Online Supplement to “Predictive Quantile Regression with Persistent Covariates: IVX-QR Approach” Ji Hyung Lee November, 2014
Abstract This online Appendix provides a supplement with additional discussion, proofs of supporting lemmas and more comprehensive numerical results.
1
Supplementary Discussion
1.1
QR Endogeneity: Derivation and Estimation
Let the innovations from (2.1) and (2.2) in the main text be (t suppressed) u= where
u
"
u0 ux
#
iid Fu
0;
u
=
1 1
!!
is a correlation matrix with corr(u0 ; ux ) = .
Our parameter of interest, the QR endogeneity, is ( )=
corr (1 (u0 < 0) ; ux ) =
corr 1 u0 < Fu01 ( ) ux =
E 1 u0t < Fu01 ( ) uxt p : (1 ) xx
For multivariate normal and t-distributions, we can show E[ux ju0 = y] =
y under the given
correlation structure (Kotz and Nadarajah, 2004), hence
E 1 u0 < Fu01 ( ) ux = E ux ju0 < Fu01 ( ) = E E [ux ju0 ] ju0 < Fu01 ( )
=
= E u0 ju0 < Fu01 ( ) = E u0 ju0 < Fu01 ( ) ! Z Fu 1 ( ) Z Fu 1 ( ) 0 0 fu0 (y) y dy = yfu0 (y) dy: 1 Fu 0 F u 0 ( ) 1 1
Assistant Professor of Economics, University of Washington. Address: 336 Savery Hall, Box 353330, Seattle, WA98115. Phone: 206-543-4532. Email:
[email protected].
1
Table A.1 shows the computed values of
( ) = ( (1
)
xx )
1
E 1 u0t < Fu01 ( ) uxt ,
where the truncated means of t-distributions are calculated by the numerical integration using R software (Nadarajah and Kotz, 2007). Table A.1: Various
( ) ’s with
=
0:95
bivariate t-distribution with degrees of freedom =
3
4
10
200
1000
Normal
= 0:5
-0.6050
-0.6717
-0.7347
-0.7570
-0.7578
-0.7580
= 0:25
-0.5843
-0.6395
-0.6844
-0.6969
-0.6970
-0.6972
= 0:05
-0.4875
-0.4936
-0.4694
-0.4506
-0.4498
-0.4496
Now using the ordinary QR regression residuals u ^0t and the autoregression residuals u ^x from (4.1) in the main text, we estimate ^( ) = q
1 n 1
P
1 (^ u0t < 1) u ^xt
Vd ar (1 (^ u0t < 1)) Vd ar (^ uxt )
where Vd ar ( ) is the usual sample variance. Table A.2 con…rm the reliable performances of ^ ( ). Table A.2: Estimation Performances of ^ ( ) with = 0:95 (n = 700, S = 1000) Empirical mean of ^ ( ) from 1000 replications; true ( ) ’s are in Table 1 =
1.2
3
4
10
200
1000
Normal
= 0:5
-0.6222
-0.6736
-0.7340
-0.7557
-0.7566
-0.7566
= 0:25
-0.6008
-0.6409
-0.6823
-0.6957
-0.6952
-0.6952
= 0:05
-0.4974
-0.4913
-0.4679
-0.4506
-0.4506
-0.4506
Look Up Table for IVX Filtering
Table A.3 here provides a look-up table for c ( ; n) according to the various values of j ( )j.
2
Table A.3: Look-Up Table for IVX Filtering Parameters
1.3
Discussion on linear predictive QR model
Here I discuss predictive mean and quantile regression. Then I show the conditional heteroskedasticity (CHE) of asset return could be treated di¤erently between the mean and quantile predictive regressions. One proper accommodation of CHE in predictive QR models is discussed, and some earlier empirical results are revisited. 1.3.1
Predictive mean and quantile regression
The model (2.6) in the main text analyzes other quantile predictability as well as median of yt . Under the assumption of symmetrically distributed errors, (2.6) with
3
= 0:5 reduces to (2.1) in the
main text since Qyt (0:5jFt yt
0 1;
0;
xt
1,
1)
= E (yt jFt
one obtains
yt =
0
1
= (1; x0t
0 1) ,
0 1;
+
0;
= where Xt
Letting the QR innovation u0t := yt Qyt ( jFt
1 ).
Xt
1
xt
1
+ u0t
1)
=
(1.1)
+ u0t ;
and Qu0t ( jFt
1)
=0
(1.2)
by construction. In fact, (1.2) is the only moment condition so innovation with in…nite mean or variance may be allowed. The robustness of QR to thick-tailed errors enables better econometric inference than mean regression for …nancial applications using heavy-tailed data. To see the relation of (2.1) of main text and (1.1), recenter the innovation of (2.1) as u0t = u0t
Fu01 ( ), then (1.2)
follows, and yt =
0;
0 1 xt 1
+
+ u0t and
=
0;
0
+ Fu01 ( ) ;
which is the construction of Xiao (2009). 1.3.2
Predictive mean regression under CHE
The mean regression with CHE can be stylized by the following model: yt =
0
+
1
0
xt
+
1
t 1 "0t ;
"0t
and it describes CHE of asset return (yt ); under the null ( V art
1 (yt )
= Et
2 2 t 1 "0t
1
mds (0; 1) 1
(1.3)
= 0) ;
=
2 t 1:
The conditional variance process can be, for example, GARCH/ARCH or stochastic volatility models. The CHE a¤ects the variance estimation of
= (
0;
1
0 )0 ,
thus we will need a proper
treatment such as a lag kernel HAC type variance estimation in this mean predictive regression. 1.3.3
Predictive QR under CHE
A linear predictive QR model is de…ned as: yt =
0;
+
1;
0
xt
1
+ u0t
(1.4)
with Qu0t ( jFt The restriction (1.5) leads to
(u0t )
describes the heterogeneous e¤ect of 1; x0t
1)
= 0:
mds (0; (1 1
0
(1.5) )) : The parameter
across the quantiles.
4
=
0;
;
1;
0 0
CHE in (1.3) can be embedded in the model (1.4). Taking the conditional quantile on (1.3); Qyt ( jFt and Q"0t ( jFt
1)
1)
=
0
+
1
0
xt
1
+
t 1 Q"0t
( jFt
1) ;
6= 0; hence is not correctly centred to satisfy (1.5). De…ning F"0 1 ( ) ;
"0t = "0t one has Q"0t ( jFt
1)
= 0 (adding CHE into Xiao(2009)’s construction). The model (1.3) now
becomes
yt = : Now
t 1 "0t
0
+
=
0
1
0
+
xt
1
0
xt
1
+ F"0 1 ( ) 1
+
t 1 t 1
;
+
t 1 "0t
;
(1.6)
+ u0t :
(1.7)
:= u0t satis…es (1.5), so (1.6) can be a special case of (1.4). Therefore, (1.4) can
accommodate CHE by including a proxy of (square root of) conditional stock variance ^ t
1
as one
of the regressors. Note that CHE process for u0t (hence u0t ) is allowed in our framework by Assumption 2.1-(i). For exposition, let’s assume a Normal GARCH(1,1) process (suppressing u0t = ut ), then ut = t
=
t 1 "t ;
q
0
"t
+
iidN (0; 1);
2 1 ut 1
+
2 2 t 1;
t
> 0 w.p.1
With standard normal CDF( ) and pdf ( ) notation, for any a 2 R; Fu;t
1 (a)
= Pr "t <
a t 1
fu;t fu;t
1 (a) 1 (0)
= =
1
a
t 1
t 1
1 t 1
hence fu;t
1 (0)
(0) '
jFt
1 t 1
1.4
t=1
; and
t 1
; and 0:4 t 1
=q
0:4 0
+
2 1 ut 2
+
2 2 t 2
is clearly not constant across time (Bfu0 (r) in Assumption 2.1-(i) is not degenerate).
The conventional weak dependence condition on n
a
=
1
2 t
(e.g., strong mixing) still allows FCLT for
, hence satisfying Assumption 2.1 -(i).
