Threshold

Sokbae Lee

Myung Hwan Seo

SEMIPARAMETRIC ESTIMATION OF A BINARY RESPONSE MODEL WITH A CHANGE-POINT DUE TO A COVARIATE

Keywords: Binary response model, maximum score estimation, nonlinear random utility models, semiparametric estimation, threshold regression

's maximum score estimator. The model in this paper is irregular because of a change-point due to an unknown threshold in a covariate. This irregularity coupled with the discontinuity of the objective function of the maximum score estimator complicates the analysis of the asymptotic behavior of the estimator. Sufficient conditions for the identification of parameters are given and the consistency of the estimator is obtained. It is shown that the estimator of the threshold parameter, γ 0 , is n -1 -consistent and the estimator of the remaining regression parameters, θ 0 , is n -1/3 -consistent. Furthermore, we obtain the asymptotic distribution of the estimator. It turns out that both estimators γ and θ are oracle-efficient in that n(γ n -γ 0 ) and n 1/3 ( θn -θ 0 ) converge weakly to the distributions to which they would converge weakly if the other parameter(s) were known.

Introduction

A binary response model is very commonly used in a number of fields. In this model, an observable binary outcome Y is modelled typically as Y = 1(Y * ≥ 0), where 1(A) denotes the indicator function, i.e., 1(A) = 1 if A is true and zero otherwise, and Y * is an unobservable continuous variable that determines the binary outcome Y (see e.g. [START_REF] Manski | Identification of binary response models[END_REF]. In most applications, Y * has the following form:

(1.1)

Y * = X β 0 + U,
where X is a vector of observed random variables including an intercept term, β 0 is a vector of unknown parameters, and U is an unobserved random variable.

In this paper, we consider a threshold regression model for Y * instead of the linear regression model (1.1). In particular, the model is

(1.2) Y * = W 1 β 10 + W 2 β 20 + U, if D ≤ γ 0 , W 1 β 10 + W 2 β 30 + U, if D > γ 0 ,
or equivalently

(1.3) Y * = W β 0 + Z δ 0 1 {D > γ 0 } + U,
where W = (W 1 , W 2 ) , Z = W 2 , β 0 = (β 10 , β 20 ) , and δ 0 = β 30 -β 20 . In particular, D is the observed threshold variable and γ 0 is the threshold parameter. We denote by X the set of regressors (W , D) and by θ 0 = (β 0 , δ 0 ) the set of regression parameters excluding γ 0 .

The threshold variable D can be an element of W. The literature on the threshold model (also called change-point model, two-phase regression, or sample splitting) is vast. It has been studied for autoregressive models (e.g. [START_REF] Tong | of Oxford Statistical Science Series[END_REF][START_REF] Chan | Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model[END_REF], for linear regression models (e.g. [START_REF] Hansen | Sample splitting and threshold estimation[END_REF][START_REF] Hira | Asymptotics of maximum likelihood estimator in a twophase linear regression model[END_REF], for nonparametric models (e.g. [START_REF] Delgado | Nonparametric inference on structural breaks[END_REF], and more recently for transformation models (e.g. [START_REF] Pons | Estimation in a Cox regression model with a change-point according to a threshold in a covariate[END_REF][START_REF] Michael | Inference under right censoring for transformation models with a change-point based on a covariate threshold[END_REF].

Threshold binary regression models have a wide variety of applications. For example, in economics, finance, and management, random utility models that are nonlinear in income and/or price are commonly employed (see, e.g. [START_REF] Herriges | Nonlinear income effects in random utility models[END_REF][START_REF] Dagsvik | Compensating variation and hicksian choice probabilities in random utility models that are nonlinear in income[END_REF]. [START_REF] Terui | Estimating heterogeneous price thresholds[END_REF] adopted an asymmetric market response model with the threshold due to price based on consumer behavior theory. The effect of financial contagion (see, e.g. [START_REF] Forbes | No contagion, only interdependence: Measuring stock market co-movements[END_REF] can also be modelled as a discontinuous threshold effect as in [START_REF] Pesaran | Econometric issues in the analysis of contagion[END_REF]. In biostatistics, dose-response models are modelled with threshold parameters (see, e.g. [START_REF] Cox | Threshold dose-response models in toxicology[END_REF][START_REF] Schwartz | Threshold models for combination data from reproductive and development experiments[END_REF]. In epidemiology, logistic regressions with change-points are used to model the relationship between the continuous exposure variable and disease risk (see [START_REF] Pastor | Use of tow-segmented logistic regression to estimate change-points in epidemiologic studies[END_REF][START_REF] Pastor-Barriuso | Transition models for changepoint estimation in logistic regression[END_REF]. While the threshold model is easier to interpret than more complex nonlinear models, the irregular feature of the model makes estimation of the unknown change-point complicated. To our best knowledge, all existing methods for estimating a change-point according to a covariate threshold in binary response models assumes that the distribution of U in (1.2) belongs to a family of parametric distributions.

The purpose of this paper is to develop a method for estimating (θ 0 , γ 0 ) in (1.2) without imposing the parametric distribution on U . First, we establish identification of (θ 0 , γ 0 ), allowing for conditional heteroskedasticity of unknown form, under the conditional median independence assumption. This is substantially weaker than the typical assumption such that the distribution of U is parametric and U and X are independent. The conditional heteroskedasticity is particularly important in random utility models as shown in [START_REF] Bryan | The random utility hypothesis and inference in demand systems[END_REF]. [START_REF] Manski | Identification of binary response models[END_REF] has shown that if the distribution of X has sufficiently rich support, the finite-dimensional parameters in a binary regression model with (1.1) are identified up to scale. As in [START_REF] Manski | Identification of binary response models[END_REF], (θ 0 , γ 0 ) in (1.2) is identified (up to scale with respect to θ 0 ) under some regularity conditions, which will be given in Section 2.

Once (θ 0 , γ 0 ) is identified via the conditional median independence assumption, a natural estimation strategy for (θ 0 , γ 0 ) in (1.2) is to apply [START_REF] Manski | Maximum score estimation of the stochastic utility model of choice[END_REF][START_REF] Manski | Semiparametric analysis of discrete response. Asymptotic properties of the maximum score estimator[END_REF]'s maximum score estimator to the threshold model (1.2). Section 3 describes the corresponding maximum score estimator and section 4 establishes the consistency of the estimator.

We also develop convergence rates and asymptotic distributions, which are highly nonstandard due to mixed irregularities of the model and maximum score estimation. In Section 5, it is shown that γn is n -1 -consistent and θn is n -1/3 -consistent. It is interesting to find that the fast rate of convergence for the threshold estimator is preserved for the maximum score estimation. Section 6 gives the asymptotic distributions of n(γ n -γ 0 ) and n 1/3 ( θn -θ 0 ). It turns out that both estimators γ and θ are oracle-efficient in that n(γ n -γ 0 )

and n 1/3 ( θn -θ 0 ) converge weakly to the distributions to which they would converge weakly if the other parameter(s) were known. Section 7 discusses the subsampling inference that enables us to carry out inference, although the distributions of n(γ n -γ 0 ) and n 1/3 ( θn -θ 0 )

are nonstandard and cannot be tabulated. Section 8 reports some simulation results that investigate the finite sample performance of the subsampling confidence intervals for the regression parameters. Concluding remarks are given in Section 9. The proofs of theorems are given in the Appendix.

2 Identification of θ 0 and γ 0

This section provides regularity conditions under which θ 0 and γ 0 are identified. Let F U |X (u|x) and f U |X (u|x), respectively, denote the cumulative distribution function and probability density function of U conditional on X = x. We make the following assumptions.

Assumption 1. Assume that θ 0 = 1 and δ 0 = 0.

Assumption 2. The distribution of U conditional on X = x is absolutely continuous with respect to Lebesgue measure and the corresponding conditional density is uniformly continuous and positive everywhere with probability one. In addition, F U |X (0|x) = 0.5 for almost every x.

Assumption 3. The support of W ∈ R q is not contained in any proper linear subspace of R q under the conditional distribution of W given D = d for almost every d.

Assumption 4. There exists at least one k ∈ [1, . . . , q] such that the k-th element of β 0 is non-zero and such that, for almost every value of w ≡ (w 1 , . . . , w k-1 , w k+1 , . . . , w q ) and k) and other elements W .

d, Pr[a 1 < W (k) < a 2 | W = w, D = d] > 0 for all open intervals (a 1 , a 2 ) ∈ R, where W is decomposed into the k-th element W (
Assumption 5. D is continuously distributed with support containing γ 0 .

Assumption 1 imposes restrictions on parameters. As in the linear binary response model

(1.1), the scale of θ 0 is not identified since the distribution of U conditional on X = x is unknown. In this paper, the scale normalization on θ 0 is given by θ 0 = 1. If δ 0 is zero, then γ 0 is unidentified. Assumption 2 allows for an arbitrary form of dependence between U and X as long as the conditional median independence assumption is satisfied. This can be extended easily to any other quantile independence assumption as well (see, Assumption QI in [START_REF] Manski | Identification of binary response models[END_REF]. Assumptions 3 and 4 are simply the restatement of Conditions X1 and X3 of [START_REF] Manski | Identification of binary response models[END_REF], respectively. Assumption 3 imposes a usual full rank condition on the distribution of X and Assumption 4 requires a rich support of one of elements of W with a non-zero coefficient conditional on other regressors. Assumption 5 ensures the identification of γ 0 . If γ 0 were not contained in the support of D, then the threshold model (1.2) would not have a two-phase feature in observed data.

The following theorem shows that θ 0 and γ 0 are identified.

Theorem 2.1. Let Assumptions 1-5 hold. Then θ 0 and γ 0 are identified.

The Maximum Score Estimator

We consider estimation with an independent and identically distributed sample {(Y i , X i ) :

i = 1, . . . , n} of (Y, X). To describe the maximum score estimator, for any θ = (β , δ ) , γ and x = (w, z, d), define

(3.1) S n (θ, γ) = n i=1 (2Y i -1)1 {G(X i , θ, γ) ≥ 0} , where G(x, θ, γ) = w β + z δ1(d > γ).
Then [START_REF] Manski | Maximum score estimation of the stochastic utility model of choice[END_REF][START_REF] Manski | Semiparametric analysis of discrete response. Asymptotic properties of the maximum score estimator[END_REF]'s maximum score estimator ( θn , γn ) of (θ 0 , γ 0 ) can be obtained in two steps. In the first stage, for fixed γ, obtain θn (γ) = argmax θ S n (θ, γ)

with the scale normalization on θ with θ = 1. In the second stage, let Ŝn (γ) = S n ( θn (γ), γ).

Obtain γn = inf{γ n : γ n = argmax γ Ŝn (γ)}. Then the maximum score estimator of θ 0 is θn = θn (γ n ). Since there could be infinitely many γ's at which Ŝn (γ) is maximized, γn is defined as the infimum of those.

The discontinuity of S n with respect to (θ, γ) complicates the analysis of the asymptotic behavior of the maximum score estimator. There exist two distinct sources of discontinuity:

on the one hand, the discontinuity arises because of the presence of the indicator function 1 {G(X i , θ, γ) ≥ 0} in (3.1); on the other hand, it may occur because of a possible changepoint in (1.2) due to an unknown threshold γ 0 . We now turn to the asymptotic properties of the maximum score estimator.

Consistency of the Estimator

This section gives conditions under which the maximum score estimator is consistent.

Assumption 6. Assume that θ 0 and γ 0 are in a compact subset of R p , where p is the dimension of (θ 0 , γ 0 ).

Assumption 7. Assume that {(Y i , X i ) : i = 1, .
. . , n} is independent and identically distributed.

These are standard assumptions in the literature. The following theorem gives the consistency result.

Theorem 4.1. Let Assumptions 1-7 hold. Then ( θn , γn ) → p (θ 0 , γ 0 ).

This section establishes the rates of convergence in probability of ( θn , γn ) to (θ 0 , γ 0 ). First, we show that γn is n -1 -consistent for γ 0 . We make the following additional assumptions.

Assumption 8. Assume that D is continuously distributed with full support on R and its probability density function is strictly positive, bounded and continuous in a neighborhood of γ 0 .

Assumption 9. With positive probability, Z δ 0 = 0 at D = γ 0 .

Assumption 8 is needed to prove a useful lemma (see Lemma A.1 in the Appendix).

Roughly speaking, this condition ensures that there exist enough data points around γ 0 for sufficiently large n. Assumption 9 imposes that the regression function in (1.2) is discontinuous. When the regression function is continuous, then we would have different rates of convergence and asymptotic distributions. We will make a brief comment on this at the end of this section.

The following theorem gives the n -1 -consistency of γn to γ 0 .

Theorem 5.1. Let Assumptions 1-9 hold. Then, γn = γ 0 + O p n -1 .

We now turn to the rates of convergence of θn to θ 0 . Given the result of Theorem 5.1

and [START_REF] Kim | Cube root asymptotics[END_REF], it is not surprising that we establish the n -1/3 -consistency for θn .

Assumption 10. As a function of θ, E [(2Y -1)1 {G(X, θ, γ 0 ) ≥ 0}] has a strictly negative definite second derivative matrix at θ 0 .

