Dan Pilbauer
email: pilbauer@protonmail.com

Delphine Bresch-Pietri

Florent Di Meglio

Christophe Prieur

Tomáš Vyhlídal

Input shaping for infinite dimensional systems with application on oil well drilling

We present an application of the input shaping technique on infinite dimensional systems such as systems described by PDEs. The presented method is a feedforward scheme which allows to target multiple modes of a flexible system and also provide robustness with respect to parameters variations. The method is based on recently developed input shaper with multiple degrees of freedom that are needed to meet all constraints for the given task. We show that, even though the system consists of infinitely many modes, it is required to target only dominant ones. In addition, the method is illustrated on a case example of oil well drilling.

I. INTRODUCTION

Time delay based input shaping is a well known feedforward technique for pre-compensation of oscillatory modes of flexible mechanical systems. A common application is pre-shaping the trajectory of a crane trolley so that the suspended load does not oscillate when moved from one position to another [START_REF] Vaughan | Control of tower cranes with double-pendulum payload dynamics[END_REF]. Another typical applications are flexible manipulators and industrial robots [START_REF] Pereira | Adaptive input shaping for manoeuvring flexible structures using an algebraic identification technique[END_REF], see also an application in orientation and pointing of solar panels of satellites [START_REF] Singhose | Extra-insensitive input shapers for controlling flexible spacecraft[END_REF]. A control scheme with an input shaper is shown in Fig. 1. The original idea of applying an input shapers S(s) in a serial interconnection with a controlled system is to fully or partially compensate its oscillatory modes, determined by one or more couples of complex poles of the flexible structure F (s). From the channel between the shaped system reference w to the output of the flexible structure z, which is not measured as a rule, T zw (s) = C(s)P (s)S(s) 1 + C(s)P (s) F (s) (1)

1 0 ODE ω 0 ODE ω 1 u(x, t) v(x, t) x U (t) K r(t)
-ω 0 (t) e(t) S w(t) P F Fig. 2. Left: A scheme of the drill, Right: a block scheme of the drilling system with feedback controller K and feedforward input shaper S it can clearly be seen that these flexible poles of F (s) can be fully or partially compensated by zeros of the shaper S(s). Thus, the shaper performs the task of a notch filter.

The key advantage of the input shapers compared to the classical notch filters is that via involvement of time delays, it is easier to distribute the response in time. This allows to form the responses of monotonously increasing (or decreasing) characters, which are advantageous with respect to the energy demands. This positive aspect of time delay filters has already been recognized by O. Smith in 1950s, who introduced the so-called posi-cast [START_REF] Smith | Posicast control of damped oscillatory systems[END_REF], also known as zero vibration (ZV) shaper. Consequently, extensive studies mainly motivated by demands on increasing robustness and decreasing response time have been made proposed in [START_REF] Singer | Preshaping command inputs to reduce system vibration[END_REF], [START_REF] Singhose | Residual vibration reduction using vector diagrams to generate shaped inputs[END_REF]. For an extensive review on input shaping see [START_REF] Singhose | Command shaping for flexible systems: A review of the first 50 years[END_REF].

A. Extension to infinite dimensional systems

Even though a single dominant flexible mode can be identified in most of the cases, there are applications where multiple modes need to be compensated, see e.g., [START_REF] Magee | Filtering micro-manipulator wrist commands to prevent flexible base motion[END_REF], [START_REF] Singh | Input-shaped control of three-dimensional maneuvers of flexible spacecraft[END_REF]. The intuitive approach to handle this task is to design a single mode shaper for each of the modes and convolve them by a serial interconnection. However, this leads to a long duration of the shaper action time [START_REF] Singhose | Convolved and simultaneous two-mode input shapers[END_REF]. In order to mitigate this inefficiency, direct methods have been designed to handle this task [START_REF] Hyde | Using input command pre-shaping to suppress multiple mode vibration[END_REF], [START_REF] Díaz | Concurrent design of multimode input shapers and link dynamics for flexible manipulators[END_REF], where the requirement to address multiple modes is performed within a single design procedure. Due to complexity of the design task, where additional constraints on the shaper performance can also be imposed, optimization based approach has been identified as a promising tool to handle it [START_REF] Goubej | Robust motion control of flexible electromechanical systems[END_REF], which will also be utilized in this paper. However, compared to [START_REF] Goubej | Robust motion control of flexible electromechanical systems[END_REF] where the shaper involves many lumped delays, our objective is to optimize a shape of a single delay of distributed nature. This will be done by adapting the recently proposed technique [START_REF] Pilbauer | Distributed delay input shaper design by optimizing smooth kernel functions[END_REF] to the multimode problem, see also [START_REF] Pilbauer | Spectral methods in vibration suppression control systems with time-delays[END_REF] and [START_REF] Vyhlídal | Parameterization of input shapers with delays of various distribution[END_REF].

Despite the fact that the state of the art of the input shapers is mainly focused on finite, low order flexible structures, it has been recently shown that it is possible to apply this technique to infinite order systems with a similar performance. Typical feature of his class of systems covering multi-body mechanical systems, flexible structures described with finite-element models [START_REF] Díaz | Concurrent design of multimode input shapers and link dynamics for flexible manipulators[END_REF], spatially-discretized models of long electrical transmission lines [START_REF] Dhaene | Selection of lumped element models for coupled lossy transmission lines[END_REF], or platoon vehicles [START_REF] Seiler | Disturbance propagation in vehicle strings[END_REF] is a large or even infinite number of oscillatory modes. Thus the potential to apply the multi-mode shaper structure is obvious. Such an attempt has recently been done in [START_REF] Martinec | On zero-vibration signal shapers and a wave-absorbing controller for a chain of multi-agent dynamical systems[END_REF] in the application to multi-agent dynamical systems.

B. Oil drilling

Motivated by the above outlined potential and preliminary results on compensation of flexible modes of infinite dimensional systems by input shaping, and utilizing the recent results on optimization of distributed delay shapers, the objective of this paper is to design a multi-mode shaper to compensate the dominant oscillatory modes of an oil drilling system. The scheme of a standard device is depicted in Fig. 2. It consists of a rotary table at the top, which is usually attached to a drilling rig located on an onshore or offshore platform or on a drilling ship. Next, it consists of a drill string and a bottom hole assembly. Drill sting vibrations and the so-called stick-slip phenomena bring negative effects on the lifespan of the drill string and equipment. The model's complexity is additionally increased due to its internal coupling and infinite spectrum with neutral distribution. In this paper, we propose to apply input shaping to avoid the oscillations of the drill string by modifying the reference trajectory and providing smooth changes.

The paper is organized as follows. In Section II, background and recent results on input shaping are given. Application of the input shaper together with numerical results and simulation experiments are described in Section III. Section IV concludes the presented results and summarizes the contributions.

II. INPUT SHAPER WITH DISTRIBUTED DELAY

A. Problem statement

The scheme of the proposed control is in Fig. 2 where the main goal is to design a feedforward controller (input shaper) S for an infinite dimensional plant P with low-damped oscillatory modes which does not excite modes of flexible structure F . The plant P is considered to be stabilized by controller C. The input shaper S shapes the reference signal w in such a way it provides non-oscillatory response of the flexible structure F while maintaining its fast response. Usually, the price for the shaped response is response time, which will be always longer. To summarize, the objective in this approach design is the choice of parameters of the input shaper which give minimal response time.

B. Input shapers with distributed delay

An input shaper with distributed delay can be described in general form as

r(t) = Aw(t) + (1 -A) T 0 w(t -η)dh(η), (2)
where the parameter A ∈ [0, 1] and the delay distribution h(η) over the finite length segment η ∈ [0, T ] satisfies h(η) = 1, η ≥ T . We can write the transfer function of the distributed delay (2) as

G(s) = A + (1 -A)F (s, T ), (3)
where

F (s, T )w(s) = L T 0 w(t -η)dh(η) (4)
is the transfer function of the delay. The method of input shaping is based on spectral theory, where the idea is to place a couple of dominant complex zeros of the shaper (2) on a location of poles of the flexible structure. As the input shaper is delay-based, its spectrum consists of infinitely many zeros. However, besides the dominant couple the infinitely many zeros of (3) follows the exponential asymptotic curves departing from the stability boundary with increasing amplitude.

