Reversible time travel with freedom of choice

General relativity allows for the existence of closed time-like curves, along which a material object could travel back in time and interact with its past self. This possibility raises the question whether certain initial conditions, or more generally local operations, lead to inconsistencies and should thus be forbidden. Here we consider the most general deterministic dynamics connecting classical degrees of freedom defined on a set of bounded space-time regions, requiring that it is compatible with arbitrary operations performed in the local regions. We find that any such dynamics can be realised through reversible interactions. We further find that consistency with local operations is compatible with non-trivial time travel: three parties can interact in such a way to be all both in the future and in the past of each other, while being free to perform arbitrary local operations.

Introduction
One of the most baffling aspects of general relativity is that certain solutions to the Einstein equations contain closed time-like curves (CTCs) [1][2][3][4][5][6][7], where an event can be both in its own future and past. Although it is not known whether CTCs are actually possible in our universe [8][9][10][11][12], their mere logical possibility poses the challenge to understand what type of dynamics could be expected in their presence.
The first systematic studies of the subject concentrated on space-time geometries where CTCs appear only in the future of some space-like surface [13][14][15] (figure 1(a)). This makes it possible to set initial conditions on that surface, i.e. in the pre-CTCs era, and to look for the corresponding solutions to the equations of motion. A prime case study is that of a billiard ball thrown in the direction of a wormhole: the initial position and velocity is chosen such that, if undisturbed, the ball comes out of the second mouth of the wormhole in the past and kicks its younger self off course, so the ball cannot reach the wormhole and kick itself. Since classical physics is clearly at odds with such 'inconsistent' dynamics, the corresponding initial conditions would have to be 'forbidden'. This, however, is itself at odds with the local nature of ordinary physical laws: What local mechanism prevents an experimenter from throwing the ball along the 'impossible' trajectory?
The surprising result is that (possibly multiple) self-consistent solutions were found for all cases studied. The ball does not enter the wormhole undisturbed: it is kicked softly, comes out the wormhole at a slightly different angle than expected and gives its younger self just the right soft kick. Even including friction, exploding bombs, and the like, solutions for any considered initial condition were found [17,18].
The existence of consistent solutions for every initial condition suggests a type of 'no new physics' principle [13]: the presence of CTCs should not modify the local laws of physics, nor the range of possible initial states. It is then meaningful to ask whether the validity of the principle can be extended to the region where CTCs are already present ( figure 1(b)). In such a case, there are no sufficiently regular space-like surfaces to set 'global' initial conditions. Furthermore, there exist classes of space-times, aimed at simulating time machines, where geodesics are 'reflected' by the CTC region, making it problematic to impose global initial conditions even in its past [19]. However, the local nature of physical laws would imply that global features, such as the presence of CTCs, should not constrain the possible actions of an agent 7 in any sufficiently localised region which itself does not contain CTCs.
Here we explore whether CTCs are compatible with such an extended requirement of 'no new physics'. Rather than considering a specific type of system (billiard balls, fields, etc), we work within an abstract framework describing classical, deterministic dynamics that does not assume any particular causal structure. It is formally a deterministic version of the formalism of 'classical correlations without causal order' [20], which in turn is the classical limit of the quantum 'process matrix' formalism [21] (see also [22][23][24][25][26][27][28][29][30][31][32]). This latter formalism [21] can be used to study correlations among parties where the causal order among them is not specified a priori. The (often implicit) assumption of a causal order, instead, is replaced by the assumption that the probabilities of measurement outcomes are well defined. It has been shown that this formalism is more general than quantum theory: correlations arise that cannot be explained with a predefined causal order of the parties. One can also consider the underlying state space to be classical (random variables) as opposed to quantum [20]. Strikingly, also in that special case such 'non-causal' correlations arise. Here, and as mentioned above, instead of dealing with quantum states or random variables, we restrict the model to deterministic dynamics; a limitation that is known to still allow for 'non-causal' correlations [20].
We prove that all classical, deterministic processes compatible with the free choice of local operations represent the evolution of classical systems via reversible dynamics in a suitable topology. Surprisingly, this is not true in the quantum case, where certain processes are incompatible with reversible dynamics [31]; it is neither true if the underlying states are random variables [33]. Yet, for deterministic dynamics, every process can be turned into a reversible one, as is shown later.
We further provide a complete characterisation of up to tripartite processes, including continuous-variable systems, identifying a broad class of processes that can only be realised in the presence of CTCs. Our results show that CTCs in general are logically compatible with classical, deterministic dynamics, where agents are free to perform any classical operation in local regions and, outside such regions, systems evolve according to classical, reversible dynamics.
Our approach departs from previous models of CTCs directly based on the quantum formalism [34][35][36][37][38][39][40][41][42][43][44][45][46]. In those models, if the underlying systems that travel on the CTCs are classical, then the 'no new physics' principle and the assumption of free choice cannot be uphold simultaneously. Rather, it was believed that one needs to invoke quantum mechanics in order to restore these desired features. In this article, we finally discuss the extension of the presented formalism to quantum CTCs and discuss it in the light of the above mentioned models. A detailed comparison among these models will be provided in a forthcoming paper [47].

Approach
It is customary to understand physical laws as rules for predicting future events based on initial conditions. This is often seen as an essential aspect for extracting testable predictions: one can always conceive, at least in principle, a controlled experiment where the initial conditions are set independently of any other relevant aspect of the experiment, and the final state is measured. CTCs undermine this view: in a general scenario, where events can be in their own  [8,9,16]. (a) Events with equal proper times along the world lines of the two mouths of the wormhole are identified. Accelerating the right mouth produces time dilation, resulting in CTCs in the future of the surface S. An experimenter acting in the past of S should be able to prepare arbitrary initial states on a space-like surface P . (b) An experimenter in a localised region L, which does not contain but is traversed by CTCs, should be able to perform arbitrary local operations.
future and global space-like surfaces are not available, it is unclear what independent variables an experimenter could try to control. Solutions to the dynamics have to be determined 'all at once', without a clear place for external interventions.
Our approach to a predictive framework for not globally hyperbolic space-times 8 , possibly containing CTCs, is to move to a more general notion of intervention, partly inspired by the methodology of classical causal models [49,50]. The key assumption is that physical laws retain their modular character, namely it is possible to separate the physical properties relevant for the dynamics of the system of interest from those responsible for the 'ntervention'. For example, when studying the trajectory of a billiard ball, one can typically ignore the particular mechanism setting the ball in motion; or when estimating the radiation produced by moving charges, one can abstract away the force causing the charge's movement. In other words, we wish to retain a notion of 'freedom of choice' in manipulating the relevant variables 9 .
To be more precise, we will identify interventions with localised operations in CTC-free regions of space-time. Following the above assumption, we assume that it is possible-at least in principle-to engineer any local operation on the system of interest without significantly affecting the dynamics outside the intervention regions. The role of the dynamics is then to predict how a system will respond to certain operations. In other words, given the specification of the operations in a set of space-time regions where intervention takes place, we should be able to calculate the state of the system observed in any other space-time region. We will call 'process' the function providing such a prediction.
Note that a 'process', in general, includes both information about the dynamics as well as additional boundary conditions not set by intervention. To clarify this point, consider an ordinary, CTC-free, scenario where an intervention sets the value of some variable (e.g. a field) on a small portion of a space-like surface and we want to estimate the resulting field somewhere in its causal future. In general, this requires fixing additional boundary conditions, for example on an extension of the initial region to a Cauchy surface. The 'process' is the function mapping the field values in the small region to the observed final field, once the remaining boundary conditions are fixed. In a limiting case, where no intervention is made, the 'process' is simply a specification of an admissible boundary condition in the region where an observation is made.
According to the above scheme, a particular dynamical law generates a class of processes, describing how interventions in arbitrary regions can influence observations in other regions, as well as all possible boundary conditions consistent with the laws. In the case of not globally hyperbolic space-times, boundary conditions might be subject to non-trivial constraints. Our working assumption is that such constraints should not affect the type of operations available in small regions that do not contain CTCs. This might at first seem at odds with 'grandfather paradox'-type arguments: denote by x the value a physical property takes on the future boundary of our region, a the value on the past, and let the 'CTC dynamics' be the identity, a = x. This is incompatible with any local operation x = f (a) other than identity.
The problem with the argument above is that it simply assumes that the identity backwards in time is a possible solution of the dynamics, based on the intuition that such evolution would be possible if a were in the future of x, without CTCs. The studies mentioned in the introduction suggest that such an assumption is typically incorrect: the system's evolution typically finds a way to 'adjust itself', preserving the consistency of 'free interventions'. The upshot is that we should not expect all conceivable functions to appear as possible solutions of the dynamics. Our approach is to take consistency with local operations as a starting point and explore the consequences: possible dynamics and their respective predictions are calculated based on the 'freedom of choice' of local operations. Whether this assumption is valid would depend on the particular geometry and dynamical equations. The main result is that it is logically possible, at least in principle, to have a predictably sound theory, compatible with local interventions, where the presence of CTCs can be made manifest through appropriate experiments.

