Pierre-Éric Treyens

Asymptotic refinements of bootstrap tests in a linear regression model; A CHM parametric bootstrap using the first four moments of the residuals. 1

Keywords: J.E.L classification : C10, C12, C13, C15 Non parametric bootstrap, Parametric Bootstrap, Cumulants, Skewness, Kurtosis

We consider linear regression models and we suppose that disturbances are either Gaussian or non Gaussian. Then, by using Edgeworth expansions, we compute the exact errors in the rejection probability (ERPs) for all one-restriction tests (asymptotic and bootstrap) which can occur in these linear models. More precisely, we show that the ERP is the same for the asymptotic test as for the classical parametric bootstrap test it is based on as soon as the third cumulant is nonnul. On the other side, the non parametric bootstrap performs almost always better than the parametric bootstrap. There are two exceptions. The first occurs when the third and fourth cumulants are null, in this case parametric and non parametric bootstrap provide exactly the same ERPs, the second occurs when we perform a t-test or its associated bootstrap (parametric or not) in the models y = µ + σu t and y = α 0 x t + σu t where the disturbances have nonnull kurtosis coefficient and a skewness coefficient equal to zero. In that case, the ERPs of any test (asymptotic or bootstrap) we perform are of the same order.

Finally, we provide a new parametric bootstrap using the first four moments of the distribution of the residuals which is as accurate as a non parametric bootstrap which uses these first four moments implicitly. We will introduce it as the parametric bootstrap considering higher moments (CHM), and thus, we will speak about the CHM parametric bootstrap.

0 Introduction [START_REF] Beran | Prepivoting test statistics: A bootstrap view of asymptotic refinements[END_REF] asserts that when a test statistic is an asymptotic pivot, bootstrapping this statistic leads to an asymptotic refinement. A test statistic is said to be asymptotically pivotal if its asymptotic distribution is the same for all data generating processes (DGPs) under the null, and by asymptotic refinement we mean that the error in the rejection probability (ERP) of the bootstrap test, or the size distortion of the bootstrap test, is a smaller power of the sample size than the ERP of the asymptotic test that we bootstrap. As [START_REF] Mackinnon | Bootstrap hypothesis testing[END_REF] notes, although there is a very large literature on bootstrapping, only a small proportion of it is devoted to bootstrap testing, which is the purpose of the present paper. Instead, the focus is usually on estimating bootstrap standard errors and constructing bootstrap confidence intervals.

Recall that a single bootstrap test may be based on a statistic τ in an asymptotic p-value form. Rejection by an asymptotic test at level α is then the event τ < α. Rejection by the bootstrap test is the event τ < Q(α, µ * ), where µ * is the bootstrap data-generating process and Q(α, µ * ) is the (random) α-quantile of the distribution of the statistic τ as generated by µ * . Now, let us consider a bootstrap test computed from a t-statistic which tests H 0 : µ = µ 0 against H 1 : µ < µ 0 in a linear regression model where disturbances may be non-Gaussian but i.i.d., which is the framework of this paper. Moreover, we suppose that all regressors are exogenous. Under these assumptions, a t-statistic is an asymptotic pivot. In order to decrease ERP, it is obvious that the bootstrap test must use extra information compared with the asymptotic test. So the question becomes: where does this additional information come from?

By definition, we can write a t-statistic T as T = √ n(μ -µ 0 )σ -1 μ where n is the sample size and µ a parameter connected with any regressor of the linear regression model. Therefore computation of a t-statistic requires only μ, the estimator of µ, and the estimator of its variance σμ . Moreover, as the limit in distribution of a t-statistic is N (0, 1), we can find an approximation of the CDF of T by using an Edgeworth expansion. In fact, the main part of the asymptotic theory of the bootstrap is based on Edgeworth expansions of statistics which follow asymptotically standard normal distributions; see [START_REF] Hall | The bootstrap and the Edgeworth expansion[END_REF]. Hence we can provide an approximation to the α-quantile of T . Next we can obtain the α-quantile of the bootstrap distribution by replacing the true values of the higher moments of the disturbances by their random estimates as generated by the bootstrap DGP. The extra information in the bootstrap test comes from these higher moments which the bootstrap DGP should estimate correctly.

A number of authors have stressed this point, including [START_REF] Parr | A Note on the Jackknife, the Bootstrap and the Delta Method Estimators of Bias and Variance[END_REF] who shows the effect of having a skewness coefficient equal to zero or not in the context of first order asymptotic theory for the jackknife and the bootstrap. See also [START_REF] Hall | The bootstrap and the Edgeworth expansion[END_REF] or [START_REF] Andrews | Higher-order Improvements of the Parametric Bootstrap for Markov Processes, Identification and Inference for Econometric Models: A Festschrift in Honor of Thomas[END_REF]. For instance, quoting Buhlmann (1998), "the key to the second order accuracy is the correct skewness of the bootstrap distribution". Intuitively, we can explain this remark in the framework of a t-test. If we first consider a parametric bootstrap test, the bootstrap DGP uses only the estimated variance of residuals σ, which is directly connected to the estimated variance of the parameter tested. Indeed, bootstrap error terms are generated following a Gaussian distribution. So it is clear that we use the same information as when computing the t-statistic. More precisely, the α-quantile of the parametric bootstrap distribution depends only on the higher moments of a centered Gaussian random variable, which is completely defined by its first two moments. So, the α-quantile of the parametric bootstrap distribution is not random anymore. If we now consider a non parametric bootstrap, we implicitly use extra information which comes from higher moments of the distribution of the residuals. Indeed, when we resample the estimated residuals we provide random consistent estimators of these moments and therefore the α-quantile of the parametric bootstrap distribution is random.

To clarify this example, consider a new parametric bootstrap using estimated higher moments; in this way we can provide a bootstrap framework which provides as much information as a non parametric bootstrap framework. We will refer to this new bootstrap as the CHM parametric bootstrap, from the acronym for "Considering Higher Moments". [START_REF] Hall | The bootstrap and the Edgeworth expansion[END_REF] provides the ERP of a non parametric bootstrap which tests the sample mean and only the magnitude of the ERP for the test of a slope parameter, but he does not derive ERPs for other tests in a linear regression model (asymptotic and parametric bootstrap tests, or tests for a slope parameter with or without an intercept in the model). In the present paper we go further and provide the exact ERPs in the following models; y = µ 0 + σu where we will test H 0 : µ = µ 0 against H 1 : µ < µ 0 , y = µ 0 + β 0 x + σu where we will test H 0 : β = β 0 against H 1 : β < β 0 , and y = β 0 x + σu where we will test β = β 0 against H 1 : β < β 0 . These three cases all include one-restriction tests which can occur in a linear regression model. For each of these three models, we will use a classical parametric bootstrap, a non parametric bootstrap (or residual bootstrap) and finally, a CHM parametric bootstrap which will use the estimators of the first four moments of the residuals. Our method is slightly different from Hall's, so we also use it in the case of a simple mean, although Hall did establish the exact ERP of the non parametric bootstrap test in the model y t = µ + σu t .

Moreover, we stress the role of the fourth cumulant for the bootstrap refinements in a linear regression model. When disturbances are Gaussian, we show that ERPs of the different tests are of lower order compared with non-Gaussian disturbances. In fact, disturbances do not need to be Gaussian, but rather to have third and fourth cumulants equal to zero, as is the case for Gaussian distributions. Finally, in the models y = µ 0 + σu and y = β 0 x + σu, we introduce a new family of standard distributions defined by its third and fourth cumulants that theoretically provides the same ERPs for the non parametric bootstrap and CHM parametric bootstrap as the Gaussian distribution; this family admits the Gaussian distribution as a special case.

