We consider long horizon regressions where the predictor with unknown degree of persistence follows a process of moderate deviations from a unit root. Some asymptotic properties of OLS estimator and of the t statistic are presented.

Jin Lee

English

Research Papers in Economics

Long horizon regressions with moderate deviations from a
unit root 
Jin Lee
National University of Singapore
Abstract
We consider long horizon regressions where the predictor with unknown degree of
persistence follows a process of moderate deviations from a unit root. Some asymptotic
properties of OLS estimator and of the t statistic are presented.
This paper was motivated by the seminar of Peter Phillips, May, 2005, Singapore.
Citation: Lee, Jin, (2005) "Long horizon regressions with moderate deviations from a unit root." Economics Bulletin, Vol. 3,
No. 52 pp. 1−11
Submitted: August 29, 2005.  Accepted: December 6, 2005.
URL: http://www.economicsbulletin.com/2005/volume3/EB−05C20063A.pdf1. Introduction
Long horizon regression models have been popular in economics and ﬁnance, particularly
in the context of return predictability (Ang and Bekaert (2001), Fama and French (1988),
Campbell and Shiller (1988) to name a few). In this paper, we consider the predictive
regression model where the predictor have unknown degree of persistence. In doing so,
we assume that the predictor follows a process of moderate deviation from a unit root
(Phillips and Magdalinos (2005)). Moderate deviations process can generate varying degree
of persistence, depending on how far the predictor deviates from unit root. In addition,
as we construct long horizon variables by taking rolling summation of the regressand and
the regressor, it is known that the long horizon variables are nothing but the partial sum
processes. Thus, one can view this long horizon regression as the regression of non-stationary
regressand on non-stationary predictor.
Given the above setup, we present asymptotic distributions of the OLS estimator and of
the t statistic. Standard functional limit theories are applied to obtain the desired conver-
gences (e.g., Phillips and Solo (1992), Valkanov (2003)). It is found that the OLS estimator
is consistent only when both the regressor and the regressand are overlapped. In such case,
the convergence rate depends on the deviation parameter of the predictor, which determines
the degree of persistence. Also, we ﬁnd that the t test statistic needs to be normalized by
T1/2, to have a well-deﬁned limit, which results from a property of spurious regression. We
provide a brief simulation studies, and investigate the eﬀects of persistence on the ﬁnite
sample bias of the OLS estimator.
2. Long Horizon Predictive Regression Model
We write a short horizon predictive regression,
yt+1 = α + βxt + ut+1, (1)
where yt+1 is the regressand at t+1, and xt is the predictor at t. We assume that ut follows
martingale diﬀerence or i.i.d. sequence (e.g., Campbell and Shiller (1988), Hodrick (1992),
Ang and Bekaert (2001)). This conventional assumption implies that, for example, the
regressand yt is short memory process under the null of no predictability, α = β =0 . On
the other hand, the predictor variable xt exhibits a certain degree of persistence. Typical
examples of such predictor include the dividend-price ratio or the yield spread in the bond
market (e.g., logarithm of the long rate divided by the short rate). As the predictor carries
1unknown degree of persistence, we assume that xt is a process of moderate deviation from a
unit root (Phillips and Magdalinos (2005))
xt =( 1−
c
Tα)xt−1 + vt,c < 0, α ∈ (0,1), (2)
where vt is i.i.d. sequence. The parameter α governs the degree of persistence. As α gets
close to zero, xt becomes stationary AR(1) process. When α approaches to one, xt behaves
as a local to unity process. In the context of testing predictability, the predictor xt is often
modelled as a local to unity process (e.g., Valkanov (2003), Rossi (2005)).On the other hand,
(2) provides a ﬂexible model for unknown persistence in the predictor, where the local to
unity process is a special case of moderate deviations for xt. For the detailed asymptotic
limit theories regarding the correlation coeﬃcient ρ =( 1− c/Tα) in (2), see Phillips and
Magdalinos (2005).
We state the assumption in (1) and (2).
Assumption 1: Given the model (1) and (2),
zt =
Ã
ut
vt
!
is mean-zero i.i.d. and Eztz0
t =
Ã
σ2
u σuv
σuv σ2
v
!