This assumption is necessary to apply Corollary 4.2 of [START_REF] Kim | Cube root asymptotics[END_REF], who also give sufficient conditions for the maximum score estimator of the binary response model with (1.1) (see Section 6.4 in [START_REF] Kim | Cube root asymptotics[END_REF]. Assumption 10 imposes a local identification at θ 0 and is a weak smoothness condition. The following theorem gives the n -1/3 -consistency of θn to θ 0 .

ACCEPTED MANUSCRIPT

Theorem 5.2. Let Assumptions 1-10 hold. Then, θn = θ 0 + O p n -1/3 .

We end this section by making a comment on the case when the model (1.2) is a continuous threshold model, in which z δ becomes zero when d = γ and G (x, θ, γ) =

w β + δ (d -γ) 1 {d > γ} .
This violates Assumption 9, invalidating Theorem 5.1. Instead, [START_REF] Kim | Cube root asymptotics[END_REF] applies to both parameters θ and γ, since the limit objective function can now be assumed to be twice differentiable in γ as well as in θ. 1

The Asymptotic Distribution of the Estimator

To obtain the asymptotic distribution of the maximum score estimator, write

S n (θ, γ) = S n1 (γ) + S n2 (θ) + S n3 (θ, γ), (6.1) where S n1 (γ) = S n (θ 0 , γ) -S n (θ 0 , γ 0 ), S n2 (θ) = S n (θ, γ 0 ) -S n (θ 0 , γ 0 ), and S n3 (θ, γ) = [S n (θ, γ) -S n (θ, γ 0 )] -[S n (θ 0 , γ) -S n (θ 0 , γ 0 )] .
Since γ is n -1 -consistent and θ is n -1/3 -consistent, it suffices to consider a n -1 -neighborhood of γ 0 and a n -1/3 -neighborhood of θ 0 . Then we obtain the following lemma. Lemma 6.1. For every A > 0, as n → ∞:

sup θ,γ |S n3 (θ, γ)| : n 1/3 θ -θ 0 ≤ A, n|γ -γ 0 | ≤ A = O p (n -1/6 ).
To establish the asymptotic distribution of n(γ -γ 0 ), we now consider the weak convergence of S n1 as a random variable on the space of cadlag functions, denoted by D, equipped with the Skorohod topology and on its restrictions to the space of cadlag functions on [-A, A], denoted by D A , for any A > 0. This approach is similar to that taken in the recent 1 A sufficient condition for this would be the conditional density f D|W of D given W is differentiable twice. Then, assuming c = -w β/δ and δ > 0, we note that [START_REF] Chan | Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model[END_REF][START_REF] Hira | Asymptotics of maximum likelihood estimator in a twophase linear regression model[END_REF][START_REF] Pons | Estimation in a Cox regression model with a change-point according to a threshold in a covariate[END_REF][START_REF] Michael | Inference under right censoring for transformation models with a change-point based on a covariate threshold[END_REF]. Now define ν = n(γ -γ 0 ). Using arguments identical to prove (A.1) in the Appendix, write a rescaled version of S n1 (γ) as

Pr {G ≥ 0|W = w} = Pr {(D -γ) 1 {D > γ} ≥ c|W = w} = ∞ c∨0 f D|W =w (x + γ) dx has second derivative. A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT literature (for example,
Sn1 (ν) = S+ n1 (ν)1(ν > 0) + S- n1 (ν)1(ν < 0),
where

S+ n1 (ν) = n i=1 1 γ 0 < D i ≤ γ 0 + n -1 ν ζ + i (θ 0 ), S- n1 (ν) = n i=1 1 γ 0 + n -1 ν < D i ≤ γ 0 ζ - i (θ 0 ), ζ + i (θ) = I + 1i (θ) -I + 2i (θ) + I + 3i (θ) -I + 4i (θ) with I + 1i (θ) = 1 W i β ≥ 0, W i β + Z i δ < 0, W i β 0 + Z i δ 0 ≥ -U i I + 2i (θ) = 1 W i β ≥ 0, W i β + Z i δ < 0, W i β 0 + Z i δ 0 < -U i I + 3i (θ) = 1 W i β < 0, W i β + Z i δ ≥ 0, W i β 0 + Z i δ 0 < -U i I + 4i (θ) = 1 W i β < 0, W i β + Z i δ ≥ 0, W i β 0 + Z i δ 0 ≥ -U i ,
and

ζ - i (θ) = I - 1i (θ) -I - 2i (θ) + I - 3i (θ) -I - 4i (θ) with I - 1i (θ) = 1 W i β ≥ 0, W i β + Z i δ < 0, W i β 0 < -U i I - 2i (θ) = 1 W i β ≥ 0, W i β + Z i δ < 0, W i β 0 ≥ -U i I - 3i (θ) = 1 W i β < 0, W i β + Z i δ ≥ 0, W i β 0 ≥ -U i I - 4i (θ) = 1 W i β < 0, W i β + Z i δ ≥ 0, W i β 0 < -U i .
To describe the asymptotic behavior of Sn1 (ν), let f D (d) denote the probability density function of D, let ν+ and ν-denote two independent jump processes on R such that ν+ (s) is a Poisson random variable with parameter sf D (γ 0 ) for s > 0 and ν+ (s) = 0 for s ≤ 0 and ν-(s) is a Poisson random variable with parameter -sf D (γ 0 ) for s < 0 and ν-(s) = 0 for s ≥ 0. In addition, let { ζ+ k : k = 0, 1, . . .} and { ζk : k = 0, 1, . . .} be independent sequences of i.i.d. random variables with characteristic functions

ϕ + (t) = E exp(itζ + )|D = γ + 0 and ϕ -(t) = E exp(itζ -)|D = γ - 0 ,
respectively, and let ζ+

0 = ζ- 0 = 0. Let S1 (s) = S+ 1 (s)1{s > 0} + S- 1 (s)1{s < 0} be a right-continuous jump process on R, where S+ 1 (s) = 0≤k≤ν + (s)
ζ+ k and S-

1 (s) = 0≤k≤ν -(s) ζ- k .
The following lemma establishes the weak convergence of Sn1 .

Lemma 6.2. As n → ∞, Sn1 converges weakly to S1 in D A for every A > 0.

Let

ν * S1 = inf{ν * : ν * = argmax ν S1 (ν)}.
As in the literature (see, e.g., [START_REF] Chan | Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model[END_REF][START_REF] Hira | Asymptotics of maximum likelihood estimator in a twophase linear regression model[END_REF][START_REF] Pons | Estimation in a Cox regression model with a change-point according to a threshold in a covariate[END_REF][START_REF] Michael | Inference under right censoring for transformation models with a change-point based on a covariate threshold[END_REF] . In addition, n 1/3 ( θ -θ 0 ) and argmax µ n -1/3 S n2 θ 0 + n -1/3 µ converge weakly to the same limiting distribution.

Theorem 6.1 implies that the limiting distributions of n(γ -γ 0 ) and n 1/3 ( θ -θ 0 ) are the same as those they would be if the other parameters were known. The limiting distribution of n 1/3 ( θ -θ 0 ) is not shown here since it is already given in Kim and Pollard (1990, Example 6.4).

Inference

The asymptotic distributions obtained in this paper are nonstandard and cannot be tabulated as they depend on nuisance parameters in complex manners. It is known from

A C C E P T E D M A N U S C R I P T
ACCEPTED MANUSCRIPT [START_REF] Abrevaya | On the bootstrap of the maximum score estimator[END_REF] that the bootstrap does not estimate the asymptotic distribution of θ consistently. While the m out of n bootstrap in Lee and Pun ( 2006) is more general and allows for the standard maximum score estimator, their regularity conditions do not permit threshold models. However, subsampling provides a consistent inferential method for the asymptotic distributions of both θ and γ, as in [START_REF] Delgado | Subsampling inference in cube root asymptotics with an application to manski's maximum score estimator[END_REF] and [START_REF] Gonzalo | Subsampling inference in threshold autoregressive models[END_REF]. Confidence intervals can be constructed following the standard subsampling procedure, see e.g. [START_REF] Politis | Subsampling. Springer Series in Statistics[END_REF].

As our asymptotic development is based on the discontinuity of the regression function (Assumption 9), the above asymptotics breaks down when the model comes close to a continuous one. In this case, the subsampling confidence intervals may be problematic as their convergence is not uniform around the continuity point, see e.g. [START_REF] Donald | The limit of finite sample size and a problem with subsampling[END_REF].

Another interesting inferential approach can be based on the smoothing of the objective function. This can be done by replacing the indicator functions in (3.1) with integrated kernels with smoothing parameters that converge to zero. [START_REF] Horowitz | A smoothed maximum score estimator for the binary response model[END_REF] and [START_REF] Seo | A smoothed least squares estimator for the threshold regression[END_REF] have developed the smoothing approach for the maximum score estimation and threshold estimation, respectively. This method enables the standard normal inference for both γ 0 and θ 0 while it causes the convergence rate of θ to increase and that of γ to decrease. The smoothing, however, demands more stringent smoothness conditions than the ones in this paper as demonstrated in [START_REF] Horowitz | Optimal rates of convergence of parameter estimators in the binary response model with weak distributional assumptions[END_REF].

Monte Carlo Experiments

We examine the finite sample performance of the subsampling inference for the regression parameters θ 0 = (β 0 , δ 0 ) and the threshold parameter γ 0 . The samples of size n = 500 are generated independently from

Y * = β 0 X 1 + δ 0 X 2 1 {D > γ 0 } + U Y = 1 {Y * > 0} ,
where β 0 = δ 0 = 1/ √ 2, γ 0 = 2, X 1 , X 2 , and D are independently distributed as N (0, 1) , 2N (0, 1) , and N (0, 1) + 2, respectively. U is independent of all the other variables and generated from either logistic distribution (Dist 1) with median 0 and variance 0.1 or student's t distribution (Dist 2) with 3 degrees of freedom and with variance 0.1. These two classes of error distributions are considered by [START_REF] Horowitz | A smoothed maximum score estimator for the binary response model[END_REF].

The model parameters β 0 , δ 0 and γ 0 are estimated by the grid search. 2 The grid for δ is set as {0, .01, .02., ..., 1} and that for γ as the data points of D after trimming at lower and upper 10 percentiles. We focus on δ and γ as the normalization restriction yields β 2 +δ 2 = 1.

Let δ and γ denote the estimates.

We describe subsampling hypothesis tests for θ 0 . Let the null hypothesis of interest be H 0 : g (θ 0 ) = 0 against the alternative hypothesis H 1 : g (θ 0 ) = 0, where g is a smooth function whose first derivative is of full rank at θ = θ 0 . We may consider a test statistic In case of the hypothesis test for γ 0 , the rate of convergence n 1/3 and m 1/3 above need to be replaced by n and m, respectively.

T n = n 1/3 g θ ,
We examine the rejection rates of the test for the hypothesis δ 0 = 1/ √ 2 with the two nominal levels α = 0.05 and 0.1. Also considered is the test for the hypothesis γ 0 = 2.

All the simulations are based on 500 repetitions with the subsampling repetition B = 500.

As we do not pursue an automatic selection rule for the block length m, 3 we experimented with several values of m to see how the choice affects the performance of subsampling. The results are reported in Table 1.

The results look promising. The rejection frequencies of the subsampling test for the threshold parameter γ 0 are close to their nominal levels with a broad range of block size m. The subsampling test for the slope parameter δ 0 appears rather conservative for larger values of m and liberal for smaller values of m. Furthermore, we note that the block length m needs to be different for the threshold and slope parameters for best results. We also experimented the subsampling test for the hypothesis for δ 0 with γ fixed at the true value 2. The results are quite similar to the one reported in Table 1 and thus omitted.

This reflects a finding in Theorem 6.1, which is that the asymptotic distribution of δ is not affected by the estimation of γ 0 .

Conclusions

This paper has considered the maximum score estimator of a binary response model with a change-point according to the unknown threshold of a covariate, allowing for an arbitrary from of heteroskedasticity. We have obtained the rates of convergence and asymptotic

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

distribution of the estimator. It turns out that the estimator of the threshold parameter is n -1 -consistent, the estimator of the remaining regression parameters is n -1/3 -consistent, and the limiting distributions of the estimators normalized by the rates of convergence are the same as if the other parameters were known. Therefore, an important practical implication of this paper is that the unknown threshold parameter can be estimated precisely with a small or moderate sample size and that there is no loss of efficiency of not knowing other parameters, which are difficult to estimate precisely without a large sample. For future research, it is an interesting but challenging research topic to develop a linearity test of the maximum score model against threshold alternatives.

A Appendix: Proofs of Theorems [START_REF] Manski | Identification of binary response models[END_REF]. Then, given Assumption 5, there is a unique value of d at which the slope coefficient changes. This change point is identified as γ 0 .

