07350015%2E2014%2E887013
Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
A Nonparametric Test of the Predictive Regression
Model
Ted Juhl
To cite this article: Ted Juhl (2014) A Nonparametric Test of the Predictive Regression Model,
Journal of Business & Economic Statistics, 32:3, 387-394, DOI: 10.1080/07350015.2014.887013
To link to this article: http://dx.doi.org/10.1080/07350015.2014.887013
Published online: 28 Jul 2014.
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A Nonparametric Test of the Predictive
Regression Model
Ted JUHL
University of Kansas, 415 Snow Hall, Lawrence, KS 66045 (juhl@ku.edu)
This article considers testing the significance of a regressor with a near unit root in a predictive regression
model. The procedures discussed in this article are nonparametric, so one can test the significance of a
regressor without specifying a functional form. The results are used to test the null hypothesis that the entire
function takes the value of zero. We show that the standardized test has a normal distribution regardless of
whether there is a near unit root in the regressor. This is in contrast to tests based on linear regression for
this model where tests have a nonstandard limiting distribution that depends on nuisance parameters. Our
results have practical implications in testing the significance of a regressor since there is no need to conduct
pretests for a unit root in the regressor and the same procedure can be used if the regressor has a unit root
or not. A Monte Carlo experiment explores the performance of the test for various levels of persistence of
the regressors and for various linear and nonlinear alternatives. The test has superior performance against
certain nonlinear alternatives. An application of the test applied to stock returns shows how the test can
improve inference about predictability.
KEY WORDS: Finance; Kernel regression; Near unit root; Time series.
1.
INTRODUCTION
The predictive regression model has received much attention
in recent years. In particular, the model where the regressors are
persistent (with a unit root or near unit root) has been covered
in the work of Campbell and Yogo (2006), Jansson and Moreira
(2006), Lewellen (2004), and Stambaugh (1999), among others.
The statistical estimation and testing of such models requires a
treatment beyond the traditional normal approximations for the
regression parameters. The persistence of the regressors directly
affects the distribution of the standard t-ratio of the estimated
regression parameters. The resulting distribution of the regression parameter is related to the Dickey–Fuller distribution but
with a nuisance parameter from the correlation of innovations to
the unit root series in the future. The distribution is nonnormal
so that standard asymptotic theory does not apply. Moreover, as
the amount of correlation between innovations in the persistent
series and the innovations in the regressand series increases,
the distribution shifts to the right so that a type of “spurious”
regression further distorts standard inference. These facts indicate the need for knowledge of the persistence of the regressor,
or the need to develop new procedures that are “robust” to the
unknown level of persistence in the regressor.
Given the above problems, Campbell and Yogo (2006),
Jansson and Moreira (2006), and Lewellen (2004) each
proposed solutions to inference when the level of persistence
of the regressor is unknown and possibly contains a unit root
(or near unit root). The approaches are related in that they
attempt to find an optimal test in some class of statistics that are
invariant to location and scale transformations of the data. The
test we propose in this article shares the invariance properties
of the above statistics. However, our test is designed to have
power against linear and nonlinear alternatives. To accomplish
this, we use nonparametric kernel estimation of the regression
function. The estimated function is then tested to see if it is
significantly different from a constant.
The nonparametric procedure that we use is based on two
steps. First, we use nonparametric regression to estimate the
relationship between the two series. The advantage of such an
approach is that we are not required to posit a functional form
for the relationship. One may have a theoretical basis for what
the functional form may look like, but it could be that some
unknown financial distortions may exist. By employing a nonparametric approach, we may detect an unforeseen relationship.
For our purposes, we use kernel regression, and we employ a
convenient reweighting procedure. Next, we see if the estimated
regression function is correlated with the demeaned data. Such
a procedure results in a U-statistic, and is known to be effective in specification testing for stationary data. The novelty of
using U-statistics for this problem is that we are able to test
whether the entire function is a constant. Testing for the significance of the function at a given number of points would then
require some sort of Bonferroni argument which would become
more conservative as we estimate more points of the function.
By using the U-statistic approach, it is possible to test the null
hypothesis that the entire function is a constant.
When testing the significance of regressors using nonparametric techniques, we have power against a much wider class of
alternative hypotheses. Of course, with such generality comes
a price in that power is not as high if there truly exists a linear
relationship. We explore these costs via a Monte Carlo experiment. We find that the nonparametric test does outperform tests
based on the linear for certain nonlinear alternatives.
The remainder of the article is organized as follows. In Section 2, we describe the predictive regression model. Section 3
provides an outline of the nonparametric test. The asymptotic
results are derived in Section 4 and explored in a Monte Carlo
387
© 2014 American Statistical Association
Journal of Business & Economic Statistics
July 2014, Vol. 32, No. 3
DOI: 10.1080/07350015.2014.887013
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388
Journal of Business & Economic Statistics, July 2014
experiment detailed in Section 5. An empirical illustration using
stock returns is treated in Section 6 and Section 7 concludes.
d
Notation is standard with weak convergence denoted by →. All
limits in the article are taken as the sample size T → ∞.
2.
E [yt E(yt |xt−1 )p(xt−1 )] = E (E [yt E(yt |xt−1 )p(xt−1 )|xt−1 ])
= E (E [yt E(yt |xt−1 )|xt−1 ] p(xt−1 ))
PREDICTIVE REGRESSION MODEL
Let the variable yt denote the dependent variable in period t
and let xt−1 denote a predictor variable observed at time t − 1.
We are interested to see if there is any relationship between yt
and xt−1 . We consider a model of the form
yt = μy + g(xt−1 ) + ǫt
xt = μx + ut
ut = ρT ut−1 + et .
with μy being the unconditional mean of yt , μx the mean of xt ,
and ρT = 1 − c/T . Note that in this case, the variable xt has
a “near” unit root. This is an extreme form of dependence in
the predictor variable but one that serves as an approximation
for the behavior of many financial ratios used in this type of
analysis. Several studies have versions of this model where
g(xt−1 ) is considered a linear function, and the findings suggest
that the distribution of OLS in the linear case is nonstandard
and depends on the correlation between ǫt and et . In light of
this result, Jansson and Moreira (2006), Campbell and Yogo
(2006), Lewellen (2004), and others have proposed tests that
are not based on the usual t-ratio of the OLS squares estimator
of the slope coefficient for xt−1 . The Jansson and Moreira (2006)
test is optimal in the class of conditionally unbiased tests, and
the distribution of their test is also nonstandard, requiring a
numerical integration routine to calculate p-values for the test
for each new dataset.
In this article, we propose testing the predictive regression
model using the nonparametric tests described in the next
section.
3.
for some value of xt−1 . Let p(xt−1 ) be the density function of
xt−1 . To simplify the discussion in this section, suppose that yt
has mean zero. Then using iterated expectations, we have
NONPARAMETRIC TESTS FOR STATIONARY
REGRESSORS
In this section, we summarize the type of nonparametric tests
that we employ in this article. This type of test has been used
when the data are assumed to be stationary as in Fan and Li
(1996, 1999) and Zheng (1996) among others. The idea is to
check if two series are related without specifying how they are
related. That is, we allow for the series to have an unknown,
perhaps nonlinear, relationship. To be specific, suppose that yt
and xt−1 are stationary series and that
yt = μy + g(xt−1 ) + ǫt ,
almost everywhere versus the alternative
H1 : g(xt−1 ) = 0
E E(yt |xt−1 )2 p(xt−1 ) ≥ 0.
(3.2)
There is equality to zero in (3.2) if and only if E(yt |xt−1 ) =
g(xt−1 ) = 0 almost everywhere. The above relation suggests
that we should find an estimate of E [yt E(yt |xt−1 )p(xt−1 )] and
test whether the estimate is significantly different from zero.