The formalism
The core assumption of our model is that any classical operation that is possible in an ordinary space-time should also be possible in the presence of CTCs, as long as the operation takes place in a localised region of space-time that does not contain CTCs. This states that localised regions are ignorant of CTCs. Let us elaborate more on this core assumption.
We consider N non-overlapping space-time regions (henceforth local regions) which, individually, cannot be distinguished from regions in ordinary, globally hyperbolic space-time. These are the regions in which 'interventions' can be made; i.e. evolution inside the regions is assumed to be arbitrary, while outside the regions it is fixed by some given dynamics and boundary conditions. We impose no restriction on the space-time in which the regions are embedded, except that it is a Lorentzian manifold fixed independently of any dynamical degree of freedom of interest. Details regarding the causal structure of space-time will not play a prominent role in our analysis, but we point to [51] for a recent review.
To simplify the analysis, we restrict to compact, simply connected regions that have only space-like boundaries, which we decompose into a past boundary and a future boundary (these local regions are therefore space-time local), see figure 2(a). Furthermore, and for the same purpose, we assume that, for each local region, any time-like curve that enters through the past (future) boundary exits through the future (past) boundary, and that the region contains no CTCs. These assumptions ensure that we can treat the local regions as 'closed laboratories' [21]-where a system can only enter once through the past boundary and exit once through the future boundary, without exchanging information with the exterior in between. Under such conditions, the local operations can be simply identified with transformations from an input to an output space. The assumption of space-like boundaries has the further role of preventing, in a globally hyperbolic space-time, that a region intersects both the future and past light cone of another region. In other words, causal order among regions would form a partial-order relation. Therefore, any departure from partial order reveals a not globally hyperbolic space-time and some non-trivial form of time travel.
As we are interested in classical systems, we can assign classical state spaces I R (input) and O R (output) respectively to the past and future boundaries of a local region R. States will be denoted as i R ∈ I R , o R ∈ O R . For example, in a field theory, a state would be a function on the corresponding boundary surface and the state spaces would be appropriate spaces of functions.
A deterministic local operation in the local region is described by a function f R from input to output space (figure 2(a)). We denote by D R := { f R : I R → O R } the set of all possible operations in region R. We drop the index to refer to collections of objects for all regions, as 10 . Local operations are not required to be reversible, i.e. the local functions f R need not be invertible. This corresponds to the assumption that the local experimenters and devices have the ability to erase information by accessing some reservoir, not included in the description of the physical degrees of freedom of interest. Furthermore, input and output state spaces need not be isomorphic, as degrees of freedom may be added or removed during the operation. We will also consider the special case in which either input or output state space is the empty set. An 'output only' region-called a source-can be identified with a space-like region on which an agent (acting somewhere in its past) can prepare an arbitrary state (figure 2(b)), while an 'input only' region-called a sink-can be identified with a space-like region where an agent (somewhere in its future) can only observe the state (figure 2(c)). Ordinary dynamics is concerned with the evolution from a source (state preparation on a space-like surface) to a sink (state observation on a space-like surface).
We want to define a generalised type of dynamics for an arbitrary number of regions-in which arbitrary classical operations can be performed-possibly embedded in a space-time with CTCs. The basic requirement of such a model is that it must be able to predict the state observed on the past boundary of each local region 11 , which in general can depend on all local operations. (In a CTC-free space-time, the input state on a space-like region would only depend on operations in its past light-cone). For a deterministic model, such a dependence is encoded in a function ω ≡ {ω 1 , . . . , ω N } : D → I that determines the state on the past boundary of each region as a function of all local operations.
We define consistency with arbitrary choices of local operations in the following way: given a set of operations f ∈ D, let i R = ω R ( f ) be the input state of region R. This is transformed into the output o R = f R (i R ) by the local operation in that region. However, there are several different functions that yield the same output o R given the input i R . Since the experimenter is free to choose any operation, and it should not matter when that choice is made (in particular, it can be made after she already knows the input i R ), the external dynamics should not distinguish between all such operations. In particular, let us define the constant operation C o (i) = o yielding outcome o for any input i. Consistency with arbitrary local operations is then formulated as (1) In words, the input states i = ω( f ) produced by the process given local operations f should be the same as the states produced when all parties perform the constant operations that yield the same output states as the original operations f applied to ω( f ). We can simplify the representation of a process ω by noticing that it defines a unique , which we will call process function. This provides an alternative view for defining a process as a function that maps the states in the future boundary of the local regions to states in the past boundaries. This is indeed what we expect for local dynamical equations relating states at different points in space-time.
By writing ω( f ) = i, we see that condition (1) implies the following condition for w: In other words, if w is a process function, then w • f has a fixed point for every local operation f . Condition (2) is in fact both necessary and sufficient: a function w satisfying (2) uniquely defines a process. This is because of the uniqueness of the fixed points: We prove this theorem in the appendix. As opposed to the proof of the analogous theorem in the probabilistic version of the formalism [33], our proof also holds for continuous and not only discrete variables 12 .
Because of theorem 1, every function w : O → I that satisfies condition (2) defines a unique function ω : D → I, with ω( f ) equal to the unique fixed point of w • f . It is furthermore easy to see that condition (2) implies the consistency condition (1). Therefore, we can identify a process with its process function w. The interpretation is that dynamics in the presence of CTCs is described by a function that maps the states on the future boundaries of all regions to states on the past boundaries of each region. Condition (2) imposes that such a dynamics is compatible with arbitrary operations in each region; theorem 1 further guarantees that specifying the operations performed in each region is sufficient to predict a unique state on each of the past boundaries.

Reversibility
Reversible classical dynamics is associated with invertible functions, such that the role of 'preparation' and 'measurement' can be swapped. Not all process functions are invertible; for example, the process function for a single 'sink' region (with trivial output) reduces to the specification of a state on that region and it is clearly not invertible. However, such a process function can be extended to a reversible one by introducing a 'source' region (with trivial input), in the past of the sink, so that the state on the sink can now be calculated as a function of the state prepared by the source, and this function can be invertible.
Crucially, we can prove that every process function can be extended to an invertible one, as expressed by the following theorem, proved in the appendix. This theorem shows that all process functions can be interpreted in terms of reversible dynamics: the source describes a space-like region 'in the past' of all other regions, while the sink is a space-like region 'in the future'. The process determines the state of the sink as well as the states on the past boundaries of all regions as a function of the states on the future boundaries of all regions and of the source. Because it is reversible, the process can be read in the opposite direction: given the states on the sink and on the past boundaries, it allows calculating the state on all future boundaries and of the source. The time-reversed process is then compatible with arbitrary reversed local operations that map local outputs to local inputs. In [47], it is further proven that the presence of a source and a sink is necessary in order to define a reversible process.

Characterization of process functions
The causal relations encoded by process functions can be better understood in terms of signalling. In general, given a function h : A × B → C, we say there is no signalling We say there is signalling if the opposite is true, i. e. h(a, b) = h(ã, b) for some a, ã, b. The terminology applies to process functions in the obvious way: a region S does not signal to a region R if the Rth component of the process function does not depend on the output of S.
Simple examples of process functions are causally ordered ones, namely those compatible with CTC-free dynamics, for which signalling is only possible in one direction 13 . For example, for regions R, S, T, . . . a process function w where the state on each past boundary is independent of future states 14 . It is easy to see that condition (2) is satisfied in such cases, i.e. a fixed point exists for every choice of local operations (it is given by i R =ī R , i S = w S • f R (ī R ), and so on). The interesting question is whether more general processes are possible, once CTCs are allowed. To answer this question, we will first give a complete characterisation of all process functions for up to three regions. The detailed proofs can be found in the appendix.
For a single local region, a process function has to be a constant: w(o) =ī ∀o. Thus, an observer acting in a localised region cannot send information back to herself; her observations are fully compatible with her region being embedded in a CTC-free space-time. A direct consequence is that, for an arbitrary number of regions, the input of each region R cannot depend on that region's output: w R (o) = w R (o \R ), where o \R is the set of outputs of all regions except R.
Bipartite process functions are characterized by the following conditions: In other words, deterministic process functions can only allow one-way signaling. Again, two observers in distinct localised regions would not be able to verify the presence of CTCs outside their regions. (Remarkably, this is not true for the quantum version of the framework [21].) Consider now three regions R, S, T. For simplicity, we denote input and output variables as a ∈ A, b ∈ B, c ∈ C and x ∈ X, y ∈ Y, z ∈ Z, respectively. A process function has then three component functions: a = w R (y ,z), b = w S (x,z), c = w T (x,y ) (where we used the fact that 13 Formally, a set of regions is causally ordered if their causal relations in space-time form a partial order: any region R is either in the causal past, causal future or space-like from any other region S. A process function is compatible with such a structure if signalling is only possible from a region to its causal future. We call such a process function causally ordered. 14 For the sake of presentation, equations like w S (o R , o S ) = w S (o R ) express the independence of the function value the input of each region cannot depend on its own output, as seen above). We give a simple characterization of process functions as functions where the output variable of one region 'switches' the direction of causal influence between the two other parties.

process function if and only if each of the three 'reduced functions' defined as
is a bipartite process function for every z ∈ Z, x ∈ X, y ∈ Y respectively.
Recall that a bipartite process function is at most one-way signaling. Theorem 3 thus says that w is a tripartite process function if and only if, for every fixed value for the outcome of one of the regions, only one-way signaling is possible between the other two. It is an open question whether a similar condition characterises arbitrary multipartite processes.