In [START_REF] Vyhlídal | Signal shaper with a distributed delay: Spectral analysis and design[END_REF], a zero vibration shaper with equally distributed delay (DZV) was proposed, considering the transfer function of the delay F (s) = 1-e -sT sT . Next to the retarded spectrum, a smoother transition and filtering effect are the main benefits in comparison with classical ZV shaper. On the other hand, the price to pay is the increased response time of the input shaper.

In order to increase robustness, i.e. provide distributed delay alternatives to the classical extra-insensitive (EI), zeroderivative-vibration (DZV) shapers, and additionally a least squares approach for distributed delay shapers have been proposed in [START_REF] Vyhlídal | Parameterization of zero vibration shapers with delays of various distribution[END_REF].

A technique using smooth kernel function was proposed by [START_REF] Pilbauer | Distributed delay input shaper design by optimizing smooth kernel functions[END_REF] where transfer function of the shaper is given by

G(s) = A + T 0 g(θ)e -sθ dθ. (5)
The kernel function g(θ) is chosen as the polynomial g(θ) = Np i=0 a i θ i , where N p ∈ N is the degree of the polynomial and A, a i are gains to be assigned. We can rewrite (5) as

G(s) = A + Np i=0 a i g i (s), (6)
where the functions g i (s) ("moments" of e -sθ ) are given by

g i (s) = T 0 θ i e -sθ dθ. (7)
Note, that the shaper is still linear in parameters A, a i , which significantly simplifies the design procedure. This approach has more degrees of freedom in the design. This flexibility allows the designer to select more constraint in view of satisfying the requirements of the application under consideration.

In order to remove undesirable oscillations, e.g., achieve zero-pole cancellation, we firstly define constraints on placing zeros of the shaper at the expected position of the oscillatory mode to be compensated denoted as

ŝn = -α n ± jβ n ,
where α n , β n ≥ 0, ŝn ∈ C and n ∈ [1, N ] for a given number N ∈ N of oscillatory modes to be compensated. By placing zeros of the shaper, we achieve zero-pole cancellation, which removes entirely the undesired oscillatory modes. These requirements corresponds to

G(ŝ n ) = 0 ⇒ A + Np i=0 a i g i (ŝ n ) = 0, (8)
which can be turned into two real equations for the case of placing a complex zero,

{G(ŝ n )} = 0, (9)
{G(ŝ n )} = 0. (10)
As will be shown for the case of infinite dimensional systems we need to place multiple zeros. However, each additional zero decreases the number of degrees of freedom. In order to arrive at a feasible solution, this may lead to an increase of the time delay of the shaper. Additionally, input shaper (5) requires more constraints that are common for every shaper [START_REF] Pilbauer | Spectral methods in vibration suppression control systems with time-delays[END_REF]. First requirement is that the static gain equals one, which can be written in from

G(0) = 1 ⇒ A + Np i=0 a i g i (0) = 1. (11)
The additional linear equality constraint that might be necessary corresponds to the requirement of continuity of both the step response and its derivatives at times t = 0 or (and) t = T . At t = 0 it is expressed by

A = 0, g(0) = a 0 = 0, (12)
and for t = T by

g(T ) = Np i=0 a i T i = 0. (13)
The next fundamental requirement that needs to be considered is the non-decreasing step response [START_REF] Singhose | Residual vibration reduction using vector diagrams to generate shaped inputs[END_REF], [START_REF] Singer | Preshaping command inputs to reduce system vibration[END_REF], or equivalently, the non-negative impulse response, which can be formulated as 26)- [START_REF] Vyhlídal | Qpmr -quasi-polynomial rootfinder: algorithm and examples[END_REF]. Blue circles: spectrum of flexible part of the system F , where the dynamics of ODE ω 0 at the top is not considered Condition ( 14) is a semi-infinite polynomial inequality, i.e., a requirement to be satisfied for a continuum of α values. The requirement can be solved via Pólya's relaxation (for more details see [START_REF] Pilbauer | Distributed delay input shaper design by optimizing smooth kernel functions[END_REF]). Optionally, next constraints can be added, such as continuity of both the step response and its derivatives or limiting the jerk and jounce, which would be important in case where additional requirements on the system are needed, e.g., less wear on mechanical parts or smoother run of motors. Since this is not the main aim of this paper we omit details, and refer the interested readers to [START_REF] Pilbauer | Distributed delay input shaper design by optimizing smooth kernel functions[END_REF].

g(α) ≥ 0, ∀α ∈ [0, T ]. (14)
We may now define an optimization problem as

min x T, subject to A 1 (T )x = b 1 (T ), A 2 (T )x ≥ b 2 (T ). (15)
where gain vector x = [A a 0 . . . a Np ] and equality constraints A 1 (T )x = b 1 (T ) come from ( 9)-( 12) and inequality constraints are given by [START_REF] Pilbauer | Distributed delay input shaper design by optimizing smooth kernel functions[END_REF]. The dependance of the matrices A 1 , A 2 on the parameter T is here solved by fixing the total delay length and looking for a feasible solution. If the solution is not found, the total time or the number of parameters N p is increased.

III. APPLICATION TO DRILLING

In order to show applicability of the proposed method, a case example is provided in the following. We apply the method on an oil drilling model. Avoiding oscillations of such systems is crucial because they can cause damage to the equipment and increase the non-productive time. A degenerate version of this behavior is often called stick-slip, an highly undesirable oscillatory limit cycle.

A. System description

We use a model similar to [START_REF] Aarsnes | Linear stability analysis of selfexcited vibrations in drilling using an infinite dimensional model[END_REF] where we consider dynamics of the drive at the top of the rig [START_REF] Navarro-López | Modelling and analysis of stickslip behaviour in a drillstring under dry friction[END_REF], [START_REF] Canudas-De-Wit | A new model for control of systems with friction[END_REF], corresponding to the ODE governing ω 0 . The model also encompasses the linearized dynamics of the interaction of the bit with the surface at the bottom of the well, corresponding to the ODE Step change in reference velocity ω 0 without input shaper (blue) and with input shaper (red), Bottom: Response of the velocity at the bottom ω 1 to the step response of the reference to the top velocity ω 0

governing ω 1 , and the drill itself as a distributed system described by two PDEs. It writes

ω0 (t) = -a 0 ω 0 (t) + b 0 v(0, t) + U (t), (16) u
(0, t) = q 0 v(0, t) + c 0 ω 0 (t), ( 17
) u t (x, t) = -λu x (x, t), ( 18
) v t (x, t) = µv x (x, t), (19) v(1, t) = q 1 u(1, t) + c 1 ω 1 (t), ( 20
) ω1 (t) = -a 1 ω 1 (t) + b 1 u(1, t), (21)
where ω 0 , ω 1 are velocities of the drill at the top and at the bottom, respectively. u(x, t), v(x, t) ∈ [0, ∞) × [0, 1] are describing propagation of the velocity through the drill with parameters λ and µ denoting speed of propagation through the drill. Parameters a 0 , a 1 , b 0 , b 1 , c 0 , c 1 , q 0 , q 1 are constants of the system. The ODE-PDE-ODE system ( 16)-( 21) can be transformed into delay differential equations (DDEs) as follows. We first notice that from ( 17)-( 19)

u(1, t) =q 0 v (0, t -τ 1 ) + c 0 ω 0 (t -τ 1 ) = q 0 v (1, t -τ 3 ) + c 0 ω 0 (t -τ 1 ) , (22)
where τ 1 = 1/λ, τ 2 = 1/µ and τ 3 = 1/λ + 1/µ. Then, by substituting v (1, tτ 3 ) by [START_REF] Vyhlídal | Signal shaper with a distributed delay: Spectral analysis and design[END_REF], shifted by time tτ 3 , we can write

u(1, t) = q 0 (q 1 u (1, t -τ 3 ) + c 1 ω 1 (t -τ 3 )) +c 0 ω 0 (t -τ 1 ) . ( 23
)
Again, we can substitute u (1, tτ 3 ) by ( 21), shifted by time tτ 3 . Following a few algebraic operations and Step change in reference velocity ω 0 without input shaper (blue) and with input shaper (red), Bottom: Response of the velocity at the bottom ω 1 to the step response of the reference to the top velocity ω 0 plugging the result into (21), the final DDE is