The nonparametric tests estimate E(yt |xt−1 )p(xt−1 ) by kernel
regression of yt on xt−1 multiplied by the density estimate of
p(xt−1 ). That is,
T
xt−1 − zs
1
1
Ê(yt |xt−1 ) =
K
ys
p̂(xt−1 ) T h s=t−1
h
and
T
xt−1 − xs
1
Ê(yt |xt−1 )p̂(xt−1 ) =
ys ,
K
T h s=t−1
h
where K(·) is a kernel function and h is a bandwidth parameter.1 The estimate of E(yt |xt−1 )p(xt−1 ) is particularly tractable
because the term p̂(xt−1 ) cancels from the denominator, eliminating the need for restricting the estimated density away from
zero. This “reweighting” of the conditional mean was also employed in Zheng (1996) and Fan and Li (1996). Moreover, we
weight the estimated function by the estimated density to account for the fact that even if g(xt−1 ) is nonzero, it may only
happen on a set with measure zero, hence the product of the
function and the density will be zero. The final estimate of
E(yt E(yt |xt−1 )p(xt−1 )) is the statistic
T
T
1
xt−1 − xs
yt ys .
K
T 2 h t=1 s=t−1
h
Inference is based on half of this quantity, which is given by
T t−1
xt−1 − xs
1
yt ys .
K
UT = 2
T h t=2 s=1
h
When the statistic UT is properly scaled, it is normally distributed under the null hypothesis that g(xt−1 ) = 0. Due to its
nonparametric construction, the test has power against a wide
variety of alternatives, both linear and nonlinear.
(3.1)
where E(ǫt |xt−1 ) = 0 which implies that xt−1 and ǫt are uncorrelated and that E(ǫt ) = 0. The model then implies that
E(ǫt g(xt−1 )) = 0, which are the conditions used for nonlinear
least squares. The hypothesis that we wish to test is
H0 : g(xt−1 ) = 0
which is equal to
4.
DISTRIBUTION THEORY
The discussion in the last section is based on the assumption
that the variables involved are stationary. Such results have been
shown to hold when the independent variables satisfy some type
of limit to the dependence, typically a condition stated in terms
of β-mixing (absolute regularity), such as in Fan and Li (1999).
1For
an introduction to kernel regression and nonparametric regression, see
Härdle (1989a).
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Juhl: A Nonparametric Test of the Predictive Regression Model
In this section, we provide the distribution theory when the
independent variable is nonstationary in the form of a near unit
root.
The model is
yt = μy + g(xt−1 ) + ǫt
xt = μx + ut
ut = ρT ut−1 + et .
The assumptions are listed below.
Assumption 1.
ut = ρT ut−1 + et
et =
∞
φi wt−i
i=0
ρT = 1 −
c
T
u0 = 0
∞ 1+δ
with i=0 φi = 0 and i=0 i |φi | < ∞ for some δ > 0. The
sequence (wt , ǫt )⊤ is iid with mean zero and matrix . The
variance of wt is 1, and ǫt has finite fourth moments. The characteristic function of wt is given by ϕ(t) such that |t|ϕ(t) → 0
as |t| → ∞.
∞
Assumption 2. K(x) is nonnegative and bounded with
|x|K(x)dx < ∞.
Assumption 3. T → ∞, h → 0, T h2 → ∞, and T h4 log2
T → 0.
These assumptions imply those used by Wang and Phillips
(2012) for their cointegrated model.2 Assumption 1 limits the
amount of dependence in the innovations of the process driving
xt , so that the resulting U-statistic itself is a martingale which
is subject to the new central limit theorem. Assumption 2 is
standard for kernels and Assumption 3 limits the rate at which
the bandwidth goes to zero. Based on these assumptions, we can
apply the model given in Wang and Phillips (2012) applies and
gives the following result.
Theorem 4.1. Given Assumptions 1–3, under the null hypothesis that g(xt−1 ) = 0 almost everywhere, the statistic
√
T 5/4 hÛT d
→ N(0, 1),
T 1/2 V̂T
where
T t−1
xt−1 − xs
1
(yt − ȳ)(ys − ȳ)
K
ÛT = 2
T h t=2 s=1
h
389
It is important to note that the standardized version of the
U-statistic is exactly the same statistic analyzed in Fan and Li
(1999), where xt is stationary with limited dependence (β mixing). Because of this result, there is no need to change the statistical analysis of predictive regression models using U-statistics
if one is unsure about the possibility of a unit root in the variable
xt−1 . This fact is unusual in that standard OLS tests in predictive regression models have a nonstandard limiting distribution
if xt−1 has a unit root or near unit root. Moreover, when using
OLS, the distribution depends on nuisance parameters such as
the correlation between ǫt and et . In our case, the limiting distribution does not change if the series xt−1 is stationary or has a
unit root.
The result is different from existing results for nonparametric
analysis of models with unit roots since all existing results are
given by estimating the functions pointwise. One must have a
method to test that the estimated function is constant almost everywhere. As stated in the introduction, it is possible to construct
a Bonferroni argument using pointwise results. However, as the
number of estimated points increases, the confidence bands for
the estimated function become more and more conservative. The
advantage of the U-statistic approach is that we directly test the
null hypothesis that the entire function is a constant.
From a practical point of view, the test is a good complement
to the existing tests based on the linear model in that it may
detect dependence in xt−1 and yt that is nonlinear, and the tests
are not affected by assumptions about the dependence in xt−1 .
One important consideration of nonparametric regression is
the choice of bandwidth parameters. The assumptions used in
Theorem 4.1 only require that h goes to zero at some prescribed
rate. No optimal bandwidth is proposed in this article, but we
suggest using h = d σ̂x T −1/5 where d is a scalar and σ̂x is the
sample standard deviation of xt−1 . In our simulations, we try
various values of d to gauge the sensitivity of the tests to the
bandwidth parameter. One motivation for using a multiple of
σ̂x is that the statistic will become scale invariant. The test
is always location invariant but using a bandwidth that is a
multiple of σ̂x ensures scale invariance. Although this procedure
is similar to methods used in practice for stationary data, there
3
The sample
is a difference when xt−1 is possibly integrated.
√
standard deviation will diverge at rate T as T → ∞, so that
h will diverge at rate T 3/10 . An alternative is to use the sample
standard deviation of
xt−1 , denoted σ̂
x , which would be
appropriate if xt−1 contains a unit root, for example, and would
also remain scale invariant. In finite samples, it remains to be
seen how the possible integration properties of xt−1 affect the
bandwidth, and, ultimately, the performance of the U-statistic.
In the Monte Carlo section, we explore the performance of the
U-statistic using the sample standard deviation of both xt−1
and
xt−1 as a basis for the bandwidth.
and
T t−1
xt−1 − xs 2
1
V̂T = 2
K
(yt − ȳ)2 (ys − ȳ)2 .
T h t=2 s=1
h
5.
MONTE CARLO EVIDENCE
We compare the nonparametric test with the two leading
tests developed by Campbell and Yogo (2006) and Jansson and
2In
particular, the partial sums of (wt , ǫt )⊤ satisfy a multivariate invariance
principal from Assumption 1. I thank an anonymous referee for suggesting a
more lucid exposition.
3We
thank an anonymous referee for pointing out the divergence and the potential problem with the resulting bandwidth.
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390
Journal of Business & Economic Statistics, July 2014
Moreira (2006). The experiment uses the following model:
yt = μy + g(xt−1 ) + ǫt
xt = ρT xt−1 + et .
We set the parameter σǫe = −0.75 and −0.95, values that are
consistent with the empirical regularities found in financial
data. For example, if xt−1 represents a dividend–price ratio
and yt represents a stock return, then a positive shock to
returns in the form of a large value of ǫt would then cause
a concurrent decrease in the dividend–price ratio, so that the
correlation between et and ǫt should be negative. Lewellen
(2004) estimated values of σǫe ranging from −0.88 to −0.96
for various financial ratios used for xt−1 .