Dequantiling Problem
I show, the dequantiled dependent variable in (3.1) of the main text, asymptotically does not cause any problem in the IVX-QR implementation. This is a special feature of IVX instrumentation. To see this, …rst consider the ordinary nonstationary quantile regression, if we use the pre-estimated
5
p
n-consistent estimator b 0; , n
1X xt n t=1 n X
1 n
=
1
n
yt
0;
1;
1X xt n
1
b
yt
1
t=1
n 1 1 u0t
xt
t=1
b
0;
and by taking conditional expectation Et n
1X xt n
=
xt
1 Et
t=1
fu
n
b
1
t=1
hence the e¤ect of replacing
0;
1(
o 0) ;
1;
xt
1
):
n 1 1 u0t
1X (0) xt n
1 (u0t
0;
b
0;
0;
0;
n 1 X
=O
0;
o 0)
1 (u0t
n3=2
xt
1
t=1
!
= Op (1);
by b 0; is still present in the limit.
0;
On the other hand, in the IVX-QR procedure: n X
t=1 n X
=
t=1
Assume Et
1(
et Z
et Z
yt
1
1
t=1
n 1 1 u0t
b
0;
et Z
1 (u0t
0;
b
yt
1
o 0) :
0;
1;
xt
1
(1.8)
): 1 n
1+ 2
n X
Pn
~t 1 = t=1 z Pn (1+ =2) ~t 1 t=1 z
z~t
n 1 1 u0t
1 Et
t=1
n X
1
fu (0)
n
1;
n X
= 1 for simplicity, since other cases will be similar. By taking conditional expectation
=
since
0;
xt
n
1+ 2
1
t=1
1
Op n 2 + = Op n
z~t
b
b
0;
0;
0;
=O
0;
o 0)
1 (u0t 1 n1+ 2
n X
z~t
t=1
1
!
= op (1);
from the signal strength of mildly integrated process (MPa ), so (1
)=2
= op (1). Now it is easy to show the predictable quadratic
variation is degenerate, i.e., 1 n1+
n X
Et
1
~t0 1 2t 1z
z~t
t=1
1 n1+
sup Et
t=1
!
0;
1 (u0t
0)
n X
z~t
~t0 1 1z
1
2 t
= op (1);
where t
= 1 u0t
b
0;
p using the consistency of b 0; . Therefore (1.8) is op (1) and using the pre-estimated n-consistent 6
estimator b 0; does not have any e¤ect asymptotically in the IVX-QR implementation.
1.5
Intercept Inference
Assume
= 1 for simplicity. De…ne ^
= arg min
0;
0
X
^ IV XQR 0 xt
yt
1;
1
;
0
then it is easy to show ^ IV XQR 0;
=
0;
"
n X
fu0t
#
1( n X
;t 1 (0)
t=1
t (u0t
)+
^ IV XQR 1;
n X
1;
t=1
fu0t
;t 1 (0) xt 1
t=1
)
hence n
^ IV XQR 0;
2
= n
0;
n
^ IV XQR 1;
1+ 2
fu (0)
1;
^ IV XQR
1+ 2
Z
1;
1;
n 1 X
n3=2
1
#
n
1X fu0t n
;t 1 (0)
t=1
0;
;n
^ IV XQR
1+ 2
1;
1;
available and the inference can be done through the self-normalization.
1.6
1
+ op (1)
Jxc .
n IV XQR n 2 ^ 0;
Therefore, joint mixed normal limit theory for
xt
t=1
!"
o
is
Empirical Motivation of Predictive QR
To understand the empirical motivation for predictive QR, I begin with some selected stylized facts on …nancial assets. Cont (2001) notes that …nancial asset returns often have (i) conditional heavy tails, (ii) time-varying volatility, and (iii) asymmetric distributions. In stock return prediction tests, these stylized facts may motivate the use of QR, as now discussed. To illustrate I consider a simple model for an asset return yt : yt = so that
t 1
and
uses
t 1
=
0
+
1)
= 0, then E (yt jFt
1 xt 1
with predictor xt
suggest that the magnitude of
1
volatility model:
1 xt 1 .
t 1
+
t 1 "t
with "t
iid F" ;
(1.9)
signify the location (mean) and scale (volatility) conditional on Ft
t 1
we assume E ("t jFt
t 1
=
0
+
1)
1.
=
t 1
1.
If
and the predictive mean regression model
Inconclusive empirical results from mean regressions
is small with most predictors xt If a predictor xt
stock returns yt , as many economic factors do, then
1 1
1.
Now also assume a stochastic
signi…cantly in‡uences the volatility of 6= 0. In fact, the conditional variance
often shows a greater systematic (or cyclical) variation than the conditional mean of stock returns, suggesting that j 1 j > j
1 j.
The predictive QR now reduces to
Qyt ( jFt
1)
=
0
+
1 xt 1
+(
0
+ 7
1 xt 1 ) Q"t
( )=
0;
+
1;
xt
1
where =
+
0 Q"t
When "t is close to symmetric,
1;
martingale stock return,
'
1 Q"t
0;
1;
'
0
( ) and with
1
=
1;
1
+
1 Q"t
= 0:5. When
1
( ) and the absolute value of
( ): is small, such as for a near1;
increases as
! 0 (or
! 1). This tendency may explain why the location (mean/median) of stock returns is more
di¢ cult to predict than other statistical measures, such as the scale (dispersion). Even when not negligible, the magnitude of when "t
1 Q"t
1
is
( ) would help in …nding the predictability. Note also that
iid t ( ) in (1.9), the absolute value of
1;
with
6= 0:5 increases as
decreases. The
location of stock return predictability varies depending on the magnitude and sign of (
1,
1 ),
but
QR can e¤ectively locate it. Therefore, QR provides forecasts on stock return quantiles where the actual predictability is more likely to exist. This illustrates the rationale for using QR to predict stock returns with thick tails (smaller ) and cyclical movements in conditional volatility (non-zero 1 ).
Lastly, consider an asymmetrically distributed "t in (1.9) with jQ"t ( )j > jQ"t (1
< 0:5. In this case, predictor variables with nonneglible (
1,
1)
)j for some
may predict lower quantiles of
stock returns better than upper quantiles. These asymmetric predictable patterns and quantilespeci…c predictors can be detected through predictive QR. Although the above example is stylized, we clearly see the advantages of QR in a stock return regression framework. This motivation is along the same line with the discussion in Xiao (2009) and Maynard et al. (2012). Many of the results hypothesized above are con…rmed in the empirical results in the main text.
1.7
Mild Integration, Long Memory and IVX
It is well known that the stationary long memory property in the form of (1
L)d xt = uxt , d 2 (0; 1=2)
(1.10)
is frequently observed in economic time series. In this section, I show the general form of (MI) speci…cation can nest the stationary long memory process, thereby validating IVX …ltering procedure over the stationary long memory regressor. The key feature of (1.10) is the polynomial decay rate of impulse response, i.e., @xt @uxt
=O j
1 j1 d
;
and the impulse response of the general (MI) process is @xt @uxt
= j
1+
Cn n
j
; Cn < 0 and
8
1 Cn + = o(1): Cn n
By equating the two quantities, we have 1+
j
Cn n
' exp Cn
j n
'
1 j1 d
;
j ' (d 1) log j; n n Cn ' (d 1) log j j
Cn
and using j = bnrc ' nr for large n with r 2 (0; 1) ; Cn =
(d
1)
log nr = O (log n) :
r
As a result, (MI) speci…cation with Rn =
1+
C0 n= log n
=
C0 kn
1+
with kn = o(n) essentially
captures the stationary long memory behavior of regressors. From the discussion of Section 3.1, the IVX persistence Rnz = 1 + 1+
C0 n= log n
since
n log n n
= o(1) for any
Cz n
is less persistent than
2 (0; 1). Therefore, IVX …ltering can successfully
reduce the persistence of long memory regressor. It con…rms the validity of IVX-QR procedure over (I0)-(ME) regions which nicely includes the long memory parameter space.
