0.174 0.213 0.260 0.313 0.346
p = 2
0.025
RMSE 0.045 0.051 0.057 0.067 0.080 0.098 0.121 0.151 0.189 0.233 0.286 0.348 0.386
p = 3
0.025
RMSE 0.043 0.049 0.056 0.067 0.083 0.106 0.137 0.176 0.226 0.289 0.367 0.468 0.535
p = 4
0.024
RMSE 0.041 0.047 0.055 0.067 0.086 0.115 0.154 0.208 0.282 0.386 0.546 0.862 1.365
aT reon (abMSticse arams) -Vund OLS VAR
scond, and foms. Trage ae reorreoet is ρ 0. 0.15, 0.2),
he Table
timation of firs
ports M
t, se
te Carlo
third,
solute) bia
urth orde
s and R
r syste
E statis
he ave
(averag
bsolut
cross pa
siduals c
eters and e
lation c
quation
fficien
from PC
= (0.00,
AR and
05, 0.10,
restricte
0.20,e
the temporal sample size is T = 100, the cross-sectional sample size is n = 25, and the number of replications is 10,000. The estimated models are the PC-VAR
model, considering r principal components, r = 2, 4, ···, 24, and the unrestricted VAR model, equivalent to the PC-VAR model with r = n (25) principal com-
ponents.
5, 0.30
Copyright © 2012 SciRes. OJS
C. MORANA
256
Table 2. Monte Carlo results (ratio of PC-VAR to OLS figures).a
# PC (explained total variance)
ρ = 0.00
2 (0.17) 4 (0.31) 6 (0.43) 8 (0.53) 10 (0.62) 18 (0.88) 20 (0.93) 22 (0.96) 24 (0.99) 25 (1.0)12 (0.70) 14 (0.77) 16 (0.83)
p = 1
Bis 2.044 1.854 1.683 1.529 1.386 1.259 1.047 0.974 0.933 0.928 0.962 1.000
R
Bis 1.722 1.590 1.463 1.342 1.227 1.119 0.943 0.886 0.861 0.878 0.945 1.000
R
Bis 1.249 1.160 1.073 0.989 0.910 0.838 0.735 0.718 0.735 0.795 0.911 1.000
R
Bis 0.512 0.478 0.444 0.411 0.382 0.357 0.333 0.347 0.389 0.479 0.677 1.000
R
a1.144
MSE 0.461 0.509 0.542 0.571 0.599 0.629 0.663 0.703 0.749 0.804 0.869 0.950 1.000
p = 2
a1.022
MSE 0.355 0.400 0.435 0.469 0.504 0.542 0.585 0.633 0.691 0.758 0.839 0.938 1.000
p = 3
a0.777
MSE 0.247 0.282 0.313 0.344 0.377 0.415 0.459 0.512 0.577 0.659 0.763 0.903 1.000
p = 4
a0.338
MSE 0.094 0.109 0.122 0.137 0.154 0.174 0.199 0.232 0.276 0.339 0.442 0.653 1.000
# P) C (explained total variance
ρ = 0.05
2 (0.19) 4 (0.33) 6 (0.45) 8 (0.55) 10 (0.64)(0.89) 20 (0.93) 22 (0.96) 24 (0.99) 25 (1.0)12 (0.72) 14 (0.78) 16 (0.84) 18
p = 1
Bias 2.199 2.061 1.889 1.711 1.539 1.378 1.108 1.013 0.952 0.934 0.965 1.000
Bias 1.815 1.712 1.587 1.454 1.323 1.198 0.988 0.915 0.876 0.884 0.946 1.000
Bias 1.299 1.231 1.146 1.057 0.969 0.887 0.760 0.733 0.740 0.796 0.910 1.000
Bias 0.533 0.504 0.470 0.435 0.402 0.372 0.340 0.351 0.391 0.478 0.678 1.000
1.232
RMSE 0.441 0.492 0.527 0.557 0.586 0.617 0.652 0.693 0.741 0.797 0.865 0.948 1.000