Examples
Given the above characterisation, it is simple to find process functions that cannot arise in ordinary, causal space time: it is sufficient that each party can signal non-trivially to the other two, while satisfying the condition of theorem 3. We present a continuous-variable example, based on a similar process for 'bits', first found by Araújo and Feix and published in [20]. Consider a tripartite scenario as above, where x, y, z, a, b, c ∈ R. We define w : where Θ(t) = 1 for t > 0, Θ(t) = 0 for t 0. In this process, the sign of the output of each region determines the direction of signaling between the other two. For example, for y 0 we have a = Θ(z) (T can signal to R) but c = 0 (R cannot signal to T), while for y > 0 the opposite direction of signaling holds (and similarly for the other pairs of regions). By theorem 2, we can extend w to a reversible process function w . To this end, we introduce source and sink spaces, both isomorphic to R 3 , with variables e 0 , e 1 , e 2 and s 0 , s 1 , s 2 , respectively. The extended process function w : R 6 → R 6 is given by This process allows three observers in regions R, S, T to perform arbitrary deterministic operations on the system they receive from the respective past boundary, sending the result out the respective future boundary. The outgoing systems then enter the CTC region and undergo some reversible transformation, interacting with each other and with the output of the source , eventually determining the state in the past of each region and of the sink ( figure 3). Crucially, the input state of each region depends non-trivially on the output state of the other two, thus each observer can communicate to every other. Thus, we have a situation where three observers can experimentally verify to be each both in the past and in the future of each other, they can perform arbitrary local operations, and no contradiction ever emerges.

Quantum CTCs
The above framework of classical, reversible dynamics can be extended to quantum systems. It is then interesting to compare the resulting model with existing quantum models for CTCs. We briefly present here the main results, and refer to [47] for a detailed analysis.
A classical system can be 'quantised' by associating to each state a distinct orthogonal state in a Hilbert space, with quantum superpositions represented by linear combinations. Thus, in the quantum version of the formalism, the boundary of each region is associated with a Hilbert space. A classical, reversible process defines a permutation of basis elements and can be extended by linearity to the entire Hilbert space, defining a unitary map from the future to the past boundaries. It is not a priori guaranteed that such a unitary defines a valid quant um process: observers in the local regions should now be able to perform arbitrary quantum operations. The resulting constraints on quantum processes can be conveniently formalised using the process matrix formalism of [21]. Using the characterisation of tripartite quantum processes of [23], it is proven in [47] that the quantisation of a finite-dimensional version of equation (3) indeed defines a valid unitary quantum process.
The two most studied models of quantum systems in the presence of CTCs are the so-called post-selected CTC model (P-CTC) [35-37, 39, 40, 42] and the Deutsch model (D-CTC) [34,38,[43][44][45]. Both models assume that CTCs are only present in a limited portion of spacetime. At some time before the CTCs, a chronology-respecting (CR) system is prepared. Then, the CR system interacts with a chronology-violating (CV) one, which travels along a CTC. The models prescribe how to calculate the state of the CR system obtained after interaction with the CV one. Within such frameworks, we can model the multi-region scenarios considered here by introducing a CR and a CV systems per region, and interpreting the interaction between each pair as our local operation in the corresponding local region. The CV systems then interact according to the unitary process and are later sent back in time, with the backward Figure 3. The output of three local regions fall into a CTC where they undergo a joint interaction with the state prepared by the source. The CTC outputs the input states to the three local regions and the sink. evolution described according to the specific model. We can then compare the evolution of the CV system predicted by each model.
As it turns out, the P-CTC model gives the same predictions as ours for any valid unitary process. The crucial difference is that the P-CTC model allows the CV system to evolve according to arbitrary unitaries, generically resulting in a non-linear evolution for the CR system and in a restriction on the local operations that can be performed. By contrast, our model imposes additional constraints, which effectively enforce the CR system to evolve linearly. The D-CTC model, on the other hand, allows arbitrary operations to be performed locally. However, it predicts non-linear evolution of the CR system, even when the CV system evolves according to a process subject to the constraints introduced here [52].

Conclusions
We developed a framework for deterministic dynamics in the presence of CTCs. The framework extends the ordinary concept of time evolution-where a future state is calculated as a function of a past one-to the more general scenario of a number of space-time regions, where the state in the past of each region is calculated as a function of the state in the future of all regions. Requiring that arbitrary operations must be possible in each region imposes strong constraints on the allowed dynamics. Our main result is that it is possible to have reversible dynamics, compatible with arbitrary local operations, where the state observed in each region depends non-trivially on the states prepared in all other regions. Because such a functional relation is reversible, it can always be realised by some physical system subject to local dynamical laws, e.g. in terms of a system of bouncing billiard balls [53].
The main message of our result is that CTCs are not necessarily in conflict with local physics, nor with the 'freedom of choice' associated with the possibility of performing arbitrary operations. The latter furthermore implies that the choice of the operations together with the CTC uniquely determines the states on the past boundaries. Importantly, and contrarily to several previous approaches [34][35][36][37][38][39][40][42][43][44][45][46], quantum mechanics need not be invoked to 'solve' paradoxes of classical time travelling. A quantum version of the framework can be developed within the so-called 'process matrix' formalism [21], where it would be natural (in analogy to classical determinism and reversibility) to impose unitarity of the process [31]. The precise connection between classical and quantum frameworks remains however an open questionwe provided a brief discussion nevertheless-for example, it is unclear weather all classical, deterministic processes can be 'quantised' to give a valid unitary quantum process. the authors and do not necessarily reflect the views of the John Templeton Foundation. We acknowledge the traditional owners of the land on which the University of Queensland is situated, the Turrbal and Jagera people.

Appendix A. Properties of process functions
Here we derive a set of properties of process functions, which will be needed in later proofs. For convenience, we will use the term process function to denote any function w that satisfies condition (2) in the main text, namely that a fixed point of w • f exists for each f ∈ D, even though the equivalence between this condition and the main-text definition of process function, equation (1), is due to theorem 1, which is proved in the next section.
Let us first fix some notation. As in the main text, an object without index refers to a collection of objects: I = I 1 × · · · × I N , etc. We will also use the notation The first property we need is a necessary condition for process functions:

Proof. For some set of local operations
and some fixed ī \R , let us define h R : We define then the function g R : It is then easy to see that h R • g R has no fixed point.
has no fixed point and w R is not a component of a process function. As this must hold for every set of local operations f and every ī \R , we conclude that each component w R of a process function must be a constant over O R . Intuitively, if we fix the operation performed in one of the N regions, we should obtain a process for the remaining N − 1 regions. This intuition will also play an important role in the proofs below. The first step to formalise such an intuition is the following definition: We will need the following fact: We can now prove two crucial properties.
To prove (ii), we can set R = 1 without loss of generality. We then have to prove that, if w f1 is a process function for every f 1 , it follows that w is also a process function, i.e. we have to find a fixed point of w • f for an arbitrary f . As the reduced function w f1 is a process function by assumption, there exists a fixed point i \1 of w f1 • f \1 . Choosing i 1 = w 1 • f \1 i \1 as input state for region 1, we see that i := i 1 ∪ i \1 is a fixed point of w • f . Indeed this is true, by definition of i 1 , for the component w 1 . For S > 1, the definition of reduced function, equation (A.1), gives where in the last equality we used the fact that i \1 is the fixed point of w f1 S • f \1 . □

Appendix B. Uniqueness of the fixed point
Here we prove theorem 1, namely the following N-dependent proposition. We prove this by induction, namely we first prove P [1] and then the implication P[N − 1] ⇒ P [N] for N > 1.

