Eric Duchene

Michel Rigo

Eric Duch

Invariant games

In the context of 2-player removal games, we define the notion of invariant game for which each allowed move is independent of the position it is played from. We present a family of invariant games which are variations of Wythoff's game. The set of P -positions of these games are given by a pair of complementary Beatty sequences related to the irrational quadratic number α k = (1; 1, k). We also provide a recursive characterization of this set.

We assume that the reader has some knowledge in combinatorial game theory. Basic definitions can be found in [START_REF] Berlekamp | Winning ways for your mathematical plays[END_REF]. The set of nonnegative (resp. positive) integers is denoted by N (resp. N ≥1 ).

Given an infinite sequence S = (A n , B n ) n≥0 of nonnegative integers with (A 0 , B 0 ) = (0, 0), a 2-player removal game on two heaps having S as set of P -positions can always be defined. Indeed, the following naïve rules can be chosen: from any position (x, y) not in S, there is a unique allowed move (x, y) → (0, 0). And from any position (A n , B n ) ∈ S, any move is allowed except those leading to another position in S. Such a definition for the rules is not satisfying. In general, game rules that are considered are those which can "easily be understood by a child". These considerations are fuzzy. Nowadays, there is no clear formal framework to decide the quality of given game rules.

We here propose an answer to this issue by introducing the notion of invariant games. An invariant game has rules that are independent of the actual position of the game. Definition 1. Consider a two-player impartial removal game G played on ℓ ≥ 1 piles of tokens. Positions and moves are thus coded by ℓ-tuples of nonnegative integers. For two ℓ-tuples x = (x 1 , . . . , x ℓ ) and y = (y 1 , . . . , y ℓ ), we write x ≺ y if x i ≤ y i for all i = 1, . . . , ℓ. The game G is invariant, if for all positions p = (p 1 , . . . , p ℓ ) and q = (q 1 , . . . , q ℓ ) and any move x = (x 1 , . . . , x ℓ ) such that x ≺ p and x ≺ q then, the move p → p -x is allowed if and only if the move q → q -x is allowed.

Otherwise stated, a game is invariant if the same moves can be played from any position, with the only restriction that enough tokens on the different piles are available.

We denote by M G ⊆ N ℓ the set of moves of G. If G is invariant, then the knowledge of M G is enough to play the game. On the other hand, a game for which at least one move depends on the actual position is called variant. In a variant game, some positions are associated with specific subsets of M G .

Example 1. For instance, the game of Nim [START_REF] Bouton | Nim, a game with a complete mathematical theory[END_REF] or Wythoff's game [START_REF] Wythoff | A modification of the game of Nim[END_REF] are invariant games. In Wythoff's game W , we have

M W = {(i, 0) | i ≥ 1} ∪ {(0, j) | j ≥ 1} ∪ {(i, i) | i ≥ 1}
and for the game of Nim on ℓ piles,

M N = {(i, 0, . . . , 0) | i ≥ 1} ∪ {(0, i, 0, . . . , 0) | i ≥ 1} ∪ • • • ∪ {(0, . . . , 0, i) | i ≥ 1}.
Other invariant games are given in [START_REF] Duchêne | Geometrical extensions of Wythoff's game[END_REF][START_REF] Fraenkel | Heap games, numeration systems and sequences[END_REF][START_REF] Fraenkel | A generalization of Wythoff's game[END_REF][START_REF] Fraenkel | A new heap game[END_REF] or also the subtraction games found in [START_REF] Berlekamp | Winning ways for your mathematical plays[END_REF].

Example 2. Games like the Raleigh game [START_REF] Fraenkel | The Raleigh game, Combinatorial number theory[END_REF], the Rat and the Mouse game [START_REF] Fraenkel | The Rat and the Mouse game[END_REF],

Tribonacci game [START_REF] Duchêne | A morphic approach to combinatorial games : the Tribonacci case[END_REF] or Cubic Pisot games [START_REF] Duchêne | Cubic Pisot Unit Combinatorial Games[END_REF] are variant. Nevertheless, these games remain appealing, since the dependence of the game rules to the actual positions is restricted to some simple logical formula.

One can however wonder if there exist invariant games having the same sets of P -positions as those in Example 2. More generally, for any sequence S : N → N ℓ , is there an invariant game having S as set of P -positions? The answer is negative. As an example, consider any sequence S = (A n , B n ) n≥0 starting with (0, 0), (1, 2), [START_REF] Bouton | Nim, a game with a complete mathematical theory[END_REF][START_REF] Duchêne | Geometrical extensions of Wythoff's game[END_REF], [START_REF] Duchêne | Extensions and restrictions of Wythoff's game preserving wythoff's sequence as set of P positions[END_REF][START_REF] Duchêne | A morphic approach to combinatorial games : the Tribonacci case[END_REF] and such that {A n | n ≥ 1} and {B n | n ≥ 1} make a partition of N ≥1 . There is no invariant game having S as set of P -positions because from the N -position (1, 1), one must play to (0, 0). Hence the move (1, 1) belongs to the set of rules. But playing from [START_REF] Duchêne | Extensions and restrictions of Wythoff's game preserving wythoff's sequence as set of P positions[END_REF][START_REF] Duchêne | A morphic approach to combinatorial games : the Tribonacci case[END_REF] to [START_REF] Bouton | Nim, a game with a complete mathematical theory[END_REF][START_REF] Duchêne | Geometrical extensions of Wythoff's game[END_REF] is not allowed (there is no move between two P -positions).

We already know that Wythoff's game is invariant. Notice that its set of Ppositions is given by a pair of complementary (homogeneous) Beatty sequences [START_REF] Beatty | Problem 3173[END_REF]. A pair of complementary homogeneous Beatty sequences is of the form (⌊nα⌋, ⌊nβ⌋) n≥1 , with α > 1 an irrational number, and β = α/(α -1). Non-homogeneous Beatty sequences are those of the form (⌊nα + a⌋, ⌊nβ + b⌋) n≥1 , with a and b any two real nonzero numbers.

Indeed, it is proved in [START_REF] Wythoff | A modification of the game of Nim[END_REF] that the nth P -position of Wythoff's game is (⌊nτ ⌋, ⌊nτ 2 ⌋), where τ is the golden ratio. In [START_REF] Fraenkel | How to beat your Wythoff games' opponent on three fronts[END_REF], Fraenkel investigates an invariant extension of Wythoff's game where the set of P -positions is also given by a pair of complementary Beatty sequences build over the quadratic irrational number having (1; k), with k ∈ N ≥1 , as continued fraction expansion.

In this paper, we consider the sequence S = (A n , B n ) n≥0 build over the quadratic irrational number α k having (1; 1, k) as continued fraction expansion and then show that there exists an invariant game having S as set of P -positions. This result is a step towards the following general conjecture. In the rest of the paper, assume that k is fixed once and for all.

Conjecture 1. Given a pair of complementary Beatty sequences S = (A n , B n ) n≥1 , there exists an invariant game having S ∪ {(0, 0)} as set of P -positions.

The converse does not hold. As shown in [START_REF] Duchêne | Geometrical extensions of Wythoff's game[END_REF][START_REF] Fraenkel | Heap games, numeration systems and sequences[END_REF], there are invariant games whose set of P -positions cannot be described with a pair of Beatty sequences. Notice that the invariant game discussed in [START_REF] Fraenkel | A generalization of Wythoff's game[END_REF] has a set of P -positions given by a pair of non-homogeneous Beatty sequences.

This paper is articulated as follows. In the next section, we study some particular properties of the sequence (⌊nα k ⌋) n≥0 using the fact that (⌊(n + 1)α k ⌋ -⌊nα k ⌋) n≥0 is a Sturmian sequence, see Definition 2. In a second part, we present a family of invariant games which are variations of Wythoff's game. We obtain two characterizations of the set of P -positions. The first one is recursive (Theorem 3). The other one is based on the results given in the first section and it expresses the set of P -position using a pair of complementary Beatty sequences based on α k (Theorem 4). From the point of view of combinatorics on words, these two theorems provide a recursive definition of a family of Sturmian words.

Some technical results

Let k ≥ 1 be an integer. Let α k be the quadratic irrational number having (1; 1, k) as continued fraction expansion and β k be such that α -1 k + β -1 k = 1. We have thus defined [START_REF] Beatty | Problem 3173[END_REF]

α k = 1+ √ k 2 + 4k -k 2 ∈ [ 1 + √ 5 2
, 2) and

β k = 3 2 + √ k 2 + 4k 2k ∈ (2, 3 + √ 5 2
].

which are represented as functions of k in Fig. [START_REF] Beatty | Problem 3173[END_REF]. Notice that for k = 1, α 1 is exactly the golden ratio.

Definition 2. For any positive real number γ, we write

∆ γ (n) := ⌊(n + 1)γ⌋ -⌊nγ⌋.
It is well-known (see for instance [START_REF] Lothaire | Algebraic Combinatorics on Words[END_REF]) that for any irrational number γ the first difference sequence (∆ γ (n)) n≥1 is a Sturmian sequence over {⌊γ⌋, ⌊γ⌋ + 1}: for all ℓ ≥ 0, there are exactly ℓ + 1 distinct blocks of ℓ consecutive elements in the sequence, i.e., for all ℓ ≥ 0

#{∆ γ (i + 1) • • • ∆ γ (i + ℓ) | i ≥ 0} = ℓ + 1.
The results of this section describe some properties of the sequences (∆ α k (n)) n≥1 and (∆ β k (n)) n≥1 . The proofs use only elementary methods but some caution is needed.

Remark 1. The sequence (∆ α k (n)) n≥1 is a Sturmian sequence over {1, 2}. As an example, for k = 2, the first elements in (∆ α2 (n)) n≥1 are 22122212221221222122212221221222122212221221222122 • • •
For all k ≥ 1, the three factors 21, 12 and 22 appear in the sequence (∆ α k (n)) n≥1 but 11 does not occur. Proceed by contradiction and assume that there exists n such that ∆ α k (n) = 1 and ∆ α k (n + 1) = 1. Adding these two relations gives

k 2 + 4k -k = {(n + 2) α} -{n α}.
The l.h.s. is greater than 1 for all k ≥ 1 (α k is increasing with respect to k) but the r.h.s. is in (-1, 1) providing the contradiction. (See Remark 2 to have the description of the k + 1 factors of length k.)

The same observation can be made for all k ≥ 1, the sequence (∆ β k (n)) n≥1 over {2, 3} for which the factor 33 does not occur. For k = 2, the first elements in

(∆ γ2 (n)) n≥1 are 23223223232232232232322322322323223223232232232232 • • •
In what follows, we assume that k is implicitly understood as a parameter for α k and β k and we omit reference to k in the corresponding notation α and β.