p = 2
1.085
RMSE 0.339 0.385 0.422 0.456 0.492 0.531 0.574 0.625 0.683 0.752 0.834 0.937 1.000
p = 3
0.814
RMSE 0.236 0.271 0.302 0.334 0.368 0.406 0.451 0.505 0.571 0.654 0.760 0.902 1.000
p = 4
0.350
RMSE 0.090 0.105 0.118 0.133 0.151 0.171 0.197 0.229 0.274 0.338 0.442 0.654 1.000
# P) C (explained total variance
ρ = 0.10
2 (0.23) 4 (0.39) 6 (0.51) 8 (0.60) 10 (0.68)(0.91) 20 (0.94) 22 (0.97) 24 (0.99) 25 (1.0)12 (0.75) 14 (0.81) 16 (0.86) 18
p = 1
Bias 2.262 2.217 2.059 1.865 1.668 1.477 1.157 1.040 0.962 0.934 0.961 1.000
Bias 1.836 1.788 1.670 1.530 1.387 1.249 1.010 0.928 0.881 0.883 0.945 1.000
Bias 1.308 1.276 1.198 1.105 1.010 0.919 0.776 0.738 0.740 0.794 0.909 1.000
Bias 0.529 0.515 0.485 0.449 0.413 0.381 0.342 0.349 0.388 0.477 0.678 1.000
RMSE 0.082 0.095 0.109 0.125 0.143 0.164 0.190 0.224 0.269 0.334 0.439 0.654 1.000
1.306
RMSE 0.402 0.453 0.492 0.525 0.557 0.591 0.629 0.674 0.726 0.786 0.858 0.946 1.000
p = 2
1.122
RMSE 0.308 0.352 0.392 0.429 0.467 0.509 0.555 0.608 0.670 0.742 0.828 0.934 1.000
p = 3
0.839
RMSE 0.215 0.248 0.280 0.313 0.349 0.389 0.436 0.492 0.561 0.646 0.755 0.900 1.000
p = 4
0.356
Copyright © 2012 SciRes. OJS
C. MORANA 257
Continued
# lal v) PC (expined totaariance
ρ = 0.15
2 (0.29) 4 (0.46) 6 (0.58) 8 (0.67) 10 (0.74) 12 (0.80) 14 (0.85) 16 (0.89)18 (0.92) 20 (0.95) 22 (0.98) 24 (0.99) 25 (1.0)
p = 1
Bias 2.149 2.165 2.073 1.911 1.717 1.518 1.334 1.173 1.045 0.959 0.932 0.962 1.000
RMSE 0.354 0.399 0.439 0.476 0.513 0.553 0.647 0.704 0.770 0.848 0.942 1.000
RMSE 0.273 0.311 0.350 0.390 0.432 0.478 0.586 0.652 0.730 0.820 0.931 1.000
RMSE 0.190 0.219 0.250 0.284 0.322 0.366 0.475 0.547 0.635 0.748 0.897 1.000
RMSE 0.072 0.083 0.097 0.113 0.132 0.154 0.216 0.263 0.329 0.434 0.650 1.000
0.597
p = 2
Bias 1.736 1.737 1.657 1.539 1.404 1.266 1.135 1.020 0.934 0.880 0.880 0.942 1.000
0.529
p = 3
Bias 1.229 1.227 1.174 1.094 1.005 0.916 0.835 0.772 0.735 0.737 0.792 0.906 1.000
0.416
p = 4
Bias 0.493 0.490 0.468 0.437 0.403 0.372 0.347 0.335 0.343 0.382 0.471 0.672 1.000
0.181
# P) C (explained total variance
ρ = 0.20
2 4 6 8 10 12 14 16 18 20 22 24 25 (0.35) (0.54) (0.66) (0.74)(0.80) (0.85) (0.89) (0.92)(0.94) (0.97) (0.98)(1.0) (1.0)
p = 1
Bias 1.903 1.944 1.901 1.801 1.655 1.475 1.297 1.137 1.012 0.931 0.913 0.955 1.000
RMSE 0.301 0.337 0.372 0.410 0.452 0.499 0.609 0.674 0.749 0.835 0.938 1.000
RMSE 0.232 0.263 0.298 0.338 0.384 0.435 0.556 0.629 0.712 0.810 0.927 1.000
