Gilles De Truchis
email: detruchis.gilles@parisnanterre.fr

Elena-Ivona Dumitrescu
email: elena.dumitrescu@parisnanterre.fr

Narrow-band Weighted Nonlinear Least Squares Estimation of Unbalanced Cointegration Systems

Keywords: Unbalanced cointegration, Long memory, Stationarity, Generalized Least Squares, Nonlinear Least Squares JEL: C22, G10

We discuss cointegration relationships when covariance stationary observables exhibit unbalanced integration orders. Least squares type estimates of the long run coefficient are expected to converge either to 0 or to infinity if one does not account for the true unknown unbalance parameter. We propose a class of narrow-band weighted non-linear least squares estimators of these two parameters and analyze its asymptotic properties. The limit distribution is shown to be Gaussian, albeit singular, and it covers the entire stationary region in the particular case of the generalized non-linear least squares estimator, thereby allowing for straightforward statistical inference. A Monte Carlo study documents the good finite sample properties of our class of estimators. They are further used to provide new perspectives on the risk-return relationship on financial stock markets. In particular, we find that the variance risk premium estimated in an appropriately rebalanced cointegration system is a better return predictor than existing risk premia measures.

Introduction

Cointegration has attracted increasing attention since the seminal paper of [START_REF] Granger | Some properties of time series data and their use in econometric model specification[END_REF] as a large part of economic theory involves long-run equilibrium relationships between economic and/or financial variables. Initially, most of the analyses have focused on the estimation of the long-run slope coefficient in the I(1)/I(0) paradigm (single unit-root in observables and short memory in cointegration errors) using OLS type regressions. There is evidence, however, that in many economic and financial applications the variables and the residuals are fractionally integrated. In the bivariate case, if y t and x t are I(δ 2 ), δ 2 ∈ R + , and there exists β 0 such that

y t = βx t + u t , (1)
where u t is I(δ 1 ) with δ 2 > δ 1 , then y t and x t are said to be fractionally cointegrated. In particular, if δ 2 ∈ (0, 1/2), y t and x t are covariance stationary with long memory. However, the risk of spurious regression when estimating β is present even for such processes as long as their orders of integration sum up to a value greater than 1/2 (see [START_REF] Tsay | The spurious regression of fractionally integrated processes[END_REF]. This strong result has led to a vast literature on stationary fractional cointegration, which generally uses frequency domain analysis. Indeed, the autocovariance function of a stationary long memory variable has a frequency domain representation known as spectral density, f y (λ), so that

E y t -E(y t ) y t+h -E(y t ) = π -π
e ikλ f y (λ)dλ,

with λ = 2πj/n, the Fourier frequencies for j = 1, . . . , n. As in a cointegration framework interest lies in the long-run dynamics of the process, i.e. the estimation of β, one can rely on the power law approximation of f y (λ) in the vicinity of the origin

f y (λ) ∼ gλ -2δ 2 , as λ → 0 + , ( 2
)
where 0 < g < ∞ and "∼" means that the ratio of the left and right sides converges to 1 in the limit, to avoid specifying the short-run dynamics explicitly. Robinson (1994a) exploits this representation to propose a narrow-band version of the frequency domain least-squares estimator of β, also known as NBLS. Its asymptotic properties are investigated by [START_REF] Robinson | Semiparametric frequency domain analysis of fractional cointegration[END_REF] and [START_REF] Christensen | Asymptotic normality of narrow-band least squares in the stationary fractional cointegration model and volatility forecasting[END_REF]. It is found that the convergence rate of β depends on the cointegration strength, δ 2δ 1 , and statistical inference based on asymptotically Gaussian distribution is valid as long as the collective memory of x t and u t does not exceed 1/2. Otherwise, the limit distribution is non-standard and presumably of Rosenblatt-type as shown by Robinson (1994b) for the OLS estimator. Motivated by such observations, [START_REF] Robinson | Time series regression with long-range dependence[END_REF] introduce the frequency domain (full-band) weighted least squares estimator. It is shown to encompass the frequency domain generalized least squares estimator and to converge at conventional √ n rate to a Gaussian limit distribution if y t is covariance stationary without further assumptions regarding the long memory of u t . As the weight function, φ(λ), is unknown, it can be either set to one, in which case the estimator comes down to the OLS in frequency domain,

or parametrically estimated by relying on the inverse of the spectral density of u t , f u t (λ) -1 , in which case one obtains the feasible generalized least squares estimator. Since the latter hinges on the correct specification of f u t (λ), a non-parametric estimator of the weight function was introduced by [START_REF] Hidalgo | Adapting to unknown disturbance autocorrelation in regression with long memory[END_REF]. Finally, [START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF] adopts an intermediate semi-parametric approach based on the approximation of the weight function in the neighborhood of the origin, i.e. φ(λ) = λ 2δ as λ → 0 + with δ the weighting parameter. For δ = 0 (δ = δ1 ), the narrow-band weighted least squares (hereafter NBWLS)

of [START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF] reduces to the feasible narrow-band (generalized) least squares.

All these studies assume that y t and x t are stationary and integrated of the same order δ 2 . Never-theless, economic theory may sometimes suggest the existence of a long run equilibrium relationship between variables, while econometric tests conclude against the equality of integration orders. [START_REF] Hualde | Unbalanced Cointegration[END_REF] analyzes this unbalanced situation and shows that one can retrieve a nontrivial balanced hidden cointegration by estimating an additional parameter, ξ, which characterizes the level of unbalance. He summarizes the asymptotic behavior of the OLS and NBLS estimators of β in a nonstationary framework by considering different balanced and unbalanced cointegrating situations. The accurate estimation of the unbalance parameter appears to be crucial for the limit properties of the long run parameter as otherwise the least-square-type estimates of β are not consistent. Indeed, they converge to 0 or diverge to infinity depending on the sign of ξ (see e.g. [START_REF] Robinson | Narrow-band analysis of nonstationary processes[END_REF]. In this vein, [START_REF] Hualde | Estimation of long-run parameters in unbalanced cointegration[END_REF] proposes a joint nonlinear least squares (NLS) estimator of β and ξ for δ 2 > 1/2 and shows that its limit distribution is very complex and depends on a modified type II fractional Brownian motion (see [START_REF] Hualde | Weak convergence to a modified fractional Brownian motion[END_REF]. Besides, he conjectures that his time domain nonlinear estimator would be inconsistent in the stationary region of the long memory parameters. More generally, to the best of our knowledge, no estimator specifically designed for stationary unbalanced triangular systems exists to date.

This paper contributes to this literature by proposing a class of narrow-band weighted nonlinear least squares (NBWNLS) estimators specifically designed for stationary unbalanced fractionally cointegrated relationships. Note that one recovers Nielsen (2005)'s NBWLS linear estimator when the true unbalance parameter is null. Although nonlinear, our joint estimators of β and ξ inherit the desirable asymptotic properties of the linear weighted least squares estimator of [START_REF] Robinson | Time series regression with long-range dependence[END_REF]. When the weight function is set to one, the NBWNLS comes down to the the narrow-band nonlinear least squares (NBNLS). Even if the NBNLS estimates are always feasible, they are generally asymptotically less efficient than the NBWNLS unless the optimal weighting parameter is δ = 0. Still, one can use them to estimate the optimal weighting parameter that further leads to the feasible narrow-band generalized nonlinear least squares (NBGNLS) estimator. We show that, under regular assumptions, our class of estimators is consistent and asymptotically Gaussian (over the entire stationary region for the NBGNLS). The convergence rates of both β and ξ depend on the cointegration strength. Most importantly, that of the unbalance estimator also depends on the true long run parameter β, while the converse does not hold. Consequently, their joint limit distribution is singular, although both estimators are identifiable. Our class of estimators can be seen as an extension of [START_REF] Robinson | Gaussian semiparametric estimation of long range dependence[END_REF] and [START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF] to a non-linear framework, which, in this particular case, accounts for the possible unbalance of the system and as a complementary approach to [START_REF] Hualde | Estimation of long-run parameters in unbalanced cointegration[END_REF] in the sense that it is particularly designed for stationary regions. These theoretical findings are subsequently corroborated by Monte Carlo experiments for different levels of cointegration strength. Our class of estimators and their main (balanced) competitors are evaluated in terms of bias, variance and root-mean-square error criteria. The simulation results reveal the good small-sample properties of our estimators.

To illustrate the usefulness of our approach, we revisit the risk-return relationship by drawing on [START_REF] Bollerslev | Risk and return: Long-run relations, fractional cointegration, and return predictability[END_REF] who show that the difference between implied and realized variance measures, i.e. the variance risk premium, is a good return predictor in the long run. As our framework accounts for possible unbalance of integration orders of the variance measures, our approach provides more accurate estimates of the variance risk premium, which results in better return predictability for four main stock market indices (S&P500, Dow Jones, Russell, NASDAQ).

The rest of the paper is organized as follows. In Section 2 we introduce our narrow-band weighted nonlinear least squares estimator, while in Section 3 we discuss its consistency and asymptotic normality.

The particular case of the feasible NBGNLS is investigated in Section 4. Section 5 presents the results of the Monte Carlo studies. An empirical application is proposed in Section 6 and then, finally, we conclude.

All proofs are gathered in Appendix A, B and C.

Narrow-band weighted nonlinear least squares

Let the unbalanced version of Equation (1) be

y t = βx t (ξ) + u t (3)
where

x t (ξ) = (1 -L) ξ x t = ∑ ∞ k=0 a k (ξ)L k , with a k (ξ) := Γ(k -ξ)(Γ(-ξ)k!) -1 , x t ∼ I(δ 2 + ξ), y t ∼ I(δ 2
), and u t ∼ I(δ 1 ). To allow for the presence of common stationary long-run dynamics in (3), we introduce a regularity assumption on the memory parameters.

Assumption 1. y t , x t and y tβx t (ξ) are covariance stationary processes integrated of orders δ 2 , δ 2 + ξ and δ 1 respectively with β 0, and satisfying

0 ≤ δ 1 < δ 2 < δ 2 + |ξ| < 1/2, ( 4
)
where |ξ| < k, with k an arbitrary real number that is small compared to δ 2 .

