Romain Couillet
email: romain.couillet@supelec.fr

Merouane Debbah
email: merouane.debbah@supelec.fr

Jack Silverstain

Mérouane Debbah

Jack W Silverstein

Asymptotic Capacity of Multi-User MIMO Correlated Channels

This paper introduces a new formula to derive explicit capacity expressions of a class of communication schemes, including single-cell multi-user MIMO and multi-cell point-topoint MIMO, when the wireless channels have separable variance profiles and the system dimensions grow large. As an introductory example, we study point-to-point MIMO channels with multicell interference, in downlink. In this setting, we provide new asymptotic capacity expressions when single-user decoding or MMSE decoding are used. Simulations are shown to corroborate the theoretical claims, even when the number of transmit/receive antennas is not very large.

I. INTRODUCTION

In the last years, while mobile networks were expected to run out of power and frequency resources, Foschini [START_REF] Foschini | On Limits of Wireless Communications in a Fading Environment when Using Multiple Antennas[END_REF] and Telatar [START_REF] Telatar | Capacity of multi-antenna Gaussian channels[END_REF] introduced the notion of MIMO (multiple input multiple output) systems and predicted a growth of capacity performance of min(n R , n T ) times the single antenna capacity for an n T -antenna transmitter and an n R -antenna receiver. However, this tremendous multiplexing gain can only be provided for large SINR (signal-to-interference plus noise ratio) and without signal correlation. In case of correlation due to antenna spacing or to poorly scattering environments, these results are still an open issue. In present multi-cell wireless mobile networks, neither base stations nor users cooperate; this leaves the device manufacturers with the dilemma of increasing the signal processing capabilities of the transmit/receive units to result into non significant throughput gains when adjacent cells interfere one another. Moreover, due to limited computational constraints, suboptimal linear techniques such as MMSE (minimum mean square error) decoding are used at the receiver [START_REF] Paulraj | Introduction to Space-Time Wireless Communications[END_REF], in place of optimal single-user decoders.

In this work, we derive the channel capacity of MMSE receivers against optimal single-user decoders in multi-cell networks, when the number of antennas at the transmitters and receivers is large. The capacity here is defined as the supremum of the achievable rates between a base station and a specific user (in uplink or in downlink) interfered by other cells. We model all transmission channels by the well-spread Kronecker model [START_REF] Chuah | Capacity Scaling in MIMO Wireless Systems under Correlated Fading[END_REF]. Few major contributions propose to study the capacity performance of point-to-point communications with interference. In [START_REF] Shamai | Information-Theoretic Considerations for Symmetric, Cellular, Multiple-Access Fading Channels -Part I[END_REF], the authors carry out the performance analysis of TDMA-based networks with inter-cell interference. In [START_REF] Zaidel | Multicell Uplink Spectral Efficiency of Coded DS-CDMA With Random Signatures[END_REF], a random matrix approach is used to study large CDMA-based networks with inter-cell interference. In the MIMO context, [START_REF] Moustakas | MIMO Capacity Through Correlated Channels in the Presence of Correlated Interferers and Noise: A (Not So) Large N Analysis[END_REF] provides an analytic solution to our problem, using replica methods [START_REF] Guo | Multiuser Detection and Statistical Physics[END_REF]. These methods are however tedious since they require heavy combinatorial calculus. We propose in the following a more direct approach, based on analytical tools of random matrix theory [START_REF] Tulino | Random Matrix Theory and Wireless Communications[END_REF]. In particular, we introduce a new theorem, related to the Stieltjes transform of a specific class of random matrices, which generalizes a similar result in [START_REF] Silverstein | On the empirical distribution of eigenvalues of a class of large dimensional random matrices[END_REF].

Although this specific work is dedicated to the study of point-to-point MIMO systems with multi-cell interference, the method we introduce covers a larger class of problems, in which channel capacities express as the log determinant of a sum of Gram matrices X i X H i , where X i is a large matrix modelled as Kronecker. For example, aside from uplink/downlink multi-cell single-user MIMO, this method encompasses single-cell multi-user MIMO communications in the uplink, evaluation of the capacity region of multiple access channels and dirty paper coding in broadcast channels [START_REF] Couillet | Asymptotic capacity of MIMO correlated broadcast channels[END_REF] etc.

The remainder of this work is structured as follows: in Section II, we provide mathematical preliminaries and we introduce a new theorem, for which we provide a sketch of the proof. In Section III, we introduce the system model. In Section IV, the point-to-point capacity of the channel between a base station and a user, interfered by other cells, is derived when optimal single-user decoding or MMSE decoding are performed at the receiver. In Section V, we provide simulation results of the previously derived theoretical formulas. Finally, in Section VI, we give our conclusions.

Notation: In the following, boldface lower-case symbols represent vectors, capital boldface characters denote matrices (I N is the N × N identity matrix). X ij denotes the (i, j) entry of X. The Hermitian transpose is denoted (•) H . The operators tr X, |X| and X represent the trace, determinant and spectral norm of matrix X, respectively. The symbol E[•] denotes expectation. The notation F Y stands for the empirical distribution of the eigenvalues of the Hermitian matrix Y.

II. MATHEMATICAL PRELIMINARIES

Part of this work is dedicated to the introduction of a novel theorem, from which the multi-cell downlink and uplink capacities will be given compact expressions. This theorem gen-eralizes Silverstein and Bai's formula [START_REF] Silverstein | On the empirical distribution of eigenvalues of a class of large dimensional random matrices[END_REF] to random matrices with separable variance profiles, i.e. following the Kronecker model, and unfolds as follows, Theorem 1: (Stieltjes Transform) Let K, N ∈ N be some positive integers. Let

B N = K k=1 R 1 2 k X k T k X H k R 1 2 k (1)
be an N × N matrix with the following hypothesis for all k ∈ {1, . . . , K},

1) T k is n k × n k Hermitian nonnegative definite, n k ∈ N * , 2) R 1 2
k is the N × N Hermitian nonnegative definite square root of the nonnegative definite matrix R k , 3) The sequences {F T k } n k ≥1 and {F R k } N ≥1 are tight, i.e. for all ε > 0, there exists

M 0 > 0 such that M > M 0 implies F T k ([M, ∞)) < ε and F R k ([M, ∞)) < ε for all n k , N . 4) X k is N × n k with i.i.d. complex Gaussian entries with variance 1/n k . For k ∈ {1, . . . , K}, let c k = n k /N . Also denote, for z ∈ C \ R + , m N (z) = 1 N (B N -zI N ) -1 .
Then, as all n k and N grow large (while

K is fixed), with ratio c k m N (z) -m (0) N (z) a.s. -→ 0 (2)
where

m (0) N (z) = 1 N tr K k=1 τ k dF T k (τ k ) 1 + τ k c k e k (z) R k -zI N -1 (3)
and the set of functions {e i (z)}, i ∈ {1, . . . , K}, form the unique solution to the K equations

e i (z) = 1 N tr R i K k=1 τ dF T k (τ k ) 1 + τ k c k e k (z) R k -zI N -1 (4) such that sgn(ℑ[e i (z)]) = sgn(ℑ[z]).
The function m N (z) is the Stieltjes transform [START_REF] Tulino | Random Matrix Theory and Wireless Communications[END_REF] of the random variable with cumulative distribution function F BN . The complete proof of a more general expression of this theorem is given in an extended version of the present article [START_REF] Couillet | A deterministic equivalent approach for the capacity analysis of multi-user MIMO correlated channels[END_REF]. In the following, we give a sketch of the essential steps of the proof Proof: First note that, when K = 1 and, for all i, R i = I N , the theorem is already known from [START_REF] Silverstein | On the empirical distribution of eigenvalues of a class of large dimensional random matrices[END_REF]. We consider here K = 1 (and drop the useless indexes), the general case being a trivial extension, see [START_REF] Couillet | A deterministic equivalent approach for the capacity analysis of multi-user MIMO correlated channels[END_REF]. Also, we assume here T diagonal, which does not restrict generality since the Gaussian matrix X is unitarily invariant.

• A first truncation and centralization step makes it possible to bound the entries of the random matrix X and the en-

tries of R, T to X ≤ k log(N ), k > 2, R ≤ log(N ), T ≤ log(N ).
It is shown first that these truncations and centralizations do not restrict the generality of the final result. Now deterministic bounds can then be used. From the resolvent identity (Equation (4.1) in [START_REF] Silverstein | On the empirical distribution of eigenvalues of a class of large dimensional random matrices[END_REF]) and the matrix inversion lemma (Equation (2.2) in [START_REF] Silverstein | On the empirical distribution of eigenvalues of a class of large dimensional random matrices[END_REF]),

• Denote D = -zI N -zp(z)R, with p(z) = -1/(nz)
(1/N ) tr D -1 -m N (z) = (1/n) n j=1 τ j d j with d j = x H j R 1 2 (B (j) -zI N ) -1 D -1 R 1 2 x j 1 + τ j y H j (B (j) -zI N ) -1 y j (5)
-

(1/N ) tr R(B N -zI N ) -1 D -1 1 + τ i e(z)/c (6)
where x j is the j th column of X, y j the j th column of R

1 2 X and B (j) is B N with j th column removed. Observ- ing that d j → 0 when N → ∞, we have (1/N ) tr D -1 - m N (z) → 0 and similarly (1/N ) tr D -1 R -e(z) → 0.
• The rest of the proof consists in proving the existence and uniqueness of a solution to Equation ( 4), from which we have the existence of a unique solution m

N in Equation (3). It is then proven that, with the initial hypothesis for X, T and R, m N (z)m -→ 0.

• Applying finally Vitali's theorem to the analytic function m

(0)
N , the theorem is shown to hold for all z ∈ C \ R + .

Remark 1: This theorem allows us to derive Stieltjes transforms of large matrices independently of the realization of the X k matrices. In wireless communications, this provides a characterization of a multi-user or multi-cell communication based only on the transmit and receive correlations R k and T k . This further helps to estimate channel capacity thanks to the Shannon transform, Theorem 2: (Shannon Transform) Let B N be a random Hermitian matrix as defined in Theorem 1 with the additional assumption that there exists M 0, such that, for all N , n k , max( T k , R k ) < M , and let x > 0. Then, for large N ,

n k , V(x) -V (0) (x) a.s.
-→ 0, where

V(x) = log 2 1 + b x dF BN (b) (7)
and

V (0) (x) = +∞ x 1 w -m (0) N (-w) dw (8)
A proof of this result is provided in [START_REF] Couillet | A deterministic equivalent approach for the capacity analysis of multi-user MIMO correlated channels[END_REF].

III. SYSTEM MODEL

In this work we derive capacity expressions of wireless channels between a multi-antenna transmitter and a multiantenna receiver, the latter of which is interfered by several multi-antenna transmitters. This scheme is well-suited to multi-cell wireless networks with orthogonal intra-cell and interfering inter-cell transmissions, both in downlink and in uplink. The following scenarios encompass in particular

• multi-cell uplink: the base station of a cell indexed by i ∈ {1, . . . , K} receives data from one user in this cell 1and is interfered by K -1 users transmitting on the same physical resource from remote cells indexed by j ∈ {1, . . . , K}, j = i. • multi-cell downlink: the user being allocated a given time/frequency resource in a cell indexed by i ∈ {1, . . . , K} receives data from its dedicated base-station and is interfered by K -1 base stations in neighboring cells indexed by j ∈ {1, . . . , K}, j = i. This situation is depicted in Figure 1. In the following, in order not to confuse both scenarios, only the downlink scheme is considered. However, one must keep in mind that the provided results can easily be adapted to the uplink case.