We consider several possible levels of persistence in xt
through the parameter ρ. In particular, we parameterize ρ as
c
ρT = 1 −
T
so that ρT is “local to unity.” The values of c used in the simulation are 0, 5, 10, 15, and 20. In this first experiment, we examine
the size properties of the test. That is, we generate the data so
that g(xt−1 ) = 0 and we tabulate the percentage of rejections in
10,000 replications. The sample sizes considered are T = 100,
T = 500, and T = 1000. There are four different bandwidth
parameters that are examined for the UT statistic. The statistics
denoted by U1 and U2 use h = σ̂x T −1/5 and h = 2σ̂x T −1/5 ,
respectively. The statistics Ud1 and Ud2 use bandwidth parameters that are based on the standard deviation of the differences
of xt , so that the respective bandwidth parameters are given by
h = σ̂
x T −1/5 and h = 2σ̂
x T −1/5 The statistic of Jansson and
Moreira (2006) is a uniformly most powerful conditionally unbiased test in the linear model, so it is denoted and UMPCU. The
Campbell and Yogo test uses a Bonferroni bound and we denote
this test as CYB. For comparison purposes, we include an OLS
t-test that the linear regression coefficient on xt−1 is zero and
denote this as TSTAT. This test is incorrect in the case when xt
has a unit root or root local to unity, but is included to illustrate
the potential problems of ignoring the extreme dependence in
xt−1 . The sizes of the tests appear in Table 1.
From Table 1, we see that the UMPCU and CYB tests have
size close to the nominal size in all cases. The U-statistic is
conservative and gets more conservative for a larger bandwidth.
As discussed earlier, the t-statistic is not normally distributed
and we included this statistic to show the costs of ignoring the
Table 1. Size of competing tests
Size
T = 100
c
UMPCU
U1
U2
Ud1
Ud2
TSTAT
CYB
σǫe = −0.95
c=0
c=5
c = 10
c = 15
c = 20
0.046
0.051
0.055
0.057
0.052
0.031
0.028
0.031
0.031
0.029
0.019
0.015
0.019
0.019
0.020
0.047
0.040
0.036
0.039
0.037
0.045
0.033
0.032
0.027
0.026
0.395
0.178
0.125
0.106
0.087
0.052
0.050
0.045
0.049
0.046
σǫe = −0.75
c=0
c=5
c = 10
c = 15
c = 20
0.051
0.050
0.051
0.055
0.051
0.033
0.029
0.030
0.033
0.029
0.020
0.017
0.017
0.018
0.018
0.048
0.037
0.043
0.039
0.038
0.042
0.030
0.030
0.032
0.030
0.287
0.135
0.110
0.090
0.085
0.040
0.034
0.036
0.033
0.039
c=0
c=5
c = 10
c = 15
c = 20
0.044
0.053
0.059
0.063
0.060
0.038
0.032
0.035
0.031
0.035
0.025
0.021
0.023
0.023
0.025
0.053
0.046
0.047
0.047
0.052
0.042
0.042
0.050
0.044
0.047
0.414
0.191
0.138
0.116
0.104
0.035
0.031
0.032
0.028
0.023
c=0
c=5
c = 10
c = 15
c = 20
0.044
0.056
0.058
0.061
0.060
0.035
0.035
0.034
0.035
0.036
0.024
0.021
0.025
0.024
0.026
0.053
0.043
0.041
0.046
0.053
0.043
0.053
0.036
0.052
0.049
0.290
0.155
0.119
0.102
0.092
0.023
0.028
0.025
0.023
0.022
c=0
c=5
c = 10
c = 15
c = 20
0.043
0.056
0.067
0.070
0.063
0.030
0.025
0.028
0.037
0.045
0.015
0.020
0.023
0.022
0.032
0.047
0.031
0.049
0.053
0.050
0.051
0.037
0.048
0.044
0.051
0.426
0.188
0.132
0.102
0.115
0.039
0.036
0.034
0.020
0.029
c=0
c=5
c = 10
c = 15
c = 20
0.046
0.052
0.075
0.061
0.067
0.026
0.029
0.046
0.035
0.043
0.015
0.025
0.039
0.032
0.027
0.052
0.041
0.036
0.051
0.049
0.057
0.030
0.046
0.049
0.050
0.292
0.143
0.134
0.105
0.106
0.017
0.021
0.023
0.022
0.025
σǫe = −0.95
T = 500
σǫe = −0.75
σǫe = −0.95
T = 1000
σǫe = −0.75
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Juhl: A Nonparametric Test of the Predictive Regression Model
391
Figure 1. Linear power.
Figure 2. Quadratic power.
unit root behavior of xt−1 . At a nominal size of 5%, we reject the
null hypothesis of no predictability approximately 40% of the
time when the innovations from the regressors and regressand
are highly correlated. When the parameter c increases (near
unit root), the normal approximation becomes less damaging.
However, even when c = 5 with an autoregressive parameter
of ρT = 0.95, we reject the null roughly 18% of the time. The
sample size T appears to have a minor affect on the performance
of all the tests considered in the experiment. In general, the
bandwidth associated with the standard deviation of xt−1 (U1
and U2) are more conservative than those associated with the
standard deviation of
xt−1 (Ud1 and Ud2).
To determine the sensitivity of the U-statistic to possible
asymmetric distributions, we generate the same data using
a centered χ12 distribution which is clearly skewed. The
experiment produces results that are very similar to those
in Table 1, and suggest that the normal distribution is still a
reasonable approximation for the test statistic even if the errors
are not from a symmetric distribution.4
For power comparisons, we now consider three alternative
functions. In the first case, the alternative is linear so that
the relationship involving 3-month Treasury bill rates exhibits
a similar shape over the middle region of historical rates.
The power curves are presented in Figure 2. As expected, the
U-statistics have the highest power. One phenomenon that
we found for quadratic alternatives was that the linear based
tests (UMPCU and CYB) had power functions that reached
a plateau at around 30% and never increased. This power
function shows the possible gains from using a U-statistic in
predictive regression models.
For a different alternative, we consider a nonlinear function
that is monotonic. The quadratic alternative gives a distinct
advantage to the U-statistic approach since a linear regression
estimate will be near zero. If we choose an alternative that
remains monotonic, such an advantage may disappear so that
the tests based on linear alternatives will again have higher
power. Our alternative is given by
3
.
g(xt−1 ) = βxt−1
The power curve is shown in Figure 3. The power of the U
statistic is less than the competing tests since for this cubic
alternative, the relationship is monotonic.
g(xt−1 ) = βxt−1 .
For purposes of comparison with Jansson and Moreira (2006),
2 γ , with γ = b/T , and let b increase. The
we set β = 1 − σeǫ
power functions for linear alternatives are shown in Figure 1.5
We notice that the Bonferroni test procedure of Campbell and
Yogo (2006) dominates all other tests for linear alternatives.
Even though our U-statistics are less powerful than the tests
based on linear alternatives, they still have power against linear
alternatives
For an example of a nonlinear alternative, we let
2
,
g(xt−1 ) = βxt−1
so that the alternative is a quadratic. The quadratic is included
as an example of a nonmonotonic function where linearity
would have little power. Moreover, in the empirical section,
4The
5We
results are available upon request.
use T = 100 for all of the power figures.
Figure 3. Cubic power.
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392
Journal of Business & Economic Statistics, July 2014
The final alternative is a step function that depends on a
“threshold” value. In particular, the function is given by
g(xt−1 ) = β1(xt−1 > 0)
so that the threshold is zero. From the figure, the power for the U
statistics is much higher than the tests based on linear models,
and we see the value of considering nonlinear alternatives in
predictive regressions. The tests based on linear models have
power less than size and power decreases as the size of the step
increases.
6.
EMPIRICAL APPLICATIONS
The procedures discussed in this article can be used to test the
significance of the regressor xt−1 without specifying a functional
form. We use a subset of the data discussed in Campbell and
Yogo (2006) and a full description of the data and sources can
be found there. We let yt represent excess monthly returns from
the NYSE/AMEX value-weighted index from 1952 to 2002.
Moreover, the excess returns are calculated by subtracting the
one-month T-bill rate for each month. The series xt represents
four different series taken in turn; the log of the dividend–price
ratio computed using dividends over the past year divided by
price, the log of earnings price ratio, the 3-month T-bill rate, and
the long–short yield spread.6 We have 612 observations. The unit
root test of Elliott, Rothenberg, and Stock (1996) is calculated
using the modified SIC and Akaike criteria developed in Ng and
Perron (2001) to select the number of lags in the test. One lag
of xt is selected by both criteria and the test value is −0.243
which is not significant at the 10% level. Hence, we fail to
reject the unit root hypothesis for the log of the dividend-price
ratio. This suggests that a standard t-ratio in a regression of yt
on xt−1 would not be well approximated by a standard normal
distribution.7 In light of the persistence in xt−1 , we employ
the procedure in Campbell and Yogo (2006) using the Q-test.