1.8
Mixed roots case
In (2.2), we could allow di¤erent
0s
and c0 s between predictors, as long as they are within I(0), MI
and I(1) cases. For I(0), MI andI(1) multiple predictors with di¤erent mechanism (through
i
the automatic correction
^ ) will reduce to the IVX AR(1) parameter of (
Rnz = IK + diag(n where
0 s,
1^
)
cz1 ; n
(
1^
)
cz2 ; :::; n
(
1^
)
czK )
could be 0 (I(0)), in such case j1 + ci j < 1 is imposed. IVX-QR limit theory will still
hold with some adjustment of notation, such as moment matrix Vcxz (having a similar Rieman integration form as long as ( ni ^ n ~ n which could be now matrices D
)
! 1 for all i, which is true by construction) or normalizing
~ n = diag n D
(
1^
)=2
;n
(
2^
)=2
; :::; n
(
K^
)=2
and so on. However, including (ME) case with other categories together is more delicate because IVX …ltering is essentially nulli…ed for (ME) case. We have drastically di¤erent signal strengths between ME predictor and others from I(0), MI and I(1), even after the IVX …ltering. This case in the mean regression is carefully treated in Section 3 of Phillips and Lee (2014), where some rate condition is provided. The same result is expected to hold for QR.
9
2
Selected Proofs
I rely on a few results from Magdalinos and Phillips (2012a,b; MPa and MPb , respectively) for selected proofs. Proof of Lemma 7.1. All proofs for (I0) cases are standard so omitted, see e.g., Koenker (2005, Theorem 4.1). 1. For (I1) case, from the functional law (2.4) and (2.5) from the main text, Gx;n = Dn 1
n X
Xt
(u0t ) =
1
t=1
"
1 n
For (MI) case, Dn 1
n X
Xt
2
(u0t ) = 4
1
t=1
where Vxx :=
Dn
1
n X
E
t=1
R1
erC
0
h
p1 n
rC xx e dr,
2
(u0t )
Pn
Pn
1
Pn
t=1
t=1 xt
1+ n 2
Pn
p1 n
(u0t )
t=1
t=1 xt 1
(u0t )
(u0t ) (u0t )
1
#
3
5 =) N
=)
"
R
B
#
(1)
Jxc (r)dB
0; (1
)
"
1
0
0 Vxx
:
#!
;
since
Xt 1 Xt0 1 jFt 1
i
Dn
1
=
"
(1 ) Pn 0 (1 ) 1+ 2 t=1 xt 1 n (1 ) Pn (1 ) Pn 0 t=1 xt 1 t=1 xt 1 xt n1+ n1+ 2 " # 1 0 (1 ) ; 0 Vxx
p !
1
#
using the result of MPa : 1 n
1 + 2
n X
x0t 1
= Op (1) and
t=1
1 n1+
n X
xt
p 0 1 xt 1 ! Vxx :
t=1
Therefore, the stability condition for Martingale (MG) CLT is established (e.g., Hall and Heyde, 1980). The conditional Lindeberg condition is easy to show considering that
(u0t ) is mds, so is
omitted. Now for (ME) case, Dn
1
n X
Xt
(u0t ) =
1
t=1
where V~xx :=
R1 0
e
"
Pn p1 (u0t ) t=1 n P n 1 n (u0t t=1 xt 1 n Rn
rC Y Y 0 e rC dr C C
and YC
N 0;
similar to (MI) case so is omitted.
10
R1 0
) e
#
"
=) M N 0; (1
rC
xx e
rC dr
)
"
1
0 ~ 0 Vxx
##
:
as in MPa : The proof is
2. For (I1) case, Mx
Dn 1
=
;n
2 4
=
"
=)
n X
fu0t
0 1 ;t 1 (0) Xt 1 Xt 1 Dn
t=1 1 n
1 3 n2
fu0
Pn
Pn
1
t=1 fu0t ;t 1 (0) 1 n2
t=1 fu0t ;t 1 (0) xt 1
3 n2
Pn
Pn
0 t=1 fu0t ;t 1 (0) xt 1
0 t=1 fu0t ;t 1 (0) xt 1 xt 1
# R fu0 (0) Jxc (r)0 fu0 (0) R R ; (0) Jxc (r) fu0 (0) Jxc (r)Jxc (r)0
3 5
where, from the standard FCLT and CMT (Phillips, 1987), n 1 X
n
3 2
fu0t
0 ;t 1 (0) xt 1
n 1 X
=
n
t=1
3 2
(fu0t
fu0 (0)) x0t
;t 1 (0)
t=1
= fu0 (0)
n X
x0t p
t=1
= fu0 (0)
n X
x0t p
t=1
n
1 1 X (fu0t +p n n t=1
1
n
1 + Op n
1
n
1 p n
Other elements can be shown similarly, hence omitted. For (MI) case, using the same method and the fact n Mx
;n
= Dn 1 =
"
n X t=1
1 n
1 1+ 2
n
fu0t
;n
=
Pn
1
t=1 fu0t ;t 1 (0) Pn t=1 fu0t ;t 1 (0) xt 1
=
=)
Dn 1 2 4
"
n X
fu0t
t=1
1 1 n2+
0 t=1 xt 1
x0t
1
t=1
fu0 (0))
xt p
n
=) fu0 (0)
Pn
3 2
Z
1
n
Jxc (r)0 :
= Op (1) again,
n 1
n1+
1 n
Pn
0 t=1 fu0t ;t 1 (0) xt 1 Pn 0 t=1 fu0t ;t 1 (0) xt 1 xt 1
1+ 2
#
!p
"
fu0 (0)
0
0
fu0 (0) Vxx
0 1 ;t 1 (0) Xt 1 Xt 1 Dn
Pn
t=1 fu0t ;t 1 (0) Pn n Rn t=1 fu0t ;t 1 (0) xt 1
fu0 (0)
0
0
fu0 (0) V~xx
Proof of Lemma 7.2.
1=2
n
;t 1 (0)
p
n 1 X
0 1 ;t 1 (0) Xt 1 Xt 1 Dn
(ME) case will be shown in the exact same way, Mx
1 + fu0 (0)
#
:
1
Pn
0 n t=1 fu0t ;t 1 (0) xt 1 Rn P 1 Rn n nt=1 fu0t ;t 1 (0) xt 1 x0t 1 Rn n n2 1 n2+
1. (i) For (MI) and (I1) cases with 0 <
11
3 5
< min( ; 1), from MPb
#
:
(Proposition A2 and lemma 3.1) n X
1 n
1+ 2
z~t
t=1
n
1+ 2
n
1+ 2
;n
=
Pn
t=1 nt
zt
t=1 n X
1
= then G
n X
1
(u0t ) =
1
(u0t ) +
1
zt
C n
1+ 2
+
n X
nt 1
(u0t )
t=1
(u0t ) + op (1)
1
t=1
+ op (1) where n X
:=
nt
t=1
n X
1 n
t=1
zt
1+ 2
(u0t ) :
1
The Stability condition for MGCLT is easy to show: n X
0 nt nt jFt 1
E
t=1
For any
)
n1+
zt
0 1 zt 1
t=1
!p (1
x )Vzz :
>0 n X
=
= (1
n X
1
t=1 n X
t=1 n X t=1
h E k
h E k
h E k
n1+ +
n1+
= op (1) + = op (1);
2 nt k 1 j
t=1
1
nt k > ) jFt
2 nt k 1 kzt
n X
1
=
2 nt k 1 (k
kzt
n X t=1
(u0t )k > n
1
1+ 2
(u0t )j > n 4 jFt
2 1k
kzt
1
i
!
h E j
2 1k 1
n
1 X zt 1 n n2 t=1
jFt 1
i
(u0t )j2 1 j
kzt 2
1k
2
>
zt
1
1
+
1
i
n X t=1
h E k
(u0t )j2 > !
2 1+ 2
n
2
> n
1
2
n2
h E j
!!
1
2 nt k 1 kzt 2
n 2 jFt
(u0t )j2 jFt (1
+ 1 k > n 2 4 jFt
1
i
1
1
i
i
)
con…rming the conditional Lindeberg condition. Thus, from MGCLT: G (ii) For (MI) and (I1) cases with 1 n
1+ 2
n X t=1
z~t
1
;n
=) N (0; (1
x )Vzz ):
2 (0; ), again from MPb (Proposition A2 and lemma 3.5) (u0t ) =
1 n
1+ 2
12
n X t=1
xt
1
(u0t ) + op (1);
and by the similar procedure above, G
=
;n
n X
1 n
t=1
(iii) If xt
1
;n
xt
(u0t ) =) N (0; (1
1
n
1 X =p z~t n t=1
1 X )= p xt n t=1
(u0t
1
2. (i) For (MI) and (I1) cases with 0 < M
;n
=
n X
fu0t
;t
~ 1 (0) Zt
~0 1;n Zt
1
n1+
n X
=
1;n
0 ;t 1 (0) zt 1 zt 1
fu0t
t=1
= fu0 (0)
1 n1+
n X
zt
0 1 zt 1
t=1
(u0t ) + op (1) =) N (0; (1
1
!