Consider a wireless mobile network with K ≥ 1 cells indexed from 1 to K, controlled by non-physically connected base stations. On a particular time/frequency resource, each base station only one user; therefore the base station and the user of cell j will also be indexed by j. Without loss of generality, we focus our attention on user 1, equipped with n R ≫ K antennas and hereafter referred to as the user or the receiver. Every base station j ∈ {1, . . . , K} is equipped with n Tj ≫ K antennas. We additionally denote c j = n Tj /n R .

Denote s j ∈ C nT j , E[s j s H j ] = I nT j , the signal transmitted by base station j, y ∈ C nR the signal received by the user and n ∼ CN(0, σ 2 I nR ) the noise vector received by the user. The fading MIMO channel between base station j and the user is denoted H j ∈ C nR×nT j . Moreover we assume that H j has a separable variance profile, i.e. can be decomposed as

H j = R 1 2 j X j T 1 2 j (9)
with R j ∈ C nR×nR the (Hermitian) correlation matrix at the receiver with respect to the channel H j , T j ∈ C nT j ×nT j the correlation matrix at transmitter j and X j ∈ C nR×nT j a random matrix with Gaussian independent entries of variance 1/n Tj .

Remark 2: Note that in this model, and contrary to what is often assumed, R j , the correlation matrix at the receiver, depends on j. In the uplink scenario, this assumption is of particular relevance in the sense that base stations are usually placed in areas clear of scatterers. In these circumstances, the solid angle from which the signals from user j originate influences the signal correlation at the receive antenna array.

Hence the dependence of the receive correlation matrices on j. Note moreover that, in this model, the transmit power assumption E[s j s H j ] = I nT j is not restrictive in the sense that the transmit power correlation of base station j can be included into the matrix T j . However, the Kronecker model has two major drawbacks: (i) the inner matrix X implicitly assumes a high density of scatterers 2 in the communication link and (ii) the correlations on both sides must be inter-independent and independent of the realizations of X , which is inaccurate to some extent.

With the assumptions above, the communication model unfolds

y = H 1 s 1 + K j=2 H j s j + n ( 10
)
where s 1 is the useful signal (from base station 1) and s j , j ≥ 2, constitute interfering signals.

IV. MULTI-CELL MIMO CAPACITY

A. Optimal Single-User Decoding

If the receiving user considers the signals from the K -1 interfering transmitters as correlated Gaussian noise and knows the value of the SNR (signal-to-noise ratio) σ -2 , then base station 1 can transmit with arbitrarily low decoding error at a rate per-receive antenna C opt (σ 2 ) given by [START_REF] Shannon | A mathematical theory of communications[END_REF]

C opt (σ 2 ) = 1 n R log 2 |I nR + 1 σ 2 K j=1 H j H H j | - 1 n R log 2 |I nR + 1 σ 2 K j=2 H j H H j | (11)
Assume that n R and the n Ti , i ∈ {1, . . . , K}, are large compared to K and such that no eigenvalue of R i or T i is too large. As in Theorem 1, we define the function m (0) as the asymptotic Stieltjes transform of

B N = K j=1 H j H H j , m (0) (z) = 1 n R tr   K j=1 t j dF Tj (t j ) 1 + tj cj e j (z) R j -zI nR   -1 (12)
where, for all i ∈ {1, . . . , K}, e i (z) is solution of the fixedpoint equation

e i (z) = 1 n R tr R i   K j=1 t j dF Tj (t j ) 1 + tj cj e j (z) R j -zI nR   -1 (13)
From Theorem 2, applied to B N = K j=1 H j H H j , we then have approximately

1 n R log 2 |I nR + 1 σ 2 K j=1 H j H H j | = +∞ σ 2 1 w -m (0) (-w) dw (14)
2 the number and distance between scatterers must be of the same order as the number and distance between the transmit and receive antennas.

A similar result is obtained for the second right-hand side term of Equation [START_REF] Shamai | Information-Theoretic Considerations for Symmetric, Cellular, Multiple-Access Fading Channels -Part I[END_REF]. The per-receive antenna capacity C opt (σ 2 ) is therefore well approximated, for large number of antennas, by

C opt (σ 2 ) = (15) 1 n R +∞ σ 2   tr   K j=1 t j dF Tj (t j ) 1 + tj cj e j (w) R j -wI nR   -1 -tr   K j=2 t j dF Tj (t j ) 1 + tj cj f j (w) R j -wI nR   -1    dw (16)
where, e i , i ∈ {1, . . . , K}, and f i , i ∈ {2, . . . , K}, verify

e i (z) = 1 n R tr R i   K j=1 t j dF Tj (t j ) 1 + tj cj e j (z) R j -zI nR   -1 (17)
f i (z) = 1 n R tr R i   K j=2 t j dF Tj (t j ) 1 + tj cj f j (z) R j -zI nR   -1 (18)

B. MMSE Decoder

Achieving C opt requires non-linear processing at the receiver, such as MMSE successive interference cancellation. A suboptimal linear technique, the MMSE decoder, is often used instead. The communication model in this case reads

y =   k j=1 H j H H j + σ 2 I nR   -1 H H 1   k j=1 H j s j + n   (19)
and each entry of y will be processed individually. This technique makes it possible to transmit data reliably at any rate inferior to the per-antenna MMSE capacity C MMSE ,

C MMSE (σ 2 ) = 1 n R nT 1 i=1 log 2 (1 + γ i ) (20)
where, denoting h j ∈ C nT j the j th column of H 1 and R 1 2

1 x j = h j , the individual SINR γ i 's express as

γ i = h H i K j=1 H j H H j + σ 2 I nR -1 h i 1 -h H i K j=1 H j H H j + σ 2 I nR -1 h i (21) = h H i   K j=1 H j H H j -h i h H i + σ 2 I nR   -1 h i (22) = x H i R 1 2 i   K j=1 H j H H j -h i h H i + σ 2 I nR   -1 R 1 2 i x i (23)
where Equation ( 22) comes from a direct application of the matrix inversion lemma. With these notations, x i has i.i.d. complex Gaussian entries with variance T 1ii /n Ti and the inner matrix of the right-hand side of (23) is independent of x i (since the entries of H 1 H H 1h i h H i are independent of the entries h i ). Applying Lemma 3.1 of [START_REF] Silverstein | On the empirical distribution of eigenvalues of a class of large dimensional random matrices[END_REF], for n Ti large, approximately

γ i = T 1ii n T1 tr R 1   K j=1 H j H H j -h i h H i + σ 2 I nR   -1 (24)
From Lemma 2.1 of [START_REF] Bai | On the signal-to-interference-ratio of CDMA systems in wireless communications[END_REF], the rank 1 perturbation (-h i h H i ) does not affect asymptotically the trace in (24). And therefore, approximately,

γ i = T 1ii n T1 tr R 1   K j=1 H j H H j + σ 2 I nR   -1 (25)
Observing that e 1 (z) in Section IV-A corresponds to the normalized trace in Equation (25) (this is shown precisely in the proof of Theorem 1 [START_REF] Couillet | A deterministic equivalent approach for the capacity analysis of multi-user MIMO correlated channels[END_REF]), we finally have the compact expression for C MMSE ,

C MMSE (σ 2 ) = 1 n R nT 1 i=1 log 2 1 + 1 c 1 T 1ii e 1 (-σ 2 ) (26)
In practice, when no power allocation strategy is applied, T 1ii = P the average power per transmit symbol, and the capacity becomes

C MMSE = c 1 • log 2 (1 + P c1 e 1 (-σ 2 )).

V. SIMULATION AND RESULTS

In the following, we apply the results (15) and (26) to the downlink of a two-cell network. The capacity analyzed here is the achievable rate on the link between base station 1 and the user, the latter of which is interfered by base station 2. The relative power of the signal received from base station 2 is on average Γ times that of base station 1. Both base stations are equipped with linear arrays of n T antennas and the user with a linear array of n R antennas. The correlation matrices T i at the transmission and R i at the reception, i ∈ {1, 2}, are modeled thanks to a generalization of Jake's model including solid angles of transmit/receive power, i.e. for instance,

T i ab = θ (i) max θ (i) min exp 2π • i • d Ti ab λ cos(θ) dθ (27)
with d Ti ab the distances between antennas indexed by a, b ∈ {1, . . . , n Ti } for transmitter i, (θ (i) min , θ (i) max ) the angles over which useful power (i.e. power that will be received by the user) is transmitted, and λ the wavelength.

In Figure 2, we took n R = 16, Γ = 0.25 and we consider optimal single-user decoding at the receiver. For every realization of T i , R i , 1000 channel realizations are processed to produce the simulated ergodic capacity and compared to the theoretical capacity (15). Those capacities are then averaged over 100 realizations of T i , R i , varying in the random choice of θ In Figure 3, with the same assumptions as previously, we apply MMSE decoding at the receiver. Here, a slight difference is observed in the high SNR regime between theory and practice. This was somehow expected, since the large n R approximations in Silverstein's lemmas [START_REF] Silverstein | On the empirical distribution of eigenvalues of a class of large dimensional random matrices[END_REF] are very loose for σ 2 close to R -in the sense of the Euclidean distance. To cope with this gap, many more antennas must be used. We also observe a significant difference in performance between optimum single-user and linear MMSE decoders, especially in the high SNR region. Therefore, in wireless networks, when interfering cells are treated as correlated Gaussian noise at the cell-edge, i.e. where the interference is maximum, the MMSE decoder provides tremendous performance loss.

VI. CONCLUSION

In this paper, we introduced a theorem relating the Stieltjes transform of a class of large matrices to deterministic approximates. Based on this formula, we provided compact capacity expressions for the optimal single-user decoder and MMSE decoder in point-to-point MIMO systems with intercell interference and random channel matrices with separable variance profile, both in downlink and in uplink. The simulations show perfect match with the theoretical formulas in the low-to-medium SNR region, even if fewer antennas are used at the transmitters and receivers. As for the high SNR region, a large number of antennas must be used to reach an accurate match between theory and Monte-Carlo simulations.

Fig. 1 .

 1 Fig. 1. Downlink multi-cell scenario

  max with constraint θ (i) maxθ (i) min = π/2, while d Ti ab = 10λ|a -b| at the transmitters, d R ab = 2λ|a -b| at the receiver. The SNR ranges from -5 dB to 30 dB, and n T ∈ {8, 16}. We observe here that Monte-Carlo simulations perfectly match the capacity obtained from Equation (15).

Fig. 2 .

 2 Fig.2. Capacity of point-to-point MIMO in two-cell uplink, optimal singleuser decoding, n R = 16, n T ∈ {8, 16}, Γ = 25%.

  VII. ACKNOWLEDGMENT Silverstein's work is supported by the U.S. Army Research Office under Grant W911NF-05-1-0244. Debbah's work is partially supported by the European Commission in the framework of the FP7 Network of Excellence in Wireless Communications NEWCOM++.

Fig. 3 .

 3 Fig. 3. Capacity of point-to-point MIMO in two-cell uplink, with MMSE decoder, n R = 16, n T ∈ {8, 16}, Γ = 25%.

this user is allocated a given time/frequency resource, which is orthogonal to time/frequency resources of the other users in the cell; e.g. the multi-access protocol is OFDMA.


Yacine Belarbi

Fayçal Hamdi

Abderaouf Khalfi

Saïd Souam

Growth, institutions and oil dependence: a buffered threshold panel approach

Keywords: Economic growth, oil dependence, quality of institutions, buffered threshold panel data model, nonlinear modelling of panel data

We examine the combined effects of oil dependence and the quality of institutions on economic growth. To do so, we introduce a new buffered threshold panel data model and apply it to 19 oil rent-dependent countries over the period 1996-2017. We show that the relationship between growth and oildependence is not linear. More precisely, three categories of oil-dependent countries are identified. Only countries with high-quality institutions are very stable. All the other countries have experienced a transition into a buffer zone and are potentially in a transition between two different regimes. When considering oil dependence as a threshold variable, it appears that the quality of institutions has a positive and significant effect on growth when dependence is either low or high. More interestingly, for countries with intermediate levels of oil-dependence, the quality of the institutions negatively impacts growth. Some of these countries have experienced something of an oil-dependence trap.