Lemma 1. Let α, β given in (1) and n ≥ 1 be such that ∆ α (n) = 1. Then ∆ β (n) = 2.
Proof. We assume that α -{(n + 1) α} + {n α} = 1 and we have to show that β -{(n + 1) β} + {n β} = 2. For the sake of simplicity, we set [START_REF] Berlekamp | Winning ways for your mathematical plays[END_REF] γ

:= √ k 2 + 4k 2 . Since (k + 1) 2 < k 2 + 4k < (k + 2) 2
, it is easy to deduce that ⌊γ⌋ = ⌈k/2⌉. Notice that if k is even, then for all n ≥ 1, {n α} = {n γ}. If k is odd, then for all n ≥ 0, we have

{2n α} = {2n γ} and {(2n + 1) α} = {(2n + 1) γ} + 1 2 .
Now let's turn our attention to expressions involving β. For all n ≥ 0, we have

{2n β} = {2n γ/k} and {(2n + 1) β} = {(2n + 1) γ/k} + 1 2 .
For all n ≥ 1, if we consider the Euclidian division ⌊n γ⌋ = q n k + r n with r n ∈ {0, . . . , k -1}, we get

⌊n γ/k⌋ = q n and {n γ/k} = 1 k {n γ} + r n k .
Notice that

(3) β = 2 + α -1 k .
Case 1. Assume first n odd and k even. We have (4)

β -{(n + 1)β} + {nβ} = 2 + α-1 k -{(n + 1) γ/k} + {n γ/k} + 1 2 = 2 + 1 k α -1 -{(n + 1) γ} -r n+1 + 1 k {n γ} + rn k + 1 2 .
Notice that to obtain the above formula, we have only used the fact that n is odd.

1.a) If {n γ} + r n < k/2, then (5) β -{(n + 1)β} + {nβ} = 2 + 1 k α -1 -{(n + 1) γ} + {n γ} =0 +r n -r n+1 + 1 2 .
We have to show that r n -r n+1 = -k/2 when {n γ} + r n < k/2 and ∆ α (n) = 1. Since, for k even we have ⌊γ⌋ = k/2, one can notice that

(n + 1) γ = q n k + r n + {n γ} + k 2 + {γ}.
Observe that under the hypothesis ∆ α (n) = 1 and k even, we have ( 6)

{(n + 1) γ} -{n γ} = α -1 = γ -k/2 = {γ}.
Therefore, {(n + 1) γ} = {n γ} + {γ} and since r n + k/2 < k, we conclude that r n+1 = r n + k/2 and ∆ β (n) = 2. In every cases we are dealing with, we always have to check that the quantity r n+1 belongs to {0, . . . , k -1}.

1.b) If {n γ} + r n ≥ k/2, then (7) β -{(n + 1)β} + {nβ} = 2 + 1 k α -1 -{(n + 1) γ} + {n γ} =0 +r n -r n+1 - 1 2 .
Since r n is an integer and {nγ} < 1, we have in this case that r n ≥ k/2. Using the same arguments as in the previous case, we obtain

(n + 1) γ = q n k + k 2 + r n + {(n + 1) γ}
and we conclude that q n+1 = q n + 1 and

r n+1 = r n -k/2.
Case 2. Assume both n and k even. We have ( 8)

β -{(n + 1)β} + {nβ} = 2 + α-1 k -{(n + 1) γ/k} + 1 2 + {n γ/k} = 2 + 1 k α -1 + {n γ} + r n -1 k {(n + 1) γ} + rn+1 k + 1 2 . 2.a) If {(n + 1) γ} + r n+1 < k/2, then β -{(n + 1)β} + {nβ} = 2 + 1 k α -1 -{(n + 1) γ} + {n γ} =0 +r n -r n+1 - 1 2 .
With the same reasonings as before, we get ( 9)

n γ = (n + 1) γ -γ = q n+1 k + r n+1 + {(n + 1) γ} - k 2 -{γ}
and (6) holds. Since r n+1 < k/2, we deduce that q n = q n+1 -1 and r n = r n+1 +k/2.

2.b)

If {(n + 1) γ} + r n+1 ≥ k/2, then as in case 1.b) we know that r n+1 ≥ k/2.
Moreover, we have

β -{(n + 1)β} + {nβ} = 2 + 1 k α -1 -{(n + 1) γ} + {n γ} =0 +r n -r n+1 + 1 2 .
From ( 9), we deduce that r n = r n+1 -k/2.

Case 3. Assume n and k odd. We have again [START_REF] Duchêne | Extensions and restrictions of Wythoff's game preserving wythoff's sequence as set of P positions[END_REF]. Observe that a main difference with the first case is that

{(n + 1) γ} -{n γ} = {(n + 1) α} -{n α} + 1 2 .

3.a)

If {n γ} + r n < k/2 and {nα} < 1/2, then (5) becomes here

β -{(n + 1)β} + {nβ} = 2 + 1 k α -1 -{(n + 1) α} + {n α} =0 + 1 2 + r n -r n+1 + 1 2 .
Since k is odd, we have ⌊γ⌋ = (k + 1)/2 and (10)

(n + 1) γ = q n k + r n + {n γ} + k + 1 2 + {γ}.
Observe that under the considered hypothesis, we have

{(n + 1) γ} -{n γ} = α -3/2 = γ - k + 1 2 = {γ}.
Since {nα} < 1/2, then {nγ} > 1/2 and r n < (k -1)/2. We conclude that r n+1 = r n + (k + 1)/2 < k.

3.b)

If {n γ} + r n < k/2 and {nα} > 1/2, then (5) becomes here

β -{(n + 1)β} + {nβ} = 2 + 1 k - 1 2 + r n -r n+1 + 1 2 and {(n + 1) γ} -{n γ} = α -1/2 = γ - k + 1 2 + 1 = {γ} + 1.
Therefore, we have

(11) (n + 1) γ = q n k + r n + k -1 2 + {(n + 1) γ}
and we get r n+1 = r n + (k -1)/2 < k.

3.c)

If {n γ} + r n ≥ k/2 and {nα} < 1/2, then ( 7) becomes here

β -{(n + 1)β} + {nβ} = 2 + 1 k 1 2 + r n -r n+1 - 1 2
and (10) holds. Since r n is an integer and k is odd, we deduce from {n γ}+r n ≥ k/2 that r n ≥ (k-1)/2. Consequently r n +(k+1)/2 ≥ k and it follows that q n+1 = q n +1 and r n+1 = r n -(k -1)/2.

3.d)

If {n γ} + r n ≥ k/2 and {nα} > 1/2, then ( 7) becomes here

β -{(n + 1)β} + {nβ} = 2 + 1 k - 1 2 + r n -r n+1 - 1 2
and (11) holds. Since {nα} > 1/2, we have {nγ} < 1/2 and r n > (k -1)/2. Therefore, r n + (k -1)/2 > k -1 and it follows that q n+1 = q n + 1 and r n+1 = r n -(k + 1)/2.

Case 4. Assume n even and k odd. Here (8) holds and one proceeds following the same scheme as in the previous case but here, since n is even, one has to use the fact that

{(n + 1) γ} -{n γ} = {(n + 1) α} + 1 2 -{n α}.
The following result is equivalent to the previous one in the sense that any of these two results implies directly the other one.

Lemma 2. Let α, β given in (1) and n ≥ 1 be such that ∆ β (n) = 3. Then ∆ α (n) = 2.
Proof. Proceed by contradiction. Assume that ∆ β (n) = 3 and that ∆ α (n) = 1. Using the previous lemma, this latter equality implies that ∆ β (n) = 2 which is a contradiction.

In the following two statements, we are considering blocks of consecutive elements of some sequence. These blocks are written using the concatenation of symbols as product operation. Therefore, notation like 2 k means a repetition of k occurrences of the symbol 2.

Remark 2. We can easily show that for any n, the block ∆

α (n) • • • ∆ α (n + k -1)
of length k is one of the k + 1 elements (recall that we are dealing with a Sturmian sequence) of the set

{2 k } ∪ {2 i 12 k-i-1 | i = 0, . . . , k -1}. Indeed, D := ∆ α (n) + • • • + ∆ α (n + k -1) = ⌊(n + k) α⌋ -⌊n α⌋ = k α -{(n + k) α} + {n α}
and as a function of k, it is easy to check that k α > 2k -1 for all k ≥ 1 (see Fig. 2). Therefore, the integer D is such that D > 2k -2 and also D ≤ 2k, the maximal This means that any block of length k contains at most one 1. From this, one can also deduce that any block of length k + 1 in (∆ α (n)) n≥1 contains 0, 1 or 2 occurrences of 1.

In the same way, for all n ≥ 1, the block ∆

β (n) • • • ∆ β (n + k -1) of length k is one of the k + 1 elements in (12) {2 k } ∪ {2 i 32 k-i-1 | i = 0, . . . , k -1}.
Moreover, 2 k+1 is not a block of length k + 1 occurring in the sequence and 32 k-1 3 is the only block of length k + 1 containing two occurrences of 3. Indeed, assume that ∆

β (n) + • • • + ∆ β (n + k) = 2 k+1 . This implies (k + 1)β = 2k + 2 + {(n + k + 1) β} -{n β} ≤ 2k + 3.
But as a function of k, we have (k + 1)β > 2k + 3 (see Fig. 2). This proves that the block 2 k+1 does not occur. Moreover, any block of length k + 1 containing two occurrences of 3 in another configuration than 32 k-1 3 would lead to a block of length k with two occurrences of 3.

Lemma 3. Let α, β given in (1) and k ≥ 2. If ∆ α (n) • • • ∆ α (n + k -1) = 2 k , then ∆ β (n) • • • ∆ β (n + k -1) ∈ {2 i 32 k-i-1 | i = 0, . . . , k -1}. In particular, if ∆ α (n) • • • ∆ α (n + k -1) = 2 k and ∆ β (n) • • • ∆ β (n + k -2) = 2 k-1 , then ∆ β (n + k -1) = 3.
Proof. We set

A := 1 k (k α -{(n + k) α} + {n α})
and

B := α -(n + k) α -1 k + n α -1 k .
The first assumption can be written kA = ⌊(n + k) α⌋ -⌊n α⌋ = 2k, i.e., A = 2, and in view of ( 12), we have to show that ⌊(n + k) β⌋ -⌊n β⌋ = 2k + 1. Using (3), we get ⌊(n + k) β⌋ -⌊n β⌋ = 2k -1 + B. In particular, this implies that B is an integer. To conclude the proof, it is enough to show that for all n ≥ 1,

A -B ∈ {- 1 k , 0, 1 - 1 k }.
If this can be shown, then whenever A = 2, B being an integer and for k ≥ 2, -1 k and 1 -1 k being not integer, we must have A = B and ⌊(n + k) β⌋ -⌊n β⌋ = 2k + 1. We have

A -B = (n + k) α -1 k -n α -1 k - 1 k {(n + k) α} + 1 k {n α}.
For any m ≥ 1, we consider the Euclidian division of ⌊m (α -1)⌋ by k:

m (α -1) = q m k + r m + {m (α -1)} to define q m ∈ N and r m ∈ {0, . . . , k -1}. Therefore, for all m ≥ 1, ( 13
) m α -1 k = 1 k {m α} + r m k .
Now, observe that

(n + k) (α -1) = (q n + ⌊α -1⌋) k + r n + {n (α -1)} + k {α -1}
and we have to study k{α -1} = k{γ -k/2} using the definition [START_REF] Berlekamp | Winning ways for your mathematical plays[END_REF]. Since ⌊γ⌋ = ⌈k/2⌉, for k even, we have

k{γ -k/2} = k{γ} = k(γ -⌊γ⌋) = k √ k 2 + 4k -k 2
and for k odd, since ⌊γ⌋ = (k -1)/2 and ⌊γ + 1/2⌋ = (k + 1)/2, we get

k{γ -k/2} = k{γ + 1/2} = k(γ + 1/2 -⌊γ + 1/2⌋) = k(γ -k/2) = k √ k 2 + 4k -k 2 .
In both cases, we have the same function f (k) which can easily be seen (see Fig. 3 for a sketch of f (k) -k + 1) to satisfy

k -1 < k{γ -k/2} < k. So k{α -1} = k -1 + {k{α -1}} and (n + k) (α -1) = (q n + ⌊α -1⌋)k + r n + k -1 + {n (α -1)} + {k{α -1}} :=C .
The conclusion follows easily. We have three cases to consider • If C ∈ [0, 1) and r n = 0, then q n+k = q n + ⌊α -1⌋ and r n+k = k -1.