Proof of Theorem 4.1. Define

H * (x) = E [1 {G(X, θ 0 , γ 0 ) + U ≥ 0} -1 {G(X, θ 0 , γ 0 ) + U < 0} |X = x] and ∆ * (θ, γ) = n -1 E[S n (θ, γ) -S n (θ 0 , γ 0 )].
Then it can be shown that

∆ * (θ, γ) = E H * (X) 1 {G(X, θ, γ) ≥ 0 > G(X, θ 0 , γ 0 )} -1 {G(X, θ 0 , γ 0 ) ≥ 0 > G(X, θ, γ)} .
Note that

H * (x) = 1 -2F U |X [-G(x, θ 0 , γ 0 )|x].
By the assumption that F U |X [0|x] = 0.5, note that H * (x) ≥ 0 when G(x, θ 0 , γ 0 ) ≥ 0 and that H * (x) < 0 when G(x, θ 0 , γ 0 ) < 0. Define

Q(θ, γ) = x ∈ supp(X) : {G(x, θ, γ) ≥ 0 > G(x, θ 0 , γ 0 )} ∪ {G(x, θ 0 , γ 0 ) ≥ 0 > G(x, θ, γ)} .
By arguments identical to those used to prove Proposition 2 of [START_REF] Manski | Identification of binary response models[END_REF], (θ 0 , γ 0 ) are identified if and only if Pr(X ∈ Q(θ, γ)) > 0 for any (θ, γ) = (θ 0 , γ 0 ). Therefore, ∆ * (θ, γ)

is non-positive everywhere and is equal to zero only when (θ, γ) = (θ 0 , γ 0 ). Also, notice that G(S, θ, γ) is continuous at each θ and γ with probability one. Thus, S n (θ, γ) converges in probability to E[S n (θ, γ)] uniformly over (θ, γ) by Lemma 2.4 of [START_REF] Newey | Large sample estimation and hypothesis testing[END_REF]. Then the consistency of the estimator follows from Theorem 2.1 of [START_REF] Newey | Large sample estimation and hypothesis testing[END_REF].

To obtain the converge rates for γn , we begin with the following lemma, which is similar to Claim 2 in the proof of Proposition 1 of [START_REF] Chan | Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model[END_REF] and Lemma 3.2 of [START_REF] Hira | Asymptotics of maximum likelihood estimator in a twophase linear regression model[END_REF]. The proof of the lemma is given in Appendix B.

Lemma A.1. For any random variables

(V, D) satisfying E[V |D] = 0 and E[V 2 |D] < C <
∞ almost surely for some constant C, assume that {(V i , D i ) : i = 1, . . . , n} is a random sample of (V, D) and that D is continuously distributed and has a bounded, continuous, positive density in a neighborhood of x ∈ R. Then, for each δ > 0 and ε > 0, there exists a positive constant B < ∞ such that for all 0 < ∆ < 1 and for all n > B/∆,

(a) Pr    sup B n <r<∆ 1 n n i=1 1 {x < D i < x + r} Pr {x < D < x + r} -1 > δ    < ε (b) Pr    sup B n <r<∆ 1 n n i=1 V i 1 {x < D i < x + r} Pr {x < D i < x + r} > δ    < ε. Proof of Theorem 5.1. Define ∆ S n (θ, γ) = S n (θ, γ) -S n (θ, γ 0 ) = n i=1 (2Y i -1) (1 {G (X i , θ, γ) ≥ 0} -1 {G (X i , θ, γ 0 ) ≥ 0}) . A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT Suppose that γ > γ 0 . Then if D i > γ or D i ≤ γ 0 , 1 {G (X i , θ, γ) ≥ 0} -1 {G (X i , θ, γ 0 ) ≥ 0} = 0. Suppose now that γ < γ 0 . Then if D i > γ 0 or D i ≤ γ, 1 {G (X i , θ, γ) ≥ 0} -1 {G (X i , θ, γ 0 ) ≥ 0} = 0.
Using these, we may write

∆ S n (θ, γ) = n i=1 1 {γ 0 < D i ≤ γ} ζ + i (θ) + n i=1 1 {γ < D i ≤ γ 0 } ζ - i (θ), (A.1)
where ζ + i (θ) and ζ - i (θ) are defined in the main text. The following lemma, which is proved in Appendix B, is useful to prove the theorem.

Lemma A.2. For any ∆ > 0, define Θ ∆ = {θ : θ -θ 0 < ∆}. There exists a sufficiently small ∆ > 0 such that

sup θ∈Θ ∆ E ζ + i (θ)|D i < 0 and sup θ∈Θ ∆ E ζ - i (θ)|D i < 0 almost surely. Now write ∆ Sn (θ, γ) = n i=1 1 {γ 0 < D i ≤ γ} E[ζ + i (θ)|D i ] + n i=1 1 {γ < D i ≤ γ 0 } E[ζ - i (θ)|D i ] + n i=1 1 {γ 0 < D i ≤ γ} ζ + i (θ) -E[ζ + i (θ)|D i ] + n i=1 1 {γ < D i ≤ γ 0 } ζ - i (θ) -E[ζ - i (θ)|D i ] .
Then apply Lemmas A.1 and A.2 to obtain ∆ S n (θ, γ) < 0 with probability approaching one for all |γ -γ 0 | > B/n and θ ∈ Θ ∆ and with some sufficiently large B and sufficiently small ∆. As θ, γ is the maximizer of S n (θ, γ) , we conclude that γ = γ 0 + O p (1/n) .

To obtain the converge rates for θn , we first give the following lemma, which is proved in Appendix B.

Lemma A.3. For every A > 0, as n → ∞:

sup θ,γ {|S n3 (θ, γ)| : n|γ -γ 0 | ≤ A} = O p (1).
Proof of Theorem 5.2. In view of Corollary 4.2 of [START_REF] Kim | Cube root asymptotics[END_REF], it suffices to show that

n -1 S n ( θ) ≥ sup θ n -1 S n (θ) -O p (n -2/3 ),
which in turn follows from the same Corollary, Lemma A.3, and the fact that S n1 does not depend on θ.

Proof of Theorem 6.1. Note that n(γ -γ 0 ) is uniformly tight by Theorem 5.1. Since S n2 depends on only θ, Lemmas 6.1 and 6.2 ensure that conditions of the argmax continuous mapping theorem of van der Vaart and Wellner (1996, Theorem 3.2.2, p.286) are satisfied.

Then the first conclusion follows from the argmax continuous mapping theorem. Similarly, n 1/3 ( θ -θ 0 ) is uniformly tight by Theorem 5.2. Since S n1 depends on only γ, by Lemma 6.2 and again by the argmax continuous mapping theorem, the second conclusion follows.

B Appendix: Proofs of Lemmas

Proof of Lemma A.1. For any r ∈ (0, ∆], there exist 0 < m < M < ∞ such that (B.1) mr < Pr {x < D < x + r} < M r.

Consider a partition of the interval (B/n, ∆) ≡ ∪ K-1 k=0 J k , where K is the largest integer such that b K-1 B/n < ∆ for some b > 1, and J k = (b k B/n, b k+1 B/n] , k = 0, ..., K -2 and

J K-1 = (b K-1 B/n, ∆).
Here we prove part (b) only as part (a) can be proved in the same but simpler way. For a r ∈ J k , we have

1 n n i=1 V i 1 {x < D i < x + r} ≤ 1 n n i=1 |V i | 1 x + b k B/n < D i ≤ x + b k+1 B/n + 1 n n i=1 V i 1 x < D i ≤ x + b k B/n and Pr {x < D < x + r} ≥ Pr x < D < x + b k B/n . Thus, sup B n <r<∆ 1 n n i=1 V i 1 {x < D i < x + r} Pr {x < D < x + r} ≤ max 0≤k<K 1 n n i=1 |V i | 1 x + b k B/n < D i ≤ x + b k+1 B/n Pr {x < D i < x + b k B/n} + max 0≤k<K 1 n n i=1 V i 1 x < D i ≤ x + b k B/n Pr {x < D ≤ x + b k B/n} . (B.2)
First, consider the second term in (B.2). It follows from the Markov and Cauchy-Schwarz inequalities that for any δ 1 > 0,

Pr max 0≤k<K 1 n n i=1 V i 1 x < D i ≤ x + b k B/n Pr {x < D ≤ x + b k B/n} > δ 1 ≤ δ -2 1 K-1 k=0 E 1 n n i=1 V i 1 x < D i ≤ x + b k B/n Pr {x < D < x + b k B/n} 2 ≤ δ -2 1 K-1 k=0 E[V 2 ] n • Pr {x < D ≤ x + b k B/n} ≤ E[V 2 ] mδ 2 1 B (1 -b -1 ) . (B.3)
Next, note that due to (B.1),

var |V | 1 x + b k B/n < D ≤ x + b k+1 B/n ≤ Cb k (b -1) B/n,
for some constant C. Thus, for the first term in (B.2), we have that Pr max

0≤k<K 1 n n i=1 |V i | 1 x + b k B/n < D i ≤ x + b k+1 B/n Pr {x < D < b k B/n} > δ 2 ≤ K-1 k=0 1 n var |V i | 1 x + b k B/n < D i ≤ x + b k+1 B/n Pr {x < D < x + b k B/n} δ 2 + max 0≤k<K E |V | 1 x + b k B/n < D ≤ x + b k+1 B/n Pr {x < D < x + b k B/n} δ 2 ≤ K-1 k=0 1 n Cb k (b -1) B/n (δ 2 mb k B/n) 2 + E[V 2 ] (b -1) /δ 2 m = C (b -1) (δ 2 m) 2 (1 -b -1 ) B + E[V 2 ] (b -1) /δ 2 m.
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Therefore, we can choose b close enough to 1 and then B large enough so that these two quantities and (B.3) can be made arbitrarily small for any δ 1 , δ 2 > 0.

Proof of Lemma A.2. We first show that E sup θ∈Θ ∆ I + 1i (θ) -I + 2i (θ)|D i < 0 for ∆ sufficiently small. Let ε > 0, 1/2 < ξ < 1, and 0 < η < (4εξ -2ε) / (1 + 4εξ -2ε) . In view of Assumption 2, Choose ∆ to be small enough so that |W

i (β 0 -β) + Z i (δ 0 -δ)| ≤ a < ∞ for all θ ∈ Θ ∆ with probability 1 -η, for a > 0 such that Pr {U i < a|X i } ≤ 1/2 + ε and Pr {U i > -a|X i } ≤ 1/2 -ε almost surely and that Pr {A i,a |A i , D i } = ξ > 1/2, where A i = 1 W i β > 0, W i β + Z i δ ≤ 0 A i,a = 1 W i β > 0, W i β + Z i δ ≤ -2a < 0 .
Then for all θ ∈ Θ ∆ , we have

E I + 1i (θ)|A i , D i ≤ η + (1 -η) ((1/2 -ε) ξ + (1/2 + ε) (1 -ξ)) E I - 2i (θ)|A i , D i ≥ (1 -η) ((1/2 + ε) ξ + (1/2 -ε) (1 -ξ)) ,
almost surely, which in turn yields that

E I + 1i (θ) -I + 2i (θ)|A i , D i ≤ η + (1 -η) (-2εξ + 2ε (1 -ξ)) = η + (1 -η) (2ε -4εξ) < 0 almost surely. Similarly, E sup θ∈Θ ∆ I + 3i (θ) -I + 4i (θ)|D i < 0 almost surely for ∆ sufficiently small. Therefore, sup θ∈Θ ∆ E ζ + i (θ)|D i < 0 almost surely for ∆ sufficiently small. The proof for ζ - i (θ) is similar. Proof of Lemma A.3. It follows from (A.1) that S n3 (θ, γ) = n i=1 1 {γ 0 < D i ≤ γ} ζ + i (θ) -ζ + i (θ 0 ) + n i=1 1 {γ < D i ≤ γ 0 } ζ - i (θ) -ζ - i (θ 0 ) . Since ζ + i (θ) and ζ - i (θ) consists of just several indicator functions, there exists a universal constant C < ∞ such that |S n3 (θ, γ)| ≤ C n i=1 1 {γ 0 < D i ≤ γ} + n i=1 1 {γ < D i ≤ γ 0 }
uniformly over θ. Thus, it suffices to show that for every A > 0,

(B.4) sup γ:n|γ-γ 0 |≤A n i=1 1 {γ 0 < D i ≤ γ} = O p (1) and sup γ:n|γ-γ 0 |≤A n i=1 1 {γ < D i ≤ γ 0 } = O p (1).
To show the first claim of (B.4), consider a class of functions indexed by γ,

M A = {1 {γ 0 < D ≤ γ} : n|γ -γ 0 | ≤ A}. Then M A is
γ:n|γ-γ 0 |≤A n i=1 [1 {γ 0 < D i ≤ γ} -E1 {γ 0 < D ≤ γ}] ≤ Cn 1/2 J(1, M A )(EM 2 A ) 1/2 ,
where J(1, M A ) is the uniform entropy integral defined in van der Vaart and Wellner (1996, p.239)

. Since J(1, M A ) is bounded for a VC class and (EM 2 A ) 1/2 = O(n -1/2 ), sup γ:n|γ-γ 0 |≤A n i=1 [1 {γ 0 < D i ≤ γ} -E1 {γ 0 < D ≤ γ}] = O p (1).
The desired result follows since

(B.5) sup γ:n|γ-γ 0 |≤A E1 {γ 0 < D ≤ γ} = O(n -1 ).
The proof of the second claim of (B.4) is similar.

