Emmanuel Jeandel
email: emmanuel.jeandel@lif.univ-mrs.fr

Pascal Vanier
email: pascal.vanier@lif.univ-mrs.fr

Turing degrees of multidimensional SFTs

come

Wang tiles have been introduced by Wang [Wang(1961)] to study fragments of first order logic. Independently, subshifts of finite type (SFTs) were introduced to study dynamical systems. From a computational and dynamical perspective, SFTs and Wang tiles are equivalent, and most recursive-flavoured results about SFTs were proved in a Wang tile setting.

Knowing whether a tileset can tile the plane with a given tile at the origin (also known as the origin constrained domino problem) was proved undecidable by Wang [Wang(1963)]. Knowing whether a tileset can tile the plane in the general case was proved undecidable by Berger [Berger(1964), Berger(1966)].

Understanding how complex, in the sense of recursion theory, the points of an SFT can be is a question that was first studied by Myers [Myers(1974)] in 1974. Building on the work of Hanf [Hanf(1974)], he gave a tileset with no computable tilings. Durand/Levin/Shen [START_REF] Durand | Complex tilings[END_REF] Durand, Levin, and Shen] showed, 40 years later, how to build a tileset for which all tilings have high Kolmogorov complexity.

A Π 0 1 class (of sets) is an effectively closed subset of {0, 1} N , or equivalently the set of oracles on which a given Turing machine does not halt. Π 0 1 classes occur naturally in various areas in computer science and recursive mathematics, see e.g. [START_REF] Cenzer | 0 1 classes in mathematics[END_REF], Simpson(2011a)] and the upcoming book [START_REF] Cenzer | Effectively Closed Sets[END_REF]]. It is easy to see that any SFT is a Π 0 1 class (up to a computable coding of Σ Z 2 into {0, 1}

N ). This has various consequences. As an example, every non-empty SFT contains a point which is not Turing-hard (see Durand/Levin/Shen [START_REF] Durand | Complex tilings[END_REF] Durand, Levin, and Shen] for a selfcontained proof). The main question is how different SFTs are from Π 0 1 classes. In the one-dimensional case, some answers to these questions were given by Cenzer/Dashti/King/Tosca/Wyman [Dashti(2008), [START_REF] Cenzer | Computable symbolic dynamics[END_REF] Cenzer, Dashti, and King, Cenzer et al.(2012) Cenzer, Dashti, Toska, and Wyman].

The main result in this direction was obtained by Simpson [Simpson(2011b)], building on the work of Hanf and Myers: for every Π 0 1 class S, there exists a SFT with the same Medvedev degree as S. The Medvedev degree roughly relates to the "easiest" Turing degree of S. What we are interested in is a stronger result: can we find for every Π 0 1 class S a SFT which has the same Turing degrees? We prove in this article that this is true if S contains a computable point but not always when this is not the case. More exactly we build (Theorem 4.1) for every Π 0 1 class S a SFT for which the set of Turing degrees is exactly the same as for S with the additional Turing degree of computable points. We also show that SFTs that do not contain any computable point always have points with different but comparable degrees (Corollary 5.11), a property that is not true for all Π 0 1 classes. In particular there exist Π 0 1 classes that do not have any points with comparable degrees.

As a consequence, as every countable Π 0 1 class contains a computable point, the question is solved for countable sets: the sets of Turing degrees of countable Π 0 1 classes are the same as the sets of Turing degrees of countable sets of tilings. In particular, there exist countable sets of tilings with some non-computable points. This can be thought as a two-dimensional version of Corollary 4.7 in [START_REF] Cenzer | Computability of countable subshifts in one dimension[END_REF].

This paper is organized as follows. After some preliminary definitions, we start with a quick proof of a generalization of Hanf, already implicit in Simpson [Simpson(2011b)]. We then build a very specific tileset, which forms a grid-like structure while having only countably many tilings, all of them computable. This tileset will then serve as the main ingredient to prove the result on the case of classes with a computable point in section 4. In section 5 we finally show the result on classes without computable points.

Preliminaries

SFTs and tilings

We give here some standard definitions and facts about multidimensional subshifts, one may consult Lind [Lind(2004)] for more details. Let Σ be a finite alphabet, the d-dimensional full shift on Σ is the set

Σ Z d = c = (c x ) x∈Z d ∀x ∈ Z d , c x ∈ Σ . For v ∈ Z d , the shift functions σ v : Σ Z d → Σ Z d , are defined locally by σ v (c x ) = c x+v . The full shift equipped with the distance d(x, y) = 2 -min{ v |v∈Z d ,xv =yv}
is a compact, perfect, metric space on which the shift functions act as homeomorphisms. An element of Σ Z d is called a configuration.

Every closed shift-invariant (invariant by application of any σ v ) subset X of Σ Z d is called a subshift. An element of a subshift is called a point of this subshift.

Alternatively, subshifts can be defined with the help of forbidden patterns. A pattern is a function p : P → Σ, where P is a finite subset of Z d . Let F be a collection of forbidden patterns, the subset X F of Σ Z d containing only configurations having nowhere a pattern of F . More formally, X F is defined by

X F = x ∈ Σ Z d ∀z ∈ Z d , ∀p ∈ F, x z+P = p .
In particular, a subshift is said to be a subshift of finite type (SFT) when the collection of forbidden patterns is finite. Usually, the patterns used are blocks or n-blocks, that is they are defined over a finite subset P of Z d of the form 0, n -1 d . Given a subshift X, a block or pattern p is said to be extensible if there exists x ∈ X in which p appears, p is also said to be extensible to x.

In the rest of the paper, we will use the notation Σ X for the alphabet of the subshift X.

A subshift X ⊆ Σ Z 2 X is a sofic shift if and only if there exists a SFT Y ⊆ Σ Z 2 Y and a map f : Σ Y → Σ X such that for any point x ∈ X, there exists a point y ∈ Y such that for all z ∈ Z d , x z = f (y z ).

Wang tiles are unit squares with colored edges which may not be flipped or rotated. A tileset T is a finite set of Wang tiles. A coloring of the plane is a mapping c : Z 2 → T assigning a Wang tile to each point of the plane. If all adjacent tiles of a coloring of the plane have matching edges, it is called a tiling.

In particular, the set of tilings of a Wang tileset is a SFT on the alphabet formed by the tiles. Conversely, any SFT is isomorphic to a Wang tileset. From a recursivity point of view, one can say that SFTs and Wang tilesets are equivalent. In this paper, we will be using both indiscriminately. In particular, we denote by X T the SFT associated to a set of tiles T .

We say a SFT (tileset) is origin constrained when the letter (tile) at position (0, 0) is forced, that is to say, we only look at the valid tilings having a given letter (tile) t at the origin.

More information on SFTs may be found in Lind and Marcus' book [START_REF] Lind | An introduction to symbolic dynamics and coding[END_REF]].

The notion of Cantor-Bendixson derivative is defined on set of configurations. This notion was introduced for tilings by Ballier/Durand/Jeandel [START_REF] Ballier | Structural aspects of tilings[END_REF]. A configuration c is said to be isolated in a set of configurations C if there exists a pattern p such that c is the only configuration of C containing p. The Cantor-Bendixson derivative of C is denoted by D(C) and consists of all configurations of C except the isolated ones. We define C (λ) inductively for any ordinal λ:

• C (0) = S • C (λ+1) = D C (λ) ) • C (λ) = γ<λ C (γ) when λ is a limit ordinal.
The Cantor-Bendixson rank of C, denoted by CB(C), is defined as the first ordinal λ such that

C (λ) = C (λ+1) . If C is countable, then C CB(C) is empty. An element x is of rank λ in C if λ is the least ordinal such that x ∈ C (λ) .
A configuration x is periodic, if there exists n ∈ N * such that σ nei (x) = x, for any i ∈ {1, . . . , d}, where the e i 's form the standard basis. A vector of periodicity of a configuration is a vector v ∈ Z d \ {(0, . . . , 0)} such that σ v (x) = x. A configuration x is quasiperiodic (see Durand [Durand(1999)] for instance) if for any pattern p appearing in x, there exists N such that this pattern appears in all N d cubes in x. In particular, a periodic point is quasiperiodic. A configuration is strictly quasiperiodic if it is quasiperiodic and not periodic. A subshift is minimal if it is non-empty and contains no proper non-empty subshift. Equivalently, all its points have the same patterns. In this case, it contains only quasiperiodic points. It is known that every subshift contains a minimal subshift, see e.g. Durand [Durand(1999)].

Computability background

A Π 0 1 class P ⊆ {0, 1}
N is a class of infinite sequences on {0, 1} for which there exists a Turing machine that given x ∈ {0, 1}

N as an oracle halts if and only if

x ∈ P . Equivalently, a class S ⊆ {0, 1}

N is Π 0 1 if there exists a computable set L so that x ∈ S if and only if no prefix of x is in L. An element of a Π 0 1 class is called a member of this class.

We say that two sets S, S are recursively homeomorphic if there exists a bijective computable function f : S → S . That is to say there are two Turing machines M (resp. M ) such that given a member of S (resp. S ) computes a member of S (resp. S). Furthermore, for any s ∈ S, s ∈ S such that s is computed by M from s, M computes s from s .

The Cantor-Bendixson rank of S, is well defined similarly as for subshifts. See Cenzer/Remmel [START_REF] Cenzer | 0 1 classes in mathematics[END_REF]] for Π 0 1 classes and Kechris [Kechris(1995)] for Cantor-Bendixson rank and derivative.

For x, y ∈ {0, 1} N we say that x is Turing-reducible to y if y is computable by a Turing machine using x as an oracle and we write y ≤ T x. If x ≤ T y and y ≤ T x, we say that x and y are Turing-equivalent and we write x ≡ T y. The Turing degree of x ∈ {0, 1}

N , denoted by deg T x, is its equivalence class under the relation ≡ T .

Subshifts and Π 0 1 classes

As is clear from the definitions, SFTs in any dimension are Π 0 1 classes. More generally, effective subshifts, see e.g. Cenzer/Dashti/King [START_REF] Cenzer | Computable symbolic dynamics[END_REF]), that is subshifts defined by a computable (or equivalently, in this case, by a computably enumerable) set of forbidden patterns are Π 0 1 classes. As such, they enjoy similar properties. In particular, there exist many "basis theorems", i.e.theorems that assert that any Π 0 1 (non-empty) class has a member with some specific property.

