1. Introduction
People routinely examine the predictability problem, for example, the mutual fund performance, the conditional capital asset pricing, and the optimal asset allocations. For the predictability of stock returns, various lagged financial variables are used, for example, the log dividend-price ratio, the log earning-price ratio, the log book-to-market ratio, the dividend yield, the term spread, default premium, and the interest rates [3]. Since many of the predictive financial variables are highly persistent and even non-stationary, it is challenging econometrically to explore the predictability of asset returns.
Predictability issues are generally addressed in parametric regressions in which rates of returns are regressed against the lagged values of stochastic explanatory variables. In predictive linear structure model [1,2], excess stock return is the predictable variable at time t, innovations 
are independently and identically distributed bivariate normal and the log dividend-price ratio is a financial variable at time
, which is modelled by an AR(1) model.
There are three limitations. At first, two innovations are unfortunately correlated in real applications [3,4]. The second difficulty arises from the unknown parameter for financial variable regression, for stationary case, see [4,5,7,8], for unit root or integrated, see [9-11], and for local-to-unity or nearly integrated, see [3,12-16]. The third difficulty comes from the instability of the predictive regression model. It concluded from many evidences on the dividend and earnings yield and the sample from the second half of the 1990s that the coefficients should change over time, see, for example [4,5,7,17-19].
In finite samples, the ordinary least squares estimate of the slope coefficient and its standard errors are substantially biased if explanatory variable is highly persistent, not really exogenous, and even non-stationary, see [20]. To avoid over-rejecting the null of non-predictability, some improvements arise, such as the first order biascorrection estimator [2], the two-stage least squares estimator [8], and the conservative bias-adjusted estimator [21], but the instability difficulty was kept silent. To deal with this issue, some predictive regression models were analyzed, for example, excess return predictive regression model on international equity indices [4], equity return predictive regression model [5] with random coefficients generated from a unit root process, asset regression model with varying coefficients [22]. A predictive functional regression model has not touched, though not only interesting in its applications to finance and economics, but also enriching the econometric theory.
The rest of this paper runs as follows. Section 2 proposes basic functional regression model. Section 3 is for nonparametric estimation. Section 4 derives the consistency for the proposed estimator. Section 5 concludes the paper.
2. Basic Model
We propose a functional regression model to capture the stability of asset returns. It is well known that a nonlinear function would better to characterize dynamic relationship between the stock return and the related financial variables, the two innovations may have a time dependent nonlinear relationship, and the log dividend-price ratio
, is a integrated or nearly integrated process [3, 22]. Our model runs as follows.
(1)
(2)
where innovation 
is exogenous.
To remove the endogeneity, we project 
onto 
by
, which is strictly increasing in 
and 
is uncorrelated with 
and
. See, for example, [23] for endogenous variable. Thus the model becomes
(3)
(4)
The function f can be estimated once function h is estimated due to the strict increasing of 
with respect to 
and the Equation (4). Indeed if the functions f and g are linear, the model reduces to [22].
3. Nonparametric Estimation
Once parametric structures are not specified for the functions h in the economic model, the function h is nonadditive in
. If the function is additive in unobservable random term
, one can interpret this added unobservable random term as being a function of the observable and other unobservable variables, which is hard to estimate this function of the observable and unobservable variables. Here we estimate a nonparametric function h, not necessarily additive.
To estimate the regression function h in the basic model (3), we will derive its expression in terms of the distribution of the vector of the observable variables. Once the unknown regression function is expressed in terms of the distribution of
, we will derive its nonparametric estimator for the unknown regression function by substituting the distribution of the observable variables. Though any type of nonparametric estimator for this distribution can be used, we present here the details and asymptotic properties for the case in which the conditional cumulative distributed functions are estimated by the method of kernels. To express the unknown function in terms of the distribution of the observable variables, we need the following assumptions [24].
Assumption 1 
is independent of 
and
, and
.
Assumption 2 For all values of 
and
, the function h is strictly increasing in
.
Assumption 1 guarantees that the distribution of 
is the same for all values of 
and
. Assumption 2 guarantees that the distribution of 
can be obtained from the conditional distribution of 
given 
and
.
Theorem 3 Under Assumptions 1 and 2, the mapping between the unknown regression function h and
, the distribution of the observable variables 
is given by
(5)
for all 
with
.
Proof.
(6)
(7)
(8)
(9)
According to the theorem above, the following four cases hold. 