.
The Assumption 1 is suﬃcient to apply the functional central limit theorem (e.g., Brown
(1971, Theorem 2), Phillips and Solo (1992)).
Here, we note that the short horizon predictive regression in (1) is problematic. First, as
the predictor behaves as local to unity process (or as α is close to one), the regression makes
little sense due to imbalance of the order of integration between xt and yt. Second, it is
widely known in empirical studies that the underlying variables in (1) are quite noisy, which
typically cause insigniﬁcant results of the estimate of β and low value of the R2. To partly
overcome this problem, one can consider a long horizon regression model by aggregating yt,
which can strengthen the signal and make the noise relatively negligible. In doing so, the
predictability is interpreted as that of k-th period continuously compounded regressand out
of the predictor at the current period. Write
y
k
t = δ + βxt + u
k
t, (3)
where yk
t =
Pk
i=1 yt+i, and uk
t =
Pk
i=1 ut+i.
2In order to understand the convergence of the OLS estimator b β, we treat the long horizon
variable as the diﬀerence in two partial sum processes as in Valkanov (2003), say,
u
k
t =
Pt+k
j=1 uj −
Pt
j=1 uj. (4)
Then, we can apply functional limit theorem to obtain convergence results of the partial sum
processes. Here, we also assume that the number of overlapping observations grows with T.
Assumption 2: k is a portion of overlapping summations; k =[ λT]f o rλ ∈ (0,1),
where T i st h es a m p l es i z ea n d[ z] is the closest integer to z. In practice, the fraction λ can
be chosen as 0.1o r0 .2.
We establish the following lemma useful to deal with some convergences. Denote uk
t =
uk
t − uk, where uk =( T − k)−1 PT−k
t=1 uk
t.
Lemma 1: Suppose Assumptions 1 and 2 hold under the model (3).
(a) σ−1
e T−1/2uk
t → U(r + λ) − U(r)=U(r,λ),
(b) σ−1
e T−1/2uk
t → U(r,λ) − (1 − λ)−1 R 1−λ
0 U(r,λ)dr = U∗(r,λ),
(c) σ−1
w T−α/2xt → Jc(r)
(d) σ−1
w T−α/2xt → Jc(r) − (1 − λ)−1 R 1−λ
0 Jc(r)dr = J∗
c(r)
where Jc(r)i sal i n e a rd i ﬀusion process deﬁned as
Jc(r)=W(r)+c
R r
0 e
c(r−z)dW(z), for c<0,
and (U(r),W(r))0 are bivariate standard Brownian motions with covariance δ = σuv/(σuσv).
Under the null of β =0 , the lemma 1 (a) and (b) are applied to {yk
t } process. The lemma
(c) and (d) make use of the results in Phillips and Magdalinos (2005).
We obtain convergence results for OLS estimator b β,tstatistic, and R2.
Theorem 1: Suppose Assumptions 1 - 2 hold under the model (3). For H0 : β =0 ,
(1) T(α−1)/2b β →
³
σe
R 1−λ
0 U∗(r,λ)J∗
c(r)dr
.
σw
R 1−λ
0 J∗
c(r)2dr
´
,
(2) T−1/2t →
hR 1−λ
0 U∗(r,λ)J∗
c(r)dr
±
G[U∗(r,λ),J∗
c(r)]1/2
i
, where
G[U∗(r,λ),J∗
c(r)] =
³R 1−λ
0 U∗(r,λ)2dr
R 1−λ
0 J∗
c(r)2dr − [
R 1−λ
0 U∗(r,λ)J∗
c(r)]2
´
(3) R2 →
h
[
R 1−λ
0 U∗(r,λ)Jc(r)dr]2
i2 .nhR 1−λ
0 U∗(r,λ)2dr
ihR 1−λ
0 J∗
c(r)2dr
io
.
3First thing to note is that OLS estimator b β is not consistent for all values of α ∈ (0,1).
Thus, we need to consider an alternative modelling to obtain consistent estimate, which will
be covered in the Theorem 2 below. Second, the t statistic diverges at the rate of T1/2,
which is expected since the long horizon regressions generate spurious regressions. Thus, we
need to consider a scaled test statistic T−1/2t to obtain a well-deﬁned limit, which depends
on unknown locality parameter c. For reference, see Valkanov (2003) for detailed analysis.