Appendix C. Reversibility
Here we prove that every process function can be extended to an invertible process function (theorem 2).
Proof. Given a process function w : O → I over N local regions, we add two extra regions, and , a source and a sink, respectively. We take the output space of the source to be isomorphic with the entire input space of the N regions, e ∈ O ∼ = I , while the input space of the sink is isomorphic to the output space of the regions, s ∈ I ∼ = O. For each R = 1, . . . , N and each e R ∈ I R , we introduce a function T eR R : I R → I R such that there exists ẽ R for which Tẽ R R (i R ) = i R and, for each i R ∈ I R , T (·) R (i R ) is invertible. For a state space with a linear structure, we can take T eR R (i R ) = i R + e R for concreteness (although the proof does not rely on this) 15  The function w is invertible, with the inverse given by i R = w R (s) − e R , o R = s R . To show that it is a process function, we have to prove that its composition with arbitrary local operations has a fixed point, condition (2) in the main text. Note that this condition is equivalent to the existence of output fixed points: f • w (o ) = o , o ≡ (o, e). Since local operations for are functions ∅ → O , where ∅ is the empty set, they can be identified with a state f (∅) ≡ e ∈ O , interpreted as 'state preparation'. The fixed-point condition for the source components is then trivially satisfied by any e ∈ O . As the sink has no output space, the fixed-point condition reduces to which should be satisfied for every f ∈ D and e ∈ O . This is true because f R • T eR R is a local operation and, as w is a process function, a fixed point o ∈ O exists for every local operation. □ 15 If |I R | = c R < ∞, we can use T eR R = i R ⊕ e R , where ⊕ is addition modulo c R .
non-null intersection.) In a similar way, we write X = X S ∪ X T and Y = Y R ∪ Y T , where w x S is constant for x ∈ X S , and so on. Thus we have w R (y R , z) = w R (y, z R ) = a 0 ∀ y R ∈ Y R , y ∈ Y, z R ∈ Z R , z ∈ Z; Now consider an arbitrary local operation f R : A → X . We want to show that the reduced function w fR , defined as in equation (A.1), is a bipartite process function. To this end, we need to prove: (i) w fR S (y, z) = w fR S (z) , (ii) w fR T (y, z) = w fR T (y) , (iii) At least one of the two components is constant.
Let us start with point (i). For z S ∈ Z S , we have w fR S (y, z S ) = b 0 , independently of y . Let us then consider z R ∈ Z R . By definition, w fR S (y, z R ) = w S ( f R • w R (y, z R ), z R ). But w R (y, z R ) = a 0 for z R ∈ Z R , thus w fR S (y, z R ) = w S ( f R (a 0 ), z R ), which is again independent of y . By a similar argument, w fR T (y, z) is independent of z. We are thus left with proving point (iii). We shall prove the equivalent implications w fR S not constant ⇒ w fR T constant, w fR T not constant ⇒ w fR S constant.
Say that w fR S is not constant. Then, there is a z R ∈ Z R such that w fR S (z R ) = b 0 . As we have seen above, for z R ∈ Z R , w fR S (z R ) = w S ( f R (a 0 ), z R ), so we need f R (a 0 ) / ∈ X S to have w fR S (z R ) = b 0 . This means that f R (a 0 ) ∈ X T . But then, for y R ∈ Y R we have w fR T (y R ) = w T ( f R (a 0 ), y R ) = c 0 , and also w fR T (y T ) = c 0 for y T ∈ Y T . This means that w fR T is constant. To recapitulate, we have proven that, if w x , w y , w z are bipartite process functions for arbitrary x ∈ X, y ∈ Y, z ∈ Z (as per hypothesis), then w fR is a bipartite process function for an arbitrary operation f R . Point (ii) of lemma A.3 finally implies that w is a process function, concluding the proof. □


THE LONG-RUN EFFECT OF INWARD AND OUTWARD FOREIGN DIRECT INVESTMENT ON ECONOMIC GROWTH: EVIDENCE FROM DEVELOPING ECONOMIES

In this paper, we investigate impact of inward and outward foreign direct investment on economic growth in developing economies over time period 2005-2014 annually. From 2005-2014, had inward and outward foreign direct investment had significant long-term effects on economic growth in developing countries. In this paper, we have used a different approach to examine the impact of FDI outflows and inflows on economic growth using Ordinary Least Squares (OLS) and Generalized Method of Moments (GMM) on the basis of macroeconomics panel data in developing economies over time span 2005-2014 annually. Our findings are as follows: (1) there is positive and significant impact of foreign direct investment outflows and inflows on economic growth in the long run among developing economies and (2) The positive and significant effects of FDI inflows and outflows on economic growth are highly robust by applying different econometric techniques (3) From our econometric results, we infer that control variables play significant role in defining exact relationship between inward FDI, outward FDI and economic growth. The control variables should be relevant and related with the econometric model to derive accurate relationship between inward FDI, outward FDI and economic growth.

INTRODUCTION
foreign direct investment (fdi) outflows from developing countries have grown faster in last thirty years than those from developed countries. according to united nations conference on Trade and development (uncTad) data, the share of developing countries in total world fdi outflows increased more than thirty-fold from 0.5% in the early 1970s to about 16% in 2008. fdi outflows from developing countries reached approximately 300 billion us dollars in 2008, which is more than three times the value of world fdi outflows in 1970. 1 developing countries' outward foreign direct investment flows have reached $426 billion, a record 31 per cent of the world total outflows (uncTad, 2013).
The relationship between foreign direct investment and economic growth has been subject of large academic literature over last decades. Borensztein et al. (1998) analyze effect of fdi on economic growth based on following periods 1970-1979 and 1980-1989.They concluded that fdi effect economic growth positively and fdi and domestic investment are complementary. Bengoa and sanchez-robles (2003) conclude that fdi influenced economic growth positively only when these countries have sufficient human capital, liberalized markets and economic stability using panel data of 18 countries in latin america. soltani and ochi (2012) found causal relationship between economic growth and fdi in Tunisia using data for time span  in Tunisia. They concluded that fdi causes economic growth significantly. de mello (1997) concluded that effects of fdi on economic growth of the host country depends on the degree of efficiency of local firms. The long run growth effects depend on rate of time preference and on productivity of domestic capital and foreign capital as well as degree of complementary behavior between domestic and foreign capital. recent studies by herzer (2008) for industrialized countries suggest significant and positive effect of fdi outflows on domestic output in the long-run. The effects of inward and outward fdi may differ between developed and developing countries. firstly, financial markets are under developed in many developing countries and many firms do not have access to foreign capital markets in developing countries, developing country multinational companies are generally more likely to face financial constraints than developed country multinationals. secondly, fdi outflows can reduce domestic capital, and thus reduce domestic output, when outward investors have scarce domestic resources. Thirdly, ability to absorb knowledge from abroad depends on absorptive capacity of investing firm, firms with low levels of technological capacity are likely to be unable to effectively access knowledge through fdi outflows (uncTad, 2006). anwar and sun (2011) found that fdi and domestic capital have positive and significant effects on economic growth. adams (2009) analyzed effects of fdi on economic growth in sub-saharan africa and found that fdi is positively and significantly correlated with economic growth. azman-saini et al. (2010) explored relationship between fdi and economic growth and concluded that fdi has no direct effects on output growth. Tang et al. (2008) found that there is uni-directional causality running from fdi to economic growth in china, while causal relationship between domestic capital and economic growth is bilateral. here in this research study, we will analyze impact of inward and outward fdi on economic growth in developing economies. Best to our knowledge, no previous studies have tried to find the impact of fdi outflows and inflows on economic growth in developing economies. Thus, we have tried to bridge the gap in the previous research studies by exploring impact of fdi outflows and inflows on economic growth in developing countries. Therefore, we have found it interesting to find impact of fdi outflows and inflows on economic growth in developing economies by introducing interesting and new research findings in the existing research literature.
in this paper, we examine the impact of fdi outflows and inflows on economic growth using ordinary least squares (ols) and generalized method of moments (gmm) on the basis of macroeconomics panel data in developing economies over time span 2005-2014 annually. our findings are as follows: (1) there is positive and significant impact of foreign direct investment outflows and inflows on economic growth in the long run among developing economies and (2) The positive and significant effects of fdi inflows and outflows on economic growth are highly robust by applying different econometric techniques (3) governance indicators show significant and negative effects on economic growth by applying different techniques as well as control variables show significant effects on economic growth in developing economies.(4) from our econometric results, we infer that control variables play significant role in defining exact relationship between inward fdi, outward fdi and economic growth. The control variables should be relevant and related with the econometric model to derive accurate relationship between inward fdi, outward fdi and economic growth.
The paper is organized as follows: section 2. describes data and sample selection; section 3. explains econometric model; section 4. estimation method; section 5. empirical results and section 6. conclusion

Waqar Ameer, Helian Xu
The long-run effecT of inward and ouTward foreign direcT invesTmenT stability. di is the domestic investment of country i in year t; ifdi is foreign direct investment inflows of the country i in year t; ofdi is outward foreign direct investment of the country i in year t; and ε it is the error term. The starting period of this data set is determined by the earliest available data. we have used net ofdi rather than the gross ofdi because the gross inward and outward fdi figures reflect the sum of the absolute outflow and inflow values in the balance of payment financial accounts and thus do not take into account disinvestment. Because the net inflows and outflows have negative values in some years, it is not possible to use logarithms. Thus, it is common practice in research to use net fdi as a percentage of the gdP to derive economically interpretable results. data on the net fdi outflows as a percentage of the gdP is taken from the uncTad fdi database. gdP, trade (% gdP), gdPg (% annual), gross capital formation (% gdP), labor force; total, inflation, gdP deflator (annual %), governance and the gdP deflator are taken from world Bank, world development indicators and world governance indicators database. This study uses a governance dataset compiled by worldwide governance indicators (wgi) over time span 2005-2014 annually for six dimensions of governance, i.e., control of corruption, government effectiveness index, Political stability and absence of violence, regulatory Quality index, rule of law, and voice and accountability (Kaufmann, Kraay, and mastruzzi, 2007). The control of corruption index captures perceptions of corruption including both petty and grand forms of corruption. The government effectiveness index captures the quality of bureaucracy, the competency of civil servants and government's commitment to policies. Political stability and absence of violence measures perceptions of likelihood of social unrest, terrorism, violent demonstrations, and security risk rating, etc. The regulatory Quality index measures price controls, inadequate bank supervision and perceptions of burdens imposed by excessive regulations such as foreign trade, business development etc. The rule of law index captures enforceability of contracts and the effectiveness of judiciary. voice and accountability captures different aspects of political process, civil liberties and independence of the media. The correlation matrix for governance indicators are displayed in Table 2. globerman and shapiro (2002) have illustrated that these indices are highly correlated with each other; therefore, it is very difficult to use all in single regression model. from an econometric point of view, the high correlation between the variables can cause multicollinearity and might reduce the extent to which the relevance of each individual governance indicator can be measured. daude & stein (2007) note that the standard solution is to group the variables into one aggregate component that measures similar dimensions. as a result, we follow globerman and shapiro (2002) by extracting the first principal component of six governance indicators by employing factor analysis. as displayed in Table 1, the governance indicator is used in our econometric model ranges from -2.38547 to 2.84429. The observed mean value of 0 and standard deviation is 1.0 is very similar with globerman and shapiro (2002) estimates. all independent variables are drawn from the world development indicators (wdi) and world governance indicators (wgi) database.The countries included are afghanistan, albania, angola, armenia, azerbaijan, Bahamas, The, Bangladesh, Belarus, Belize, Benin, Bolivia, Bosnia and herzegovina, Botswana, Brazil, Bulgaria, Burkina faso, Burundi, cambodia, cameroon, chile, china, colombia, congo, rep., costa rica, cote d'ivoire, dominican republic, egypt, arab rep., el salvador, ga-(5 -24)