ω1 (t) -q 0 q 1 ω1 (t -τ 3 ) = -a 1 ω 1 (t)+ (q 0 q 1 a 1 + b 1 c 1 q 0 ) ω 1 (t -τ 3 ) + b 1 c 0 ω 0 (t -τ 1 ) (24)
Following the same procedure for the second ODE of the variable ω 0 , we obtain ω0 (t)q 0 q 1 ω0 (tτ 3 ) = a 0 ω 0 (t) + (q 0 q 1 a 0 + b 0 c 0 q 1 ) ω 0 (tτ 3 )

+ b 0 c 1 ω 1 (t -τ 2 ) + U (t) -q 0 q 1 U (t -τ 3 ) (25)
If we define a new state vector as Ω = [ω 1 ω 0 ] T , the system in the matrix form is

Ω(t) + -q0q1 0 0 -q0q1 Ω(t -τ3) = -a1 0 0 -a0 Ω(t) + q0q1a1 + b1c1q0 0 0 q0q1a0 + b0c0q1 Ω (t -τ3) + 0 b1c0 0 0 Ω (t -τ1) + 0 0 b0c1 Ω (t -τ2) + 0 1 U (t) + 0 -q0q1 U (t -τ3) (26)
The system is commonly controlled via PI or PID controller.

Here, we compare the results with a PI controller tuned to achieve fast tracking at the top velocity, which is the case in the field. The controller is in the form

K : U (t) = kP (r(t) -ω0(t)) + kI t 0 (r(ξ) -ω0(ξ)) dξ (27)
We can check the spectrum of the closed loop ( 26)-( 27) which is given in Fig. 3 (red dots). However, for application of input shaping we must mention that the targeted system (flexible structure) corresponds only to the part F shown in Fig. 2 bounded by dashed box. The zeros of the shaper Step change in reference velocity ω 0 without input shaper (blue) and with shaper (red), Bottom: Response of the velocity at the bottom ω 1 to the step response of the reference to the top velocity ω 0 should target only the poles of the flexible structure and not the overall system. The poles of the flexible part of the system is shown in Fig. 3 (blue circles). Spectra of the systems have been computed by QPmR algorithm [START_REF] Vyhlídal | Qpmr -quasi-polynomial rootfinder: algorithm and examples[END_REF].

B. Simulation results

This section presents simulation results for the following set of parameters: c 0 = c 1 = 2, q 0 = q 1 = -0.9, a 0 = 0.17,

a 1 = 1.8, λ = µ = 1, b 0 = a 0 , b 1 = a 1 .
Four rightmost poles of the flexible structure are shown in Tab. I. Firstly, we design the input shaper with one zero compensating the rightmost pole of the flexible system. The result is shown in Fig. 4. The response is significantly slower than without the input shaper but the residual vibrations after the response time are much smaller. However, there are still significant residual oscillations after the response time which is due to the fact that the other rightmost poles of the flexible may play a role. If we now place two poles, as shown in Fig. 5, it is easy to see that the response is much smoother and with almost no visible residual vibrations. If we go further, we can place three or four zeros of the shaper, see Figs. 6 and7. If we place three the response is totally smooth without any oscillation after the response time. If we place four zeros, we can observe that the response is very similar. Step change in reference velocity ω 0 without input shaper (blue) and with input shaper (red), Bottom: Response of the velocity at the bottom ω 1 to the step response of the reference to the top velocity ω 0

The results can be also confirmed by looking at the so called residual vibration function, i.e., amplitude of the oscillation after the response time T . However, this function has been always evaluated for cases where we target only one pole of the flexible structure [START_REF] Pilbauer | Distributed delay input shaper design by optimizing smooth kernel functions[END_REF], [START_REF] Vyhlídal | Parameterization of zero vibration shapers with delays of various distribution[END_REF], [START_REF] Singhose | Extra-insensitive input shapers for controlling flexible spacecraft[END_REF]. Therefore, we need to modify it for multimode input shaper. We can express this function by substituting complex variable s = ω and using ( 5)- [START_REF] Singhose | Command shaping for flexible systems: A review of the first 50 years[END_REF]. Then we can write

V (ω) = |G(ω)|e ωT ( 28
)
where ω is defined as continuous function crossing each zero of the input shaper that has been placed. This is visually shown in Fig. 8, where points are connected with lines. The starting point is at the origin of the complex plane ω(0) = 0 + j0 and the last point goes to infinity with constant real part given by the last placed zero. Fig. 9 shows how the residual vibration function changes for shapers with one, two, three and four zeros placed exactly on the desired locations. As can be seen, when only one zero is placed, the amplitude for the second, third and fourth poles of the flexible structure is still significant. When two zeros are placed, the residual vibrations decreases much faster with increasing ω. Placing more zeros improves performance even more.

IV. CONCLUSIONS

As the main contribution, the input shaper with distributed delay is applied to an infinite dimensional system. The input shaper is designed in such a way that it compensates multiple dominant poles of the flexible structure while maintaining fast response. The proposed design of the parameters of the shaper is based on constrained optimization and solved We illustrated that it is not necessary to target all of the modes of the flexible structure to obtain a non-oscillatory response but it is necessary to target more poles than the dominant ones. Moreover, as the real part of shaper's zeros tends to go to minus infinity with increasing moduli, it appears it will not excite any high frequency modes, even though they are not directly targeted.

Fig. 1 .

 1 Fig. 1. The classical feedforward application of input shaper

Fig. 3 .

 3 Fig.3. Red dots: Spectrum of the closed loop system (26)-[START_REF] Vyhlídal | Qpmr -quasi-polynomial rootfinder: algorithm and examples[END_REF]. Blue circles: spectrum of flexible part of the system F , where the dynamics of ODE ω 0 at the top is not considered

Fig. 4 .

 4 Fig. 4. Top: Spectrum of the shaper with one exact zero placed, Middle:Step change in reference velocity ω 0 without input shaper (blue) and with input shaper (red), Bottom: Response of the velocity at the bottom ω 1 to the step response of the reference to the top velocity ω 0

Fig. 5 .

 5 Fig. 5. Top: Spectrum of the shaper with two exact zeros placed, Middle:Step change in reference velocity ω 0 without input shaper (blue) and with input shaper (red), Bottom: Response of the velocity at the bottom ω 1 to the step response of the reference to the top velocity ω 0

Fig. 6 .

 6 Fig. 6. Top: Spectrum of the shaper with three exact zeros placed, Middle:Step change in reference velocity ω 0 without input shaper (blue) and with shaper (red), Bottom: Response of the velocity at the bottom ω 1 to the step response of the reference to the top velocity ω 0

Fig. 7 .

 7 Fig. 7. Top: Spectrum of the shaper with four exact zeros placed, Middle:Step change in reference velocity ω 0 without input shaper (blue) and with input shaper (red), Bottom: Response of the velocity at the bottom ω 1 to the step response of the reference to the top velocity ω 0

Fig. 8 .Fig. 9 .

 89 Fig. 8. Spectrum of an input shaper with four placed zeros (black dots) and a line representing values of ω where the residual function is evaluated

*This work was supported by ANR project MACS-Drill ANR-15-CE23-0008-01 and Czech Science Foundation under the project 16-17398S


Yonas Alem
email: yonas.alem@economics.gu.se.

Mathilde Maurel
email: maurel@univ-paris1.fr.

Katrin Millock
email: millock@univ-paris1.fr.