Third, the R2 does not converge to zero under the null, which is also expected due to the
spurious regressions.
Now, we consider a long horizon regressions when both yt and xt are overlapped.
y
k
t = δ + βx
k
t + u
k
t, (5)
where xk
t = xk
t − xk, where xk
t =
Pk−1
i=0 xt+i, and xk =( T − k)−1 PT−k
t=1 xk
t.
In addition to Lemma 1, we further have following convergence results for overlapped
predictor.
Lemma 1 (continued):
(e) σ−1
w T−(α+2)/2xk
t →
R r+λ
r Jc(z)dz ≡ Jc(r,λ)
(f) σ−1
w T−(α+2)/2xk
t → Jc(r,λ) − (1 − λ)−1 R 1−λ
0 Jc(r,λ)dr = J∗
c(r,λ)
Given this, we are led to have convergence results as follows.
Theorem 2: Suppose Assumptions 1 - 2 hold under the model (5). For H0 : β =0 ,
(1) T(α+1)/2b β →
³
σe
R 1−λ
0 U∗(r,λ)J∗
c(r,λ)dr
.
σw
R 1−λ
0 J∗
c(r,λ)2dr
´
,
(2) T−1/2t →
hR 1−λ
0 U∗(r,λ)J∗
c(r,λ)dr
±
G[U∗(r,λ),J∗
c(r,λ)]1/2
i
, where
G[U∗,J∗
c]i sd e ﬁn e da si nT h e o r e m1
(3) R2 →
h
[
R 1−λ
0 U∗(r,λ)J∗
c(r,λ)dr]2
i2 .nhR 1−λ
0 U∗(r,λ)2dr
ihR 1−λ
0 J∗
c(r,λ)2dr
io
.
It is clear that b β is consistent for α ∈ (0,1). The convergence rate depends on α in a
sense that the greater the value of α, the faster the convergence rate. Given the correlation
between disturbance and the predictor, we expect that as α increases, the ﬁnite sample bias
decreases. This is shown through simulations in the later section. Convergences of the t
statistic and of the R2 are pretty similar to those in Theorem 1.
43. Simulation
Finally we investigate a ﬁnite sample bias of the OLS estimator and empirical distribu-
tions of scaled t statistic through simulations. In the long horizon regressions in (5), {ut}T
t=1
is generated from i.i.d. N(0,1). The predictor {xt}T
t=1 is generated from (2), where c = −1
and vt ∼i.i.d. N(0,1). We denote a contemporaneous correlation between ut and vt as δ.
The values of deviation parameter α are chosen from 0.1t o0 .9t oa l l o wd i ﬀerent degree
of persistence. Further, the portion of overlapping summation k is set as k =[ 0 .1T]a n d
=[ 0 .2T], to see the eﬀect of k on the ﬁnite sample performance. Two sample sizes n =2 0 0
and 500 are considered and 5,000 replications are conducted.
Table 1 shows the ﬁnite sample bias of OLS estimator according to diﬀerent values of α,
and of the number of horizon, given δ = −0.9. First, as α increases, the bias reduces, which
is well expected from Theorem 2. Thisb i a sr e d u c t i o nw i t hi n c r e a s ei nα is more pronounced
for the larger sample size. Second, the eﬀect of k on the bias is negligible, particularly for
larger sample size of T =5 0 0 .
Next, we present the empirical distributions of the scaled t test T−1/2t. In practice,
empirical distributions using ﬁnite sample are more useful than the asymptotic distributions,
specially when there exist correlations between disturbance and the predictor (See Mishkin
(1995)). The distributions are simulated under the null of β =0w i t hk =[ 0 .1T]a n d
T = 500 in Tables 2. Selective values of percentiles are given according to diﬀerent values of
α and δ. We note some ﬁndings. First, when δ =0 , the distribution is shown to be nearly
symmetric around zero. As α gets larger, the variance of scaled t test increases, so the
distribution becomes more ﬂattened. Second, negative correlation δ shifts the distribution
to the negative range as expected. The eﬀect of negative correlations, however, are mitigated
as the persistence of the predictor increases. We do not report empirical distributions with
other choices of k,δ, and T, which are available upon request.