ECONOMETRIC MODEL
in this section, we construct the econometric model and explain it in detail. following previous studies and economic intuition, we formulate an econometric model where we assume that level of economic growth depends upon the level of economic growth in the previous years, on outward fdi, inward fdi and a list of control variables that captures economic conditions in developing economies. i,t is a vector of control variables. we have added this list of control variables, namely, governance (gov), domestic investment (di), inflation and Trade. we consider governance (gov) very important control variable, given that pivotal and significant role of governance at the macroeconomic level. Thus, we consider governance as very important factor of economic growth.
where i = 1, 2, 3,...,N; t = 1, 2, 3,…T, i is the home country, t is the time, α s and β are unknown parameters to be estimated, η is the unobserved country-specific effects, and ε is the random disturbance term. The primary interest of our analysis is the sign and magnitude of the estimated coefficient of fdi inflows (ifdi) and outward fdi (ofdi). The control variables are selected based on existing research literature. The past values of economic growth are expected to have positive effects on current economic growth because it may be a sign of good and healthy investment environment. each country's economic stability plays significant role in economic development. macroeconomic instability is measured by inflation rate and is generally considered to have a negative association (greene and villanueva, 1991; serven and solimano, 1993; oshikoya, 1994; ndikumana, 2000).
The relationship between economic growth and outward foreign domestic investment as well as economic growth and foreign direct investment inflows, has been subject of large academic research over the past few decades. nguyen and nguyen (2010) have identified bi-directional relationship between fdi and economic growth in which fdi promotes economic growth and, in turn, economic growth is regarded as tool to attract fdi. Tsai (1994) applied a simultaneous equation system to examine bi-directional relationship between fdi and economic growth for 62 countries between 1975-1978 and 51 countries for period [1983][1984][1985][1986].his work supports that two-way relationship exists between fdi and economic growth. anwar and nguyen (2010) study bi-directional relationship between economic growth and fdi in 61 provinces of vietnam over time span [1996][1997][1998][1999][2000][2001][2002][2003][2004][2005].They support view that two-way relationship between fdi and economic growth exists in vietnam and explored relationship between fdi and economic growth across seven regions of vietnam. The empirical results suggest that bi-directional relationship exist between fdi and economic growth only in four regions.
anwar and sun (2011) have also shown that foreign direct investment and domestic capital have positive significant impact on economic growth. adams (2009) analyzed that impact of fdi on economic growth in sub-saharan africa and found that fdi and economic growth are positively and significantly correlated with each other. azmansaini et al. (2010) found linkage between fdi and economic growth and proved that fdi by itself has no positive direct effect on output growth. on the contrary, Tang el al. (2008) found that there is one-way causality from fdi to economic growth in china, while relationship between domestic investment and economic growth is bilateral.

ESTIMATION METHOD
we use the system-generalized method of moments (gmm) one-step and two-step estimator developed by arellano and Bover (1995) and Blundell and Bond (1988) for our estimates. The arellano-Bover/Blundell-Bond estimator is referred to as a-B-B estimator. gmm is generally used to study dynamics of adjustment using samples with relatively large cross-sections and short time periods. in order to measure the effects of fdi inflows and fdi outflows on economic growth in the home country, this research study uses the system-gmm estimator developed by arellano and Bover (1995) and Blundell and Bond (1998), which yield consistent and efficient estimates by addressing two key econometric issues.
considering equation (1): this includes one of the explanatory variables of the lagged level of economic growth. firstly, the presence of a lagged dependent variable would yield biased estimates because ordinary least square estimates (ols) leads to auto-correlation because of the correlation between error terms and lagged dependent variable (i.e., explanatory variable). using ordinary least squares (ols) would make estimations inconsistent and bias the coefficient of lagged terms upwards, while using the fixed-effects would cause a downward bias in estimated results. The system-gmm estimator controls for unobserved country-specific factors and the estimated coefficients would not be biased from an omitted variable. secondly, fdi inflows and fdi outflows are endogenous and jointly determined with economic growth. Thus, there is a two-way causality running between economic growth and fdi inflows as well as economic growth and outward fdi. it is very difficult to find appropriate instrument for inward fdi and (5 -24)

Waqar Ameer, Helian Xu
The long-run effecT of inward and ouTward foreign direcT invesTmenT outward fdi and thus system gmm estimator resolves the endogeneity issue by using instruments based on lagged values of dependent and independent variables.
To resolve these issues, arellano and Bond (a-B) (1991) recommend a first difference a-B gmm estimator. one advantage of this is that endogenous regressors and the lagged dependent variable can be instrumented using its lagged levels. The other advantage is that it also removes fixed country-specific effects by taking first differences of equation (1), thus removing individual specific effects, as reported below in equation (3).
Blundell and Bond (1998) point out that the first-differenced gmm estimator developed by arellano and Bond (1991) has poor finite sample bias and poor precision when lagged levels of series are weak instruments for the first differences, specifically for variables that are close to a random walk. The system-gmm model overcomes this problem by combining in one system the regression in differences with the regression in levels under the assumption. in equation (3), given assumption of no autocorrelation between error terms and regressors or regressors and error terms, the minimum lag level of dependent variables must be two or greater.
a key limitation of the first difference gmm estimator is that it does not necessarily remove first order serial correlation in the residuals because instruments used to control endogeneity are weakly exogenous in the regression. Thus, in our study, we use one step and two-step system-gmm to control for weak instrument problems by using a level equation to obtain a system of two equations. The first equation includes instruments in first differences, while the second equation includes instruments in levels. By inclusion of the second equation, the variables in the first differences are instruments for variables in levels, which make estimates more efficient and consistent. The use of two-step gmm makes standard covariance matrix robust to panel-specific autocorrelation and heteroscedasticity. for testing the validity of the one step and two-step system-gmm model, the system-gmm estimator checks for validity of instruments using the sargan/hansen test for over-identifying restrictions. The second-order serial correlation of the differenced error term is also tested for the null hypothesis that there is no serial correlation.

OLS Results
some studies employ cross-sectional regressions to test the relationship between fdi inflows, fdi outflows and economic growth. This study starts with crosssectional analysis in order to test relationship between fdi inflows, fdi outflows and economic growth over time span 2005 to 2014 annually. The ols cross-sectional results are reported in Table 3. Source: authors' results note: system-gmm is applied for estimation. The t-statistics are in brackets. *,** and *** indicate 10%, 5%, and 1% level of significance respectively. ar1 and ar2 are tests for first and second order serial correlation, respectively. The variables are lagged gdP (Previous year gdP growth), ifdi (inward fdi), ofdi (outward fdi), Trade, inflation, gov(governance), Trade openness (Trade), gdP (gdP growth), and domestic investment.