Migration as an Adaptation Strategy to Weather Variability: An Instrumental Variables Probit Analysis

Keywords: climate change, drought, Ethiopia, household survey, migration, rainfall. JEL: O15, Q54, R23

There is solid scientific evidence predicting that a large part of the developing world will suffer a greater incidence of extreme weather events, which may increase the incidence of displacement migration. We draw on the new economics of migration to model migration decisions of smallholder and rain-dependent farm households in rural Ethiopia and investigate both the ex-ante and ex-post impacts of climate variables. Using detailed household survey panel data matched with rainfall data, we show that weather variability -measured by the coefficient of variation of rainfall -has a strong positive impact on the probability of sending a migrant. This implies that households engage in migration to cope with risk ex-ante. We also find evidence suggesting that rainfall shocks have ex-post impact on households' likelihood of migration, but the effect is not statistically significant at the conventional levels. Instrumental variables probit regression results also show that controlling for endogeneity of income using a credible instrument is important to identify its impact on the decision to send a migrant. Our findings have important implications for policies aiming to improve the capacity of vulnerable households to adapt to climate change.

Introduction

Global environmental threats, in particular climate change, have been identified as potential causes of large migration flows in the future (UNFPA, 2009; IPCC, 2014). Although the climate is a global good, available projections indicate that climate change damages will differ significantly according to region and will affect developing countries more than developed countries (IPCC, 2014). There is now a large degree of certainty that parts of the developing world will suffer a greater incidence of extreme weather events, which may increase the necessity of displacement migration [START_REF] Hulme | African Climate Change: 1900-2100[END_REF]. There are several reasons for this: vulnerability of coastal zones, greater dependence on agriculture, and less adaptive capacity [START_REF] Tol | Distributional Aspects of Climate Change Impacts[END_REF]Mendelsohn et al., 2006). The term "environmental refugees" was popularized by Myers (see for example [START_REF] Myers | Environmental Refugees[END_REF], or Bates, 2002) 1 . The term has since been criticized, in particular because what matters in predicting future migration flows is ultimately identifying vulnerable populations, which is a result of the joint influence of household characteristics, social networks, access to infrastructure, political environment, etc. [START_REF] Raleigh | Assessing the Impact of Climate Change on Migration and Conflict[END_REF][START_REF] Black | Migration as Adaptation[END_REF]. More recent credible reports, such as the Stern Report (2007), warn that, by 2050, 200 million people could be at risk of climate events that may induce migration.

Certainly, environmental conditions have always influenced habitat, and increased stress on natural resources constitutes an important factor in migration. Some of the most prevalent examples are drought and increased water scarcity and soil erosion, which lead farmers to move away from arid land. Specific types of migration form an integral part of human adaptation to changes in the environment: seasonal or circular migration have very different drivers than displacement migration following a disaster [START_REF] Adger | Social and Ecological Resilience: Are They Related?[END_REF][START_REF] Perch-Nielsen | Exploring the Link between Climate Change and Migration[END_REF]. The most recent literature studies climate change and uses weather factors that are strictly exogenous. In this paper, we investigate both the ex-ante and ex-post impacts of climate variables on the decision to engage in migration by smallholder farm households in Ethiopia. We use the coefficient of variation of rainfall and objectively defined rainfall shock variables to test our hypotheses on the relationship between weather and migration.

Rural Ethiopia provides a valuable setting to investigate our hypothesis. About 82% of its population depends on rain-fed agriculture, access to irrigation is insignificant (CIA, 2015), and the country has been affected by frequent climatic shocks in recent decades [START_REF] Dercon | Growth and shocks: evidence from rural Ethiopia[END_REF]. Future projections show a significant impact on African migration flows, in particular, following climate change-induced water stress [START_REF] Le Blanc | The Relationship between Rainfall and Human Density and its Implications for Future Water Stress in Sub-Saharan Africa[END_REF]. According to the IPCC's 5th Assessment Report, Sub-Saharan Africa is the region with the highest exposure to drought in the world, in terms of the share of the exposed population [START_REF] Niang | Africa. In: Climate Change 2014: Impacts, Adaptation, and Vulnerability. Part B: Regional Aspects. Contribution of Working Group II to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change[END_REF]. In this regard, our analysis provides important insights on the possible impacts of climate change on migration flows.

There is a small but growing literature that investigates the link between climate change and migration. [START_REF] Barrios | Climatic Change and Rural-Urban Migration: The Case of Sub-Saharan Africa[END_REF], one of the first studies measured the impact of rainfall on the distribution of population among rural and urban areas in Sub-Saharan Africa, and found a significant effect of rainfall patterns on the population distribution. Some recent studies analyze macro-level data on international migration flows to examine the effect of rainfall and temperature on out-migration from Sub-Saharan Africa [START_REF] Marchiori | The impact of weather anomalies on migration in sub-Saharan Africa[END_REF], and on in-migration to OECD countries from developing countries [START_REF] Coniglio | Climate Variability and International Migration: An Empirical Analysis[END_REF]. Using yearly migration data from the OECD, Coniglio and Pesce (2015) find that inter-annual rainfall variability increases out-migration from developing countries to OECD countries, especially from countries with large agricultural sectors. [START_REF] Beine | Climatic Factors as Determinants of International Migration[END_REF] include both climate factors and natural disasters in an analysis of international migration flows derived from 10-year interval migrant stock data from the World Bank from 1960 to 2000. On that standardized data, which also include South-South migration, they find no statistically significant long-run average effects of neither climate factors nor natural disasters on international migration flows. They find a statistically significant effect only of natural disasters on internal migration flows, proxied by the rate of urbanization. This result seems compatible with the argument of [START_REF] Marchiori | The impact of weather anomalies on migration in sub-Saharan Africa[END_REF], who find that rainfall and temperature anomalies first increase migration from rural to urban areas, and then, in a second step, also increase international out-migration from Sub-Saharan Africa over the period 1960-2000, depending on the size of urban agglomeration externalities. Recently, [START_REF] Maurel | Climate Instability, Urbanisation and International Migration[END_REF] also find a similar effect on international bilateral migration, using the same standardized data as in [START_REF] Beine | Climatic Factors as Determinants of International Migration[END_REF].

Our paper uses household data to analyze the migration decision in the "new economics of migration" framework [START_REF] Stark | The New Economics of Labour Migration[END_REF][START_REF] Stark | The Migration of Labour[END_REF]. While a study from one country hardly can be used to infer more general conclusions, compared to the studies on international migration flows, the use of detailed household data enables us to disentangle the many factors that interact to explain migration. Early evidence of environmentally induced migration are found in geographical studies from Burkina Faso (Henry et al. 2004a(Henry et al. , 2004b)), Ethiopia [START_REF] Ezra | Rural Out-Migration in the Drought Prone Areas of Ethiopia: A Multi-Level Analysis[END_REF][START_REF] Meze-Hausken | Migration Caused by Climate Change: How Vulnerable are People in Dryland Areas? Mitigation and Adaptation Strategies for[END_REF] and Mali [START_REF] Findley | Does Drought Increase Migration? A Study of Migration from Rural Mali during the 1983-1985 Drought[END_REF]. Most of these studies use proxies for climate and its consequences -vulnerability to famines, for example, according to an NGO assessment in the Ezra and Kiros (2001) study on Tigray, Northern Ethiopia. Very few of them can therefore separate the impact of household and community characteristics from the possibility of exogenous environmental push factors, such as rainfall variability and rainfall shocks. Our key contribution is thus to assess both the ex-ante and the ex-post impacts of one of the most relevant climate variables for Sub-Saharan Africa -rainfall -in addition to household socio-economic characteristics, based on detailed and representative household panel data from Ethiopia.