As an example, every countable Π 0 1 class has a computable member, see e.g. Cenzer/Remmel [START_REF] Cenzer | Effectively Closed Sets[END_REF]]. For subshifts, we can say a bit more: every countable subshift has a periodic (hence computable) member.

Some basis theorems for Π 0 1 classes can be easily reproven in the context of subshifts: The proof that every Π 0 1 class has a point of low degree (the formal definition is not important here, but it can be interpreted as "nearly computable") [Jockusch and Soare(1972b)] was reproven for subshifts (actually tilings) in Durand/Levin/Shen [START_REF] Durand | Complex tilings[END_REF].

Π 0 1 classes and origin constrained tilings

A straighforward corollary of Hanf [Hanf(1974)] is that every Π 0 1 class is recursively homeomorphic to an origin constrained SFTs and conversely. This is stated explicitly in Simpson [Simpson(2011b)].

Theorem 2.1. Given any Π 0 1 class P ⊆ {0, 1} N , there exists a SFT X and a letter t ∈ Σ X such that each origin constrained point corresponds to a member of P .

Proof. Let P be a Π 0 1 class, and M the Turing machine that proves it, that is M given x ∈ {0, 1}

N as an oracle halts if and only if x ∈ P . We use the classic encoding of Turing machines as Wang tiles, see fig. 1. We modify all tiles containing a symbol from the tape, to allow them to contain a second symbol. This symbol is copied vertically. All these second symbols represent the oracle.

Then the SFT constrained by the tile starting the computation contains exactly the runs of the Turing machine with members of P on the oracle tape.

Corollary 2.2. Any Π 0 1 class P of {0, 1} N is recursively homeomorphic to an origin constrained SFT.

Producing a sparse grid

The main problem in the previous construction is that points which do not have the given letter at the origin can be very wild: they may correspond to configurations with no computation (no head of the Turing Machine) or computations starting from an arbitrary (not initial) configuration. A way to solve this problem is described in Myers' paper [Myers(1974)] but is unsuitable for our purposes (It was however used by Simpson to obtain a weaker result on Medvedev degrees, see [Simpson(2011b)

]). (a) (b) (c) (d)
Figure 2: The tiling in which the Turing machines will be encoded.

Our idea is as follows: we build a SFT which will contain, among other points, the sparse grid of Figure 2c. The interest being that all other points will have at most one intersection of two black lines. This means that if we put computation cells of a given Turing machine in the intersection points, every point which is not of the form of Figure 2c will contain at most one cell of the Turing machine, and thus will contain no computation.

To do this construction, we will first draw increasingly big and distant columns as in Figure 2a and then superimpose the same construction for rows as in Figure 2b, thus obtaining the grid of Figure 2c.

It is then fairly straightforward to see how we can encode a Turing machine inside a configuration having the skeleton of Figure 2c by looking at it diagonally: time increases going to the north-east and the tape is written on the northwest/south-east diagonals1 .

Our set of tiles T of Figure 3 gives the skeleton of Figure 2a when forgetting everything but the black vertical borders. We will prove in this section that it is countable. We set here the vocabulary:

• tile 30 is the corner tile • tile 20 and 27 are the top tiles

• tiles 30, 32, 33, 34 are the bottom tiles • a vertical line is formed of a vertical succession of tiles containing a vertical black line (tiles 5, 6, 7, 17, 21, 24, 25, 26, 31, 35, 36, 37), which may be ended by bottom and/or top tiles.

• a horizontal line is formed of a horizontal succession of tiles containing a horizontal black line (tiles 13, 14, 15, 16, 18, 19, 22, 23, 28, 32, 33, 34, 38), and may be ended by tiles 5,6,7,25,26,36,37, thus forcing a vertical line at this end, • a diagonal is a diagonal succession (positions (i,j),(i+1,j+1),...) of tiles among 4,12,11,

• a square is a 0, k 2 valid tiling such that {0}× 1, k-1 and {k}× 1, k-1 are vertical lines, and 1, k -1 × {0} and 1, k -1 × {k} are horizontal lines. Remark that the color on the right of the first column and on the left of the last one force the existence of a counting signal inbetween and of a diagonal tile on each of the (i, i) positions for 0 < i < k.

• the increase signal is formed by a path of tiles among 7, 14,22,23,24,25,26,27,30,31,36,38, such that the blue signal is connected, this signal will force the squares to increase in size by exactly one in each column.

• the counting signal is formed by a path of tiles among 3 ,7, 10, 12, 14, 19, 22, 32, 33, 38, such that the counting signal is connected. It may be ended only by tiles 30,32 and 7,21. This signal will force the number of squares in each column to be at most the size of these squares.

Note that whenever the corner tile appears in a point, it is necessarily a shifted copy of the point on Figure 4: the corner tile forces the tiles on its right to be bottom tiles and the first above to be tile 33, then on top of it must be tile 31 and then tile 27. This forces the existence of the first square, i.e.the first column. Then the increase signal forces the second column to start with a square of size increased by one, and thus to have exactly one more square (increase signal), and so on.

Lemma 3.1. The SFT X T admits at most one point, up to translation, with two or more vertical lines. This point is drawn on Figure 4.

Proof. The idea of the construction is to force that whenever there are two vertical lines, then the point is a shifted copy of the one in Figure 4.

Suppose that we have a tiling in which two vertical lines appear. They may be ended on their bottom only by a bottom tile 30 or 32, but when a bottom tile appears, it forces all tiles to its right to be bottom tiles. Because the color on each side of the vertical lines is not the same they necessarily are connected by horizontal lines, which must form squares due to the diagonal. Suppose the two vertical lines are at distance k, then there are exactly k squares between them vertically, because of the counting signal: it must appear in each square, and is shifted to the right every time it crosses a horizontal line, it may be ended in each column only by tiles 32 (or 30 if it is the leftmost column) in the bottommost square and by 21 (resp. 7) in the topmost.

The bottommost square must have an increase signal as its top horizontal line, since the lower left corner 32 (or 30 in case it is the leftmost square) forces the left side to be formed of a succession of tiles 24 ended by tile 26 (resp. only one tile 31), then the top left corner is necessarily a tile 25 (resp. 27). This forces the size of the squares on its left to be k -1 and on its right to be k + 1.

If we focus only on the bottommost squares, they are of decreasing size when going left, the last one is of size 1, and necessarily has the corner tile as its lower left corner. Lemma 3.2. X T is countable. Proof. Lemma 3.1 states that there is one point, up to shift, that has two or more vertical lines. This means that the other points have at most one such line.

• If a point has exactly one vertical line, then it can have at most two horizontal lines: one on the left of the vertical one and one on the right. Otherwise a square would appear and the configuration would be α. A counting signal may then appear on the left or the right of the vertical line arbitrarily far from it. There is a countable number of such points.

• If a point has no vertical line, then it has at most one horizontal line. A counting signal can then appear only once. There is a finite number of such points, up to shift.

There is a countable number of points that can be obtained with the tileset T . All types of obtainable points are shown in Figures 4 and5.

By taking our tileset T = {1, . . . , 40} and mirroring all the tiles along the south-west/north-east diagonal, we obtain a tileset T = {1 , . . . , 40 } with the exact same properties, except it enforces the skeleton of Figure 2b. Remember that whenever the corner tile appeared in a point, then necessarily this point was a shifted of α. Analogously, the corner tile of T appearing in a point means that this point is a shifted of α . We hence construct a third tileset τ = (T \ {30} × T \ {30 }) ∪ {(30, 30 )} which is the superimposition of T and T with the restriction that tiles 30 and 30 are necessarily superimposed to each other. The corner tile (30, 30 ) of τ has the property that whenever it appears, the tiling is the superimposition of the skeletons of Figures 2a and2b with the corner tiles at the same place: there is only one such tiling up to shift, we call it β.

The skeleton of Figure 2c is obtained from β if we forget about the parts of the lines of the T layer (resp. T ) that are superimposed to white tiles, 29' (resp. 29), of T (resp. T ).

As a consequence of Lemma 3.2, X τ is also countable. And as a consequence of Lemma 3.1, the only points in x τ in which computation can be embedded are the shifts of β. The shape of β is the one of Figure 2c, the coordinates of the points of the grid are the following (supposing tile (30, 30 ) is at the center of the grid):

{(f (n), f (m)) | f (m)/4 ≤ f (n) ≤ 4f (m)} {(f (n), f (m)) | m/2 ≤ n ≤ 2m} where f (n) = (n + 1)(n + 2)/2 -1.
Lemma 3.3. The Cantor-Bendixson rank of X τ is less than or equal to 13. 5. The rank of a product being the sum of the rank (when it is finite), (X τ ) is at most of rank 12.

Proof. It is clear that (X τ ) ⊂ (X T -{α} × X T -{α }). X T -{α} is of rank 6 as depicted in Figure

Π 0 1 classes with computable members and SFTs

The SFT constructed before will allow us to prove a series of theorems on Π 0 1 classes with computable points. The foundation of these is Theorem 4.1 which establishes a recursive homeomorphism between SFTs and Π 0 1 classes, up to a computable subset of the SFT. This recursive homeomorphism is the best we can hope for, as will be shown in section 5. Then from this "partial" homeomorphism, we will be able to deduce results on the set of Turing degrees of SFTs and Π 0 1 classes.

Theorem 4.1. For any Π 0 1 class S of {0, 1} N there exists a tileset τ S such that S × Z 2 is recursively homeomorphic to X τ S \ O where O is a computable set of configurations.

Proof. This proof uses the construction of section 3. Let M be a Turing machine such that M halts with x as an oracle iff x ∈ S. Take the tileset τ of section 3 and encode, as explained earlier, in configuration β the Turing machine M having as an oracle x on an unmodifiable second tape. This defines a new tiling system τ M , and we define O as the set of all points which were not constructed from a shift of β. To each (x, z) ∈ S × Z 2 we associate the β tiling having a corner at position z and having x on its oracle tape. O is computable, because it contains a countable number (Lemma 3.2) of computable points (none of these points can contain more than one step of computation).

Corollary 4.2. For any Π 0 1 class S of {0, 1} N with a computable member, there exists a SFT X with the same set of Turing degrees.