Waqar Ameer, Helian Xu
The long-run effecT of inward and ouTward foreign direcT invesTmenT Table 3. ols cross-sectional results show that estimated coefficient of outward fdi is positive and significant at 10 % level of significance across columns (1) -(7). ols results show that estimated coefficient of fdi inflows have also positive and significant at 5 % and 10 % level of significance. The positive and significant effects of fdi inflows and outflows on economic growth remain unchanged using aggregate governance and individual governance indicators along with other control variables, as reported in Table 3 in models 1-7. The results show that lagged value of gdP (Previous year of economic growth) has strongly positive and significant effects on current gdP (current year of economic growth). across models 1-7 in Table 3, our general findings show that economic growth in previous years have consistently highly positive and significant effects on current economic growth, particularly at the 1 %, 5 % and 10 % level of significance, and a 1% increase in economic growth in previous years increases current economic growth in range of 35.55% -36.64%. as results reported in Table 3, a one percent increase in the lagged dependent variable (economic growth in previous years) leads to an increase in current economic growth by 35.55% in model 1, 35.84% in model 2, 35.97% in model 3, 36.64% in model 4, 35.74% in model 5, 36.06% in model 6, and 35.97% in model 7. The other control variables, i.e. (domestic investment and inflation) have positive and highly significant effects on economic growth but trade have insignificant effects on economic growth. it implies from statistical results that increase in domestic investment stimulates economic growth as well as inflation motives economic growth positively. indeed, the cross-sectional regression results show that there is a strong positive relationship between fdi inflows and its economic growth as well as outward fdi and economic growth are strongly positively associated. Yet, cross-sectional analysis ignores time-series fluctuations and only tests the relationship between outward fdi, inward fdi and economic growth in the long run. such econometric methodology cannot capture or control for unobserved country-specific effects that can vary across countries and may be correlated with inward fdi and outward fdi.

GMM-One
Step Table 4. contains the principal empirical result findings and reports results for one step system gmm in columns (1) -(7). across columns (1) -(7) in Table  4, our general result findings are that, in all cases, fdi inflows has consistently positive and significant effects on economic growth particularly at 5 % and 10 % level of significance, a 1% increase in fdi inflows increases economic growth in range of 18.54 % -22.36% in models (1) -(7). our general result findings are that, in all cases, fdi outflows have consistently positive and significant effects on economic growth particularly at 5% and 10 % level of significance, a 1% increase in fdi outflows increases economic growth in range of 73.25% -76.70% across columns (1) -(7) in Table 4. our result findings show across models (1) -(7) in Table 4, in all cases, governance have significant and negative effects on economic growth. our result findings show that governance is negatively associated with economic growth. Perhaps this negative relationship between governance and economic growth may be due to weak and poor governance indicators, i.e. weak control of corruption; government ineffectiveness; poor law and order; Political instability etc. governance indicators should be strong enough to offset negative effects on economic growth in the long run and have positive and significant effects on economic growth as a result stimulate economic growth. strong governance indicators can result in increased fdi inflows and outflows and stimulate economic growth.

Waqar Ameer, Helian Xu
The long-run effecT of inward and ouTward foreign direcT invesTmenT Source:authors' results note: system-gmm is applied for estimation. The t-statistics are in brackets*,** and *** indicate 10%, 5%, and 1% level of significance respectively. ar1 and ar2 are tests for first and second order serial correlation, respectively. The variables are lagged gdP (Previous year gdP growth), ifdi (inward fdi), ofdi (outward fdi), Trade, inflation, gov(governance), Trade openness (Trade), gdP (gdP growth), and domestic investment.
as results reported in Table 4., the estimated results for fdi inflows is statistically significant and positive at the 1% level of significance, which complies with prior research studies. The estimated coefficients are stable and robust with different model specifications. from our findings, in all cases, fdi inflows have consistently significant and positive impacts on economic growth at the 1% level of significance. regarding ifdi in Table  4, a one percent increase in fdi inflows lead to increase in economic growth by 21.64% in model 1, 21.70% in model 2, 18.54 % in model 3, 22.36% in model 4, 19.37% in model 5, 20.17 % in model 6, and 21.40 % in model 7. from our findings, in all cases, fdi outflows have consistently significant and positive impacts on economic growth at the 1% level of significance. regarding ofdi in Table 4, a one percent increase in fdi inflows lead to increase in economic growth by 73.88% in model 1, 75.03% in model 2, 73.25% in model 3, 76.70% in model 4, 75.73 % in model 5, 75.67 % in model 6, and 74.46 % in model 7.The positive and significant effects of outward and inward fdi on economic growth still remain unchanged using the aggregate governance variable and individual governance indicators along with other control variables, as displayed in Table 4. in models 1-7.

GMM-Two
Step as results reported in Table 5., the estimated results for outward fdi is statistically significant and positive at 5 % and 10 % level of significance. The estimated coefficients are stable and robust with different model specifications. from our findings, in all models, outward fdi has significant and positive effects on economic growth at the 10% level of significance. regarding ofdi in Table 5, a one percent increase in outward fdi increases economic growth by 75.14% in model 1, 75.94% in model 2, 74.75% in model 3, 77.30 % in model 4, 76.89 % in model 5, 77.06 % in model 6 and 74.87% in model 7. The significance of positive relationship between outward fdi and economic growth remains unchanged, even after using aggregate and individual governance indicators along with independent and control variables in Table 5. in models 1-7. regarding ifdi in Table 5., a one percent increase in fdi inflows increases economic growth by 20.48 % in model 1, 20.48% in model 2, 16.62% in model 3, 21.92% in model 4, 17.82% in model 5, 18.62% in model 6 and 21.33% in model 7. The positive and significant effects of fdi inflows and outflows on economic growth still remain unchanged using the aggregate governance variable and individual governance indicators along with other control variables, as displayed in Table 5 in models (1) - (7). now, we discuss other two very important variables in our model: economic growth and governance. economic growth in previous years (lagged dependent variable) have positive and significant effects on current economic growth in all models. The result findings displayed in Table 5. show that previous year economic growth robustly enhances the current economic growth rate. across models 1-7 in Table 5, our general findings show that economic growth in previous years have consistently highly positive and significant effects on current economic growth, particularly at the 1% level of significance, and a 1% increase in economic growth in previous years increases current economic growth in range of 13.27% -15.96%. as results reported in Table 5, a one percent increase in the lagged dependent variable (economic growth in previous years) leads to an increase in current economic growth by 13.91% in model 1, 13.27% in model 2, 15.96% in model 3, 13.43% in model 4, 14.18% in model 5, 14.16% in model 6, and 14.02% in model 7 in Table 5, we report results of seven econometric models, referred to as models 1-7, respectively. Based on equation (3), our core model specification comprises of the previous year's economic growth, ifdi, ofdi, and governance. in order to control for endogeneity between economic growth, ifdi and ofdi, we include control variables in our econometric model. we include set of control variables, namely, governance, inflation, trade and domestic investment, given their strong influence found in previous research studies. domestic investment has positive and significant effects on current economic growth across all models (1) -(7) at 1 % level of significance. inflation and Trade have also positive and significant effects on current economic growth across all models (1) -(7) at 5% and 10% level of significance. our statistical results show that increase in domestic investment stimulates economic growth as well as inflation and Trade have positive effects on economic growth. (5 -24)

Waqar Ameer, Helian Xu
The long-run effecT of inward and ouTward foreign direcT invesTmenT The sargan test and serial correlation test results are displayed in Table 5. across all seven models, the sargan tests suggest that the null hypothesis of validity of instruments cannot be rejected. The serial correlation test results suggest that there are first-order serial correlations, which are usually expected, but there is no evidence of second-order serial correlation in the differenced error terms. we also report differences in hansen tests to confirm validity of each subset of instruments in Table 5. again, the hansen tests do not reject the null hypothesis of the joint validity of all the instruments.

CONCLUSION
This paper has analyzed long run effects of inward and outward fdi on economic growth in developing economies over time span 2005-2014 annually. in this paper, we have used a different approach to examine the impact of fdi outflows and inflows on economic growth using ordinary least squares (ols) and generalized method of moments (gmm) on the basis of macroeconomics panel data in developing economies over time span 2005-2014 annually. The result findings are robust by using different econometric techniques. our findings are as follows: (1) there is positive and significant impact of inward and outward fdi on economic growth in the long run among developing economies and (2) The positive and significant effects of fdi inflows and outflows on economic growth are highly robust by applying different econometric techniques (3) governance indicators show significant and negative effects on economic growth by applying different techniques as well as control variables show significant effects on economic growth in developing economies.(4) from our econometric results, we infer that control variables play significant role in defining exact relationship between inward fdi, outward fdi and economic growth. The control variables should be relevant and related with the econometric model to derive accurate relationship between inward fdi, outward fdi and economic growth.
Perhaps this negative relationship between governance and economic growth may be due to weak and poor governance indicators, i.e. weak control of corruption; government ineffectiveness; poor law and order; Political instability etc. governance indicators should be strong enough to offset negative effects on economic growth in the long run and have positive and significant effects on economic growth as a result stimulate economic growth. strong governance indicators can result in increased fdi inflows and outflows and stimulate economic growth. in sum, our result findings support that inward and outward fdi have positive and significant impact on economic growth in developing economies in long run. in developing economies where inward outward fdi are very high in last two decades, it is expected that fdi inward and outward fdi have significant and positive effects on economic growth in the long run. outward and inward fdi

Waqar Ameer, Helian Xu
The long-run effecT of inward and ouTward foreign direcT invesTmenT have increased rapidly in developing countries in last decade which strongly support our result findings. our results strongly agree with herzer (2011) result findings that fdi outflows have positive and significant impact on domestic output in developing economies in the long run.