Under Assumption 1, δ 2 > δ 1 such that β is identifiable, and β 0 to ensure the identification of ξ.

It follows that f z (λ j ), the joint spectral density of z t = (u t , x t ) , exists, where λ j denotes the Fourier frequencies, λ j = 2πj/n, with j = 1, . . . , m and m is the bandwidth parameter. Setting m = o(n), we avoid a parametric treatment of f z (λ) by specifying the spectral density only locally around the zero frequency

f z (λ) ∼ Λ(λ) -1 G Λ(λ) * -1 , Λ(λ) = diag λ δ 1 , λ δ 2 +ξ , as λ → 0 + (5)
where the superscript " * " denotes the conjugate transpose. As [START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF] we assume that G is diagonal, thereby ruling out the possibility of any endogeneity in the vicinity of the origin but without imposing any further restrictions on frequencies away from the origin in contrast to [START_REF] Hualde | Estimation of long-run parameters in unbalanced cointegration[END_REF]. Now, let I uu (λ j ) = w u (λ j )w u (λ j ) * be the periodogram of u t with w u (λ j ) = (2πn) -1/2 ∑ n t=1 u t e itλ j the Fourier transform of u t . In view of (3), I uu (λ j ) actually comes down to

I uu (λ j ) = I yy (λ j ) -Re 2βλ ξ I xy (λ j ) + β 2 λ 2ξ I xx (λ j ) .
Note that the presence of λ ξ j corrects for the fact that the long memory parameters of y t and x t are unbalanced. In this context, we identify two natural ways to estimate the long-run and unbalance parameters and show that they are particular cases of a more general class of NBWNLS estimators. A first approach to estimate θ = (β, ξ) consists in the NBNLS method,

θ(0) = arg min θ∈Θ L m (θ), where L m (θ) = m -1 m ∑ j=1 I uu (λ j ) (6)
and

Θ = Θ β × Θ ξ is a compact subset of R 2 .
However, a more efficient alternative appears naturally from the local Whittle approximation to the likelihood associated with Equation (3),

Q m (θ, G uu ) = m -1 m ∑ j=1 log f u (λ j ) + I uu (λ j ) f u (λ j ) , (7)
where G uu ∈ Θ G uu , the set of real positive numbers. The objective function Q m is minimized over Θ G uu by Ĝuu (θ;

δ 1 ) = Re m -1 m ∑ j=1 λ 2δ 1 I uu (λ j ) , (8)
where λ 2δ 1 arises as an implicit weight function. Substituting (8) in ( 7) and solving for θ, we obtain the local Whittle quasi maximum likelihood estimator (QMLE) of β and ξ,

θ(δ 1 ) = arg min θ∈Θ R m (θ), (9)
where R m (θ) = log Ĝuu (θ; δ 1 ) is the concentrated likelihood function. Notice that θ(δ 1 ) is implicitly a NBGNLS estimator and in the sequel we will retain this label for the QMLE so as to maintain a clear connection with the NBNLS estimator in (6). Finally, we relax the definition of the weighting parameter and introduce the general class of NBWNLS estimators

θ(δ) = arg min θ∈Θ W m (θ) with W m (θ) = log Ĝuu (θ; δ). ( 10
)
Said otherwise, the NBNLS and NBGNLS estimators are particular cases of the NBWNLS estimator that correspond to δ = 0 (i.e. no weight) and δ = δ 1 , respectively.

Limit theory

In this section we present the limit theory of the general class of infeasible NBWNLS estimators θ(δ)

and discuss the particularities of the NBNLS and NBGNLS estimators when necessary. As [START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF] and [START_REF] Christensen | Asymptotic normality of narrow-band least squares in the stationary fractional cointegration model and volatility forecasting[END_REF], we work under several assumptions commonly used in this literature.

Assumption 2. The elements of the spectral density f z (λ) satisfy

| f ab z (λ) -G 0 ab λ -ϑ 0a -ϑ 0b | = O(λ α-ϑ 0a -ϑ 0b ), a, b = {1, 2},
as λ → 0 + and for some α ∈ (0, 2] with ϑ 0 = (δ 01 , δ 02 + ξ 0 ) , matrix G 0 finite, real, symmetric and where

G 0 ab = G 0 ba = 0.
Assumption 2 imposes a smoothness condition on the spectral density matrix f z (λ) and specifies the properties of G 0 , which imply a zero-coherence condition that applies in the vicinity of the origin. As argued in [START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF], G 0 ab = G 0 ba = 0 is a less restrictive assumption than the traditional orthogonality condition encountered in linear (see e.g. [START_REF] Robinson | Time series regression with long-range dependence[END_REF]) and non-linear (see [START_REF] Hualde | Estimation of long-run parameters in unbalanced cointegration[END_REF] least squares theory. In particular, it allows for the presence of correlation in the errors as we move away from the origin, i.e. they can share a common short-and/or medium-term dynamics.

Assumption 3. The sequence z t = (u t , x t ) is a linear process defined as

z t -E(z t ) = A(L)ε t = ∞ ∑ j=0 A j ε t-j , ∞ ∑ j=0 ||A j || 2 < ∞,
with ||.|| the Euclidean norm, so that A j is a causal square summable matrix filter. Moreover, ε t satisfies, almost surely, E(ε t |F t-1 ) = 0 and E(ε t ε t |F t-1 ) = I 2 and we further impose that the matrices

µ 3 = E(ε t ⊗ ε t ε t |F t-1 ) and µ 4 = E(ε t ε t ⊗ ε t ε t |F t-1
) are non-stochastic, finite and do not depend on t, with F t a σ-field generated by {ε s , s ≤ t}.

Assumption 4. In a neighborhood of the origin, A(λ) = ∑ ∞ j=0 A j e ijλ is differentiable and

∂ ∂λ A a (λ) = O(λ -1 ||A a (λ)||) as λ → 0 +
where A a (λ) is the a-th row of A(λ).

Assumptions 3 and 4 are standard in this literature since [START_REF] Robinson | Gaussian semiparametric estimation of long range dependence[END_REF]. The former imposes uniformly square integrable martingale-difference innovations with constant conditional variance, while the latter implies ∂A a (λ)/∂λ = O(λ -ϑ 0a -1 ) by the Cauchy inequality

||A a (λ)|| ≤ (A a (λ)A * a (λ)) 1/2 = (2π f aa (λ)) 1/2 .
Thereby, under Assumptions 3 and 4 we have

f z (λ) = (2π) -1 A(λ)A(λ) * .
Assumption 5. As n → ∞, the bandwidth parameter m = o(n) and α ∈ (0, 2] jointly satisfy

1 m + m 1+2α (log m) 2 n 2α → 0.
The bandwidth expansion rate is restricted by Assumption 5 as one needs m to tend to ∞ as n → ∞ but at a slower rate so as to remain in a neighborhood of the origin where the behaviour of the spectral density is of interest. Theoretically, the upper bound on the bandwidth parameter m is n 4/5 (i.e. α = 2) but in practice a too small bandwidth increases the variance of the estimator while a too large m generally increases the bias.

Assumption 6. The weight function φ(λ) = λ 2δ as λ → 0 + and the weighting parameter δ satisfies

1 2 δ 2 + δ 1 -1/2 < δ ≤ δ 1 . ( 11
)
Assumption 6 restricts the weighting parameter as a function of the long memory parameters as discussed in [START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF]. When δ 1 + δ 2 < 1/2, δ = 0 is covered by our theory and consequently the NBWNLS estimator is equivalent to the NBNLS one and should be uniformly Gaussian over the weighting parameter space. When δ = δ 1 , this assumption restricts the cointegration strength to δ 2δ 1 < 1/2, which is consistent with Assumption 1. Under slightly weaker versions of the assumptions above, which are generally stated in the literature, we establish the consistency of our NBWNLS class of estimators.

Theorem 1. Let Assumptions (1)-( 5) be satisfied and assume 0 ≤ δ ≤ δ 1 . Then, as n → ∞,

λ δ 01 -δ 02 m   1 log λ m     β(δ) -β 0 ξ(δ) -ξ 0   p -→   0 0  
where θ(δ) = ( β(δ) , ξ(δ) ) .

In addition, the central limit theorem for β(δ) and ξ(δ) is stated below.

Theorem 2. Under Assumptions (1)-( 6), as n → ∞,

√ mλ δ 01 -δ 02 m   1 log λ m     β(δ) -β 0 ξ(δ) -ξ 0   d -→   1 β -1 0   N (0, E -1 FE -1 ).
where θ(δ) = ( β(δ) , ξ(δ) ) and

E = 2G xx G uu (1 -2δ 02 + 2δ) , F = 2G xx G uu (1 + 4δ -2δ 01 -2δ 02 )
.