Corollary 4.3. For any countable Π 0 1 class S of {0, 1} N , there exists a SFT X with the same set of Turing degrees.

Proof. We know, from Cenzer/Remmel [START_REF] Cenzer | 0 1 classes in mathematics[END_REF]], that countable Π 0 1 classes have 0 (computable elements) in their set of Turing degrees, thus the SFT X τ M described in the proof of Theorem 4.1 has exactly the same set of Turing degrees as S.

Theorem 4.4. For any countable Π 0 1 class S of {0, 1} N there exists a SFT X with the same set of Turing degrees such that CB(X) = CB(S) + c for some constant c ≤ 13. This theorem holds when CB(S) is any ordinal, finite or infinite.

Proof. Lemma 3.3 states that X τ is of Cantor-Bendixson rank c ≤ 13 In the tileset τ M of the previous proof, the Cantor-Bendixson rank of the contents of the tape is exactly CB(S), hence CB(X τ S ) = CB(S) + c.

From Ballier/Durand/Jeandel [START_REF] Ballier | Structural aspects of tilings[END_REF] we know that for any subshift X, if CB(X) ≤ 2, then X has only computable points. Thus an optimal construction would have to augment the Cantor-Bendixson rank by at least 2.

Corollary 4.5. For any countable Π 0 1 class S of {0, 1} N there exists a sofic subshift X with the same set of Turing degrees such that CB(X) = CB(S) + 2.

Proof. Take a projection that just keeps the symbols of the Turing machine tape in the SFT τ M from the proof of Theorem 4.1 and maps everything else to a blank symbol. Recall the Turing machine tape cells are the intersections of the vertical lines and horizontal lines. This projection leads to 3 possible configurations (up to shift):

• A configuration with a white background and points corresponding to the intersections in the sparse grid of Figure 2c. This is an isolated point, of rank 1

• A completely blank configuration with only one symbol somewhere. This configuration is isolated once we remove the previous one(s), hence of rank 2.

• A completely blank configuration, of rank 3.

Note that a similar theorem in dimension one for effective rather than sofic subshifts is proved in Cenzer/Dashti/Toska/Wyman [Cenzer et al.(2012)Cenzer, Dashti, Toska, and Wyman, Theorem 4.6].

Π 0 1 classes without computable members and subshifts

In this section we prove that two-dimensional SFTs containing only non-computable points have the property that they always have points with different but comparable degrees, this is Corollary 5.11. But we first prove this result for onedimensional subshifts, not necessarily of finite type, in Theorem 5.3, the proof for two-dimensional SFTs needing only a bit more work.

One interest of these proofs, lies in the following theorem, proved by Jockusch and Soare: Theorem 5.1 (Jockusch, Soare). There exist Π 0 1 classes containing no computable member, such that any two different members are Turing-incomparable.

The proof of this result can be found in Cenzer and Remmel's upcoming book [START_REF] Cenzer | 0 1 classes in mathematics[END_REF]] or in the original articles by Jockusch and Soare [Jockusch and Soare(1972b), Jockusch and Soare(1972a)].

This means that one cannot expect a full recursive homeomorphism, i.e. without removal of the computable points. Furthermore, this shows that in general, when a Π 0 1 class P has no computable member, it is not true that one can find a SFT with the same set of Turing degrees.

The main idea of the proof is that any subshift contains a minimal subshift. If the subshift has no computable points (actually, no periodic points), this minimal subshift contains only strictly quasiperiodic points. We will then use some combinatorial properties of this minimal subshift to obtain our results.

One-dimensional subshifts

We start with a technical lemma that will allow us to prove the theorem: Lemma 5.2. Let x be a strictly quasiperiodic point of a minimal one-dimensional subshift A and ≺ be an order on Σ A . For any word w extensible to x, there exist two words w 0 and w 1 such that:

• w appears exactly twice in both w 0 and w 1 ,

• let a and b (resp. c and d) be the first differing letters in the blocks directly following the first and second occurrence of w in w 0 (resp. w 1 ), then a ≺ b (resp. d ≺ c).

See Figure 6 for an illustration of w 0 and w 1 .

Proof. By quasiperiodicity of x, w appears infinitely many times in x. By non periodicity, any two occurrences of w must be followed by eventually distinct words. Let y be the largest word so that whenever w appears in x, it is immediately followed by y. Note that w appears only once in wy, otherwise x would be periodic. By definition of y, the letters after each occurrence of wy cannot be all the same. So there exist two consecutive occurrences of wy with differing next letters a, b with, e.g., a ≺ b (the other case being similar). w 0 is then defined as the smallest word containg both occurrences of wy and these letters a, b. Now x is quasiperiodic, hence some occurrence of wyb must also appear before some occurrence of wya, so we can find between these two positions two occurrences of wy with differing next letters c, d with d ≺ c. We can then define w 1 similarly.

Theorem 5.3. Let A be a minimal one-dimensional subshift containing only strictly quasiperiodic points and x a point of A. Then for any Turing degree d such that deg T x ≤ d, there exist a point y ∈ A with Turing degree d.

Proof. To prove the theorem, we will give two computable functions f : A × {0, 1}

N → A and g : A → {0, 1} N such that for any x ∈ A and s ∈ {0, 1} N we have g(f (x, s)) = s. This means in terms of Turing degrees:

deg T s ≤ deg T f (x, s) ≤ sup T (deg T x, deg T s) (1)
That is to say, we give two algorithms, one (f ) that given a point x of A and a sequence s of {0, 1}

N reversibly computes a point of A that embeds s, the second (g) retrieves s from the computed point. Let us now give f . Let ≺ be an order on Σ A . Given a point x ∈ A and a sequence s ∈ {0, 1}

x C i C i e f C i+1
N , f recursively constructs another point of A: it starts with a block C -1 = x 0 and recursively constructs bigger and bigger blocks C i such that C i+1 has C i in its center. Furthermore these blocks are each centered in 0. So that the sequence

C -1 → C 0 → C 1 → • • • → C i → . . . converges to a point c of A having all C i 's in its center. It is sufficient to show then how C i+1 is constructed from C i .
f works as follows: It searches for two consecutive occurrences of C i in x, where the two first differing letters satisfy a ≺ b if s i+1 = 0 and b ≺ a if s i+1 = 1. We know that f will eventually succeed in finding these occurrences due to Lemma 5.2. Now we define C i+1 as the word in x where we find these two occurrences, correctly cut so that the first occurrence of C i is at its center, and its last letter is the differing letter of the second occurrence. See Figure 7.

We thus have f , which is clearly computable. We give now g.

Given C i and c, one can compute s i+1 easily: we just have to look for the second occurrence of C i in c, the first one being in its center. We then check whether the first differing letters between the blocks following each occurrence are such that e ≺ f or f ≺ e. This also gives us C i+1 .

This means that from c, one can recover s. We know C -1 = c 0 and from this information, we can get the rest: from c and C i , one computes easily C i+1 and s i . We have constructed our function g.

So now if we take a sequence s such that deg T s > deg T x, we can take y = c = f (x, s). It follows from inequality 1 that it has the same Turing degree as s since deg T s = sup T (deg T x, deg T s).

Corollary 5.4. Every non-empty one-dimensional subshift S containing only non computable points has points with different but comparable degrees.

Proof. Take any minimal subshift of S. It must contain only strictly quasiperiodic points, so the previous theorem applies.

For effective subshifts, we can do better: Lemma 5.5. Every non-empty one-dimensional effective subshift S contains a minimal subshift S whose language is of Turing degree less than or equal to 0 . 0 is the degree of the Halting problem.

Proof. Let F be the computable set of forbidden patterns defining S. Let w n be a (computable) enumeration of all words. Define F n as follows:

F -1 = ∅. Then if F n ∪ F ∪ {w n+1 } defines a non-empty subshift, then F n+1 = F ∪ {w n+1 } else F n+1 = F n . Now take F = ∪ n F n .
It is clear from the construction that F is computable given the Halting problem. Moreover F defines a non-empty, minimal subshift S. More exactly the complement of F is exactly the set of patterns appearing in S. This lemma cannot be improved: an effective subshift is built in Ballier/Jeandel [START_REF] Ballier | Computing (or not) quasi-periodicity functions of tilings[END_REF]] for which the language of every minimal subshift is at least of Turing degree 0 . Now it is clear that any minimal subshift S has a point computable in its language, so that:

Corollary 5.6. Every non-empty one-dimensional effective subshift with no computable point contains configurations of every Turing degree above 0 .

We do not know if this can be improved. While it is true that all minimal subshifts in [START_REF] Ballier | Computing (or not) quasi-periodicity functions of tilings[END_REF]] have a language of Turing degree at least 0 , this does not mean that their configurations have all Turing degree at least 0 . In the construction of [START_REF] Ballier | Computing (or not) quasi-periodicity functions of tilings[END_REF]], there indeed exist computable minimal points. The construction of Myers [Myers(1974)] has noncomputable points, but points of low degree.

Two-dimensional SFTs

We now prove an analogous theorem for two dimensional SFTs. We cannot use the previous result directly as it is not true that any strictly quasiperiodic configuration always contain a strictly quasiperiodic (horizontal) line. Indeed, there exist strictly quasiperiodic configurations, even in SFTs with no periodic configurations, where some line in the configuration is not quasiperiodic (this is the case of the "cross" in Robinson's construction [Robinson(1971)]) or for which every line is periodic of different period (such configurations happen in particular in the Kari-Culik construction [START_REF] Culik | [END_REF], Kari(1996)]).

We will first try to prove a result similar to Lemma 5.2, for which we will need an intermediate definition and lemma.

Definition 5.7 (line).

A line or n-line of a two-dimensional configuration x ∈ Σ Z 2 is a function l : Z × H → Σ, with H = h + 0; n -1 , h ∈ Z, such that x |Z×H = l.
Where n is the width of the line and h the vertical placement.

One can also define a line in a block by simply taking the intersection of both domains. The notion of quasiperiodicity for lines is exactly the same as the one for one dimensional subshifts. We need this notion for the following lemma, that will help us prove the two-dimensional version of Lemma 5.2. We also think that this lemma might be of interest in itself.

Lemma 5.8. Let A be a two-dimensional minimal subshift. There exists a point x ∈ A such that all its lines are quasiperiodic.