RMSE 0.161 0.185 0.212 0.245 0.286 0.333 0.452 0.529 0.622 0.739 0.894 1.000
RMSE 0.061 0.070 0.082 0.097 0.117 0.140 0.206 0.254 0.321 0.429 0.647 1.000
0.551
p = 2
Bias 1.537 1.565 1.521 1.437 1.330 1.214 1.097 0.991 0.910 0.864 0.870 0.940 1.000
0.492
p = 3
Bias 1.086 1.096 1.066 1.013 0.945 0.870 0.798 0.742 0.712 0.721 0.781 0.902 1.000
0.388
p = 4
Bias 0.436 0.438 0.425 0.403 0.378 0.352 0.331 0.321 0.330 0.370 0.460 0.663 1.000
0.169
# P) C (explained total variance
ρ = 0.25
2 4 6 8 10 12 14 16 18 20 22 24 25 (0.41) (0.63) (0.75) (0.83)(0.88) (0.91) (0.94) (0.96)(0.97) (0.98) (0.99)(1.0) (1.0)
p = 1
Bias 1.417 1.461 1.453 1.416 1.347 1.262 1.158 1.044 0.946 0.890 0.885 0.945 1.000
RMSE 0.205 0.229 0.254 0.285 0.326 0.379 0.517 0.601 0.696 0.802 0.925 1.000
RMSE 0.163 0.184 0.208 0.240 0.282 0.336 0.478 0.566 0.667 0.781 0.917 1.000
RMSE 0.114 0.129 0.148 0.174 0.211 0.260 0.394 0.481 0.587 0.717 0.885 1.000
RMSE 0.043 0.049 0.057 0.069 0.086 0.110 0.181 0.232 0.304 0.416 0.638 1.000
0.442
p = 2
Bias 1.134 1.160 1.144 1.106 1.055 0.998 0.937 0.877 0.831 0.818 0.845 0.928 1.000
0.402
p = 3
Bias 0.812 0.825 0.813 0.789 0.755 0.718 0.684 0.659 0.656 0.687 0.761 0.896 1.000
0.321
p = 4
Bias 0.316 0.319 0.314 0.304 0.293 0.281 0.274 0.277 0.296 0.346 0.440 0.649 1.000
0.141
Copyright © 2012 SciRes. OJS
C. MORANA
Copyright © 2012 SciRes. OJS
258
Continued
# lal v) PC (expined totaariance
ρ = 0.30
2 (0.44) 4 (0.67) 6 (0.79) 8 (0.86) 10 (0.88) 12 (0.91) 14 (0.94) 16 (0.96)18 (0.97) 20 (0.98) 22 (0.99) 24 (1.0) 25 (1.0)
p = 1
Bias 1.004 1.036 1.031 1.009 0.979 0.934 0.891 0.849 0.815 0.800 0.827 0.925 1.000
RMSE 0.142 0.158 0.176 0.198 0.230 0.274 0.409 0.504 0.617 0.751 0.906 1.000
RMSE 0.117 0.131 0.149 0.173 0.206 0.253 0.392 0.488 0.603 0.740 0.903 1.000
RMSE 0.081 0.092 0.105 0.126 0.155 0.198 0.329 0.423 0.540 0.686 0.874 1.000
RMSE 0.030 0.035 0.040 0.049 0.063 0.084 0.153 0.207 0.283 0.400 0.631 1.000
0.333
p = 2
Bias 0.824 0.844 0.834 0.814 0.785 0.754 0.725 0.709 0.709 0.732 0.798 0.916 1.000
0.314
p = 3
Bias 0.599 0.609 0.602 0.586 0.569 0.552 0.541 0.543 0.569 0.629 0.734 0.887 1.000
0.255
p = 4
Bias 0.232 0.235 0.231 0.225 0.219 0.215 0.216 0.229 0.259 0.319 0.423 0.644 1.000
0.113
aThe ree rntebsias SEs (acrme eq froARtiot,
sed, ath orAR systes, red Oation figres. Teragete reorr coeft is ρ
0. 0. thraless-l sae i anmb 10,e
e strength of the con-
e Tablports thatio Mo Carlo (aolute) band RM statisticverage aoss paraters anduations)m PC-V estiman of firs
cond, thir
05, 0.10, 0.