These results are given in the second and third sections of this paper; the first section is devoted to preliminaries. In the fourth section, we proceed to simulations and explain how to apply the CHM parametric bootstrap. The fifth section concludes.

Preliminaries

In this paper, we show how higher moments of the disturbances in a linear regression model influence either asymptotic and bootstrap inferences. In this way, we have to consider non Gaussian distributions whose skewness and/or kurtosis coefficients are not zero. For any centered random variable X, if we define its characteristic function f c by f c (u) = E (e iux ) we can obtain, by using a MacLaurin expansion,

ln (f c (u)) = κ 1 (iu) + κ 2 (iu) 2 2! + κ 3 (iu) 3 3! + • • • + κ k (iu) k k! + . . . (1.1)
In this equation, the κ k are order k cumulants of the distribution of X. Moreover, for a centered standardised random variable the four first cumulants are

κ 1 = 0, κ 2 = 1, κ 3 = E(X 3 ), and κ 4 = E(X 4 ) -3 (1.2)
In particular, κ 3 and κ 4 are the skewness and kurtosis coefficients. One of the main problems when we deal with higher moments is how we can generate centered standardised random variable fitting these coefficients. Treyens (2006) provides two methods to generate random variables in this way and we use them because they are the existing fastest methods. Let us consider three independent random variables p, N 1 and N 2 where N 1 and N 2 are two Gaussian variables of expectations µ 1 and µ 2 and of standard error σ 1 and σ 2 and define X = pN 1 + (1 -p)N 2 . If p is a uniform distribution U (0, 1), the set of admissible couples (κ 3 , κ 4 ) this method can provide is Γ as showed on the figure 1.1 and it will be called the unimodal method. If p is a binary variable and if 1 2 is the probability that p = 1, the set of admissible couples is Γ 1 and this method will be called the bimodal method. On figure 1.1, the parabola and the straight line are just structural constraints which connect κ 4 to κ 3 . Now, if a centered standardised random variable X has κ 3 and κ 4 as skewness and kurtosis coefficients, we will write X → ∆(0, 1, κ 3 , κ 4 ). In this paper, all disturbances will be distributed as u t → ii∆(0, 1, κ 3 , κ 4 ).

In order to estimate the error in the rejection probability, we are going to use Edgeworth expansions. With this theory, we can express the error in the rejection probability as a quantity of the order of a negative power of n, where n is the size of the sample from which we compute the test statistic. Let t be a test statistic which asymptotically follows a standard normal distribution, and F (.) be the CDF of the test statistic. Almost with a classic Taylor expansion, we can develop the function F (.) as the CDF Φ(.) of the standard normal distribution plus an infinite sum of its successive derivatives that we always can write as a polynomial in t multiplied by the PDF φ(.) of the standard normal distribution. Precisely, we have

F (t) = Φ(t) -n -1/2 φ(t) ∞ i=1 λ i He i-1 (t) (1.3)
In this equation, He i (.) is the Hermite polynomial of degree i and the λ i are coefficients which are at most of the order of unity. Hermite polynomials are implicitly defined by the relation φ (i) (x) = (-1) i He i (x)φ(x), as a function of the derivatives of φ(.) i! E (He i (t)). Moreover, we will use in computations several random (or not) variables which are functions of the disturbances and of parameters of the models, all these variables (w, q, s, k, X, Q, m 1 , m 3 and m 4 ) are described in the Appendix.

Testing a simple mean

Let us consider the model y t = µ 0 + σu t with u t → ii∆(0, 1, κ 3 , κ 4 ). In order to test H 0 : µ = µ 0 against H 1 : µ < µ 0 , we use a Student statistic and we bootstrap it. The t-statistic is obviously

T = √ n μ-µ 0 σµ
, where μ is the estimator of the mean of the sample and σ is the unbiased estimator of the standard error of the OLS regression. So, we can give an approximate value at order o p (n -1 ) of the test statistic T :

T = w 1 -n -1/2 q 2 + n -1 w 2 2 - 1 2 + 3q 2 8 + o p n -1 (2.1)
In order to give the Edgeworth expansion F 1,T (.) of the CDF of this test statistic, we have to check two points. First, the asymptotic distribution of the test statistic must be a standard normal distribution. Secondly, all expectations of its successive power must exist. The first point is easy to check, indeed, by applying the central limit theorem on w, we see that the asymptotic distribution of w is a standard normal distribution. Moreover, the limit in probability of the right hand of the equation 2.1 divided by w is deterministic and equal to 1. In order to check the second, we will just compute successive expectations of powers of T . This will allow us to deduce easily expectations of Hermite polynomials of T and in that way we obtain an estimate of F 1,T (.) at order n -1 . Now, we can compute an approximation q α of the α-quantile Q (α, Θ) of the test statistic at order n -1 , where Θ is the DGP which generated the original data y. To find this approximation, we introduce q α = z α +n -1/2 q 1α +n -1 q 2α , the Cornish-Fisher expansion of Q (α, Θ), where z α is the α-quantile of a standard normal distribution, with actually,

Q (α, Θ) = z α + n -1/2 q 1α + n -1 q 2α + o (n -1
). If we now evaluate F 1,T (.) in q α , then we can find the expression of q 1α and q 2α .

q 1α = - κ 3 (1 + 2z 2 α ) 6
(2.2)

q 2α = - z 3 α (6κ 4 -20κ 2 3 -18) + z α (-18κ 4 + 5κ 2 3 -18) 72 (2.3)
And so, the ERP of the asymptotic test is obviously at order n -1/2 . Indeed, estimating F 1,T (.) in z α , we can provide the ERP of the asymptotic test

ERP 1 as = n -1/2 φ(z α ) κ 3 (1 + 2z 2 α ) 6 (2.4) +n -1 φ(z α ) κ 2 3 (3z α -2z 3 α -z 5 α ) 18 + κ 4 (z 3 α -3z α ) 12 - z α (1 + z 2 α ) 4
Now, to compute the ERP of a bootstrap test, we have first to find q * α the α-quantile of the bootstrap statistic's distribution. So, we replace κ 3 by κ * 3 which is its estimate as generated by the bootstrap DGP. For κ 4 , the bootstrap DGP just needs to provide any consistant estimator because it does not appear in q 1α but only in q 2α . The rejection condition of the bootstrap test is T < q * α at the order we consider and this condition is equivalent to T -q * α + q α < q α . Now, we just have to compute the Edgeworth expansion F * (.) of T -q * α + q α to provide an estimate of its CDF and to eveluate it at q α to find the ERP of the bootstrap test. If we consider a non parametric bootstrap or a CHM parametric bootstrap, we obtain exactly the same estimator κ3 of κ 3 . Indeed, the first uses the empirical distribution of the residuals and the second the estimate of κ 3 provided by these residuals. Both these methods lead us to κ * 3 = κ 3 + n -1/2 s -3w -3 2 κ 3 q + o p n -1/2 which is random because s, w and q are. Then, we compute the Edgeworth expansion as described earlier and we obtain an ERP equal to zero at order n -1/2 and nonnull at order n -1 . More precisely, we obtain

ERP 1 BTnonpar = n -1 z α φ(z α ) (1 + 2z 2 α )(3κ 2 3 -2κ 4 ) 12 (2.5)
Actually, [START_REF] Hall | The bootstrap and the Edgeworth expansion[END_REF] already obtained this result with a method quite different. Then, the parametric bootstrap DGP just uses a centered normal distribution to generate bootstrap samples. So, its third and fourth cumulants are zero and thus, they are not random. By using exactly the same method as previously, we obtain an ERP at order n -1/2 as for an asymptotic test. Actually, this ERP is