Proof. Let {(a i , b i )} i∈N be an enumeration of Z × N and H i = a i + 0; b i .

If x is a configuration, denote by p i (x) : Z × H i → Σ the restriction of x to Z × H i . We will often view p i as a map from A to (Σ Hi ) Z . A horizontal subshift is a subset of Σ Z 2 which is closed and invariant by a horizontal shift.

We will build by induction a non-empty horizontal subshift A i of A with the property that every configuration x of A i has the property that every line of support H j , for any j < i, is quasiperiodic. More precisely, p j (A i ) will be a minimal subshift.

Define

A -1 = A. If A i is defined, consider p i+1 (A i )
. This is a non-empty subshift, so it contains a minimal subshift X. Now we define the horizontal subshift A i+1 = p -1 i+1 (X) ∩ A i . By construction p i+1 (A i+1 ) is minimal. Furthermore, for any j < i, p j (A i+1 ) is a non-empty subshift, and it is included in p j (A j ), which is minimal, hence it is minimal.

To end the proof, remark that by compactness ∩ i A i is non-empty, as every finite intersection is non-empty.

Lemma 5.9. Let A be a two-dimensional minimal subshift where all points (equivalently, some point) have no horizontal period.

Let x be a point of A and ≺ be an order on Σ A . For each n ∈ N, for any n-block w extensible to x, there exist two blocks w 0 and w 1 , of the same size, both extensible to x such that:

• w appears exactly twice in both w 0 and w 1 , each on the n-line of vertical placement 0.

• the first differing letters e and f in the blocks containing w in their center are such that e ≺ f in w 0 and f ≺ e in w 1 .

Here the word "first" refers to an adequate enumeration of N × Z.

Proof. As the result is about patterns rather than configurations, and all points of a minimal subshift have the same patterns, it is sufficient by Lemma 5.8 to prove the result when all lines of x are quasiperiodic. Since w appears in x, it appears a second time on the same n-line in x. Since x is not horizontally periodic, both occurrences are in the center of different blocks. (The place where they differ may be on a different line, though, if this particular n-line is periodic)

Now we use the same argument as lemma 5.2 on the m-line containing both occurrences of w and the first place they differ. (Note that we cannot use directly the lemma as this m-line might itself be periodic, but the proof still works in this case) Theorem 5.10. Let A be a two-dimensional minimal subshift where all points (equivalently, some point) have no horizontal period and let x be a point of A. Then for any Turing degree d such that deg T x ≤ d, there exists a point y ∈ A with Turing degree d.

Proof. The proof is almost identical to the one of Theorem 5.3, Lemma 5.9 being the two-dimensional counterpart of Lemma 5.2, the only difference being that we have to search simultaneously all lines for the presence of two occurences of C i in order to construct C i+1 . One can see in Figure 8 how the C i 's are contructed in this case. Corollary 5.11. Every two-dimensional non-empty subshift X containing only non-computable points has points with different but comparable Turing degrees.

Proof. X contains a minimal subshift A, which cannot be periodic since it would otherwise contain computable points. There are now two possibilities:

• If A contains a point with a horizontal period, then all points of A have a horizontal period, and the result follows from Theorem 5.3, since all points are strictly quasiperiodic in the vertical direction.

• Otherwise, it follows from Theorem 5.10.

Lemma 5.5 is still valid in any dimensions so that we have:

Corollary 5.12. Every two-dimensional non-empty effective subshift (in particular any non-empty SFT) with no computable points contains points of any Turing degree above 0 .

We conjecture that a stronger statement is true: The set of Turing degrees of any subshift with no computable points is upward closed. To prove this, it is sufficient to prove that for any subshift S and any configuration x of S (which is not minimal), there exists a minimal configuration in S of Turing degree less than or equal to the degree of x. We however have no idea how to prove this, and no counterexample comes to mind.

Figure 1 :

 1 Figure 1: A set of Wang tiles, encoding computation of a Turing machine: the states are in the circles and the tape is in the rectangles. The bottom right tile starts computations. A tiling containing this tile contains the space-time diagram of some run of the Turing machine.

Figure 3 :

 3 Figure 3: Our set of Wang tiles T .

Figure 4 :

 4 Figure 4: Tiling α: the unique valid tiling of T in which there are 2 or more vertical lines. This tiling has Cantor-Bendixson rank 1.

Figure 5 :

 5 Figure 5: The configurations of X T \ {α}: the A -ZZ configurations are unique (up to shift), and the configurations with subscripts i, j ∈ N, k ∈ Z 2 represent the fact that distances between some of the lines (red, horizontal, vertical) can vary. The configurations are classified according to their Cantor-Bendixson rank. Note that configuration ZZ cannot have a counting signal on its left, because it would force another vertical line.

Figure 6 :

 6 Figure 6: Two nearest w blocks, the first differing letter a, b and c, d in their following blocks, and how they form the w i . Note that the first differing letter might in some cases be inside the second occurrence of w, as illustrated on the right with c, d.

Figure 7 :

 7 Figure 7: How we construct c i+1 from c i . When s i+1 = 0, we have e ≺ f and f ≺ e otherwise. The words of Lemma 5.2 are completed on the left with the block preceeding them in x.

Figure 8 :

 8 Figure 8: How C i+1 is constructed inductively from C i . C i is in the center of C i+1 . The letters e and f are the first differing letters in the blocks containing the C i 's. Whether e ≺ f of f ≺ e depends on what symbol we want to embed, 0 or 1.

Note that we have to wait for the diagonal to increase to have a new step of computation, in order to have enough space on the tape.


Caroline Mothe
email: caroline.mothe@univ-savoie.fr

Uyen Nguyen
email: thithucuyen.nguyen@ceps.lu

Firm technological innovation persistence: Organizational innovation

Keywords: Organizational Innovation, Technological Innovation, Persistence JEL codes: L 21, 0 32, 0 33

Organizational innovation favors technological innovation, but does it also influence technological innovation persistence? This article investigates empirically the pattern of technological innovation persistence and tests the potential impact of organizational innovation using firm-level data from three waves of the French Community Innovation Surveys. Evidence shows a positive effect of organizational innovation on technological innovation persistence, according to various measures of organizational innovation. Moreover, this impact is more significant for complex innovators (i.e., those who innovate in both products and processes). These results highlight the complexity of managing organizational practices with regard to the firm's technological innovation. They also add to comprehension of the drivers of innovation persistence, through a focus on an often forgotten dimension of innovation in a broader sense.

Résumé

Les innovations organisationnelles favorisent les innovations technologiques mais influencent-elles la persistance à l'innovation technologique ? Cet article analyse empiriquement un modèle de persistance de l'innovation et teste sur des données de panel l'impact potentiel des innovations d'organisation. La base de données est issue de trois vagues d'enquêtes communautaires sur l'innovation réalisées sur un échantillon d'entreprises françaises. Les résultats montrent un impact positif de plusieurs mesures des innovations organisationnelles sur la capacité des firmes à innover de façon persistante. Toutefois, l'impact est plus significatif pour les innovateurs complexes (innovant dans les produits et les procédés). Les résultats mettent en lumière la complexité des pratiques de management organisationnel. Ils aident aussi à comprendre plus finement les déterminants de la persistance à l'innovation à travers une analyse centrée sur une dimension souvent oubliée : l'innovation organisationnelle.

innovation persistence, we consider the specific role of organizational innovation, which clearly is important for corporate performance but has not been researched with regard to its potential impact on technological innovation persistence. To fill this gap, we begin by establishing our focal research question, based on a survey of extant literature. We then describe our data set before outlining our methodology and empirical models. Next, we discuss our results and conclude with some avenues for further research.

Organizational innovation and technological innovation persistence

The neglected role of organizational innovation

Innovation refers to the adoption of an idea, behavior, system, policy, program, device, process, product, or service that is new to the organization [START_REF] Damanpour | Organizational size and innovation[END_REF]. Although [START_REF] Damanpour | Organizational size and innovation[END_REF] considers the general concept of organizational innovation as related to all parts of the organization, most approaches divide innovation into technological and organizational versions. [START_REF] Phillips | Innovation and firm performance in Australian manufacturing, industry commission[END_REF] separates technological and non-technological innovation to include new marketing strategies and changes to management techniques or organizational structures in the latter category. Most literature in innovation management and economics instead concentrates on technological innovation, without clear guidelines for how firms should address the types of innovation that may lead to technological innovation (e.g., 'administrative innovation', [START_REF] Damanpour | Organizational innovation: a meta-analysis of effects of determinants and moderators[END_REF]. The expanded definition of innovation in the Oslo Manual (OECD 2005) treats organizational innovation as an innovation type, separate from the technological innovation type. Yet the question remains: How does organizational innovation affect technological innovation and thus global firm performance? [START_REF] Pavitt | Innovation Processes[END_REF] argues that technological innovation consists of three overlapping processes: the production of scientific and technological knowledge, the translation of this knowledge into working artifacts (products, systems, processes, services), and response to market needs.

Generally, technological innovation encompasses product and process innovation. Product innovation is the introduction of goods or services that are new or significantly improved with respect to their specifications or intended uses, such as significant improvements in technical specifications, components and materials, incorporated software, user friendliness, or other functional characteristics (OECD 2005). Process innovation is defined as the implementation of a new or significantly improved production or delivery method, including significant changes in techniques, equipment, and/or software. Process innovations might attempt to decrease unit costs of production or delivery, increase quality, or produce or deliver new or significantly improved products (OECD 2005). [START_REF] Lam | Organizational innovation[END_REF] also proposes two meanings for organizational innovation: the creation or adoption of an idea new to the organization (organizational innovation lato sensu) and changes in managerial practices or kinds of organizational forms (organizational innovation stricto sensu). Theoretically, organizational innovation is a broad concept that encompasses strategies, structural, and behavioral dimensions, yet there is no consensus about its definition.