1 n1+
n X
1
=
n 2 + t=1 n X fu0t t=1
hence 1 n1+
n X
(fu0t
(fu0t
+
1 n1+
;t 1 (0)
;t 1 (0)
p
;t 1 (0)
fu0t
~t 1 z~t0 1 ;t 1 (0) z
t=1
n X
(fu0t
zt 1 n =2
fu0 (0)) zt
fu0 (0)
n
fu0 (0)) zt
t=1
13
;t 1 (0)
fu0 (0)) zt
0 1 zt 1
+ op (1)
t=1
where we used Assumption 2.1-(i) and the result supt n X
xx ) :
+ op (1) (from MPb , lemma 3.1-(iii))
x = fu0 (0) Vzz + op (1)
1
)
< min( ; 1), note that
t=1
=
)Vxx ) :
belongs to (I0), using Lemma A.2 in Kostakis et al. (2012), it is easy to show that n
G
1+ 2
zt 1 n =2
0 1 zt 1
= Op (1) from MPa so that 0 1 zt 1
zt 1 n =2
0
= Op (1);
1 = Op ( p ) = op (1): n
(ii) For (MI) and (I1) cases with M
=
;n
n X
fu0t
2 (0; ) ;
~0 ~ ;t 1 (0) Zt 1;n Zt 1;n
=
t=1
n X
1
=
n1+
fu0t
0 ;t 1 (0) xt 1 xt 1
n X
1 n1+
= fu0 (0)
n1+
n 1 X +p n t=1
n X
+ op (1) (from MPb , lemma 3.5-(ii))
0 ;t 1 (0) xt 1 xt 1
fu0t
t=1
fu0t
fu0 (0)
;t 1 (0)
p
~t 1 z~t0 1 ;t 1 (0) z
t=1
t=1
1
fu0t
!
xt 1 n =2
n
xt 1 n =2
0
+ op (1)
= fu0 (0) Vxx + op (1): (iii) If xt
1
belongs to (I0), using Lemma A.2 in Kostakis et al. (2012) again, n
M
;n
=
n
1X fu0t n
;t
~t0 1z
~t 1 (0) z
1
1X fu0t n
=
t=1
t=1
3. (i) For (MI) and (I1) cases with 0 < M
;n
=
n X
fu0t
0 ;t 1 (0) xt 1 xt 1
= = =) =
n1+
0 ~ ;t 1 (0) Zt 1;n Xt 1;n
n X
( (
~t 1 x0t 1 ;t 1 (0) z
fu0t
= fu0 (0)
t=1
fu0 (0)
xx
< min( ; 1),
t=1
1
+ op (1) !p fu0 (0)
(
1 n1+
fu0 (0)
n X
zt
t=1
Cz
1
R
fu0 (0) fu0 (0)
Cz
1
fu0 (0)
1C
Cz n1+
0 1 xt 1
dBx Jxc0 +
(
n X
n1+
xt
Cz 1 C
1
1 CV
R
Cz xx Cz xx ; R R c0 c c0 xx + dBx Jx + C Jx Jx Cz
1f
xx
+ CVxx g ;
z~t 1 x0t 1
t=1
0 1 xt 1
t=1
xx
n X
1
)
dBx Jxc0
+C
Z
Jxc Jxc0
14
=
Z
+ op (1) as earlier,
+ op (1) (from MPb , lemma 3.1-(ii))
Jxc Jxc0
;
if if
;
if if
and using the fact dJxc = dBx + CJxc from Ito calculus: Z
)
dJxc Jxc0 :
=1 <
=1 <
<1
<1 := fu0 (0)
cxz .
(ii) For (MI) and (I1) cases with M
n X
=
;n
2 (0; ),
0 ~ ;t 1 (0) Zt 1;n Xt 1;n
fu0t
t=1
n X
1
=
n1+
(
1 n1+
=) fu0 (0) Vxx . (iii) If xt
1
n X
xt
0 1 xt 1
;n
=
)
+ op (1)
t=1
(
1 n1+
n X
z~t 1 x0t 1
t=1
)
+ op (1) as earlier,
+ op (1) (from MPb , lemma 3.5-(ii))
belongs to (I0), using Lemma A.2 in Kostakis et al. (2012) again, n
M
= fu0 (0)
t=1
fu0 (0)
=
fu0t
~t 1 x0t 1 ;t 1 (0) z
1X fu0t n
n
~t 1 (0) z
;t
0 1 xt
=
1
t=1
1X fu0t n
0 ;t 1 (0) xt 1 xt 1
t=1
+ op (1) !p fu0 (0)
xx
Proof of Lemma 7.3. WLOG, we assume K = 1: I use the following standard argument: sup =
fj
j n
S
j=1;:::;J
=
g sup
(1+ )=2 C
fj j n
max @
fj
max @
fj
max @
j=1;:::;J
j=1;:::;J
=
where j j
n
0
j=1;:::;J
0
0
(1+ )=2 C
jGn ( )
(1+ )=2 C(
Gn (0)j
j )g
jGn ( )
sup j n
(1+ )=2 C(
j n
(1+ )=2 C(
fj j n
(1+ )=2 C(
j)
g
sup j)
g
sup j )g
( j ) : j = 1; :::; J
Gn (0)j
jGn ( )
1
Gn (0)jA
jGn ( )
Gn (lj )j + jGn (lj )
jGn ( )
1
1
Gn (0)jA
Gn (lj )jA + max jGn (lj ) j=1;:::;J
Gn (0)j
is a collection of closed intervals with a length
union is the original shrinking neighborhood
j j
n
(1+ )=2 C
j
whose
(see below). lj is the lowest
vertex of each interval. Therefore, we show (i) maxj=1;:::;J jGn (lj )
Gn (0)j = op (1); and (ii)
maxj=1;:::;J supfj j n (1+ )=2 C( j )g jGn ( ) Gn (lj )j = op (1). (i) maxj=1;:::;J jGn (lj ) Gn (0)j = op (1): It su¢ ces to show that
Gn (cn ) for cn =
c 1+ n 2
argument: if (
with any …xed jcj
nt ; Ft )
Gn (0) = op (1)
(2.1)
C; hence a pointwise convergence. I will use the following
is martingale di¤erence array (mda), and the (predictable) quadratic variation
15
Pn
0 nt nt jFt 1
t=1 E
= op (1); then n X
= op (1):
nt
t=1
Let
nt
1
:= n t
1+ 2
z~t
:=
1 t
where
(u0t
cn xt
1)
Et
1[
(u0t
cn xt
then Gn (cn )
Gn (0) =
1 )]
n X
(u0t ) + Et
1[
(u0t )]
nt
t=1
and by construction
t
are mds and
nt
are mda.
Note that t
= 1 (u0t < 0)
Given Ft
1;
1 (u0t < cn xt
(a) assume cn xt t
=
1
1)
+ Pr (u0t < cn xt
1 jFt 1 )
Pr (u0t < 0jFt
1) :
> 0; then
1 (0
u0t < cn xt
= 1 (0
u0t < cn xt
1)
+ Pr (0
u0t < cn xt
1 jFt 1 ) ;
+ fPr (0
u0t < cn xt
and 2 t
2 Pr (0
1)
u0t < cn xt
1 jFt 1 ) 1 (0
1 jFt 1 )g
u0t < cn xt
2
1) ;
hence E
2 t jFt 1
= fPr (0
u0t < cn xt
1 jFt 1 )g f1
fPr (0 = Op thus supt E
2 t jFt 1
u0t < cn xt 1 jFt 1 )g p 1 cn n = Op ( ) for all t; n2
t=1
E
fu0t
u0t < cn xt
;t 1 (0)cn xt 1
1 jFt 1 )g
+ op (1)
= op (1):
It is easy to show for the case (b) cn xt n X
Pr (0
2 nt jFt 1
=
1 n1+ a:s
0 given Ft
1
n X
z~t2 1 E
t=1
t
in a similar way. Therefore
2 t jFt 1
2 t jFt 1
sup E
1
1 n1+
n X t=1
con…rming (2.1). (ii) maxj=1;:::;J supfj
j n
(1+ )=2 C(
j)
g jGn ( ) 16
Gn (lj )j = op (1).
z~t2
1
!