• If C ∈ [0, 1) and r n ∈ {1, . . . , k -1}, then q n+k = q n + ⌊α -1⌋ + 1 and

r n+k = r n -1. • If C ∈ [1,
2), then q n+k = q n + ⌊α -1⌋ + 1 and r n+k = r n . Consequently, using [START_REF] Fraenkel | A new heap game[END_REF],

A -B reduces to (r n+k -r n )/k ∈ {-1/k, 0, 1 -1/k}. Lemma 4. Let α, β given in (1) and k ≥ 1. If ∆ β (n) • • • ∆ β (n + k) = 32 k-1 3, then ∆ α (n) • • • ∆ α (n + k) = 2 k+1 .
Proof. We assume that [START_REF] Lothaire | Algebraic Combinatorics on Words[END_REF] k⌊(n + k + 1) β⌋ -k⌊n β⌋ = 2k(k + 2).

We proceed by contradiction and assume that ∆ α (n) • • • ∆ α (n + k) contains 1 or 2 occurrences of 1 (from Remark 2, these are the only cases to consider), i.e.,

⌊(n + k + 1) α⌋ -⌊n α⌋ ∈ {2k, 2k + 1}. Using (3), ⌊(n + k + 1) α⌋ -⌊n α⌋ is written (15) (k + 1)(kβ -2k + 1) -{(n + k + 1)kβ} + {nkβ} ∈ {2k, 2k + 1}.
Set β ′ = kβ and subtract ( 15) from ( 14). We have that

(k + 1)β ′ -k{(n + k + 1)β ′ /k} + k{nβ ′ /k} -(k + 1)β ′ -{(n + k + 1)β ′ } + {nβ ′ } ∈ {k, k + 1}
Proceeding as in the proof of the previous lemma, one can show that this expression divided by k:

1 k ({(n + k + 1)β ′ } -{nβ ′ }) -{(n + k + 1)β ′ /k} + {nβ ′ /k} belongs to {-1 k , 0, 1 -1 k } by noticing that ⌊β ′ ⌋ = 2k and {β ′ } > k/(k + 1)
. But this expression divided by k must belong to {1, 1 + 1 k } leading to a contradiction.

An invariant game

For all k ≥ 1, let us present a variation of Wythoff's game that we call G(α k ), this notation will become clear in a few lines (see Theorem 4). The set of moves of the invariant game G(α k ) is

M W \ {(2i, 2i) | 0 < i < k} ∪ {(2k + 1, 2k + 2), (2k + 2, 2k + 1)}
where M W is the set of Wythoff's moves. In other terms, this game can be described as follows: either take a positive number from a single pile, or take (2i, 2i), i < k from both, or 2k + 1, k > 0 from one and 2k + 2 from the other. The first player unable to move loses. Note that the rules of the game G(α k ) are due to computer experiments, so as to fit the Beatty sequence (⌊nα k ⌋, ⌊nβ k ⌋), as it will be shown in Theorem 4. Since the moves are symmetric on the two piles of tokens, we can restrict ourselves to positions (x, y) with x ≤ y. Recall that k was given once and for all, thus the P -positions will be denoted (A n , B n ). For instance, the first ones in the case k = 2 are n 0 1 2 3 4 5 6 7 8 . . . A n 0 1 3 5 6 8 10 12 13 . . . B n 0 2 4 7 9 11 14 16 18 . . .

Remark 3.

Observe that for k = 1, we have M W ∪ {(3, 4), (4, 3)} as set of moves for G(α 1 ) and it is shown in [START_REF] Duchêne | Extensions and restrictions of Wythoff's game preserving wythoff's sequence as set of P positions[END_REF] that adding such a move to M W does not change the set of P -positions of Wythoff's game. So the game G(α 1 ) has exactly the same set of P -positions as the classical Wythoff's game. Note that in [START_REF] Duchêne | Extensions and restrictions of Wythoff's game preserving wythoff's sequence as set of P positions[END_REF], it is also proved that adding any move of the form (k, k + 1) with k = 1, 2 does not change the set of P -positions of Wythoff's game.

Definition 3. Let k ≥ 2.
We define recursively a sequence (A n , B n ) n≥0 as follows.

(A 0 , B 0 ), (A 1 , B 1 ), . . . , (A k , B k ) = (0, 0), (1, 2), (3, 4), . . . , (2k -1, 2k),

A n = Mex{A i , B i | i < n}.
For all n ≥ k, if the following condition holds true

A n+1 -A n = 2 ∧ (B n -A n = B n-k+1 -A n-k+1 + 1 ∧ A n+1 -A n-k = 2k + 1) ∨ B n-k -A n-k = B n -A n -1 then B n+1 -A n+1 = B n -A n , otherwise B n+1 -A n+1 = B n -A n + 1.
The sequence (A n ) n≥0 is obviously increasing and the sequence (B n -A n ) n≥0 is non-decreasing. Therefore the sequence (B n ) n≥0 is also increasing. By the Mex rule defining (A n ) n≥0 , we assert that {A n | n ≥ 1} and {B n | n ≥ 1} make a partition of N ≥1 . Lemma 5. Let k ≥ 2 (the same as in Definition 3). For the sequence (A n , B n ) n≥0 given in Definition 3, we have A n+1 -A n ∈ {1, 2} for all n ≥ 0.

Proof. We proceed by induction on n ≥ 0. We have A n+1 -A n ∈ {1, 2} for all n < k. Assume now that A j+1 -A j ∈ {1, 2} for all j ≤ n. The definition of the sequence implies that, for all j ≤ n,

(16) B j+1 -B j ∈ {2, 3}.
If A n+1 = A n + 1, then the result holds. Otherwise, there exists i ≤ n such that A n + 1 = B i . From (16), we deduce that B i+1 ≥ A n + 3 and from the Mex rule defining (A n ) n≥0 , we conclude that A n+1 = A n + 2.

The next corollary follows directly from the definition of the sequence and the above lemma.

• m ≤ k < n. Since B n -A n ≥ 2 > B m -A m ,
A n -A m = B n -B m = 2(n -m).
Hence n-m ≥ k and the n-m consecutive differences

(A m+1 -A m , B m+1 - B m ), . . . , (A n -A n-1 , B n -B n-1 )
A i -A i-1 , B i -B i-1
), which is equal to [START_REF] Beatty | Problem 3173[END_REF][START_REF] Berlekamp | Winning ways for your mathematical plays[END_REF]. By Lemma 9, we deduce that m + 1 < i < n. Hence we have that

(A n -A n-1 , B n -B n-1 ) = (2, 2). Notice that B n-k-1 -A n-k-1 = B n-1 -A n-1 -1 and A n -A n-k-1 = 2k + 1, which contradicts Definition 3.
As in the previous case, one can check that the case (A n -A m , B n -B m ) = (2k + 2, 2k + 1) does not occur.

Let (a, b) be a game position which is not in the sequence (A n , B n ) n≥0 . We now show that it is always possible to play from (a, b) to a position in (A n , B n ) n≥0 .

Without loss of generality, assume that a ≤ b. If a = 0, then we can play from (a, b) to (0, 0). Now assume that a > 0. Since (A n , B n ) n≥1 makes a partition of N ≥1 , there exists n > 0 such that a = A n or a = B n . If a = B n , then we play b → A n and leave the other pile unchanged. If a = A n , then we consider three cases:

• b > B n . Then play b → B n and leave the other pile unchanged.

• b < B n and there exists 0 < i < n such that b = B i . Then play a → A i and leave the other pile unchanged. • b < B n and there exists j > n such that b = A j . Since A j -A n < B n -A n , there exists i < n such that B i -A i = A j -A n . We choose the smallest i having this property. If A n -A i = 2p for all 0 < p < k, then we play A n → A i and A j → B i . Otherwise, there exists 0 < p < k such that A n -A i = A j -B i = 2p. According to Corollary 2 and Lemma 7, the n -

i differences A n -A n-1 , . . . , A i+1 -A i equal 2 and n -i = p < k. Since B n - A n > B i -A i and by Lemma 6, we have (A i+1 -A i , B i+1 -B i ), . . . , (A n - A n-1 , B n -B n-1
) are equal to (2, 2) except for one of them. There exists a unique t ∈ {i + 1, . . . , n} such that (A t -A t-1 , B t -B t-1 ) = (2, 3). Now, by minimality of i, we have that

(A i -A i-1 , B i -B i-1 ) ∈ {(1, 2
), (2, 3)}, and by Lemma 6, (A

i -A i-1 , B i -B i-1 ) = (1, 2). According to Lemma 8, all the differences (A t-k -A t-k-1 , B t-k -B t-k-1 ), . . . , (A i-1 -A i-2 , B i-1 -B i-2 )
are equal to [START_REF] Berlekamp | Winning ways for your mathematical plays[END_REF][START_REF] Berlekamp | Winning ways for your mathematical plays[END_REF]. Notice that t -k ≤ n -k < i. We can conclude below to cover all the situations, we do not pay attention to some of the four conditions. In fact, it should be understood that if the conditions are compatible, then we show how to get the expected thesis. In case of incompatible conditions, there is nothing to show. Table 1 enumerate all the possible situations and give the corresponding case where we are considering it. Each of the conditions (H1) to (H4) can be either True or False. C.5 The last case not yet considered is when we have (H1), (¬H2), (H3) (and as a consequence of (H1) and (H3), we get (¬H4)). As in the first case, we have that ∆ α (n -i) = 2 for all i = 0, . . . , k and (17) also holds: ∆ β (n -k) • • • ∆ β (n) is equal to either 32 k-1 3 or 2 i 32 k-i , for some i ∈ {1, . . . , k -1}. But from (¬H2), we get ⌊nβ⌋ -⌊(n -k + 1) β⌋ = 2k -1 so, ∆ β (n -k + 1) • • • ∆ β (n -1) = 2 k-1 and therefore ∆ β (n -k) = ∆ β (n) = 3. This concludes the last case: ∆ β (n) = ∆ α (n) + 1.

Figure 2 .

 2 Figure 2. The functions k α -2k + 1 and (k + 1) β -2k -3. value being reached in case of the block 2 k . Consequently, for all n ≥ 1, we have D ∈ {2k -1, 2k}.

Figure 3 .