Sketch of proof. The proof of Theorem 2 relies on Theorem 1 and the usual Taylor series expansion for extremum estimators

∂W m (θ) ∂θ θ = ∂W m (θ) ∂θ θ 0 + ∂ 2 W m (θ) ∂θ∂θ θ ( θ -θ 0 ) = 0,
where || θθ 0 || ≤ || θθ 0 ||. Since the joint limit distribution of β and ξ is singular, it is the same up to a constant term β -1 0 . Focusing on β, one can show that

√ mλ δ 01 -δ 02 m ( β -β 0 ) = λ 2(δ 02 -δ) m ∂ 2 W m (θ) ∂β∂β θ -1 √ mλ δ 01 +δ 02 -2δ m ∂W m (θ) ∂β θ 0 has the stated distribution, √ mλ δ 01 -δ 02 ( β -β 0 ) d -→ N (0, E -1 FE -1 ), if by application of Cramer-Wold theorem √ mλ δ 01 +δ 02 -2δ m ∂W m (θ) ∂β θ 0 d -→ N (0, F) and λ 2δ 02 -2δ m ∂ 2 W m (θ) ∂β∂β θ p -→ E.
A similar reasoning allows one to show that √ mλ

δ 01 -δ 02 m log(λ m ) -1 ( ξ -ξ 0 ) d -→ β -1 0 N (0, E -1 FE -1
). The complete proofs of Theorems 1 and 2 are given in Appendix A and B respectively. Unsurprisingly, we recover the standard convergence rate √ mλ

δ 01 -δ 02 m
for β(δ) since its limit theory does not depend on ξ 0 (see e.g. [START_REF] Robinson | Semiparametric frequency domain analysis of fractional cointegration[END_REF][START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF][START_REF] Christensen | Asymptotic normality of narrow-band least squares in the stationary fractional cointegration model and volatility forecasting[END_REF]. Besides, its convergence rate approaches √ n when the cointegration strength is close to 1/2 and it remains higher than the standard semi-parametric √ m rate in all other cases. The limit theory of ξ(δ) is however non-standard. It always converges faster than β(δ) and can be superconsistent when δ 02δ 01 is close to 1/2. This is a peculiar result for memory parameters in stationary semi-parametric frameworks where the maximum achievable rate is √ m and even for parametric frameworks in time and frequency domains (where the maximum rate is √ n). Note also that the limit distribution of ξ(δ) depends on β 0 .

Consequently, although Gaussian and hence simple to use for inference, the joint limit distribution of β(δ) and ξ(δ) is singular. In contrast, the limit distribution of Hualde (2014)'s estimator, which is also singular, involves a modified version of the type II fractional Brownian motion, which makes it hardly suitable for inference.

When Assumption 6 holds, one can compare the relative asymptotic efficiency of θ(0) and θ(δ) . Our NBWNLS estimators are always asymptotically more efficient than the NBNLS ones except for the case δ 1 = 0 as their variances become identical. The maximum gain in efficiency is actually reached when δ = δ 1 , i.e. for the NBGNLS estimator. In this case, the asymptotic relative efficiency of the weighted and unweighted estimators comes down to

V( θ(0) ) V( θ(δ 1 ) ) = (1 -2δ 2 ) 2 (1 -2δ 2 ) 2 -4δ 2 1 .
A second advantage of the NBGNLS estimator over its two competitors is that simple inference is possible for the whole stationary parameter space (see Assumption 6).

Feasible narrow-band generalized nonlinear least squares

The main issue with the NBWNLS approach is that θ(δ) is infeasible in practice since δ is unknown.

Feasible estimates δ are available without knowledge of the memory parameters, e.g. based on residuals from (3), but they have to fulfill Assumption 6 as well as the following requirement Assumption 7. As n → ∞, the estimate of the weighting parameter δ satisfies

(log n)( δ -δ) p -→ 0.
In particular, in view of Theorem 2, the most asymptotically efficient feasible NBWNLS estimator is θ( δ1 ) provided that one can consistently estimate the weighting parameter δ 1 such that Assumption 7 is fulfilled. In the balanced cointegration framework, [START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF] relies on results from [START_REF] Velasco | Gaussian semi-parametric estimation of fractional cointegration[END_REF] to state that δ1δ 1 = O(m -1/2 ) for carefully chosen m and hence Assumption 7 holds. But unlike [START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF], in order to estimate δ 1 in our unbalanced cointegration framework one requires consistent estimators not only for β but also for ξ. For this reason, we propose to use in a first step the NBNLS estimator θ(0) , whose consistency is proven in Appendix A as a Corollary of Theorem 1.

Corollary 1. Let Assumptions (1)-( 5) be satisfied and assume δ = 0. Then, as n → ∞,

λ δ 1 -δ 2 m   1 log λ m     β(0) -β 0 ξ(0) -ξ 0   p -→   0 0   where θ(0) = ( β(0) , ξ(0) ) .
Since ξ(0) converges faster than β(0) , it is innocuous for δ1 's rate of convergence. Consequently, for carefully chosen m, Assumption 7 will always be satisfied if one uses the NBNLS residuals to estimate semi-parametrically, in a second step, δ 1 in an unbalanced framework. The asymptotic distribution of the feasible NBWNLS estimator is given in Theorem 3.

Theorem 3. Under Assumptions 1-6 and 7, as n → ∞, Theorem 2 holds when the weighting parameter δ is replaced by δ1 .

For a proof of Theorem 3 see Appendix C. It indicates that the error arising from the estimation of the weighting parameter does not impact the limit distribution of the parameters of interest stated in Theorem 2.

Monte Carlo experiments

This section discusses the finite sample performance of the NBWNLS class of estimators by means of Monte Carlo simulations. We generate a fractionally cointegrated system according to (3) with β = 1, ξ = 0.1, u t ∼ N (0, 1), and analyze three levels of cointegration strength ν 0 = {0.35, 0.15, 0.05} where δ 2 is fixed to 0.35. We generate 10000 replications for sample sizes n = {256, 512, 1024} and fix the bandwidth parameter to m = {[n 0.6 ], [n 0.75 ]}. Besides, in the case of the long run parameter, β, we compare our three specifications (NBNLS, NBGNLS and NBWNLS) with two misspecified competitors, i.e. the NBLS estimator of Robinson (1994a) and the NBGLS estimator of [START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF]. The NBGNLS estimator involves a weighting parameter δ1 obtained in two steps. First, one gathers the residuals from the NBNLS estimation of β and then estimates δ 1 by a simple local Whittle procedure à la [START_REF] Robinson | Gaussian semiparametric estimation of long range dependence[END_REF]. 2 The same bandwidth is used for the entire estimation procedure. At the same time, for the NBWNLS estimator the weighting parameter is arbitrarily fixed to δ 1 /2.

Tables 1 and2 report the bias, variance and root-mean-square error (RMSE) for all the estimators under analysis. First, we discuss the estimates of β. Notice that in this unbalanced setup our NBWNLS class of estimators performs better than the existing balanced estimators. The NBLS and NBGLS estimators (1994a) and the NBGLS estimator of [START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF], respectively. The NBWNLS estimators are obtained by fixing the weighting parameter to δ 1 /2. Besides, ν 0 = 0.35 corresponds to δ 1 = 0 and in this case the NBWNLS estimators come down to the NBNLS ones (indicated by '-'). (1994a) and the NBGLS estimator of [START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF], respectively. The NBWNLS estimators are obtained by fixing the weighting parameter to δ 1 /2. Besides, ν 0 = 0.35 corresponds to δ 1 = 0 and in this case the NBWNLS estimators come down to the NBNLS ones (indicated by '-').

= 0.1 m = [n 0.6 ] βLS β GLS β(0) ξ(0) β( δ1 ) ξ( δ1 ) β(δ) ξ(δ) β( δ) ξ( δ) n = 256 ν 0 = 0.
= 0.1 m = [n 0.6 ] βLS β GLS β(0) ξ(0) β( δ1 ) ξ( δ1 ) β(δ) ξ(δ) β( δ) ξ( δ) n = 256 ν 0 = 0.
assume that ξ = 0 and exhibit a strong negative bias. This follows from their limit behaviour in an unbalanced framework, where the bias of βGLS is of order

O p (λ ξ 0 m ) as m → ∞ since βGLS = β ∑ m j λ 2δ 01 +ξ 0 j I xx (λ j ) ∑ m j λ 2δ 01 j I xx (λ j ) + O p (λ ν 0 +ξ 0 m ) ∼ O p (λ ξ 0 m ) + O p (λ ν 0 +ξ 0 m ). ( 12
)
By a similar reasoning and setting δ 1 to zero in the previous equation one can show that the asymptotic bias of βLS is of the same order. The finite-sample bias of these estimators is increasing in the sample size and consequently so is the RMSE. By contrast, our estimators exhibit very small finite sample bias.

It diminishes with n and so does the RMSE. At the same time, as expected given our limit theory, the variance of all estimators increases when the cointegration strength reduces and diminishes with the sample size.

Focusing on our three estimators, one notices that for ν 0 = 0.35, i.e. δ 1 = 0, there is almost no difference between them regardless of the sample size. As ν 0 decreases, the NBGNLS and NBWNLS estimators become more efficient than the NBNLS one. The performance of these two estimators also improves relatively to that of the NBNLS when n increases. In particular, the NBGNLS estimator appears to be slightly more efficient than the NBWNLS estimator for large sample sizes and low cointegration strength consistently with the asymptotic theory.

We now focus on the estimates of ξ. All three estimators exhibit similar minor bias levels but the variances of ξ( δ1 ) and ξ (δ) are smaller relatively to that of ξ (0) . The variance of the unbalance parameter estimates is always much lower than that of the corresponding long-run parameter estimates. As expected, this finite-sample finding goes along the lines of the rates of convergence identified in the asymptotic theory (see Theorem 2). All results are robust to the choice of the bandwidth parameter.

Empirical illustration

There is a huge literature in finance that focuses on return predictability. On one hand, multiyear predictability patterns have been studied in association with the default spread, the P/E ratio and the consumption-wealth ratio. But there is extensive evidence that the predictive power of such traditional indicators has been decreasing since the 1990s (see e.g. [START_REF] Fama | Dividend Yields and Expected Stock Returns[END_REF], Campbell and Shiller 1988[START_REF] Lettau | Consumption, AggregateWealth, and Expected Stock Returns[END_REF]. On the other hand, the role of stock market volatility in return forecasting, also known as volatility feedback effect, has been deeply investigated in the existing literature using both conditional and realized measures to proxy risk (see [START_REF] Ghysels | There is a risk-return trade-off after all[END_REF], and the references therein). Merton's Intertemporal CAPM suggests that market's conditional variance should have a positive impact on the conditional expected excess return. Indeed, if volatility is priced, an anticipated increase in volatility would raise the required rate of return, in turn necessitating an immediate stock-price decline to allow for higher future returns. Existing empirical results are however often inconclusive and sometimes conflicting (see [START_REF] Nelson | Conditional heteroskedasticity in asset returns: a new approach[END_REF][START_REF] Campbell | No news is goodnews: an asymmetric model of changing volatility in stock returns[END_REF], Bali and Peng 2006, among others).