ERP 1 BTpar = -n -1/2 φ(z α ) κ 3 (1 + 2z 2 α ) 6 (2.6) +n -1 φ(z α ) κ 4 4 - 7κ 2 3 36 - κ 4 z 2 α 12 - κ 2 3 z 4 α 18
So, for any κ 3 and κ 4 , the ERP of the non parametric or of the CHM parametric bootstrap test is at order n -1 . If the disturbances are Gaussian, κ 3 = κ 4 = 0 then the ERP of the non parametric bootstrap test or, in an equivalent way, of the CHM parametric bootstrap is now at order n -3/2 . On the other hand, by considering 2.4 and 2.6, we see that if κ 3 = 0 then the dominant term of both ERP is the same and it is at order n -1/2 . Thus, if disturbances are asymmetrical, the parametric bootstrap fails to decrease ERP. However, if both κ 3 and κ 4 are null, i.e. if the first four moments of their distribution are the same as for a standard normal distribution, whichever bootstrap we use, then we obtain the same accuracy at order n -3/2 . Now if κ 3 = 0 and κ 4 = 0, the three tests have the same accuracy. This result is quite surprising, indeed it contradicts the Beran's assertion recalled in the introduction for asymptotic pivotal statistics. Actually, it just occurs because the ERP of the non parametric bootstrap test at order n -1 depends on the kurtosis coefficient κ 4 and not only on the skewness coefficient. Another special case appears in equation 2.5 , when we have 3κ 2 3 = 2κ 4 , the ERP of the non parametric bootstrap test is now at order n -3/2 . In the next part we find this condition again in the model y = β 0 x + σu. We will test this special case in the simulation part. In the next part, we will use a Student not on the intercept but on other variables. We will consider two cases, with or without an intercept in addition to this variable.

A linear model

With intercept

Now, we consider linear models y , t = µ , 0 + α 0 x , t + Z t γ + σu , t with u , t → ii∆(0, 1, κ 3 , κ 4 ) and where Z is a n × k matrix. By projecting both the left and the right hand side of the defining equation of the model on M ιZ 2 and by using Frisch-Waugh-Lovell Theorem (FWL Theorem), we obtain the model M ιZ y , t = αM ιZ x , t + residuals with n t=1 M ιZ y t = n t=1 M ιZ x t = 0. Obviously, in this last model, if we want to test the null H 0 : α = α 0 the test statistic is a Student with k + 2 degrees of freedom. In this part, we just consider the model y t = µ 0 + αx t + σu t with u t → ii∆(0, 1, κ 3 , κ 4 ). Or in an equivalent way, the model y t = αx t + σ u t -n -1/2 w with n t=1 y t = n t=1 x t = 0 and two degrees of liberty. Moreover, we suppose that V ar(x) = 1 without loss of generality. We obtain the asymptotic test

T = X 1 -n -1/2 q 2 + n -1 X 2 2 + w 2 2 -1 + 3q 2 8 (3.1)
The limit in probability of T is a standard normal distribution and we use Edgeworth expansions to provide an approximation of the CDF F (.) of T at order n -1 . Following the same framework as in the previous part, we compute an approximation q α = z α + n -1/2 q α1 + n -1 q α2 of the α-quantile of T .

q α1 = κ 3 m 3 (z 2 α -1) 6 (3.2) q α2 = z 3 α (3κ 4 + 9)m 4 -4κ 2 3 m 2 3 -9κ 4 + 18 72 (3.3) +z α (-9κ 4 -27)m 4 + 10κ 2 3 m 2 3 + 27κ 4 + 18 72
And so, we obtain the ERP of the asymptotic test

ERP 2 as = n -1/2 κ 3 m 3 φ(z α ) 1 + z 2 α 6 + o n -1 2 (3.4)
We recall that the rejection condition at order n -1 of the bootstrap test is T < q * α where q * α is the approximation of the α-quantile of the bootstrap distribution and now, we can write this condition as T < q * α -q α + q α . In order to obtain q * α , we just replace κ 3 by its estimate as generated by the bootstrap DGP in q α1 and q α2 . If we deal with the non parametric or CHM parametric bootstrap, we obtain the same estimator as in the last part. We recall that κ *

3 = κ 3 + n -1/2 s -3w -3 2 κ 3 q + o p n -1/2
. Now, we just use exactly the same framework as in the previous part and in this case, the CDF F * (.) of T -q * α + q α is the same as F (.) the CDF of T at order n -1 . So when we evaluate F * (.) in q α we obviously find an ERP at order n -3/2 .

Intuitively, we thought we would find an ERP of the bootstrap test at order n -1 as in the previous part. But according to [START_REF] Davidson | Improving the reliability of bootstrap tests with the fast double bootstrap[END_REF], independence between the test statistic and the bootstrap DGP improves bootstrap inferences by an n -1/2 factor, this is the reason why we have F (.) = F * (.) up to order n -1 and why we find an ERP of the bootstrap test at order n -3/2 . Actually, we do not have independence but a weaker condition. Let B be the bootstrap DGP, it comes from the random part of κ * 3 , and so the random part of B is the same as the random part of κ * 3 . Here, we just have E T k B = o n -1/2 for all k ∈ ℵ. But this is enough for F * (.) to be equal to F (.) at order n -1 . In fact, we obtain this result just because we have m 1 = 0 by introducing the intercept in linear model. Then, for the parametric bootstrap the estimator of κ 3 is still zero. We proceed in the same way as for the non parametric bootstrap or CHM parametric bootstrap and we obtain an ERP at order n -1/2 which is exactly the same than ERP 2 as as defined in equation 3.4. This result is natural, at least when we consider the order of the ERP, indeed we use as much information to perform asymptotic and parametric bootstrap tests, the estimators of the parameter α and of the variance; and so, no more information, no more accuracy. Now, let us consider κ 3 = 0 and κ 4 = 0, i.e. symmetrical distributions for the disturbances. The ERP of both the non parametric bootstrap and CHM parametric bootstrap are still at order n -3/2 . This is logical, indeed whether κ 3 = 0 or not, the κ * 3 we use in the bootstrap DGP is a random variable with the true κ 3 as expectation. So, we do not use more information coming from the true DGP which generates the original data. However, the ERP of asymptotic and parametric bootstrap tests are now at order n -1 but they are no longer equal. Indeed, when κ 3 = 0 the ERP of the parametric bootstrap test is

ERP 2 BTpar = n -1 φ (z α ) κ 4 (z 3 α -z α ) (3 -m 4 ) 24 + o n -1 (3.5)
So the distribution of these two statistics are not the same, which we could have thought by only considering the case κ 3 = 0 and so, the parametric bootstrap still fails to improve the accuracy of inferences. The last case we have to consider is κ 3 = κ 4 = 0. Here, the parametric bootstrap estimates κ 3 and κ 4 perfectly because for a centered Gaussian distribution they are both equal to zero and it decreases its ERP at order n -3/2 exactly as the non parametric bootstrap and CHM parametric bootstrap. Such a result just occurs because we use extra information by chance, using a bootstrap DGP very close to the original DGP. So, the parametric bootstrap test is better than the asymptotic test only when disturbances have skewness and kurtosis coefficients equal to zero whereas the non parametric bootstrap and CHM parametric bootstrap always improve the quality of the asymptotic t test. In particular, when disturbances are Gaussian, the parametric bootstrap has the same accuracy as the non parametric bootstrap and the CHM parametric bootstrap.