Feed Efficiency Estimates in Cattle: The Economic and Environmental Impacts of Reranking

This paper proposes the application of hierarchical models to the assessment of feed efficiency in beef cattle. Using a large dataset comprising 5600 cattle assembled from different experimental studies, feed efficiency rankings of cattle were estimated using the proposed approach. This was compared to more commonly used linear, and nonlinear estimators. A phenotypic selection scheme that selects cattle at the means of different percentiles was developed to illustrate potential economic and environmental outcomes resulting from changes in feed efficiency rankings. The former involved the specification of a multi-year stochastic farm simulation model. In general, our results show that improved feed efficiency is associated with positive economic and environmental benefits. A unit reduction in feed intake (kg as fed/day) is associated with an average increase of $13.23 in net returns and 33.46 tonnes reduction in emission at the end of the feeding period. We also find that feed efficiency ranking of cattle is sensitive to estimation approach. The within percentile mean estimates of the hierarchical model were comparable to the conventional linear estimator. There were, however, deviations at the tails of feed efficiency distributions where selection is most likely to occur.

Introduction
Profit margins in cattle feeding are thin and feed cost constitutes a major component of total production cost (Dilenzo and Lamb, 2012). In fragmented production systems where cattle breeding and feeding are distinct operations, reduced margins from higher feed costs at the feeder end means lower valuation of weaned calves. Feed utilization by cattle is also linked with methane from enteric fermentation. Methane is an important greenhouse gas with an estimated global warming potential 1 of 28-36 times that of carbon (Environmental Protection Agency, 2016). This implies that improvement in feed efficiency has important environmental and economic implications for cattle producers. This notwithstanding, improvements in the trait has occurred at a much slower pace as compared to other traits. Figure 1 shows genetic trends in birthweight, weaning weight, feed intake and feed efficiency (measured as residual average daily gain) in Angus cattle. Further, the pork and poultry sectors have seen much higher improvements in the trait as compared to cattle (Basarab, Baron, López-Campos, Aalhus, Haugen-Kozyra, & Okine, 2012).  Beef, 2015) Improvements in feed efficiency through breeding requires the identification of high performing individuals and the linking of this information back to their parents (bulls and dams) and/or their genomic information. In breeding programs this requires both genotypic and phenotypic data. The latter refers to actual feed intake and performance data. Feed intake is affected by both genetic and environmental factors. This suggests that accurately accounting for these different sources of heterogeneity is critical for several reasons. The most important being that errors in the feed efficiency estimate of an individual can distort its ranking within a cohort of cattle. This can, in turn, affect the valuation of performance of the parents of these animals leading to errors in selection. The intertemporal nature of breeding decisions means the economic and environmental implications of these errors can be far-reaching.
Studies (e.g. Robinson, 2005a;Robinson 2005b;Berry & Crowley, 2013) have evaluated aspects of RFI estimation. Robinson (2005a) examined the impact of measurement errors in covariates in the context of residual feed intake estimation. Errors in covariates were found to be a significant source of bias in the resulting coefficient estimates. Approaches such as inverse regression were suggested as a means of correcting for the effect of these errors (Robinson 2005a). Robinson (2005b) compared different estimates of weight gain using feed efficiency data. Berry and Crowley (2013) noted possible nonlinear relationships between feed intake, average daily gain (ADG) and metabolic mid-weight (MWT) amongst diverse populations and animals with inferior genetic merit for these traits. Basarab, Price, Aalhus, Okine, Snelling and Lyle (2003) found evidence of the possibility of re-ranking cattle based on efficiency estimates from the different linear models used. The use of more robust empirical techniques and the impact of errors in estimation, as it pertains to the economic and environmental wellbeing of beef farms, have not been previously explored.
In the light of the foregoing, this paper has two objectives. First, we estimate and rank a large dataset of cattle by feed efficiency estimator using a multi-level estimation technique and compare rankings with that of more commonly used approaches. Second, we evaluate the economic and environmental consequences resulting from changes in the feed efficiency ranking of cattle by the different estimation approaches. From this analysis, we can quantify the implications of model accuracy. In the economic analysis, we assess the impact of selection at different percentiles on profits using a stochastic farm model. We also track methane from enteric fermentation sources and evaluate differences in emissions by feed efficiency estimator.
The rest of the paper proceeds as follows. A brief overview of the existing literature on the measurement of feed efficiency is first presented. This is followed by a description of empirical approach and data sources. The results of the analysis are then presented. The final section provides concluding remarks.

Feed Efficiency, Measurement and Improvement
Feed efficient cattle consume less feed per unit measure of beef produced. Commonly used estimates of feed efficiency include: feed conversion ratios (FCR) (Brody, 1945) and residual feed intake (RFI) (Koch, Swiger, Chambers, & Gregory, 1963). Although extensively used in the past, the former has been less favored in recent years as a result of unfavorable correlations between FCR and its component traits (Nkrumah, Basarab, & Arthur, 1999). Residual feed intake (RFI), estimated as the difference between actual animal feed intake and predicted feed intake, is the prevalent measure of feed efficiency across the different livestock production systems (Koch, Swiger, Chambers, & Gregory1963;Archer, Richardson, Herd & Arthur, 1999;Rauw, Soler, Tibau, Reixach & Raya, 2006). Differences in RFI capture variations in the efficiency with which individual animals utilize feed for maintenance and production (Basarab et al., 2003;Johnson, Chewning & Nugent, 1999). Lower RFI values are more desirable since they imply more efficient conversion of feed into meat per unit of intake. Predicted feed intake is typically estimated as a linear regression of actual animal feed intake on a set of covariates such as average daily gain, mid metabolic body weight and measures of meat composition such as back fat (Hoque, Kadowaki, Shibata, Oikawa & Suzuki, 2009).
Previous studies (e.g. Arthur, Archer, Johnston, Herd, Richardson, & Parnell, 2001;Herd & Bishop, 2000;Basarab et al., 2003) have examined residual feed intake for different categories of beef cattle. This residual portion of feed intake forms the basis for identifying relatively more efficient-cattle with lower (negative) RFIs. It has been suggested that RFI could represent variations in metabolic processes which determine feed efficiency (Brelin & Brannang, 1982;Korver, 1988). As errors from the regression of covariates that capture size and growth, residual feed intake is phenotypically independent of these production traits. This implies that there will be little correlated effects of the selection for feed efficiency.  in a study of feed efficiency in Angus cattle, estimated a linear regression model of feed intake on metabolic weight and ADG controlling for group and sex effects. Residual feed intake was derived as the difference between an animal's actual feed intake and predicted feed intake given its' size and growth. Additionally, the study found evidence of genetic and phenotypic independence between RFI and component traits. Herd and Bishop (2000) assessed the existence of genetic variation in RFI in young bulls and the phenotypic and genotypic relationships between RFI and other important production traits such as mature cow size. The predicted feed equation was estimated as a linear regression of feed intake (FI) on MWT and ADG. Archer (1999) estimated RFI as residuals from the linear regression of feed intake on ADG and MWT. To capture possible heterogeneity across animals resulting from gender and treatment, separate models were estimated for each gender within each test cohort. Arthur, Renand and Krauss (2001) examined genetic and phenotypic relationships between different feed efficiency and growth measures in young Charolais bulls.
While measurement is a crucial aspect of the assessment of feed efficiency an equally important consideration is improvements in trait. Improvements in feed efficiency can be achieved in a variety ways. Table 1 is a summary of selected approaches. These include: through selection, nutrition, the use of growth promotants and management practices. Nutrition related practices include changes in diet composition. For example, the inclusion of higher levels of concentrate in feed is associated with higher feed conversion rates. Meyers (1999) reported a 25% increment in feed efficiency from cattle fed 90-100% concentrate as compared to 60% concentrate diets. The use of growth promotants such as implants, ionophores and beta-agonists is associated with higher levels of feed efficiency (4-23%). In addition to these, evidence from other studies (Myers, Faulkner, Ireland, Berger, & Parrett, 1999;Schoonmaker, Fluharty & Loerch, 2004;Barker-Neef, Buskirk, Black, Doumit, & Rust, 2001) that examined the effect of management practices on the feed efficiency report a negative relationship between weaning time and feed efficiency. In general, feed efficiency is higher in early weaned calves-9-8%. The reported rate of improvement in feed efficiency from selection (i.e. 0.75-1%) is relatively low in magnitude as compared to the other practices outlined. However, the genetic change is per year and is cumulative and permanent.