Proof of Lemma 6.1. The proof of this lemma is similar to that of Lemma A.3. It suffices

to show that n i=1 1 {γ 0 < D i ≤ γ} ζ + (θ) -ζ + (θ 0 ) = O p (n -1/6 ) (B.6) and n i=1 {γ < D i ≤ γ 0 } ζ -(θ) -ζ -(θ 0 ) = O p (n -1/6 ). (B.7)
uniformly over (θ, γ) satisfying n 1/3 |θ -θ 0 | ≤ A and n|γ -γ 0 | ≤ A. We will prove only (B.6) since the proof of (B.7) is similar. Consider a class of functions indexed by (θ, γ)

, L A = {1 {γ 0 < D ≤ γ} [ζ + (θ) -ζ + (θ 0 )] : n 1/3 |θ -θ 0 | ≤ A and n|γ -γ 0 | ≤ A}. Then L A is a VC class of functions with an envelope function L A = sup θ,γ {1 {γ 0 < D ≤ γ} [ζ + (θ) -ζ + (θ 0 )] : n 1/3 |θ -θ 0 | ≤ A and n|γ -γ 0 | ≤ A}.
Then as in the proof of Lemma A.3,(B.6) follows from Theorem 2.14.1 of Van der Vaart and Wellner (1996, p.239), the fact that (EL 2 A ) 1/2 = O(n -1/2 n -1/6 ), (B.5), and

sup θ:n 1/3 |θ-θ 0 |≤A E ζ + (θ) -ζ + (θ 0 ) = O(n -1/6 ).
Proof of Lemma 6.2. Since S+ n1 and S-n1 are independent and similarly defined, we prove the convergence of S+ n1 . Define S+ n1 (ν) = S n1 (γ 0 + n -1 ν). We first prove the tightness of S+ n1 (ν). By the D-tightness criterion of Billingsley (1968, equation (15.21)), it suffices to show that there exists a finite, universal constant C < ∞ such that (B.8) E S+ n1 (ν) -S+ n1 (ν 1 ) S+ n1 (ν 2 ) -S+ n1 (ν) ≤ C(ν 2 -ν 1 ) 2

for any ν, ν 1 and ν 2 satisfying 0 < ν 1 < ν < ν 2 . Note that

E S+ n1 (ν) -S+ n1 (ν 1 ) S+ n1 (ν 2 ) -S+ n1 (ν) ≤ 4E n i=1 n j=1 1 γ 0 + ν 1 n < D i ≤ γ 0 + ν n 1 γ 0 + ν n < D j ≤ γ 0 + ν 2 n ≤ 4E n i=1 n j=1,j =i 1 γ 0 + ν 1 n < D i ≤ γ 0 + ν 2 n 1 γ 0 + ν 1 n < D j ≤ γ 0 + ν 2 n ≤ 4n(n -1) F D γ 0 + ν 2 n -F D γ 0 + ν 1 n 2 ≤ 4C F D n(n -1) n 2 (ν 2 -ν 1 ) 2 ,
where the first inequality comes from the fact |ζ + i (θ 0 )| ≤ 2 for all i, the second inequality follows since 1{γ 1 < D i ≤ γ} and 1{γ < D j ≤ γ 2 } are disjoint, and 1{γ 1 < D i ≤ γ 2 } is nondecreasing as γ 1 decreases or as γ 2 increases, the third inequality is obvious since We now consider the weak convergence of the finite-dimensional distributions of S+ n1 (ν). To use the Cramér-Wold device, let 0 < ν 1 < . . . < ν J for J ∈ N and q 1 , . . . , q J be constants.

Instead of obtaining the weak convergence of J j=1 q j S+ n1 (ν j ), we will show below the weak convergence of Sn1 , where Sn1 is defined as Sn1 = J j=1 q j S+ n1 (ν j ) -S+ n1 (ν j-1 ) .

As shown in the proof of Theorem 4 of [START_REF] Pons | Estimation in a Cox regression model with a change-point according to a threshold in a covariate[END_REF], it is easier to deal with Sn1 . Specifically, since Sn1 is a linear combination of { S+ n1 (ν 1 ), . . . , S+ n1 (ν J )}, the weak convergence of the finite-dimensional distributions of S+ n1 (ν) follows if we show that its characteristic function converges to the characteristic function of J j=1 q j S+ 1 (ν j ) -S+ 1 (ν j-1 ) .

We now consider the characteristic function of Sn1 , which has the form

  , S1 tends to -∞ almost surely as |ν| → ∞, thereby implying that the distribution of ν * S1 is tight. An application of the argmax continuous mapping theorem of van der Vaart and Wellner (1996, Theorem 3.2.2, p.286), combined with Lemmas 6.1 and 6.2, gives the main theorem of this paper. Theorem 6.1. Let Assumptions 1-10 hold. Then, as n → ∞, n(γ -γ 0 ) converges weakly to ν * S1

  where • is the Euclidean norm. Now, let m < n be the block length for the subsampling and randomly draw m samples with replacement from a given data set and estimate δ 0 and γ 0 from this subsample. Let δ * and γ * denote subsample estimates and construct T * n = m 1/3 g θ * -g θ . Repeat this many times, say B times, and obtain empirical distributions of T * n . Denote the 1 -α quantile of the distribution as c 1-α . Then, the nominal level α test rejects H 0 if and only if T n > c 1-α .

D

  i are i.i.d. with the common distribution F D , and the fourth inequality follows from the assumption that F D is Lipschitz continuous with the constant C F D . Then (B.8) follows immediately by taking C = 4C F D .

.

  Note that as in the proof of Theorem 4 of[START_REF] Pons | Estimation in a Cox regression model with a change-point according to a threshold in a covariate[END_REF], using the fact that for each i, there is at most one index j such thatI nj (D i ) = 0itq j I nj (D)ζ(θ 0 )) -1] , ν j -ν j-1 )f D (γ 0 ) E exp (itq j ζ(θ 0 )) -1|D = γ + 0 + o(1) .

Table 1 :

 1 Rejection rates of subsampling hypothesis tests

		Threshold parameter γ 0
		Nominal Level		Block Size (m)
			30	40	50	60
	Dist 1	0.05	0.064 0.042 0.030 0.028
		0.1	0.126 0.106 0.096 0.082
	Dist 2	0.05	0.080 0.058 0.050 0.050
		0.1	0.134 0.104 0.116 0.100
		Slope parameter δ 0
		Nominal Level		Block Size (m)
			10	15	20	30
	Dist 1	0.05	0.090 0.046 0.030 0.014
		0.1	0.152 0.114 0.094 0.052
	Dist 2	0.05	0.070 0.030 0.022 0.006
		0.1	0.138 0.080 0.058 0.048

Alternatively, one could use a random search algorithm based on[START_REF] Manski | Operational characteristics of maximum score estimation[END_REF] rather than the simple grid search.

See Delgado et al (2001) for an automatic selection rule, while it is computationally demanding to perform a Monte Carlo experiment.

* Lee thanks the Economic and Social Research Council (ESRC) and the Leverhulme Trust through the funding of the Centre for Microdata Methods and Practice (http://cemmap.ifs.org.uk ) and the research programme Evidence, Inference and Inquiry (http://www.evidencescience.org). Seo thanks the Economic and Social Research Council for financial support through ESRC Grant RES-061-23-0059. We would like to thank an associate editor, an anonymous referee, Jean-Pierre Florens, Joel Horowitz, Charles Manski, Kyungchul Song, and participants at several seminars for helpful comments.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

Then using the fact that A n = exp(n log A) for any A > 0 and that log(1

Then the lemma follows from the fact that the characteristic function ϕ S1 (s) (t) of S1 (s) is

as in Lemma 5 of [START_REF] Pons | Estimation in a Cox regression model with a change-point according to a threshold in a covariate[END_REF].


Valentina Corradi
email: vcor-radi@warwick.ac.uk.y

Emma M Iglesias
email: iglesia5@msu.edu.

Emma M Iglesias Yz

Silvia Gonçalves

Lutz Kilian

Oliver Linton

Nour Meddahi

Paolo Za¤aroni

Tim Vogelsang

,1) parameters

Keywords: block bootstrap, Edgeworth expansion, higher order re…nements, Quasi Maximum Likelihood, GARCH

.

Introduction

It is well known that the quasi likelihood function of a GARCH (Generalized Autoregressive Conditional Heteroskedasticity) model, introduced by Bollerslev (1986), depends on the entire past history of the observables. In this case, resampling blocks of observations is not equivalent to resampling blocks of the likelihood function. This point has been lucidly pointed out by [START_REF] Gonçalves | Maximum Likelihood and the Bootstrap for Nonlinear Dynamic Models[END_REF], who indeed suggest to construct bootstrap estimators of GARCH models based on resampling blocks of the likelihood function. We go a step further, and we investigate the higher order properties of such estimators. This is accomplished in two steps. First, we establish the necessary conditions for the existence of an Edgeworth expansion up to the k th term. This collapses to the existence of a minimum number of moments of the innovation process. Second, we provide su¢ cient conditions for higher order improvements of equally tailed and symmetric t-tests based on Quasi Maximum Likelihood Estimators (QMLE) of GARCH(1,1) parameters. This is done by providing the su¢ cient number of moments of the innovation process needed for the moments of the actual and bootstrap statistics to approach each other at an appropriate rate. Broadly speaking, this allows to control the rate at which the di¤erence between the Edgeworth expansion of the actual and bootstrap statistic approaches zero. [START_REF] Linton | An Asymptotic Expansion in the GARCH(1,1) Model[END_REF] calculated the Edgeworth-B distribution function for the GARCH(1,1), and we extend his setting to the Edgeworth expansion in the context of the existence of re…nements of the bootstrap. If we were willing to assume that all the moments of the innovation process exist, then all the assumptions in [START_REF] Andrews | Higher-Order Improvements of a Computationally Attractive k Step Bootstrap for Extremum Estimators[END_REF] would be satis…ed and higher order re…nements would then follow from his Theorem 2. On the other hand, for the GARCH case we can prove re…nements for QML estimators under somewhat weaker assumptions on the existence of moments of the score and Hessian derivatives, than those required in [START_REF] Andrews | Higher-Order Improvements of a Computationally Attractive k Step Bootstrap for Extremum Estimators[END_REF].

In nutshell, we need (i) conditions on the parameter space of the process in order to ensure that the GARCH process is exponentially mixing (see e.g. [START_REF] Carrasco | Mixing and Moment Properties of Various GARCH and Stochastic Volatility Models[END_REF]), (ii) smoothness of the density of the innovation process (iii) the existence of a given number of moments of the innovation process. First, we show that under (i)-(iii) the [START_REF] Götze | Asymptotic Distributions of Statistics in Time Series[END_REF] conditions for

the existence of the Edgeworth expansion for weakly dependent observations are satis…ed. Then, we show that (i)-(iii) su¢ ce to obtain the same higher order re…nements of [START_REF] Andrews | Higher-Order Improvements of a Computationally Attractive k Step Bootstrap for Extremum Estimators[END_REF]. Thus, the block bootstrap, based on resampling the likelihood, leads to an error in rejection probability and con…dence interval coverage probability of smaller order than T 1=2 for equal-tailed t-tests for GARCH parameters and of order smaller than T 1 for symmetric t-tests, for > 0; and such that < and + < 1=2; where is the parameter controlling the lengths of the block l; i.e. l ' T :

Needless to say, an advantage of using bootstrap estimators is that of obtaining more accurate inference on the GARCH(1,1) parameters. Gonçalves andKilian (2004, 2007) focused on bootstrap inference for conditional mean that is robust to GARCH or other unspeci…ed forms of conditional heteroskedasticity. However, there are many cases in practice where we are interested on the GARCH parameters themselves. From a more empirical perspective, one of the most popular application of bootstrap estimation of GARCH parameters is in the context of risk management, and more precisely in the evaluation of Value at Risk and Expected Shortfall. For example, [START_REF] Christo¤ersen | Estimation Risk in Financial Risk Management[END_REF] rely on bootstrapping GARCH parameters in order to take into proper consideration the contribution of parameter estimation error when evaluating Value at Risk; [START_REF] Mancini | Robust Semiparametric Bootstrap Methods for Value at Risk Prediction under GARCH-type Volatility Processes[END_REF] address the same issue using an estimation and bootstrap procedure robust to the presence of outliers. Both papers rely on the residual based bootstrap approach outlined by [START_REF] Pascual | Bootstrap Prediction for Returns and Volatilities in GARCH Models[END_REF].

If one knew the data generating process (DGP), then a residual based bootstrap approach, which makes direct use of the structure of the model, seems to be more natural than a nonparametric bootstrap approach, such as the block bootstrap. [START_REF] Hidalgo | A Goodness of Fit Test for ARCH(1 )[END_REF] show the …rst order validity of the residual-based bootstrap for ARCH(1) process, which indeed include …nite GARCH(1; 1).

Hence, it is interesting to investigate whether the residual-based bootstrap provides higher order improvements over asymptotic normality, and whether these improvements are sharper than those provided by the block bootstrap. This is a quite challenging task, and we leave it for future research.

Nevertheless, we outline how the arguments used by Hidalgo and Za¤aroni to show …rst order validity cannot be directly extended to show the higher order properties of the residual-based bootstrap.

In the case of possible nonlinear Markov processes, recent papers by [START_REF] Andrews | High Order Improvements of the Parametric Bootstrap for Markov Processes[END_REF][START_REF] Horowitz | Bootstrap Methods for Markov Processes[END_REF] have established that higher-order re…nements, very close to those attainable in the iid case, can be achieved via the use of the Markov bootstrap, even in the case the underlying transition density is unknown, and replaced by a nonparametric estimator. Though a GARCH process is not Markov, under certain conditions is approximate Markov, as in the de…nition of Horowitz. Thus, we have analyzed the applicability of the Markov process for the estimation of the conditional variance parameters. If the marginal density of the innovation process is known, then the results of [START_REF] Andrews | High Order Improvements of the Parametric Bootstrap for Markov Processes[END_REF] apply and the Markov bootstrap provides re…nements arbitrarily close to those attainable in the iid case. On the other hand, whenever the density of the innovation is unknown and has unbounded support, as in the gaussian case for example, then some of the assumptions for the existence of higher order re…nements are violated. This point has been already outlined in [START_REF] Horowitz | Bootstrap Methods for Markov Processes[END_REF].