Without intercept

Let us consider linear models

y , t = λ 0 x , t + Z t β + σ , v t with v t → ii∆(0, 1, κ , 3 , κ , 4
) and Z a matrix n × k. In order to test H 0 : λ = λ 0 , we can use the FWL theorem to test this hypothesis in the model M Z y t = λM Z x t + residuals in an equivalent way. So, any Student test can be seen as a particular case of the Student test connected to α 0 in the model y t = α 0 x t + σu t with u t → ii∆(0, 1, κ 3 , κ 4 ) and with k + 1 degrees of freedom rather than only one. Now, we just consider this last model with only one degree of freedomand where we impose n -1 n t=1 x 2 t = 1 without loss of generality. The t statistic we compute is given by

T = X 1 -n -1/2 q 2 + n -1 X 2 2 - 1 2 + 3q 2 8 (3.6)
As the limit in probability of T is still a standard normal distribution, we can follow exactly the same procedure as in the previous part in order to obtain the approximation of the CDF F (.) of T at order n -1 by using Edgeworth expansions and then an approximation q α = z α + n -1/2 q α1 + n -1 q α2 at order n -1 of the α-quantile of T. Computations provide

q α1 = (κ 3 m 3 -3κ 3 m 1 ) z 2 α -κ 3 m 3 6 (3.7) q α2 = z 3 α (3κ 4 + 9) m 4 -4κ 2 3 m 2 3 + 6κ 2 3 m 1 m 3 + 18κ 2 3 m 2 1 -9κ 4 + 18 72 (3.8) +z α (-9κ 4 -27) m 4 + 10κ 2 3 m 2 3 -6κ 2 3 m 1 m 3 -9κ 2 3 m 2 1 + 27κ 4 -18 72
As previously, the ERP of the asymptotic test T is at order n -1/2 because q α1 = 0. Moreover, estimating F (.) in z α and if φ(.) is the PDF of a standard normal distribution, then we find that

ERP 3 as = n -1/2 κ 3 φ(z α ) m 3 + z 2 α (m 3 -3m 1 ) 6 + o n -1 2 (3.9)
Considering this last equation, we see that if m 1 = 0, we have ERP 3 as = ERP 2 as Now, whatever the bootstrap we consider, we have to center the residuals to provide a valid bootstrap DGP because the intercept does not belong to the model. We recall again that the rejection condition at order n -1 of the bootstrap test is T < q * α where q * α is the approximation of the α-quantile of the bootstrap distribution and now, we can write this condition as T < q * α -q α + q α . In order to obtain q * α , we just replace κ 3 by its estimate as generated by the bootstrap DGP in q α1 and q α2 .

If we deal with the non parametric or CHM parametric bootstrap, we obtain the same estimator as in the last part. We recall that κ * 3 = κ 3 + n -1/2 s -3w -3 2 κ 3 q + o p n -1/2 . Now,we just use exactly the same framework as in the previous part. In this part, we do not have anymore m 1 = 0, so we do not obtain E T k B = o n -1/2 for all k ∈ ℵ and in this case, the CDF F * (.) of T -q * + q is not equal to F (.) the CDF of T . Now, by estimating F * (.) in q α , we find the ERP of both non parametric and CHM parametric bootstrap

ERP 3 BTnonpar = n -1 φ (z α ) m 1 z α (2κ 4 -3κ 2 3 ) (m 3 (z 2 α -1) -3m 1 z 2 α ) 12 + o n -1 (3.10)
Considering last result, we see this ERP is at order n -3/2 if κ 3 = κ 4 = 0. However, there is an other way to obtain this order for the ERP, this is the special case we already obtained by testing a simple mean in the previous chapter, i.e. when 2κ 4 -3κ 2 3 = 0. We will consider this case in the simulation part in order to know if we can find the same accuracy as for Gaussian distributions or if it is just a theoritical result. Now, let us consider a parametric bootstrap test, the estimator of κ 3 used by bootstrap DGP is still zero, we proceed exactly in the same way as previously and we obtain an ERP which is the same than ERP 3 as as defined in equation 3.9 but when, distribution of the disturbances is symmetrical, i.e. when κ 3 = 0, we now have

ERP 3 BTpar = n -1 κ 4 z α (m 4 -3) (z 2 α -3) 24 + o n -1 (3.11)
And now, explanations are the same as at the end of part 3.1

Simulation evidence

In the different figures provided in appendix, we seek to estimate the power of these four tests when the level of significance is α = 0.05. For the asymptotic test, there are 100000 repetitions and for the bootstrap tests we limit the number of repetitions to 20000 and bootstrap repetitions to 999. We want to examine the convergence rate Then, in order to estimate the level of CHM parametric bootstrap tests, a new problem arises in generating bootstrap samples. Indeed, even if we generate disturbances following a standard distribution belonging to the set Γ 1 , then the estimated standardised residuals do not always provide an estimate (κ 4 , κ3 ) which belongs to Γ 1 . So, we cannot directly use the bimodal method to generate bootstrap samples. This problem happens because estimates of higher moments are not very reliable for small sample size. We correct it by multiplying (κ 4 , κ3 ) by a constant k ∈ [0, 1]. In our algorithm, we choose k = 10-i 10 with i the first integer in [0, 10] which satisfies (kκ 4 , kκ 3 ) ∈ Γ 1 \ F r(Γ 1 ). Actually, this homothetic transformation respects the signs of both estimated cumulants κ3 and κ4 and never provides a couple on the frontier of Γ 1 . Indeed, on this frontier, the distributions connected with the couple (κ 4 , κ 3 ) which defines it are not continuous. We provide an example in figure 4.1 with (κ 4 , κ3 ) = (4, 2), here we have k = 5 and we obtain the couple (2, 1). Actually, we prefer this method rather than a method projecting directly on to a subset very close to the frontier of Γ 1 , as described in figure 4.1, because it projects in the direction of the cumulants of a standard normal distribution, i.e. (κ 4 , κ 3 ) = (0, 0).

A last problem can occur when κ 3 is very close to 0. It is not a theoretical problem because solutions always exist in the set Γ 1 ; it is just a computational problem. So, if κ3 < 2.10 -2 , then we fit κ 3 to zero, in order to suppress all the algorithmic problems which can occur.

y

t = µ 0 + u t
Here, we suppose that µ 0 = 0 and u t ∼ iid(0, 1). Then, we test H 0 : µ = 0 against H 1 : µ < 0 because we deal with unilateral tests. We consider the following couples κ 3 and κ 4 . 3 ;κ 4 ) (0,8;0) (0,4;0) (0;4;8;0) Couples (κ 3 ;κ 4 ) (0,8;1) (0,4;1) (0;4;8;1) By considering figures 7.1 to 7.4, we check that parametric bootstrap tests and asymptotic ones provide the same rejection probabilities, in agreement with the theoretical results. In fact, even the signs of the ERP are the ones predicted by our computations. Then, as soon as κ 3 = 0, we check that asymptotic and parametric bootstrap tests have the same accuracy. Thus, we check that the parametric bootstrap test does not use more information than the asymptotic test when κ 3 = 0. Now, if we consider the next four figures, we first observe that the nonparametric bootstrap and CHM parametric bootstrap have the same convergence rates and they are better than parametric bootstrap or asymptotic tests. Thus, at the order we consider, we do not use more information than what is contained in the first four moments. Moreover, we can observe under-rejection and over-rejection phenomena; these are in agreement with theory. Actually, when κ 3 > 0, the tails of distributions are thicker on the left, so we have more chance to find a realization in the rejection area and to obtain over-rejection. Then, when κ 3 < 0, it is exactly the reverse.

Couples (κ

y

t = µ 0 + α 0 x t + u t
Here, we suppose that µ 0 = 2 and α 0 = 0 with u t ∼ iid(0, 1) and V ar(x) = 1. Then, we still have an unilateral test and we test H 0 : α = 0 against H 1 : α < 0. We consider the following couples κ 3 and κ 4 .