 3 Figure 3. f (k) -k + 1 > 0.

CC. 2 C. 3 C. 4

 234 T T T T T T F F F F F F F F H2 T T T T F F F F T T T T F F F F H3 T T F F T T F F T T F F T T F F H4 T F T F T F T F T F T F T F T FTable1. The possible truth values and the corresponding case. Assume now that we have (H1), (¬H3) and (H4). We have to show thatB ′ n+1 -A ′ n+1 = B ′ n -A ′ n , i.e., ∆ β (n) = 2.From (H1) and (¬H3), we getA ′ n -A ′ n-k = 2k -1 and ∆ α (n -k) • • • ∆ α (n -1) contains exactly one occurrence of 1. (H4) can be written as B ′ n -B ′ n-k = 2k. Therefore ∆ β (n -k) • • • ∆ β (n -1) contains exactly one occurrence of 3, say ∆ β (n -j) = 3 for some j ∈ {1, . . . , k}. If j < k, from Remark 2, ∆ β (n -k + 1) • • • ∆ β (n) contains at most one occurrence of 3, we deduce that ∆ β (n) = 2. If j = k, ∆ β (n -k) = 3 and we proceed by contradiction. Assume that ∆ β (n) = 3, so from Lemma 4, we deduce that ∆ α (n -k) • • • ∆ α (n) = 2 k+1which is a contradiction (we know that this block contains an occurrence of 1). Assume now that (H1) is not satisfied. This means that ∆ α (n) = 1 and we have to show thatB ′ n+1 -A ′ n+1 = B ′ n -A ′ n + 1, i.e., that ∆ β (n) = ∆ α (n) + 1 = 2.It is an immediate consequence of Lemma 1. Assume now that we have (H1), (¬H3) and (¬H4). We have to show thatB ′ n+1 -A ′ n+1 = B ′ n -A ′ n + 1, i.e., that ∆ β (n) = 3.As in the second case, from (H1) and (¬H3), ∆ α (n -k) • • • ∆ α (n -1) contains exactly one occurrence of 1. From (¬H4), we get ∆ β (n -k) • • • ∆ β (n -1) = 2 k . Proceed by contradiction and assume that ∆ β (n) = 2. Therefore, we would have a block of length k + 1 of the form 2 k+1 in (∆ β (n)) n≥1 . This is impossible in view of Remark 2.

  Figure 1. α k and β k as functions of k (⌊nβ k ⌋) n≥1 are complementary Beatty sequences giving a partition of N ≥1

								1. The sequences (⌊nα k ⌋) n≥1 and
	0 Α	10	20	30	40	50	k	2.6 Β
	1.95
	1.9							2.5
	1.85							2.4
	1.8							2.3
	1.75							2.2
	1.7							2.1
	1.65							10	20	30	40	50	k

  the only possible move is such that (A n -A m , B n -B m ) is equal to (2k + 1, 2k + 2) or (2k + 2, 2k + 1). If the latter case occurs, we would haveB m -A m > B n -A n . This is impossible because the sequence (B i -A i ) i≥0 is non-decreasing. Assume that (A n -A m , B n -B m ) = (2k+1, 2k+2). Since B m -A m < 2, this implies B n -A n = 2. In the sequence (A i , B i ) i≥0 there is a unique pair (A n , B n ) satisfying B n -A n = 2. It is (A k+1 , B k+1 , B k+1 ) leads to (0, 1) which is not in the sequence (A i , B i ) i≥0 . • k < m < n. Assume that the move (A n -A m , B n -B m ) is of the form (x, x) with x = 2i, for all 0 < i < k. Hence B m -A m = B n -A n .According to Corollary 2, this implies that

	k+1 ) = (2k + 1, 2k + 3) because
	(A k+2 , B k+2 ) = (2k +2, 2k +5). Therefore, playing the move (2k +1, 2k +2)
	from (A

  are equal to[START_REF] Berlekamp | Winning ways for your mathematical plays[END_REF][START_REF] Berlekamp | Winning ways for your mathematical plays[END_REF]. This contradicts Lemma 9. Now assume that the move (A n -A m , B n -B m ) is of the form (2k + 1, 2k + 2). Then according to Corollary 2, all the k + 1 pairs of differences (A m+1 -A m , B m+1 -B m ), . . . , (A n -A n-1 , B n -B n-1 ) are equal to (2, 2) except for one of them (

Acknowledgment

We would like to thank Professor A.S. Fraenkel for having considered our work with much interest, and also thank the referees for their remarks about the paper.

Corollary 2. Let k ≥ 2 (the same as in Definition 3). For the sequence (A n , B n ) n≥0 given in Definition 3, we have, for all n ≥ 0, [START_REF] Berlekamp | Winning ways for your mathematical plays[END_REF][START_REF] Berlekamp | Winning ways for your mathematical plays[END_REF], (2, 3)}.

The next result is correlated with the observation made in Remark 2. Lemma 6. Let k ≥ 2 (the same as in Definition 3). For the sequence (A n , B n ) n≥0 given in Definition 3, any subsequence of k consecutive differences of the kind

contains at most one occurrence of [START_REF] Berlekamp | Winning ways for your mathematical plays[END_REF][START_REF] Bouton | Nim, a game with a complete mathematical theory[END_REF].

Proof. Assume to the contrary that there exists n ≥ 0 and i ∈ {1, . . . , k -1} such that

In particular, this means that [START_REF] Berlekamp | Winning ways for your mathematical plays[END_REF][START_REF] Bouton | Nim, a game with a complete mathematical theory[END_REF], the following 3-terms condition should be satisfied

By the definition the sequence (

The last term in this condition should be satisfied. Therefore all the pairs of differences

From the above discussion, we can also conclude that B n+k -A n+k = B n -A n + 1. Therefore the first two terms of the condition are never satisfied which is a contradiction.

Lemma 7. Let k ≥ 2 (the same as in Definition 3). For the sequence (A n , B n ) n≥0 given in Definition 3, any subsequence of k + 1 consecutive differences

contains at most one occurrence of (1, 2).

Proof. Assume that A n = j and A n+1 = j + 1 for some n, j ≥ 2 (this means that A n+1 -A n = 1). Let us show that the next k differences A n+2 -A n+1 , . . . , A n+k+1 -A n+k are all equal to 2. Observe that there exists m such that j -1 = B m and j + 2 = B m+1 because otherwise, we would have B m+1 -B m > 3 which contradicts Corollary 2. In particular, B m+1 -B m = 3. By Lemma 6, we conclude that B m+2 , . . . , B m+k = j + 4, . . . , j + 2k and since B m+k+1 -B m+k ≥ 2, we also have B m+k+1 ≥ j + 2k + 2. From the Mex rule defining the sequence, we get that

Lemma 8. Let k ≥ 2 (the same as in Definition 3). For the sequence (A n , B n ) n≥0 given in Definition 3, any subsequence of k + 1 consecutive differences

contains at most two elements in {(1, 2), (2, 3)}.

Proof. Assume to the contrary that there exist i ∈ {1, . . . , k-2} and j ∈ {2, . . . , k-1} such that i < j and

. By Lemma 7, we get that [START_REF] Bouton | Nim, a game with a complete mathematical theory[END_REF]. By Lemma 6, we then get that (A n+i+1 -

. But these two occurrences of (1, 2) contradict Lemma 7.

Lemma 9. Let k ≥ 2 (the same as in Definition 3). For the sequence (A n , B n ) n≥0 given in Definition 3, any subsequence of k consecutive differences

contains at least one element in {(1, 2), (2, 3)}.

Proof. Assume to the contrary that there exists n ≥ 0 such that

We choose the smallest possible such n. Hence A n+k -A n = 2k. According to Definition 3, we should have either

And by minimality of n, we have B n-1 -A n-1 = B n -A n -1, contradicting the second condition.

Theorem 3 (Recursive characterization). The P -positions of G(α k ), k ≥ 2, are given by the sequence (A n , B n ) n≥0 in Definition 3.

Proof. We first show that there is no move from a position (A n , B n ) to some position (A m , B m ) with 0 ≤ m < n. Assume that such a move exists. Then this move is necessarily of the form

and no rule of G(α k ) allows to play a move (x, y) with |x -y| > 1 and xy = 0. Notice that, since the four numbers A n , A m , B n , B m are pairwise distinct, the only moves to be considered are those played on both piles. We now consider the three possible cases about n and m:

) are of the form (2i, 2i) with 0 < i < k, which are forbidden moves according to the rules of G(α k ).

that A n -A n-k = 2k + 1 and A j -A n-k = 2k + 2. Hence we can play

Theorem 4 (Algebraic characterization). Let α := α k , β := β k given in (1) for some k ≥ 2. If for all n ≥ 0, we set (A ′ n , B ′ n ) := (⌊nα⌋, ⌊nβ⌋), then the set of

Proof. We simply have to show that the sequence (A ′ n , B ′ n ) n≥0 is verifying the recursive characterization given in the previous theorem.

A. We first check the initial conditions. It is obvious that (A ′ 0 , B ′ 0 ) = (0, 0) and

As a function of k, one can easily check that 2k -1 < kα < 2k (see Fig. 2) and in the same way, 2k < kβ < 2k + 1.

B.

As it was already observed by A. Fraenkel in [START_REF] Fraenkel | How to beat your Wythoff games' opponent on three fronts[END_REF], the facts that the sequences (A ′ n ) n≥1 and (B ′ n ) n≥1 are complementary Beatty increasing sequences and that A ′ n < B ′ n for all n ≥ 1 imply that we necessarily have

We now turn our attention to the determination of B ′ n+1 -A ′ n+1 with respect to the value of

From (H1) and (H3), we get ⌊nα⌋ -⌊(n -k) α⌋ = 2k -1 so from Remark 2, we deduce that ⌊nα⌋ -⌊(n -k) α⌋ = 2k. Consequently, ∆ α (n -i) = ⌊(n -i + 1)α⌋ -⌊(n -i)α⌋ = 2 for i = 0, . . . , k. Using Lemma 3, we know that the two overlapping blocks

contain exactly one occurrence of 3. This is not enough to conclude, at this stage it could be possible that ∆

contains the only 3 occurring in x and y. Consequently, ∆ β

Notice that the remaining condition occurring in the recursive definition of Theorem 3:

(H4) B ′ n-k -A ′ n-k = B ′ n -A ′ n -1 has not been considered in our discussion. It is never satisfied whenever (H1) and (H3) are satisfied. Indeed, assume that (H1), (H3), (H4) are satisfied. We show that such a situation never occurs. With (H1) and (H3), we know that ∆ α (n -i) = 2 for i = 0, . . . , k and that we have If condition (Hi) is not satisfied, i = 1, 2, 3, 4, we write (¬Hi) as a shorthand. Therefore (H1), (H2), (H3) and (¬H4) are incompatible: it is impossible to have simultaneously (H1), (H2), (H3) and (¬H4). In all the other cases that we consider


Martin Stojanovikj
email: martin_stojanovik@yahoo.com

Goran Petrevski

Ss Cyril

Adopting Inflation Targeting in EMEs: Exploring the Factors Behind the Decision *

Keywords: Inflation Targeting, Binary Response Models, Emerging Market Economies. JEL codes: C25, E42, E52, E58

This paper examines the macroeconomic, financial, and institutional factors which affect the adoption of inflation targeting as a monetary policy strategy. We estimate a panel binary response model for 44 emerging market economies (EMEs) during 1990-2017. The main findings from our empirical investigation suggest that it is inflation and output growth volatility rather than inflation and output which matters for the adoption of IT in EMEs. In addition, we provide evidence that financial development, central bank independence, and capital mobility are associated with higher likelihood to adopt IT, whereas public debt has opposite effects. Finally, we show that, when deciding whether to adopt IT, policy makers take into consideration only mediumterm macroeconomic, financial, and institutional conditions, while the longer-run historical performance becomes less relevant in the decision-making process.