The rest of this paper is organized as follows. Section 2 describes the implementation of the bootstrap procedure based on the resampling of the likelihood function. Section 3 establishes the higher order improvement of the bootstrap approximation of the coverage error of con…dence intervals for GARCH parameters, and summarizes the main theoretical results. Section 4 outlines the di¢ culties involved in the establishment of possible higher order re…nements for the residual-based bootstrap. Section 5 brie ‡y outlines the use of the Markov bootstrap, whenever the marginal density of the innovation is unknown and has unbounded support. Section 6 reports Monte Carlo simulation results, which provide some evidence of the improved accuracy of bootstrap, for the error in the coverage probability. All the proofs are collected in the Appendix.

2 The Block Bootstrap: Set-Up Suppose y t is generated by the GARCH(1,1) process,

(1)

y t = p h t t ; h t = y 1 + y 2 y 2 t 1 + y 3 h t 1 ; t = 1; :::; T
where t is iid and y = y 1 ; y 2 ; y 3 is the true parameter vector. We de…ne the log-likelihood function as if yt p ht were normally distributed, thus using a Gaussian likelihood. The quasi maximum likelihood estimator, QMLE, is then de…ned as:

(2) b T = arg max 2 1 T T X t=1 L t ( ) where 1 T T X t=1 L t ( ) = 1 2 T X t=1 ln h t ( ) 1 2 T X t=1 y 2 t h t ( ) :
Note that after a few manipulations, and imposing h 0 = 0 = 0; we have that1

h t = 1 + 2 2 t 1 + 3 h t 1 = 1 " 1 + t 1 X j=1 j Y i=1 2 2 t i + 3 # : (3)
From (3) it is immediate to see that resampling blocks of the observable series y t is not equivalent to resampling blocks of the log-likelihood. Thus, we need to resample b blocks of length l from the loglikelihood L t ; setting bl = T: Hereafter, let I i ; i = 1; :::; b denote identical and independent draws from a discrete uniform on 0; 1; :::; T l: Thus, for each i = 1; ::; b I i = j, with j = 0; 1; :::; T l; with equal probability 1=(T l+1): Now, for all 2 ; de…ne L 1 ( ) ; L 2 ( ) ; :::; L T l+1 ( ) ; :::; L T ( ) to be equal to L I 1 +1 ( ) ; L I 1 +2 ( ) ; ::::; L I b +1 ( ) ; :::; L I b +l ( ) :

Note, that we use the same random draws I i for any 2 : In general, resampling the data or resampling the likelihood is equivalent. However, in the GARCH case, the (quasi) likelihood function

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depends on the entire past history of the observables. In this case, resampling blocks of observations is not equivalent to resampling blocks of the likelihood function. This point has been lucidly pointed out by [START_REF] Gonçalves | Maximum Likelihood and the Bootstrap for Nonlinear Dynamic Models[END_REF], who indeed suggest to construct bootstrap estimators of GARCH model based on resampling blocks of the likelihood function. We use the moving blocks bootstrap (MBB) of [START_REF] Künsch | The Jackknife and the Bootstrap for General Stationary Observations[END_REF] as in [START_REF] Gonçalves | Maximum Likelihood and the Bootstrap for Nonlinear Dynamic Models[END_REF].

It should be pointed out that, if we were just interested in …rst order validity, then we could have set l = 1 and relied on the iid nonparametric bootstrap. In fact, as the score is a martingale di¤erence sequence, bootstrap samples based on iid resampling of the log-likelihood would have ensured that the …rst two bootstrap moments properly mimic the correspondent sample moments. However, as for re…nements, we need to match higher moments, and in this case the fact that the score is a martingale di¤erence sequence does not help.

We now use the resample log-likelihood in order to construct the bootstrap estimator b T , that is:

b T = arg max 2 1 T T X t=1 L t ( ) E rL t ( b T ) 0 :
Note that the recentering term, E rL t ( b T ) 0

; ensure that the score, evaluated at b T ; has zero mean.

De…ne the Hessian and the variance of the score as B y and A y respectively, where

B y = E r 2 L t ( y )
and

A y = V ar 1 p T T X t=1 r L t ( y ) ! = E r L t ( y )r L t ( y ) 0 :
Also, de…ne the sample analogs of B y and A y ; that is

b B T = 1 T T X t=1 r 2 L t ( b T ) (4) b A T = 1 T T X t=1 r L t ( b T )r L t ( b T ) 0 :
Our objective is to provide higher order re…nements of the approximation of the coverage error for the t-statistic

(5)

t i ;T = p T b B 1 b A b B 1 1=2 ii b i;T y i ;
for i = 1; 2; 3:

B T = 1 T T X t=1 r rL t ( b T ) E rL t ( b T ) and (6) b A T = 1 T T X t=1 rL t ( b T ) E rL t ( b T ) rL t ( b T ) E rL t ( b T ) 0 :
The bootstrap analog of t i ;T is then given by:

(7) e t i ;T = p T b B 1 T b A T b B 1 T 1=2 ii b i;T b i;T : As rL t ( y ) is a martingale di¤erence sequence, b A T is a consistent estimator of V
e A T = b T (T l + 1) T l X i=0 l X j=1 l X m=1 rL i+j ( b T ) E rL i+j ( b T ) r L i+m ( b T ) E rL i+m ( b T ) 0 : (8) Now, let b T;ii = b B 1 T b A T b B 1 T 1=2 ii ; e T;ii = b B 1 T e A T b B 1 T 1=2
ii and following [START_REF] Hall | Bootstrap Critical Values for Tests based on Generalized-Method-of-Moments Estimators[END_REF], de…ne the adjustment term as i;T = b T;ii =e T;ii ;

and the adjusted t-statistics as:

(9)

t i ;T = p T b B 1 T b A T b B 1 T 1=2 ii b i;T b i;T i;T :
3 The Block Bootstrap: Re…nements

De…ne the vectors containing the centered score, the centered outer product and all their derivatives up to order d 1;k as:

(10)

f i;t;d 1;k ( ) = r i L t ( ); :::; r d 1;k i r L t ( ) ; r i L t ( )r i L t ( ) 0 ; :::; r d 1;k i r L t ( )r L t ( ) 0 ;
where In what follows, we rely on the following assumption:

r i L t ( ) = r i L t ( ) E (r i L t ( )) ; r i L t ( )r i L t ( ) 0 = r i L t ( ) r i L t ( ) 0 E r i L t ( ) r i L t ( ) 0 ;
Assumption A: (i) The parameter space is a compact subset of R 3+ such that = f 1 ; 2 ; 3 ; 1 ; 2 ; 3 > 0; 2 + 3 < 1g. (ii) y = y 1 ; y 2 ; y 3 ; b T = b 1T ; b 2T ; b 3T are in the interior of (iii) t is iid(0; 2 )
and it has a positive continuous density:

Assumption A ensures that y t and h t ( y ) are strictly and covariance stationary mixing processes, with exponentially decaying mixing coe¢ cients (see e.g. [START_REF] Carrasco | Mixing and Moment Properties of Various GARCH and Stochastic Volatility Models[END_REF]). Let N y be a neighbourhood of y ; then for any 2 N y ; f i;t;d 1;k ( ) is also mixing process with exponential decay.

In Lemma A1 in the Appendix, we show that Assumption A ensures that the [START_REF] Götze | Asymptotic Distributions of Statistics in Time Series[END_REF] Cramer type conditions hold. Hence, in order to prove the existence of the k th term of the Edgeworth expansion for the statistic in ( 5) we need to show the existence of enough moments for the components of f i;t;d 1;k ( ) ; for i = 1; 2; 3: This is accomplished in Theorem 1, where we provide necessary and su¢ cient conditions for the existence of the k th term of the Edgeworth expansion for the t-statistic in (5).

Hereafter, let 1 + P k i=1 T i=2 i (@=@z) (z) denote the Edgeworth expansion of t i up to order k (see e.g. p.142 in [START_REF] Andrews | Higher-Order Improvements of a Computationally Attractive k Step Bootstrap for Extremum Estimators[END_REF]. We have:

Theorem 1: Let Assumptions A hold. Then (11) lim T !1 T k=2 sup z2R P (t i z) 1 + k X i=1 T i=2 i (@=@z) ! (z) = 0 (i) only if E 2(k+2)(k+4) t < 1 (ii) and if E 2(k+3)(k+5) t < 1:
Theorem 1 provides a set of necessary and a set of su¢ cient conditions for the existence of the k-th term of the Edgeworth expansion of the statistics. Needless to say, for k = 2; 3 (i) is also a necessary condition for obtaining higher order re…nements for equally-tailed and symmetric t-tests respectively.

Hereafter, z t i ;T ; =2 and z jt i ;T j; denote the (1 )=2 bootstrap quantile for t i ;T and the (1 ) quantile for t i ;T respectively. As usual, z t i ;T ; =2 and z jt i ;T j; are constructed from the empirical distribution of the bootstrap statistic t i ;T : We have:

Theorem 2: Let Assumption A hold. Let + < 1=2 and < ; where is the parameter controlling the block size, i.e. l ' T :

If E 2(d 1;k +d 2;k +2)(k+2)p t < 1; where d 1;k k + 3; d 2;k k + 1; and A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT p > k=2 2(1=2 ) ; then: (i) for k = 2; P t i ;T < z t i ;T ; =2 or t i ;T > z t i ;T ;1 =2 = + o T (1=2+ ) (ii) for k = 3; P jt i ;T j < z jt i ;T j; = + o T (1+ ) :
From Theorem 2, it is immediate to say that we obtain the same higher order re…nements for the error in rejection probability (ERP), as in [START_REF] Andrews | Higher-Order Improvements of a Computationally Attractive k Step Bootstrap for Extremum Estimators[END_REF]. As an immediate corollary, we also have that the bootstrap error in the coverage probability (ECP) is of order smaller than T (1=2+ ) for the equal-tailed test and of T (1+ ) for the symmetric test, with < 1=4: It is interesting to note that, in the exponentially mixing GARCH(1,1) case, the only conditions we need for higher order re…nements are in terms of smoothness of the density and of existence of a su¢ cient number of moments of the innovation process.

The Residual-Based Bootstrap

The residual-based bootstrap is based on iid resampling of the centered residuals.

Let b t = y t =h 1=2 t b T ;
we resample, with replacement, from the empirical distribution of b t ; and obtain the sequence t ;

where 1 is equal to b t T 1 P T t=1 b t ; t = 1; :::; T with equal probability 1=T: 1 ; 2 ; :::; T are de…ned analogously, so that t ; is iid conditional on sample. We then proceed as follows:

Fix y 0 ; h 0 , =) b h 1 = b 1;T + b 2;T y 2 0 + b 3;T h 0 y 1 = b h 1=2 1 1 b h 2 = b 1;T + b 2;T y 2 1 + b 3;T b h 1 y 2 = b h 1=2 2 2
and so on, until sequences y t ; h t ; t = 1; :::; T are formed. Now, de…ne the bootstrap QML estimator as:

e T = arg max 2 T X t=1 L (RS) t ( ) = arg max 2 1 2 T X t=1 ln h t ( ) 1 2 T X t=1 y 2 t h t ( ) ! ;
where the superscript (RS) denotes the fact that the bootstrap likelihood is based on the residual bootstrap approach.

This residual based bootstrap procedure has been recently suggested by [START_REF] Pascual | Bootstrap Prediction for Returns and Volatilities in GARCH Models[END_REF]. See also [START_REF] Christo¤ersen | Estimation Risk in Financial Risk Management[END_REF], for an application to Value at Risk evaluation with GARCH models.

In the case of correctly speci…ed ARMA models, the asymptotic …rst order validity, as well as re-…nements, of the residual-based bootstrap have been already established (see e.g. [START_REF] Inoue | Bootstrapping Autoregressive Processes with Possible Unit Roots[END_REF], [START_REF] Kreiss | Bootstrapping Stationary Autoregressive Moving-Average Models[END_REF], and [START_REF] Bose | Edgeworth Correction by Bootstrap in Autoregressions[END_REF]). [START_REF] Robinson | Pseudo Maximum Likelihood Estimation of ARCH(1) Models[END_REF] establish conditions for strong consistency and asymptotic normality of QML estimators of ARCH(1) models: Now ARCH(1) models encompass GARCH(1,1) models.

In particular, the ARCH(1) model allows for a higher degree of memory, including also long-memory behavior. [START_REF] Hidalgo | A Goodness of Fit Test for ARCH(1 )[END_REF] hereafter HZ) take a step further and establish the …rst order validity of their residual-based bootstrap analogs.

Below, we outline the key steps in the HZ proof for the residual-based bootstrap …rst order validity, and sketch the further steps one should follow in order to establish higher order re…nements. In fact, the interesting question is whether the residual based-bootstrap may provide sharper re…nements than the block-bootstrap does. This is a rather involved issue, and we leave it for further research.