Couples (κ 3 ; κ 4 ) (0, 8; 0) (0, 8; 1) (0; 0) (0; 1)

In this subsection, we observe quite the same results. When κ 3 = 0, convergence rates are less fast for both asymptotic and parametric bootstrap tests and they are the same in the both cases. On the other hand, non parametric bootstrap and CHM parametric bootstrap provide exactly the same results and these two methods provide better convergence rates especially when κ 3 is very different from zero.

y

t = α 0 x t + u t
Here, we suppose that α 0 = 0 with u t ∼ iid(0, 1) and V ar(x) = 1. Then, we still have an unilateral test and we test H 0 : α = 0 against H 1 : α < 0. We consider the following couples κ 3 and κ 4 . Couples (κ 3 ; κ 4 ) (0, 8; 0) (0, 8; 1) (0; 0) (0; 1) Finally, in this last subsection, we still obtain the same results with convergence rates faster when κ 3 = 0 for asymptotic and parametric bootstrap tests than when κ 3 = 1. Moreover, convergence rates are the same for both methods. Then, for non parametric bootstrap and CHM parametric bootstrap, convergence rates are the same and we still observe over-rejection when κ 3 > 0.

4.4 The case 2κ 4 -3κ 2 3 = 0

In the parts 2 and 3.2, we show we theoretically can obtain an extra refinement for the non parametric bootstrap and the CHM parametric bootstrap if the third and fourth cumulants of the disturbances check 2κ 4 -3κ 2 3 = 0. On the figure 4.2, we see that the distributions of this family have the same quantiles 0.275 and 0.825. In order to check if these special cases provide a refinment, we just simulate ERPs for the nonparametric bootstrap framework and for the model y t = µ + u t . In this model, we use four different disturbances; the gaussian distribution as a special case satisfying the equality 2κ 4 -3κ 2 3 = 0, an other distribution which satisfies it, (κ 3 ; κ 4 ) = ( 2/3; 1) and two other distributions defined by (κ 3 ; κ 4 ) = ( 2/3; 0) and (κ 3 ; κ 4 ) = ( 2/3; 2). Then, we observe on the figure 4.3 that we obtain a very slight refinment compared to the case where κ 4 = 2. When κ 4 = 0, the ERP is almost the same and it is difficult to conclude against an improvement other than theoritical.

Conclusion

In this paper, we provide the ERPs for all one-restriction tests which can occur in a linear regression model for asymptotic tests and different bootstrap tests by using Edgeworth expansions. These results clarify how the bootstrap DGP needs to estimate correctly not only the third cumulant but also the fourth, at least at order of the unity, to provide first and second order refinements. So, we introduce a new parametric bootstrap method which uses the four first moments of the estimated residuals. Asymptotically, this method has the same convergence rates as the non parametric bootstrap and they are better than asymptotic and parametric bootstrap when κ 3 = 0. Actually, the accuracy of a test is directly linked to the information it uses. As both asymptotic and parametric tests use the same information coming from the first two moments (except when κ 3 = κ 4 = 0), they provide the same convergence rates. On the other hand, non parametric bootstrap and CHM parametric bootstrap use extra information from third and fourth moments and they provide better convergence rates. We resume the different results of this paper in the tables A and B.

Test

κ 3 = 0 = κ 4 κ 3 = 0 = κ 4 κ 3 = κ 4 = 0 Asymptotic O n -1 2 O (n -1 ) O (n -1 ) Parametric bootstrap O n -1 2 O (n -1 ) O n -3 2 Non parametric and CHM bootstrap O (n -1 ) O (n -1 ) O n -3 2 Table A. "Models y t = µ 0 + σ 0 u t and y t = α 0 x t + σ 0 u t "
Actually, even if we did not do the simulations, it seems logical to think that the results would be the same for any test in a linear regression model with exogenous explicatives and disturbances which are iid. Obviously, it could be very different for other models. Now, let us imagine another model with rejection probabilities such as those in the figure 5.1.

Test

κ 3 = 0 = κ 4 κ 3 = 0 = κ 4 κ 3 = κ 4 = 0 Asymptotic O n -1 2 O (n -1 ) O (n -1 ) Parametric bootstrap O n -1 2 O (n -1 ) O n -3 2 Non parametric and CHM bootstrap O n -3 2 O n -3 2 O n -3 2 Table B. "Model y t = µ 0 + α 0 x t + σ 0 u t "
Figure 5.1: Hypothetical rejection probabilities

In this example, it would be obvious that other cumulants than the first four ones appear in the dominant term of the rejection probability. Actually, if we could develop other methods to control more than the first four cumulants of a distribution, it would be possible to know the information a bootstrap test uses because non parametric bootstrap always uses all the estimated moments of the residuals. Now, the obvious question is : "Does a non parametric bootstrap test always use the information contained in the first four moments?". CHM parametric bootstrap could help to answer this question. Then, our simulations show that if disturbances are normal, parametric bootstrap can provide better results than non parametric bootstrap or CHM parametric bootstrap, however, it is quite impossible in small samples to know if the distribution is normal or not. Moreover, even if CHM parametric bootstrap and non parametric bootstrap have almost the same rejection probabilities, the first can reject H 0 when the second does not and conversely. We show with the help of the figure 5.2 why the CHM parametric bootstrap can be more accurate than the non parametric bootstrap. In this figure, there are 5000 points which are estimated couples (κ 3 , κ4 ) from a distribution for which (κ 3 , κ 4 ) = (1, 1.5). By considering this figure, we immediately see that a lot of couples (κ 3 , κ4 ) are outside the set Γ 1 as defined in the figure 4.1. In these cases, CHM parametric bootstrap, by projecting towards normality uses a DGP closer to the true distribution than the non parametric bootstrap. And so, CHM parametric bootstrap will provide better inferences than the non parametric bootstrap.

So, we think that we must use a principle of precaution using bootstrap and compute the three bootstrap tests. Actually, this procedure (using the three tests) could be seen as a very restricted maximized Monte-Carlo test (MMC). Instead of using a grid of values for the couple (κ 3 ; κ 4 ) which are the nuisance parameters of the model, we only use these three bootstrap tests and so three couples (κ 3 , κ 4 ). If one of the three tests does not reject the null, we do not reject it. Thus, this method would decrease the computing times compared with a MMC test, a current research seek to know if the rejection probabilities are the same in both cases.

Actually, we try to go further and to connect more closely the MMC and the bootstrap by using the framework of the CHM parametric bootstrap. Let us suppose we can provide a 1 -α 1 confidence region for the couple (κ 3 ; κ 4 ) where α 1 must be selected correctly. The main difference with the classical MMC is that the confidence region depends on the estimates of κ 3 and κ 4 as obtained by estimating the initial model and which belong to the confidence region. Then, we build a grid on this confidence region and for each point of this grid we use the framework introduced for the CHM parametric bootstrap and we compute a p-value. Finally, we maximize the p-value on this grid and we do not reject the null if the result of this maximization 7 Appendix 7.1 y t = µ 0 + u t

Different variables.

We give all variables we use to compute the ERPs of this paper. Here, µ i denotes the uncentered moment of the disturbances distribution at order i. (u t x t ) with p lim n→∞ X = N (0, m 2 ) (7.9)

Q ≡ n -1/2 n t=1
(u 2 t -1)x t with p lim n→∞ Q = N (0, (2 + κ 4 )m 2 ) (7.10)

Figure 1

 1 Figure 1.1: Sets of admissible couples Γ and Γ 1

Figure 4

 4 Figure 4.1: Methods of projection inside of Γ 1

Figure 4

 4 Figure 4.2: Examples of distributions such as 2κ 4 -3κ 2 3 = 0

Figure 5

 5 Figure 5.2: Distribution of couples (κ 3 ; κ 4 ) estimated.

Figure 7

 7 Figure 7.1: RP of asymptotic tests with κ 4 = 0 and κ 3 varying.

Figure 7

 7 Figure 7.2: RP of asymptotic tests with κ 4 = 1 and κ 3 varying.

Figure 7

 7 Figure 7.3: RP of parametric bootstrap tests with κ 4 = 0 and κ 3 varying.

Figure 7

 7 Figure 7.4: RP of parametric bootstrap tests with κ 4 = 1 and κ 3 varying.

Figure 7

 7 Figure 7.5: RP of non-parametric bootstrap tests with κ 4 = 0 and κ 3 varying.

Figure 7

 7 Figure 7.6: RP of non-parametric bootstrap tests with κ 4 = 1 and κ 3 varying.

Figure 7

 7 Figure 7.7: RP of CHM parametric bootstrap tests with κ 4 = 0 and κ 3 varying.

Figure 7

 7 Figure 7.8: RP of CHM parametric bootstrap tests with κ 4 = 1 and κ 3 varying.

  Figure 7.9: RP of the asymptotic tests.