Introduction

Dependence on natural resources is the subject of a wide debate in the analysis of economic growth in rentier states. However, in the empirical studies, there is no clear consensus on the negative impact of resource rents on long-term growth [START_REF] Havranek | Natural resources and economic growth: A meta-analysis[END_REF]. In practice, rentier states are characterized by important heterogeneity in their economic performance. The quality of institutions is one major explanation that has been advanced in the literature to explain these disparities. As is now well documented in the literature, natural resource dependence has given rise to some negative phenomena that could hinder growth (rent-seeking behaviours, the contraction of non-resource production activities, corruption, the voracity effect, civil conflicts, social pressure for additional redistribution, increases in public spending in less productive sectors, etc.). In fact, a diversification of the economy and an improvement in the quality of the institutions in natural resource exporting countries seem to be efficient tools for enhancing their growth performance. Indeed, these countries could reach such high levels of dependency that it would become very difficult to sustain good economic or institutional reforms. From an economic policy point of view, it is thus important to understand how institutional reforms could impact economic growth while interacting with natural resource dependence. Indeed, the economic cost of ameliorating the quality of the institutions could be very high before having a positive effect on economic growth.

The empirical literature studying the relationships among natural resources, the quality of institutions and economic growth has not brought about a consensus. Such studies can be roughly classified into three categories. In the first category, natural resources are found to have a negative effect on growth when they are associated with weak institutions (see for example, Leite and Weidman, 1999[START_REF] Acemoglu | The Colonial Origins of Comparative Development: An Empirical Investigation[END_REF][START_REF] Göcke | Various concepts of hysteresis applied in economics[END_REF], and Salai-Martin and Subramanian, 2013). The second category shows that natural resources interact with the quality of institutions, and their combined effects on growth depend on the nature of their combination (Mehlum et al. 2006a, b, Boschini et al., 2007[START_REF] Arezki | Do natural Resources depress income per capita?[END_REF]. The third and last category shows that the observed heterogeneity in economic growth between rentier states is not explained by institutions [START_REF] Sachs | The Big Rush, Natural Resource Booms And Growth[END_REF][START_REF] Brunnschweiler | Cursing the Blessings? Natural Resource Abundance, Institutions, and Economic Growth[END_REF][START_REF] Alexeev | The Elusive Curse of Oil[END_REF]. It is worth noting that this literature generally assumes linearity in the dynamics to address these rather complex relationships. Only a few contributions have insisted on their nonlinearity [START_REF] Leite | Does Mother Nature corrupt? Natural resources, corruption and economic growth (Working Paper of the International Monetary Fund[END_REF][START_REF] Leite | Does Mother Nature corrupt? Natural resources, corruption and economic growth (Working Paper of the International Monetary Fund[END_REF]Sala-i-Martin and[START_REF] Sala-I-Martin | Addressing the natural resource curse: An illustration from Nigeria[END_REF].

Since their introduction by [START_REF] Tong | On a Threshold Model in Pattern Recognition and Signal Processing[END_REF], threshold models have been considered to be a very useful and sophisticated way to take into account the nonlinearity exhibited in several financial and macroeconomic phenomena. Indeed, they provide a simplified formulation to mimic nonlinear stylized facts and, more precisely, the dynamics of regime changes. Their structure has been widely used by econometricians in time series analysis. However, many extensions and mathematical developments of threshold models, in particular the panel data treatment framework, have been adopted for the analysis of other data structures. [START_REF] Hansen | Threshold effects in non-dynamic panels: Estimation, testing, and inference[END_REF] proposed a panel threshold regression (P T R) model for the nondynamic panel case. His main contribution lies in the possibility of allowing the individuals constituting the panel to be in different regimes during a given period. This enables the heterogeneity in the panel to be better captured and allows for a visualization of the nonlinearity in the interaction between the dependent variable and the explanatory variables for each panel's component. However, the sudden change in regime that characterizes Hansen's formulation may be problematic in some situations in which the transition is smooth. To capture the absence of sudden jumps, [START_REF] Gonzalez | Panel smooth transition regression models[END_REF] develop a non-dynamic panel smooth transition regression (P ST R) model with individual fixed effects. The parameters are allowed to change smoothly as a function of the threshold variable. The performance of this model may depend on the choice of the transition function for the studied phenomenon. Overall, this form of modelling turns out to be useful when the number of regimes is sufficiently high.

In some circumstances, an interesting phenomenon happens when a past temporary change in a relevant forcing variable leads to a change in the economic behaviour of the analysed variable but a return to the initial value of the forcing variable does not induce a return to the initial behaviour (i.e., the state of a system is dependent on its history). This so-called hysteresis phenomenon, originally stemming from physics, has been widely used in labour theory and foreign trade to explain the persistent effects of temporary stimuli (see, e.g., [START_REF] Göcke | Various concepts of hysteresis applied in economics[END_REF]. For example, in foreign trade, temporary exchange rate shocks could induce persistent consequences for quantities and prices due to sunk market-entry costs. Indeed, to sell in a foreign market, a firm incurs some entry costs that cannot be recovered after exit (e.g., distribution and service networks). If the domestic currency temporarily depreciates, entering this foreign market becomes profitable for some domestic firms. However, even if the exchange rate regains its initial level, it is still profitable to sell in the foreign market if the variable costs are recovered. This simple microeconomic hysteresis can thus be aggregated and gives rise to a continuous macro-level loop in overall exports [START_REF] Borgersen | Exchange rate overshooting and pathdependence in international trade[END_REF]. This effect has been widely documented in the empirical literature (see Belke and Kronen, 2019, for a recent study).

This smooth switching between different equilibria (finite configurations or finite states) of the studied variable (system) could thus be usefully mobilized to analyse the dynamics of its evolution. This is why in this article, we propose an alternative model based on this idea of hysteresis by defining a new smooth and flexible regime switching mechanism. To illustrate and highlight this point, let us limit ourselves, without loss of generality, to the case of a two-regime model. Instead of assuming a single threshold parameter, we consider an interval consisting of a lower and an upper threshold that acts as a buffer zone. If the threshold variable is below the lower boundary of the buffer zone, then the observation is from the first regime. Conversely, the observation comes from the second regime when the threshold variable is above the upper boundary. When the threshold variable falls within the buffer zone, the regime indicator keeps the value of its most recent past. This makes the transition dynamics smoother and more flexible than those of the classical P T R model. Even though this idea is still in its infancy, it provides a new way to understand and explain the nonlinearity observed in the data. In addition to this new modelling approach, our paper provides several interesting empirical results. First, we clearly show that the relationships between growth and dependence on oil rents are not linear: there is a gradual positive impact as the quality of institutions is enhanced. More precisely, our analysis identifies three categories of oil-dependent countries with respect to the quality of institutions. It is worth noting that except for three countries in the sample with high-quality institutions that are very stable, all the other countries have experienced a transition into a buffer zone. They are thus potentially in a transition between two different regimes, and the impact of oil resource dependence on their growth has not yet stabilized. Moreover, while considering dependence on oil rents as a threshold variable, it appears that the quality of institutions has a positive and significant effect on growth when dependence is low or high. More interestingly, it turns out that for intermediate oil-dependent countries, the quality of their institutions negatively impacts growth. Some of these countries have experienced something of an oil-dependence trap.

The remainder of the paper is organized as follows. In Section 2, we provide the analytical framework for our model: the definition of our buffered threshold panel data (BT P D) model, our estimation methodology, different test procedures and a simulation study. In Section 3, we first provide and discuss the results of our empirical study, which is devoted to the analysis of the combined interaction effects of natural resource dependence and the quality of institutions on economic growth in rentier countries. We thus compare our results to those provided by some alternative models and show how our model gives better results. Section 4 concludes.

Analytical framework

In this section, we first provide a precise definition of our BT P D model. We thus describe the general outlines of the proposed least squares estimation of the model. We afterwards lay down the general principles of our procedures for testing the number of regimes. We finally discuss the main results of our simulation study of the finite sample properties of these procedures.

Our model is inspired by the buffered process developed by [START_REF] Li | Hysteretic autoregressive time series models[END_REF] for time series analysis, the "hysteresis autoregressive time series model". It is worth noting that this approach has been used in the recent literature exclusively to analyse time series.1 It seems quite natural to extend this approach to phenomena characterized by a smooth transition in their dynamics in addition to having an individual dimension. This is the purpose of our model.

The buffered threshold panel data model

We consider the following balanced panel {Y i,t , q i,t , X i,t : 1 ≤ i ≤ N, 1 ≤ t ≤ T }, where i and t denote the individual and temporal indices, respectively. The dependent variable Y i,t is scalar. The threshold variable q i,t is scalar. X i,t is an m-vector of control variables. We first present the case of two regimes. The observed data are generated from a nondynamic tworegime BT P D model with fixed effects if they satisfy the following regression model:

Y i,t = µ i + X i,t β 1 1 (R i,t =1) + X i,t β 2 1 (R i,t =2) + ε i,t (1)
where µ i is the fixed effect for individual i and ε i,t is the error term, which is assumed to be independent and identically distributed with mean zero and positive finite variance σ 2 . β k , k = 1, 2, represents the slope coefficients, and 1 A is the indicator function for the set A. R i,t is the regime indicator defined as follows:

R i,t =    1 if q i,t ≤ r L,1 R i,t-1 if r L,1 < q i,t ≤ r U,1 2 if r U,1 < q i,t
where r L,1 and r U,1 (r L,1 ≤ r U,1 ) are the boundary parameters that constitute the buffer or hysteresis zone. In model ( 1), the transition mechanism is modelled in the same way as for the buffered threshold in a time series (see [START_REF] Li | Hysteretic autoregressive time series models[END_REF]. The originality lies in the representation of a panel as several distinct regimes. Each regime is characterized by a different regression slope (β 1 or β 2 ), and the transition between the two regimes is gradual rather than abrupt. This proposal is justified by the fact that in practice, regime R i,t may not shift immediately, and there could be a buffer region in which the regime of Y i,t depends on the regime of Y i,t-1 . In addition, the contribution of the individual dimension to the smooth transition mechanism is to group the individuals following the same linear regression model on a given date into the same regime. However, the individuals composing this regime are likely to evolve smoothly over time, since the transition variable also depends on the temporal dimension. In our empirical analysis, Y i,t is the real GDP growth rate (GDP G) of country i at time t. X i,t is a vector of explanatory variables. The threshold variable q i,t is oil rents or the quality of the institutions. µ i is the fixed effect of country i.

Note that from formulation (1), we can obtain the classical P T R model of [START_REF] Hansen | Threshold effects in non-dynamic panels: Estimation, testing, and inference[END_REF] as a special case by setting r L,1 = r U,1 = r. However, it is worth mentioning that the transition is abrupt in [START_REF] Hansen | Threshold effects in non-dynamic panels: Estimation, testing, and inference[END_REF]. Indeed, a country can switch from one regime to another if the status of the threshold variable oil rents (or quality of the institutions) q i,t crosses up or down the threshold r. More precisely, if the variable oil rents drops below the threshold (q i,t ≤ r), even slightly, GDP G is described by the first regime with slope coefficients β 1 . Conversely, when this variable exceeds r, GDP G is described by the second regime with slope coefficients β 2 . In other words, the regime indicator R i,t depends only on whether the oil rent value is smaller or larger than r. In our proposed model, the regime indicator may depend on the infinitely far away past of the regime indicators [START_REF] Li | Hysteretic autoregressive time series models[END_REF].