Household data have been used in studies on migration and climate factors on Ecuador [START_REF] Gray | Environment, land, and rural out-migration in the Southern Ecuadorian Andes[END_REF], Nigeria [START_REF] Dillon | Migratory Responses to Agricultural Risk in Northern Nigeria[END_REF], Malawi [START_REF] Lewin | Do Rainfall Conditions Push or Pull Rural Migrants: Evidence from Malawi[END_REF], Indonesia (Bohra-Mishra et al., 2014), Bangladesh (Gray and Mueller, 2012b) and Ethiopia (Gray and Mueller, 2012a), among others (see the survey in [START_REF] Millock | Migration and Environment[END_REF]. Among the few of these studies to use actual rainfall data, Bohra-Mishra et al. (2014) find a significant positive but small effect from rainfall on household permanent migration in Indonesia (and a larger significant effect of temperature); both effects are nonlinear. Below a certain threshold, an increase in rainfall reduces out-migration, whereas an increase in rainfall above that threshold increases out-migration. In an analysis of Malawi, a very poor country, [START_REF] Lewin | Do Rainfall Conditions Push or Pull Rural Migrants: Evidence from Malawi[END_REF] find that rainfall shocks lead to a lower probability of migration. This result is consistent with the hypothesis that severe weather shocks reduce a household's income and stock of capital so much that the household does not have the resources necessary to migrate. 2The paper that is the most closely related to our paper is Gray and Mueller (2012a). Using the Ethiopian Rural Household Survey (ERHS), which we use in the current paper, these authors investigate the effects of drought on farm households' mobility and document that men migrate more than women following drought. Gray and Mueller (2012a) model migration as an individual decision and use subjective indicators of exposure to drought reported by households. In this paper, we follow the new economics of migration [START_REF] Stark | The Migration of Labour[END_REF] and model migration as a household decision. We also use objectively measured rainfall data to define our key variables of interest: weather variability, as measured by the coefficient of variation of rainfall, and rainfall shocks, constructed as a one standard deviation from the long-term mean two years before migration. This is the second contribution of the current study, since it is plausible to expect that reported shocks would be correlated with household characteristics that determine the migration decision. Our study also extends the analysis of Gray and Mueller (2012a) by taking into account endogeneity of income -an important determinant of migration -using an instrumental variables estimator and the level of rainfall as an instrument.

Instrumental variables probit regression results suggest that the likelihood of sending a migrant increases with rainfall variability measured by the coefficient of variation. This can be interpreted as an ex-ante measure of the riskiness of a household's environment. 3 Results also provide suggestive evidence that households respond to rainfall shocks by sending a migrant, but this effect is not statistically significant at the conventional levels. We also show that controlling for endogeneity of income is important to clearly identify its role in migration. Our findings have important implications for possible future impacts of climate change related to migration and households' adaptation strategies.

The remaining part of the paper is organized as follows: Section 2 describes the relevance of studying the case of Ethiopia. The conceptual framework that defines our hypotheses is outlined in Section 3. The data and the econometric strategy are described in Section 4, and the variables in Section 5. The estimation results are presented and discussed in section 6. Section 7 concludes.

Rainfall and agricultural income in Ethiopia

We test the hypothesis that migration is induced by exogenous environmental factors, such as rainfall variability, in a predominantly agrarian country. It is particularly relevant to focus on Africa and especially on Ethiopia since climate change has been shown to affect the less developed countries more, in particular those that are exposed to severe water stress, and where the low level of adaptive capacity may lead to environmentally induced migration [START_REF] Black | Demographics and Climate Change: Future Trends And their Policy Implications for Migration[END_REF]. Ethiopia is one Sub-Saharan African country that has been classified as extremely vulnerable to drought and other natural disasters such as floods, heavy rains, frost and heat waves. These extreme weather events cause loss of lives and property and disrupt livelihoods. A large proportion of the population of Ethiopia are heavily dependent on rain-fed agriculture, which is affected by climate change [START_REF] Kurukulasuriya | Will African Agriculture Survive Climate Change?[END_REF]. According to the World Bank development indicators, less than 0.5 % of agricultural lands in Ethiopia were irrigated in the period 2001-2005. The livelihood of 85 % of the population depends on agriculture, which also constitutes approximately 40% of the GDP and 60% of export earnings (CIA, 2015).

Using a Ricardian approach of measuring the impacts of climate change, [START_REF] Deressa | Economic Impact of climate change on crop production in Ethiopia: Evidence from cross-section measures[END_REF] show significant negative marginal impacts of increases in temperature and decreases in rainfall on crop revenue per hectare. Their projections of damages up to 2050 and 2100 calls into question the very survival of the Ethiopian agricultural sector. In the face of such serious impacts, it is natural to expect that farmers use migration as one coping strategy, especially as alternative means of insurance are limited. Agricultural adaptation measures, such as planting trees, have been shown to reduce damages to some extent (Di Falco et al., 2012), but may not be sufficient to compensate for the expected losses. Also, the most vulnerable households are the least likely to undertake adaptation measures (Di Falco et al., 2011). Although higher rainfall levels in the preceding season increase yield and income, and thus relax credit constraints that may limit farmers' use of productivity-enhancing inputs, the effect of rainfall variability as such is different. [START_REF] Alem | Do Fertilizer Use Respond to Rainfall Variability? Panel Data Evidence from Ethiopia[END_REF] show that rainfall variability, a credible proxy for uninsured risk, decreases both the probability to apply fertilizer and the amount applied by farmers in their sample from the Ethiopian highlands. The risk induced by rainfall variability has also been shown to directly affect farmers' wellbeing [START_REF] Alem | Consumption Smoothing and the Welfare Cost of Uncertainty[END_REF]. The risk of a negative rainfall shock and its impact on consumption discourages households' adoption of a risky input such as fertilizer and can leave households in poverty traps [START_REF] Dercon | Consumption Risk, Technology Adoption and Poverty Traps: Evidence from Ethiopia[END_REF]. Poverty and market imperfections also reduce households' investments in soil quality [START_REF] Shiferaw | Soil Erosion and Smallholders' Conservation Decisions in the Highlands of Ethiopia[END_REF]. The existing evidence thus underscores the importance of rainfall variability in Ethiopia and the fact that most vulnerable smallholder households do not have the means to adapt to climate change in the most effective manner. These households may thus be trapped in poverty, unless they have some means to cope with climate variability such as sending a member as a migrant.

Conceptual Framework

We view migration as a risk-coping strategy of the household, again following the new economics of migration [START_REF] Stark | The New Economics of Labour Migration[END_REF][START_REF] Stark | The Migration of Labour[END_REF]. A household bases its migration choice on the expected utility of consumption with migration compared to no migration. The impact of rainfall variability is an indirect effect, in that it affects agricultural production, and hence consumption, should the household not be able to smooth its consumption through other strategies, such as drawing down on assets, or searching for employment in the non-agricultural sector, for example.

Drought is indeed the most commonly cited shock in rural Ethiopia and harvest failure is the most cited cause of hardship [START_REF] Dercon | Vulnerability, Seasonability and Poverty in Ethiopia[END_REF]. Using a subsample from the ERHS data from 1989 to 1997, [START_REF] Dercon | Growth and shocks: evidence from rural Ethiopia[END_REF] finds that a 10% decline in rainfall reduces food consumption by about 5% and that the effect lingers on several years afterwards. In a subsequent analysis, [START_REF] Porter | Shocks, Consumption and Income Diversification in Rural Ethiopia[END_REF] analyzes the complete ERHS survey data beyond 1997 and find that extreme rainfall realizations drive the results. Rainfall in the lowest quintile can reduce consumption by 10 to 20%. Alternative strategies, such as diversification from non-agricultural income, does not succeed in reducing the losses from particularly bad rainfall shocks, whereas idiosyncratic shocks can be insured through informal mechanisms.

The migration decision depends upon household characteristics (family size, highest education level in the household, asset holdings) and the costs of migration (in terms of distance to major destinations, or road access). Agricultural income is endogenous in the year of migration and we will use an instrumental variables approach to instrument it in the migration equation.

M ante ij,t-1 = f (F ij,t-1 , C j,t-1 , RCV j,t-1 ) (1)
Ex-post, after observing the current year's (season's) 4 rainfall outcome and its impact on the harvest, and thus on agricultural income (Y AG ), the household may decide to send a member away, but this ex-post decision will be based on the actual rainfall level and the shock it represents compared to the expected (mean) rainfall:

M post ij,t = f (F ij,t , Y AG ij,t , C j,t , SHOCK j,t ) (2)
Taken together, the final migration observed in household i in PA j in year t is a function of both rainfall variability and the rainfall shock:

M ijt = f (F ij,t , Y AG ij,t , C j,t , RCV j,t-1 , SHOCK j,t ) (3)
Note that the final specification includes controls for other coping strategies that are available to the household. After a rainfall shock, households may draw down on assets (Fafchamps et In order to investigate the role of migration as an adaption to weather variability, we use the 1999 and 2004 waves of the ERHS. Attrition in the panel has been low, at 1-2 percent of households per year [START_REF] Dercon | The Ethiopian Rural Household Surveys 1989-2009: Introduction[END_REF]. The ERHS documents detailed information on individual and household characteristics, assets, expenditures, consumption, health, agricultural production and use of agricultural inputs.