4. Conclusion
We present some asymptotic properties of OLS estimator and of the test statistic in long
horizon regressions when the predictor follows a process of moderate deviation from unity.
I ti ss h o w nt h a tO L Se s t i m a t o ri sc o n s i s t e n tu nder the regression of overlapped regressand
on the overlapped predictor. In this case, the convergence rate of the estimator depends on
how far the predictor variable deviates from unit root process. Our simulation studies verify
the theoretical conjecture.
5Appendix
Proof of Theorem 1: We write
b β − β =
PT−k
t=1 uk
txt
PT−k
t=1 (xt)2, (A.1)
where uk
t = uk
t − uk, xt = xt − x with uk =( T − k)−1 PT−k
t=1 uk
t.
Using the lemma 1, we obtain
b β − β =
T(3+α)/2[T−1 PT−k
t=1 (T−1/2uk
t)(T−α/2xk
t)]
T1+α[T−1 PT−k
t=1 (T−α/2xt)2]
= T
(1−α)/2[T−1 PT−k
t=1 (T−1/2uk
t)(T−α/2xk
t)]
[T−1 PT−k
t=1 (T−α/2xt)2]
, (A.2)
thus
T
(α−1)/2(b β − β) →
σe
R 1−λ
0 U∗(r,λ)J∗
c(r)dr
σw
R 1−λ
0 J∗
c(r)2dr
, (A.3)
under the H0 : β =0 .
For the t statistic, we can write
t =
b β[
PT−k
t=1 (xt)2]1/2
h
T−1 PT−k
t=1 (yk
t − b βxt)2
i1/2
=
b β[
PT−k
t=1 (xt)2]
h
T−1 PT−k
t=1 (yk
t)2 PT−k
t=1 (xt)2 − T−1(
PT−k
t=1 yk
txt)2
i1/2. (A.4)
Making use of lemma 1 for the denominator, we have
t =
T(3+α)/2[T−1 PT−k
t=1 (T−1/2yk
t)(T−α/2xt)]
(AT − BT)1/2 , (A.5)
where
AT = T
−1 PT−k
t=1 (y
k
t)
2 PT−k
t=1 (xt)
2
= T
2+α
n
[T
−1 PT−k
t=1 (T
−1/2y
k
t)
2][T
−1 PT−k
t=1 (T
−α/2x
k
t)
2]
o
,
and
BT = T
−1[
PT−k
t=1 y
k
txt]
2
= T
−1
n
T
(3+α)/2[T
−1 PT−k
t=1 (T
−1/2y
k
t)(T
−α/2xt)]
o2
= T
2+α[T
−1 PT−k
t=1 (T
−1/2y
k
t)(T
−αx
k
t)]
2.
6It follows that
T
−1/2t →
R 1−λ
0 U∗(r,λ)J∗
c(r)dr
hR 1−λ
0 U∗(r,λ)2dr
R 1−λ
0 J∗
c(r)2dr − [
R 1−λ
0 U∗(r,λ)J∗
c(r)]2
i1/2. (A.6)
By similar reasoning, we further obtain
R
2 =
b β
2 PT−k
t=1 (xt)2
PT−k
t=1 (yk
t)2 →
[
R 1−λ
0 U∗(r,λ)Jc(r)dr]2
[
R 1−λ
0 U∗(r,λ)2dr][
R 1−λ
0 J∗
c(r)2dr]
. (A.7)
Proof of Theorem 2: Under the lemma 1 (e)-(d), we have
b β − β =
PT−k
t=1 uk
txk
t PT−k
t=1 (xk
t)2, (A.8)
where xk
t = xk
t − xk with xk =( T − k)−1 PT−k
t=1 xk
t.
By similar fashion as in (A.1) to (A.4), we obtain
b β − β =
T(5+α)/2[T−1 PT−k
t=1 (T−1/2uk
t)(T−(α+2)/2xk
t)]
T3+α[T−1 PT−k
t=1 (T−(α+2)/2xk
t)2]
,
then
T
(α+1)/2(b β − β) →
σe
R 1−λ
0 U∗(r,λ)J∗
c(r,λ)dr
σw
R 1−λ
0 J∗
c(r,λ)2dr
, (A.9)
under the H0 : β =0 .