This two-regime BT P D model (1) can be easily extended to a threeregime model. Indeed, suppose there are two buffer zones (r L,1 , r U,1 ] and (r L,2 , r U,2 ] with

-∞ = r U,0 < r L,1 ≤ r U,1 < r L,2 ≤ r U,2 < r L,3 = +∞.
Then, the three-regime BT P D model can be defined as follows:

Y i,t = µ i + 3 k=1 X i,t β k 1 (R i,t =k) + ε i,t , (2)
where the regime indicator R i,t divides the observations into three regimes characterized by three different regression slopes β k , k = 1, 2, 3. Note that the regime indicator R i,t-1 is equal to k if the variable oil rents falls into (r U,k-1 , r L,k ], and GDP G (more generally, the data-generating process, or DGP ) will stay in the same regime at time t if the variable oil rents increases towards the buffer zone (r L,k , r U,k ] or decreases towards (r L,k-1 , r U,k-1 ]. In general, when the variable oil rents falls into the buffer zone (r L,l , r U,l ] with l > k or l < k -1, the regime indicator R i,t is set to l or l + 1, respectively (see Figure 1).

Least squares estimation

The first step of the estimation process consists of eliminating the permanent differences that exist between individuals over the period, which could skew the estimates. Eliminating individual fixed effects involves removing specific individual means. This step is standard in linear models (within transformation). However, it requires more careful processing in the context of threshold models and particularly in the BT P D model. Indeed, the individual effects depend on our knowledge about the buffer zone and must therefore be recalculated for each candidate zone. Another problem arises when the values of the threshold variables for the first observations may fall within one of the buffer zones. This can make the identification of the regime indicators difficult. To address this problem, we propose setting the regime indicator for each individual component i of the first observation as a realization of a uniform random variable on {k, k + 1} when its threshold variable value falls within the buffer zone (r L,k , r U,k ], i.e., when r L,k < q i,1 ≤ r U,k .

The 2-regime model

If q i,t²rL,1, then R i,t=1 ¹ The buffer zone If q i,t>rU,1, then R i,t=2 ¹ ?K r L,1 r U,1 +K 1st regime • If rL,1 < qi,t ² rU,1, then Ri,t = Ri,t?1 2nd regime
The 3-regime model The regime indicator sequences {R i,t , i = 1, ..., N and t = 2, ..., T } are subsequently generated.

If q i,t²rL,1, then R i,t=1
To eliminate the individual effect, we need the following compact representation of the BT P D model:

Y i,t = µ i + X i,t β + ε i,t , (3)
where

X i,t = X i,t 1 (R i,t =1) , ..., X i,t 1 (R i,t =K) and β = (β 1 , ..., β K ) , for K ∈ {2, 3}. Let us define the centred individual variables Y * i = Y * i,1 , ..., Y * i,T , X * i = X * i,1 , ..., X * i,T and ε * i = ε * i,1 , ..., ε * i,T constructed from Y i = (Y i,1 , ..., Y i,T
) , X i = X i,1 , ..., X i,T and ε i = (ε i,1 , ..., ε i,T ), respectively, by removing the corresponding individual means. We are thus able to rewrite model (3) in the following compact form

Y * = X * β + ε * , where Y * = (Y * 1 , ..., Y * N ) , X * = (X * 1 , ...X * N ) and ε * = (ε * 1 , ..., ε * N ) .
Hence, assuming that the number of regimes is known and, for a given configuration of the buffer zones γ = (r L,1 , r U,1 , ..., r L,K-1 , r U,K-1 ), that the N T × Km matrix X * (γ) has full column rank, the ordinary least squares (OLS) estimator of the slope coefficients is given by β

(γ) = (X * (γ)) X * (γ) -1 (X * (γ)) Y * .
Once β (γ) is obtained, the individual fixed effects are then estimated by

µ i = Y i -X i (γ) β (γ) , for i = 1, ..., N.
We can also obtain the residual vector as follows ε (γ) = Y * -X * (γ) β (γ) . This enables the calculation of the sum of the squared errors S (γ) = ε (γ) ε (γ). We can thus estimate the residual variance as follows σ 2 (γ) = 1 N T -N S (γ) . As previously discussed, the estimation of the slope coefficients requires knowledge of the buffer zones γ that we should estimate. To do this, we explore a set of buffer zone candidates constructed from quantiles of the observed values of the threshold variable. To ensure a minimum number of observations in each regime, we can -for example, when K = 2 -take quantiles from a% to b% of each threshold value and sample within a carefully chosen interval [a, b], and we generate all ordered 2-vectors of γ such that r L,1 < r U,1 . However, when K = 3, we require all buffer zone parameters to lie within the bounded subset [r 0 , r 1 ] of the threshold variable sample space, and we choose the 4-vector γ = ( r L,1 , r U,1 , r L,2 , r U,2 ) that minimizes the sum of the squared errors, i.e.,

γ = arg min γ∈Γ (S (γ)) , where Γ = {( r L,1 , r U,1 , r L,2 , r U,2 ) ∈ [r 0 , r 1 ] 4 | r L,1 ≤ r U,1 < r L,2 ≤ r U,2 }.
Note that our Γ is larger than the one used in [START_REF] Hansen | Threshold effects in non-dynamic panels: Estimation, testing, and inference[END_REF]. Therefore, our exploration of buffer zone candidates requires more computational effort.

Linearity test and determination of the number of regimes

We now deal with the issue of inference in BT P D models. As a priority, we present the tests of linearity and the determination of the number of regimes. These tests occupy a prominent place and guide us in the choice of specification to take into account the nonlinearities.

Consider again a BT P D model with two regimes (K = 2) as given by (2). As in [START_REF] Hansen | Threshold effects in non-dynamic panels: Estimation, testing, and inference[END_REF], one can use a likelihood ratio (LR) test to detect the effect of this nonlinearity. The null hypothesis (no buffered threshold effects) and the alternative hypothesis (the existence of buffered threshold effects) are given by

H 0 : β 1 = β 2 and H 1 : β 1 = β 2 .
This testing problem corresponds to the famous Davies problem (1977Davies problem ( , 1987) ) and has been investigated by [START_REF] Andrews | Optimal tests when a nuisance parameter is present only under the alternative[END_REF] and [START_REF] Hansen | Inference when a nuisance parameter is not identified under the null hypothesis[END_REF]. The bootstrap procedure suggested by [START_REF] Hansen | Inference when a nuisance parameter is not identified under the null hypothesis[END_REF] and used in [START_REF] Hansen | Threshold effects in non-dynamic panels: Estimation, testing, and inference[END_REF] to simulate the corresponding distribution of the LR test can be adapted for our framework.

Under H 0 , the OLS method provides an estimation β 1 for β 1 and the residuals ε * i,t . The LR test statistic is then defined by

F 1,2 = S 1 -S 2 ( γ 2 ) σ 2 2 ( γ 2 )
,

where S 1 and S 2 ( γ 2 ) are the residual sums of squared errors obtained from (1) without and with buffered threshold effects, respectively; σ 2 2 ( γ 2 ) is the residual variance of the two-regime BT P D estimation.

For the special case in which r L,1 = r U,1 , the asymptotic distribution of F 1,2 is nonstandard [START_REF] Hansen | Threshold effects in non-dynamic panels: Estimation, testing, and inference[END_REF]. Hence, the critical values and pvalues cannot be tabulated. However, this problem can be solved by referring to [START_REF] Hansen | Inference when a nuisance parameter is not identified under the null hypothesis[END_REF][START_REF] Hansen | Threshold effects in non-dynamic panels: Estimation, testing, and inference[END_REF] bootstrap methodology. The same logic can be applied to our buffered threshold model. The asymptotic p-value can be approximated with the following bootstrap procedure.

Algorithm 1.

1. Treat the regressors X i,t and the threshold jump variable q i,t as given, and their values remain fixed during repeated bootstrap simulations. 2. Recover the regression residuals obtained under the null hypothesis ε * i,t and group them by individual

ε * i = ε * i,1 , ε * i,2 , • • • , ε * i,T . Treat the sample { ε * 1 , ε * 2 , • • • , ε * N }
as the empirical distribution to be used for bootstrapping.

3. Draw (with replacement) a sample of size N from the empirical distribution and use these errors to create a bootstrap sample under H 0 . 4. Using the bootstrap sample, estimate the model under the null hypothesis and the alternative hypothesis and thereby calculate the bootstrap value of F 1,2 . 5. Repeat this procedure a large number of times and calculate the percentage of draws for which the simulated statistic exceeds the observed statistic F 1,2 . This percentage represents the p-value of F 1,2 under H 0 .

Once the linearity hypothesis is rejected, the question is whether all the nonlinearities in the observations have been taken into account or, in other words, whether we should use a buffered threshold model containing three regimes. The tests to determine the optimal number of regimes are an extension of the linearity tests and allow us to answer this question. Indeed, to test whether the model has two regimes, i.e., H 0 : β 3 = 0, or at least three regimes, i.e., H 1 : β 3 = 0, the following LR test statistic must be applied:

F 2,3 = S 2 ( γ 2 ) -S 3 ( γ 3 ) σ 2 3 ( γ 3 )
.

The null hypothesis of a single buffer zone is rejected in favour of at least two buffer zones if the value of F 2,3 is greater than the critical value simulated by bootstrapping. It is worth noting that the bootstrap approach to approximating the asymptotic critical values is very close to that presented in Algorithm 1. The only significant change is in Step 2, in which the bootstrap errors used are no longer those obtained under the null hypothesis but under the alternative hypothesis.

Simulation study

We conducted a Monte Carlo experiment to evaluate the performance of the OLS estimation under different settings. We generated samples from (2) using different values of γ and β = (β 1 , β 2 , ..., β K ) , where

β k = (β k,1 , β k,2 , ..., β k,m ) .
X i,t , q i,t and ε i,t are generated from different independent distributions (normal distribution, log-normal distribution, uniform distribution). For the number of individuals N and the time period T , we use combinations of T = 10, 20, 50, and 100 and N = 10, 20, 50, and 100. For each set of generated sample observations, we calculate the OLS estimator. We do this 1000 times with different values of the model parameters and for different choices of N and T . The finite sample properties of the estimators are summarized in Tables 10-13 provided in the appendix. For each case, we report the true values (True) of the parameters for each of the considered BT P D models, the empirical mean (Mean) and the empirical standard deviation (Std).

The estimates of the BT P D structural coefficients based on the OLS method display reasonable biases, which decrease as either N or T becomes large. Moreover, the Stds of some parameters are relatively large, but they rapidly decrease with either the number of individuals or the time period. The desirable consistency property of the OLS estimators is thus empirically satisfied, and the proposed estimation procedure provides good results.

We now examine the empirical distribution of the estimator for the slope parameters. Figures 678in the appendix depict the sample histograms for the estimated first parameters (β 1,1 or β 2,1 ) for three configurations of the BT P D model that differ by the distribution of the error ε i,t . In the first case, the error term has a uniform distribution, while in the second and third cases, the distribution is normal. It is important to note that the DGP s are the same and are given in Tables 111213. The Jarque-Bera test (JB in Figures 678) shows that when the cross-sectional and temporal dimensions of the panel are small, the empirical distribution is far from normal. Indeed, the corresponding p-values are lower than 0.1. We thus reject the normality hypothesis at the 1% significance level. However, when the number of individuals and/or the time period is at least equal to 50, we approach asymptotic normality. Indeed, according to the Jarque-Bera test, we do not reject the normality of the empirical distribution at any reasonable level. It is worth noting that the same kind of results hold for the other slope parameters.