Lemma 4 (Case 1) For all 
and some 
with
,
(10)
and Assumptions 1 and 2 hold. Then
(11)
(12)
Lemma 5 (Case 2) For all 
and some 
with
, and 
such that 
and
,
(13)
(14)
and Assumptions 1 and 2 hold. Then
(15)
(16)
Lemma 6 (Case 3) For some unknown function
, all 
and some
, some
, and some 
such that
, and
(17)
(18)
Assumptions 1 and 2 hold, and for all
, 
is strictly increasing. Then, for
,
(19)
(20)
Lemma 7 (Case 4) For some unknown function
, all 
and some
, some
, and some 
such that
, and
(21)
(22)
Assumptions 1 and 2 hold, and for all
, 
is strictly increasing. Then, for
,
(23)
(24)
Let 
denote the data, 
and
, respectively, the joint probability distribution function and cumulative distribution function of
, 
and
, respectively, their kernel estimators, and 
and 
the kernel estimators of the conditional probability distribution function and cumulative distribution function of 
given 
and
. Then, according to [6], for all
,
(25)
(26)
(27)
(28)
where 
is a kernel function and 
is the bandwidth. Hence, for case 1,
(29)
for case 2,
(30)
for case 3,
(31)
for case 4,
(32)
4. Consistency
The consistency and asymptotic normality of the estimator of the marginal or conditional distribution of 
will follow from the consistency and asymptotic normality of the kernel estimator for the conditional distribution of y given x and
. In particular, the asymptotic properties for each of the estimators for the distribution of 
given above can be derived from Theorem 13 after substituting the corresponding values of y, x, and
. For this result, we need following assumptions.
Assumption 8 The sequence 
is independently identically distributed.
Assumption 9 
has compact support 
and it is continuously differentiable on 
up to the order 
for some
.
Assumption 10 The kernel function 
is differentiable of order
, the derivatives of K of order 
are Lipschitz, 
vanishes outside a compact set, integrates to 1, and is of order 
where
.
Assumption 11 As 
and
, 
, 
, 
, and 
.
Assumption 12
.
Assumptions 8, 9, 10, 11 and 12 for 
are similar to Assumptions 
in [24] for
.
Theorem 13 Let 
denote the kernel estimator for the conditional distribution of Y conditional on x and 
evaluated at
. Assumptions 8, 9, 10, 11 and 12 hold. Then, for 
and
,
(33)
in probability, and
(34)
in distribution, where
(35)
Proof. It is the case for 
in the Theorem 1 in [24] in their notations when 
is not an argument. 
Theorem 13 states that 
converges to 
in the supremum norm, and 
is asymptotically normal with mean 
and variance equal to
.
To study the asymptotic properties of the estimator for the unknown function h, notice that Equation (3), the estimator for the unknown regression function h can be obtained by substituting the true conditional distributions of Y by their kernel estimators, the consistency and asymptotic normality of the estimator of h will follow from the consistency and asymptotic normality of the functional, 
, of the kernel estimator for the distribution of
. For this result, one more assumption is required as follows.
Assumption 14 The vectors 
and 
have at least one coordinate in common, and the values 
and 
are different at one such coordinate;
,
; and there exist 
such that
,
.
Assumption 14 is the Assumption 
if 
in their notations.
Theorem 15 Assumptions 8, 9, 10, 11 and 14 hold for 
and
. Let
,
. Then,
(36)
in probability, and
(37)
in distribution, where
(38)
Proof. It is the case for 
of the Theorem 2 in [24] in their notations when X0 is not an argument. 
Theorem 15 implies that 
is consistent and asymptotically normal with mean 
and asymptotic variance equal to
(39)
5. Conclusions
This paper studied a predictive regression model which includes the state variable of NI(1) or I(1) and allows endogeneity, where nonlinear regression function is not necessarily additive in unobservable random terms.
We develop a nonparametric method for estimating the functional regression and find that the estimators for the distribution of the unobservable random terms and the nonparametric function are consistent and asymptotically normal. The estimators are nonlinear functionals of a kernel estimator for the distribution of the observable variables. However, the model specification or stationary is not discussed here.
More investigations are worth for the predictive application of this functional regression model due to its importance in various applications in economics and finance. For example, we here keep silent of mixing of 
and 
in the context of nonparametric functional predication, though a time-varying coefficient model is valid in [22].
6. Acknowledgement
This project was sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry ([2010]609), and the Science Research Foundation, Hubei Provincial Department of Education, P. R. China (D20111508) respectively.
NOTES