= op (1);
For each j; sup fj
(1+ )=2 C(
j n
=
j)
g
jGn ( )
fj
(1+ )=2 C(
j n
j )g
n
(1+ )=2 C(
j n
+
1+ 2
j)
gn
1+ 2
fj
j n
n
j )g
(1+ )=2 C(
Now since 1 (lj xt
t=1
1+ 2
j~ zt
1 j 1 (lj xt 1
t=1
u0t < xt
1
1 f1 (lj xt 1
n X
1
sup
z~t
t=1
n X
1
sup fj
n X
1
sup
Gn (lj )j
j~ zt
1)
u0t < xt u0t < xt
1 j Pr (lj xt 1
1
(1+ )=2
n
1)
u0t < xt
1
1 jFt 1 )g
1)
u0t < xt
j xt 1
Pr (lj xt
1 jFt 1 ) :
(1+ )=2
u0t < n
j xt 1
from the monotonic-
ity, sup fj j n 1 n Let
nt
1
= n
1+ 2
n X
j~ zt
Et
1+ 2
1j 1
n
1+ 2
uniformly in
t=1
j,
j~ zt
1
=
t=1
1j
1j 1
(1+ )=2
n
1 [ nt ]
n 2Mf X
t=1
j~ zt
n
1+ 2
jn
n X t=1
1+ 2
n
j )g
(1+ )=2 C(
n X
n X
1
(1+ )=2
n
j xt 1
j~ zt
(1+ )=2
t=1
1
where we use that fu0t
1 j 1 (lj xt 1
j xt 1
n
(1+ )=2
u0t < xt (1+ )=2
u0t < n
(1+ )=2
u0t < n
1 j Pr
xt
j~ zt
j xt 1
j xt 1
!
1)
j xt 1
:
; then
u0t <
jn
(1+ )=2
xt
1 jFt 1
= Op ( j ) ;
;t 1 (
) is bounded above a.s (in the local neighborhood of
zero) by Mf from Assumption 2.1-(ii). We can also show
n X
Et
1
2 nt
= op (1);
t=1
as before. Thus, 1
sup fj j n
(1+ )=2 C(
n
j )g
1+ 2
n X t=1
j~ zt
1 j 1 (lj xt 1
u0t < xt
!
1)
= op (1);
Similarly we can show 1
sup fj
j n
(1+ )=2 C(
j)
g
n
1+ 2
n X t=1
j~ zt
1 j Pr (lj xt 1
17
u0t < xt
!
1 jFt 1 )
= op (1);
con…rming (ii).
3
Supplementary Numerical Results
This section collects some additional numerical results such as size and power performances and empirical works, that were contained an earlier working paper version of this paper. IVX-QR performances for various quantiles (Tables A.6,A.7 and A.9) and data work (Tables A.10-13) are based a particular choice of main text, where
(either 0.5 or 0.6). It will be informative to compare the results of the
is chosen by the practical rule from Section 4.1. Power comparisons (Figrues
A.1-4) employed the IVX-meant tests rather than the modi…ed CY-Q.
3.1
Size and Power Performances Table A.3: Finite sample sizes of IVX-QR and CY-Q tests (n = 100) R=1 S = 2500
c=0
R = 0:98 c=
2
R = 0:9
R = 0:8
R = 0:5
c=
c=
c=
10
20
50
R=0 c=
100
=
IVXQR
IVXQR
IVXQR
IVX-QR
IVXQR
IVXQR
0.95
0.0724
0.0624
0.0564
0.0456
0.0492
0.0376
0.85
0.0552
0.0584
0.0468
0.0472
0.0412
0.0392
0.75
0.062
0.0572
0.0396
0.0456
0.048
0.0304
0.65
0.0608
0.0556
0.0428
0.0504
0.0324
0.0412
0.55
0.0444
0.0424
0.0456
0.0484
0.0544
0.0344
CY-Q
0.05
0.046
0.0608
0.0792
N/A
N/A
Table A.4: Finite sample sizes of IVX-QR and CY-Q tests (n = 200) R=1 S = 2500
c=0
R = 0:99 c=
2
R = 0:95
R = 0:9
c=
c=
10
20
R = 0:5 c=
100
R=0 c=
200
=
IVXQR
IVXQR
IVXQR
IVXQR
IVXQR
IVXQR
0.95
0.0664
0.0616
0.052
0.0492
0.0556
0.042
0.85
0.0716
0.0604
0.0548
0.0564
0.0444
0.0376
0.75
0.0552
0.0652
0.044
0.0504
0.0416
0.0388
0.65
0.0492
0.0492
0.0492
0.0432
0.0544
0.0416
0.55
0.0472
0.0472
0.0448
0.0484
0.0412
0.0388
CY-Q
0.0492
0.0508
0.05
0.05
N/A
N/A
18
Table A.5: Size Performances(%) of Ordinary QR (n = 700, S = 1000) Normally distributed errors = c=0
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
14.5
13.5
15.0
15.5
16.3
17.8
17.0
16.3
16.0
13.1
14.5
c=
2
10.6
9.9
11.0
12.5
12.9
11.9
11.9
11.4
9.8
9.6
11.5
c=
5
8.7
8.1
8.3
8.4
7.7
9.4
8.9
9.8
7.5
11.0
10.6
c=
7
9.7
9.2
7.4
6.3
7.4
6.2
6.2
7.1
7.4
7.8
9.5
c=
70
6.7
6.1
6.0
4.2
5.2
3.8
4.2
4.6
4.6
5.6
8.2
t(4) errors = c=0
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
17.2
16.6
16.7
13.9
13.1
13.2
14.6
14.1
16.7
18.0
18.3
c=
2
13.0
13.4
11.0
9.9
9.8
9.7
8.1
10.5
13.8
12.8
16.3
c=
5
12.0
10.0
10.3
6.0
7.5
6.4
6.5
8.5
9.8
11.6
11.2
c=
7
10.5
10.4
8.3
6.8
5.7
6.7
5.6
8.1
8.1
10.8
10.1
c=
70
10.0
6.7
6.9
4.8
4.1
4.6
4.7
4.3
6.1
6.0
10.1
t(3) errors = c=0
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
18.1
17.0
13.5
12.5
12.3
11.5
11.9
12.9
14.4
17.1
17.6
c=
2
14.4
13.3
11.1
8.5
8.5
9.1
9.2
11.2
11.3
13.3
15.0
c=
5
11.8
11.5
8.1
9.1
6.6
8.1
7.4
6.5
8.4
11.4
12.9
c=
7
12.5
8.1
10.1
7.5
5.5
7.1
5.8
6.9
6.4
9.4
13.5
c=
70
10.0
8.9
5.4
4.8
5.5
3.6
4.2
4.7
6.5
7.8
9.3
t(2) errors = c=0
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
20.3
16.1
11.3
11.1
9.1
8.5
8.5
10.4
12.3
14.2
16.2
c=
2
15.8
12.4
10.7
8.3
8.6
5.8
6.7
8.1
11.8
12.0
15.7
c=
5
13.0
9.0
6.7
6.2
5.4
5.4
4.7
6.5
8.4
12.9
13.7
c=
7
12.2
10.7
7.8
5.6
4.1
3.9
5.8
5.3
7.9
10.7
13.9
c=
70
7.2
7.1
6.8
4.5
4.6
3.4
3.7
5.3
7.0
8.3
8.6
19
Table A.6: Size Performances(%) of IVX-QR with
= 0:5 (n = 700, S = 1000)
Normally distributed errors =
0.05
c=0
7.0
0.1 6.0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
6.9
4.6
5.1
4.5
6.7
6.0
6.5
5.9
8.0
c=
2
7.4
5.6
4.6
4.3
4.8
3.7
5.3
4.0
4.8
5.8
7.5
c=
5
6.6
5.5
5.1
4.2
4.8
3.2
3.7
4.5
4.7
5.2
6.8
c=
7
5.5
5.6
4.5
4.9
3.6
3.4