For the convergence of the t statistic, we write
t =
T(5+α)/2[T−1 PT−k
t=1 (T−1/2yk
t)(T−α/2xt)]
(AT − BT)1/2 ,
where
AT = T
4+α
n
[T
−1 PT−k
t=1 (T
−1/2y
k
t)
2][T
−1 PT−k
t=1 (T
−(α+2)/2x
k
t)
2]
o
,
and
BT = T
4+α[T
−1 PT−k
t=1 (T
−1/2y
k
t)(T
−(α+2)/2x
k
t)]
2.
Thus,
T
−1/2t →
R 1−λ
0 U∗(r,λ)J∗
c(r,λ)dr
hR 1−λ
0 U∗(r,λ)2dr
R 1−λ
0 J∗
c(r,λ)2dr − [
R 1−λ
0 U∗(r,λ)J∗
c(r,λ)]2
i1/2. (A.10)
The convergence of R2 is the same as that in Theorem 1.
7References
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working paper, 10026, 1-51.
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Financial Economics, 25, 23-49.
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inference and measurement” Review of Financial Studies,5 ,3 5 7 - 3 8 6 .
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evidence on the forward discount anomaly” Journal of Applied Econometrics, 16, 671-708.
Mishkin, F. (1995): “Nonstationarity of regressors and tests on real-interest-rate” Journal
of Business and Economic Statistics, 13, 47-51.
Phillips, P.C.B., and V. Solo (1992): “Asymptotics for linear processes” Annals of Statistics,
20, 971-1001.
–—, and T. Magdalinos. (2005): “Limit theory for moderate deviations from a unit root”
manuscript.
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8T a b l e1 :F i n i t es a m p l eb i a so fO L Se s t i m a t o r :
D G P :T h em o d e l( 3 )u n d e rt h en u l lβ =0 .
T=200
k\α 0.10 .30 .50 .70 .9
[0.1T] -0.2793 -0.0971 -0.0335 -0.0118 -0.0040
[0.2T] -0.2819 -0.0981 -0.0341 -0.0123 -0.0045
T=500
[0.1T] -0.2624 -0.0760 -0.0221 -0.0067 -0.0020
[0.2T] -0.2645 -0.0766 -0.0222 -0.0067 -0.0021
Note: (a) 5000 replications.
(b) k is the portion of rolling summations of xt and yt.
(c) The locality parameter c is set to −1.
(d) The correlation δ is set to −0.9.
9Table 2: Percentiles of the simulated distribution of the scaled t statistic:
DGP: The model (5) with k =[ 0 .1T],T = 500 under the null β =0 .
1% 2.5% 5% 50% 95% 97.5% 99%
αδ =0
0.1 -0.7046 -0.5796 -0.4785 0.0031 0.4843 0.5798 0.6751
0.3 -0.7402 -0.6237 -0.4950 0.0056 0.5061 0.6021 0.7114
0.5 -0.7822 -0.6625 -0.5440 0.0005 0.5349 0.6395 0.7597
0.7 -0.8419 -0.7010 -0.5821 -0.0092 0.5637 0.6891 0.8168
0.9 -0.8606 -0.7154 -0.5910 -0.0122 0.5684 0.6996 0.8273
δ = −0.9
0.1 -1.8737 -1.7000 -1.5393 -0.8378 -0.3153 -0.2274 -0.1277
0.3 -1.7559 -1.5838 -1.4287 -0.7586 -0.2496 -0.1630 -0.0556
0.5 -1.4292 -1.2728 -1.1293 -0.5441 -0.0759 0.0121 0.1051
0.7 -1.0762 -0.9238 -0.8108 -0.2929 0.1539 0.2334 0.3293
0.9 -0.8650 -0.7235 -0.6139 -0.1186 0.3285 0.3908 0.4977
Note: (a) 5000 replications.
(b) k is the portion of rolling summations of xt and yt.
(c) The locality parameter c is set to −1.
10

Long horizon regressions with moderate deviations from a unit root

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Long horizon regressions with moderate deviations from a unit root

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