Methods
The collection of actual feed intake data of cattle imbedded in different treatment cohorts often requires the pooling of data from multi-sources. A major challenge in analyzing datasets with large sources of heterogeneity is the potential to over-fit the data due to the incorporation of a large set of treatment effects. This context however, provides the ideal condition for the application of estimation techniques that are more adapt to modelling heterogeneity. Gelman (2006) noted that hierarchical models represent a more robust approach when juxtaposed against standard completely pooled and unpooled regression in the analysis of nested data. Previous applications in beef cattle include: the assessment of culling risk (Babcock, Cernicchiaro, White, Dubnicka, Thomson, Ives, Scott, Milliken & Renter, 2013); assessment of variances in average daily gain (Cernicchiaro, Renter, Xiang, White & Bello, 2013).
We propose the application of a hierarchical model approach to the assessment of cattle feed intake data. In estimating the hierarchical model, it is assumed that animals (level 1) are nested within treatment cohorts (level 2) which are nested within the location of the feeding experiment (level 3). The inclusion of level 3 is an attempt to capture region specific idiosyncratic effects common with multi-source dataset. Following Hedeker and Gibbons  (1994), the three level model is specified as: where subscripts k and u denote data source and contemporary respectively, and variance is 2  .For each animal i, the covariates in the feed intake equation are average daily gain and mid metabolic body weight. In addition to the hierarchical model both linear and quadratic functions are estimated. The breed, sex and contemporary group effects are included as dummy variables. Contemporary group refers to animals in the same treatment cohort. For each of these estimators, the corresponding RFI estimate for the ith cattle is derived as the difference between actual and predicted feed intake: where NR is net returns f p is finisher price of cattle, f y is cattle at finisher weight, w p is the price at purchase weight, w y cattle at purchased weight, FC is feed cost, DMI is dry matter intake, rfi is residual feed intake, p f is the price of feed and OC refers to other production costs(veterinary, bedding, transportation costs). Net returns are discounted at 5%.
Cattle are assumed to be initially placed on a backgrounding diet from weaning (550 lbs) to 750 lbs before being transferred to a finisher diet. Cattle are finished at 1225 lbs (Carlberg & Brown 2001). It is assumed that the feeding operation has 5600 animals. Methane from enteric fermentation emitted over the feeding period is estimated as: where Emissions is total emissions from enteric fermentation, cattle is number of cattle, DOF is days on feed, DMI is dry matter intake, GE is gross content of diet assumed to be 18.45 MJ/kg dry matter, EF is emission factor for enteric fermentation assumed to be 6.5% and EC is the energy content of methane. The quantification equation is taken from the Alberta low RFI protocol (Environment Alberta, 2012).

Data and Data Sources
Feed intake data was collected from three research projects: Agriculture and Agri-Food Canada's(AAFC) Phenomic Gap project (PG1), the University of Alberta at the Kinsella ranch (KIN) and the Elora test station at the University of Guelph (UOG Data from multiple sources are used in the simulation analysis. Information on feed requirements and entry and exit weights for the feeding phase of the model were obtained from the Alberta beef cow-calf manual (Alberta Agriculture and Food, 2008). Data on overwinter feeding cost of cows and the feeding phase (i.e. backgrounding and finisher) is obtained from Alberta Agriculture and Rural Development (2004) and the Feedlot Investment Risk Simulation decision making tool.

Results
Tables 2 captures the descriptive statistics of DFI, MWT, ADG cattle and feed prices for the overall sample. Mean feed intake for the entire sample is approximately 9.15kg/day, over the range of 4.07kg/day (min) and 15.54kg/day (max). Mean MWT and ADG were 93.51kg and 1.47kg/day respectively. In general, steer prices were higher than heifer prices for both weaned and finished prices. Cattle prices were also inversely related with weight reflecting the so-called price slide. The mean feed barley price ($/tonne) was approximately $158. The cost of barley silage (i.e. $40) was about a third of the feed barley price. Further 73% of the cattle in the sample were male, whilst heifers constituted the remaining 27%. The dominant breeds were HY (12%), AN (12%), HE (30%) and TX (22%).  Table 3 shows the results of the multi-level model. It is assumed that cattle (level 1) are nested within treatment cohorts (level 2) which are nested within the location of the feeding experiment (level 3). In addition to coefficient estimates, the variance components and the variance partition coefficients are also reported.  Table 3, ADG and MWT positively affect feed intake. A unit increase in MWT is associated with a 0.08 increase (kg/day) in feed intake. A unit change in ADG is associated with a much higher response in feed intake-1.4 kg/day. In general, it seems that the greatest heterogeneity occurs at the contemporary group level. This is followed by the individual level. The source variance was least. Consequently, the VPC estimates were 0, 65 and 35% at the source, contemporary group and animal levels respectively. This suggests that 35% of the differences in feed intake are observable at the animal level as compared to (65%) at the contemporary group level. The breed and gender effects are all significant implying that controlling for these effects was indeed appropriate.
In Table 4, the results of the linear and nonlinear model estimates are presented. In these models, the treatment, breed and sex effects are included as dummy variables. The coefficient estimates of the linear model are generally consistent with the multi-level model although the magnitudes differ slightly. The coefficient estimate for MWT is marginally greater than the corresponding estimate in the multi-level model, the reverse applies to ADG. Also from Table 4, the nonlinear specification does not improve model fit. For each of the reported estimates, the feed efficiency score of each animal is calculated as the difference between actual and predicted feed intake. Feed efficiency scores ranged from a low of -4.02 kg (as fed/day) to a higher 4.16 kg (as fed/day). This implies that the most efficient cattle consumed 4.02 kg less feed when controlled for ADG and MWT as compared to the least efficient animal which eats 4.16 kg of feed more. The distribution of feed efficiency scores was comparable across the different estimators but the rankings of cattle differed.  Research Vol. 6, No. 2;2017 43 From Figure 2, the feed efficiency rankings of approximately 98-99% of cattle assessed changed by estimation approach. This translates to changes in 5500 animals in the total sample of 5600 animals. The greatest variability was between the multi-level and quadratic estimate whilst the multi-level and linear model showed the least variation in ranking. This notwithstanding, the results imply that estimation approach significantly impacts the feed efficiency scores and rankings of cattle.
To further illustrate the effect of the changes in ranking from the different estimates, a stochastic multi-year simulation model that tracks net returns and methane emissions from enteric fermentation was estimated. The size of the operation is assumed to be 5906 calves. For each estimator three intensities of selection are assessed-1, 5 and 10%. From the results of the multi-level estimates this translates to reductions (kg/day as fed) of 2.30(1%), 1.57(5%) and 1.24(10%). The corresponding rates of reduction (kg/day as fed) in the linear model are: 2.28(1%), 1.56(5%) and 1.23(105%). The rates of reduction (kg/day as fed) for the nonlinear estimation are: 2.29(1%), 1.55(5%) and 1.23(10%). Table 5 is a summary of results. Changes in methane and net returns from the multi-level estimates were more comparable to the linear estimator than the non-linear estimator. As showed in Figure 3, the differences between these measures were 0%, 0.24% and 0.6% for the linear model as compared 0-35% in the case of the nonlinear estimator. The highest differences occur at the upper tail of the distribution. It seems that as one moves further away from the tails of the distribution, the impact of estimation approach reduces. to multi-level model

Conclusion
As concerns continue to grow about the environmental impact of beef production, producers and researchers need to evaluate methods that can help to produce beef profitably while minimizing the effect on the environment. Using RFI (a measure of feed efficiency) as another selection criteria along with production traits can be an effective tool in achieving this objective. Historically, selection in beef cattle has focused on output traits resulting in differences in genetic improvements in these traits as compared to feed efficiency. Also, cattle are relatively less efficient as compared to other livestock-hogs and poultry. Breeding programs designed to improve the trait require the inclusion of both genotypic and phenotypic information. Even in the case of new DNA-based selection programs, phenotypic measurements are required in training population to validate genotypes. It is not unlikely that this may require data on cattle raised under different conditions and treatments. Feed intake however varies under the different conditions and the rigorous modelling of the different sources of heterogeneity can impact the accuracy of feed efficiency estimates. Given the multiplicity of factors that can influence feed intake, overfitting of conventional linear estimators can occur. This can negatively impact inferences made from parameter estimates. For feed efficiency, this can translate into errors in the ranking of the feed efficiency performance of cattle and the in selection of breeding stock for breeding. In this paper, we proposed and assessed the performance of a multi-level regression estimator and compared the feed efficiency rankings of cattle. Our results show that the ranking of cattle is sensitive to estimation approach. The multi-level estimator compared favorably with the conventional linear estimator when selections outcomes were assessed at the means of different percentiles. There were however minor deviations at the upper tail of the feed efficiency distribution where selection is most likely to occur. This suggests that the multi-level estimator can at the very least yield similar outcomes as conventional estimators whilst incorporating heterogeneity more robustly. We also assessed the economic and environmental impact of selection at different percentiles for the different estimators. This analysis was situated in the context of a feeding operation comprising 5600 animals. Our results show that selection for feed efficiency can increase net returns and reduce methane emission from enteric fermentation sources. A unit reduction in feed intake (kg as fed/day) was associated with an average increase of $13.23 in net returns and 33.46 tonnes reduction in emission at the end of the feeding period. On a percent intake basis, this translates into a 2.3% increase in net returns for each percent improvement in efficiency. This is lower than the 4.3% increase in profits reported in Fox, Tedeschi and Guiroy, (2001). The modelling assumptions in the two studies are not the same.
Selection for different traits in beef cattle requires performance data on multi-traits and genetic merit values. The genetic and phenotypic relationships with other traits are also considered together in formulating a breeding objective, and used to develop selection indices. Our focus on a single trait is limiting as we do not factor the economic implications of correlations with other traits. Ultimately, progeny performance is one input amongst  Vol. 6, No. 2;2017 45 many in the selection of breeding stock and our results must be interpreted keeping this caveat in mind. This notwithstanding, as showed in this study, rigorous empirical analysis of phenotypic data can have far-reaching implications.