3.7
4.1
5.1
4.4
7.3
c=
70
4.9
4.8
4.6
4.0
3.1
3.4
3.7
4.1
3.9
4.8
7.1
c=
700
4.3
6.2
4.3
4.6
5.4
4.8
4.0
4.2
6.5
6.5
6.8
t(4) errors =
0.05
c=0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
9.1
7.5
6.0
5.9
6.4
4.5
6.0
6.1
6.5
8.8
9.5
c=
2
7.4
5.6
4.0
3.7
3.4
3.8
5.5
5.0
4.9
6.0
9.8
c=
5
7.5
7.2
5.2
5.0
3.5
3.3
3.4
3.4
5.2
6.0
7.5
c=
7
6.5
7.6
5.6
4.9
3.3
3.8
4.8
4.6
5.8
4.8
6.8
c=
70
6.5
5.6
5.2
4.1
3.5
3.9
3.2
3.8
5.4
5.6
7.1
c=
700
8.0
7.6
4.8
5.4
4.9
3.9
3.9
3.8
6.6
5.7
7.8
t(3) errors = c=0
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
9.8
10.2
7.8
6.4
5.2
5.1
4.8
4.8
6.7
7.6
11.1
c=
2
7.7
8.3
6.2
4.5
4.4
3.1
4.4
5.5
4.8
7.2
8.6
c=
5
6.5
7.0
6.0
4.6
3.7
4.1
3.5
3.0
5.2
6.0
6.5
c=
7
6.0
5.7
4.9
3.4
3.3
3.7
3.6
4.9
5.2
5.5
7.3
c=
70
7.2
7.3
6.1
4.1
4.0
2.8
4.4
5.5
5.0
6.1
5.2
c=
700
6.7
6.9
5.8
4.8
4.0
4.4
4.9
4.5
6.0
5.0
6.5
t(2) errors =
0.05
c=0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
10.5
8.9
6.3
5.8
4.7
4.6
5.3
6.4
8.0
5.9
7.7
c=
2
7.9
5.9
6.1
3.3
4.5
3.9
3.2
4.1
7.2
6.5
5.6
c=
5
5.3
5.9
6.0
6.0
3.7
3.3
5.3
4.2
7.1
6.5
6.2
c=
7
6.8
6.4
6.4
3.8
3.3
4.1
4.3
3.7
6.0
8.4
7.3
c=
70
4.9
7.9
6.6
4.6
3.9
4.3
4.1
4.8
6.3
5.9
5.2
c=
700
4.4
5.5
6.2
6.4
4.7
3.8
5.6
5.3
5.2
6.8
3.8
20
Table A.7: Size Performances(%) of IVX-QR with
= 0:6 (n = 700, S = 1000)
Normally distributed errors =
0.05
c=0
7.0
0.1 7.8
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
8.9
6.7
6.7
6.8
8.5
7.9
8.2
7.2
8.9
c=
2
7.6
6.0
5.1
5.0
4.6
4.1
5.1
4.1
5.3
5.7
7.2
c=
5
6.6
6.1
4.7
4.3
4.6
3.0
4.7
4.5
5.0
5.4
6.4
c=
7
7.1
5.9
4.9
5.3
4.1
4.0
4.1
4.4
4.7
4.9
6.5
c=
70
5.1
5.5
4.3
3.3
3.3
3.1
3.5
4.8
4.4
5.0
7.7
c=
700
4.7
5.7
4.4
4.6
5.5
4.7
3.7
4.1
7.0
7.4
6.8
t(4) errors = c=0
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
10.8
8.4
8.2
7.7
7.6
5.6
6.8
8.0
9.7
9.1
9.7
c=
2
8.0
6.7
5.8
4.4
4.4
4.8
5.4
6.2
6.3
6.5
10.3
c=
5
8.4
7.9
5.4
4.5
3.9
3.3
4.4
3.7
4.7
6.2
7.7
c=
7
7.5
8.1
5.5
5.0
2.5
4.0
4.8
4.8
6.2
5.5
7.4
c=
70
6.9
6.4
5.0
4.2
3.7
4.5
3.2
4.1
5.6
6.0
6.9
c=
700
8.0
7.2
4.9
5.6
4.7
4.1
3.7
4.1
6.7
6.0
7.7
t(3) errors = c=0
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
10.6
11.4
9.4
7.1
7.1
7.2
5.6
7.7
7.7
9.1
10.7
c=
2
9.1
7.6
6.0
5.8
4.8
3.9
4.9
5.4
5.7
7.8
8.2
c=
5
5.8
7.4
5.6
5.1
4.1
3.8
3.5
3.7
5.6
6.9
7.2
c=
7
8.2
6.5
5.1
4.2
4.3
4.2
3.5
4.3
5.7
5.0
7.8
c=
70
7.6
7.4
5.8
3.2
4.0
3.1
4.6
4.6
5.2
5.8
6.7
c=
700
7.0
6.8
5.5
3.9
3.9
4.3
5.3
4.4
6.5
5.3
6.5
t(2) errors = c=0
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
13.5
9.7
7.3
6.6
5.6
5.2
6.5
7.6
9.7
7.4
8.8
c=
2
8.1
6.6
6.6
4.4
4.3
3.9
3.1
4.7
8.6
7.9
5.5
c=
5
6.5
7.3
6.4
5.3
4.9
3.8
5.4
4.7
7.1
7.6
7.1
c=
7
8.0
7.7
6.2
4.6
3.3
3.8
4.4
4.3
6.7
8.2
7.5
c=
70
5.1
7.2
6.1
4.5
4.1
3.5
3.9
4.8
6.8
6.0
6.5
c=
700
3.8
5.1
5.6
6.2
4.8
3.9
6.5
5.4
5.7
6.8
4.1
21
Table A.8: Size Performances(%) of Ordinary QR (n = 700, S = 1000) Normally distributed errors = c1 = c2 = 0 c1 =
2, c2 = 0
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
14.0
12.5
13.2
14.6
13.7
13.7
13.2
12.1
13.4
11.4
15.4
12.7
11.5
10.1
10.9
9.2
9.8
11.6
13.0
11.4
10.5
12.8
c1 =
5, c2 =
0
13.5
10.4
9.0
8.1
9.6
6.9
8.1
8.5
8.0
10.1
13.2
c=
7, c2 =
0
9.6
9.9
7.2
8.4
7.7
7.4
7.0
7.5
8.4
9.2
10.1
c1 =
2, c2 =
2
11.3
10.4
9.3
9.2
9.5
10.5
9.6
8.8
10.1
10.4
12.6
c1 =
5, c2 =
2
10.5
9.5
8.1
8.5
6.4
7.2
9.3
9.9
7.9
9.9
11.4
c1 =
7, c2 =
2
12.7
10.9
8.5
6.3
6.7
5.6
7.7
6.8
7.5
9.8
11.9
c1 =
5, c2 =
5
10.2
9.0
7.8
6.8
7.0
8.7
8.1
8.5
8.0
9.1
11.1
c1 =
7, c2 =
5
11.0
9.5
6.7
7.7
5.9
6.3
8.4
8.1
7.0
8.9
11.3
c1 =
7, c2 =
7
10.1
7.6
6.9
6.7
6.3
7.8
8.2
8.1
6.6
8.4
11.0
t(4) errors = c1 = c2 = 0 c1 =
2, c2 = 0
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
21.8
16.0
14.2
10.5
11.6
9.6
11.3
12.4
12.3
15.2
19.2
15.4
12.5
10.2
8.1
8.6
9.7
9.6
9.2
10.3
12.8
16.4
c1 =
5, c2 =
0
15.1
12.3
8.9
8.8
7.6
6.7
7.6
7.6
9.8
12.0
14.1
c=
7, c2 =
0
13.5
12.2
8.2
6.7
5.3
6.1
6.7
7.5
8.2
12.4
17.3
c1 =
2, c2 =
2
17.6
12.9
9.8
8.5
7.8
7.4
8.7
8.2
10.5
13.2
18.2
c1 =
5, c2 =
2
13.5
11.8
8.8
6.0
6.4
8.1
7.8
8.8
8.5
11.1
14.7
c1 =
7, c2 =
2
15.2
11.1
6.7
7.9
6.5
6.0
6.8
7.1
9.4
10.8
13.3
c1 =
5, c2 =
5
15.5
11.5
9.9
7.2
5.7
5.6
6.0
7.9
8.6
10.7
16.4
c1 =
7, c2 =
5
14.2
11.4
8.6
5.9
6.0
6.7
7.5
8.6
9.1
10.1
13.3
c1 =
7, c2 =
7
14.7
11.1
9.9
7.3
4.7
5.1
5.8
7.7
7.5
9.6
15.9
t(3) errors = c1 = c2 = 0 c1 =
2, c2 = 0
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
21.1
16.3
13.0
11.4
8.7
9.0
9.4
10.6
12.3
14.4
19.5
20.9
12.4
12.5
9.1
8.1
7.3
8.6
8.6
11.0
15.7
18.5
c1 =
5, c2 =
0
17.1
12.9
9.3
7.2
7.4
6.4
6.8
7.5
9.8
13.9
17.9
c=
7, c2 =
0
17.3
11.9