Figure 7 .

 7 Figure 7.10: RP of the parametric bootstrap tests.

Figure 7 .

 7 Figure 7.11: RP of the non-parametric bootstrap tests.

Figure 7 .

 7 Figure 7.12: RP of the CHM parametric bootstrap tests.

  Figure 7.13: RP of the asymptotic tests.

Figure 7 .

 7 Figure 7.14: RP of the parametric bootstrap tests.

Figure 7 .

 7 Figure 7.15: RP of the non-parametric bootstrap tests.

Figure 7 .

 7 Figure 7.16: RP of the CHM parametric bootstrap tests.

I thank the GREQAM and the Université de la Méditerranée for its financial and research support. Helpful comments, suggestions and corrections from Russell Davidson, John Galbraith and Jean-François Beaulnes.

M ιZ is a projection matrix defined byM ιZ = I -[ιZ] [ιZ] ⊥ [ιZ] -1 [ιZ] ⊥ with ι a n × 1 vectorof 1, Z a matrix n × k of explicatives and I the matrix identity (n + 1) × (n + 1).


Kathia Bahloul Zekkari
email: kathia.zekkari@univ-amu.fr.

Thomas Seegmuller
email: thomas.seegmuller@univ-amu.fr.

Asset bubble and endogenous labor supply: a clarification

Keywords: JEL classification: E22, E44, J22 Asset bubbles, crowding-in effect, endogenous labor, overlapping generations

This paper analyzes the link between asset bubbles, endogenous labor and capital. The question is whether endogenous labor, per se, can explain a crowding-in effect of the bubble, i.e. higher levels of capital and labor. With respect to the existing literature, our contribution is twofold. First, we explicitly and theoretically derive the conditions to have a crowding-in effect of the bubble. Second, the utility function we consider allows us to show that this result does not require an arbitrarily high elasticity of intertemporal substitution in consumption. Our result still holds for a unit value of this elascticity (Cobb-Douglas utility).

Introduction

Some recent episodes of asset bubbles, measured by the difference between the market price and the market fundamental, are characterized by the sharp increase of asset prices followed by a contraction, causing a great recession of economic activity and waves of job destructions. Many countries have witnessed this phenomenon, for instance, the United States during 2000-2007 and Japan during [1980][1981][1982][1983][1984][1985][1986][1987][1988][1989][1990]. Asset bubble periods are associated to high levels of employment and capital stock, and boom in economic activity [START_REF] Martin | Economic growth with bubbles[END_REF], [START_REF] Miao | Stock market bubbles and unemployment[END_REF], [START_REF] Hashimoto | Asset bubbles, labour market frictions and R& D-based growth[END_REF]). After the collapse of housing and asset prices in the U.S in 2007, the average annual labor hours recorded a decline of 1.83% in 2009 (OECD database). Despite these observations, the relationship between rational bubbles and employment has not been extensively investigated in macroeconomics. The purpose of this paper is to address this relationship.

In their seminal studies, [START_REF] Tirole | Asset bubbles and overlapping generations[END_REF] and [START_REF] Weil | Confidence and the real value of money in an overlapping generations economy[END_REF] show that deterministic and stochastic bubbles have a recessionary effect on GDP, by absorbing a share of over-saving which leads to lower capital and increase the interest rate. This is the so-called crowding-out effect of bubble, but is hard to reconcile with the observations. Thus, recent works exhibit different mechanisms explaining that bubbly episodes are characterized by the boom of productive capital and its bursting causes depression, meaning that there is a crowding-in effect of the bubble. For instance, [START_REF] Martin | Economic growth with bubbles[END_REF] or [START_REF] Hirano | Asset bubbles, endogenous growth, and financial frictions[END_REF] rely on the existence of heterogeneous investment projects, Fahri and Tirole (2012) focus on the liquidity role of bubbles, [START_REF] Kocherlakota | Bursting bubbles: consequences and cures[END_REF] assumes that the bubble plays the role of collateral in the credit constraint, while Raurich and Seegmuller (2019) make a distinction between liquid bubbles and illiquid capital. However, these studies are not interested in the link between asset bubbles and employment.

Our paper contributes to the few literature that investigates the relationship between bubbles and employment provided some mechanism explaining that asset bubbles raise employment. Among the others, [START_REF] Miao | Stock market bubbles and unemployment[END_REF] incorporate endogenous credit constraints. Under optimistic beliefs, the firms can borrow more because their value used as a collateral is higher. In that case, they are hiring more. [START_REF] Kunieda | Asset bubbles, unemployment, and a financial crisis[END_REF] rely on the existence of heterogeneous investment projects. The mechanism is close to the one introduced in Martin and Ventura (2012), but is extended to a model with unemployment. Investors having the projects with the higher return sell the bubble to those which have projects with lower returns to increase their investment, capital and employment. Finally, [START_REF] Kocherlakota | Bubbles and unemployment[END_REF] combines overlapping generations to a matching model on the labor market. Following a crash of the bubble, the increase of unemployment is due to the zero lower bound of the interest rate imposed by the monetary authority, which restricts the liquidity in periods of recession. Hence, in all these papers, the positive link between the bubble and employment is driven by some form of credit or liquidity constraint.

Shi and Suen (2014) adopt another strategy, which does not rely on the exis-tence of a credit constraint. They extend the [START_REF] Tirole | Asset bubbles and overlapping generations[END_REF] model to endogenous labor and highlight that asset bubbles can promote capital and labor. A higher interest rate at the bubbly steady state may be in accordance with higher capital when labor is higher too. The main mechanism goes through the labor supply. It should strongly increase with respect to the interest rate. Considering a separable utility function over consumptions, Shi and Suen (2014) highlight, using a calibration, that it requires a high elasticity of intertemporal substitution in consumption. However, the empirical estimations show that realistic values of this elasticity are between 1 and 2 (Gruber (2013), [START_REF] Mulligan | Capital interest and aggregate intertemporal substitution[END_REF], Vissing-Jorgensen and Attanasio (2003)), while Shi and Suen (2014) calibration is 6.66. In addition, they don't provide an explicit and theoretical proof of their result. 1This paper fills these gaps. We consider an overlapping generations model with a non-separable utility function over consumptions, that allows to disentangle the elasticity of intertemporal substitution in consumption and the degree of homogeneity over consumptions when young and old, and endogenous labor. Of course, our utility function admits as a particular case the Shi and Suen (2014) specification. At the first period of life, each agent works, consumes and saves, and at the second period, she only consumes. There is a portfolio choice between the investment in productive capital and in a purely speculative asset.