In this context, recent studies have theoretically shown that the variance risk premium, i.e. the difference between the implied and realized variance series, should be a useful return predictor (see e.g. [START_REF] Bollerslev | Expected stock returns and variance risk premia[END_REF][START_REF] Drechsler | What's vol got to do with it[END_REF][START_REF] Bollerslev | Volatility in equilibrium: Asymmetries and dynamic dependencies[END_REF]. Interestingly, the long-run risk model developed by [START_REF] Bollerslev | Volatility in equilibrium: Asymmetries and dynamic dependencies[END_REF] accounts for the persistent nature of the unobserved integrated variance, IV t . At the same time, the variance swap rate, mirroring the definition of the squared VIX volatility index, is defined as the risk-neutralized expectation of IV t and inherits the long memory property of volatility. A direct implication of this fractionally integrated framework is the presence of cointegration between V IX 2 t and IV t , with the variance risk premium, VRP t = V IX 2 t -IV t , exhibiting less memory than the volatility series. 3 VRP t is expected to increase if uncertainty about volatility increases.

Hence, the model theoretically predicts that the cross-correlations between the variance risk premium and future returns should be positive, reflecting the premium for bearing volatility risk. These insights are corroborated by empirical evidence on the SP500 index mainly. Indeed, [START_REF] Bollerslev | Risk and return: Long-run relations, fractional cointegration, and return predictability[END_REF] exploit this long-run relationship to extract the variance risk premium and show that it results in nontrivial return predictability over within-year horizons. Further empirical evidence that variance risk premium results in strong return predictability is documented in [START_REF] Bollerslev | Stock return predictability and variance risk premia: statistical inference and international evidence[END_REF][START_REF] Bollerslev | Stock return and cash flow predictability: The role of volatility risk[END_REF].

The theoretical and empirical results aforementioned hinge on model-free measures of integrated and implied (unobserved) variances. But depending on the accuracy of these measurements, the long-run dynamics of the two variance proxies might spuriously diverge. Indeed, empirical differences in the integration orders of the volatility series inherent to the finite sample context may occur and pollute the estimate of the long-run relationship. We therefore conjecture that our general unbalanced cointegration framework is more suitable than existing ones as it implicitly avoids such biases that are likely to appear when estimating the variance risk premium.

To estimate the risk-return relation we essentially follow [START_REF] Bollerslev | Risk and return: Long-run relations, fractional cointegration, and return predictability[END_REF] but rely on data sampled at monthly frequency. Our data set runs from January 2014 to March 2019 and consists in monthly realized measures of volatility for four stock market indexes, S&P500, Dow Jones 30, Russell 2000 and NASDAQ 100. For each index, the realized variance (RV), the realized kernel (RK), the median realized variance (medRV) and the bipower variation (BV) are used as variance proxies to control for the presence of microstructure noise or jumps that might affect the results. 4 As we construct a forwardlooking measure of VRP t , we align beginning of month VIX CBOE indexes, intended to represent a 30-days target framework for expected variance, with the monthly realized measures. 5

3 The fast exponential decay in the autocorrelation structure of VRP t and the slow hyperbolic decay in the autocorrelations of volatility series are consistent with the empirical findings of [START_REF] Bandi | Long memory and the relation between implied and realized volatility[END_REF] and [START_REF] Nielsen | Local Whittle Analysis of Stationary Fractional Cointegration and the Implied-Realized Volatility Relation[END_REF] among others. 4 All realized measures are based on 5-minutes intraday prices and collected from Oxford-Man Institute's realized library website. 5 All VIX data are collected from the CBOE website http://www.cboe.com/.

More appropriate for Whittle estimation and linear regressions, the log transforms vix 2 t = log(V IX 2 t ) and rm t = log( IV t ) are used in the following, with IV standing for any of the four realized measures of variance. Figure 1 reports the vix 2 t versus two rm t measures, RV and BV, in the S&P500 case. A visual inspection clearly reveals that all series exhibit long term swings and similarities in their dynamics. A local Whittle estimation of the long memory parameters confirms that all series are fractionally integrated with estimates confined in the stationary region between 0.3 and 0.5. The bandwidth retained for the local Whittle estimator (and all narrow-band estimations hereafter) by using the informal " optimal" bandwidth selection procedure in Robinson (2008b) is m = n 0.6 where n = 253 for all stock market indexes. This intermediate bandwidth should ensure that our estimates are not contaminated by high frequencies and that they exhibit a reasonable finite sample variance. These results suggest that a long-run relationship may exist between the implied and the realized volatility in the form of rm t = βvix 2 t (ξ) + u t . In contrast to [START_REF] Bollerslev | Risk and return: Long-run relations, fractional cointegration, and return predictability[END_REF], our general unbalanced fractional cointegration framework allows for (possibly small) differences between vix 2 t and rm t long memory parameters. We also implement the (possibly misspecified) NBLS and GNBLS estimators of β. The results are reported in Table 3.

All the estimates of the long-run parameter β are statistically significant apart from the cases where Assumption 6 is violated and hence the standard errors are not available. Our NBNLS and NBGNLS estimates are quite similar, with values close to one, as expected from economic theory. However, the NBLS and NBGLS estimates are systematically lower. This finding appears to be a consequence of the integration order unbalance between the two types of variance proxy. Indeed, the estimates of the unbalance parameter ξ are always positive and significant, indicating that an unbalanced cointegration framework is more appropriate. In line with equation ( 12), we can hence conjecture that even a quite small unbalance (estimated to be around 0.1 here) may have a large impact on the long-run parameter β (a drop of about 0.15). Besides, the integration order of the long-run regression errors does not seem to be statistically different from zero, supporting the fact that vrp t = βvix 2 t (ξ)rm t exhibits less persistence than the (rebalanced) variance series.

We can now express the risk-return relationship in regression form as

r t+1 = α 0 + α 1 vrp t + u t+1 ,
where r t+1 is the one-step-ahead monthly stock market log-return. As one might expect the association between risk and return to be more of a long-run than a short-run phenomenon, we draw on [START_REF] Bollerslev | Risk and return: Long-run relations, fractional cointegration, and return predictability[END_REF] and focus on the long-run component of each variable. Using time-domain band-pass filters, we extract from the observed series the specific low-frequency region up to the Fourier frequency λ = 0.65, which corresponds to the chosen bandwidth and equivalently to a periodicity of about 9 months. For this, we set the truncation parameter to k = 12, resulting in a loss of one year of observations at both ends of the sample. Each filtered series is given by y low t = k ∑ i=-k a i L i y t , where

a i =      sin(iλ) iπ -λ π + 2 ∑ k j=1 sin(jλ) jπ -1 (2k + 1) -1 , i = ±1, . . . , ±k, 1 -∑ -1 h=-k a h -∑ k h=1 a h , i = 0.
Figure 2 displays the low-pass filter returns and variance risk premium based on the realized kernel variance proxy for S&P500. This illustration emphasizes the comovement between the series and motivates the following low-frequency regression analysis.

The slope estimates of the regressions fitted with low-pass filter variables are reported in Table 4. The upper panel shows that the long-run component of the variance risk premia estimated in the unbalanced fractional cointegration (UFC) framework explains a nontrivial fraction of the low-pass returns at an intermediate 9.5 months horizon for all realized measures. More importantly, the slope parameters are positive and consistent with the trade-off between returns and variance risk premium as documented in [START_REF] Bollerslev | Volatility in equilibrium: Asymmetries and dynamic dependencies[END_REF][START_REF] Hualde | A simple test for the equality of integration orders[END_REF][START_REF] Hualde | Estimation of long-run parameters in unbalanced cointegration[END_REF]. α1 estimates are reported in Figure 3 for the Fourier frequency λ ranging in [0.01, 0.9] in the spirit of a robustness analysis. The predictive power of the UFC variance risk premia is stable and reaches its maximum when the low-pass filter is defined in terms of frequencies between 0.6 and 0.8. This covers a horizon of 8 to 10.5 months and supports the usefulness of our approach for return prediction at long horizons where existing literature fails to find significant predictive power.

The middle panel reports the slope parameters in the case where the variance risk premia are computed following the naive approach in [START_REF] Bollerslev | Risk and return: Long-run relations, fractional cointegration, and return predictability[END_REF], i.e. imposing β = 1 and ξ = 0. Neglecting the possible unbalance of the system, their estimates of the long run parameter are close to unity and hence vrp t = vix 2 trm t . The low-pass filtered naive vrp t seems, however, less informative for the returns of the four indices in our monthly sample (α 1 estimates are systematically lower than those estimated with the UFC approach). In contrast, the results reported in the lower part of the table correspond to more conventional regressors, i.e. the low-pass variance risk measures. The estimates are now negative and rarely significant, in line with existing literature on the risk-return puzzle. All in all, the long-run component of variance risk premia estimated in an unbalanced framework seems to carry more useful information about future returns than competing risk proxies.

Conclusion

This paper introduces a class of narrow band weighted non-linear least squares estimators that allow for the joint estimation of the long run and unbalance parameters in a bivariate unbalanced fractionally cointegrated equation. Our estimates are shown to be asymptotically Gaussian with singular distributional properties. Moreover, they exhibit faster than regular √ m rate of convergence. A partic-ular estimator of our class is the feasible narrow-band generalized nonlinear least squares. It offers the same appealing limit properties over the whole stationary region, thereby allowing for straightforward statistical inference. In contrast, existing balanced estimators are inconsistent in presence of unbalance in the integration orders of the observables. This supports the need for a general to specific approach to cointegration as the balanced framework is a specific case of the unbalanced one. Our contribution meets this challenge. By means of Monte Carlo simulations we further show the good finite sample properties of the proposed estimators. Finally, they are used to provide new perspectives on the risk-return relationship on financial stock markets. In particular, we find that the variance risk premium estimated in an appropriately balanced cointegration system is a better return predictor than existing risk premia measures. Robinson (1994a) and the NBGLS estimator of [START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF], respectively. β(0) and ξ(0) correspond to the NBNLS estimator, while β( δ1 ) and ξ( δ1 ) correspond to the NBGNLS estimator. Standard errors are reported in parentheses.