Introduction

Since New Zealand's switch to inflation targeting (IT) in 1990, it has become an increasingly popular monetary policy strategy in both advanced and emerging market economies (EMEs). In contrast to alternative strategies, such as monetary and exchange rate targeting, IT does not rely on intermediate targets. In these regards, [START_REF] Bernanke | Inflation Targeting: A New Framework for Monetary Policy?[END_REF] describes IT as a "rulelike strategy" or "constrained discretion", which enables the central bank to be focused on price stability while at the same time being able to deal with short-run macroeconomic fluctuations. The practical implementation of IT involves the following common features: announcements of numerical inflation targets, an institutional commitment to price stability as the primary monetary policy objective, short-term flexibility, and central bank independence accompanied by accountability and transparency [START_REF] Bernanke | Inflation Targeting: A New Framework for Monetary Policy?[END_REF]Mishkin 1997, Mishkin 2004).

Undoubtedly, the popularity of IT is related to the flexibility in the conduct of monetary policy. On the one hand, it is associated with increased credibility and accountability of central bank, which has favourable effects on inflation expectations; on the other hand, the flexibility of this framework enables the central bank to approach the inflation target gradually over time while offsetting adverse shocks. In addition, [START_REF] Thornton | Inflation Targeting and the cyclicality of monetary policy[END_REF] show that inflation targeting facilitates the implementation of countercyclical monetary policy in EMEs.

The literature suggests that the adoption of IT requires the fulfilment of several economic and institutional prerequisites, such as: the absence of fiscal dominance, strong external position, relatively low inflation, developed financial markets and sound financial system, central bank independence, structural characteristics (price deregulation, low dollarization, and low sensitivity to supply shocks), the absence of de facto exchange rate targets, and well developed technical infrastructure. However, these preconditions are not stringent so that they have not been met in many EMEs [START_REF] Agénor | Monetary Policy under Flexible Exchange Rates: An Introduction to Inflation Targeting[END_REF][START_REF] Amato | Inflation targeting in emerging market and transition economies: Lessons after a decade[END_REF], Battini and Laxton 2006[START_REF] Carare | Establishing Initial Conditions in Support of Inflation Targeting[END_REF][START_REF] Carare | Inflation Targeting Regimes[END_REF], Eichengreen et al. 1999, IMF 2006[START_REF] Masson | The Scope for Inflation Targeting in Developing Countries[END_REF][START_REF] Mishkin | Inflation Targeting in Emerging Market Countries[END_REF][START_REF] Mishkin | Monetary Policy Strategies for Emerging Market Countries: Lessons from Latin America[END_REF][START_REF] Mishkin | A Decade of Inflation Targeting in the World: What Do We Know and What Do We Need to Know? Inflation Targeting: Design, Performance, Challenges[END_REF].

The specific institutional and macroeconomic environment prevailing in EMEs often complicates the design and implementation of monetary policy regimes. For instance, both the presence of fiscal dominance undermines the effectiveness of monetary policy. Similarly, the weak banking system precludes the use of market-based monetary policy instruments. Also, the poor technical infrastructure (data availability, lack of systematic forecasting process, low understanding of the transmission mechanism etc.) hampers the day-to-day implementation of IT. Further on, the long historical experience with high inflation reduces the credibility of the central bank, requiring at least partial disinflation before the introduction of IT. Finally, given the crucial importance of the exchange rate channel in small open economies as well as the large-scale dollarization, simultaneously with inflation targets, central banks in EMEs must be concerned with exchangerate fluctuations [START_REF] Amato | Inflation targeting in emerging market and transition economies: Lessons after a decade[END_REF][START_REF] Mishkin | Inflation Targeting in Emerging Market Countries[END_REF][START_REF] Mishkin | Can Inflation Targeting Work in Emerging Market Countries[END_REF][START_REF] Mishkin | Monetary Policy under Inflation Targeting: An Introduction[END_REF]).

As mentioned above, an increasing number of EMEs have adopted IT during the past two decades. Yet, many more EMEs still rely on other strategies for controlling inflation. Therefore, it would be interesting to investigate the most important factors that induce policymakers in the EMEs to choose IT vis-a-vis the other monetary policy strategies. In theory, the choice of optimal monetary policy is analyzed within a well specified (usually, a small-scale) macroeconomic model by comparing the central bank's loss function under alternative policy rules. Here, a number of papers demonstrate that IT outperforms the alternative monetary policy strategies in terms of inflation/output variability (Ball 1999a, 1999b[START_REF] Haldane | Forward-Looking Rules for Monetary Policy[END_REF][START_REF] Rudebusch | Policy Rules for Inflation Targeting[END_REF], Svensson 1999a, 1999b, 2000). At the same time, there is an accumulated empirical evidence on the macroeconomic effects of IT, but their findings are inconclusive: while some studies suggest that IT is associated with both lower average inflation and improved inflation/output variability, others show that IT does not produce superior macroeconomic benefits or, at most, they are quite modest (For instance, see: [START_REF] Amira | Growth effects of inflation targeting revisited: empirical evidence from emerging markets[END_REF][START_REF] Ardakani | Re-evaluating the effectiveness of inflation targeting[END_REF][START_REF] Ball | Does Inflation Targeting Matter[END_REF][START_REF] Barnebeck Andersen | Inflation targeting and macroeconomic performance since the Great Recession[END_REF][START_REF] Batini | Under What Conditions Can Inflation Targeting Be Adopted? The Experience of Emerging Markets[END_REF][START_REF] Brito | Inflation targeting in emerging economies: Panel evidence[END_REF][START_REF] De Carvalho Filho | Inflation Targeting and the Crisis: An Empirical Assessment[END_REF][START_REF] Gonçalves | Inflation Targeting in Emerging Economies: What Do the Data Say[END_REF][START_REF] Lee | Comparative case studies of the effects of inflation targeting in emerging economies[END_REF][START_REF] Lin | Does inflation targeting make a difference in developing countries[END_REF][START_REF] Thornton | Inflation targeting in developing countries revisited[END_REF][START_REF] Vega | Inflation Targeting and Inflation Behavior: A Successful Story[END_REF]. Therefore, the increasing adoption of IT by EMEs is not based on strong empirical evidence with respect to the macroeconomic performance of this monetary regime. In addition, the experience of advanced countries may not be relevant for EMEs due to their specific features.

In these regards, the aim of this paper is to provide additional evidence on the set of preconditions required for adopting IT as a monetary policy strategy in EMEs. Compared with the existing empirical studies, we adopt a slightly different methodological approach by employing a static panel data logistic fixed-effects regression, controlling for time-invariant heterogeneity among the countries, which has not been done previously. Additionally, instead of controlling directly for the time specific effects, we use a modified logistic fixed-effects regression with analytical adjustment for the incidental parameter bias [START_REF] Fernandez-Val | Individual and time effects in nonlinear panel models with large N, T[END_REF]. Finally, in our baseline model we work with transformed data by means of three-year moving averages, thus allowing for both forward-and backward-looking approach in conducting monetary policy in the medium term.

The main findings from our empirical investigation are as follows: we cannot provide evidence that macroeconomic performance as measured by inflation and output growth affects the decision of whether to adopt IT. Instead, we show that higher inflation and output growth volatility lowers the likelihood to adopt IT. The above results suggest that it is macroeconomic instability which matters for the adoption of IT in EMEs. This finding can be explained by the unfavorable macroeconomic environment prevailing in EMEs, which are traditionally exposed to large and persistent exogenous shocks. Under these conditions, central banks have to be engaged frequently in offsetting the adverse shocks, thus, undermining the successful implementation of IT. In addition, we confirm that financial development, central bank independence, and capital mobility are associated with higher likelihood to adopt IT, whereas public debt has opposite effects. Finally, we show that, when deciding whether to adopt IT, policy makers take into consideration only medium-term macroeconomic, financial, and institutional conditions, while the longer-run historical performance becomes less relevant in the decision-making process.

The remainder of this paper is organized as follows: Section 2 reviews the relevant empirical literature, Section 3 presents the data issues, research methodology, estimation results, and robustness checks, while the concluding remarks are provided in Section 4.

Literature Review

As mentioned above, although the literature points to a number of macroeconomic, institutional, and technical preconditions for successful implementation of IT. While theoretically sound, the experience shows that many EMEs have not met all of these preconditions, at least in the initial phase of adopting IT. Therefore, since the mid-2000s, the empirical literature has begun to investigate the most important determinants behind the adoption of IT in EMEs. As expected, the empirical research has not led to firm conclusions reflecting the fact that EMEs represent a heterogeneous group with specific institutional and macroeconomic characteristics. In what follows, we provide a brief overview of the empirical literature on the determinants of IT.

The first attempt to address the factors behind the decision to adopt IT has been provided by [START_REF] Mishkin | A Decade of Inflation Targeting in the World: What Do We Know and What Do We Need to Know? Inflation Targeting: Design, Performance, Challenges[END_REF], who conduct an empirical analysis on 27 developed and developing countries during 1990s, based on cross-section data with 10 year-averages. Their probit model indicate that the adoption of the inflation targeting regime is positively associated with inflation and central bank instrument independence, whereas the existence of money targets and central bank goal independence reduce the probability of adopting IT. [START_REF] Carare | Inflation Targeting Regimes[END_REF] review the global experience with IT regimes by focusing on the factors affecting the evolution between various variants of IT ("lite", full-fledged, and eclectic). They find that higher level of economic and financial development, and lower central bank restrictions are significant factors affecting the increase in overall central bank credibility and, thus, the choice of IT regime. They, also, discuss the experience of EMEs and show that the likelihood to move from "lite" to full-fledged IT is higher for countries with higher stock market capitalization, lower public debt, and restrictions on government financing.

Employing a panel logit regression on a sample of 66 developed and developing countries over 1980-2000[START_REF] Hu | The choice of inflation targeting -An empirical investigation[END_REF] finds that the probability to adopt IT increases with the following conditions: lower output growth, lower inflation, higher real interest rates, balanced fiscal position, de facto floating exchange rate regime, central bank independence, and higher pressure on exchange rates.

Working with a panel of 49 countries between 1987 and 2003, [START_REF] Mukherjee | Monetary Institutions, Partisanship, and Inflation Targeting[END_REF] show that countries are more likely to adopt IT when the government and the central bank share the same preferences for tight monetary policy. More specifically, the combination of a rightleaning government and a central bank without bank regulatory authority is likely to be associated with the adoption of IT. Also, they show that GDP growth variability, floating exchange rates, real interest rates, inflation have positive effects on the likelihood for adopting IT, while current account deficits reduce the probability to adopt IT.