Based on inverting the Q-test, a 90% confidence interval for the
regression parameter is (−0.004, 0.010) which contains zero.
Hence, there is no evidence for linear predictability of returns
using the log of the dividend–price ratio.
To consider the possibility of a nonlinear relationship, we
conduct a nonparametric kernel regression of yt on xt−1 . As
noted in Bandi (2004), kernel regression can be standardized so
that the normal approximation still holds if xt−1 has a unit root
or near unit root. We tried several bandwidth parameters, and the
results were not sensitive within a large range of bandwidths.
A representative graph of the estimated function as well as
95% pointwise confidence bands is shown in Figure 5 using a
bandwidth of h = σx T −1/5 .
There are several features to note about the estimated regression function in Figure 5. Although the estimated function looks
highly nonlinear, we notice that the 95% pointwise confidence
bands contain zero when the log dividend–price ratio is low,
so that the null hypothesis of no effect cannot be rejected.
6The
Figure 4. Threshold power.
However, at higher levels of the log dividend–price ratio, zero is
not in the confidence bands, and so one would presume that we
can reject the null of no effect of xt−1 on yt for higher levels of
xt−1 . However, this is not a valid statistical argument. The width
of each confidence interval is based on a pointwise argument
which means at each point, the confidence interval is valid if
we consider the function at that point in isolation. However,
in this graph, we have estimated the function at 612 points.
Therefore, the confidence bands shown in the graph are not
confidence bands for the entire function. One adjustment is to
use a Bonferroni argument to make the confidence bands wider.
Such an adjustment is often extremely conservative, especially
when many points of the function are estimated, as in this case.
Eubank and Speckman (1993) compared the pointwise bands
with the conservative Bonferroni bands to verify the increased
width. An alternative approach is to construct asymptotically
accurate confidence bands using the approach of Härdle
(1989b). Such an approach involves constructing confidence
bands based on the maximal deviation of the Gaussian process
long yield is the Moody’s seasoned Aaa corporate bond yield.
7From
Table 1, we see that if the null hypothesis of no relationship between yt
and xt−1 is true, we would still reject the null hypothesis around 40% of the
time.
Figure 5. Dividend price ratio function.
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Juhl: A Nonparametric Test of the Predictive Regression Model
393
Figure 6. Earnings price ratio function.
Figure 8. Yield spread function.
associated with the kernel estimator over the range of the data.
Moreover, it is unknown if such an approach is valid when the
data have any type of dependence.
An alternative to correcting the confidence bands for the nonparametric regression function is to simply construct the Ustatistic developed in this article. We calculate the test and get a
value of 0.33. The U-statistic is normally distributed and rejects
for large values. Hence, we fail to reject the null hypothesis of
no predictability (g(xt−1 ) = 0). In fact, the p-value for a test
statistic of 0.33 is 0.38, so that there is very little evidence of
predictability. Inference using the U-statistic changes the conclusion found using a nonparametric function with pointwise
confidence bands.
Next, consider the relationship between excess returns and the
log of the earnings price ratio, one of the variables considered in
the forecasting model of Lettau and Ludvigson (2001). As noted
in Campbell and Yogo (2006), the unit root hypothesis is not
rejected for this data series. We plot estimated nonparametric
function and along with the 95% pointwise confidence bands
in Figure 6. Even using the pointwise confidence bands, there
appears to be no evidence of predictability. The U-statistic takes
a value of −0.28 with a p-value of 0.61, which confirms the
graphical evidence for lack of predictability.
The relationship between excess returns and 3-month Treasury bill rates is explored in Figure 7. Again, there is little
evidence of predictability, and the U-statistic of −0.54 with a
p-value of 0.71 confirms.
Finally, as in Boudoukh, Richardson, and Whitelaw (1997),
the predictive regression is estimated using the yield spread as a
predictor. The 95% confidence interval for the largest root of the
series does not contain one, but Campbell and Yogo (2006) noted
that the upper limit is 0.968. We plot the estimated relationship
between excess returns and the yield spread in Figure 8. The
estimated function and pointwise confidence bands suggest predictability that is significantly different from zero in the middle
range of the spread values. The function is nonmonotonic with
the predictability disappearing is either tail of the distribution of
the spread. The U-statistic takes a value of 2.19 with a p-value
of 0.02, so that we reject the null hypothesis of no predictability.
The use of the U-statistic shows that the predictability is not just
an artifact of the pointwise confidence bands being too narrow.8
7.
CONCLUSION
In this article, we have examined nonparametric tests applied
to predictive regression models. In particular, we have applied
recent results of Wang and Phillips (2012) for cointegrated models to tests based on U-statistics for the predictive regression
model. The most useful feature of the tests is the lack of the
so-called razor’s edge in asymptotic theory typically associated
with unit root models. The limiting distribution for the properly
8I
Figure 7. T-bill function.
also test the null of linearity using the U-statistic proposed in Zheng (1996).
The test is based on residuals from a linear model and uses the same bandwidth
as the nonparametric tests proposed in this article. The variable used is the yield
spread and find a p-value of 0.0985, so that there is some evidence that the
relationship is nonlinear.
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394
Journal of Business & Economic Statistics, July 2014
standardized statistic is standard normal if the predictor variable
is stationary or if it has a unit root.
A Monte Carlo experiment shows that the test has good size
properties and has power against some nonlinear alternatives
that tests based on linear models tend to miss. In particular, if
the alternative is nonmonotonic, the U-statistic based tests have
a distinct advantage.
An empirical section is used to show the usefulness of the
test. In one example, we fail to find evidence for predictability of returns using the log dividend–price ratio as a predictor.
The example shows how the test is an important complement
to tests based on a linear model and standard nonparametric
estimation. In particular, the nonparametric U-statistic provides
a way to circumvent the currently intractable problem of finding accurate (not conservative) uniform confidence bands for
nonparametric regression functions if the regressor has a unit
root. In our example, pointwise confidence bands suggest the
existence of predictability of returns at very high levels of the
dividend–price ratio. However, this is due to the liberal nature
of pointwise confidence bands. In the example using the yield
spread, we find some evidence of predictability where the significance of the predictability is apparent only in the middle
range of the data. Moreover, there is evidence that we can reject a linear model for this data. These applications serve to
illustrate that the proposed U-statistic provides a simple way
to test the significance of a nonparametric regression function
uniformly across all observed data in a predictive regression
model.
ACKNOWLEDGMENTS
We thank an Associate Editor, two anonymous referees,
Roger Koenker, Peter Phillips, Zhongjun Qu, and Zhijie Xiao
for comments on earlier versions of this article. I thank Michael
Jansson for providing the code for his procedures.
[Received May 2013. Revised November 2013.]
REFERENCES
Bandi, F. M. (2004), “On the Persistence and Nonparametric Estimation With an Application to Stock Return Predictability,” Graduate
School of Business, University of Chicago, available at http://faculty.
chicagobooth.edu/federico.bandi/research/b04.pdf. [392]
Boudoukh, J., Richardson, M., and Whitelaw, R. F. (1997), “Nonlinearities
in the Relation Between the Equity Premium and the Term Structure,”
Management Science, 43, 371–385. [393]
Campbell, J. Y., and Yogo, M. (2006), “Efficient Tests of Stock Return Predictability,” Journal of Financial Economics, 81, 27–60.