Comparing the bubbleless and bubbly steady states, we theoretically show that the bubble may have a crowding-in effect on capital and labor, i.e. the levels of capital and labor are higher when there is a positive bubble. The main idea is the following. When there is a bubble, the interest rate is higher because the bubble compensates the shortage of asset and reduces over-savings. In that case, capital can increase only if labor increases too. This happens when the labor supply is positively correlated with the interest rate. Under a separable utility function, [START_REF] Shi | Asset bubbles in an overlapping generations model with endogenous labor supply[END_REF] emphasizes that this requires a high elasticity of intertemporal substitution in consumption. We theoretically show that this is no more the case when one considers a more general family of non-separable utility functions. A bubble has a crowding-in effect on labor and capital for a realistic interval of elasticities of intertemporal substitution in consumption, which encompasses the Cobb-Douglas case. Therefore, we show that under realistic parameterizations, endogenous labor per se is a channel which may explain the crowding-in effect of bubbles.

Our paper is organized as follows. Next section presents the model and defines an intertemporal equilibrium. Sections 3 describes the steady state equilibrium with and without bubble and determines the condition for the existence of asset bubble. Section 4 figures out the condition under which the asset bubble has a crowding-in effect, i.e. raises employment and capital. The last section concludes.

Model

We consider an economy with two types of agents, households and firms. Time is discrete, t = 0, 1, 2... + ∞. We describe the behavior of this two types of agents and, finally, define an intertemporal equilibrium.

Households

We consider an overlapping generations (OLG) model in which consumers live two periods: young and old ages. In each period of time t ≥ 0, a new generation of identical consumers is born with a population size N t = (1 + n) t growing at the rate n > -1. Each young households supplies elastically labor (l t ). At the first period of life, each household works and earns a wage income w t , consumes c 1t and saves through two types of assets. Each saver makes a portfolio choice between investing in productive capital, s t , and holding m t units of an intrinsically useless paper asset, for instance "money", which has a positive value P t 0. In the second period, the household consumes c 2t+1 , that is, the returns on productive investment (R t+1 s t ) and useless asset (P t+1 m t ), where R t+1 and P t+1 denote the interest factor on capital and the price of the intrinsically useless asset at time t + 1, respectively. 2The utility over the life-cycle is given by:

U (c 1t , c 2t+1 , l t ) = 1 π c θ-1 θ 1t + γc θ-1 θ 2t+1 πθ θ-1 -V l 1+µ t 1 + µ (1)
where π ∈ (0, 1] denotes the degree of homogeneity of the utility for consumption, V > 0 a constant parameter of disutility of work, µ ≥ 0 the inverse of the Frisch elasticity of labor supply, γ ∈ (0, 1) the subjective discount factor and θ 1 the intertemporal elasticity of substitution in consumption. Note that we restrict our attention to configurations where θ is not lower than 1 to focus on saving functions which do not decrease with the interest factor R t+1 . This utility function allows us to consider several well-known utilities as particular cases. Especially, when π = (θ -1)/θ, the utility is separable between consumption in both periods, with an elasticity of intertemporal substitution of consumption equal to θ. This is the specification considered in [START_REF] Shi | Asset bubbles in an overlapping generations model with endogenous labor supply[END_REF]. When π = 1, the utility over consumptions is CES. It is Cobb-Douglas if in addition θ = 1.

Taking (w t , P t , P t+1 , R t+1 ) as given, the consumer's problem is to choose an allocation (c 1t , c 2t+1 , s t , m t , l t ) that maximizes her lifetime utility (1) subject to the budget constraints:

c 1t + s t + P t m t = w t l t (2)
c 2t+1 = R t+1 s t + P t+1 m t (3)
The first-order conditions for this problem are given by:

c -1 θ 1t = γR t+1 c -1 θ 2t+1 and V l µ t = c -1 θ 1t w t c θ-1 θ 1t + γc θ-1 θ 2t+1 πθ θ-1 -1 (4)
and the no arbitrage condition between capital and the useless asset:

P t R t+1 = P t+1 (5)
When there is a bubble on the useless asset, P t > 0 and the returns on productive capital is equal to the returns of holding the speculative bubble. In other words, the bubble must grow at the rate of interest. Using these equations, we obtain:

c 1t = w t l t 1 + γ θ R θ-1 t+1 , c 2t+1 = γ θ R θ t+1 w t l t 1 + γ θ R θ-1 t+1 , (6)
s t + P t m t = γ θ R θ-1 t+1 1 + γ θ R θ-1 t+1 w t l t (7)
and, substituting ( 6) into (4), we get:

l t = [V -1 π w t (1 + γ θ R θ-1 t+1 ) 1 θ-1 ] π 1+µ-π (8)
Note that the second order condition for the labor choice requires 1 + µ > π, which implies that the labor supply is increasing with the wage w t and the interest factor R t+1 .

Firms

In the economy, there is a continuum of identical and competitive firms of unit mass. Each firm rents capital K t and hires labor L t at interest rate R t and wage w t , respectively. Production is given by the following Cobb Douglas technology:

Y t = AK β t L 1-β t
with A > 0 and 0 < β < 1. Profit maximization yields:

w t = (1 -β)K β t L -β t and R t = βK β-1 t L 1-β t (9)

Intertemporal equilibrium

Labor marker clears in each period, so that l t = L t /N t . Using ( 9), the equilibrium wage and interest rate can be expressed as:

w t = (1 -β)k β t l -β t and R t = βk β-1 t l 1-β t ( 10
)
where k t = K t /N t denotes the capital stock per worker.

Equilibrium on the asset market requires that aggregate savings of young households is used to buy the useless bubble and the future capital stock.3

K t+1 + P t m t N t = N t S t (11)
The total supply of the intrinsically useless asset is constant over time and is denoted by M > 0. Since this asset has no fundamental value, it is a (pure) bubble when its price is strictly positive, i.e. P t > 0. Each young household buy m t units of asset bubbles, meaning that at equilibrium,

N t m t = M (12)
Let b t = P t m t be the value of financial asset at time t hold by an household. Using the no-arbitrage condition between bubble and capital stock (5), we have:

b t+1 = R t+1 (1 + n) b t ( 13
)
where b t > 0 if there is a bubble. Using equations ( 7) and ( 10), the equilibrium on the asset market (11) rewrites:

(1 + n)k t+1 = (1 -β) γ θ R θ-1 t+1 1 + γ θ R θ-1 t+1 k t l t β l t -b t (14)
while using equations ( 8) and ( 10), we get:

l t = V -1 π (1 -β) k t l t β (1 + γ θ R θ-1 t+1 ) 1 θ-1 π 1+µ-π (15)
Definition 1 Given the initial capital stock k 0 = K 0 /N 0 > 0, an intertemporal equilibrium is a sequence (k t , l t , R t , b t ) ∈ R 4 ++ satisfying (10), ( 13), ( 14) and (15)

Stationary equilibria

A steady state is a solution k t = k, l t = l, R t = R and b t = b for all t ≥ 0 satisfying:

l = V -1 π (1 -β) β R β 1-β (1 + γ θ R θ-1 ) 1 θ-1 π 1+µ-π (16) k = l β R 1 1-β (17) b = k 1 -β β γ θ R θ 1 + γ θ R θ-1 -(1 + n) (18) b = R (1 + n) b (19)
There are two types of stationary equilibria, the bubbleless ones with b = 0 and the bubbly ones with b > 0.

We first study the existence and uniqueness of a bubbleless steady state. Using equation ( 18), the value of the interest rate at such a steady state R is determined by:

δ( R) ≡ γ θ Rθ 1 + γ θ Rθ-1 = β(1 + n) 1 -β (20)
where the function δ(R) is strictly increasing with δ(0) = 0 and lim R→+∞ δ(R) = +∞. We easily deduce that there exists a unique R > 0 that solves (20). The following proposition summarizes this result:

Proposition 1 There exists a unique bubbleless steady state given by (k, l, R, b) = ( k, l, R, 0).

The stationary values l and k are determined by substituting R in ( 16) and [START_REF] Weil | Confidence and the real value of money in an overlapping generations economy[END_REF].