Appendix A : Proof of Theorem 1

Proof. Let θ be the vector of admissible parameter values and θ 0 the vector of true parameter values. For any b > 0 and e > 0, define the neighborhoods

Θ n ξ (e) = {ξ : |ξ -ξ 0 | < e}, Θ n β (b) = {β : |β -β 0 | < b}, and their complements Θ c ξ = Θ ξ \Θ n ξ and Θ c β = Θ β \Θ n β such that Θ n (ε) = Θ n ξ (ε -1 λ ν 0 m log(λ m ) -1 ) × Θ n β (ε -1 λ ν 0 m ), and Θ c (ε) = Θ\Θ n (ε), with ν 0 = δ 02 -δ 01 the true cointegration strength. Since θ 0 ∈ Θ n (ε), it follows that Pr θ ∈ Θ c (ε) = Pr inf θ∈Θ c (ε) W m (θ) ≤ inf θ∈Θ n (ε) W m (θ) ≤ Pr inf θ∈Θ c (ε) W m (θ) -W m (θ 0 ) ≤ 0 ,
where θ stands for θ(δ) . Accordingly, to prove Proposition 1 it suffices to show that as n → 0,

S(θ) = log Ĝuu (θ) -log G uu + log G uu -log Ĝuu (θ 0 ) = S 1 (θ) + S 2 (θ),
is positive and bounded away from 0 uniformly on Θ c (ε) so that Pr inf

θ∈Θ c (ε) S(θ) ≤ 0 → 0 (13)
where S(θ) = W m (θ) -W m (θ 0 ) and Ĝuu (θ) replaces Ĝuu (θ; δ) to simplify notations. From the analysis of [START_REF] Robinson | Gaussian semiparametric estimation of long range dependence[END_REF](Robinson , p. 1635) and the fact that

Pr (|S 2 (θ)| ≤ ε) = Pr | log Ĝuu (θ 0 ) -log G uu | ≤ ε ≤ Pr Ĝuu (θ 0 ) -G uu G uu ≤ ε/2 (14) one can immediately show that S 2 (θ) is o p (1)
. It remains to study S 1 (θ). A similar reasoning to that in ( 14), but in terms of S 1 (θ), shows that its treatment is equivalent to the analysis of Ĝuu (θ)

-G uu = m -1 m ∑ j=1 λ 2δ j I uu (λ j ) -G uu . ( 15
)
Accounting for the measurement errors arising from estimating β and ξ,

I uu (λ) -I 0 uu (λ) = λ 2ν 0 m λ 2ξ 0 j β2 I jxx (λ) -2λ ν 0 m λ ξ 0 j β Re(I 0 jux (λ)),
where

λ ν 0 m λ ξ 0 j β = (β 0 λ ξ 0 j -βλ ξ j ), it follows that Ĝuu (θ) -G uu = 1 m m ∑ j=1 λ 2δ j I 0 uu (λ j ) -2 β G ux m m ∑ j=1 λ 2δ j Re(I 0 ux (λ j )) G ux λ -ν 0 m λ -ξ 0 j + β2 G xx m m ∑ j=1 λ 2δ j I xx (λ j ) G xx λ -2ν 0 m λ -2ξ 0 j -G uu = 1 m m ∑ j=1 λ 2δ j I 0 uu (λ j ) -λ ν 0 m 2 βG ux m m ∑ j=1 λ -ν 0 +2δ-2δ 01 j Re(I 0 ux (λ j )) G ux λ -δ 01 -δ 02 -ξ 0 j + λ 2ν 0 m β2 G xx m m ∑ j=1 λ -2ν 0 +2δ-2δ 01 j I 0 xx (λ j ) G xx λ -2(δ 02 +ξ 0 ) j .
By Proposition 1 of [START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF], as n → ∞, the first term satisfies

1 m m ∑ j=1 λ 2δ j I 0 uu (λ j ) - G uu 1 + 2δ -2δ 1 p -→ 0 since m -1 ∑ m j=1 λ 2δ j f uu (λ j ) -G uu /(1 + 2δ -2δ 1 ) = o λ 1+2δ-2δ 1 m
is valid under our Assumptions. By similar manipulations of the other two terms using Lemma 1 of [START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF] and the analysis of [START_REF] Robinson | Gaussian semiparametric estimation of long range dependence[END_REF]Robinson ( , p. 1638)), one can show that under Assumption 5 Ĝuu (θ)

-G uu = G uu 1 + 2δ -2δ 01 - 2 βG ux 1 + 2δ -δ 01 -δ 02 + β2 G xx 1 + 2δ -2δ 02 -G uu + o p (1).
By Assumptions 1, 2 and 6, the first term is positive and greater than (or equal to) G uu , the second term is null, whereas the last term is bounded away from 0 when

{ θ ∈ Θ c } ∩ { ξ ∈ Θ ξ } or { θ ∈ Θ c } ∩ { β ∈ Θ β } ∀ 0 ≤ δ ≤ δ 01 .
This completes the proof of ( 13) and Theorem 1. Note also that the consistency of the NBNLS emerges in the limit case when δ = 0 and that of the NBGNLS when setting δ = δ 1 .

Appendix B : Proof of Theorem 2

As an implication of the consistency result proved in Theorem 1, θδ satisfies

∂W m (θ) ∂θ θ = ∂W m (θ) ∂θ θ 0 + ∂ 2 W m (θ) ∂θ∂θ θ ( θ -θ 0 ) = 0 where || θ -θ 0 || ≤ || θ -θ 0 ||.
Then, we can show that the NBWNLS estimator converges to the stated distribution

√ mλ δ 1 -δ 2 diag(1, log(λ m ) -1 )( θ -θ 0 ) d -→   1 β -1 0   N (0, E -1 FE -1 ).
if by application of the Cramer-Wold theorem

η √ mλ δ 1 +δ 2 -2δ m diag(1, log(λ m ) -1 ) dW m (θ) dθ θ 0 d -→   1 β -1 0   N (0, η Fη) (16)
and

λ 2δ 2 -2δ m diag(1, log(λ m ) -2 ) ∂ 2 W m (θ) ∂θ∂θ θ p -→   1 β -1 0   E. (17)

B.1. Limit of the score

In this section we investigate the limit of the score in ( 16), while (17) will be analysed in the next section. Note also that the subscript 0, indicating the true parameter values, will be omitted unless its absence causes confusion.

Proof. The derivative with respect to β is

∂W m (θ 0 ) ∂β = - 2 m m ∑ j=1 λ 2δ-δ 1 -δ 2 j
Re Ĝuu (θ 0 ) -1 λ δ 2 +ξ+δ 1 j (I jxyβλ ξ j I jxx ) .

where Ĝuu (θ) is used instead of Ĝuu (θ; δ) to simplify notation. The derivative with respect to ξ is

∂W m (θ 0 ) ∂ξ = - 2 m β m ∑ j=1 λ 2δ-δ 1 -δ 2 j log λ j Re Ĝuu (θ 0 ) -1 λ δ 2 +ξ+δ 1 j (I jxy -βλ ξ j I jxx ) .
We proceed as in Robinson (2008a), albeit in a unidimensional framework, and after rearrangements implying only negligible errors (see Lobato 1999, Appendix C) arising from the replacement of (Λ j I j Λ j ) by P j I ju P * j := Λ j A(λ j )I ju A(λ j ) * Λ j and Ĝuu (θ 0 ) by G as | Ĝuu (θ 0 ) -G| = o p (1), we obtain

η 1 √ mλ δ 1 +δ 2 -2δ m ∂W m (θ) ∂β θ 0 = -η 1 λ δ 1 +δ 2 -2δ m 2 √ m m ∑ j=1 γ j Re G -1 uu P j1 I ju P * j2 + o p (1). ( 18
)
with γ j = λ 2δ-δ 1 -δ 2 j . In (18) we decompose

I ju = (2πn) -1 | ∑ n t ε t | 2 such that η 1 √ mλ δ 1 +δ 2 -2δ m ∂W m (θ) ∂β θ 0 = -η 1 λ δ 1 +δ 2 -2δ m 1 π √ m m ∑ j=1 γ j Re G -1 uu P j1 P * j2 (19) -η 1 λ δ 1 +δ 2 -2δ m 1 π √ m m ∑ j=1 γ j Re G -1 uu P j1 1 n n ∑ t=1 ε t ε t -I 2 P * j2 (20) -η 1 λ δ 1 +δ 2 -2δ m 2 √ m m ∑ j=1 γ j Re G -1 uu P j1 1 2πn n ∑ t=1 ∑ s t ε t ε s e i(t-s)λ j P * j2 , (21)
As f z (λ) = (2π) -1 A(λ)A(λ) * , by Assumptions 2 and 5, ( 19) is

O 1 √ m λ δ 1 +δ 2 -2δ m m ∑ j=1 f 12 (λ j )λ δ 1 +δ 2 +ξ+2δ-δ 1 -δ 2 j = O 1 √ m λ δ 1 +δ 2 -2δ m m ∑ j=1 λ α+2δ-δ 1 -δ 2 j = O(n -α m 1/2+α log m) → 0, (20) is O p 1 √ m λ δ 1 +δ 2 -2δ m m ∑ j=1 1 √ n f 12 (λ j )λ δ 1 +δ 2 +ξ+2δ-δ 1 -δ 2 j = O p λ 1/2+α m log m p -→ 0
by the law of large numbers and the remaining term ( 21) can be rearranged as

- n ∑ t=1 ε t t-1 ∑ s=1 η 1 πn √ m λ δ 1 +δ 2 -2δ m m ∑ j=1 γ j Re G -1 uu P j 1 e i(t-s)λ j Pj2 ε s (22)
where Pj denotes the conjugate of P j .