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Autoregressive Unit Root,” Econometrica, 64, 813–836. [392]
Eubank, R., and Speckman, P. (1993), “Confidence Bands in Nonparametric
Regression,” Journal of the American Statistical Association, 88, 1287–
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Fan, Y., and Li, Q. (1996), “Consistent Model Specification Tests: Omitted
Variables and Semi-Parametric Functional Forms,” Econometrica, 64, 865–
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——— (1999), “Central Limit Theorem for Degenerate U-Statistics of Absolutely Regular Processes With Applications to Model Specification Testing,”
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Härdle, W. (1989a), Applied Nonparametric Regression, Cambridge: Cambridge
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——— (1989b), “Asymptotic Maximal Deviation of M-Smoothers,” Journal
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Jansson, M., and Moreira, M. J. (2006), “Optimal Inference in Regression
Models With Nearly Integrated Regressors,” Econometrica, 74, 681–714.
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Lettau, M., and Ludvigson, S. (2001), “Consumption, Aggregate Wealth, and
Expected Stock Returns,” Journal of Finance, 56, 815–849. [393]
Lewellen, J. (2004), “Predicting Returns with Financial Ratios,” Journal of
Financial Economics, 74, 209–235. [387,388,390]
Ng, S., and Perron, P. (2001), “Lag Length Selection and the Construction of
Unit Root Tests With Good Size and Power,” Econometrica, 69, 1519–1554.
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Stambaugh, R. F. (1999), “Predictive Regressions,” Journal of Financial Economics, 54, 375–421. [387]
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ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
A Nonparametric Test of the Predictive Regression
Model
Ted Juhl
To cite this article: Ted Juhl (2014) A Nonparametric Test of the Predictive Regression Model,
Journal of Business & Economic Statistics, 32:3, 387-394, DOI: 10.1080/07350015.2014.887013
To link to this article: http://dx.doi.org/10.1080/07350015.2014.887013
Published online: 28 Jul 2014.
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A Nonparametric Test of the Predictive
Regression Model
Ted JUHL
University of Kansas, 415 Snow Hall, Lawrence, KS 66045 (juhl@ku.edu)
This article considers testing the significance of a regressor with a near unit root in a predictive regression
model. The procedures discussed in this article are nonparametric, so one can test the significance of a
regressor without specifying a functional form. The results are used to test the null hypothesis that the entire
function takes the value of zero. We show that the standardized test has a normal distribution regardless of
whether there is a near unit root in the regressor. This is in contrast to tests based on linear regression for
this model where tests have a nonstandard limiting distribution that depends on nuisance parameters. Our
results have practical implications in testing the significance of a regressor since there is no need to conduct
pretests for a unit root in the regressor and the same procedure can be used if the regressor has a unit root
or not. A Monte Carlo experiment explores the performance of the test for various levels of persistence of
the regressors and for various linear and nonlinear alternatives. The test has superior performance against
certain nonlinear alternatives. An application of the test applied to stock returns shows how the test can
improve inference about predictability.
KEY WORDS: Finance; Kernel regression; Near unit root; Time series.
1.
INTRODUCTION
The predictive regression model has received much attention
in recent years. In particular, the model where the regressors are
persistent (with a unit root or near unit root) has been covered
in the work of Campbell and Yogo (2006), Jansson and Moreira
(2006), Lewellen (2004), and Stambaugh (1999), among others.
The statistical estimation and testing of such models requires a
treatment beyond the traditional normal approximations for the
regression parameters. The persistence of the regressors directly
affects the distribution of the standard t-ratio of the estimated
regression parameters. The resulting distribution of the regression parameter is related to the Dickey–Fuller distribution but
with a nuisance parameter from the correlation of innovations to
the unit root series in the future. The distribution is nonnormal
so that standard asymptotic theory does not apply. Moreover, as
the amount of correlation between innovations in the persistent
series and the innovations in the regressand series increases,
the distribution shifts to the right so that a type of “spurious”
regression further distorts standard inference. These facts indicate the need for knowledge of the persistence of the regressor,
or the need to develop new procedures that are “robust” to the
unknown level of persistence in the regressor.
Given the above problems, Campbell and Yogo (2006),
Jansson and Moreira (2006), and Lewellen (2004) each
proposed solutions to inference when the level of persistence
of the regressor is unknown and possibly contains a unit root
(or near unit root). The approaches are related in that they
attempt to find an optimal test in some class of statistics that are
invariant to location and scale transformations of the data. The
test we propose in this article shares the invariance properties
of the above statistics. However, our test is designed to have
power against linear and nonlinear alternatives. To accomplish
this, we use nonparametric kernel estimation of the regression
function. The estimated function is then tested to see if it is
significantly different from a constant.
The nonparametric procedure that we use is based on two
steps. First, we use nonparametric regression to estimate the
relationship between the two series. The advantage of such an
approach is that we are not required to posit a functional form
for the relationship. One may have a theoretical basis for what
the functional form may look like, but it could be that some
unknown financial distortions may exist. By employing a nonparametric approach, we may detect an unforeseen relationship.
For our purposes, we use kernel regression, and we employ a
convenient reweighting procedure. Next, we see if the estimated
regression function is correlated with the demeaned data. Such
a procedure results in a U-statistic, and is known to be effective in specification testing for stationary data. The novelty of
using U-statistics for this problem is that we are able to test
whether the entire function is a constant. Testing for the significance of the function at a given number of points would then
require some sort of Bonferroni argument which would become
more conservative as we estimate more points of the function.
By using the U-statistic approach, it is possible to test the null
hypothesis that the entire function is a constant.
When testing the significance of regressors using nonparametric techniques, we have power against a much wider class of
alternative hypotheses. Of course, with such generality comes
a price in that power is not as high if there truly exists a linear
relationship. We explore these costs via a Monte Carlo experiment. We find that the nonparametric test does outperform tests
based on the linear for certain nonlinear alternatives.
The remainder of the article is organized as follows. In Section 2, we describe the predictive regression model. Section 3
provides an outline of the nonparametric test. The asymptotic
results are derived in Section 4 and explored in a Monte Carlo
387
© 2014 American Statistical Association
Journal of Business & Economic Statistics
July 2014, Vol. 32, No. 3
DOI: 10.1080/07350015.2014.887013
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388
Journal of Business & Economic Statistics, July 2014
experiment detailed in Section 5. An empirical illustration using
stock returns is treated in Section 6 and Section 7 concludes.
d
Notation is standard with weak convergence denoted by →. All
limits in the article are taken as the sample size T → ∞.
2.
E [yt E(yt |xt−1 )p(xt−1 )] = E (E [yt E(yt |xt−1 )p(xt−1 )|xt−1 ])
= E (E [yt E(yt |xt−1 )|xt−1 ] p(xt−1 ))
PREDICTIVE REGRESSION MODEL
Let the variable yt denote the dependent variable in period t
and let xt−1 denote a predictor variable observed at time t − 1.
We are interested to see if there is any relationship between yt
and xt−1 . We consider a model of the form
yt = μy + g(xt−1 ) + ǫt
xt = μx + ut
ut = ρT ut−1 + et .
with μy being the unconditional mean of yt , μx the mean of xt ,
and ρT = 1 − c/T . Note that in this case, the variable xt has
a “near” unit root. This is an extreme form of dependence in
the predictor variable but one that serves as an approximation
for the behavior of many financial ratios used in this type of
analysis. Several studies have versions of this model where
g(xt−1 ) is considered a linear function, and the findings suggest
that the distribution of OLS in the linear case is nonstandard
and depends on the correlation between ǫt and et . In light of
this result, Jansson and Moreira (2006), Campbell and Yogo
(2006), Lewellen (2004), and others have proposed tests that
are not based on the usual t-ratio of the OLS squares estimator
of the slope coefficient for xt−1 . The Jansson and Moreira (2006)
test is optimal in the class of conditionally unbiased tests, and
the distribution of their test is also nonstandard, requiring a
numerical integration routine to calculate p-values for the test
for each new dataset.
In this article, we propose testing the predictive regression
model using the nonparametric tests described in the next
section.
3.
for some value of xt−1 . Let p(xt−1 ) be the density function of
xt−1 . To simplify the discussion in this section, suppose that yt
has mean zero. Then using iterated expectations, we have
NONPARAMETRIC TESTS FOR STATIONARY
REGRESSORS
In this section, we summarize the type of nonparametric tests
that we employ in this article. This type of test has been used
when the data are assumed to be stationary as in Fan and Li
(1996, 1999) and Zheng (1996) among others. The idea is to
check if two series are related without specifying how they are
related. That is, we allow for the series to have an unknown,
perhaps nonlinear, relationship. To be specific, suppose that yt
and xt−1 are stationary series and that
yt = μy + g(xt−1 ) + ǫt ,
almost everywhere versus the alternative
H1 : g(xt−1 ) = 0
E E(yt |xt−1 )2 p(xt−1 ) ≥ 0.