The key point is that we can express h t ; the score, and the hessian, as well as their bootstrap analogs, as a function on entire past of squared innovations. For notational simplicity we just focus on h t : Recalling equation (3); it's bootstrap analog is given by:

h t ( ) = 1 + 2 2 t 1 + 3 h t 1 = 1 " 1 + t 1 X j=1 j Y i=1 2 2 t i + 3 # h t ( ) = 1 + 2 2 t 1 + 3 h t 1 = 1 " 1 + t 1 X j=1 j Y i=1 2 2 t i + 3 # (12)
Thus one can write h t ( y ); as h t ( y ) = h 2 t 1 ; 2 t 2 ; :::; 2 1 ; y ; Now,

E 1 T T X t=1 h t ( y ) ! ' 1 T T T X i 1 =1 T X i 2 =1
:::

T X i T =1 h b 2 i 1 ; b 2 i 2 ; :::; b 2 i T ;
as any permutation and combination can occur with equal probability 1=T: 2 2 Note the abuse of notation. As we are resampling from centered residuals, more correctly we should have b i1 T 1 P t b t 2 ; though for notational simplicity we simply write b 2 i1 :

Let h t;M ( y ) be a truncated version of h t ( y ); i.e. h t;M ( y ) = h 2 t 1 ; 2 t 2 ; :::; 2 t M ; 0; :::; 0 ; HZ proof (see Lemmas 6.13-6.15) for …rst order asymptotic validity is based on three arguments. The …rst argument is that the process can be truncated, and the truncation error approaches zero fast enough, i.e.

E 1 T T X t=1 h t ( y ) h t;M ( y ) ! = O p M d ;
where d > 1 depends on the degree of memory of the process. Note that,

E 1 T T X t=1 h t;M ( y ) ! = 1 T M T X i 1 =1 T X i 2 =1
:::

T X i M =1 h b 2 i 1 ; b 2 i 2 ; :::; b 2 i M = V T;M ;
where V T;M de…nes a V -statistic. De…ne, the U statistic U T;M as

U T;M = T M 1 X 1 i 1 ::: i M T h b 2 i 1 ; b 2 i 2 ; :::; b 2 i M :
The second argument is that V statistics are very accurately approximated by U statistics, in fact by e.g. Theorem 5.1 in [START_REF] Grams | Convergence Rates for U-statistics and Related Statistics[END_REF],

E (jV T;M U T;M j r ) = O(T r ); for r > 0:
Finally, the third argument is that, by the law of large numbers for U statistics,

E U T;M 1 T T X t=1 h t;M ( y ) ! = O M T :
The same applies if we replace y with b T : The point is that, as for …rst order validity, it su¢ ces to match the Hessian and the variance of the score. As the score is a martingale di¤erence sequence, it all collapses to match sample and bootstrap sample …rst moments. In fact, as we outline below, things get more complicated once we move to higher moments.

The question is whether we can also obtain re…nements for the residual-based bootstrap and whether they are sharper than those obtained via the block-bootstrap.

A crucial condition for bootstrap re…nements is that the "distance" between the Edgeworth expansion of the statistic and that of its bootstrap counterpart approaches zero at fast enough rate, which in turn de…nes the "sharpness" of re…nements. This collapses to check the speed at which the distance between higher moments of the statistic and its bootstrap counterpart approaches zero.

Hereafter, let f i;t;d 1;k b T be the bootstrap analog of f i;t;d 1;k y evaluated at b T ; de…ned as in Section 3. Also, with an abuse of notation, let f t b T be a generic element of f i;t;d 1;k b T :

A C C E P T E D M A N U S C R I P T

ACCEPTED MANUSCRIPT

In a nutshell, to have an improvement in the error in coverage probability (ECP) of T ; we need that (e.g. Andrews 2002):

(13)

T a P T (m) E 1 T 1=2 T X t=1 f t b T ! m T (m) E 1 T 1=2 T X t=1 f t y ! m > T ! ! 0;
where (m) = 0 if m is even and (m) = 1=2 if m is odd, a = 3=2 and m = 4 for equally tailed test, and a = 2; m = 5 for symmetric tests. Needless to say, the residual-based block bootstrap will provide sharper re…nements if > ; where is de…ned in Theorem 2 above. Now, as convergence in r mean implies converges in probability, the statement in ( 13) is implied by ( 14) 1); for some r > 0:

T a E T T (m) E 1 T 1=2 T X t=1 f t b T ! m T (m) E 1 T 1=2 T X t=1 f t y ! m r ! = o(
Now, f t b T writes as

f t b T = f 2
t 1 ; 2 t 2 ; :::; 2 1 ; b T ; and de…ne its truncated version as

f t;M b T = f 2 t 1 ; 2 t 2 ; :::; 2 t M ; b T :
The left hand side of ( 14) is majorized by I T + II T + III T ; where

I T = T a E T T (m) E 1 T 1=2 T X t=1 f t b T ! m T (m) E 1 T 1=2 T X t=1 f t;M b T ! m r ! II T = T a E T T (m) E 1 T 1=2 T X t=1 f t;M b T ! m T (m) E 1 T 1=2 T X t=1 f t;M y ! m r ! III T = T a T T (m) E 1 T 1=2 T X t=1 f t;M y ! m T (m) E 1 T 1=2 T X t=1 f t y ! m r ! :
It is immediate to see that the case of m = 1 can be treated along the same lines of HZ. Now, given

(3) and ( 10), for 0 < < 1;

I T = T a O T (m)(r m) T & M ,
so that it is enough to set M = T " ; for " > 0 arbitrarily small to ensure that I T = o(1): The same order of magnitude applies to III T : The di¢ cult part is II T :

We need to compute E 1 T 1=2 P T t=1 f t;M b T m
: For sake of simplicity, we begin by considering the case of m = 2: By noting that E f 2 t 1 ; 2 t 2 ; :::

; 2 t M ; b T f 2 t 1 ; 2 t 2 ; :::; 2 t M ; b T = 0 for j j > M; A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT E 1 T 1=2 T X t=1 f t;M b T ! 2 ' 1 T M X = M T M X t=2M E f 2 t 1 ; 2 t 2 ; :::; 2 t M ; b T f 2 t 1 ; 2 t 2 ; :::; 2 t M ; b T = 1 T M X = M T M X t=2M 1 T 1 T M T M X i 1 ::: T M X i ::: T M X i M ::: T M X i +M f b 2 i 1 ; :::; b 2 i ; :::; b 2 i M ; b T f b 2 i ; :::; b 2 i M ; :::; b 2 i M + ; b T : Hereafter, let (1) min i M j i j ; (m) max i M j i j ; (1) (2)
:::

(m) : Thus, for a generic m;

E 1 T 1=2 T X t=1 f t;M b T ! m ' 1 T m=2 1 M X 1 = M ::: M X m= M E f 2 2M 1 ; :::; 2 M 1 ; b T ::: f 2 2M m ; :::; 2 M m ; b T 1 max i m i min i m i M = 1 T m=2 1 0 X 1 = M ::: 0 X m= M 1 T M + (m) T M X i 1 =2M ::: T M X i (1) =2M ::: T M X i (m) =2M
::: ::: It is immediate to see that, contrary to the case of m = 1;

T M X i M =2M ::: T M X i M + (1) =2M ::: T M X i M + (m) =2M
E 1 T 1=2 P T t=1 f t;M b T m cannot be
expressed as a V statistic, and thus the arguments used by HZ do no longer directly apply.

In fact, we can write

E 1 T 1=2 T X t=1 f t;M b T ! m = 1 T m=2 1 0 X 1 = M ::: 0 X m= M e V 1;T;M + (m) + 1 T m=2 1 M X 1 =1 ::: M X m=1 e V 2;T;M + (m) ;
where for i = 1; 2 e V i;T;M + (m) is not a V statistic with M + (m) elements, as di¤erent indexes appear with di¤erent frequency, as for example b 2 i (m) ; :::; b 2 i M appears m times, while b 2 i 1 ; :::; b 2 i (1) appears only once. Hence, we cannot directly rely on the same argument used by Hidalgo and Za¤aroni for showing …rst order validity.

As m is …nite, we believe it is possible to …nd a sharp enough bound for II T ; however this is a rather challenging task and we leave it for future research.

The Markov Bootstrap

For nonlinear, …nite order markov processes it has been already established that the Markov bootstrap can provide higher order re…nements very close to those available for iid observations. Andrews (2005) has considered the case in which the transition density is known in closed form, so that the Markov bootstrap is indeed a parametric bootstrap, while [START_REF] Horowitz | Bootstrap Methods for Markov Processes[END_REF] has considered the case in which the transition density is unknown. More precisely, [START_REF] Andrews | High Order Improvements of the Parametric Bootstrap for Markov Processes[END_REF] suggests to recursively resample the data from the likelihood evaluated at the estimated parameters. Thus, the bootstrap sample is generated by the same conditional distribution as the original sample, but with the "true" parameters replaced by the estimated parameters. However, this approach is not directly applicable in our context, as it is well known that y t ; as de…ned in (1) is non markovian, though (y t ; 2 t ) are jointly markovian. Nevertheless, if we knew the marginal density of the ; say ( ) ; then we could draw from it T iid observations, and use them to recursively construct h t b T and y t in the same manner outlined in the previous section. In this case, the only di¤erence between the DGP and the bootstrap DGP is the latter is generated using b T instead of y : As a consequence, we could obtain higher order re…nements along the same lines as in Andrews. Needless to say, this is not particularly useful, as in general we do not know the marginal density of the errors.

For the case in which the transition density of a Markov process is unknown, [START_REF] Horowitz | Bootstrap Methods for Markov Processes[END_REF] has suggested to draw the observations from a kernel density estimator. For one-dimensional Markov processes of order q; Horowitz shows that the error in the bootstrap estimate of one-sided and symmetrical probabilities are O P (T 1+" ) and O P T 3=2+" ; respectively, where " > 0 can be set arbitrarily small.

The key point in Horowitz result, stated in his Lemma 14, is that the cumulants of the original and bootstrap statistics di¤er only of a term of order T 1=2+" ; which re ‡ects the uniform rate at which the estimated conditional density converges to the true one, provided one uses a "enough" higher order kernel. In fact, because of the markov property, all moments, as well as smooth functions of them, can be computed via the transition density. Thus it is enough to control the error between the estimated and the true conditional density, in a uniform manner.

It is easy to see that mixing GARCH processes, with exponentially decaying mixing coe¢cients, are approximate Markov, according to the de…nition of Horowitz (2003, Section 4.3).

Though, one of the assumption in Horowitz, i.e. Assumption 4, requires that y t has bounded support. In the GARCH context, if the innovation has unbounded support, as in the case of normal innovation e.g., then also y t has unbounded support and Assumption 4 is violated. Needless say, the boundedness of the support is part of a set of su¢ cient conditions, and therefore we cannot claim it is indeed necessary. We leave the study of possible higher order re…nements via the Markov bootstrap for future research.

Simulation results

Table 2 in Gonçalves and White (2004, p. 209) reports the coverage rates of nominal 95% symmetric percentile intervals for an ARCH(1) process, both using asymptotic theory and the block bootstrap.

They consider sample sizes T equal to 200 and 500 and, for = 0 (in their notation, implies that t in (1) follows a iidN (0; 1) distribution). We extend their simulation experiment to the case of a GARCH(1,1) model as in ( 1), where the true parameter vector is given by y 1 ; y 2 ; y 3 0 = (0:8; 0:1; 0:8) 0 ; and t is either iidN (0; 1) or a t distribution with 5 degrees of freedom (t 5 ): We use the MBB of [START_REF] Künsch | The Jackknife and the Bootstrap for General Stationary Observations[END_REF]. Since our model is a GARCH(1,1), a more complicated process than the simulated ARCH(1) of Gonçalves and White (2004, page 209, Table 2), we report results for sample sizes 300 and 500. The number of Monte Carlo replications is set equal to 4000. Table 1 reports the results of coverage rates of nominal 95% symmetric percentile-t intervals both using the QML asymptotic approximation (ASYM) and our block bootstrap procedure (Bboot). We report the con…dence intervals (C. I.) for 2 and 3 : For the block bootstrap procedure (Bboot), we have used 999 bootstrap replications. We provide the results for block sizes l = 1; 5; 10; 25 and 50: When the block size is equal to 1; the block bootstrap only ensures …rst order validity; and then the di¤erence in the empirical coverage rate will illustrate the degree to which higher-order improvements help3 .

Pascual, Romo and Ruiz (2006, Tables 1 and2, p. 2299-2300) provide simulation results for a normal and a t 5 innovation process in the GARCH(1,1) context. Their simulation results provide evidence of the good performance of the residual based bootstrap procedure, although as far as we know, there are no theoretical results available in the literature that support the existence of re…nements with this type of bootstrap. From Table 1, it is immediate to see that when t is iidN (0; 1); the empirical coverage of ASYM is substantially below the nominal one, specially for 3 ; even with samples of 500 observations. Also, for sample of 300 observations, the empirical coverage of ASYM is substantially below the nominal one for both 2 and 3 :

The bootstrap empirical coverages for 2 and 3 seem to be rather robust to the choice of the block size, provided l 10: The improvements in coverage of the block bootstrap relative to …rst order asymptotics are noticeable, specially for 3 for 500 observations, and for both 2 and 3 in the case of 300 observations. This shows the usefulness of the block bootstrap in this context. The simulation results of Table 1 when t is drawn from iidN (0; 1) innovations, all the moments of t exist, and thus conditions in Theorem 2 are satis…ed. Moreover, when l = 1; the bootstrap only ensures …rst order validity. Indeed, as expected, the improvements over asymptotic normality, are smaller than in the case of l 10: This demonstrates once again the usefulness of the block-bootstrap in providing re…nements when the block size is carefully chosen.