In addition to the ERHS panel data, annual rainfall data have been collected from the Ethiopian meteorology agency for the weather stations nearest to each village. 7 The rainfall data have been matched with the ERHS villages using the geo codes of the villages to compute our key weather variables.

Empirical Strategy

We use a random effects probit framework to model the decision by households in rural Ethiopia to send at least one household member as a migrant in response to weather variability. Let the latent model of migration be specified as

m * it = x it β + ε it i = 1, 2, ..., N ; t = 1, ..., T (4)
ε it = c i + u it (5)
where m * it is a latent dependent variable; m it is the observed binary outcome variable defined as

m it =    1 if m * it > 0; 0, otherwise. (6)
x it represents a vector of time-varying and time-invariant variables which influence m * ; β represents a vector of parameters to be estimated; and ε it is a composite error term which can be decomposed into c i , a term capturing unobserved individual (household in our case) heterogeneity, and u it ∼ IN (0, σ 2 u ), a random error term. The subscripts i and t refer to households and time periods respectively. One can marginalize the likelihood function by assuming that, conditional on the x it , the unobserved individual heterogeneity term c i ∼ IN (0, σ 2 c ), is independent of the x it s and u it . 8 Assuming that the distribution of the latent variable m * , conditioned on c i , is independent normal [START_REF] Heckman | The incidental parameters problem and the problem of initial conditions in estimating a discrete time-discrete data stochastic process[END_REF], the vector of parameters, i.e., the βs can be estimated easily. Thus,

P r(m it = 1|c i , x it ) = P r( u it σ u > -x it β -c i σ u ) = Φ(v it ) (7)
where

v it = -(x it β + c i )/σ u (8)
8 A straightforward implication of this assumption is that the equicorrelation between the composite error terms in two successive periods for household i is constant and given by: λ = corr(εit, εit-1) =

σ 2 c σ 2 c +σ 2 u .
and Φ is the distribution function of the standard normal variatevariable. Consequently, the likelihood function to be maximized (which is marginalized with respect to c) is given by

i { ∞ -∞ T t=1 [1 -Φ(x it β * + λ 1 -λ c * )] 1-m it × [Φ(x it β * + λ 1 -λ c * )] m it φ(c * )dc * }
where β * = β/σ u and c * = c/σ c . A standard software can be used to estimate β * and λ, which are normalized on σ u . 9

The time-varying and time-invariant variables captured in the vector x it include weather variability (as measured by the coefficient of variation), income from agriculture, wealth (as measured by the value of livestock owned and the size of land), human capital, membership in informal risk-sharing institutions, and village-level variables such as access to a good road and to town. A detailed description and motivation of these variables is provided in the next section.

It is plausible to argue that some of the explanatory variables (e.g., income) would be endogenous.

We attempt to take care of endogeneity of income using the level of rainfall in the previous production year as a credible instrumental variable (IV), and estimate a binary instrumental variables model of migration. We specifically estimate the instrumental variables probit model using the ivprobit command in Stata. This estimator identifies the parameters of a model with a binary dependent variable and an endogenous explanatory variable (or variables). As in the linear instrumental variables estimator, the instrumental variables probit model is estimated in a two-stage process. Consistent estimation is based on the assumption that the error terms of the two equations (in both the first and second-stage) are independently and identically distributed multivariate normal. If this assumption is not fulfilled, one could use clustered standard errors to control for the lack of independence [START_REF] Maddala | Limited Dependent and Qualitative Variables in Econometrics[END_REF]).

Explanatory variables

We discuss the ex-ante hypotheses on each explanatory variable used in the migration equation and in the agricultural income equation. The rainfall variability and shock measures are discussed separately in section 5.3.

9 One other applicable binary choice panel data estimator is the fixed effects logit model [START_REF] Cameron | Microeconometrics Using Stata[END_REF]. However, because it is based on a within transformation (which also drops any time-invariant observable variables in xit) and is based on variation in the dependent variable over time (which limits the number of observations to be used for estimation and consequently reduces our sample size significantly), we preferred not to use it.

The migration decision

Agricultural income

Migration depends first upon agricultural income (in logarithms). The standard hypothesis is that droughts decrease agricultural income and push households to migrate [START_REF] Munshi | Networks in the Modern Economy: Mexican Migrants in the US labor market[END_REF]. The higher the agricultural income of the household, the less likely it is to have to resort to migration to cope with climatic shocks. However, studies on Burkina Faso (Henry et al., 2004a(Henry et al., , 2004b) and Ecuador [START_REF] Gray | Environment, land, and rural out-migration in the Southern Ecuadorian Andes[END_REF] found the opposite result. A positive correlation is consistent with the so-called humpshaped pattern of migration [START_REF] Hatton | What Fundamentals Drive World Migration?[END_REF], which refers to the cost of migration that is too much for households to afford below a critical income threshold. The probability of migration by poor households may thus vary positively with agricultural income. The measure of agricultural income in the ERHS comprises total agricultural income, from both crop and livestock production. Income is given in Ethiopian birr, adjusted for spatial price differences using carefully constructed price deflators. 10

Household assets

The more assets the household possesses, the less likely it is to have to rely upon migration as a risk coping strategy. For example, the landless have a higher propensity to migrate as a response to shocks, as demonstrated by [START_REF] Jayachandran | Selling Labour Low: Wage Responses to Productivity Shocks in Developing Countries[END_REF]. Asset ownership is often proxied by the number of livestock units owned [START_REF] Kazianga | Consumption Smoothing? Livestock, Insurance and Drought in Rural Burkina Faso[END_REF]. Livestock sales can be used as one strategy to smooth consumption when there is an environmental shock [START_REF] Hoddinott | Shocks and their Consequences across and within Households in Rural Zimbabwe[END_REF]. 11 [START_REF] Rogg | Precautionary Saving and Portfolio Management in Uncertain Environments: Evidence from Rural Ethiopia[END_REF] provides an in-depth study of the asset portfolio responses of the ERHS households to adverse shocks during the first four rounds of the survey, and also on portfolio responses to ex ante uncertainty.

He finds that households in more risky environments hold significantly less livestock. He also shows that households use buffer-stock assets such as crop/food stocks and some types of livestock to smooth consumption when income fluctuates. In order to reduce endogeneity problems, we use the value of livestock in t -1 as a control variable for migration in t.

Household size

We control for household size before migration, which would enter as a factor of production in working the field to produce agricultural income. Household size also takes into account our assumption that the migration decision is part of a household-level optimization strategy. Larger households are more likely to send more migrants away. Ideally, we would have preferred to use the dependency ratio, defined as the number of dependent persons (under 15 and above 65 years of age) over the number of working persons. The rationale is that a higher dependency ratio may be an obstacle to migration, as migration would result in fewer adults available to take care of the youngest and oldest. This control variable has not been retained, however, for two main reasons.

First, the number of working persons over non-working persons is endogenous to migration, as the latter has been shown to reduce child and female work (see [START_REF] Acosta | School attendance, child labour, and remittances from international migration in El Salvador[END_REF], for an application to El Salvador). Second, age data is incomplete on former members, so the variable would very likely be biased.

Education

Education is generally considered to have a positive influence on internal migration [START_REF] Lucas | Internal Migration in Developing Countries[END_REF][START_REF] Taylor | Human Capital: Migration and Rural Population Change[END_REF] and positive evidence of this is found in many studies (Henry et al., 2004a;[START_REF] Konseiga | Household Migration Decisions as Survival Strategy: The Case of Burkina Faso[END_REF][START_REF] Tsegai | Migration as a Household Decision: What are the Roles of Income Differences? Insights from the Volta Basin of Ghana[END_REF][START_REF] Beegle | Migration and Economic Mobility in Tanzania: Evidence from a Tracking Survey[END_REF]. On the one hand, it should increase the employment possibilities of a member who migrates. On the other hand, it also increases the opportunities to find non-agricultural work within the same PA, thus decreasing the probability of sending a member away. The first effect seems to dominate in [START_REF] Beegle | Migration and Economic Mobility in Tanzania: Evidence from a Tracking Survey[END_REF], which highlights a strong positive and non-linear effect of education on migration. The variable used here (human capital) measures the highest education level in the household, in the following categories: some primary education or adult literacy program, completed primary education, secondary education, some university education.