Based on a survey of 31 central banks in EMEs, [START_REF] Batini | Under What Conditions Can Inflation Targeting Be Adopted? The Experience of Emerging Markets[END_REF] assess whether some preconditions must be met before adopting IT, such as: technical infrastructure, financial system, institutional central bank independence, and economic structure. They construct an extensive list of parameters and, by quantifying each of them, conclude that EMEs had not satisfy these required preconditions when adopting IT. Also, they insert these parameters in their Differencein-Difference regression and find that none of them is significant, concluding that adopting IT does not depend on meeting some strict initial pre-conditions. [START_REF] Leyva | The Choice of Inflation Targeting[END_REF] studies the institutional and macroeconomic factors affecting the adoption of IT in 28 countries over 1975-2005, and finds that inflation rate, financial development, GDP per capita, and trade openness are the most factors affecting the choice of IT. [START_REF] Lucotte | The Choice of Adopting Inflation Targeting in Emerging Economies: Do Domestic Institutions Matter[END_REF] investigates the role of institutional and political factor in adopting IT for a sample of 50 countries during 1986-2005. His findings imply that a number of determinants influence the decision to adopt IT, such as: per capita GDP, exchange rate flexibility, trade openness, central bank independence, the number of veto players in the political system, political stability, and the degree of federalism (decentralization). Lin andYe (2007, 2009), and de Mendonca and de Guimaraes e Souza (2012) point to the negative association of the inflation rate with the likelihood of adopting IT. Comparing IT with exchange rate pegs for a large set of countries, [START_REF] Rose | Surprising Similarities: Recent Monetary Regimes of Small Economies[END_REF] show that IT is a preferred monetary regime for larger countries as well as for those with more developed democratic institutions. [START_REF] Samarina | Right on Target: Exploring the Factors Leading to Inflation Targeting Adoption[END_REF] show that the determinants affecting the choice of IT differ between OECD and non-OECD countries as well as between soft and full-fledged inflation targeters. Specifically, among the full list of the macroeconomic, external, financial, and institutional factors, they show that only flexible exchange rate regime and central bank independence matters for the adoption of IT in non-OECD countries. On the other hand, the most important factors affecting the adoption of IT in OECD countries are the following: inflation, government debt, flexible exchange rates, financial development, openness, and high central bank independence. As for the choice between soft and full-fledge, they find that flexible exchange rate regimes, higher exchange rate volatility, higher central bank independence, lower external debt, and lower financial development are associated with higher probability to adopt soft IT; on the other hand, lower inflation, lower output growth, lower public debt, and lower financial development increase the likelihood to adopt full IT. [START_REF] Ismailov | Choice of inflation targeting: Some international evidence[END_REF] investigate the determinants of adopting IT for a cross-section of 82 developed and developing countries in 2010, taking into account the exchange rate regime. They divide the sample into treatment (inflation targeters) and the control group (non-targeters) for both high-income and low-income countries. As for the latter, they are only able to show that the size of public debt reduces the likelihood to adopt IT while the other determinants (most notably, inflation and political risk) are not significant. [START_REF] Thornton | Inflation Targeting and the cyclicality of monetary policy[END_REF] investigate the likelihood of adopting IT in 90 industrial and developing countries. Their results indicate that developing countries with lower inflation rates, higher rates of output growth, better fiscal position, developed financial markets, and higher exchange rate flexibility are more likely to adopt IT as their policy strategy.

Empirical Investigation

Data Description

In our empirical study, we work with annual data for a panel of 44 EMEs during the period of 1990-2017. Due to data availability, we work with an unbalanced panel. Before conducting the empirical analysis, we have transformed the raw data into three-year moving averages. Though this is not a conventional approach in the empirical literature, we provide the following rationale for our procedure: first, decision to adopt a particular monetary strategy is usually based on the expected macroeconomic benefits over the medium term; second, when deciding on the adoption of IT, the central bank is both forward-and backward-looking, both of which are captured by the transformation of the data into three-year moving averages; third, within the IT framework, the central bank targets the medium-term inflation rate, extending for a period of 2-3 years. At the same time, we are aware that the above transformation does come at a cost, by reducing the number of available observations, which coupled with the unbalanced panel dataset, creates some issues regarding the use of a non-linear estimator, which does not have a closed-form solution, and it uses sample information to achieve convergence. Due to this problem, we allow a broader significance level in the presentation and interpretation of our results.

Our sample consists of 17 inflation targeters and 27 EMEs with alternative monetary regimes (See Table 1). 2 Since the group of EMEs is not strictly defined and it is very heterogeneous, we take an eclectic approach and include all countries that have been characterized as EMEs in the empirical literature in this field. Concerning the adoption date of IT, we rely on the classification provided by [START_REF] Hammond | State of the Art of Inflation Targeting[END_REF], by taking the year when the central banks decided to adopt this strategy regardless on the particular month of that respective year when the move to IT occurred. In addition, as inflation targeting can take several flexible forms of implementation, we follow [START_REF] Calvo | The mirage of exchange rate regimes for emerging market countries[END_REF] in regarding the switch to full-fledged IT as the adoption date. We choose 1990 as the starting year for the empirical analysis as it coincides with the year when New Zealand adopted IT as a monetary strategy [START_REF] Mishkin | A Decade of Inflation Targeting in the World: What Do We Know and What Do We Need to Know? Inflation Targeting: Design, Performance, Challenges[END_REF]. While the concept of IT was already present in the theoretical literature, we assume that its practical implementation only became valid in 1990, once it was officially adopted by the Central Bank of New Zealand. Hence, we treat the period prior to this year as irrelevant to our analysis. We find this assumption to be reasonable given that before 1990 the central banks in EMEs were not in a position to decide whether or not to adopt IT as their monetary strategy. In other words, IT as a policy option for EMEs emerged only after its practical implementation in New Zealand. As a result, including the pre-1990 period in the analysis would only bias the estimation results by increasing the likelihood not to adopt IT. However, we also report the results concerning the full dataset available for the 1970-2017 period.

We work with CPI-based annual inflation rate because it is a target variable in all the countries within our sample. Here, a potential source of concern is the high inflation rates prevailing in the early years in our sample, which might bias both the volatility measure derived from it and the overall estimation results. Therefore, following [START_REF] Cukierman | Measuring the Independence of Central Banks and its Effect on Policy Outcomes[END_REF] and [START_REF] Cukierman | Central Bank Reform, Liberalization and Inflation in Transition Economies -An International Perspective[END_REF] we adopt a common approach in the empirical literature and transform the inflation rate by the following formula: infltr = infl / (1+ infl), where infl denotes the actual inflation rate. It is expected that this approach will reduce heteroskedasticity and potential bias in the estimation process, thus improving the efficiency of parameter estimates. Output growth is expressed as the annual growth of real GDP while inflation and output growth volatility are measured by the standard deviation of transformed inflation rate and real GDP growth, respectively. Net capital flows (NCF) are measured in billions of US dollars whereas both public debt (Debt) and credit to private sector (CPS) are expressed as ratios to GDP. As for the latter, we use the weighted CBI index provided by [START_REF] Gariga | Central Bank Independence in the World: A New Data Set[END_REF] for a large set of countries, including many which are part of our sample, and covering the same period as our sample, thus making it compatible to our analysis. The descriptive statistics along with the simple correlation coefficients among the variables are presented in Table 2.

Model Specification

An increasing number of EMEs have adopted IT during the past two decades. In this regard, it would be interesting to investigate the most important factors behind the decision of policymakers in the EMEs to switch to IT. Specifically, our primary research interest lies in explaining the likelihood of adopting IT, based on a set of explanatory variables. Consequently, we can express the IT variable as a latent variable model:

(1),

where 1(•) is an indicator function mapping non-negative to 1 if country i chooses IT in period t, and 0 otherwise. This restricts our analysis to a non-linear specification of the model within the framework of binary discrete choice models. Equation ( 1) indicate that the choice of IT as a monetary regime is a complex decision.

Additionally, we assume an implicitly linear dynamic functional form for ,

(2), with relatively strong assumptions that and . The first of these assumptions will be relaxed further on, as our baseline specification will not include the autoregressive component. The second assumption is an arbitrary choice of the potential density function, which in our case is restricted by the modelling setup we have proposed. Specifically, simple functions of the parameters of interest, which are independent of the heterogeneity parameters are non-existent, rendering the probit fixed-effects approach computationally immensely difficult [START_REF] Hsiao | Analysis of Panel Data[END_REF]). As we work with a panel of EMEs, which represent a rather heterogeneous group, it is of crucial importance that we control for their respective country-specific characteristics, represented by . This assumption, thus, offers a manner to address this issue, without making the analysis computationally challenging.

The likelihood of adopting IT can be modelled as:

(3), where Ф (•) is an index function (in our case, the logistic density function), ( are the unknown parameters to be estimated, is the autoregressive variable, and is the set of explanatory variables. In this setup, the autoregressive term (y i, t-1 ) depicts the effect of the choice made in the previous period of the implementation of IT regime on the likelihood that the IT regime will be retained in the current period. Here, there are two issues to be addressed: 1) all the countries in our sample exhibit a highly persistent behavior in the adoption of the IT regime, that is once they have adopted IT, they do not switch to an alternative monetary regime (hence, the effect of on is absent); 2) policymakers are unlikely to incline towards changing their monetary strategy just because they are implementing a certain type; instead, they are likely to adjust or to change the existing strategy based on a closely monitoring of macroeconomic, fiscal, institutional, and other domestic and global conditions. It is for these reasons that we opt for a static baseline specification:

(4), where is the set of explanatory variables (including the constant term), and are the unobserved country-specific (time-invariant) effects.

We estimate two variants of the baseline specification for the same dependent variable (IT) by varying the macroeconomic variablesinflation, inflation volatility, output growth, and growth volatility -while at the same time controlling for the fiscal position, financial market development, supply shocks, and central bank independence. As regards the control variables, our specification is basically similar to those commonly employed in the empirical literature. Yet, we choose a slightly different set-up primarily to demonstrate that inflation and output volatility are significant factors in deciding whether to adopt IT in EMEs (once we control for the rest of the macroeconomic, financial, and institutional determinants), especially in light of the long historical experience of these countries with unstable macroeconomic environment.

In what follows we provide brief explanation of the control variables included in the empirical model: inflation, output growth, inflation variability, and output variability comprise the block of variables representing macroeconomic performance. As mentioned above, we estimate two variants of the empirical modelone with inflation and output growth and another one with inflation and growth variability. Central banks choose their monetary strategy in a response to the macroeconomic performance. In theory, inflation and output are the two standard elements in the central bank's loss function. Similarly, in practice, inflation control is the primary goal of monetary policy, but central banks often pay attention to output growth, too. In this regard, if a country has experienced unsatisfactory growth rates, then the central bank might consider switching from one strategy (say, exchange rate peg) to IT as the latter enables policy makers to take care for output, too. Similarly, the central bank might choose its monetary policy strategy based on its experience with inflation: on the one hand, if inflation is very high, the central bank might choose exchange rate peg, while if inflation is moderate, then it can adopt IT.

In EMEs, capital flows (NCF) seem to be an important factor that affects the choice of monetary policy strategy, because some strategies (such as IT) allow the central bank to offset exogenous capital flows via exchange rate depreciation, while other strategies (most notably, exchange rate pegs) do not allow for this flexibility. In this regard, capital mobility has been the main driving force behind the secular trend towards regimes with greater exchange rate flexibility [START_REF] Eichengreen | Kicking the Habit: Moving from Pegged Exchange Rates to Greater Exchange Rate Flexibility[END_REF][START_REF] Fischer | Exchange rate regimes: is the bipolar view correct[END_REF].