Some authors include all types of innovation under its umbrella (e.g., [START_REF] Daft | A dual-core model of organizational innovation[END_REF][START_REF] Kimberly | Organizational Innovation: the influence of individual, organizational, and contextual factors on hospital adoption of technological and administrative innovations[END_REF][START_REF] Damanpour | Organizational innovation and performance: the problem of organizational lag[END_REF][START_REF] Crossan | A multi-dimensional framework of organizational innovation[END_REF], whereas others, including this study, use it in contrast with technological innovation (e.g., [START_REF] Gumusluoğlu | Transformational leadership and organizational innovation: the roles of internal and external support for innovation[END_REF]) by drawing on the distinction between technological and non-technological innovation (e.g., [START_REF] Baldridge | Organizational Innovation: Individual, Organizational, and Environmental Impacts[END_REF][START_REF] Battisti | How innovative are UK firms? Evidence from the CIS4 on synergies between technological and organizational innovations[END_REF]. In the absence of a unified theoretical definition, we follow the OECD (2005), which views an organizational innovation as 'the implementation of a new organizational method in the firm's business practices, knowledge management, workplace organization or external relations that has not been previously used by the firm'.

Studies of the relationships between organizational and technological innovations often highlight that technological innovation drives organizational changes within the firm [START_REF] Henderson | Architectural innovation: the reconfiguration of existing product technologies and the failure of established firms[END_REF][START_REF] Dougherty | A practice-centered model of organizational renewal through product innovation[END_REF], because firms introducing technological innovation must reorganize their production, workforce, sales, and distribution systems.

Another research stream suggests an inverse relationship, such that organizational innovation enhances flexibility and creativity, which facilitates the development of technological innovations [START_REF] Greenan | Innovation organisationnelle, dynamisme technologique et performance des entreprises[END_REF]Lokshin et al. 2008b). Organizational (re)structuring, if it leads to structural renewal, could facilitate other types of innovations (Günday et al. 2011).

For example, [START_REF] Staropoli | Cooperation in R&D through a network, an organizational gamble? An empirical analysis of Rhône Poulenc Rorer-Gencell[END_REF] emphasizes how external relations and networks can enhance technological innovation in pharmaceutical firms, and [START_REF] Bharadwaj | Making innovation happen in organizations: Individual creativity mechanisms, organizational creativity mechanisms or both[END_REF] show that innovation is a function of individual efforts and organizational systems aimed at facilitating creativity, such that successful product innovation depends partly on organizational factors. Studying interrelations of different innovation strategies, [START_REF] Schmidt | Non-technological and technological innovation: strange bedfellows?[END_REF] indicate that a combination of technological and non-technological innovation has a positive impact on innovation performance. Similarly, Günday et al. (2011) find a positive relationship between organizational innovation and technological innovation.

With sample of fast-moving consumer goods firms in Germany, Lokshin et al. (2008a) study the effect of organizational skills on firms' innovative performance. Firms implementing a combination of customer, organizational, and technological skills tend to introduce more innovations. [START_REF] Mothe | Non-technological and technological innovations: Do services differ from manufacturing? An empirical analysis of Luxembourg firms[END_REF] find that the effects of non-technological innovations differ depending on the phase of the innovation process. Organizational innovations significantly increase the likelihood of innovation, but not its commercial success.

These studies all acknowledge the crucial role of organizational practices on competitive advantage and firm innovation, in the sense that they provide input into the firm's innovation process and innovation capability. Therefore, we argue that firms that dedicate more resources to new organizational forms should be in a better position to use new skills and technologies efficiently.

Three frameworks for technological innovation persistence

Few studies deal with organizational innovation in relation to technological innovation in the context of persistence, though [START_REF] Antonelli | Inside innovation persistence: New evidence from Italian micro-data[END_REF] find a relatively higher persistence level for product innovation than for process innovation. When a firm undertakes different types of innovation jointly (e.g., product, process, and organizational), it achieves a lower degree of state dependence, acknowledging for the fact that, for firms combining several innovations (product, process and organization), the state of the period depends less on the state of the previous period. To put it simply, 'general' innovators are less persistent. Le [START_REF] Bas | Are complex innovators more persistent than single innovators? An empirical analysis of innovation persistence drivers[END_REF], using two waves of the Community Innovation Surveys (CIS) in Luxembourg, indicate that complex innovators (product and process) are more inclined to be persistent than single innovators (product or process). Organizational innovation exerts a positive impact only on the probability of being a complex innovator.

In turn, we consider the role of organizational innovation for technological innovation persistence, according to three complementary explanations of technological innovation persistence at the firm level (Le [START_REF] Bas | Are complex innovators more persistent than single innovators? An empirical analysis of innovation persistence drivers[END_REF]. First, the knowledge accumulation hypothesis stipulates that experience in innovation is associated with dynamic increasing returns, in the form of learning-by-doing and learning-to-learn effects, which enhance knowledge stocks and the probability of future innovations [START_REF] Geroski | How persistently do firms innovate?[END_REF][START_REF] Duguet | Les fondements microéconomiques de la persistance de l'innovation: Une analyse économétrique[END_REF][START_REF] Latham | The Economics of Persistent Innovation[END_REF]. Learning in this sense pertains to a capacity to innovate later. According to the OECD (2005) definition, we might anticipate that new practices for organizing work drive the changing processes, and new methods of organizing external relations increase the level of technological knowledge exchanged (learning by interacting) and spur the emergence of improved technologies. [START_REF] Schmidt | Non-technological and technological innovation: strange bedfellows?[END_REF] provide some support for this approach. Taking into account the interrelations across different innovation strategies, they note that the combination of technological and non-technological innovation has a positive impact on effective innovation performance.

Second, the 'success breeds success' approach indicates that a firm gains locked-in advantages over other firms through successful innovations. Specifically, innovation feeds profitability, which funds subsequent innovation activities. Economic and commercial successes matter in this regard, and [START_REF] Polder | Product, process and organizational innovation: drivers, complementarity and productivity effects, United Nations University -Maastricht Economic and social Research and training centre on Innovation and Technology[END_REF] argue that organizational innovation also plays a role. Product and process innovations, in combination with organizational innovation, have positive impacts on firm productivity. [START_REF] Cordelier | Innovations et performances des sociétés des services et de l'industrie manufacturière, comparaisons[END_REF] provides similar results, obtained from the market shares of French firms after they innovate; product or process innovations are more profitable than organizational innovations in the industrial sector, but when firms combine different types of innovations, their market shares grow more.

Ultimately, the greater the combination of innovation types, the higher the market share growth. Thus a complex innovator implementing organizational innovations should perform better than a complex innovator that does not adopt new organizational practices. [START_REF] Evangelista | The economic impact of technological and organizational innovations. A firm-level analysis[END_REF] support this view by showing that firms that introduce both technological and organizational innovations have a clear competitive advantage over both non-innovating firms and those introducing technological innovations only. All these studies imply that organizational innovations, used together with technological innovations, exert a positive effect on firm economic performance.

Third, noting sunk costs in R&D activities, [START_REF] Antonelli | Inside innovation persistence: New evidence from Italian micro-data[END_REF] interpret evidence of persistence in innovation efforts as intertemporal stability in R&D efforts. The firm chooses between investing or not in R&D activities, but sunk costs encourage the continuity of R&D expenditures. That is, a firm deciding for the first time to engage in R&D activities incurs start-up costs that are not recoverable, and the resulting sunk costs represent a barrier, to both entry into and exit from R&D activity. At first glance, organizational innovation seems irrelevant in this context, but if firms implement new practices to organize the work or procedures of their R&D department or adopt new methods to interact with external relations, the cost of their R&D might decrease over time (from a resource allocation perspective), which would increase incentives to persist in their R&D activity.

Following this line of research, we highlight the effects of organizational innovation strategies on firms' technological innovation persistence. Unlike [START_REF] Antonelli | Inside innovation persistence: New evidence from Italian micro-data[END_REF], we focus on the impact of organizational innovation on technological innovation persistence. Thus, we can compare our findings, derived from three waves of France's CIS, with those recently published by Le [START_REF] Bas | Are complex innovators more persistent than single innovators? An empirical analysis of innovation persistence drivers[END_REF], using two waves of the Luxembourg CIS.

Our research thus features a larger and more representative sample, while also providing more detail about the underlying dynamic mechanisms by which organizational innovation affects technological innovation persistence, using panel econometrics (versus simple cross-sectional estimates) and more complex variables for organizational innovation that can account for continuity and intensity effects.

Data, variables, and descriptive statistics

Data collection

Community Innovation Surveys (CIS) follow a subject approach to studying innovation, with the firm as the statistical unit (rather than an individual innovation), and combine census and stratified sampling methods for each wave. The stratum variables are consistently activity and size, and the data collection includes both innovators and non-innovators. For statistical consistency, we draw on three successive waves of the French CIS: CIS4 (2002-2004, which we call t 0 ), CIS6 (2004CIS6 ( -2006, or , or These most recent surveys are homogenous in their definitions of innovation. We thus can identify the same questions that relate to product, process, and organizational innovations, though they differed in the waves prior to the 2005 CIS. For the analysis, we merged the three survey waves, such that the final data set includes only firms that responded to all three waves and excluded any that entered or exited the market midway through 2002-2008. The balanced panel consisted of 1,180 manufacturing firms with 20 or more employees.

The sector composition and size distribution of the final sample does not vary substantially from one period to another. Therefore, we describe the balanced data set for 2006, an intermediate year during our study period (see Table I). More than half the sample consists of low or medium-low technology firms (according to NACE 1 classifications), operating in sectors such as plastic products, metals (12%), food, textiles, and wood (20%). The rest of the sample features high and medium-high technology firms (40% of the total), operating in 1 NACE is the 'statistical classification of economic activities in the European Community', used uniformly by all member states. We classified manufacturing industries according to their global technological intensity with NACE Revision 1.1 for the periods t 0 and t 1 , whereas t 2 was covered by the NACE Revision 2, according to the Eurostat classification (http://epp.eurostat.ec.europa.eu/cache/ITY_SDDS/Annexes/hrst_st_esms_an9.pdf).

industries such as electronics, instruments, and chemicals. Regarding the size distribution, we find a majority (66%) of medium-sized firms (250-1000 employees) in our sample.

INSERT TABLE I ABOUT HERE

Variables and descriptive statistics i. Dependent variables

We used four dependent variables. The CIS considers a firm innovative if, in a given period of time (i.e., three years prior to the survey), it introduced a new product or process.