We finally investigate the finite sample performance of our proposed procedure for testing linearity in the BT P D framework. For that purpose, we consider two DGP s. The first is used to assess the size of the test, while the second is used to study its power. The number of replications is fixed at 1000. In each replication, the model is estimated and analysed using the proposed bootstrap-based test of linearity.

The rejection frequencies at the nominal levels of 1, 5 and 10 percent are presented in Table 14. It is clear that the rejection frequencies are quite close to the nominal sizes. Furthermore, the power of the test increases with the cross-sectional and temporal dimensions of the panel. In summary, our extension of the bootstrap-based test proposed by [START_REF] Hansen | Threshold effects in non-dynamic panels: Estimation, testing, and inference[END_REF] to investigate linearity seems to work very well in our framework.

Empirical results

The BT P D model

In this section, we study the combined effects of the interaction between natural resource dependence and the quality of institutions on economic growth for a panel of 19 countries for the period 1996-2017. Through the buffered regime switching mechanism, we analyse the heterogeneity in the studied panel and how the interaction between natural resources and the quality of institutions impacts the economic growth of rentier states. The countries in our sample are given in Table 1.

To control for dependence on natural resources and the effects of the quality of institutions, we introduce the variables 'oil rents' and 'rule of law', respectively. The interaction effect can be analysed by using these variables as explanatory and transition variables at the same time. We add the most used variables in the traditional empirical literature on the macroeconomic determinants of growth to our econometric specification. According to several studies (e.g., [START_REF] Barro | Economic Growth in a Cross Section of Countries[END_REF][START_REF] Barro | Economic growth[END_REF][START_REF] Jones | Was an Industrial Revolution Inevitable? Economic Growth Over the Very Long Run[END_REF], trade openness, fixed investment, moderate inflation and output volatility, and a better educated workforce have helped countries achieve a higher rate of growth. All the variables used in our study are taken from the World Development Indicators, the World Government-Based Indicators and the International Financial Statistics databases. A short description of these variables is provided in Table 2.

Table 3 very slightly. Moreover, we observe a significant decrease in the mean level of oil rents as a percentage of GDP , from 14.37% to 8.52%. Inflation also decreased significantly, from approximately 7.95% to 4.16%. However, the means of investment and population growth increased during this period, while openness decreased very slightly. These tendencies are globally and clearly observed in comparing the maximum and minimum values between these two years.

We investigate two models to explain the interaction between natural resources and the quality of institutions and their impact on GDP growth in our sample. In the first model, we consider the quality of institutions to be a threshold variable and dependence on natural resources to be an explanatory variable. In the second model, dependence on natural resources is considered the threshold variable, and the quality of institutions is considered an explanatory variable.

We first test for linearity against a two-regime BT P D model and then a two-regime model against a three-regime model. 2 The results of these tests are reported in Table 4. In both models, we reject linearity at the 1% level. We also reject the two-regime model in favour of the three-regime model 2 It is important to note that in practice it is sufficient to consider only the cases K = 2 and K = 3 to capture the nonlinearities due to regime switching. This is why our analysis is limited to a model with at most three regimes. In our case, this is mainly due to the computational costs of the estimation. Indeed, for our case we have N ×T = 19×22 = 418 observations and if the values of a and b are the 10th and 90th percentiles of the data respectively, we would need to estimate 324540216 candidate models when K = 4. Note that the the creation of arrays with this dimension (324540216 × 4) causes MATLAB to become unresponsive. at the 1% level. We hereafter discuss the implications of this model for explaining the evolution of the growth of these oil rent-dependent countries.

The results of our regressions are given in Table 5.

We first discuss Model (I), in which the threshold variable is the quality of institutions. Three regimes are clearly identified. 3 In the lower one (regime 1 with low values for the rule of law), it appears that oil rents and openness have no significant effect on the growth of the economy, while inflation has a negative impact and investment and population growth have a positive impact. Using a V AR model, Antonakakis et al. (2016) showed that in developing and medium-high income countries with weak political institutions, oil dependence is not growth-enhancing. In the middle regime (regime 2 with intermediate values for the rule of law), oil rents have a positive and significant impact on growth as well as investment and population growth. Openness still has no significant effect. In the upper regime (regime 3 with high values for the rule of law), oil rents have an even greater positive and significant effect on growth. Moreover, openness has a strong and significant positive effect, while population growth and inflation have a negative impact. It thus seems that the stronger the rule of law is, the higher the positive impact of oil rents on economic growth. This gradual positive impact of the * * * Significant at the 0.01 level, * * at the 0.05 level, and * at the 0.10 level.

Table 5: GDP growth, the quality of institutions and oil dependence: estimated threeregime BT P D model.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Australia 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 Bolivia 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 Brunei Darussalam 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 Cameroon 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Canada 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 Colombia 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Cote d'Ivoire 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Ecuador 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 Egypt 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Gabon 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Indonesia 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Kuwait 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 Nigeria 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Qatar 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 Norway 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 Oman 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Saudi Arabia 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Vietnam 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Australia 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Bolivia 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 1 1 Brunei Darussalam 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Cameroon 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Canada 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Colombia 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 Cote d'Ivoire 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Figure 3: GDP growth and oil dependence (as the threshold variable): regime indicator (R i,t ) values obtained from the estimated three-regime BT P D model.

quality of the institutions has also been shown in a P T R model by [START_REF] Abdulahi | Resource rents, economic growth, and the role of institutional quality: A panel threshold analysis[END_REF].

In practice, our model shows that most of the countries (14 of 19) did not experience a switch in their growth regimes during the period of study. We can classify the countries in our sample into the three regimes as follows. Regime 1: Algeria, Cameroon, Colombia, Côte d'Ivoire, Gabon, Indonesia and Nigeria. Regime 2: Egypt, Oman, Saudi Arabia and Vietnam. Regime 3: Australia, Canada and Norway. However, five countries in our sample did experience a change in their growth regime. The situation for four of them worsened: Bolivia and Ecuador (from regime 2 to regime 1) and Brunei Darussalam and Kuwait (from regime 3 to regime 2). Only Qatar improved its situation (from regime 2 to regime 3). It is worth noting that for most of the period, these countries were in buffer zones (see Figure 2). They can be considered to still be in transition, and their regime shift is not definitive. More generally, it is important to mention that except for the countries in Regime 3, which were very stable during the whole period of study, all the other countries experienced, to different degrees, a transition into the buffer zones.

For Model (II), the three regimes show very different impacts of the quality of institutions on economic growth. In the lower and upper regimes (regimes 1 and 3), the quality of institutions has a positive and significant effect on growth (even though the impact is greater in the upper regime, in which the oil rents are very high). In the middle regime (regime 2, in which the oil rents have intermediate values), the effect of the quality of institutions is negative and significant. This means that enhancing the rule of law damages the economic growth of the countries in this regime. One can assume that some of these countries have somehow experienced an oildependence trap. To the best of our knowledge, only [START_REF] Belarbi | The effects of institutions and natural resources in heterogeneous growth regimes[END_REF] mentioned such a result when considering the quality of institutions and growth in rentier states. In regime 1, we have Australia, Canada and Côte d'Ivoire (see Figure 3). The countries belonging to regime 2 are Algeria, Brunei Darussalam, Cameroon, Ecuador, Egypt, Nigeria and Norway. All the other countries experienced a shift from one regime to another during the period of study. For example, Colombia (in 2011) andVietnam (in 2000) moved from regime 1 to regime 2. In these countries, the quality of institutions exerts a negative impact on economic growth (beginning in the transition year), while their oil rents significantly increased. Other countries experienced back-and-forth changes: between regimes 1 and 2 (Bolivia and Indonesia) and between regimes 2 and 3 (Gabon, Kuwait, Oman, Qatar and Saudi Arabia). It is worth noting that no switch between the extreme regimes 1 and 3 took place during the whole period of study.

Comparison with alternative models

For the sake of comparison, we first provide the same kind of analysis with a P T R model à la [START_REF] Hansen | Threshold effects in non-dynamic panels: Estimation, testing, and inference[END_REF]. Table 6 provides the results of the tests of linearity vs. a one-threshold model and a one-threshold vs. two-threshold model. Our results clearly reject linearity in favour of the existence of a threshold at the 5% level. Moreover, the one-threshold model is rejected in favour of the two-threshold model at the 1% level. Table 7 provides the results of the estimation of a two-threshold [START_REF] Hansen | Threshold effects in non-dynamic panels: Estimation, testing, and inference[END_REF] model with a 0.1% step.

Concerning Model (I), in which the transition variable is the quality of institutions, the three regimes provide results that differ substantially from ours, particularly in terms of the impact of natural resource dependence on economic growth. Indeed, in regimes 1 and 3, the impact is positive and significant, while in regime 2, the impact is negative and significant. However, it is worth noting that countries are very rarely in regime 2 in our sample (only five times during the whole period of study; see Figure 4). Moreover, it appears that there was some switching between the extreme regimes 1 and 3, in contrast to the results of the BT P D model. This is explained by the presence of the buffer zone, which makes the transitions Linearity against a one-threshold model.

Model between the regimes smoother. More generally, it appears that our model allows us to clearly classify countries into three categories, which is not the case with simple threshold regression models. The same kind of results are found for Model (II). Regime 2 is very different from the others but is very rarely occupied (only five times, see Figure 5). Even in threshold regressions with a 1% step, regime 2 is not significant. Our model thus better captures the evolution of the dynamics in the studied sample. This PTR approach has recently been used by [START_REF] Abdulahi | Resource rents, economic growth, and the role of institutional quality: A panel threshold analysis[END_REF] for 14 resource-rich sub-Saharan countries. They have results similar to ours: a three-regime model with a central regime that is also very rarely occupied, and their estimated thresholds are very close. The use of a buffer zone is thus more strongly indicated even for their data. Fundamentally, it is worth noting that the classical P T R model is a special case of our BT P D model that is generated by setting r L,k = r U,k = r k for all k = 1, ..., K -1 (i.e., the buffering regions are absent). Therefore, when we choose the vector γ that minimizes the sum of the squared errors, we implicitly consider P T R models as candidates. Indeed, these two classes of models are nested. Therefore, minimizing the sum of squares allows us to implicitly identify whether it is better to choose our buffered model or the classical threshold model.

We can go further in the interpretation of our results. The minimization of S (γ) indicates that the optimal transition mechanism is the one given by buffered models (see Tables 4 and6). Moreover, we can strengthen our strategy for choosing between the two classes of models by calculating certain selection criteria that are very often used in the literature, such as the AIC and BIC that select models via the optimization of a penalized objective function. It turns out that based on the minimum AIC and BIC and for a fixed K, the buffered models are preferred to the classical threshold models.

As we have already mentioned, another approach to tackling sudden jumps is P ST R modelling (Gonzales et al., 2017). We thus propose a second alternative model for our data using this approach. More precisely, we consider the following model:

y i,t = µ i + X i,t β 0 + r j=1 X i,t β j (1 + exp (-γ j (q i,t -c j ))) -1 + u i,t .
Table 8 provides the results of our tests for the different numbers of regimes. In Model (I), in which the quality of institutions is the thresh- old variable, we do not reject linearity. Therefore, P ST R modelling is not suitable for explaining the observed heterogeneity in our data. Indeed, as our BT P D (and even P T R) models reject linearity, we can say that our approach fits our data better than a P ST R approach. For Model (II), in which the threshold variable is dependence on natural resources, we accept a three-regime P ST R model. Let us now compare the two models: the threeregime BT P D and the three-regime P ST R. The estimation results for the latter are provided in Table 9. In this model, we have two location parameters, c 1 = 3.98 and c 2 = 37.56. They represent the threshold values that indicate breaks in the relationship within the different regimes. However, the interpretation of the parameters is less easy than in the BT P D case. Indeed, in the latter, the beta coefficients correspond to the effect of the exogenous variables in each regime (lower, middle and upper) identified by the threshold values. In the P ST R case, the beta coefficients should be combined with the threshold functions to determine the effect of the exogenous variables on the dependent variable. Roughly speaking, we can determine two extreme regimes with certainty: for very low natural resource dependence, the effect of the exogenous variables is β 0 ; for very high dependence rates, this effect is given by β 0 + β 1 + β 2 . In between, the impact of the exogenous variables varies continuously. In practice, the interpretation of the marginal effect is more interesting (e.g., [START_REF] Belarbi | The effects of institutions and natural resources in heterogeneous growth regimes[END_REF]. BT P D modelling thus offers a more precise classification while giving, at the same time, a smooth transition between the different regimes. The effect of the quality of institutions is highly statistically significant in the three regimes, while only β 2 is significant in the P ST R model. The BT P D approach thus better captures the heterogeneity in GDP growth among the countries in our sample.