Finally, Table 1 also reports the results for the case in which the innovations are drawn from tdistribution with 5 degrees of freedom. Hence, the conditions in Theorems 1 and 2 about the existence of a su¢ cient number of moments, are clearly violated. Even if the block bootstrap performs much worse than in the case of normal innovations, it still provides some improvement over asymptotic normality.

The proof of Theorem 1 and Theorem 2 below require the following Lemma, which ensures that the Götze and Hipp conditions (1994) for the existence of the Edgeworth expansion hold for the GARCH case.

Lemma A1:

Let Assumption A hold and assume that E 4(d 1;k +2) t < 1. Then there exist constants K 1 < 1

and > 0 such that for arbitrarily large & > 1 and all integers m 2 1 ; T and 2 R

dim f t;d 1;k with < k k < T & : (15) E E exp i 0 2m+1 X t=1 f t;d 1;k y ! j j : jj mj > K 1 ! exp ( ) ;
where j is the innovation process in (1), i = p 1 and f t;d 1;k y = f 1;t;d 1;k ; f 2;t;d 1;k ; f 3;t;d 1;k and for = 1; 2; 3 f ;t;d 1;k is the vector containing the unique components of the score and the outerproduct and their derivatives through order d 1;k , evaluated at y and de…ned in (10).

Proof of Lemma A1:

We proceed in two steps. First we show that we can express f t;d 1;k y as a function of all the past innovation, i.e., we can write (16) f t;d 1;k y := i j : j > 0; y ; i 2 Z:

Then, we show that f t;d 1;k y satis…es conditions (i)-(iii) in Lemma 2.3 in [START_REF] Götze | Asymptotic Distributions of Statistics in Time Series[END_REF], and

hence the statement follows.

We begin by showing ( 16). Recall that the element of f t;d 1;k y are the derivatives of the centered score and Hessian up to order d 1;k : Now, straightforward calculations give: From Lemma 1 in Lumsdaine (1996, equations A1.1-A1.4), it follows that for i = 1; 2; 3

r L t ( y ) = r h t ( y ) h t ( y ) 2 t 1 ; r 2 L t ( y ) = 1 2 ( 2 t 1) r 2 h t ( y ) h t ( y ) r h t ( y )r h t ( y ) 0 h 2 t ( y ) 1 2 2 t r h t ( y )r h t ( y ) 0 h 2 t ( y ) r 3 L t y = 1 2 2 t 1 r 3 h t y h t y r h t y r 2 h t y h 2 t y + 2
E r i ht( y ) ht( y ) k !
is of the same order of magnitude as E

y 2k t j h k t ( y ) = E h k t j ( y ) h k t ( y ) 2k t j
for any j < t: Therefore, a necessary condition for E r j 1 ::: j k L T y < 1 is that E 2k t j < 1: Indeed, this is also a su¢ cient condition. In fact, Ling and McAleer (2003, p.304) show that a su¢cient condition for the existence of E (r ht( y ) r ht( y ))

h 2 t ( y ) is that E( 4 i ) < 1:
The same argument can be used to show a su¢ cient condition for E r j 1 :::

j k L T y < 1 is that E 2k t < 1:
From the expression for r L t ( ); r 2 L t ( ); :::; r k L t ( ) we see that they can be expressed as ratios of products of derivatives of h t y over power of h t ( y ): Recalling that,

h t ( ) = 1 h 1 + P t 1 j=1 Q j i=1 2 2 t i + 3 i
we have that,

r 1 h t = 1 + P t 1 j=1 Q j i=1 2 2 t i + 3 and that r 2 h t = 1 P t 2 k=0 k 3 P t 2 k j 1 =0 j 1 3 2 t j 1 1 +2 1 2 P t 3 k=0 k 3 P t 2 k j 2 =1 P j 1 <j 2 j 2 1 3 2 t j 1 1 2 t j 2 1 +3 1 2 2 P t 4 k=0 k 3 P t 2 k j 3 =2 P j 2 <j 3 P j 1 <j 2 j 3 2 3 2 t j 1 1 2 t j 2 1 2 t j 3 1 +::: +l 1 l 1 2 P t l 1 k=0 k 3 P t 2 k j l =l 1
P j l 1 <j l :::

P j 1 <j 2 j l (l 1) 3 2 t j
P j 1 <j 2 j t 1 (t 2) 3 2 t j 1 1 :: 2 t j t 1 1 r 1 h t =h t = 1= 1 :
and

r 3 h t = 1 2 h P t 2 k=0 k k 1 3 P t 2 k j 1 =0 j 1 3 2 t j 1 1 + P t 2 k=0 k 3 P t 2 k j 1 =0 j 1 j 1 1 3 2 t j1 1 i + 1 2 2 h P t 3 k=0 k k 1 3 P t 2 k j 3 =1 P j 1 <j 2 j 2 1 3 2 t j 1 1 2 t j 2 1 A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT + P t 3 k=0 k 3 P t 2 k j 2 =1 P j 1 <j 2 (j 2 1) j 2 2 3 2 t j 1 1 2 t j 2 1 i + 1 3 2 h P t 4 k=0 k k 1 3 P t 2 k j 3 =2 P j 2 <j 3 P j 1 <j 2 j 3 2 3 2 t j 1 1 2 t j 2 1 2 t j 3 1 + P t 4 k=0 k 3 P t 2 k j 3 =2 P j 2 <j 3 P j 1 <j 2 (j 3 2) j 3 3 3 2 t j 1 1 2 t j 2 1 2 t j 3 1 i + +:::+ + 1 l 2 h P t l 1 k=0 k k 1 3 P t 2 k j l =l 1
P j l 1 <j l :::

P j 1 <j 2 j l (l 1) 3 2 t j 1 1 ::: 2 t j l 1 + P t l 1 k=0 k 3 P t 2 k j l =l 1
P j 1 <j l :::

P j 1 <j 2 (j l (l 1)) j l l 3 2 t j
y i x y i 1 i < t i = t i > t
Recalling that E 4(d 1;k +2) < 1; it follows that E ( ; x 1 ) t < 1: Also, given assumption A, f t;d 1;k is mixing with mixing coe¢ cients decaying at an exponential rate. Hence, condition (i) in Götze and Hipp (1994, Lemma 2.3) follows, i.e.

There exists K < 1 and > 0 such that for j 2 Z and

x 1 ; x 2 2 R E ( ; x 1 ) t ( ; x 2 ) t Ke jtj jx 1 x 2 j :
As all partial derivatives of the Gaussian likelihood for GARCH(1,1) models are almost surely continuous provided the innovation process has a continuous positive density, then condition (ii) in Götze and Hipp (1994, Lemma 2.3), follows, i.e.

For t 2 Z there exists t R, P ( t ) = 1, such that for all x 0 2 t ; ; > 0; there exists > 0 satisfying P (y 2 R Z : 8x 2 R, jx x 0 j < ; @ @ 0 f t;d 1;k exists at the point (y; x) t and @ @ 0 f t;d 1;k ((y; x) t) @ @ 0 f t;d 1;k (y; x 0 ) t ) 1 :

Finally, for a Gaussian likelihood, the partial derivatives, with respect to the same argument, of the various component of f t;d 1;k are not linearly dependent; also the partial derivatives of a given component of f t;d 1;k ; with respect to generic 6d 1;k arguments, are not linearly dependent. Hence, (iii)

in Götze and Hipp (1994, Lemma 2.3) follows, i.e.

For some distinct l 1 ; :::; l 6d 1;k 0;

det 1 X t=0 @ @ l v f t;d 1;k : v = 1; :::; 6d 1;k ! 6 = 0:
on a set of positive P-probability.

Proof of Theorem 1:

(i) We begin with the "only if" part. As we are interested in necessary conditions it su¢ ces to consider T 1=2 b i;T y i : We …rst need to establish the existence of a stochastic expansion of T 1=2 b i;T y i up to a term of order o T k 1 2

: Given A, and assuming that E( 4 t ) < 1; it follows that T 1=2 b i;T y i is asymptotically normal (see e.g. Bollerslev and Wooldridge 1992), thus

T 1=2 b T y = T 1=2 T = O P (1): Let L T ( ) = 1 T P T t=1 L t ( ) ;
then the Taylor expansion of r L T b T around y ; up to the k order writes as

0 = r L T y + r 2 L T y T + 1 2 0 B B @ P 3 j 1 =1 P 3 j 2 =1 r 1 j 1 j 2 L T y T;j 1 T;j 2 P 3 j 1 =1 P 3 j 2 =1 r 2 j 1 j 2 L T y T;j 1 T;j 2 P 3 j 1 =1 P 3 j 2 =1 r 3 j 1 j 2 L T y T;j 1 T;j 2 1 C C A +::: + 1 k! 0 B B @
P 3 j 1 =1 ::: P 3 j k =1 r 1 j 1 ::: j k L T y T;j 1 T;j 2 :::

T;j k P 3 j 1 =1 :::

P 3 j k =1 r 2 j 1 ::: j k L T y T;j 1 T;j 2
:::

T;j k P 3 j 1 =1 :::

P 3 j k =1 r 3 j 1 ::: j k L T y T;j 1 T;j 2 ::: T;j k 1 C C A +e k+1;T ;
where r 1 j 1 ::: j k L T y denote the k + 1 th derivative with respect to 1 ; j 1 ; :::; j k ; and T;j k the y < 1 is that E 2k t < 1; in order to have an Edgeworth expansion for the LHS of (11) ; up to the k th term, we need the existence of the (k+2) moment of the (k+2) derivative of the score and the outerproduct, which in turn requires that E 2(k+2)(k+4) t

< 1:

(ii) We now turn to the "if" part. Given Assumption A and recalling (17), we have that there exist a constant K; and > 0; such that for m 1;

( < 1 ensures that the k + 3 th moments of all the component of f t;k+3 is …nite. Now, the latter condition, plus (15) and (18) ensures that (see e.g. Götze and Hipp (1983, p.216-217) that lim

T !1 T k=2 sup z2R P G S T;k+3 ( y ) z 1 + k X i=1 T 1=2 i (@=@z) ! (z) = 0;
where S T;k+3 ( y ) = 1 T P T t=1 f t;k+3 ( y ): Finally, by Proposition 1 in [START_REF] Hall | Bootstrap Critical Values for Tests based on Generalized-Method-of-Moments Estimators[END_REF],

lim T !1 T k=2 sup z2R P G S T;k+3 ( y ) z P (t i z) > T k=2 = 0:
The statement then follows.

Hereafter, let P denoting the probability law governing the block bootstrap, E and V ar the mean and variance operator under P ; also Hereafter, as the arguments used in the proof are the same across all i; when there is no risk on confusion, we omit the subscript i: f t;k , where for j = 1; :::; T l ; T l = T l + 1; b j = (j; :::; j + l 1) j = 1; :::; b and b j is an iid discrete uniform random variable which is equal to b j with equal probability 1=T l : As shown in [START_REF] Andrews | Higher-Order Improvements of a Computationally Attractive k Step Bootstrap for Extremum Estimators[END_REF], the most di¢ cult case is that of m = 3:

Now,

T 1=2 E 3 =1 i;T; = T 1 b X j 1 =1 b X j 2 =1 b X j 3 =1 E Y j 1 Y j 2 Y j 3 = T 1 bE Y 3 1 = T 1 bT 1 l T l X j=1 e Y 3 j :
If for some C < 1; and for i; j = 1; 2; 3 E r d 1;k +d 2;k r i L T ( ) r i L T ( ) Also, by the proof of Lemma 14 in [START_REF] Andrews | Higher-Order Improvements of a Computationally Attractive k Step Bootstrap for Extremum Estimators[END_REF], it follows that for the case of m = 5; we need p > k=(3 2 4 ); requiring + 2 < 3=2; which is implied by + < 1=2 and E f 1;k 5p+4 < 1:

Given (10) and the last line of ( 27), it su¢ ce that the …rst mp+4 -th moments of all score derivatives up to order (d 1;k + d 2;k ) exist: In the proof of Lemma A1 we have shown that E r r ) : Thus, the left hand side of ( 27) is approaching zero, provided E 2(2k+6)(k+2)p t < 1:

As for B 2 ; along the lines of proof of Lemma 14 in [START_REF] Andrews | Higher-Order Improvements of a Computationally Attractive k Step Bootstrap for Extremum Estimators[END_REF], recalling that l = T ;

T 1 bE Y 3 1 T 1=2 E 3 =1 i;T; = 1 l E l X i=1 f i;k ! 3 1 T T X i 1 =1 T X i 2 =1 T X i 2 =1 f i 1 ;k f i 2 ;k f i 3 ;k = l 1 X i 1 = l+1 l 1 X i 2 = l+1 1 (i 1 + i 2 ) l E f 0;k f i 1 ;k f i 2 ;k T 1 X i 1 = T +1 T 1 X i 2 = T +1 1 (i 1 + i 2 ) T E f 0;k f i 1 ;k f i 2 ;k : Now, T l 1 X i 1 = l+1 l 1 X i 2 = l+1 E f 0;k f i 1 ;k f i 2 ;k T 1 X i 1 = T +1 T 1 X i 2 = T +1 E f 0;k f i 1 ;k f i 2 ;k ! 0

  analogs of b B T and b A T as b B T and b A T ; where b

!