Regarding ξ, by analogy with (18) we obtain

η 2 √ m λ δ 1 +δ 2 -2δ m log λ m ∂W m (θ) ∂ξ θ 0 = -η 2 λ δ 1 +δ 2 -2δ m log λ m β π √ m m ∑ j=1 log λ j γ j Re G -1 uu P j1 P * j2 (23) -η 2 λ δ 1 +δ 2 -2δ m log λ m β π √ m m ∑ j=1 log λ j γ j Re G -1 uu P j1 1 n n ∑ t=1 ε t ε t -I 2 P * j2 (24) -η 2 λ δ 1 +δ 2 -2δ m log λ m 2β √ m m ∑ j=1 log λ j γ j Re G -1 uu P j1 1 2πn n ∑ t=1 ∑ s t ε t ε s e i(t-s)λ j P * j2 . ( 25
)
As f z (λ) = (2π) -1 A(λ)A(λ) * , by Assumptions (2) and ( 5), ( 23) is

O 1 √ m λ δ 1 +δ 2 -2δ m log λ m m ∑ j=1 log λ j f 12 (λ j )λ δ 1 +δ 2 +ξ+2δ-δ 1 -δ 2 j = O(n -α m 1/2+α log m) → 0, (24) is O p 1 √ m λ δ 1 +δ 2 -2δ m log λ m m ∑ j=1 log λ j 1 √ n f 12 (λ j )λ δ 1 +δ 2 +ξ+2δ-δ 1 -δ 2 j = O p λ 1/2+α m log m p -→ 0
and the remaining term (25) can be rearranged as

- n ∑ t=1 ε t t-1 ∑ s=1 η 2 πn √ m λ δ 1 +δ 2 -2δ m log λ m β m ∑ j=1 log λ j γ j Re G -1 uu P j 1 e i(t-s)λ j Pj2 ε s . (26)
Using ( 26), ( 22) and Euler formula, ( 16) has the same asymptotic distribution as ∑ n t=1 ε t ∑ t-1 s=1 Ξ t-s,n ε s where

Ξ t-s,n = 1 πn √ m m ∑ j=1 (θ j,1 + θ j,2 ) cos((t -s)λ j ), θ j,1 = -λ δ 1 +δ 2 -2δ m η 1 γ j θβ , θ j,2 = - λ δ 1 +δ 2 -2δ m log λ m η 2 log λ j βγ j θβ with θβ = Re G -1 uu P j 1 Pj2 + G -1 uu P j 2 Pj1 . By Assumption 2, ||θ j,1 || = O((m/j) δ 1 +δ 2 -2δ ) and ||θ j,2 || = O((m/j) δ 1 +δ 2 -2δ × (log j)/(log m)). It follows that η 1 λ δ 1 +δ 2 -2δ m √ m ∂W m (θ) ∂θ 1 θ 0 + η 2 λ δ 1 +δ 2 -2δ m log λ m √ m ∂W m (θ) ∂θ 2 θ 0 = n ∑ t=1 ζ t + o p (1)
where ζ t = ε t ∑ t-1 s=1 Ξ t-s,n ε s is a martingale difference array with respect to F t = σ({ε s , s ≤ t}). By a standard martingale central limit theorem, (16) follows if

n ∑ t=1 E(ζ 2 t |F t-1 ) - 2 ∑ a=1 2 ∑ b=1 η a η b Ω ab p -→ 0, ( 27
) n ∑ t=1 E(ζ 2 t 1(|ζ t | > d)) → 0, ∀d > 0, ( 28
)
where Ω is shown to be singular and defined as

Ω =   1 β 0 β 0 β 2 0   F, with F = 2G xx /(G uu (1 + 4δ -2δ 1 -2δ 2 )
). We first show (27). For this, we use the following decomposition

n ∑ t=1 E(ζ 2 t |F t-1 ) = n ∑ t=1 E t-1 ∑ s=1 t-1 ∑ r=1 ε s Ξ t-s,n ε t ε t Ξ t-r,n ε r |F t-1 = n ∑ t=1 t-1 ∑ s=1 ε s Ξ t-s,n Ξ t-s,n ε s + n ∑ t=1 t-1 ∑ s=1 s-1 ∑ r=1 ε s Ξ t-s,n Ξ t-r,n ε r , (29)
where the second term has mean 0 and variance

O n n ∑ s=1 ||Ξ s,n || 2 2 + n ∑ t=3 t-1 ∑ u=2 u-1 ∑ s=1 ||Ξ u-s,n || 2 u-1 ∑ s=1 ||Ξ t-s,n || 2 (30)
as shown by [START_REF] Lobato | A semiparametric two-step estimator in a multivariate long memory model[END_REF]. Following [START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF]

, when s < n/m, ||Ξ s,n || = O(1/(n √ m) ∑ n j=1 ||θ j,1 + θ j,2 ||) = O(n -1 √ m log m) and when s > n/m, ||Ξ s,n || = O(s -1 m -1/2 log m)
, where for the latter we use

| ∑ j cos(sλ j )| = O(n/s) and therefore n ∑ s=1 ||Ξ s,n || 2 = O n/m ∑ s=1 m(log m) 2 n 2 + n ∑ s= n/m +1 (log m) 2 s 2 m = O((log m) 2 n -1 )
such that the first term of ( 30) is O((log m) 4 n -1 ). Besides, the second term in (30

) is O n ∑ n s=1 ||Ξ s,n || 2 ∑ n/2 s=1 s||Ξ s,n || 2
), following the analysis in [START_REF] Robinson | Gaussian semiparametric estimation of long range dependence[END_REF], where

n/2 ∑ s=1 s||Ξ s,n || 2 = O m -1 (log m) 2 log n . It follows immediately that (30) is O(n -1 (log m) 4 + m -1 (log m) 4 log n) → 0.
To complete the proof of ( 27) we now have to show that the mean of the first term in ( 29) is equal to

∑ 2 a=1 ∑ 2 b=1 η a η b Ω ab . Since E(ε t ε t |F t-1 ) = I 2 by Assumption (3), we can rewrite E( n ∑ t=1 t-1 ∑ s=1 ε s Ξ t-s,n Ξ t-s,n ε s ) = n ∑ t=1 t-1 ∑ s=1 E tr (Ξ t-s,n Ξ t-s,n ε s ε s ) = n ∑ t=1 t-1 ∑ s=1 E tr (Ξ t-s,n Ξ t-s,n )
and decompose it as

n ∑ t=1 t-1 ∑ s=1 m ∑ j=1 m ∑ k=1 1 π 2 n 2 m tr (θ j,1 + θ j,2 )(θ k,1 + θ k,2 ) × cos((t -s)λ j ) cos((t -s)λ k ) = n ∑ t=1 t-1 ∑ s=1 m ∑ j=1 1 π 2 n 2 m tr θ j,1 θ j,1 ) cos((t -s)λ j ) 2 (31) + n ∑ t=1 t-1 ∑ s=1 m ∑ j=1 1 π 2 n 2 m tr θ j,2 θ j,2 ) cos((t -s)λ j ) 2 (32) + n ∑ t=1 t-1 ∑ s=1 m ∑ j=1 1 π 2 n 2 m 2 tr θ j,1 θ j,2 ) cos((t -s)λ j ) 2 (33) + n ∑ t=1 t-1 ∑ s=1 m ∑ j=1 m ∑ k j 1 π 2 n 2 m tr θ j,1 θ k,1 ) cos((t -s)λ j ) cos((t -s)λ k ) (34) + n ∑ t=1 t-1 ∑ s=1 m ∑ j=1 m ∑ k j 1 π 2 n 2 m tr θ j,2 θ k,2 ) cos((t -s)λ j ) cos((t -s)λ k ) (35) + n ∑ t=1 t-1 ∑ s=1 m ∑ j=1 m ∑ k j 1 π 2 n 2 m 2 tr θ j,1 θ k,2 ) cos((t -s)λ j ) cos((t -s)λ k ). (36)
For ( 31), after approximating a Riemann sum by an integral and using ∑ n t=1 ∑ t-1 s=1 cos(sλ j ) 2 = (n -1) 2 /4, we have that

n ∑ t=1 t-1 ∑ s=1 m ∑ j=1 1 π 2 n 2 m tr θ j,1 θ j,1 ) cos((t -s)λ j ) 2 ∼ η 2 1 2G xx G uu (1 + 4δ -2δ 1 -2δ 2 )
.

Among the remaining terms we first analyze (32) and observe that tr (θ j,2 θ j,2 )

4π 2 = tr η 2 2 λ 2δ 2 -2δ m 4π 2 (log λ m ) 2 β 2 θ β θβ γ 2 j = η 2 2 λ 2δ 2 -2δ m (log λ m ) 2 2β 2 G xx G -1 uu γ 2 j ,
Then, using the same approximation as for ( 31),( 32) is asymptotically equal to

n ∑ t=1 t-1 ∑ s=1 m ∑ j=1 1 π 2 n 2 m tr θ j,2 θ j,2 ) cos((t -s)λ j ) 2 ∼ η 2 2 2β 2 G xx G uu (1 + 4δ -2δ 1 -2δ 2 )
.

As the joint limiting distribution of β and ξ is singular, when analyzing (33) we find that it is asymptotically equivalent to

η 1 η 2 2β G xx G uu 1 (1 + 4δ -2δ 1 -2δ 2 )
.