8.3
7.3
4.5
5.9
5.6
6.9
8.4
12.7
14.6
c1 =
2, c2 =
2
19.0
14.0
9.8
9.0
6.8
6.1
7.8
9.7
10.8
11.9
18.1
c1 =
5, c2 =
2
19.4
12.3
8.8
6.9
7.5
5.5
6.8
6.6
10.3
15.3
15.5
c1 =
7, c2 =
2
16.0
12.5
8.2
7.0
6.0
6.2
6.4
7.5
10.0
12.5
17.2
c1 =
5, c2 =
5
15.0
12.1
6.6
7.4
6.3
5.8
5.3
8.1
9.2
8.9
17.0
c1 =
7, c2 =
5
16.9
11.2
8.8
6.8
6.3
5.1
5.8
5.3
9.4
14.7
14.5
c1 =
7, c2 =
7
14.4
11.7
6.3
6.7
6.4
4.9
4.6
7.3
8.6
9.4
16.6
22
Table A.9: Size Performances(%) of IVX-QR with
= 0:5 (n = 700, S = 1000)
Normally distributed errors = c1 = c2 = 0 c1 =
2, c2 = 0
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
10.8
8.4
8.0
7.8
6.4
6.5
7.0
6.3
8.0
6.1
10.4
10.3
7.9
6.6
3.9
5.2
5.0
5.6
5.4
5.6
6.8
7.6
c1 =
5, c2 =
0
9.9
8.3
6.5
4.8
4.5
5.2
5.4
4.2
5.3
6.6
10.4
c=
7, c2 =
0
9.8
4.7
5.0
4.9
5.3
5.0
5.2
5.3
4.7
6.6
7.6
c1 =
2, c2 =
2
8.0
5.8
5.5
4.7
5.3
5.4
4.9
4.1
7.8
4.3
8.5
c1 =
5, c2 =
2
8.6
6.3
5.0
3.3
4.4
3.9
3.9
3.5
3.9
6.6
6.8
c1 =
7, c2 =
2
8.7
6.9
6.0
3.9
3.5
3.9
4.4
5.0
4.1
4.7
7.5
c1 =
5, c2 =
5
7.3
5.3
4.6
4.8
4.4
5.3
4.6
3.8
7.1
5.6
8.4
c1 =
7, c2 =
5
7.7
6.1
4.5
3.2
4.6
4.0
3.7
3.3
4.5
6.8
5.9
c1 =
7, c2 =
7
6.7
5.7
4.7
4.9
4.0
4.6
4.2
3.9
6.3
4.9
8.0
t(4) errors = c1 = c2 = 0 c1 =
2, c2 = 0
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
16.5
13.0
8.0
6.9
6.0
5.2
6.7
8.0
6.4
11.6
13.9
11.9
9.0
7.5
5.9
4.8
5.1
4.6
5.6
6.7
8.8
13.1
c1 =
5, c2 =
0
12.3
8.0
6.9
4.9
5.3
3.5
5.3
5.1
6.3
9.0
11.2
c=
7, c2 =
0
10.9
11.4
7.5
5.7
4.6
4.6
5.0
5.1
5.6
8.3
11.8
c1 =
2, c2 =
2
13.7
10.1
6.6
4.1
3.8
3.8
4.3
5.2
5.4
7.9
11.5
c1 =
5, c2 =
2
10.3
8.0
6.4
4.1
4.0
4.0
4.6
4.7
5.1
8.0
9.6
c1 =
7, c2 =
2
10.7
7.9
5.9
3.5
3.0
3.0
3.8
5.4
5.7
7.7
10.7
c1 =
5, c2 =
5
11.7
8.9
5.5
3.9
4.0
4.0
4.3
4.8
5.0
7.4
9.5
c1 =
7, c2 =
5
9.5
7.5
5.9
4.1
3.8
3.0
3.9
4.7
4.8
8.0
9.5
c1 =
7, c2 =
7
10.8
8.5
5.6
3.1
3.7
3.7
4.3
4.5
5.0
6.3
9.2
t(3) errors = c1 = c2 = 0 c1 =
2, c2 = 0
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
15.0
12.4
7.9
7.4
6.2
7.4
6.3
7.6
7.1
10.5
14.1
15.4
9.6
8.8
4.3
6.3
4.4
4.3
7.0
7.3
12.3
13
c1 =
5, c2 =
0
11.0
11.7
6.6
6.5
4.1
4.5
4.1
6.7
7.2
9.1
14.1
c=
7, c2 =
0
10.9
10.7
6.6
5.1
3.6
3.7
4.9
4.7
6.7
9.7
9.7
c1 =
2, c2 =
2
11.5
9.3
5.7
4.5
4.1
4.8
4.2
5.2
5.4
9.0
9.2
c1 =
5, c2 =
2
12.7
8.8
6.4
3.4
5.1
3.3
3.7
5.9
5.9
9.8
10.5
c1 =
7, c2 =
2
11.7
10.6
5.4
6.2
3.6
3.9
4.0
6.5
7.0
7.4
11.8
c1 =
5, c2 =
5
11.4
7.7
4.9
4.2
3.6
4.3
3.7
4.3
5.2
7.9
8.5
c1 =
7, c2 =
5
10.6
8.0
6.8
4.0
5.2
3.6
3.3
5.7
6.0
10.5
9.5
c1 =
7, c2 =
7
11.4
6.5
4.8
4.1
3.2
3.5
3.4
4.0
5.5
7.4
7.5
23
Table A.10: p-values(%) of quantile prediction tests (1927:01-2005:12) Univariate regressions with each of the eight predictors: d/p, d/e, b/m, tbl, dfy, ntis, e/p, tms =
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
d/p
1.0*
0.1*
0.3*
14.1
76.0
50.3
17.6
0.5*
6.4
9.6
1.8*
d/e
0.0*
0.0*
0.0*
0.4*
1.8*
16.4
59.5
49.8
89.7
5.8
0.3*
b/m
0.2*
0.0*
0.1*
0.4*
22.5
83.3
56.0
2.8*
6.2
0.8*
1.7*
tbl
46.0
50.3
10.6
11.7
3.6*
12.3
6.0
19.9
3.9*
0.6*
0.3*
dfy
33.6
66.3
98.5
89.8
53.5
5.6
3.0*
0.1*
0.0*
0.0*
0.1*
e/p
75.3
89.8
42.1
51.4
96.5
71.2
89.7
47.2
55.7
33.0
32.6
ntis
23.0
96.7
24.1
13.2
10.1
73.8
71.6
74.1
65.9
94.8
32.5
tms
41.8
15.6
82.5
68.2
26.6
36.7
14.6
58.1
23.2
13.0
16.3
Table A.11: p-values(%) of quantile prediction tests (1927:01-2005:12) Multivariate regressions with two predictors among d/p, d/e, tbl, dfy and b/m =
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
d/p, tbl
0.4*
0.1*
0.8*
2.4*
1.0*
7.6
8.6
0.0*
1.2*
1.0*
0.0*
d/e, tbl
0.0*
0.0*
0.0*
0.2*
0.1*
0.7*
6.8
36.6
6.0
0.0*
0.0*
d/e, dfy
13.4
3.6*
13.0
20.2
33.0
11.0
2.6*
0.2*
0.0*
0.5*
0.2*
d/e, b/m
0.0*
0.0*
0.0*
0.1*
1.2*
57.2
40.6
16.3
2.1*
0.0*
0.0*
tbl, b/m
0.6*
0.0*
0.2*
2.7*
6.0
4.6*
0.6*
0.6*
0.1*
0.2*
2.9*
dfy, b/m
0.9*
0.2*
0.9*
9.4
12.5
14.6
19.1
0.1*
0.1*
0.0*
0.3*
Table A.12: p-values(%) of quantile prediction tests (1952:01-2005:12) Univariate regressions with each of the eight predictors: d/p, d/e, b/m, tbl, dfy, ntis, e/p, tms =
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
d/p
75.6
73.0
57.8
30.4
8.1
84.6
1.4*
0.1*
15.7
76.1
19.8
d/e
75.5
93.1
75.9
69.8
64.6
49.3
80.3
22.8
92.6
50.2
57.8
b/m
0.1*
0.0*
7.0
2.4*
7.1
48.7
68.5
18.6
33.2
20.6
19.0
tbl
37.8
7.3
0.1*
0.3*
0.1*
3.1*
1.1*
6.9
5.4
0.5*
1.1*
dfy
14.6
32.5
76.7
62.1
16.8
8.6
13.9
1.8*
0.2*
3.1*
15.1
e/p
59.2
82.0
70.7
51.8
40.9
2.5*
23.6
1.4*
98.8
80.8
6.6
ntis
57.6
8.2
28.6
43.5
52.2
12.1
9.3
22.1
18.5
21.3
8.0
tms
67.5
88.8
26.3
67.0
25.5
10.1
28.0
22.6
30.1
15.9
5.7
24
Table A.13: p-values(%) of quantile prediction tests (1952:01-2005:12),