Conclusion

In this article, we revisit the question of the relationships among growth, oil dependence and institutions by providing a new approach to address nonlinearities in the panel data framework. Our model is suitable for accounting for the problem of sudden jumps in P T R models. The results show that it is a very useful tool for studying smooth transitions between different regimes. It can also be considered a promising alternative to P ST R modelling and provides results that are more easily interpretable. Indeed, our application shows that, compared with P T R and P ST R models, BT P D model can provide a richer and more precise description of the evolution of economic growth in rentier states while taking into account the interactions between oil dependence and the quality of institutions. In our sample, it is clear that the relationships between growth and dependence on oil rents are not linear: there is a gradual positive impact as the quality of institutions increases. Only oil-dependent countries with high-quality institutions are very stable. All the other countries in our sample have experienced a transition into a buffer zone. They are potentially in a transition between different regimes. Moreover, when considering dependence on oil rents as a threshold variable, it appears that the quality of institutions has a positive and significant effect on growth when oil dependence is low or high. More interestingly, the quality of institutions negatively impacts growth in intermediate oil-dependent countries. Some of these countries experience something of an oil-dependence trap. These results are relatively new in the literature dealing with growth in rentier states. Our BT P D formulation better highlights the impacts of oil dependence and the quality of institutions on economic growth. Indeed, the BT P D model allows us to clearly classify countries into three categories, while this is not the case with classical P T R and P ST R models. Finally, our model offers a wide perspective in terms of applications to different frameworks and could enhance our understanding of the dynamics of the evolutions of various economic phenomena.

  Figure 2 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 Algeria 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Australia 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 Bolivia 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 Brunei Darussalam 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 Cameroon 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Canada 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 Colombia 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Cote d'Ivoire 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Ecuador 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 Egypt 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Gabon 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Indonesia 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Kuwait 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 Nigeria 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1Qatar 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 Norway 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 Oman 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Saudi Arabia 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Vietnam 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Figure 2 :

 2 Figure 2: GDP growth and quality of institutions (as the threshold variable): regime indicator (R i,t ) values obtained from the estimated three-regime BT P D model.

  Figure 3 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 Algeria2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Australia 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Bolivia 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 1 1 Brunei Darussalam 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Cameroon 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Canada 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Colombia 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 Cote d'Ivoire 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Ecuador 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Egypt 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Gabon 2 2 2 2 3 2 2 2 2 3 3 2 3 2 2 2 2 2 2 2 2 2 Indonesia 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 Kuwait 3 3 2 2 3 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 Nigeria 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 22 2 2 2 2 2 Qatar 2 2 2 2 3 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 Norway 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Oman 3 2 2 2 3 3 2 2 3 3 3 3 3 2 2 3 3 3 2 2 2 2 Saudi Arabia 2 2 2 2 3 2 2 3 3 3 3 3 3 2 3 3 3 3 3 2 2 2 Vietnam 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Table 1 :

 1 Sample countries.

	Algeria	Cameroon	Ecuador	Kuwait Qatar
	Australia	Canada	Egypt, Arab Rep Nigeria Saudi Arabia
	Bolivia	Colombia	Gabon	Norway Vietnam
	Brunei Darussalam Côte d'Ivoire Indonesia	Oman

  provides some descriptive statistics of the data used. It appears that the mean growth rate of GDP in our sample decreased between 1996 and 2017. The indicator of the quality of institutions (rule of law) increased

	Variable Description
	GDPG	The growth rate of GDP.
	QINST	Rule of law is a governance indicator developed by the World
		Bank. It includes several indicators that measure the confidence
		in and respect for the laws and rules of society.
		Its value varies between -2.5 and 2.5. A high value indicates
		a favourable institutional environment and a low value indicates
		the opposite.
	DEP	Dependence on natural resources is represented by oil rents as a
		percentage of GDP. Oil rents are the difference between the value
		of crude oil production at world prices and the total costs of
		production.
	INFL	Macroeconomic stability as measured by the inflation rate.
	INVEST Gross fixed capital formation (GFCF) is measured as a percentage
		of GDP.
	OPEN	Trade openness is the sum of exports and imports of goods and
		services relative to GDP.
	POPG	Population growth is measured as the annual rate of population
		growth.

Table 2 :

 2 Description of the variables used.

		Mean	Max	Min	Std
		1996	2017 1996 2017 1996 2017 1996 2017
	GDPG	4.13	2.337 9.34	7.70	0.61	-2.87	2.23	2.55
	QINST	-0.026 0.019 1.92	2.02	-1.44 -1.20	1.00	0.97
	DEP	14.37	8.52 40.96 36.61 0.60	0.21 14.25 9.86
	INFL	7.95	4.16 29.27 29.50 0.496 -0.83	8.84	7.17
	INVEST 23.45 27.26 41.31 48.40 12.11 15.27 8.14	8.77
	OPEN	0.67	0.66 1.206 2.00	0.36	0.23 0.231 0.39
	POPG	1.84	1.89	3.08	4.67	0.51	0.84 0.628 0.90
	Source: Constructed using World Bank and IMF datasets

Table 3 :

 3 Descriptive statistics.

Table 4 :

 4 Results of the tests of the BT P D models.

		Linearity against a 2-regime BT P D model.
		Model (I)	Model (II)
	γ 2	[0.1490, 0.9584]	[0.9941, 6.3392]
	S ( γ 2 )	3503.1316	3760.4068
	F 1,2	55.5199	52.6517
	p-value	0.0020	0.0000
		2-regime BT P D model against a 3-regime BT P D model.
		Model (I)	Model (II)
	γ 3	[-1.0506, 0.0720, 0.1490, 0.9584] [0.9941, 6.3392, 35.9635, 36.6427]
	S ( γ 3 )	3263.9008	3324.4110
	F 2,3	29.2451	52.3288
	p-value	0.0020	0.0000

Table 6 :

 6 Results of the tests of the P T R model.

						(I)		Model (II)
		γ 2 = r 1	0.4079			36.7892
		S ( γ 2 )	3798.6519		3855.9799
		F 1,2	20.1600		41.4572
		p-value	0.0360			0.0000
		One-threshold model against a 2-threshold model.
				Model (I)		Model (II)
		γ 3 = ( r 1 , r 2 ) [0.0757, 0.1037] [27.3327, 28.6872]
		S ( γ 3 )	3376.3984		3441.7750
		F 2,3	52.4932		48.0182
		p-value	0.0000			0.0000
	Model			Model (I)					Model (II)
	Endogenous							GDPG
	variable
	Threshold			QINST					DEP
	variable
		Lower	Middle	Upper	Lower	Middle	Upper
	QINST	-		-			-		0.4499	-99.9189 * * *	2.2351
									(0.5717)	(2.6600)	(1.5584)
	DEP	0.1035	* * *	-4.3509 * * *	0.1884	* * *	-		-	-
		(2.4978)	(3.4777)	(4.0568)
	OPEN	-1.7152	152.2385 * * *	-3.0108	-0.3238	781.1823	* * *	-0.1336
		(1.1812)	(2.5700)		(0.9635)	(0.2492)	(3.0363)	(0.0579)
	INF	-0.0576 * * *	4.3268	* * *	-0.0037	-0.0636 * * *	21.0582 * * *	0.2342
		(2.9157)	(5.6810)		(0.0319)	(3.2244)		(4.0837)	(1.6050)
	INVEST	0.1016	* * *	-3.2426 * * *	0.0731	0.0347	-21.3537 * * *	0.0647
		(2.7473)	(2.5130)		(1.6057)	(1.1076)	(3.0790)	(0.9093)
	POPG	0.6112	50.9949	* * *	0.6168	* * *	0.4377	*	13.2968 * * *	0.4620	* * *
		(0.9315)	(3.4523)	(4.4795)	(1.7740)	(3.6394)	(2.7784)
	γ 3 = ( r 1 , r 2 )			[0.0757, 0.1037]				[27.3327, 28.6872]
	AIC			2.2122					2.2313
	BIC			2.3666					2.3858
	Table 7: GDP growth, quality of institutions and oil dependence (step = 0.1%): estimated
	three-regime P T R model.

Table 8 :

 8 Results of the tests of the P ST R models. The LM k,k+1 statistic is defined as LM k,k+1 = T N (SSR k -SSR k+1 ) /SSR k , where SSR k is the panel sum of squared residuals under H 0 (P ST R model with k regimes) and SSR k+1 is the panel sum of squared residuals under H 1 (P ST R model with (k + 1) regimes).

		Linearity against a 2-regime P ST R model.
			Model (I)	Model (II)
		γ		70.2885			2.0617
		c		0.3944			37.6029
	SSR 2		3938.0969			3987.9135
	LM 1,2		5.5343			26.3876
	p-value		0.3542			0.0001
	2-regime P ST R model against a 3-regime P ST R model.
			Model (I)	Model (II)
	γ = ( γ 1 , γ 2 )			-	(25.8914, 2.1034)
	c = ( c 1 , c 2 )			-	(3.9882, 37.5683)
	SSR 3			-			3463.8230
	LM 2,3			-			42.2983
	p-value			-			0.0000
	Model	Model (I)				Model (II)
	Endogenous					GDPG
	variable
	Threshold	QINST				DEP
	variable
		β 0			β 1 β 2	β 0			β 1	β 2
	QINST				--	-0.7567	-0.7750	8.6892	* * *
						(-0.7515)	(-1.5072)	(3.6096)
	DEP	0.1639	* * *	--	-			-	-
		(5.2576)
	OPEN	-2.7269	*	--	-0.8193	1.2486	3.5827
		(-1.9166)			(-0.7292)	(1.1822)	(1.0384)
	INF	-0.0509 * *	---0.2536	* * *	0.2402	* * *	-0.0595
		(-2.4788)			(-3.7430)		(3.3965)	(-0.2948)
	INVEST	0.0908	* * *	--	0.0882	* *	-0.0747 *	0.1304
		(2.9265)			(2.1498)		(-1.6931)	(0.9377)
	POPG	0.4829	* * *	--	1.1144	* *	-0.2132	-1.6709 * * *
		(3.9210)			(2.5139)		(-0.4643)	(-4.0201)
	γ = ( γ 1 , γ 2 )			-				(25.8914, 2.1034)
	c = ( c 1 , c 2 )			-				(3.9882, 37.5683)
	AIC		2.2922				2.2797
	BIC		2.3404				2.4631

Table 9 :

 9 GDP growth, the quality of institutions and oil dependence: estimated P ST R Model.

Table 10 :

 10 Results of a simulation study for a two-regime BT P D model with ε i,t ∼ N (0, 1) , X i,t ∼ N (0, I 2 ) and q i,t ∼ N (0, 2).