(3.2)
There is equality to zero in (3.2) if and only if E(yt |xt−1 ) =
g(xt−1 ) = 0 almost everywhere. The above relation suggests
that we should find an estimate of E [yt E(yt |xt−1 )p(xt−1 )] and
test whether the estimate is significantly different from zero.
The nonparametric tests estimate E(yt |xt−1 )p(xt−1 ) by kernel
regression of yt on xt−1 multiplied by the density estimate of
p(xt−1 ). That is,
T
xt−1 − zs
1
1
Ê(yt |xt−1 ) =
K
ys
p̂(xt−1 ) T h s=t−1
h
and
T
xt−1 − xs
1
Ê(yt |xt−1 )p̂(xt−1 ) =
ys ,
K
T h s=t−1
h
where K(·) is a kernel function and h is a bandwidth parameter.1 The estimate of E(yt |xt−1 )p(xt−1 ) is particularly tractable
because the term p̂(xt−1 ) cancels from the denominator, eliminating the need for restricting the estimated density away from
zero. This “reweighting” of the conditional mean was also employed in Zheng (1996) and Fan and Li (1996). Moreover, we
weight the estimated function by the estimated density to account for the fact that even if g(xt−1 ) is nonzero, it may only
happen on a set with measure zero, hence the product of the
function and the density will be zero. The final estimate of
E(yt E(yt |xt−1 )p(xt−1 )) is the statistic
T
T
1
xt−1 − xs
yt ys .
K
T 2 h t=1 s=t−1
h
Inference is based on half of this quantity, which is given by
T t−1
xt−1 − xs
1
yt ys .
K
UT = 2
T h t=2 s=1
h
When the statistic UT is properly scaled, it is normally distributed under the null hypothesis that g(xt−1 ) = 0. Due to its
nonparametric construction, the test has power against a wide
variety of alternatives, both linear and nonlinear.
(3.1)
where E(ǫt |xt−1 ) = 0 which implies that xt−1 and ǫt are uncorrelated and that E(ǫt ) = 0. The model then implies that
E(ǫt g(xt−1 )) = 0, which are the conditions used for nonlinear
least squares. The hypothesis that we wish to test is
H0 : g(xt−1 ) = 0
which is equal to
4.
DISTRIBUTION THEORY
The discussion in the last section is based on the assumption
that the variables involved are stationary. Such results have been
shown to hold when the independent variables satisfy some type
of limit to the dependence, typically a condition stated in terms
of β-mixing (absolute regularity), such as in Fan and Li (1999).
1For
an introduction to kernel regression and nonparametric regression, see
Härdle (1989a).
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Juhl: A Nonparametric Test of the Predictive Regression Model
In this section, we provide the distribution theory when the
independent variable is nonstationary in the form of a near unit
root.
The model is
yt = μy + g(xt−1 ) + ǫt
xt = μx + ut
ut = ρT ut−1 + et .
The assumptions are listed below.
Assumption 1.
ut = ρT ut−1 + et
et =
∞
φi wt−i
i=0
ρT = 1 −
c
T
u0 = 0
∞ 1+δ
with i=0 φi = 0 and i=0 i |φi | < ∞ for some δ > 0. The
sequence (wt , ǫt )⊤ is iid with mean zero and matrix . The
variance of wt is 1, and ǫt has finite fourth moments. The characteristic function of wt is given by ϕ(t) such that |t|ϕ(t) → 0
as |t| → ∞.
∞
Assumption 2. K(x) is nonnegative and bounded with
|x|K(x)dx < ∞.
Assumption 3. T → ∞, h → 0, T h2 → ∞, and T h4 log2
T → 0.
These assumptions imply those used by Wang and Phillips
(2012) for their cointegrated model.2 Assumption 1 limits the
amount of dependence in the innovations of the process driving
xt , so that the resulting U-statistic itself is a martingale which
is subject to the new central limit theorem. Assumption 2 is
standard for kernels and Assumption 3 limits the rate at which
the bandwidth goes to zero. Based on these assumptions, we can
apply the model given in Wang and Phillips (2012) applies and
gives the following result.
Theorem 4.1. Given Assumptions 1–3, under the null hypothesis that g(xt−1 ) = 0 almost everywhere, the statistic
√
T 5/4 hÛT d
→ N(0, 1),
T 1/2 V̂T
where
T t−1
xt−1 − xs
1
(yt − ȳ)(ys − ȳ)
K
ÛT = 2
T h t=2 s=1
h
389
It is important to note that the standardized version of the
U-statistic is exactly the same statistic analyzed in Fan and Li
(1999), where xt is stationary with limited dependence (β mixing). Because of this result, there is no need to change the statistical analysis of predictive regression models using U-statistics
if one is unsure about the possibility of a unit root in the variable
xt−1 . This fact is unusual in that standard OLS tests in predictive regression models have a nonstandard limiting distribution
if xt−1 has a unit root or near unit root. Moreover, when using
OLS, the distribution depends on nuisance parameters such as
the correlation between ǫt and et . In our case, the limiting distribution does not change if the series xt−1 is stationary or has a
unit root.
The result is different from existing results for nonparametric
analysis of models with unit roots since all existing results are
given by estimating the functions pointwise. One must have a
method to test that the estimated function is constant almost everywhere. As stated in the introduction, it is possible to construct
a Bonferroni argument using pointwise results. However, as the
number of estimated points increases, the confidence bands for
the estimated function become more and more conservative. The
advantage of the U-statistic approach is that we directly test the
null hypothesis that the entire function is a constant.
From a practical point of view, the test is a good complement
to the existing tests based on the linear model in that it may
detect dependence in xt−1 and yt that is nonlinear, and the tests
are not affected by assumptions about the dependence in xt−1 .
One important consideration of nonparametric regression is
the choice of bandwidth parameters. The assumptions used in
Theorem 4.1 only require that h goes to zero at some prescribed
rate. No optimal bandwidth is proposed in this article, but we
suggest using h = d σ̂x T −1/5 where d is a scalar and σ̂x is the
sample standard deviation of xt−1 . In our simulations, we try
various values of d to gauge the sensitivity of the tests to the
bandwidth parameter. One motivation for using a multiple of
σ̂x is that the statistic will become scale invariant. The test
is always location invariant but using a bandwidth that is a
multiple of σ̂x ensures scale invariance. Although this procedure
is similar to methods used in practice for stationary data, there
3
The sample
is a difference when xt−1 is possibly integrated.
√
standard deviation will diverge at rate T as T → ∞, so that
h will diverge at rate T 3/10 . An alternative is to use the sample
standard deviation of
xt−1 , denoted σ̂
x , which would be
appropriate if xt−1 contains a unit root, for example, and would
also remain scale invariant. In finite samples, it remains to be
seen how the possible integration properties of xt−1 affect the
bandwidth, and, ultimately, the performance of the U-statistic.
In the Monte Carlo section, we explore the performance of the
U-statistic using the sample standard deviation of both xt−1
and
xt−1 as a basis for the bandwidth.
and
T t−1
xt−1 − xs 2
1
V̂T = 2
K
(yt − ȳ)2 (ys − ȳ)2 .
T h t=2 s=1
h
5.
MONTE CARLO EVIDENCE
We compare the nonparametric test with the two leading
tests developed by Campbell and Yogo (2006) and Jansson and
2In
particular, the partial sums of (wt , ǫt )⊤ satisfy a multivariate invariance
principal from Assumption 1. I thank an anonymous referee for suggesting a
more lucid exposition.
3We
thank an anonymous referee for pointing out the divergence and the potential problem with the resulting bandwidth.
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390
Journal of Business & Economic Statistics, July 2014
Moreira (2006). The experiment uses the following model:
yt = μy + g(xt−1 ) + ǫt
xt = ρT xt−1 + et .