Equations ( 34) -( 36), where j k, remain to analyze. For (34) we use that ||θ j,1 || = O((m/j) δ 1 +δ 2 -2δ ) to show its equivalence to

O n ∑ t=1 t-1 ∑ s=1 m ∑ j=1 m ∑ k j 1 n 2 m m j δ 1 +δ 2 -2δ m k δ 1 +δ 2 -2δ cos((t -s)λ j ) cos((t -s)λ k ) = O n -1 m(log m) 2
where ∑ n t=1 ∑ t-1 s=1 cos((ts)λ j ) cos((ts)λ k ) = -n/2 for λ j λ k . Similarly, using that ||θ j,2 || = O((m/j) δ 1 +δ 2 -2δ (log j)/(log m)) we show that ( 35) is equivalent to

O n ∑ t=1 t-1 ∑ s=1 m ∑ j=1 m ∑ k j 1 n 2 m m j δ 1 +δ 2 -2δ m k δ 1 +δ 2 -2δ (log j) 2 (log m) 2 cos((t -s)λ j ) cos((t -s)λ k ) = O n -1 m(log m) 2 .
For the last term, we use both ||θ j,1 || and ||θ j,1 || and find that ( 36) is bounded by

O n ∑ t=1 t-1 ∑ s=1 m ∑ j=1 m ∑ k j 1 n 2 m m j δ 1 +δ 2 -2δ m k δ 1 +δ 2 -2δ log j log m cos((t -s)λ j ) cos((t -s)λ k ) = O n -1 m(log m) 2 .
It remains to show (28) or equivalently the sufficient condition ∑ n t=1 E(ζ 4 ) → 0. As our analysis of ( 27) is similar to Lemma 4 of [START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF], this condition can be proved under Assumption (3). We then obtain ∑ n t=1 E(ζ 4 ) = O(n(∑ n t=1 ||Ξ 2 tn ||) 2 ) = O(n -1 (log m) 4 ) as in [START_REF] Nielsen | Local Whittle Analysis of Stationary Fractional Cointegration and the Implied-Realized Volatility Relation[END_REF]. This completes the proof of (16). (38)

B.2. Limit of the Hessian

Using ( 38) and by similar arguments to [START_REF] Lobato | A semiparametric two-step estimator in a multivariate long memory model[END_REF] and [START_REF] Nielsen | Local Whittle Analysis of Stationary Fractional Cointegration and the Implied-Realized Volatility Relation[END_REF] one can easily show that

λ 2δ 2 -2δ m ∂ 2 W m ( θ) ∂β∂β - ∂ 2 W m (θ 0 ) ∂β∂β p -→ 0 (39)
and implicitly

λ 2δ 2 -2δ m (log λ m ) 2 ∂ 2 W m ( θ) ∂ξ∂ξ - ∂ 2 W m (θ 0 ) ∂ξ∂ξ p -→ 0. ( 40
)
To analyze (37), we first study We decompose it as

λ 2δ 2 -2δ m ∂ 2 W m (θ 0 ) ∂β∂β = λ 2δ 2 -2δ
λ 2δ 2 -2δ m G -1 uu 2 m m ∑ j=1 λ 2δ+2ξ j (I jxx -A 2 (λ j )I ju A 2 (λ j ) * ) (41) + λ 2δ 2 -2δ m G -1 uu 2 m m ∑ j=1 λ 2δ+2ξ j A 2 (λ j ) 1 2πn n ∑ t=1 ε t ε t -I 2 A 2 (λ j ) * (42)
After approximation of the Riemann sums by integrals, we obtain

λ 2δ 2 -2δ m ∂ 2 W m (θ 0 ) ∂β∂β p -→ E = 2G xx G uu (1 -2δ 2 + 2δ) .
implying the first part of (37) in view of (39). It remains to analyze the second part of (37) and hence

λ 2δ 2 -2δ m (log λ m ) -2 ∂ 2 W m (θ 0 ) ∂ξ∂ξ = λ 2δ 2 -2δ m log(λ m ) -2 β 2 G -1 uu G xx 2 m m ∑ j=1 log λ 2 j λ 2δ-2δ 2 j + o p (1). ( 46
)
By analogy with (45), ( 46) is asymptotically equivalent to

λ 2δ 2 -2δ m log(λ m ) -2 ∂ 2 W m (θ 0 ) ∂ξ∂ξ p -→ β 2 E = 2β 2 G xx G uu (1 -2δ 2 + 2δ) ,
which completes the proof in view of (40). Besides, as the joint limiting distribution of β and ξ is singular, we omit the superfluous covariance term.

Appendix C : Proof of Theorem 3

Proof. Under Assumptions 1-6, we need to show that ( 16) and ( 17) remain true when the unknown weighting parameter δ = δ 1 is replaced by δ1 . We first discuss the case of β. Regarding the limit of the Hessian, we need to prove that λ By similar arguments to those of (47) and Assumption 2, we obtain

√ mλ δ 2 -δ 1 m G -1 uu 2 m m ∑ j=1 (λ 2 δ1 j -λ 2δ 1 j )λ ξ j Re I jux = O p √ mλ δ 2 -δ 1 m max 1≤j≤m λ 2 δ1 -2δ 1 j -1 1 m m ∑ j=1 λ 2δ 1 +ξ j I jux = O p √ mλ δ 2 -δ 1 m | δ1 -δ 1 |(log n) 1 m m ∑ j=1 λ α+δ 1 -δ 2 j = O p √ mλ α m | δ1 -δ 1 | log n .
By similar reasoning, the same bounds can be obtained for the parameter ξ, which completes the proof of Theorem 3.

Figure 1 :Figure 3 :

 13 Figure 1: S&P500 VIX (solid lines) and realized volatility measures (dashed lines, secondary axis)

  We now derive the limit of the Hessian for any estimator θ such that || θθ 0 || ≤ || θθ 0 || and prove that m ( θ)/(∂β∂β)∂ 2 W m ( θ)/(∂ξ∂ξ) As Nielsen (2007), we strengthen the approximation | Ĝuu (θ 0 ) -G uu | = o p (1) by showing that | Ĝuu ( θ) -Ĝuu (θ 0 )| = o p (1). Using the consistency Theorem 1 and the fact that || θθ 0 || ≤ || θθ 0 ||, we have Ĝuu ( θ) -Ĝuu (θ 0 ) = λ 2ν ( βλ ξβλ ξ ).Then, by Assumption 2, Ĝuu ( θ) -Ĝuu (θ 0 ) = O p λ 2ν 0

  + o p (1).

  s e i(t-s)λ j A 2 (λ j ) * λ j )A 2 (λ j ) * + o p (1).(44) By similar arguments to that of Proposition 1 in[START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF], Equations (41)-(43) are o p (1) and by Assumptions 3 and 4, the last term (44) yields toλ 2δ 2 -2δ m ∂ 2 W m (θ 0 ) ∂β∂β = λ 2δ 2 -2δ

=

  As (38) holds with δ1 , we replaceG -1 uu ( θ) by G -1 uu . Then, since | max 1≤j≤m λ 2 δ1 jλ 2δ 1 j -1| = O p (| δ1δ 1 | log n), as in[START_REF] Nielsen | Semiparametric Estimation in Time-Series Regression with Long-Range Dependence[END_REF], we haveλ O p | δ1δ 1 | log n . (47)Regarding the limit of the score, we need to show that

Table 1 :

 1 Simulation results for m = n 0.6 , β = 1 and ξ

  The results are based on I = 10000 replications. βLS and βGLS stand for the NBLS estimator of Robinson

							35
	Bias	-0.207	-0.199	0.002	0.008	0.003	0.009	0.002	0.008	0.003	0.009
	Variance	0.005	0.006	0.031	0.007	0.032	0.008	0.031	0.007	0.031	0.008
	RMSE	0.219	0.213	0.175	0.086	0.179	0.089	0.175	0.086	0.177	0.088
						ν 0 = 0.15
	Bias	-0.207	-0.175	0.014	0.013	0.022	0.017	0.017	0.014	0.024	0.018
	Variance	0.012	0.01	0.059	0.021	0.121	0.023	0.085	0.022	0.128	0.024
	RMSE	0.233	0.202	0.244	0.146	0.349	0.152	0.293	0.149	0.358	0.156
						ν 0 = 0.05
	Bias	-0.205	-0.161	0.019	0.011	0.027	0.02	0.019	0.014	0.029	0.021
	Variance	0.02	0.014	0.094	0.043	0.079	0.034	0.078	0.035	0.090	0.038
	RMSE	0.25	0.201	0.308	0.207	0.283	0.187	0.28	0.187	0.301	0.197
	n = 512
						ν 0 = 0.35
	Bias	-0.241	-0.229	-0.004	0.005	-0.003	0.006	-0.004	0.005	-0.003	0.006
	Variance	0.003	0.003	0.018	0.003	0.019	0.003	0.018	0.003	0.018	0.003
	RMSE	0.247	0.235	0.134	0.056	0.136	0.057	0.134	0.056	0.136	0.057
						ν 0 = 0.15
	Bias	-0.242	-0.202	0.008	0.009	0.009	0.01	0.008	0.009	0.010	0.010
	Variance	0.007	0.006	0.042	0.01	0.037	0.009	0.037	0.009	0.038	0.009
	RMSE	0.256	0.215	0.205	0.102	0.192	0.095	0.194	0.096	0.196	0.097
						ν 0 = 0.05
	Bias	-0.241	-0.186	0.016	0.01	0.017	0.013	0.014	0.011	0.019	0.014
	Variance	0.013	0.008	0.073	0.021	0.053	0.016	0.055	0.016	0.055	0.016
	RMSE	0.267	0.207	0.27	0.147	0.231	0.125	0.235	0.129	0.235	0.128
	n = 1024
						ν 0 = 0.35
	Bias	-0.274	-0.254	0	0.006	0	0.006	0	0.006	0.000	0.006
	Variance	0.002	0.002	0.01	0.001	0.011	0.001	0.01	0.001	0.011	0.001
	RMSE	0.277	0.257	0.102	0.037	0.103	0.037	0.102	0.037	0.103	0.037
						ν 0 = 0.15
	Bias	-0.274	-0.227	0.006	0.006	0.006	0.007	0.005	0.007	0.008	0.008
	Variance	0.004	0.003	0.031	0.006	0.026	0.005	0.027	0.005	0.027	0.005
	RMSE	0.282	0.234	0.176	0.075	0.162	0.069	0.164	0.07	0.164	0.070
						ν 0 = 0.05
	Bias	-0.274	-0.21	0.018	0.008	0.014	0.009	0.013	0.008	0.017	0.010
	Variance	0.009	0.005	0.063	0.013	0.041	0.009	0.044	0.01	0.043	0.009
	RMSE	0.29	0.221	0.252	0.114	0.203	0.094	0.211	0.098	0.208	0.096
	Note:

Table 2 :

 2 Simulation results for m = n 0.75 , β = 1 and ξ

  The results are based on I = 10000 replications. βLS and βGLS stand for the NBLS estimator of Robinson

							35
	Bias	-0.164	-0.152	-0.012	0.006	-0.012	0.006	-0.012	0.006	-0.012	0.006
	Variance	0.004	0.004	0.009	0.003	0.009	0.003	0.009	0.003	0.009	0.003
	RMSE	0.176	0.166	0.094	0.056	0.095	0.057	0.094	0.056	0.095	0.057
						ν 0 = 0.15
	Bias	-0.163	-0.111	-0.014	0.007	-0.013	0.007	-0.013	0.007	-0.012	0.008
	Variance	0.008	0.006	0.011	0.008	0.01	0.007	0.01	0.007	0.010	0.007
	RMSE	0.186	0.136	0.106	0.091	0.099	0.083	0.1	0.084	0.099	0.085
						ν 0 = 0.05
	Bias	-0.162	-0.09	-0.018	0.007	-0.013	0.009	-0.014	0.008	-0.012	0.011
	Variance	0.013	0.007	0.014	0.015	0.01	0.01	0.011	0.011	0.010	0.010
	RMSE	0.198	0.124	0.12	0.122	0.101	0.099	0.104	0.104	0.101	0.102
	n = 512
						ν 0 = 0.35
	Bias	-0.195	-0.177	-0.009	0.006	-0.009	0.006	-0.009	0.006	-0.009	0.006
	Variance	0.002	0.002	0.005	0.001	0.005	0.001	0.005	0.001	0.005	0.001
	RMSE	0.201	0.184	0.072	0.037	0.073	0.037	0.072	0.037	0.073	0.037
						ν 0 = 0.15
	Bias	-0.194	-0.13	-0.01	0.006	-0.01	0.006	-0.01	0.006	-0.009	0.006
	Variance	0.005	0.004	0.008	0.004	0.007	0.003	0.007	0.003	0.007	0.003
	RMSE	0.207	0.144	0.089	0.064	0.082	0.057	0.083	0.059	0.083	0.059
						ν 0 = 0.05
	Bias	-0.196	-0.108	-0.01	0.009	-0.009	0.008	-0.009	0.008	-0.009	0.008
	Variance	0.009	0.004	0.011	0.008	0.007	0.005	0.008	0.006	0.007	0.005
	RMSE	0.217	0.125	0.106	0.091	0.085	0.071	0.088	0.076	0.086	0.074
	n = 1024
						ν 0 = 0.35
	Bias	-0.224	-0.197	-0.007	0.005	-0.007	0.005	-0.007	0.005	-0.007	0.005
	Variance	0.002	0.001	0.003	0.001	0.003	0.001	0.003	0.001	0.003	0.001
	RMSE	0.228	0.201	0.055	0.025	0.056	0.025	0.055	0.025	0.056	0.025
						ν 0 = 0.15
	Bias	-0.223	-0.148	-0.007	0.006	-0.007	0.006	-0.007	0.006	-0.007	0.006
	Variance	0.003	0.002	0.006	0.002	0.004	0.002	0.005	0.002	0.004	0.002
	RMSE	0.231	0.154	0.075	0.048	0.066	0.041	0.067	0.043	0.067	0.043
						ν 0 = 0.05
	Bias	-0.224	-0.124	-0.008	0.007	-0.007	0.006	-0.008	0.006	-0.007	0.006
	Variance	0.006	0.002	0.009	0.005	0.005	0.003	0.006	0.003	0.005	0.003
	RMSE	0.237	0.133	0.096	0.072	0.071	0.053	0.075	0.058	0.072	0.055
	Note:

Table 3 :

 3 Fractional cointegration estimates with m = n 0.6 This table reports the balanced and unbalanced fractional cointegration estimates for the long run equation rm t = βvix 2 t (ξ) + u t , where rm t stands for realized measures of variance. βLS and βGLS are the NBLS estimator of

		βLS	βGLS	β(0)	ξ(0)	β( δ1 )	ξ( δ1 )	δ1	δ2
					S&P 500
	RV	0.875	0.890	1.061	0.099	1.053	0.095	0.051	0.362
		-	(0.022)	(0.016)	(0.037)	(0.012)	(0.028)	(0.096)	(0.096)
	RK	0.888	0.899	1.065	0.093	1.061	0.091	0.031	0.363
		-	(0.019)	(0.012)	(0.028)	(0.010)	(0.025)	(0.096)	(0.096)
	medRV	0.873	0.879	1.050	0.095	1.060	0.100	-0.087	0.344
		-	(0.014)	(0.006)	(0.014)	(0.003)	(0.007)	(0.096)	(0.096)
	BV	0.876	0.880	1.030	0.083	1.036	0.086	-0.079	0.346
		-	(0.010)	(0.005)	(0.012)	(0.002)	(0.004)	(0.096)	(0.096)
					Dow Jones
	RV	0.853	0.863	0.967	0.064	0.951	0.055	0.076	0.396
		-	(0.027)	(0.031)	(0.067)	(0.012)	(0.026)	(0.096)	(0.096)
	RK	0.865	0.874	0.968	0.057	0.948	0.046	0.091	0.407
		-	(0.027)	-	-	(0.012)	(0.026)	(0.096)	(0.096)
	medRV	0.847	0.852	0.985	0.077	0.993	0.082	-0.043	0.358
		-	(0.019)	(0.008)	(0.017)	(0.007)	(0.017)	(0.096)	(0.096)
	BV	0.850	0.854	0.960	0.062	0.965	0.065	-0.026	0.371
		-	(0.017)	(0.007)	(0.015)	(0.007)	(0.015)	(0.096)	(0.096)
					Russell
	RV	0.817	0.831	0.987	0.105	0.967	0.092	0.080	0.298
		-	(0.035)	(0.026)	(0.057)	(0.020)	(0.043)	(0.096)	(0.096)
	RK	0.851	0.855	0.957	0.064	0.962	0.067	-0.027	0.319
		(0.031)	(0.024)	(0.011)	(0.023)	(0.011)	(0.023)	(0.096)	(0.096)
	medRV	0.793	0.819	1.031	0.147	1.015	0.137	0.073	0.260
		-	(0.044)	(0.031)	(0.071)	(0.026)	(0.058)	(0.096)	(0.096)
	BV	0.823	0.833	0.984	0.098	0.987	0.100	-0.013	0.285
		-	(0.031)	(0.015)	(0.033)	(0.015)	(0.034)	(0.096)	(0.096)
					NASDAQ
	RV	0.856	0.856	0.991	0.076	1.009	0.086	-0.123	0.299
		(0.015)	(0.014)	(0.007)	(0.015)	(0.005)	(0.011)	(0.096)	(0.096)
	RK	0.871	0.872	1.002	0.073	1.015	0.080	-0.095	0.310
		(0.016)	(0.016)	(0.007)	(0.016)	(0.006)	(0.014)	(0.096)	(0.096)
	medRV	0.838	0.848	1.008	0.097	1.018	0.102	-0.043	0.305
		-	(0.027)	(0.012)	(0.027)	(0.012)	(0.028)	(0.096)	(0.096)
	BV	0.848	0.847	0.994	0.083	1.016	0.095	-0.137	0.290
		(0.015)	(0.015)	(0.007)	(0.016)	(0.004)	(0.010)	(0.096)	(0.096)
	Note:

Table 4 :

 4 Risk-return regressions This table reports the slope estimates of univariate regressions of the long run components of the returns r t on the long run components of variance risk premia and realized measures, respectively. The low-pass filtered series are obtained as in[START_REF] Bollerslev | Risk and return: Long-run relations, fractional cointegration, and return predictability[END_REF] with the long run frequency band set to λ = 0.65 and truncation parameter k = 12. The standard errors (reported in paranthesis) are based on the HAC covariance matrix estimator. Standard errors are reported in parentheses.

		S&P 500	Russell	Dow Jones	NASDAQ
			UFC approach: vrp t = βvix 2 t (ξ) -rm t
	vrp low t,RV	0.315	0.302	0.324	0.347
		(0.101)	(0.077)	(0.063)	(0.122)
	vrp low t,RK	0.327	0.291	0.415	0.356
		(0.108)	(0.075)	(0.057)	(0.130)
	vrp low t,medRV	0.342	0.300	0.278	0.290
		(0.101)	(0.077)	(0.052)	(0.102)
	vrp low t,BV	0.384	0.322	0.354	0.345
		(0.115)	(0.081)	(0.058)	(0.119)
			Naive approach: vrp t = vix 2 t -rm t
	vrp low t,RV	0.253	0.206	0.266	0.280
		(0.079)	(0.067)	(0.046)	(0.083)
	vrp low t,RK	0.274	0.209	0.340	0.295
		(0.084)	(0.068)	(0.042)	(0.090)
	vrp low t,medRV	0.265	0.204	0.225	0.237
		(0.075)	(0.060)	(0.040)	(0.070)
	vrp low t,BV	0.289	0.220	0.278	0.274
		(0.083)	(0.065)	(0.043)	(0.079)
				Realized measures
	V IX low t	-0.057	-0.060	0.005	-0.013
		(0.047)	(0.039)	(0.034)	(0.043)
	RV low t	-0.107	-0.111	-0.069	-0.070
		(0.045)	(0.036)	(0.030)	(0.048)
	RK low t	-0.106	-0.107	-0.067	-0.067
		(0.045)	(0.037)	(0.029)	(0.049)
	medRV low t	-0.112	-0.115	-0.074	-0.073
		(0.044)	(0.035)	(0.030)	(0.047)
	BV low t	-0.113	-0.115	-0.072	-0.074
		(0.044)	(0.035)	(0.030)	(0.048)
	Note:

Similarly, the NBGLS estimator uses NBLS residuals in its first step.

2006 2008 2010 2012 2014 2016 2018 2006 2008 2010 2012 2014 2016 2018