We designed dichotomous variables to measure whether the firm produced an innovation during that period, as well as to assess the type of innovation (product, process, or organization). Product innovators introduced, in the three years prior to the survey, goods or services that were 'either new or significantly improved with respect to its fundamental characteristics, technical specifications, incorporated software or other immaterial components, intended uses, or user friendliness' (OECD 2005). Process innovators implemented 'new techniques … or significantly improved production technology, new and significantly improved methods of supplying services and of delivering products' (OECD 2005).

From these definitions, to study the persistent innovation behavior of firms, we identified four types of innovators: pure product, pure process, single, and complex. The binary variable (Only_prod) takes a value of 1 if the firm is a pure product innovator; (Only_proc) takes the value of 1 if the firm is a pure process innovator; the (Single) variable equals 1 if the firm is a single innovator, such that it introduced either a product or a process innovation during the studied period; and the fourth dependent variable (Complex) takes a value of 1 if the firm is a complex innovator because it introduced both product and process innovations. For each type of innovator, we considered the dependent variable related to each of the three time periods (see Tables II andIII for definitions and descriptive statistics).

INSERT TABLES II and III ABOUT HERE

ii. Organizational innovation

Our main hypothesis relates to the impact of organizational innovation on firms' technological persistence. Several measures of organizational innovation appear in previous studies examining technological innovation [START_REF] Schmidt | Non-technological and technological innovation: strange bedfellows?[END_REF][START_REF] Armbruster | Organizational innovation: The challenge of measuring non-technical innovation in large-scale surveys[END_REF]Mothe andNguyen 2010, 2012). Generally, organizational innovations include changes in business practices (including knowledge management), in the workplace organization or the firm's external relations. Prior research has tended to concentrate on the probability of introducing new organizational practices during a reference period, a procedure that fails to account for the degree of intensity of the organizational innovation or the temporal continuity of organizational change. Thus, these approaches cannot assess some key aspects of organizational innovation, such as intensity, continuity, or the impact on the dynamics of firms' innovation behavior.

Instead, we adopted new measures of organizational innovation that could (1) handle the temporal continuity of organizational innovation (ConOrg) and ( 2) examine the degree of intensity of organizational innovation (IntOrg). Specifically, we introduced two variables in each case to control for the continuity of implementing organizational innovation across two periods of time, such that we attain four organizational innovation variables.

To construct ConOrg, we began with data about organizational innovations implemented during the reference period for each wave and created the binary composite variable of organizational innovation (Org). The CIS04 reported three organizational practices: (1) new or significantly changed corporate strategy, (2) advanced management techniques, and (3) major changes to organizational structure. The CIS06 included data on four organizational practices: (1) new business practices for organizing work and procedures, (2) new knowledge management systems, (3) new methods of workplace organization, and (4) new methods of organizing external relations. We constructed four dummy variables for each practice. Finally, CIS08 provides information about three organizational practices: (1) new business practices for organizing work and procedures, (2) new methods of workplace organization, and (3) new methods of organizing external relations. 2 The variable Org(t) (t -2, t -1) equals 1 if at least one organizational practice was implemented during t (t -2, t -1) and 0 otherwise.

In a second step, we constructed ConOrg using Org. The variable ConOrg (t -1, t), for example, depends on the firms' organizational innovation during t -1 (2004)(2005)(2006) andt (2006)(2007)(2008). It equals 0 if firms did not introduce organizational innovation in any of the two periods, 1 if organizational innovation appears only in t -1 (Org(t -1) = 1 and Org(t) = 0), 2 if it has been introduced only in t (Org(t -1) = 0 and Org(t) = 1), and 3 if it is continuously adopted throughout both periods (Org(t -1) = 1 and Org(t) = 1) (see Table III).

The second variable of organizational continuity, ConOrg(t -2, t -1), follows similar principles: It equals 0 if no organizational innovation was adopted in t -2 (2002-2004) or t -1 (2004-2006), 1 if it were adopted only in t -2, 2 if it adopted only in t -1, and 3 if it has been continuously adopted. Both indicators of organizational innovation thus are intertemporal, such that we may control for the temporal dimension of the impact of organizational innovation on the dynamics of technological innovation. Although the items pertaining to diverse organizational practices are not the same across different CIS waves, this issue does not appear to be a problem for our analysis, because we determine ConOrg on the basis of the composite organizational variable determined for each reference period.

In Table IV we provide the descriptive statistics pertaining to the relationship between organizational innovation and technological innovators' profiles. More than 10% of pure product innovators do not introduce any organizational innovation in either t -2 or t -1, 11.63% introduce organizational innovations in t -2 but not t -1, 21.52% implement organizational innovations only in t -1, and 56.42% do so in both periods.

INSERT TABLE IV ABOUT HERE

In addition, we introduced two other variables to control for the degree of intensity of organizational innovation over time. Specifically, IntOrg(t) is determined on the basis of data about the three organizational practices reported in CIS08. It equals 0 if none of the organizational practices arise in t,3 1 if only one practice is adopted; 2 if two practices have been adopted, and 3 if all three practices are adopted during t.

Similarly, the construction of IntOrg(t -1) uses information about four organizational practices reported in CIS06: (1) new business practices for organizing procedures, (2) new methods for organizing work responsibilities and decision making, (3) new methods for organizing external relations with other firms or public institutions, and (4) knowledge management procedures. Thus IntOrg(t -1) equals 0 if firms never introduce organizational practices in t -1, 1 if they adopt one practice, 2 if they introduce two practices, 3 if three practices have been adopted, and 4 if all practices are adopted.4

iii. Other explanatory variables

We added several explanatory variables, in addition to organizational innovation, to our model. Prior literature suggests that the probability of innovation depends on firm characteristics and sector-specific features. For example, external and internal R&D investments per employee raise the stock of technological knowledge in firms, because R&D increases the firm's ability to capture external knowledge [START_REF] Cohen | Absorptive capacity: A new perspective on learning innovation[END_REF] and exerts a positive impact on the propensity to innovate [START_REF] Raymond | Persistence of innovation in Dutch manufacturing: Is it spurious?[END_REF]. We included two variables to differentiate external from internal R&D. First, Int_RD(t-1) represents internal R&D intensity, measured as in-house R&D expenditures divided by the number of employees for the lagged period t -1. Second, Ext_RD(t-1) accounts for external R&D intensity, measured as external R&D expenditures divided by the number of employees for the lagged period t -1.5 Because non-innovators do not provide R&D expenses in CIS, we assume they have no R&D expenses (i.e., these variables equal 0 for non-innovators).

Regarding firms' characteristics, we introduced four variables. First, we account for firm size, which is an important determinant of innovation activities [START_REF] Cohen | Empirical Studies of Innovative Activity[END_REF], using a dichotomous ordered variable with four size classes: 1 for the firms with fewer than 50 employees, 2 for those with 50-250 employees, 3 for 250-1000 employees, and 4 for those with more than 1000 employees. Second, we account for market conditions, which provides a proxy for the geographic area of sales for each firm [START_REF] Peters | Persistence Effects of Innovation Activities[END_REF]. This qualitative ordered variable ranges from 1 to 4, according to the situation of the geographic market where the firm sells its goods and products: 1 if the market is local or regional, 2 if it is national, 3 for EU member countries, and 4 for all other countries. Most firms in our sample fall into the fourth category (70%) and about 15% sell their goods and services throughout the European Union. Third, we address ownership status, because firms that are part of a group may have more incentive for innovation activities through their easier access to financing [START_REF] Love | Location and network effects on innovation success: evidence for UK, German and Irish manufacturing plants[END_REF]. It is also important to control for the estimations at the group level, because some firms in our sample must apply the innovation strategy adopted by their headquarters [START_REF] Mairesse | Using innovations surveys for econometric analysis[END_REF]. We use Gp to represent ownership, a binary variable that equals 1 if the firm is part of a group. Most firms (approximately 80%) in our sample belong to a group. Fourth, sector controls usually involve adding dummies for each industry, but instead, to address the technological level of the industry, we control for industrial specificity with a dichotomous ordered variable (Dumsect), ranging from 1 to 4: high-tech, mediumhigh-tech, medium-low-tech, and low-tech sectors (or NACE, Rev 1 at the three-digit level of aggregation).

Estimation method

Our goal is to test for the probability of being an innovator in period t 2 , as a function of the intensity of past innovation behavior in the two previous periods. We thus needed to estimate not only past innovation behavior but also the different types of innovations that firms have adopted and the extent to which they are more persistent with organizational innovations.

However, in panel data sets, investigating the impact of observed and unobserved individual characteristics and their relation with initial conditions can be problematic [START_REF] Heckman | The incidental parameters problem and the problem of initial conditions in estimating a discrete time discrete data stochastic process[END_REF].

Empirical literature on persistent innovation resolves this issue by using dynamic panel models. Previous studies of the persistence of innovation generally use binary discrete choice modeling, out of consideration of the nature of the data sets and variables. [START_REF] Peters | Persistence Effects of Innovation Activities[END_REF] investigates the persistence effects of innovation activities using several binary dependent

) u βx (γγ Φ ) u , 1/x (y P i t i, 1 - t i, i t i, t i,    
, where x i,t is a k-dimensional row vector that includes variables that vary across individuals and time, balanced by a γ -state dependence parameter; u i captures unobservable firm-specific attributes that are assumed to be constant over time and correlated by definition with 1 -t i, y ; and ε i,t includes other time-varying, unobservable determinants.

There are two issues to resolve for proper estimations of these models. The first relates to a case of unobserved heterogeneity, which clearly occurs when u i > 0. The second problem is linked to initial conditions, which, if not properly accounted for, could lead to an overestimation of the dependent variable, such as when y io is correlated with the unobserved random effect. To handle this problem, we write u i as a function of y io and α i , as Wooldridge ( 2005) suggests, such that

    ε x y ε ψ u i i,0 0 i a i ;
where a i ~N (0,σ² a ) and is independent of y io and z i .

This joint distribution enables us to apply a conditional maximum likelihood estimator,

    α x y α α u 2 i i,0 1 0 i a i (2),
where i

x are the time averages of x it that control for unobserved heterogeneity [START_REF] Peters | Persistence Effects of Innovation Activities[END_REF].