We set the parameter σǫe = −0.75 and −0.95, values that are
consistent with the empirical regularities found in financial
data. For example, if xt−1 represents a dividend–price ratio
and yt represents a stock return, then a positive shock to
returns in the form of a large value of ǫt would then cause
a concurrent decrease in the dividend–price ratio, so that the
correlation between et and ǫt should be negative. Lewellen
(2004) estimated values of σǫe ranging from −0.88 to −0.96
for various financial ratios used for xt−1 .
We consider several possible levels of persistence in xt
through the parameter ρ. In particular, we parameterize ρ as
c
ρT = 1 −
T
so that ρT is “local to unity.” The values of c used in the simulation are 0, 5, 10, 15, and 20. In this first experiment, we examine
the size properties of the test. That is, we generate the data so
that g(xt−1 ) = 0 and we tabulate the percentage of rejections in
10,000 replications. The sample sizes considered are T = 100,
T = 500, and T = 1000. There are four different bandwidth
parameters that are examined for the UT statistic. The statistics
denoted by U1 and U2 use h = σ̂x T −1/5 and h = 2σ̂x T −1/5 ,
respectively. The statistics Ud1 and Ud2 use bandwidth parameters that are based on the standard deviation of the differences
of xt , so that the respective bandwidth parameters are given by
h = σ̂
x T −1/5 and h = 2σ̂
x T −1/5 The statistic of Jansson and
Moreira (2006) is a uniformly most powerful conditionally unbiased test in the linear model, so it is denoted and UMPCU. The
Campbell and Yogo test uses a Bonferroni bound and we denote
this test as CYB. For comparison purposes, we include an OLS
t-test that the linear regression coefficient on xt−1 is zero and
denote this as TSTAT. This test is incorrect in the case when xt
has a unit root or root local to unity, but is included to illustrate
the potential problems of ignoring the extreme dependence in
xt−1 . The sizes of the tests appear in Table 1.
From Table 1, we see that the UMPCU and CYB tests have
size close to the nominal size in all cases. The U-statistic is
conservative and gets more conservative for a larger bandwidth.
As discussed earlier, the t-statistic is not normally distributed
and we included this statistic to show the costs of ignoring the
Table 1. Size of competing tests
Size
T = 100
c
UMPCU
U1
U2
Ud1
Ud2
TSTAT
CYB
σǫe = −0.95
c=0
c=5
c = 10
c = 15
c = 20
0.046
0.051
0.055
0.057
0.052
0.031
0.028
0.031
0.031
0.029
0.019
0.015
0.019
0.019
0.020
0.047
0.040
0.036
0.039
0.037
0.045
0.033
0.032
0.027
0.026
0.395
0.178
0.125
0.106
0.087
0.052
0.050
0.045
0.049
0.046
σǫe = −0.75
c=0
c=5
c = 10
c = 15
c = 20
0.051
0.050
0.051
0.055
0.051
0.033
0.029
0.030
0.033
0.029
0.020
0.017
0.017
0.018
0.018
0.048
0.037
0.043
0.039
0.038
0.042
0.030
0.030
0.032
0.030
0.287
0.135
0.110
0.090
0.085
0.040
0.034
0.036
0.033
0.039
c=0
c=5
c = 10
c = 15
c = 20
0.044
0.053
0.059
0.063
0.060
0.038
0.032
0.035
0.031
0.035
0.025
0.021
0.023
0.023
0.025
0.053
0.046
0.047
0.047
0.052
0.042
0.042
0.050
0.044
0.047
0.414
0.191
0.138
0.116
0.104
0.035
0.031
0.032
0.028
0.023
c=0
c=5
c = 10
c = 15
c = 20
0.044
0.056
0.058
0.061
0.060
0.035
0.035
0.034
0.035
0.036
0.024
0.021
0.025
0.024
0.026
0.053
0.043
0.041
0.046
0.053
0.043
0.053
0.036
0.052
0.049
0.290
0.155
0.119
0.102
0.092
0.023
0.028
0.025
0.023
0.022
c=0
c=5
c = 10
c = 15
c = 20
0.043
0.056
0.067
0.070
0.063
0.030
0.025
0.028
0.037
0.045
0.015
0.020
0.023
0.022
0.032
0.047
0.031
0.049
0.053
0.050
0.051
0.037
0.048
0.044
0.051
0.426
0.188
0.132
0.102
0.115
0.039
0.036
0.034
0.020
0.029
c=0
c=5
c = 10
c = 15
c = 20
0.046
0.052
0.075
0.061
0.067
0.026
0.029
0.046
0.035
0.043
0.015
0.025
0.039
0.032
0.027
0.052
0.041
0.036
0.051
0.049
0.057
0.030
0.046
0.049
0.050
0.292
0.143
0.134
0.105
0.106
0.017
0.021
0.023
0.022
0.025
σǫe = −0.95
T = 500
σǫe = −0.75
σǫe = −0.95
T = 1000
σǫe = −0.75
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Juhl: A Nonparametric Test of the Predictive Regression Model
391
Figure 1. Linear power.
Figure 2. Quadratic power.
unit root behavior of xt−1 . At a nominal size of 5%, we reject the
null hypothesis of no predictability approximately 40% of the
time when the innovations from the regressors and regressand
are highly correlated. When the parameter c increases (near
unit root), the normal approximation becomes less damaging.
However, even when c = 5 with an autoregressive parameter
of ρT = 0.95, we reject the null roughly 18% of the time. The
sample size T appears to have a minor affect on the performance
of all the tests considered in the experiment. In general, the
bandwidth associated with the standard deviation of xt−1 (U1
and U2) are more conservative than those associated with the
standard deviation of
xt−1 (Ud1 and Ud2).
To determine the sensitivity of the U-statistic to possible
asymmetric distributions, we generate the same data using
a centered χ12 distribution which is clearly skewed. The
experiment produces results that are very similar to those
in Table 1, and suggest that the normal distribution is still a
reasonable approximation for the test statistic even if the errors
are not from a symmetric distribution.4
For power comparisons, we now consider three alternative
functions. In the first case, the alternative is linear so that
the relationship involving 3-month Treasury bill rates exhibits
a similar shape over the middle region of historical rates.
The power curves are presented in Figure 2. As expected, the
U-statistics have the highest power. One phenomenon that
we found for quadratic alternatives was that the linear based
tests (UMPCU and CYB) had power functions that reached
a plateau at around 30% and never increased. This power
function shows the possible gains from using a U-statistic in
predictive regression models.
For a different alternative, we consider a nonlinear function
that is monotonic. The quadratic alternative gives a distinct
advantage to the U-statistic approach since a linear regression
estimate will be near zero. If we choose an alternative that
remains monotonic, such an advantage may disappear so that
the tests based on linear alternatives will again have higher
power. Our alternative is given by
3
.
g(xt−1 ) = βxt−1
The power curve is shown in Figure 3. The power of the U
statistic is less than the competing tests since for this cubic
alternative, the relationship is monotonic.
g(xt−1 ) = βxt−1 .
For purposes of comparison with Jansson and Moreira (2006),
2 γ , with γ = b/T , and let b increase. The
we set β = 1 − σeǫ
power functions for linear alternatives are shown in Figure 1.5
We notice that the Bonferroni test procedure of Campbell and
Yogo (2006) dominates all other tests for linear alternatives.
Even though our U-statistics are less powerful than the tests
based on linear alternatives, they still have power against linear
alternatives
For an example of a nonlinear alternative, we let
2
,
g(xt−1 ) = βxt−1
so that the alternative is a quadratic. The quadratic is included
as an example of a nonmonotonic function where linearity
would have little power. Moreover, in the empirical section,
4The
5We
results are available upon request.
use T = 100 for all of the power figures.
Figure 3. Cubic power.
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392
Journal of Business & Economic Statistics, July 2014
The final alternative is a step function that depends on a
“threshold” value. In particular, the function is given by
g(xt−1 ) = β1(xt−1 > 0)
so that the threshold is zero. From the figure, the power for the U
statistics is much higher than the tests based on linear models,
and we see the value of considering nonlinear alternatives in
predictive regressions. The tests based on linear models have
power less than size and power decreases as the size of the step
increases.