Membership in iddir

The literature suggests a range of mechanisms that households living in risky environments have developed to shield their consumption from risk, including social insurance arrangements, particularly important in the absence of formal insurance or credit markets. An iddir is a kind of mutual insurance association that pays for funerals. The variation is not that large in this variable; in 2004, for example, 80 % of the households were members of an iddir. Nevertheless, we use it to control for access to credit or support if the household were to suffer a reduction in its income.

Distance to town and good road

Accessibility is measured by the distance between the PA that the household resides in and the nearest town. We also use a measure of road accessibility from the PA. Following Dercon et al.

(2009) we define road accessibility as a dummy variable equal to one if the PA has at least one road that is accessible to trucks and buses in both rainy and dry weather, and equal to zero otherwise.

The influence of distance to town and accessibility by road is a priori ambiguous. On the one hand, closer proximity to markets makes it easier to purchase inputs and sell crops, and it should also increase the possibilities of seeking alternative wage employment in the origin location as a diversification strategy. These factors all speak for a negative influence of the accessibility variables on the migration decision. On the other hand, proximity to a nearby village and road accessibility reduces the cost of migration and should thus positively influence migration, as in Henry et al.

(2004a), for example. These last two variables are measured at the level of the PA, and do not vary across households in the same PA.

The agricultural income equation

Household agricultural income is a function of the household's number of laborers, proxied by the total number of household members before migration. This variable should be refined to only include working-age people, but, since the sample shows that people start working in the field as early as 6 years old, there is no clear cut-off for the definition of useful labor. The value of livestock holdings in the current year is included as a production factor for agricultural income. Following a Ricardian production factor approach (Mendelsohn and Dinar, 2009), we control for the size of the household's landholdings, measured in standardized units (ha).

Weather variables

Rainfall variability has been shown to be a relevant climate parameter for agricultural yields in other parts of Africa [START_REF] Sultan | Multi-Scales and Multi-Sites of the Role of Rainfall in Cotton Yields in West Africa[END_REF]. The household survey data is matched with data on rainfall over the period 1967-2004 from the weather station closest to the PA. The rainfall data used in the study thus vary only across PAs and over time. We use the rainfall in the 12 months preceding the survey to account for the impact of rainfall on agricultural income. The rainfall shock variable is defined as equal to one if the annual rainfall in the 12 months preceding the survey was below one standard deviation from the long-term mean of 30 years of rainfall data, representing a negative rainfall shock. 12 Apart from average rainfall based on past observations, and deviations from this rainfall level, another relevant measure is the variability of rainfall. We use the coefficient of variation to measure the riskiness in the household's environment. The coefficient of variation is defined as the standard deviation over the mean, multiplied by 100. We will thus use the coefficient of variation of the 30 year rainfall distribution as a measure of the riskiness of the household's 12 Other definitions of extreme shocks above two standard deviations were also tested and the results were robust in terms of coefficients and significance.

environment with respect to rainfall. This should be the basis for a household's ex ante decision to send a member as a migrant, before the actual rainfall level and its impact on the harvest have been observed. Ex post, once the rainfall level of the particular year (harvest season) has been observed, it is the actual rainfall shock that will determine a migration decision. Because the survey does not enable us to separate the two types of decisions (ex ante and ex post), it is important to include both measures in the equation. A test of the hypothesis of a non-zero coefficient on any of the variables should test for its relevance in the final migration decision, which includes both ex ante and ex post considerations.

Table 1 shows the descriptive statistics of the variables used in the study.

Results

Table 2 shows random effects probit regression results and the corresponding marginal effects for the household model of migration presented in Equation [START_REF] Alem | The Impact of Safety Nets on Technology Adoption: A Difference-in-differences Analysis[END_REF]. The results show that household size and household education positively affect the decision to send a member as a migrant, while distance to the nearest town reduces the likelihood of sending a migrant. The results also show that more households sent a migrant in 2004 relative to 1999, which is the base year. We do not find any statistically significant effect of our key variables of interest: household income, weather variability, and rainfall shocks. The random effects probit treat all the right-side variables as exogenous. It is however plausible to expect that most of the household variables are endogenous. We use the level of rainfall in the main agricultural season as an instrument to take care of endogeneity of agricultural income. Other household variables, such as value of livestock, land size, household size and education are also likely to be endogenous. However, we do not find credible IVs to instrument them. Thus, we do not make causal inference between migration and these variables, but we control for them in the estimations. The first-stage and second-stage regression results from the instrumental variables probit estimator are reported in Tables 3 and4. We begin with the first-stage regression results. Table 3 shows that the first-stage relationship between rainfall and agricultural income is strongly positive: the level of annual rainfall is significantly related to agricultural income at the one percent significance level.

This relationship is robust to exclusion of the other variables 13 and the Wald test rejects the null hypothesis of no endogeneity (pvalue=0.001). In a country like Ethiopia where more than 95% of smallholder farmers are rain-dependent for their livelihood, positive rainfall typically leads to better agricultural production. The first-stage regression results also show that all the other correlates of agricultural income have the theoretically expected signs and are all statistically significant at the one percent level, except the rainfall shocks variable, which is significant at the ten percent level. Regression results from the second-stage of the instrumental variables probit model presented in Table 4 demonstrate the importance of controlling for endogeneity of agricultural income in the migration equation. Column [START_REF] Acosta | School attendance, child labour, and remittances from international migration in El Salvador[END_REF] presents the parameter estimates and Column [START_REF] Adger | Social and Ecological Resilience: Are They Related?[END_REF] presents the corresponding marginal effects. The log of agricultural income is now statistically significant at the one percent level in Column [START_REF] Acosta | School attendance, child labour, and remittances from international migration in El Salvador[END_REF] and positively and strongly determines rural Ethiopian households' decision to send a member as a migrant. From the marginal effects reported in Column [START_REF] Adger | Social and Ecological Resilience: Are They Related?[END_REF], we observe that a one percent increase in agricultural income leads to about an 11.5 percent increase in the probability of sending a member as a migrant. This is consistent with findings in earlier studies of migration (Henry et al., 2014a;[START_REF] Gray | Environment, land, and rural out-migration in the Southern Ecuadorian Andes[END_REF], where the liquidity constraints argument seems to be the dominant mechanism in explaining the effect of income on migration. Migration is costly and higher income provides more resources for households to cover the cost of migration.

The value of livestock on the other hand is negatively correlated with the probability of sending a household member as a migrant and the effect is statistically significant at the one percent level.

A one percent increase in the value of livestock is associated with a 2.5 percent decrease in the 13 Results are available from the authors upon request.

probability of sending a migrant (Column [START_REF] Adger | Social and Ecological Resilience: Are They Related?[END_REF]). Livestock are important as wealth and factors of production in a smallholder farming setups like rural Ethiopia. Consequently, other factors being constant, greater livestock wealth gives the household the opportunity to be productive in farming and get better access to other productivity-enhancing modern agricultural inputs such as chemical fertilizer (Alem and Broussard, 2016).14 We will now analyze how climate factors affect households' decisions to send a migrant. Instrumental variables probit regression results presented in Table 4 suggest that the decision to send a household member as a migrant is positively influenced by weather variability, as measured by the coefficient of variation of rainfall. A one unit increase in the coefficient of variation increases the probability of sending a migrant by about 0.3 percent. The effect is statistically significant at the one percent level. This provides strong and important evidence supporting our hypothesis that households in areas with highly variable weather are more likely to adapt through sending a migrant. This is an ex-ante effect of weather variability. Because climate change is expected to result in extreme weather events and the Sub-Saharan African region is going to experience even greater variability in rainfall (IPCC, 2014), our findings imply that there will be increased migration by household members in rural communities in the future. On the data studied here, no statistically significant evidence (probably due to a large standard error) is found that migration decisions are driven by negative rainfall shocks over the short run (24 months); rather, such decisions are taken due to assessments based on the long-term risk of variability in rainfall as given by the coefficient of variation.