= 0:5
Multivariate regressions with two predictors among b/m, d/p, dfy, e/p and tbl =
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
b/m, tbl
0.0*
0.0*
5.2
1.1*
0.6*
2.2*
1.1*
1.5*
1.3*
0.0*
2.6*
dfy, tbl
17.1
36.3
3.8*
0.5*
1.2*
0.4*
0.2*
0.8*
1.6*
2.6*
7.5
Figure A.1: Power curves of IVX-QR, IVX-mean and CY-Q tests c = 0 (n = 200; R = 1) with t(4), t(3), t(2) and t(1) innovations. t(4)
1
t(3) 1
0.8
0.8
0.6
0.6
0.4
0.4
0.2 0 0
0.01
0.02
0.03
IVX-QR IVX-mean CY-Q Nominal Size 0.04 0.05
0.03
IVX-QR IVX-mean CY-Q Nominal Size 0.04 0.05
0.03
IVX-QR IVX-mean CY-Q Nominal Size 0.04 0.05
0.2 0 0
0.01
0.02
t(2)
t(1)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2 0 0
0.01
0.02
0.03
IVX-QR IVX-mean CY-Q Nominal Size 0.04 0.05
0.2 0 0
0.01
0.02
Figure A.2: Power curves of IVX-QR, IVX-mean and CY-Q tests c=
2 (n = 200; R = 0:99) with t(4), t(3), t(2) and t(1) innovations. t(3)
t(4)
1
1
0.8
0.8
0.6
0.6 0.4
0.4 0.2 0 0
0.01
0.02
0.03
IVX-QR IVX-mean CY-Q Nominal Size 0.04 0.05
0.2 0 0
0.02
t(2)
1
0.03 t(1)
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2 0 0
0.01
IVX-QR IVX-mean CY-Q Nominal Size 0.04 0.05
0.01
0.02
0.03
IVX-QR IVX-mean CY-Q Nominal Size 0.04 0.05
IVX-QR IVX-mean CY-Q Nominal Size
0.2 0 0
25
0.01
0.02
0.03
0.04
0.05
Figure A.3: Power curves of IVX-QR, IVX-mean and CY-Q tests c=
20 (n = 200, R = 0:9) with t(4), t(3), t(2) and t(1) innovations. t(4)
t(3)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2 0 0
0.02
0.04
0.06
IVX-QR IVX-mean CY-Q Nominal Size 0.08
IVX-QR IVX-mean CY-Q Nominal Size
0.2 0 0
0.1
0.02
0.04
t(2)
1
0.06
0.08
0.1
t(1) 1
0.8
0.8
0.6
0.6
0.4
0.4
0.2 0 0
0.02
0.04
0.06
IVX-QR IVX-mean CY-Q Nominal Size 0.08
IVX-QR IVX-mean CY-Q Nominal Size
0.2 0 0
0.1
0.02
0.04
0.06
0.08
0.1
Figure A.4: Power curves of IVX-QR, IVX-mean and CY-Q tests c=
40 (n = 200; R = 0:8) with t(4), t(3), t(2) and t(1) innovations. t(4)
t(3)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2 0 0
0.05
0.1
IVX-QR IVX-mean CY-Q Nominal Size 0.15
0 0
0.2
0.05
t(2)
1
0.1
0.15
0.2
t(1) 1
0.8
0.8
0.6
0.6
0.4
0.4
0.2 0 0
IVX-QR IVX-mean CY-Q Nominal Size
0.2
0.05
0.1
IVX-QR IVX-mean CY-Q Nominal Size 0.15
IVX-QR IVX-mean CY-Q Nominal Size
0.2 0 0
0.2
26
0.05
0.1
0.15
0.2
3.2
Diagnostic Tests for Comovement between Persistent Predictors Table A.14: Diagnostic Tests for Comovement (full sample period: Jan 1927 to Dec 2005) (d/p, tbl)
(d/p, dfy)
(d/e, tbl)
(d/e, dfy)
(d/p, b/m)
(d/e, b/m)
(tbl, b/m)
(dfy, b/m)
correlation
Johansen tests for cointegration
-0.16
maximum rank
trace statistics
5% critical value
0
11.76*
15.41
1
3.46
3.76
correlation
Johansen tests for cointegration
0.44
maximum rank
trace statistics
5% critical value
0
20.74
15.41
1
2.99*
3.76
correlation
Johansen tests for cointegration
-0.52
maximum rank
trace statistics
5% critical value
0
21.54
15.41
1
7.46
3.76
correlation
Johansen tests for cointegration
0.56
maximum rank
trace statistics
5% critical value
0
33.98
15.41
1
7.57
3.76
correlation
Johansen tests for cointegration
0.82
maximum rank
trace statistics
5% critical value
0
23.74
15.41
1
3.82
3.76
correlation
Johansen tests for cointegration
0.29
maximum rank
trace statistics
5% critical value
0
21.82
15.41
1
6.78
3.76
correlation
Johansen tests for cointegration
0.07
maximum rank
trace statistics
5% critical value
0
18.37
15.41
1
7.20
3.76
correlation
Johansen tests for cointegration
0.52
maximum rank
trace statistics
5% critical value
0
25.79
15.41
1
8.64
3.76
27
Table 10: Diagnostic Tests for Comovement (subperiod: Jan 1952 to Dec 2005) (b/m, d/p)
(b/m, dfy)
(b/m, e/p)
(b/m, tbl)
(d/p, tbl)
(dfy, tbl)
(e/p, tbl)
4
correlation
Johansen tests for cointegration
0.88
maximum rank
trace statistics
5% critical value
0
10.34*
15.41
1
2.85
3.76
correlation
Johansen tests for cointegration
0.47
maximum rank
trace statistics
5% critical value
0
32.86
15.41
1
3.13*
3.76
correlation
Johansen tests for cointegration
0.89
maximum rank
trace statistics
5% critical value
0
9.70*
15.41
1
1.95
3.76
correlation
Johansen tests for cointegration
0.50
maximum rank
trace statistics
5% critical value
0
14.89*
15.41
1
2.00
3.76
correlation
Johansen tests for cointegration
0.35
maximum rank
trace statistics
5% critical value
0
15.68
15.41
1
2.05*
3.76
correlation
Johansen tests for cointegration
0.58
maximum rank
trace statistics
5% critical value
0
40.87
15.41
1
5.77
3.76
correlation
Johansen tests for cointegration
0.53
maximum rank
trace statistics
5% critical value
0
16.74
15.41
1
3.75*
3.76
References
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