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
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,
C. MORANA
OJS
253
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