Therefore, our estimated model, transforming Equation ( 1) and taking Equation (2) into account, becomes: , x xi, yi2, yi1, 1/yio, (y

P i t i,  a i ) = α x y α α β x (γγ Φ 2 i i,0 1 0 t i, 1 - t i,     + a i ). (3)

Estimation results

Econometric results

We estimate dynamic probit random models, using the approach recommended by [START_REF] Wooldridge | Simple solutions to the initial conditions problem in dynamic nonlinear panel data models with unobserved heterogeneity[END_REF], to account for unobserved heterogeneity and overcome initial condition problems [START_REF] Peters | Persistence Effects of Innovation Activities[END_REF]. With this procedure, we can examine the factors that explain the dynamics of different profiles of technological innovators, taking into account different dynamic specifications of organizational innovation.

First, to gain a better understanding of the role of organizational innovation, we estimated a set of models that each included one measure of organizational innovation and four profiles of technological innovators: pure product, pure process, single, and complex. Model 1 provides the standard model and includes the two measures of temporal continuity of organizational innovation, ConOrg(t -1, t). In Table V we present the results when ConOrg(t -1, t) is a dynamic specification of the organizational continuity between the lagged period (t -1) and the current one (t).

INSERT TABLE V ABOUT HERE

Thus we determine that the persistence parameters for single and complex technological innovators are positive and significant, but we find no evidence of persistence for simple product or process innovators. 6 Being a single or a complex innovator in the previous time period positively correlates with the probability of being a single or complex innovator in the future. The value of the estimated coefficient also indicates the strength of the persistence dynamic, that is, the degree of influence of past innovation on a current decision to innovate.

A higher coefficient indicates a stronger persistence process. The results show that complex innovators are prone to be more persistent than single innovators, and the initial conditions have positive and highly significant effects, such that firms' initial innovation status is strongly correlated with unobserved heterogeneity.

As another important result, we determine that the degree of organizational continuity is significant and positively correlated with the probability of being a single or complex innovator. Firms that occasionally implement organizational innovation during the lagged or current period and those that have continuously implemented it in both periods exhibit a higher probability of being complex innovators, compared with firms that implemented no organizational practices during the two periods. This expected result, in line with Le Bas and

Poussing's (2012) findings, confirms the crucial role of organizational innovation for generating complex innovation over time. Its effect is twice as strong for complex innovators as for single innovators, but it does not explain firms' likelihood to be pure product or process innovators.

The geographical market variable is positively and significantly correlated with the pure product and single innovators. Firms open to the international market, facing higher foreign competition, exhibit a higher probability to innovate over time compared with firms that sell products or services only in local or regional markets. We also find a positive effect of past R&D intensity variables on current pure product and single innovators. Firms that invest in internal R&D and the acquisition of external knowledge likely innovate in product or in product or process, which reflects the role of R&D in both types of innovations, in line with the sunk costs hypothesis. That is, R&D activities drive technological innovation, so the dynamics of the former induce those of the latter. Yet internal and external R&D intensity variables do not reveal any impact on current complex innovation persistence. At first sight, this result appears counterintuitive; perhaps the relatively small number of complex innovators in our sample (approximately 8%) created some estimation problems. Also surprisingly, we do not find any evidence of an impact of firm size on any technological innovation profile. In our model, being larger cannot explain persistent innovation.7

Sensitivity analysis and robustness checks

To check the robustness of the results, we ran further regressions with different specifications of our main explanatory variable: organizational innovation. We therefore introduce three new measures of organizational innovation: ConOrg(t -2, t -1), IntOrg(t),

and IntOrg(t-1), with the results reported, respectively, in Tables VI, VII, and VIII. The estimated coefficients and their level of significance are roughly the same as those reported in Model 1. The effects of other explanatory variables, such as R&D intensity and size, are similar across the various models, such that our estimations are robust for the control variables. Therefore, we report only the estimated coefficients related to the block of the main independent variables.

In Model 2 in Table VI, in which ConOrg(t -2, t -1) is specified as a dynamic measure of the organizational continuity between the periods t -2 and t -1, the results for the persistence parameters are similar to those in Model 1. Pure product and pure process innovators do not appear persistent. On the contrary, the persistence parameters of single and complex innovators are positive and highly significant. These results confirm our previous findings from Model 1: Firms with the capacity to introduce products and/or processes in the past have a higher chance of being persistent than those that have implemented only products or only processes. The effects of organizational innovation on single and complex innovators' behaviors are positive and significant, though not as strong as in Model 1. This finding seems to suggest that organizational innovation, once it has been continuously adopted over two recent periods of time (t -1 and t), is more efficient for generating a higher probability of innovating than that adopted for two preceding periods (t -2, t -1) that are more distant in time from the reference period t. That is, there is a temporal dimension to the efficiency of the effect of organizational innovation on technological innovation.

In addition to Models 1 and 2, we estimated several other models that include our new indicators of temporal intensity of organizational innovation, IntOrg(t) and IntOrg(t -1).

We therefore explore another aspect of firms' intensity, in terms of organizational innovation practices. Recall that IntOrg(t) is a proxy for the degree of organizational intensity in period t (CIS8, 2006(CIS8, -2008)), and it takes a value from 0 to 3, depending on the type of combinations of organizational practices reported in CIS8. Thus, we can determine whether, aside from firms' characteristics and R&D activities, the intensity of organizational practices affects the persistence parameters. Models 3-5 thus mirror Models 1 and 2, except in the indicators of organizational innovation.

INSERT TABLE VI ABOUT HERE

Turning to organizational innovation, the estimation results for IntOrg(t) (Model 3) in

Table VII indicate that the pure product innovation variable is significant when we control for the degree of organizational intensity in the current period, all else being equal. The fact that firms implement more than two organizational practices in the current period could change the dynamics of their product innovation behavior, compared with a case in which no organizational practices are adopted.

INSERT TABLE VII ABOUT HERE

The interpretation of these results is twofold. First, the joint implementation of organizational practices during the current period might induce a complementary effect, in terms of management and competence profitability, that enhances firms' capacity to continue to introduce new or improved products over time. Second, product innovators in general seem to achieve higher growth rates (Colombelli, Haned, and Le Bas 2011), which enables them to devote more resources to innovation activities and which could, in turn, create a higher capacity to innovate persistently, though this effect holds only after we control for the degree of organizational intensity. As for the other innovator profiles, we observe that the persistence parameters are positive and significant for complex innovators and highly significant for single innovators. With regard to the impact of organizational innovation, the organizational parameters are positive and highly significant for pure product, single, and complex innovators. The simultaneous introduction of more than one organizational practice during the three-year period t enhances firms' technological innovation capacity in that period.

Finally, we present the results for the last model with IntOrg(t -1) in Table VIII.

INSERT TABLE VIII ABOUT HERE

With these organizational innovation variables, all else being equal, single technological innovation remains persistent. In contrast, the persistence parameters for pure product and complex innovators are no longer significant. At first sight, this result might seem contradictory, but conditional on the degree of organizational intensity in the current period (IntOrg(t)), the two innovators profiles are persistent. These results may reflect the effects of lagged time returns of organizational innovation on current innovations.

Overall, the results across three models confirm that the joint implementation of organizational practices, compared with a case in which no organizational practices are adopted, has a crucial impact in terms of leading firms to innovate and enhancing their technological innovation capacity in the same period. There also could be a temporal dimension, in terms of returns on organizational strategies undertaken during the previous periods on current firms' likelihood to innovate.

Discussion and conclusion

With this study, we have attempted to explore the consequences of organizational innovation on the patterns of firm technological innovation persistence. This research complements previous literature by providing detailed statistical evidence of the impact of organizational innovations, then inserting them as regressors in different empirical models. In so doing, we provide new insights into the relationship between nontechnological and technological innovation and add to comprehension of the impact of organizational innovation on technological innovation persistence. Three waves of French CIS data enabled us to examine the determinants of four profiles of technological innovators, focusing on different dynamic specifications of organizational innovation. These findings enrich the learning approach to innovation persistence. Product, process, and organization innovation exhibit strong, systematic interactions [START_REF] Antonelli | Inside innovation persistence: New evidence from Italian micro-data[END_REF]. Implementing new practices or procedures, new methods of work responsibilities, and new external relations all have consequences for (or offer incentives to) the design of newly structured products or improved processes in general.

Two key results thus emerge from our empirical analysis among French manufacturing firms. First, we find persistence in innovation but also that this trend does not hold for all types of innovators. Our methodology builds on the idea that different kinds of innovators exist. We explicitly distinguish pure product, pure process, single (product or process), and complex (product and process) innovators. In line with another recent study using the Luxembourg CIS, we find that complex innovators are more persistent (Le Bas and Poussing 2012), likely due to the positive returns on past investments and the role of accumulating competencies during the previous period, which then enhance firms' capacity to innovate persistently in the future. These results also reaffirm the existence of system effects and synergies among alternative innovations. Competencies and knowledge gained during product development processes spill over to projects designed to improve innovation processes.

Conversely, innovation in processes enhances firm efficiency, which can improve capacities to introduce new goods or services (Le Bas and Poussing 2012). Thus, firms that have combined product and process innovations in the past are more likely to be prepared, in terms of innovations opportunities, competencies, and work procedures, to introduce complex innovations in the present and future.

Second, and perhaps even more important, our estimations reveal a positive impact of organizational innovations on technological innovation persistence. This impact is neither general nor always of the same magnitude. Organizational innovation exerts a positive impact on complex innovators but almost never on pure process innovators. We have tracked the effects of two aspects of organizational innovation: relative continuity in the implementation of organizational innovation and the level of intensity in organizational innovation behavior.

If the organizational innovation goes farther back, its effect is weak. That is, a specific organizational innovation exerts an effect on technological innovation in the short term, leaving almost no positive propagation effects in subsequent time periods. Instead, intensity matters more significantly and positively. The more practices are implemented by the firm, the higher the probability it remains an innovator (though this pattern does not apply to pure process innovators).

We also acknowledge that our approach is clearly exploratory. The three CIS surveys do not use the same questions pertaining to organizational innovation, nor has there been any standard, unanimously accepted definition of organizational innovation in academic research.

Therefore, we hope ongoing studies elaborate on the concept of organizational innovation and reach a standardized definition, similar to those that already exist for product and process innovations. In addition, technological and organizational innovations significantly help explain firm performance, but we lack proper models to track the effect of different types of innovation on firm performance over time. Thus, it is necessary to expand on our analysis of innovation, beyond technological aspects, to gain a better understanding of firms' economic performance. Further research should include qualitative, longitudinal studies that can effectively tackle the continuity and intensity aspects of organizational innovation.