nd four
15, 0.20,
der V
25, 0.30),
m
e tempo
lative to unre
sample siz
stricte
is T = 100,
LS estim
the cro
u
sectiona
he av
mple siz
absolu
s n = 25,
siduals c
d the nu
elation
er of replic
ficien
ations is
= (0.00,
000. Th
estimated models are the PC-VAR model, considering r principal components, r = 2, 4, ···, 24, and the unrestricted VAR model, equivalent to the PC-VAR
model with r = n (25) principal components.
of both lower bias and higher efficiency, independently
of the order of the system and th
temporaneous cross-sectional correlation relating the er-
ror terms. By following a bias minimization criterion,
two broad cases may however be distinguished, i.e., the
(contemporaneously) non-correlated errors (0
) and
correlated errors (0
) cases. For the former one
(0
), the optimal proportion of total variance to be
explained by the selected PCs ranges between 77% and
88%, depending on the order of the system, falling as the
order of the VAR increases. While the bias improvement
is small for the VAR(1) and VAR(2) cases, for which the
degrees of freedom are not smaller than 50% of the sam-
ple size, PC-VAR estimation yields a much more dra-
matic bias reduction for the VAR(3) and VAR(4) cases
(30% and 350%, respectively), as the degrees of free-
dom fall to 25% of the sample size and 0, respectively.
The improvement in relative efficiency is also large, be-
tween 20% and 70%, increasing with the order of the
VAR, i.e., as the number of degrees of freedom de-
creases.
On the other hand, for the latter case (0
) a higher
optimal level of explained variance would appear to be
determined, increasing with ρ, and decrth the
order of the VAR model, i.e., 93% to 99% for the VAR(1)
and VAR(2) models, 89% to 97% for the VAR(3) model
and 84% to 94% for the VAR (4) model. Large im-
provements in bias reduction (5% to 70%) and relative
efficiency (20% to 80%), increasing as the number of
Overall, Monte Carlo results point to important gains,
in terms of bias and efficiency, of PC-VAR estimation
over OLS estimation of high dimensional unrestricted
VAR models, in small samples. Concerning the cases of
main empirical interest for VAR analysis, i.e., showing a
low (0.1
easing wi
tained also for the contemporaneously correlated case.
degrees of freedom falls, would then appear to be at-
) or null average degree of contemporane-
ous correlation of the error terms, and a number of de-
grees of freedom about or below 25% of the sample size,
a target proportion of total variance, to be explained by
the selected PCs, in the range 80% to 90%, may then be
expected to yield highly satisfactory results, and there-
fore advisable for empirical applications.
4. Conclusion
In this paper principal components vector autoregressive
estimation (PC-VAR) for large scale dynamic economet-
ric models is proposed. The procedure involves a dy-
namic regression using a subset of principal components
ector time series, and the recovery of extracted from a v
the implied unrestricted VAR parameter estimates by
solving a set of linear constraints. PC-VAR and OLS
estimation of unrestricted VAR models show the same
asymptotic properties. Monte Carlo results strongly sup-
port PC-VAR estimation, yielding gains, in terms of both
lower bias and higher efficiency, relatively to OLS esti-
mation of high dimensional unrestricted VAR models in
small samples.
C. MORANA 259
5. Acknowledgements
The author is grateful to two anonymous referees of the
OJS for constructive comments.
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