The inclusion of public debt (Debt) as a control variable is motivated by the notion that fiscal discipline represents one of the basic requirements for adopting IT. Clearly, unsustainable fiscal policy undermines the central bank's efforts to keep inflation low with harmful effects on the credibility of announced inflation targets. Also, central bank independence (CBI) as an important institutional factor necessary for implementing IT. Here, the literature points to the so-called instrument independence, i.e., the autonomy of central banks in choosing their instruments to achieve inflation targets.

In addition, the literature identifies financial development as a precondition for successful implementation of IT. Therefore, we include bank credit to the private sector (CPS) as a proxy for the development of the banking sector. Moreover, this variable can be used as proxy for the anti-inflation preferences of the society. For instance, one may argue that the societies with greater preferences for price stability choose IT as a means of keeping inflation low. A special variant of this argument is presented by [START_REF] Posen | Declarations Are Not Enough: Financial Sector Sources of Central Bank Independence[END_REF], who argues that monetary policy makers are influenced by various interest groups. In particular, those sectors in the society that benefit from price stability (the banking and financial sectors) lobby for increased CBI as a means for maintaining low inflation. Since the contracts in the banking sector are predominantly expressed in nominal terms and banks hold substantial proportion of their assets in fixed-income instruments, then it is quite expected that this sector would be inflation-averse. Hence, according to this argument, higher level of financial development implies higher aversion to inflation.

Estimation Strategy

We assume that the asymptotic framework for analysis is N→∞ with T fixed, and we treat the country-specific effects as parameters to be estimated. As a result, the Maximum Likelihood (ML) estimation of parameters is consistent. However, this framework gives rise to the incidental parameters problem, which is addressed below [START_REF] Neyman | Consistent estimates based on partially consistent observations[END_REF]. In order to estimate the likelihood as well as the average partial effects of macroeconomic factors, fiscal position, domestic financial market development, central bank independence, and supply shocks on the likelihood of adopting IT in EMEs, we employ the conditional logit estimator [START_REF] Chamberlain | Panel Data[END_REF]. In this regard, we implement the ML estimation conditional on the country-specific fixed effects. This estimator allows us to control for the country-specific fixed-effects, and to obtain consistent results under the framework presented in the previous section. [START_REF] Wright | Better procedures for sample-free item analysis[END_REF] show that the ML estimation for the fixed-effects logit model when T=20 yields virtually unbiased results for the parameters. This property has been confirmed in the Monte Carlo simulations by [START_REF] Greene | The Behavior of the Maximum Likelihood Estimator of Limited Dependent Variable Models in the Presence of Fixed Effects[END_REF] where, as the T dimension increases, the bias of the parameters decreases.

It is well known that fixed-effects estimators suffer from the incidental parameters problem, which is a cause of concern particularly for non-linear panel data models, where ML estimates of and are not independent. In contrast to linear models, the failure to satisfy this assumption in non-linear models introduce a bias in the estimation results of both parameters rendering the within transformation obsolete [START_REF] Hsiao | Analysis of Panel Data[END_REF]. In order to deal with this issue, we use the analytical bias correction proposed by Fernandez-Val and[START_REF] Fernandez-Val | Individual and time effects in nonlinear panel models with large N, T[END_REF][START_REF] Cruz-Gonzalez | Bias Correction for Probit and Logit Models with Two-way Fixed Effects[END_REF], which removes the estimates of the leading terms of the bias from the estimator of . Since the baseline specification in ( 4) is static and assumes exogenous independent variables, there is no need for specifying a trimming parameter. Despite its potential shortcomings related to the strong assumptions behind it, this estimator is preferred to its alternatives (non-parametric estimators) since they converge at rates slower than the parametric estimator (which is of great importance in a panel structure as ours), making the interpretation of parameter estimates is dubious [START_REF] Hsiao | Analysis of Panel Data[END_REF]. The chosen estimation strategy restricts us to losing all the observations for which .

Finally, note that we only control for the country-specific fixed effects, i.e., we do not control for the time-specific fixed effects in our analysis. This has been motivated by the fact that the countries in our dataset exhibit smooth behavior in the main variables (inflation, inflation volatility, output growth and output growth volatility) throughout the period covered by the analysis. In addition, the design of our methodological approach allows us to control these effects indirectly by transforming the raw data into three-year moving averages, thus smoothing out the time effects across the three-year periods and reducing their impact on the estimates. This has a direct bearing on the choice of the above estimator, which enable us to correct for both country-and time-specific effects bias.

Results and Discussions

Table 3 shows the estimates of the two variants of equation ( 4) based on the sample over 1990-2017: the first regression contains inflation and output growth as dependent variables whereas the second set of estimates refers to the equation with inflation and output growth variability.

Since the regression coefficients in binary response models do not have a meaningful interpretation, we only comment the average marginal effects. As can be seen, in the first regression, most of the variables are statistically insignificant, except for inflation and net capital flows (NCF), which are significant at 15% significance level. Therefore, we cannot provide strong evidence that macroeconomic performance as measured by inflation rates and output growth affects the decision of whether to adopt IT. As for the inflation performance, our findings are consistent with the stylized facts that IT is not a feasible strategy for the countries with high inflation rates, i.e., the adoption of IT should be preceded by disinflation to relatively low level of inflation [START_REF] Mishkin | Inflation Targeting in Emerging Market Countries[END_REF]. The regression coefficient of NCF has a negative sign, too, implying that the greater exposure to capital flows reduces the likelihood to adopt IT in EMEs. This finding is contrary to our prior expectations as, when faced with higher capital mobility, countries usually tend to move towards monetary strategies with flexible exchange rate regimes, such as IT. Notes: (1) IT is the binary dependent variable; (2) logit FE estimator with an analytical bias correction for the incidental parameter bias (both time and country specific), conditional on the country specific fixed effects; (3) ***, **, and * indicate significance levels of 5, 10 and 15% respectively; (4) robust standard errors are reported in parentheses.

The results from the second regression are much more favourable. Except for financial development (CPS), all the marginal effects are statistically significant at 10% level of significance. As for macroeconomic performance, both inflation and output volatility have negative signs, suggesting that increased macroeconomic instability lowers the likelihood to adopt IT. In other words, the estimates from this variant of the regression model imply that it is macroeconomic instability (inflation and output growth volatility) which matters for the adoption of IT in EMEs. This finding can be explained by unfavorable macroeconomic environment prevailing in EMEs, which traditionally have worse performance than advanced economies due to the worse inflation-output trade-off as well as the exposure to large and persistent exogenous shocks [START_REF] Fraga | Inflation Targeting in Emerging Market Economies[END_REF]. Under these conditions, central banks have to be engaged frequently in offsetting the adverse shocks with harmful effects on the credibility of inflation targets [START_REF] Svensson | The Swedish Experience of an Inflation Target[END_REF][START_REF] Agénor | Monetary Policy under Flexible Exchange Rates: An Introduction to Inflation Targeting[END_REF].

As for the other regressors, we obtain the following results: once again, the greater exposure to capital flows (higher NCF) reduces the likelihood to adopt IT in EMEs; higher government debt significantly reduces the probability to adopt IT, thus confirming that strong fiscal position is a prerequisite for successful implementation of IT; also, we provide evidence that higher central bank independence is associated with increased likelihood to adopt IT; finally, we find that financial development does not affect the decision to adopt IT, which can be explained by the fact that EMEs have less developed financial sectors in comparison to advanced economies.

Sensitivity Analysis

In Tables 4 to 6 we present the results of various robustness checks. We examine the sensitivity of the results with respect to: estimating the model on the sample covering the whole available data from 1970 to 2017 (with yearly observations); estimation with different data transformations (three-year, five-year, and ten-year averages; and estimation based on cross-section instead of panel data. 4 presents the estimates of equation (1) based on the whole available data from 1970 to 2017. As can be seen, the estimation based on the extended sample does not affect the baseline results (regression 1 of Table 3) in any substantial way. In fact, it is quite expected that extending the sample backward could not affect the main findings from our study: on the one hand, it is obvious that distant past may not be relevant for the present choice of monetary regimes; on the other hand, IT was not available as a policy option back in the 1970s and 1980s. Here, inflation is the only statistically significant regressor at 15% significance level. Once again, the coefficient is negative implying that high inflation reduces the likelihood to adopt IT in EMEs.

The first regression in Table

In contrast to our baseline panel-data model presented in Table 3, the regressions 2 to 4 in Table 4 are estimated with cross-section data, but with different transformations of data, i.e., we work with three-year, five-year, and ten-year averages, respectively. However, for all these three regressions, the sample has been constructed as follows: for inflation targeters, the sample begins from each individual date of adopting IT, whereas for non-targeters the sample begins in 2001, which is the average adoption year for our sample. In each case, this procedure has resulted in 34 observations in total.

As can be seen, estimating the empirical model with cross-section instead of panel-data logit model does affect the results: inflation loses its statistical significance whereas CPS, CBI, and NCF become statistically significant at much higher level (10% and 5%). Also, the sign of the coefficient before NCF changed from negative to positive in the regressions based on crosssection data. According to this set of estimates, higher capital mobility increases the probability to adopt IT as a strategy that goes hand in hand with greater exchange rate flexibility. In addition, the positive signs before CPS and CBI confirm the favourable impact of financial development and central bank independence on the choice to adopt IT. Table 5 replicates the above regressions for the second specification of the empirical model, which includes inflation and output variability as measures of macroeconomic performance. Once again, the first regression in Table 4 presents the results based on the whole available data from 1970 to 2017, while regressions 2 to 4 refer to the estimates with cross-section data with three-year, five-year, and ten-year averages, respectively.

Several basic conclusions emerge from this set of estimates: inflation volatility reduces the probability to adopt IT, thus confirming the common notion that countries need to stabilize inflation before moving to IT; as for growth variability, we obtain conflicting results: the sign of the regression coefficient changes from first to second regression, but loses its statistical significance when working with longer averages; we obtain similar results for the public debt, which becomes statistically insignificant and changes its sign in the regression with cross-section data; however, this set of results, too, confirm that financial development, central bank independence, and capital mobility are all associated with higher likelihood to adopt IT; finally, note that when working with ten-year averages, only two variables retain its statistical significance, implying that, when deciding whether to adopt IT, policy makers take into consideration only medium-term macroeconomic, financial, and institutional conditions, while the longer-run historical performance becomes less relevant in the decision-making process. Finally, in Table 6 we conduct an additional sensitivity analysis. Here, we work with yearly data instead of moving average, i.e., in contrast to our baseline specification, here, we do not perform any transformation of the data, but we work with ordinary panel data. In each regression we introduce a slight modification in the model: the first regression refers to the specification with inflation and output growth, the second one includes their volatilities as regressors, while in the third regression the standard deviation of inflation is substituted by the coefficient of variation. The three regressions provide consistent results with respect to the impact of inflation on the choice of monetary strategy, i.e., both the level of inflation and its variability are associated with lower likelihood to adopt IT. Hence, once again, we provide evidence that EMEs need to lower inflation to a relatively low level before moving to IT. Similarly, greater output volatility reduces the probability of adopting IT.

Combined, these two findings suggest that adopting IT is not feasible in countries with high degree of macroeconomic instability, i.e., large macroeconomic uncertainty compromises the credibility of this monetary regime. As regards CBI and NCF, the sensitivity analysis produces similar results to our previous regressions. However, two additional findings emerge from this set of estimates: financial development loses its statistical significance as a determinant in the choice of IT in EMEs, whereas higher level of public debt decreases the likelihood to adopt IT, i.e., we are able to provide some evidence that fiscal discipline is an important precondition for successful implementation of this strategy.