6.
EMPIRICAL APPLICATIONS
The procedures discussed in this article can be used to test the
significance of the regressor xt−1 without specifying a functional
form. We use a subset of the data discussed in Campbell and
Yogo (2006) and a full description of the data and sources can
be found there. We let yt represent excess monthly returns from
the NYSE/AMEX value-weighted index from 1952 to 2002.
Moreover, the excess returns are calculated by subtracting the
one-month T-bill rate for each month. The series xt represents
four different series taken in turn; the log of the dividend–price
ratio computed using dividends over the past year divided by
price, the log of earnings price ratio, the 3-month T-bill rate, and
the long–short yield spread.6 We have 612 observations. The unit
root test of Elliott, Rothenberg, and Stock (1996) is calculated
using the modified SIC and Akaike criteria developed in Ng and
Perron (2001) to select the number of lags in the test. One lag
of xt is selected by both criteria and the test value is −0.243
which is not significant at the 10% level. Hence, we fail to
reject the unit root hypothesis for the log of the dividend-price
ratio. This suggests that a standard t-ratio in a regression of yt
on xt−1 would not be well approximated by a standard normal
distribution.7 In light of the persistence in xt−1 , we employ
the procedure in Campbell and Yogo (2006) using the Q-test.
Based on inverting the Q-test, a 90% confidence interval for the
regression parameter is (−0.004, 0.010) which contains zero.
Hence, there is no evidence for linear predictability of returns
using the log of the dividend–price ratio.
To consider the possibility of a nonlinear relationship, we
conduct a nonparametric kernel regression of yt on xt−1 . As
noted in Bandi (2004), kernel regression can be standardized so
that the normal approximation still holds if xt−1 has a unit root
or near unit root. We tried several bandwidth parameters, and the
results were not sensitive within a large range of bandwidths.
A representative graph of the estimated function as well as
95% pointwise confidence bands is shown in Figure 5 using a
bandwidth of h = σx T −1/5 .
There are several features to note about the estimated regression function in Figure 5. Although the estimated function looks
highly nonlinear, we notice that the 95% pointwise confidence
bands contain zero when the log dividend–price ratio is low,
so that the null hypothesis of no effect cannot be rejected.
6The
Figure 4. Threshold power.
However, at higher levels of the log dividend–price ratio, zero is
not in the confidence bands, and so one would presume that we
can reject the null of no effect of xt−1 on yt for higher levels of
xt−1 . However, this is not a valid statistical argument. The width
of each confidence interval is based on a pointwise argument
which means at each point, the confidence interval is valid if
we consider the function at that point in isolation. However,
in this graph, we have estimated the function at 612 points.
Therefore, the confidence bands shown in the graph are not
confidence bands for the entire function. One adjustment is to
use a Bonferroni argument to make the confidence bands wider.
Such an adjustment is often extremely conservative, especially
when many points of the function are estimated, as in this case.
Eubank and Speckman (1993) compared the pointwise bands
with the conservative Bonferroni bands to verify the increased
width. An alternative approach is to construct asymptotically
accurate confidence bands using the approach of Härdle
(1989b). Such an approach involves constructing confidence
bands based on the maximal deviation of the Gaussian process
long yield is the Moody’s seasoned Aaa corporate bond yield.
7From
Table 1, we see that if the null hypothesis of no relationship between yt
and xt−1 is true, we would still reject the null hypothesis around 40% of the
time.
Figure 5. Dividend price ratio function.
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Juhl: A Nonparametric Test of the Predictive Regression Model
393
Figure 6. Earnings price ratio function.
Figure 8. Yield spread function.
associated with the kernel estimator over the range of the data.
Moreover, it is unknown if such an approach is valid when the
data have any type of dependence.
An alternative to correcting the confidence bands for the nonparametric regression function is to simply construct the Ustatistic developed in this article. We calculate the test and get a
value of 0.33. The U-statistic is normally distributed and rejects
for large values. Hence, we fail to reject the null hypothesis of
no predictability (g(xt−1 ) = 0). In fact, the p-value for a test
statistic of 0.33 is 0.38, so that there is very little evidence of
predictability. Inference using the U-statistic changes the conclusion found using a nonparametric function with pointwise
confidence bands.
Next, consider the relationship between excess returns and the
log of the earnings price ratio, one of the variables considered in
the forecasting model of Lettau and Ludvigson (2001). As noted
in Campbell and Yogo (2006), the unit root hypothesis is not
rejected for this data series. We plot estimated nonparametric
function and along with the 95% pointwise confidence bands
in Figure 6. Even using the pointwise confidence bands, there
appears to be no evidence of predictability. The U-statistic takes
a value of −0.28 with a p-value of 0.61, which confirms the
graphical evidence for lack of predictability.
The relationship between excess returns and 3-month Treasury bill rates is explored in Figure 7. Again, there is little
evidence of predictability, and the U-statistic of −0.54 with a
p-value of 0.71 confirms.
Finally, as in Boudoukh, Richardson, and Whitelaw (1997),
the predictive regression is estimated using the yield spread as a
predictor. The 95% confidence interval for the largest root of the
series does not contain one, but Campbell and Yogo (2006) noted
that the upper limit is 0.968. We plot the estimated relationship
between excess returns and the yield spread in Figure 8. The
estimated function and pointwise confidence bands suggest predictability that is significantly different from zero in the middle
range of the spread values. The function is nonmonotonic with
the predictability disappearing is either tail of the distribution of
the spread. The U-statistic takes a value of 2.19 with a p-value
of 0.02, so that we reject the null hypothesis of no predictability.
The use of the U-statistic shows that the predictability is not just
an artifact of the pointwise confidence bands being too narrow.8
7.
CONCLUSION
In this article, we have examined nonparametric tests applied
to predictive regression models. In particular, we have applied
recent results of Wang and Phillips (2012) for cointegrated models to tests based on U-statistics for the predictive regression
model. The most useful feature of the tests is the lack of the
so-called razor’s edge in asymptotic theory typically associated
with unit root models. The limiting distribution for the properly
8I
Figure 7. T-bill function.
also test the null of linearity using the U-statistic proposed in Zheng (1996).
The test is based on residuals from a linear model and uses the same bandwidth
as the nonparametric tests proposed in this article. The variable used is the yield
spread and find a p-value of 0.0985, so that there is some evidence that the
relationship is nonlinear.
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394
Journal of Business & Economic Statistics, July 2014
standardized statistic is standard normal if the predictor variable
is stationary or if it has a unit root.
A Monte Carlo experiment shows that the test has good size
properties and has power against some nonlinear alternatives
that tests based on linear models tend to miss. In particular, if
the alternative is nonmonotonic, the U-statistic based tests have
a distinct advantage.
An empirical section is used to show the usefulness of the
test. In one example, we fail to find evidence for predictability of returns using the log dividend–price ratio as a predictor.
The example shows how the test is an important complement
to tests based on a linear model and standard nonparametric
estimation. In particular, the nonparametric U-statistic provides
a way to circumvent the currently intractable problem of finding accurate (not conservative) uniform confidence bands for
nonparametric regression functions if the regressor has a unit
root. In our example, pointwise confidence bands suggest the
existence of predictability of returns at very high levels of the
dividend–price ratio. However, this is due to the liberal nature
of pointwise confidence bands. In the example using the yield
spread, we find some evidence of predictability where the significance of the predictability is apparent only in the middle
range of the data. Moreover, there is evidence that we can reject a linear model for this data. These applications serve to
illustrate that the proposed U-statistic provides a simple way
to test the significance of a nonparametric regression function
uniformly across all observed data in a predictive regression
model.
ACKNOWLEDGMENTS
We thank an Associate Editor, two anonymous referees,
Roger Koenker, Peter Phillips, Zhongjun Qu, and Zhijie Xiao
for comments on earlier versions of this article. I thank Michael
Jansson for providing the code for his procedures.
[Received May 2013. Revised November 2013.]
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