		T	10		20		50		100
	N	True	Mean	Std	Mean	Std	Mean	Std	Mean	Std
	β 1,1 -0.5 -0.5158 0.1554 -0.5053 0.0932 -0.4994 0.0583 -0.4975 0.0405
	β 1,2	0.6	0.6244	0.1512	0.6027	0.0962	0.6004	0.0571	0.6017	0.0396
	β 2,1	0.3	0.3444	0.2263	0.3177	0.1370	0.2987	0.0798	0.2980	0.0541
	10 β 2,2 -0.2 -0.2157 0.2276 -0.2130 0.1387 -0.1980 0.0785 -0.1971 0.0543
	r L,1 0.25	0.1334	0.4536	0.2312	0.2417	0.2464	0.1105	0.2554	0.0583
	r U,1 0.75	0.7656	0.3432	0.7551	0.1700	0.7483	0.0688	0.7494	0.0339
	σ 2	1	0.9183	0.1377	0.9680	0.1038	0.9937	0.0628	0.9968	0.0438
	β 1,1 -0.5 -0.5072 0.0981 -0.4971 0.0636 -0.4994 0.0406 -0.4987 0.0283
	β 1,2	0.6	0.6085	0.1002	0.5999	0.0659	0.5996	0.0408	0.5999	0.0280
	β 2,1	0.3	0.3073	0.1450	0.3002	0.0880	0.3015	0.0568	0.2964	0.0390
	20 β 2,2 -0.2 -0.2214 0.1467 -0.2013 0.0943 -0.1972 0.0573 -0.1987 0.0380
	r L,1 0.25	0.2460	0.2664	0.2493	0.1504	0.2513	0.0598	0.2506	0.0287
	r U,1 0.75	0.7571	0.2040	0.7499	0.0832	0.7486	0.0357	0.7498	0.0213
	σ 2	1	0.9650	0.1036	0.9842	0.0732	0.9970	0.0453	1.0003	0.0320
	β 1,1 -0.5 -0.4996 0.0613 -0.4997 0.0417 -0.4981 0.0257 -0.4977 0.0179
	β 1,2	0.6	0.5988	0.0584	0.6004	0.0419	0.5987	0.0252	0.5981	0.0175
	β 2,1	0.3	0.2938	0.0862	0.2916	0.0591	0.2961	0.0351	0.2952	0.0257
	50 β 2,2 -0.2 -0.2016 0.0880 -0.1960 0.0572 -0.1963 0.0345 -0.1969 0.0249
	r L,1 0.25	0.2690	0.1415	0.2524	0.0751	0.2505	0.0249	0.2500	0.0158
	r U,1 0.75	0.7534	0.0895	0.0645	0.0422	0.7497	0.0195	0.7499	0.0138
	σ 2	1	0.9900	0.0678	0.9988	0.0449	1.0021	0.0278	1.0028	0.0199
	β 1,1 -0.5 -0.4965 0.0411 -0.4970 0.0284 -0.4980 0.0185 -0.4976 0.0125
	β 1,2	0.6	0.5994	0.0421	0.5967	0.0291	0.5975	0.0174	0.5981	0.0127
	β 2,1	0.3	0.2887	0.0603	0.2959	0.0394	0.2950	0.0245	0.2961	0.0181
	100 β 2,2 -0.2 -0.1931 0.0612 -0.1945 0.0376 -0.1953 0.0253 -0.1957 0.0183
	r L,1 0.25	0.2620	0.0914	0.2525	0.0350	0.2495	0.0158	0.2499	0.0129
	r U,1 0.75	0.7505	0.0419	0.7501	0.0240	0.7498	0.0150	0.7497	0.0133
	σ 2	1	1.0017	0.0471	1.0047	0.0330	1.0040	0.0204	1.0034	0.0142

Table 11 :

 11 Results of a simulation study for a two-regime BT P D model with ε i,t ∼ U [-2, 2] , X i,t ∼ N (0, I 5 ) and q i,t ∼ N (0, 2).

		T	10		20		50		100
	N	True	Mean	Std	Mean	Std	Mean	Std	Mean	Std
	β 1,1	0.3	0.3021	0.0453	0.3008	0.0320	0.2991	0.0203	0.3001	0.0140
	β 1,2 -0.2 -0.2003 0.0445 -0.1980 0.0316 -0.1989 0.0192 -0.1999 0.0134
	β 1,3	1.5	1.4976	0.0472	1.4977	0.0320	1.4977	0.0195	1.4989	0.0137
	β 1,4	1.3	1.2883	0.0501	1.2923	0.0317	1.2946	0.0204	1.2954	0.0137
	β 1,5 -0.2 -0.2004 0.0475 -0.2008 0.0313 -0.1999 0.0191 -0.2007 0.0139
	β 2,1	0.5	0.4960	0.0798	0.4962	0.0540	0.4966	0.0317	0.4981	0.0217
	100 β 2,2	0.2	0.1887	0.0774	0.1941	0.0521	0.1967	0.0306	0.1963	0.0226
	β 2,3	0.9	0.9183	0.0785	0.9089	0.0521	0.9054	0.0314	0.9038	0.0222
	β 2,4 -0.8 -0.7502 0.0829 -0.7666 0.0569 -0.7783 0.0363 -0.7825 0.0252
	β 2,5 -0.5 -0.4907 0.0773 -0.4964 0.0539 -0.4964 0.0318 -0.4976 0.0233
	r L,1 0.25	0.2624	0.0512	0.2529	0.0224	0.2503	0.0107	0.2498	0.0087
	r U,1 0.75	0.7505	0.0223	0.7506	0.0137	0.7498	0.0114	0.7493	0.0103
	σ 2	1.33	1.3711	0.0537	1.3620	0.0349	1.3545	0.0198	1.3510	0.0151

Table 11 :

 11 Continued.

		T	10		20		50		100
	N	True	Mean	Std	Mean	Std	Mean	Std	Mean	Std
	β 1,1	0.3	0.3351	0.2801	0.3101	0.1814	0.3067	0.0916	0.3031	0.0591
	β 1,2 -0.2 -0.1205 0.3100 -0.1743 0.1895 -0.1937 0.0971 -0.1932 0.0616
	β 2,1	0.5	0.4980	0.6182	0.5102	0.1978	0.5005	0.0617	0.4965	0.0380
	β 2,2	0.2	0.0744	0.6573	0.1836	0.2236	0.2023	0.0697	0.1970	0.0371
	β 3,1	0.3	0.2909	0.2510	0.2991	0.0936	0.3009	0.0423	0.3012	0.0279
	10 β 3,2 -0.4 -0.4013 0.2327 -0.4049 0.0876 -0.4019 0.0428 -0.3976 0.0288
	r L,1	0.3	0.4915	0.2479	0.3717	0.1539	0.3113	0.0473	0.3021	0.0200
	r U,1	0.5	0.6898	0.3283	0.5591	0.1950	0.5112	0.1015	0.5037	0.0708
	r L,2	0.8	0.9627	0.4588	0.8361	0.2258	0.8029	0.0740	0.8013	0.0264
	r U,2	1	1.3702	0.7407	1.0975	0.3838	1.0079	0.1130	0.9989	0.0338
	σ 2	2	1.7494	0.2777	1.9023	0.1952	1.9825	0.1291	2.0028	0.0941
	β 1,1	0.3	0.3062	0.1772	0.2978	0.1130	0.3058	0.0622	0.3056	0.0415
	β 1,2 -0.2 -0.1582 0.2048 -0.1851 0.1171 -0.1949 0.0616 -0.1937 0.0414
	β 2,1	0.5	0.5055	0.2046	0.5053	0.0725	0.4996	0.0391	0.4973	0.0254
	β 2,2	0.2	0.1654	0.2611	0.2014	0.0812	0.1958	0.0404	0.1933	0.0263
	β 3,1	0.3	0.2979	0.0868	0.3007	0.0485	0.3002	0.0275	0.3012	0.0191
	20 β 3,2 -0.4 -0.4000 0.1132 -0.4008 0.0493 -0.3991 0.0280 -0.3970 0.0204
	r L,1	0.3	0.3907	0.1623	0.3188	0.0593	0.3036	0.0249	0.3021	0.0156
	r U,1	0.5	0.5842	0.2107	0.5097	0.1122	0.5001	0.0739	0.4962	0.0452
	r L,2	0.8	0.8492	0.2269	0.8014	0.0616	0.8042	0.0387	0.8001	0.0205
	r U,2	1	1.1171	0.4104	1.0073	0.1334	1.0007	0.0565	1.0002	0.0249
	σ 2	2	1.8779	0.1899	1.9556	0.1500	2.0205	0.0631	2.0056	0.0611
	β 1,1	0.3	0.3001	0.1040	0.3038	0.0529	0.3075	0.0363	0.3032	0.0293
	β 1,2 -0.2 -0.2095 0.0946 -0.2022 0.0587 -0.1980 0.0359 -0.1874 0.0239
	β 2,1	0.5	0.4972	0.0599	0.4975	0.0381	0.4973	0.0252	0.4953	0.0174
	β 2,2	0.2	0.2042	0.0575	0.1977	0.0391	0.1914	0.0267	0.1898	0.0181
	β 3,1	0.3	0.3004	0.0418	0.3081	0.0273	0.3011	0.0171	0.3005	0.0116
	50 β 3,2 -0.4 -0.3953 0.0492 -0.3974 0.0298 -0.4000 0.0177 -0.3980 0.0116
	r L,1	0.3	0.3114	0.0445	0.2985	0.0180	0.2998	0.0153	0.3027	0.0161
	r U,1	0.5	0.4856	0.0812	0.4962	0.0692	0.4991	0.0387	0.4961	0.0267
	r L,2	0.8	0.7951	0.0394	0.8004	0.0292	0.7998	0.0199	0.8000	0.0189
	r U,2	1	0.9911	0.0577	0.9976	0.0347	0.9993	0.0212	1.0005	0.0196
	σ 2	2	1.9701	0.1322	1.9811	0.1026	2.0239	0.0608	2.0303	0.0394
	β 1,1	0.3	0.2999	0.0539	0.3050	0.0493	0.3098	0.0272	0.3097	0.0217
	β 1,2 -0.2 -0.1907 0.0639 -0.1931 0.0487 -0.1881 0.0292 -0.1831 0.0203
	β 2,1	0.5	0.4992	0.0357	0.4929	0.0251	0.4927	0.0155	0.4954	0.0114
	β 2,2	0.2	0.1993	0.0385	0.1979	0.0259	0.1924	0.0144	0.1912	0.0127
	β 3,1	0.3	0.2984	0.0292	0.3000	0.0202	0.3023	0.0117	0.3008	0.0073
	100 β 3,2 -0.4 -0.3969 0.0268 -0.3973 0.0228 -0.3986 0.0140 -0.3967 0.0096
	r L,1	0.3	0.3014	0.0211	0.3016	0.0195	0.3031	0.0159	0.3060	0.0158
	r U,1	0.5	0.5052	0.0737	0.5012	0.0353	0.4984	0.0222	0.5022	0.0164
	r L,2	0.8	0.8011	0.0260	0.8030	0.0200	0.7987	0.0212	0.8011	0.0200
	r U,2	1	1.0082	0.0394	1.0018	0.0252	0.9986	0.0205	1.0002	0.0166
	σ 2	2	2.0036	0.0904	2.0251	0.0692	2.0331	0.0399	2.0378	0.0298

Table 12 :