Conclusion

The main research goal of this paper is to investigate the most important determinants behind the adoption of IT in a sample of 44 emerging market economies (EMEs) during 1990-2017. Specifically, we estimate a fixed-effects logit model, controlling for several macroeconomic, financial, and institutional factors affecting the choice of monetary strategies in EMEs. Based on our empirical investigation, we cannot provide evidence that macroeconomic performance as measured by inflation and output growth affects the decision of whether to adopt IT. Instead, we show that higher inflation and output growth volatility lowers the likelihood to adopt IT. Therefore, the above results suggest that it is macroeconomic instability which matters for the adoption of IT in EMEs. This finding can be explained by the unfavorable macroeconomic environment prevailing in EMEs, which are traditionally exposed to large and persistent exogenous shocks. Under these conditions, central banks must be engaged frequently in offsetting the adverse shocks, thus, undermining the successful implementation of IT. In addition, we confirm that financial development, central bank independence, and capital mobility are associated with higher likelihood to adopt IT, whereas public debt has opposite effects. Finally, we show that, when deciding whether to adopt IT, policy makers take into consideration only medium-term macroeconomic, financial, and institutional conditions, while the longer-run historical performance becomes less relevant in the decision-making process.

Table 1 .

 1 Dates of inflation targeting adoption in a sample of 17 EMEs

	Country	Year of
		adoption of IT
	Brazil	1999
	Chile	1999
	Colombia	1999
	Czech Republic	1998
	Hungary	2001
	Israel	1997
	Mexico	1999
	Peru	2002
	Philippines	2002
	Poland	1998
	South Africa	2000
	South Korea	1998
	Thailand	2000
	Turkey	2006
	Ghana	2007
	Indonesia	2005
	Serbia	2009

Table 2

 2 Inflation Targeting Regime and Related Variables for 44 Countries (1990-2017): Descriptive Statistics and Simple Correlations All the variables in our dataset have been extracted from the World Bank's World Development Indicators database, except for the IT dummy and the index of central bank independence (CBI).

				Descriptive Statistics
		IT	Inflation Output	Output	Inflation	CBI NCF Debt	CPS
				growth	growth	volatility
					volatility
	Mean	0.244 0.088	0.040	0.004	0.0004	0.540 -0.18 0.528 0.475
	St. Dev.	0.43 0.113	0.032	0.004	0.0015	0.209 2.155 0.303 0.356
	Min	0	-0.009	-0.164 0.002	0.0000	0.121 -34.6 0.046 0.017
	Max	1	0.936	0.308	0.047	0.021	0.904 3.71 1.90	1.559
				Correlation Matrix
	Variable	IT	Inflation Output	Output	Inflation	CBI NCF Debt	CPS
				growth	growth	volatility
					volatility
	IT	1
	Inflation -0.21	1
	Output	-0.09	-0.29	1
	growth
	Output	-0.14	0.23	-0.214	1
	growth
	volatility
	Inflation	0.003	0.71	-0.34	0.255	1
	volatility
	CBI	0.18	0.01	-0.06	0.015	-0.086	1
	NCF	0.08	0.06	-0.22	-0.019	0.045	0.038	1
	Debt	-0.12	-0.02	-0.06	-0.045	0.016	-0.21 0.018	1
	CPS	0.10	-0.29	0.21	-0.063	-0.088	-0.38 -0.24 -0.02	1

Table 3 .

 3 Estimates of the likelihood of adopting IT

		Logit FE		Logit FE
			(1)			(2)
		Likelihood	Marginal effect		Likelihood	Marginal effect
	Inflation	-161.077***	-3.75*	Inflation	-858.08***	-47.423*
		(33.132)	(2.538)	volatility	(405.512)	(32.419)
	Output growth	47.418***	1.105	Output growth	-464.79***	-25.687**
		(22.732)	(0.887)	volatility	(130.620)	(14.598)
	Debt	-13.423***	-0.313	Debt	-8.027***	-0.444**
		(5.093)	(0.234)		(2.395)	(0.256)
	CPS	2.430	0.056	CPS	0.633	0.035
		(2.855)	(0.076)		(1.827)	(0.102)
	CBI	31.228*	0.727	CBI	58.714***	3.245**
		(21.422)	(0.685)		(13.963)	(1.779)
	NCF	-2.998***	-0.070*	NCF	-1.344***	-0.074**
		(0.800)	(0.049)		(0.370)	(0.042)
	Pseudo R	0.8279		Pseudo R	0.487
	squared			squared
	Log-	-42.174		Log-	-90.115
	likelihood			likelihood

Table 4 :

 4 Estimates of the likelihood of adopting IT: Sensitivity analysis of Logit FE (1)

		Logit FE (1)	Logit FE (2)	Logit FE (3)	Logit FE (4)
		Full sample	Cross-section data	Cross section data	Cross section data
		(1970-2017)
		Likelihood Marginal	Likelihood Marginal	Likelihood Marginal	Likelihood Marginal
			effect		effect		effect		effect
	Inflation	-161.21***	-2.561*	-0.941	-0.135	1.760	0.238	1.803	0.497
		(32.484)	(1.800)	(8.702)	(1.242)	(14.109)	(1.905)	(3.086)	(0.866)
	Output	38.434**	0.611	-10.234	-1.472	0.438	0.059	-5.984	-1.649
	growth	(20.291)	(0.522)	(30.065)	(4.492)	(25.532)	(3.452)	(10.607)	(2.825)
	Debt	-15.788***	-0.251	0.161	0.023	0.570	0.077	-0.002	-0.0006
		(5.335)	(0.189)	(2.018)	(0.289)	(1.752)	(0.236)	(0.886)	(0.244)
	CPS	4.813**	0.076	4.949***	0.712***	3.592***	0.485***	1.421*	0.391**
		(2.615)	(0.066)	(1.648)	(0.188)	(1.749)	(0.192)	(0.910)	(0.230)
	CBI	49.832***	0.792	8.236	1.185**	8.482***	1.146***	3.441***	0.948***
		(20.587)	(0.625)	(5.900)	(0.622)	(3.878)	(0.459)	(1.608)	(0.350)
	NCF	-3.203***	-0.051	4.114**	0.592***	5.459***	0.737***	2.829**	0.779***
		(0.791)	(0.036)	(2.101)	(0.185)	(2.331)	(0.199)	(1.623)	(0.375)
	constant	-	-	-7.154	-	-7.524***	-	-2.678***
				(5.191)		(3.015)		(1.302)
	Pseudo R	0.8279		0.3776		0.4014		0.2957
	squared
	Log-	-42.174		-14.632		-14.071		-16.558
	likelihood
	Notes: (1) IT is the binary dependent variable; (2) ***, **, and * indicate significance levels of 5, 10 and 15%
	respectively; (3) robust standard errors are reported in parentheses.

Table 5 :

 5 Estimates of the likelihood of adopting IT: Sensitivity analysis of Logit FE (2)

		Logit FE (1)	Logit FE (2)	Logit FE (3)	Logit FE (4)
		Full sample	Cross section data	Cross section data	Cross section data
		(1970-2017)
		Likelihood Marginal	Likelihood	Marginal	Likelihood Marginal	Likelihood Marginal
			effect		effect		effect		effect
	Inflation	-1142.1***	-41.627*	-3606.4**	-427.6***	-2009.7**	-258.8***	-21.456	-3.617
	volatility	(390.295)	(26.375)	(2064.53)	(194.506)	(1040.56)	(112.61)	(272.545)	(45.605)
	Output	-497.31***	-18.126**	5.127***	0.608**	0.843	0.108	0.909	0.153
	growth	(126.818)	(10.719)	(2.549)	(0.322)	(2.931)	(0.379)	(1.593)	(0.263)
	volatility
	Debt	-7.034***	-0.256**	0.807	0.096	0.782	0.101	0.080	0.013
		(2.146)	(0.157)	(1.938)	(0.225)	(1.717)	(0.221)	(1.430)	(0.241)
	CPS	3.632***	0.132*	5.930***	0.703***	3.515**	0.453***	1.562	0.263
		(1.365)	(0.086)	(2.624)	(0.232)	(1.893)	(0.202)	(1.511)	(0.243)
	CBI	58.582***	2.135**	13.537***	1.605***	10.035***	1.292***	5.588***	0.942***
		(12.169)	(1.222)	(6.314)	(0.432)	(3.427)	(0.304)	(2.521)	(0.359)
	NCF	-1.406***	-0.051**	6.013***	0.713***	6.242***	0.804***	5.176*	0.872***
		(0.370)	(0.030)	(2.512)	(0.165)	(2.209)	(0.162)	(3.347)	(0.415)
	constant	-	-	-12.489***	-	-8.305***		-4.487***
				(4.927)		(3.040)		(2.190)
	Pseudo R	0.6189		0.474		0.4248		0.2720
	squared
	Log-	-93.532		-12.356		-13.521		-17.113
	likelihood
	Notes: (1) IT is the binary independent variable; (2) ***, **, and * indicate significance levels of 5, 10 and 15%
	respectively; (3) robust standard errors are reported in parentheses.

Table 6 :

 6 Estimates of the likelihood of adopting IT: Sensitivity analysis : (1) IT is the binary independent variable; (2) logit FE estimator with an analytical correction for the incidental parameter bias (both time and country specific), conditional on the country specific fixed effects; (3) ***, **, and * indicate significance levels of 5, 10 and 15% respectively; (4) robust standard errors are reported in parentheses.

		Logit FE (1)		Logit FE (2)	Logit FE (2)
		(1990-2017)		(1990-2017)	(1990-2017)
		Yearly data		Yearly data
		Likelihood	Marginal		Likelihood Marginal	Likelihood	Marginal
			effect			effect		effect
	Inflation	-59.507***	-2.436**	Inflation	-484.3***	-34.069*	-	-
		(9.871)	(1.356)	volatility	(218.274)	(21.584)
				Coefficient			-1.967**	-0.112*
				of variation			(1.053)	(0.080)
	Output	-2.842	-0.116	Output	-1.347***	-0.095*	-455.403***	-
	growth	(9.514)	(0.394)	growth	(0.622)	(0.061)	(129.444)	25.581**
				volatility				(14.465)
	Debt	-9.305***	-0.381**	Debt to GDP -5.873***	-0.413**	-8.107***	-0.455**
		(2.732)	(0.231)		(1.631)	(0.217)	(2.350)	(0.259)
	CPS	0.997	0.041	CPS	1.129	0.079	0.174	0.010
		(1.685)	(0.072)		(1.268)	(0.096)	(1.824)	(0.102)
	CBI	30.162***	1.235*	CBI	37.697***	2.652***	59.807***	3.359**
		(10.648)	(0.788)		(9.063)	(1.342)	(14.316)	(1.828)
	NCF	-1.249***	-0.051**	NCF	-0.664***	-0.047**	-1.287***	-0.072**
		(0.346)	(0.031)		(0.209)	(0.025)	(0.361)	(0.041)
	Pseudo R	0.6099		Pseudo R	0.3672		0.479
	squared			squared
	Log-	-76.503		Log-	-124.097		-91.532
	likelihood			likelihood

Notes