We switch now to the bubbly steady state where b > 0. When there is a bubble, b > 0, equation ( 19) implies that R * = (1 + n), where R * represents the interest rate at a bubbly steady state. Substituting this last expression in (18), a positive bubble b > 0 means:

γ θ (1 + n) θ 1 + γ θ (1 + n) θ-1 > β(1 + n) 1 -β (21)
which is equivalent to δ(1 + n) > δ( R) using equation (20), i.e. 1 + n > R.

Proposition 2 There exists a unique bubbly steady state

(k, l, R, b) = (k * , l * , 1+ n, b * ) if R < (1 + n) or equivalently: γ θ (1 + n) θ-1 1 + γ θ (1 + n) θ-1 > β 1 -β (22)
By inspection of inequality (22), we see that Proposition 2 will be satisfied if β is low enough, γ and n are sufficiently high. In that case, equations ( 16)-(18) give the stationary values l * , k * and b * substituting R by 1 + n.

The bubble exists if the interest rate at the bubbleless steady state R is lower than the sum of the population growth rate and the depreciation of capital 1+n, which is also the return of capital at the bubbly steady state. The increase of the capital return occurs because the bubble compensates the shortage of asset and over-saving. When labor is exogenous, this means a lower capital stock at the bubbly than at the bubbleless steady state [START_REF] Tirole | Asset bubbles and overlapping generations[END_REF]). When labor is endogenous, it is the capital-labor ratio which is lower at the bubbly steady state. Therefore, as we will see, if labor is higher, capital can be higher too at the bubbly steady state, which means that there is a crowding-in effect of the bubble.

Labor supply and crowding-in effect of bubbles

We now examine whether the existence of asset bubble raises employment and capital comparing the bubbly and and bubbleless steady states. 4 By assessing the elasticity of labor supply and capital with respect to the interest rate, we will be able to deduce whether the bubbly steady state is characterized by higher levels of capital and labor supply.

When the bubble exists, the interest rate increases from R to R * . Such an increase will push up employment. Indeed, using equation ( 16), the elasticity of l with respect to R, ξ l,R , is determined by:

ξ l,R = π 1 + µ -π γ θ R θ-1 1 + γ θ R θ-1 - β 1 -β (23)
Using ( 18), b 0 implies:

ξ l,R π 1 + µ -π β 1 -β 1 + n R -1 0 ( 24
)
because all the steady states we consider are characterized by R 1 + n. This means that l * > l.

The capital stock per worker depends positively on the labor supply and negatively on the interest rate. According to equation [START_REF] Weil | Confidence and the real value of money in an overlapping generations economy[END_REF], the elasticity of k with respect to R, ξ k,R , is given by:

ξ k,R = ξ l,R - 1 1 -β = π 1 + µ -π γ θ R θ-1 1 + γ θ R θ-1 -Ω ( 25
)
where Ω ≡ 1+µ-π(1-β)

π(1-β) > 0. We note that Ω < 1 if and only if:

π > 1 + µ 2(1 -β) (26)
which is satisfied for µ and β low, and π sufficiently close to 1. Using equation (25), ξ k,R > 0 if and only if:

R > Ω γ θ (1 -Ω) 1 θ-1 = 1 + µ -π(1 -β) γ θ (2π(1 -β) -(1 + µ)) 1 θ-1 ≡ R (27)
We will have

k * > k if R * > R > R. Using (20), it requires δ(R) < (1 + n)β/(1 -β), which is equivalent to: [1 + µ -π(1 -β)] θ θ-1 πβ [γ θ (2π(1 -β) -(1 + µ))] 1 θ-1 < 1 + n (28)
This last inequality is satisfied for µ and β low, π sufficiently close to 1, γ and n high enough. This result is summarized in the following proposition: Proposition 3 Assume that inequality (22) holds.

1. Labor is higher at the bubbly steady state, i.e. l * > l.

2. Capital is higher at the bubbly steady state, i.e. k * > k, if inequalities ( 26) and (28) are satisfied, which requires µ and β low, π sufficiently close to 1, γ and n high enough.

When there is a bubble, the return of capital raises. Despite the decrease of the wage it implies, labor supply always increases. This raise allows to generate a higher level of capital in accordance with the higher interest and, therefore, the lower level of capital-labor ratio. In such a case, the bubble is productive and has a crowding-in effect. Of course, this implies a higher level of production (per capita).

Our model generalizes the result obtained by Shi and Suen (2014). They consider the particular case of a separable utility function between consumptions when young and old. In our setting, it corresponds to the particular case where π = θ-1 θ . Using a calibration, they explain that the bubble has a crowding-in effect if the elasticity of intertemporal substitution in consumption θ is sufficiently high. They calibrate the value of θ to 6.66. This result is inconsistent with the empirical estimates that show that the value of θ is rather between 1 and 2 (Gruber (2013), [START_REF] Mulligan | Capital interest and aggregate intertemporal substitution[END_REF], Vissing-Jorgensen and Attanasio ( 2003)).

In contrast to Shi and Suen (2014), we explicitly and theoretically prove that the bubble may have a crowding-in effect on capital and labor. The utility function we consider is more general than their and disentangles the degree of homogeneity π from the elacticity of intertemporal substitution in consumption θ. This degree of homogeneity should be sufficiently close to one, but the bubble may have a crowding-in effect for smaller and more realistic values of the elacticity of intertemporal substitution in consumption than in [START_REF] Shi | Asset bubbles in an overlapping generations model with endogenous labor supply[END_REF].

To reinforce our argument, let us consider the case of a Cobb-Douglas utility over the consumptions in both periods. We obtain this case for π = 1 and θ = 1. Then, the utility function (1) writes c 1t c γ 2t+1 -V l 1+µ t 1+µ . According to Proposition 2, equations ( 23) and (25), the bubble exists and has a crowding-in effect on labor and capital, i.e. k * > k and l * > l, if:

γ 1 + γ > µ + β 1 + β
This proves that contrary to what is agued by Shi and Suen (2014), a bubble exists and has a crowding-in effect for an elasticity of intertemporal substitution in consumption not arbitrarily large. As we show, the bubble can enhance capital and labor under a Cobb-Douglas utility over consumptions, which corresponds to a unit value of this elasticity. It requires an elasticity of labor supply 1/µ and a saving rate γ/(1 + γ) sufficiently high, and a share of capital β low enough.

Conclusion

In this paper, we have highlighted in a more general way than previous papers the interaction between bubbles, labor and capital. We have thus considered an OLG model with elastic labor supply. Asset bubble can appear if the interest rate at the bubbleless equilibrium is lower than the growth rate of population. We have found some conditions such that the bubble has a crowding-in effect on capital and labor. In contrast to Shi and Suen (2014), our result is theoretically shown and does not require an elasticity of intertemporal substitution in consumption too high. Considering a non-separable utility function between consumptions when young and old, the crowding-in effect of the bubble holds for values of that elasticity close or equal to 1, which are empirically relevant values. Therefore, this paper shows that endogenous labor is a relevant channel that may contribute to explain that bubble episodes are characterized by higher levels of capital and employment.

In fact, the reader does not know if the condition in their Proposition 3 is satisfied or not.

As it is often assumed in overlapping generations economy where the period is long, we assume complete depreciation of capital after one period of use, which means that R t+1 represents equivalently the gross return of capital or the interest rate.

At time 0, all assets are owned by a generation of "initial-old" consumers, which sell these assets to young households born at this period.

For the sake of brevity, we do not analyze dynamics, which have been incidentally studied in previous papers with or without bubbles[START_REF] Benhabib | On competitive cycles in productive economies[END_REF],[START_REF] Nourry | Stability of equilibria in the overlapping generations model with endogenous labor supply[END_REF]).

‡ This work was supported by French National Research Agency grants ANR-17-EURE-0020 and ANR-15-CE33-0001-01.

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