  denote convergence in probability and in distribution under P ; conditionally on the sample. and let v i;k = lim T !1 v i;T;k ; and (23) e v i;T;k = E T (m) m =1 i;k;T;j where (m) = 1=2 if m is odd and 0 if m even, and 2 m k + 2: We need to show that

:

  in order to show(24), one has to show that B 1 and B 2 below approach zero, where By Markov inequality and a double application of Yokoyama-Doukan inequality(Doukhan 1995, p.25-30), there exists a constant C and > 0 such that,

Table 1 :

 1 Empirical coverage rates of nominal 95% symmetric percentile-t intervals for GARCH(1,1)

	Bboot

  where t is iid(0; 2 ) with positive and continuous density; is a measurable function : R Z ! R 3 2d 1;k : For y 2R Z and x 2 R de…ne (y;x) t as the sequence with coordinates,

							i
							1 1 ::: 2 t j l 1	+
	+:::+ 1 t 1 2	P t 2 k j t 1 =t 2	P j t 2 <j t 1 ::	P j 1 <j 2 (j t 1 (t 2))	j t 1 (t 1) 3	2 t j 1 1 :: 2 t j t 1 1 :
	Thus, we can write f t;d 1;k	y = f 1;t;d 1;k	y ; f 2;t;d 1;k	y ; f 3;t;d 1;k	y	as
	(17)				f t;d 1;k :=		t i : i 0 ; t 2 Z
							8 > > <
					i =	> > :

  Let f i;t;d 1;k y and f i;t;d 1;k b T be de…ned as (10) and (19), respectively. De…ne f t;k = f i;t;d 1;k

	also let Y j =	P t2b j f t;k ; e Y j =	P t2b j f t;k ; and Y j =	P t2b j	b T	E f i;t;d 1;k	b T ;

y E f i;t;d 1;k y and f t;k = f i;t;d 1;k

  by the same argument, and as a straightforward consequence of the chain rule E r r 1

					1 i L T	y r 2 <
	1 if and only if E Now, recalling that 2	2(r 1 +r 2 ) t j m	< 1; i L T k + 2; setting = 1=4; it follows that E f 1;k y L T y 0 r 2 < 1 if and only if E	2(r 1 +r 2 +1) t j mp+4 < 1 and < 1:
	E r
	d 2;k	k + 1; and p >	k=2 2(1=2

d 1;k +d 2;k r i l t ( ) r i l t ( ) 0 mp < 1 if E 2(d 1;k +d 2;k +2)(k+2)p t < 1; where d 1;k k + 3;

Note that if either h 0 or 0 are di¤erent from zero, then the expression in (3) holds up to a term converging to zero exponentially as t ! 1:

We are very grateful to one of the referees for pointing this out.

for very useful comments and suggestions. Both authors thank the …nancial support from an ESRC grant (Award number: T026 27 1238) and Intramural Research Grants Program (IRGP) at Michigan State University. Corradi also gratefully acknowledge …nancial support from ESRC grant code RES-062-23-0311.

A C C E P T E D M A N U S C R I P T

ACCEPTED MANUSCRIPT j k th component of T : Thus,

P 3 j 2 =1 r 1 j 1 j 2 L T y T;j 1 T;j 2 P 3 j 1 =1 P 3 j 2 =1 r 2 j 1 j 2 L T y T;j 1 T;j 2 P 3

P 3 j 1 =1 :::

:::

T;j k P 3 j 1 =1 :::

:::

T;j k P 3 j 1 =1 :::

+T 1=2 e k+1;T :

Hereafter, r k L T y is the matrix with generic element r j 1 ::: j k L T y ; j i = 1; 2; 3 for all i: The expression for r L t y ; r 2 L t y ; r 3 L t y ; etc. have been already provided in the proof of the Lemma.

Recalling that, E r 1 j 1 ::: j k L T

Proof of Theorem 2:

It is immediate to see that our set-up can be casted into [START_REF] Andrews | Higher-Order Improvements of a Computationally Attractive k Step Bootstrap for Extremum Estimators[END_REF] set-up, with L t ( ) playing the same role as (X t ; ) and = 1; as the score is a martingale di¤erence sequence. Now, given (18), Assumption 1 in [START_REF] Andrews | Higher-Order Improvements of a Computationally Attractive k Step Bootstrap for Extremum Estimators[END_REF] holds, and given Lemma A1 his Assumption 4 is satis…ed too. As all the derivatives of the gaussian likelihood are continuous, and recalling that, because of Lemma 1 in [START_REF] Lumsdaine | Asymptotic Properties of the Maximum Likelihood Estimator in GARCH(1,1) and IGARCH(1,1) Models[END_REF], E r j 1 :::

Assumptions 2, 3 and 5 in Andrews were satis…ed if we assumed that the innovation process has all moments …nite. On the other hand, we have only required that E 2(d 1;k +d 2;k +2)(k+2)p t

< 1; where

The moment conditions on the likelihood derivatives of the elements of f i;t;d 1;k y play a crucial role in the proof of Lemma 14 of Andrews. Here, we proceed by showing that the moments conditions imposed on the innovation term ensure that the statement in his Lemma 14 applies also in our context. The rest of the proof comes straightforwardly from [START_REF] Andrews | Higher-Order Improvements of a Computationally Attractive k Step Bootstrap for Extremum Estimators[END_REF].

De…ne the bootstrap analog of (10), that is:

where

; and

Given A, d 1 k + 2; > 0; there exists an in…nite di¤erentiable function G( ); with G(S i ) = 0 and =1 i;k;T;j by the Doukhan mixing inequality (1995, p.9), provided > : Also,

by mixing inequality, and

(i 1 + i 2 ) E f 0;k f i 1 ;k f i 2 ;k ! 0;

provided < : The statement in (24) follows by the same argument used to show that B 2 ! 0:

Finally, from ( 24) and ( 25) it follows that lim T !1 T k=2 P je v i;T;k v i;k j > T = 0:

The statement in Lemma 14 in Andrews then follows.


Amemiya A Ronald Takeshi

John F Gallant

Cheng Geweke

Peter Hsiao

Arnold Robinson

Zellner

Aït-Sahalia Yacine

H Badi

M W Baltagi

Marcus J Brandt

Songnian Chambers

Manfred Chen

Miguel A Deistler

Jean-Marie Delgado

Sylvia Dufour

Eric Fruhwirth- -Schnatter

John C Ghysels

Javier Ham

Han Hidalgo

Yongmiao Hong

Bo E Hong

Maxwell L Honoré

Yuichi King

G M Kitamura

Chung-Ming Koop

Naoto Kuan

Kajal Kunitomo

Q Lahiri

Tong Li

Oliver Li

James G Linton

Robert Mackinnon

Rosa L Mcculloch

Franz C Matzkin

Dale J Palm

Nicholas Poirier

B M Polson

Ingmar Pötscher

Peter C Prucha

Eric Reiss

Frank Renault

Robin Schorfheide

Fallaw Sickles

G J Sowell

Herman Van Den Berg

Dijk Van

H Quang

Edward Vuong

Tom Vytlacil

Andrew Wansbeek

Weiss

Professor Marcus J Chambers

    Corrigendum to "Testing for unit roots with flow data and varying sampling frequency" [Journal of Econometrics 119(1) (2004) 1-18]

Marcus J.

A c c e p t e d m

a n u s c r i p t [START_REF] Chambers | Testing for unit roots with flow data and varying sampling frequency[END_REF] considers the properties of tests for unit roots in flow data satisfying dy(t) = γy(t)dt + σdw(t), t > 0,

where w(t) denotes a standard Wiener process and y(0) is assumed to be fixed. Observations on y(t) are made at intervals of length h > 0 over the interval 0 < t ≤ N and take the form

y th = th th-h y(r)dr, t = 1, . . . , T, (2)
where the sample size T = N/h. Theorem 1 shows that, under the maintained/alternative hypothesis γ = 0, the observations satisfy y th = α h y th-h + u th , where α h = e γh , while under the null hypothesis γ = 0 they satisfy y th = y th-h + u th . It is claimed that these relations hold for t = 1, . . . , T , implying that the first observation satisfies y h = α h y(0) + u h under the maintained hypothesis and y h = y(0) + u h under the null. In fact, the stated solutions only hold for t = 2, . . . , T , because integrating the solution to (1), given by

y(t) = e γt y(0) + σ t 0 e γ(t-s) dw(s), t > 0,
over the interval (0, 1] shows that under the alternative hypothesis the first observation satisfies y h = γ -1 (e γh -1)y(0) + u h , while under the null y h = hy(0) + u h . In each case the coefficient on y(0) is O(h) rather than O(1), and here this correct treatment of the initial condition is incorporated to amend the relevant results presented in [START_REF] Chambers | Testing for unit roots with flow data and varying sampling frequency[END_REF]. Two least squares regressions are considered in [START_REF] Chambers | Testing for unit roots with flow data and varying sampling frequency[END_REF] based on data indexed by n of the form y nt = y thn . The first is given by y nt = α n y nt-1 + u nt while the second includes an intercept and is of the form y nt = μ + αn y nt-1 + ũnt . All results in [START_REF] Chambers | Testing for unit roots with flow data and varying sampling frequency[END_REF] concerning α n when y(0) = 0 are correct as are all the results for αn because the second regression eradicates the effects of y(0) regardless of its value. It is therefore only the results for the regression without an intercept when y(0) = 0 that need amending i.e. those for α n in Case (2).

Under the null hypothesis γ = 0 the data satisfy y nt = y nt-1 + u nt , t = 2, . . . , T n ; y n1 = h n y(0) + u n1 ,

(3) implying that y nt = t j=1 u nj + h n y(0); the original article used y nt = t j=1 u nj + y(0). It is

  The asymptotic analysis considers three scenarios in which the sampling frequency increases (h ↓ 0) while the sample size T ↑ ∞ in each case; these are further classified into two cases concerning y(0), namely: (1) y(0) = 0; and (2) y(0) = 0. It is also convenient to index each of h, N and T by a generic index n ↑ ∞, in which case T n = N n /h n . The three scenarios are: (a) N n = N (fixed) and h n ↓ 0; (b) N n ↑ ∞ and h n ↓ 0 such that h n N 1/2 n ↑ ∞; and (c) N n ↓ 0 and h n ↓ 0. The motivation for considering these is discussed in the original article.

Acknowledgement

I thank the Co-Editor, Peter Robinson, for his feedback which has helped to improve the presentation of the results.

The amended parts of Theorem 2, and Lemma A.1 upon which it is based, are:

Theorem 2. Let {{y nt } Tn t=1 } ∞ n=1 denote a triangular array of random variables generated by (3). Then, as n ↑ ∞ and h n ↓ 0, when y(0) = 0:

denote a triangular array of random variables generated by (3). Then, as n ↑ ∞ and h n ↓ 0, when y(0) = 0:

n ↑ ∞, the limits in Case (1) apply;

(c) If N n ↓ 0, 1

Proof of Lemma A.1. The objective is to express the sample moments in terms of the random function

where

The results follow straightforwardly from these expressions. 2

Under the alternative hypothesis γ = 0 the data satisfy

implying that y nt = t j=1 e (t-j)γhn u nj + γ -1 e tγhn (1 -e -γhn )y(0); the original article used

A c c e p t e d m a n u s c r i p t

y nt = t j=1 e (t-j)γhn u nj + e tγhn y(0). It is convenient to define the functionals Θ(N, y(0), J) = σ 2 N J(1) 2 -2 3 + 2σN 1/2 e γN y(0)J( 1)

where J(r) = r 0 e (r-s)γN dW (s). The amended parts of Theorem 3 and Lemma A.2 are:

Theorem 3. Let {{y nt } Tn t=1 } ∞ n=1 denote a triangular array of random variables generated by (6). Then, as n ↑ ∞ and h n ↓ 0, when y(0) = 0:

denote a triangular array of random variables generated by (6). Then, as n ↑ ∞ and h n ↓ 0, when y(0) = 0:

A c c e p t e d m a n u s c r i p t

Proof of Lemma A.2. Defining y n0 = γ -1 (e γhn -1)y(0), some algebra yields where

)e γNn y(0). The results then follow by examining each case in turn and also utilising Lemma A.3 (taking c = γN n ).

2

The results in Theorems 2 and 3 enable the consistency properties of tests for unit roots based on the normalised estimators to be established. The amended part of Theorem 4 is:

Theorem 4. Let {{y nt } Tn t=1 } ∞ n=1 denote a triangular array of random variables generated by ( 6), and consider testing the null hypothesis γ = 0 against a fixed alternative γ = 0. Then, as n ↑ ∞ and h n ↓ 0, when y(0) = 0:

From a qualitative point of view it is now clear that it is not possible to consistently test for a unit root in cases (a) and (c). As shown by [START_REF] Perron | Test consistency with varying sampling frequency[END_REF] for the case of a stock variable, an increasing span of data is also required with flow variables for consistent unit root testing based on the statistics considered here.