 12 -0.7 -0.6928 0.6169 -0.7005 0.2058 -0.6988 0.1081 -0.6954 0.0752 β 3,1 -0.4 -0.4005 0.2695 -0.4002 0.1427 -0.3967 0.0852 -0.3798 0.0597 β 3,2 -0.3 -0.3103 0.2699 -0.2951 0.1493 -0.2982 0.0868 -0.3004 0.0703 -0.5 -0.5024 0.2800 -0.5035 0.1482 -0.4922 0.0871 -0.4864 0.0588 β 3,5 -0.8 -0.8176 0.2673 -0.7993 0.1499 -0.7996 0.0875 -0.8070 0.0591 r L,1 -0.2 -0.3261 0.3386 -0.2104 0.1183 -0.2023 0.0575 -0.1920 0.0365 r -0.7 -0.7032 0.2040 -0.6916 0.1199 -0.6912 0.0761 -0.6893 0.0519 β 3,1 -0.4 -0.4027 0.1621 -0.3989 0.0990 -0.3948 0.0589 -0.3952 0.0418 β 3,2 -0.3 -0.3020 0.1498 -0.3001 0.1000 -0.2983 0.0609 -0.2987 0.0426 -0.5 -0.4977 0.1511 -0.4932 0.0980 -0.4893 0.0634 -0.4888 0.0455 β 3,5 -0.8 -0.7923 0.1426 -0.7987 0.0976 -0.8000 0.0577 -0.7985 0.0427 r L,1 -0.2 -0.2199 0.1308 -0.2028 0.0702 -0.1994 0.0413 -0.1999 0.0333 r

		T	10		20		50		100
	N	True	Mean	Std	Mean	Std	Mean	Std	Mean	Std
	β 1,1 -0.5 -0.5145 0.1988 -0.5122 0.1108 -0.4998 0.0646 -0.5044 0.0478
	β 1,2	0.6	0.6179	0.1899	0.6062	0.1145	0.6012	0.0673	0.5963	0.0450
	β 1,3	1.2	1.2051	0.1992	1.1917	0.1074	1.2026	0.0663	1.2036	0.0434
	β 1,4	1.5	1.5021	0.1869	1.5027	0.1119	1.5011	0.0701	1.5032	0.0480
	β 1,5	0.2	0.2114	0.1945	0.2004	0.1150	0.1988	0.0648	0.1922	0.0459
	β 2,1	0.3	0.2764	0.5906	0.3053	0.1967	0.2904	0.1077	0.2932	0.0723
	β 2,2 -0.2 -0.1877 0.5329 -0.2029 0.1941 -0.1946 0.1078 -0.1890 0.0636
	β 2,3	1.5	1.5075	0.8154	1.4991	0.1918	1.4905	0.1075	1.4853	0.0645
	β 2,4	0.9	0.9473	0.5113	0.8804	0.1985	0.9000	0.1130	0.8963	0.0663
	10 β 2,5 β 3,3	0.9	0.9090	0.2625	0.9052	0.1529	0.8989	0.0850	0.9111	0.0623
	β 3,4 U,1	0.1	0.0700	0.2503	0.1021	0.1049	0.0965	0.0571	0.0953	0.0370
	r L,2	0.6	0.4744	0.2554	0.5554	0.1360	0.5914	0.0711	0.5952	0.0509
	r U,2	0.8	0.8441	0.2732	0.8194	0.1099	0.7993	0.0518	0.8022	0.0440
	σ 2	1	0.7856	0.1295	0.9113	0.0982	0.9857	0.0639	1.0107	0.0457
	β 1,1 -0.5 -0.5085 0.1174 -0.4987 0.0746 -0.4966 0.0488 -0.4958 0.0325
	β 1,2	0.6	0.5997	0.1196	0.5974	0.0735	0.5969	0.0462	0.5946	0.0326
	β 1,3	1.2	1.2010	0.1130	1.1998	0.0795	1.2010	0.0464	1.2011	0.0330
	β 1,4	1.5	1.4981	0.1145	1.4983	0.0747	1.4969	0.0455	1.4971	0.0325
	β 1,5	0.2	0.1999	0.1116	0.1980	0.0772	0.1941	0.0483	0.1955	0.0334
	β 2,1	0.3	0.3175	0.2106	0.2955	0.1271	0.2822	0.0737	0.2784	0.0548
	β 2,2 -0.2 -0.2036 0.2019 -0.1986 0.1207 -0.1912 0.0735 -0.1881 0.0517
	β 2,3	1.5	1.4954	0.2039	1.4944	0.1194	1.4888	0.0749	1.4885	0.0512
	β 2,4	0.9	0.9015	0.2133	0.8987	0.1282	0.8912	0.0759	0.8920	0.0559
	20 β 2,5 β 3,3	0.9	0.8952	0.1543	0.8997	0.1017	0.9013	0.0613	0.9070	0.0424
	β 3,4 U,1	0.1	0.0975	0.1201	0.0991	0.0648	0.0995	0.0378	0.0965	0.0308
	r L,2	0.6	0.5516	0.1420	0.5883	0.0824	0.6000	0.0518	0.6014	0.0396
	r U,2	0.8	0.8152	0.1166	0.8062	0.0602	0.8026	0.0409	0.7989	0.0367
	σ 2	1	0.9103	0.1024	0.9736	0.0720	1.0117	0.0488	1.0220	0.0350

Results of a simulation study for a three-regime BT P D model with ε i,t ∼ N (0, 2) , log (X i,t ) ∼ N (0, I 2 ) and log (q i,t ) ∼ N (0, 2).

Table 13 :

 13 Results of a simulation study for a three-regime BT P D model with ε i,t ∼ N (0, 1) , X i,t ∼ N (0, I 5 ) and q i,t ∼ N (0, 2). -0.7 -0.6903 0.1113 -0.6899 0.0843 -0.6814 0.0489 -0.6852 0.0311β3,1 -0.4 -0.4019 0.0869 -0.4007 0.0612 -0.3995 0.0394 -0.3950 0.0249 β 3,2 -0.3 -0.2962 0.0893 -0.2922 0.0561 -0.2961 0.0397 -0.3010 0.0297 -0.5 -0.4858 0.0862 -0.4896 0.0676 -0.4864 0.0438 -0.4903 0.0309 β 3,5 -0.8 -0.8103 0.0844 -0.7983 0.0578 -0.8072 0.0402 -0.7995 0.0260 r L,1 -0.2 -0.2127 0.0746 -0.1955 0.0383 -0.1959 0.0290 -0.2016 0.0288 r -0.7 -0.6883 0.0839 -0.7034 0.0521 -0.6877 0.0356 -0.6934 0.0235 β 3,1 -0.4 -0.3979 0.0669 -0.3938 0.0387 -0.3928 0.0270 -0.3882 0.0201 β 3,2 -0.3 -0.3031 0.0699 -0.3013 0.0421 -0.2987 0.0243 -0.3010 0.0183 -0.5 -0.4908 0.0610 -0.4918 0.0441 -0.4826 0.0341 -0.4845 0.0263 β 3,5 -0.8 -0.7883 0.0626 -0.7939 0.0436 -0.7995 0.0233 -0.8005 0.0195 r L,1 -0.2 -0.1938 0.0404 -0.1978 0.0321 -0.1997 0.0289 -0.1899 0.0250 r

		T	10		20		50		100
	N	True	Mean	Std	Mean	Std	Mean	Std	Mean	Std
	β 1,1 -0.5 -0.5187 0.0661 -0.4942 0.0477 -0.4940 0.0293 -0.4982 0.0189
	β 1,2	0.6	0.6032	0.0705	0.5993	0.0446	0.5949	0.0315	0.5954	0.0195
	β 1,3	1.2	1.1992	0.0686	1.1982	0.0483	1.2048	0.0290	1.2040	0.0232
	β 1,4	1.5	1.4961	0.0658	1.4964	0.0523	1.4937	0.0324	1.5012	0.0182
	β 1,5	0.2	0.2149	0.0740	0.1987	0.0416	0.1956	0.0323	0.1971	0.0212
	β 2,1	0.3	0.2820	0.1035	0.2908	0.0768	0.2779	0.0485	0.2723	0.0367
	β 2,2 -0.2 -0.1894 0.1131 -0.1877 0.0785 -0.1954 0.0507 -0.1926 0.0351
	β 2,3	1.5	1.4794	0.1140	1.4852	0.0707	1.4835	0.0466	1.4829	0.0316
	β 2,4	0.9	0.9172	0.1141	0.8904	0.0779	0.8851	0.0515	0.8791	0.0427
	50 β 2,5 β 3,3	0.9	0.8996	0.0911	0.8911	0.0631	0.9055	0.0366	0.9033	0.0243
	β 3,4 U,1	0.1	0.0833	0.0595	0.0947	0.0369	0.0998	0.0301	0.0939	0.0270
	r L,2	0.6	0.5825	0.0672	0.6019	0.0554	0.5975	0.0383	0.6067	0.0357
	r U,2	0.8	0.7958	0.0590	0.8021	0.0413	0.8049	0.0388	0.8053	0.0362
	σ 2	1	0.9924	0.0679	1.0151	0.0532	1.0249	0.0283	1.0350	0.0241
	β 1,1 -0.5 -0.4995 0.0476 -0.4956 0.0365 -0.4974 0.0196 -0.4966 0.0152
	β 1,2	0.6	0.6047	0.0536	0.5988	0.0366	0.5959	0.0213	0.5952	0.0164
	β 1,3	1.2	1.2025	0.0461	1.2089	0.0348	1.2000	0.0196	1.2004	0.0148
	β 1,4	1.5	1.5037	0.0440	1.4923	0.0312	1.5009	0.0202	1.4969	0.0160
	β 1,5	0.2	0.2027	0.0478	0.1903	0.0308	0.1960	0.0197	0.1935	0.0141
	β 2,1	0.3	0.2765	0.0786	0.2823	0.0478	0.2786	0.0351	0.2685	0.0274
	β 2,2 -0.2 -0.2053 0.0771 -0.1985 0.0521 -0.1957 0.0341 -0.1926 0.0252
	β 2,3	1.5	1.4926	0.0757	1.4927	0.0516	1.4847	0.0385	1.4808	0.0221
	β 2,4	0.9	0.8907	0.0840	0.8781	0.0576	0.8897	0.0372	0.8757	0.0358
	100 β 2,5 β 3,3	0.9	0.9037	0.0656	0.9089	0.0419	0.9066	0.0284	0.9066	0.0215
	β 3,4 U,1	0.1	0.0949	0.0392	0.0990	0.0353	0.0989	0.0261	0.0910	0.0192
	r L,2	0.6	0.6041	0.0486	0.6012	0.0392	0.6031	0.0371	0.6112	0.0346
	r U,2	0.8	0.8049	0.0422	0.8067	0.0359	0.8034	0.0348	0.8056	0.0332
	σ 2	1	1.0047	0.0479	1.0211	0.0307	1.0331	0.0249	1.0365	0.0179

Table 13 :

 13 Continued.

[START_REF] Lo | On buffered threshold GARCH models[END_REF] used buffered threshold GARCH models to study closing prices, adjusted for dividends and splits.[START_REF] Chen | On double hysteretic heteroskedastic model[END_REF] developed a double hysteretic heteroscedastic model to analyse the relationship between the underlying stock markets and index futures markets.[START_REF] Zhu | Buffered autoregressive models with conditional heteroscedasticity: An application to exchange rates[END_REF] used a buffered autoregressive model with conditional heteroscedasticity to study exchange rates.

Figures2 and 3give the different regimes of the countries according to both models. A number in black means that the country is clearly in a given regime. When the colour is different, the country is in a buffer zone (i.e., in a possible transition to another regime).

Acknowledgements

We would like to thank the Co-Editor Angus Chu, an Associate Editor and two anonymous referees for their very helpful comments and suggestions that highly improved the quality of the paper.

Figure 4: GDP growth and the quality of institutions (as the threshold variable): regime indicator (R i,t ) values obtained from the estimated P T R model.

Figure 5: GDP growth and oil dependence (as the threshold variable): regime indicator (R i,t ) values obtained from the estimated P T R model.