We finally note that distance to the nearest town negatively affects the decision to send a migrant.

This may sound counterintuitive, as proximity to towns should make migration easier. However, it is also plausible that households close to towns have the option of participating in off-farm activities without the need to leave home. This last result thus yields support to the proximity to markets argument: better access to markets for inputs and outputs enables the household to be more productive in agriculture and also to find alternative employment in urban centers [START_REF] Bezabih | Participation in Off-Farm Employment, Rainfall Patterns and Rate of Time Preferences: The Case of Ethiopia[END_REF].

Conclusions

Climate change is predicted to impact society and ecosystems by resulting in extreme weather events, changing precipitation and declining agricultural productivity in large parts of the world (IPCC 2014). As a result of its dependence on climate variables, agriculture is more vulnerable than other sectors to the effects of climate change. Sub-Saharan Africa, whose agricultural sector is known for its low productivity, is one of the most vulnerable regions. This paper attempts to shed light on whether households in rural areas of Ethiopia use migration as a strategy to adapt to weather variability. We use Ethiopian Rural Household Survey panel data combined with village-level rainfall data to investigate both the ex-ante and ex-post impact of climate variables on households' likelihood to send a migrant. Compared to earlier studies, the key contributions of the paper are in using objective measures of rainfall variability and rainfall shocks, which are exogenous to households, and in teasing out the ex-ante and ex-post effects. We also control for endogeneity of income -an important determinant of migration -using a credible instrument.

Instrumental variables probit regressions suggest that households in rural Ethiopia adapt to weather variability by sending a migrant. Smallholder farm households that live in places with higher rainfall variability are likely to send a household member as a migrant. This effect is economically important and statistically significant at the one percent level. We also find suggestive evidence on the ex-post impacts of weather, proxied by the prevalence of negative rainfall shocks between one and two standard deviations above the long-run mean, in the year preceding migration. The variable positively affects the decision to send a migrant, but it is not statistically significant at the conventional levels, most probably due to large standard errors. We also find that income (instrumented by the level of rainfall) has a significant positive impact on the decision to migrate, while wealth (proxied by the value of livestock owned by the household) is negatively associated with the likelihood of engaging in migration. Regression results also show that living close to a town reduces the likelihood of sending a migrant.

Extensions to this analysis could study the role of irrigation, not included in our study because the measurement errors in the variable are large in the ERHS data. 15 Improving water access would be a useful policy objective, given the strong impact that we find from rainfall variability.

Developing rainfall insurance products that could cushion the impact of rainfall variability also seems an important policy implication of the results, in line with previous research on the negative effects of rainfall variability on small holder farm households. Because several factors enter into the migration decision, it is also clear that some migration is unavoidable -and even desirable from a household viewpoint. Policy measures should also be directed toward supporting development in the urban sector of the country that will receive the out-migrants from the rural areas that are subject to high rainfall variability. 15 Water harvesting by traditional means is often interpreted as irrigation in the survey.

  , we separate rainfall variability as a determinant of ex-ante migration decisions and rainfall levels -and hence shocks -as a determinant of ex-post migration decisions. Prior to observing the current year's (season's) rainfall, household i in PA j may take an ex-ante deci-

		Kiremt
	t-1		t
	Ex ante	rainfall	Ex post
	RCV j	R j	SHOCK j
		Figure: Timeline
	Following Rose (2001)

sion to send a migrant as a function of household characteristics (F ij ), costs of migration from the PA (C j ) and rainfall variability based on past observations of the rainfall distribution. A relevant measure for rainfall variability is the rainfall coefficient of variation measured at the village level (RCV j ). This ex-ante decision can be expressed as

  ij controls for household assets, represented by land owning and livestock, and human capital, which increases employability in the non-agricultural sector. We also use membership in an iddir -

	an informal risk-sharing institution in Ethiopia -to control for access to informal insurance against
	shocks.
	4 Data and Empirical Strategy
	4.1 Data

[START_REF] Fafchamps | Drought and Saving in West Africa: Are Livestock a Buffer Stock[END_REF][START_REF] Hoddinott | Shocks and their Consequences across and within Households in Rural Zimbabwe[END_REF]

;

[START_REF] Kazianga | Consumption Smoothing? Livestock, Insurance and Drought in Rural Burkina Faso[END_REF][START_REF] Porter | Shocks, Consumption and Income Diversification in Rural Ethiopia[END_REF]

or may increase their activities in the non-agricultural sector

[START_REF] Bezabih | Participation in Off-Farm Employment, Rainfall Patterns and Rate of Time Preferences: The Case of Ethiopia[END_REF]

. In particular, the vector of household characteristics F We use data from the Ethiopia Rural Household Survey (ERHS), a longitudinal survey implemented by the department of Economics of Addis Ababa University, Ethiopia, in collaboration with the International Food Policy Research Institute (IFPRI), and the Centre for the Study of African Economies (CSAE) at the University of Oxford. Data were collected for the first time in 1989 from 6 villages that suffered drought in 1984. The survey was further expanded in 1994 to encompass 15 peasant associations (PA) 5 across the four major regions of Ethiopia (Tigray, Amhara, Oromia and Southern Nations and Nationalities and people's region), constituting 1477 households, and subsequent rounds took place in 1995, 1997, 1999, 2004 and 2009. The sample of villages and households were chosen randomly to represent the major agro-ecological zones of the country, excluding the nomadic population. The data thus give a representative sample of Ethiopia apart from nomadic pastoral lands. 6

Table 1 :

 1 Descriptive statistics of variables over time

	[1999]		[2004]
	Mean	SD	Mean	SD

Source: Authors' computation from ERHS 1999 and 2004.

Table 2 :

 2 A Household Model of Migration: Random Effects Probit Regression Results

	[RE-PROB]	[ME]
	[1]	Coeff.	SE

Table 3 :

 3 A Household Model of Migration: IV Probit -First Stage Regression

	[First]
	Coeff.	SE

Table 4 :

 4 A Household Model of Migration: IV Probit -Second Stage Regression

	[1]		[2]
	[IV-Prob]	[ME]
	Coeff.	SE	Coeff.	SE

Notes: * * * p < 0.01, * * p < 0.05, * p < 0.1.

See UNFPA (2009) for estimations of the numbers of displaced people due to environmental degradation, and CRED (2015) for the numbers of displaced people due to natural disasters.

The analysis relies on cross-sectional data, though, so caution should be taken in interpreting the results.

[START_REF] Dillon | Migratory Responses to Agricultural Risk in Northern Nigeria[END_REF] also separate ex ante and ex post risk, but focus on risk from temperature variations, and they could not control for income endogeneity, as they were lacking data on income for the 185 households in Northern Nigeria that were studied.

In Ethiopia, there are two main rainy seasons, kiremt or meher, which is the long rainy season from mid-June to mid-September, and belg, which is the short one from February to May.

A peasant association is the lowest administrative unit in Ethiopia and normally consists of several villages.

Refer to Dercon and Hoddinott (2011) for a detailed description of the ERHS panel data set.

We thank Catherine Porter for sharing the rainfall data with us.

In 1999, 1 ETB equaled approximately 0.17 USD.

Whether livestock is used to smooth consumption depends on the particular context. For example, Kazianga and Udry (2006) find that livestock is not sensitive to idiosyncratic household shocks, based on a sample of households in Burkina Faso.

Due to the many restrictions in the land market in Ethiopia, livestock plays an important role in farm productivity in rural Ethiopia. It is an important wealth indicator and it can serve as collateral for credit.

We thank seminar participants at the 18th European Association of Environmental and Resource Economics (EAERE) annual conference, and Catherine Porter for helpful input in earlier versions of this paper. We gratefully acknowledge financial support from the French National Research Agency grant ANR-JCJC-0127-01, as well as the Swedish Research Council Formas through the program Human Cooperation to Manage Natural Resources (COMMONS) through the Environmental Economics Unit of the Department of Economics, University of Gothenburg. All errors and omissions are ours.