Finally, our study provides several new insights regarding tools to support innovation policies. The extant targets of regional and national innovation policies have been product and process innovations; we show that organizational innovation matters, perhaps even more.

New routines and organizational practices by the firm not only affect its current technological innovation but also exert lasting effects on its innovation activities. Thus organizational innovation should be a more important feature in the design of new types of public support. ConOrg(t-2,t-1) DO Equals 0 if no organizational innovation was adopted in t-2 and t-1; 1 if it is adopted only in t-2; 2 if it is adopted only in t-1; and 3 if it is continuously adopted during the two periods.

IntOrg(t) DO Equals 0 if none of the organizational practices are adopted in t; 1 if only one practice was adopted; 2 if two practices were adopted; and 3 if both three practices were adopted IntOrg(t-1) DO Equals 0 if firms did not introduce any organizational practices in t-1; 1 if only one practice was adopted; 2 if only two practices were adopted; 3 if 3 practices were adopted and 4 if all practices were adopted.

  variables that express the innovation behavior of firms as a function of past innovation and other explanatory variables, indicating variation across individuals and time. She then uses a second set of explanatory variables that are time constant and implements a probit model with the Wooldridge estimation method to control for unobserved heterogeneity. Raymond et al. (2010) use a panel of CIS-derived Dutch manufacturing firms to study the persistence of innovation with a maximum likelihood dynamic tobit model that accounts for individual effects and initial conditions. Antonelli et al. (2012) use a dynamic probit random effect model to evaluate the persistence of innovation among a set of Italian manufacturing firms. We retained a dynamic probit random model. With our panel, we can differentiate the sources of persistence from past innovation behavior versus those due to individual heterogeneity. In our basic model, we explain current innovation status (measured with alternative innovation variables)attempt to measure observable and unobservable firm-specific attributes. Thus our dynamic probit model is as follows: T 2,.

Table I . Sectoral composition and size distribution, 2006 Branches NACE two-digit codes: NACE Rev 1.1 Number Percentage

 I

	Table II. Variables
	Variables	Type		Description
				Alternative endogenous variables of innovation performance indicators all
	Pharmaceuticals Only_prod	B	24.41-24.42 displayed for the year 2008 (present period, t) Equals 1 for firms that are "pure product innovators": this category includes the 47 3.98
	Computers, office machinery and electronics-communication firms that introduce a new or significantly improved good or service with respect 30 and 32 30 2.54 to its capabilities, user friendliness, components or sub-systems
	Medical, precision, and optical Only_proc B Equals 1 for firms that are "pure process innovators": this category includes firms 33 22 1.86 that at least one type of one of the three process innovations regarding any new or instruments, watches and clocks significantly improved (1) methods of manufacturing or producing goods or
	Aerospace			35.3 services (2) logistics, delivery or distribution methods for your inputs, goods or 27 2.29 services (3) supporting activities for your processes, such as maintenance,
				systems or operations for purchasing, accounting, or computing
	Chemicals Single		B	24 (excluding 24.41 and 24.42) Equals 1 for firms that have introduced a product or a process innovation 92	7.80
	Machinery and equipment Complex B	29 Equals 1 for firms that have introduced both product and process innovations 106	8.98
	Electrical machinery		31	72	6.10
	Motor vehicles and transport equipment	34-35 (excluding 35.1 and	99	8.39
					35.3)
	Petroleum refining	23 Varying across individuals and time	10	0.85
	Rubber and plastic products and other Organizational innovation (several variables) and R&D expenses 25-26 146 non-metallic mineral products Metals 27-28 133 ConOrg(t-1,t) DO Equals 0 if firms did not introduced organizational innovation in either of the 12.37 11.27 Shipbuilding 35.1 4 two periods; 1 if organizational innovation is introduced only in t-1; 2 if it is 0.34 introduced only in t; 3 if it is continuously adopted during t-1 and t.
	Other manufacturing		36.2-36.6	29	2.46
	Food				15	194	16.44
	Textiles				17-19	59	5.00
	Wood, paper and furniture		20-21 and 36.1	88	7.46
	Printing and reproduction of recorded	22 (excluding 22.3)	22	1.86
	media
	Total				1180	100
	Technological intensity
	Low-technology			363	30.76
	Medium low-technology		322	27.29
	Medium high-technology		369	31.27
	High-technology			126	10.68
	Total				1180	100
	Size class
	Less than 50				54	4.58
	50-250				154	13.05
	250-1000				786	66.61
	More than 1000			186	15.76
			Totals		1180	100

Table III . Summary statistics

 III

	Variable	Obs.	Mean	Std. Dev.	Min	Max
	Dependent variables
	Only_prod	1180	0,63	0,483	0	1
	Only_proc	1180	0,10	0,302	0	1
	Single	1180	0,74	0,440	0	1
	Complex	1180	0,08	0,272	0	1
	Organizational variables
	ConOrg(t-1,t)	1180	1.01	1.286	0	3
	ConOrg(t-2,t-1)	1180	.98	1.242	0	3
	IntOrg(t)	1180	1.11	1.139	0	3
	IntOrg(t-1)	1180	1.33	1.245	0	4
	Explanatory variables
	Int_RD(t-1)	1180	4.90	12.333	0	167.310
	Ext_RD(t-1)	1180	1.96	6.896	0	91.767
	Size	1180	2.943	0.688	1	4
	Market	1180	3.624	0.736	1	4
	Gp	1180	0.887	0.316	0	1
	Dumsect	1180	2.761	1.025	1	4

Table IV . Descriptive statistics for technological and organizational innovations (%)

 IV

		Values	Only_Prod(t)	Only_Proc(t)	Single(t)	Complex(t)
	ConOrg(t-2,t-1)	0	10.43	18.33	11.56	2.11
		1	11.63	12.50	11.78	5.26
		2	21.52	23.33	21.74	21.05
		3	56.42	45.83	54.92	71.58
			100	100	100	100
	ConOrg(t-1,t)	0	10.03	15.00	10.76	6.32
		1	21.52	26.67	22.08	5.26
		2	12.03	15.83	12.59	1.05
		3	56.42	42.50	54.58	87.37
			100	100	100	100
	IntOrg(t-1)	0	30.11	32.56	31.80	9.00
		1	19.56	18.44	17.28	12.10
		2	20.83	22.64	22.10	20.34
		3	18.60	14.00	16.12	23.06
		4	11.00	12.36	13.70	34.50
			100	100	100	100
	IntOrg(t)	0	33.56	42.50	34.67	6.32
		1	17.11	18.33	17.16	11.58
		2	27.67	27.50	27.92	32.63
		3	21.66	11.67	20.25	49.47
			100	100	100	100
	Observations		748	120	874	95

Table V . Dynamic RE Probit estimations (Model 1)

 V , **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. Standard errors in parentheses. Random effects estimates are computed by adaptive Gausse-Hermite quadrature.

		Only_Prod	Only_Proc	Single	Complex
	Lagged Innovation
	Only_prod(t-1)	0.230
		(0.177)
	Only_proc(t-1)		0.143
			(0.220)
	Single(t-1)			0.421**
				(0.198)
	Complex(t-1)				0.507**
					(0.221)
	Organizational Innovation
	ConOrg(t-1,t)	0.0412	-0.0193	0.0521*	0.116***
		(0.030)	(0.035)	(0.031)	(0.031)
	Explanatory variables
	Int_RD(t-1)	0.005*	0.009	0.146***	-7.47e-05
		(0.003)	(0.005)	(0.019)	(0.002)
	Ext_RD(t-1)	0.022***	-0.008	0.0418***	-0.002
		(0.007)	(0.005)	(0.013)	(0.005)
	Size	-0.401	0.593	0.009	0.142
		(0.297)	(0.363)	(0.297)	(0.358)
	Market	0.224**	0.169	0.219**	0.216
		(0.106)	(0.129)	(0.102)	(0.142)
	Gp	0.141	0.348	0.256	0.315
		(0.282)	(0.336)	(0.271)	(0.344)
	Dumsect	-0.168	0.219	-0.101	-0.140
		(0.209)	(0.273)	(0.212)	(0.263)
	Individual heterogeneity
	Only_prod(0)	1.225***
		(0.233)
	Only_proc(0)		0.834***
			(0.240)
	Single(0)			0.589***
				(0.219)
	Complex(0)				0.419**
					(0.209)
	Gpmean	0.248	-0.163	0.063	-0.370
		(0.320)	(0.367)	(0.293)	(0.371)
	Dumsectmean	-0.013	-0.165	0.012	0.030
		(0.214)	(0.277)	(0.217)	(0.267)
	Sizemean	0.798**	-0.759**	0.253	0.139
		(0.313)	(0.374)	(0.306)	(0.368)
	Marketmean	0.0910	-0.235	-0.042	-0.017
		(0.127)	(0.145)	(0.116)	(0.158)
	Intercept	-2.581***	-1.370***	-1.721***	-3.049***
		(0.389)	(0.369)	(0.296)	(0.493)
	ρ	0.436	0.346	0.160	0.045
		(0.088)	(0.117)	(0.150)	(0.164)
	-2lnL	1114.85	697.47	912.34	546.07
	Percent correctly	82.5	71.9	87.9	76.6
	predicted
	Observations	2360	2360	2360	2360
	Notes: *

Table VI . Dynamic RE Probit estimations (Model 2)

 VI , **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. Standard errors in parentheses.

		Only_Prod	Only_Proc	Single	Complex
	Lagged Innovation
	Only_Prod(t-1)	0.184
		(0.179)
	Only_Proc(t-1)		0.151
			(0.222)
	Single(t-1)			0.352*
				(0.200)
	Complex(t-1)				0.381*
					(0.230)
	Organizational Innovation
	ConOrg(t-2,t-1)	-0.019	0.005	0.008	0.063*
		(0.029)	(0.034)	(0.031)	(0.033)
	ρ	0.466	0.341	0.214	0.143
		(0.086)	(0.119)	(0.145)	(0.104)
	-2lnL	1115.55	697.60	918.06	576.45
	Percent correctly predicted 82.0	71.0	86.3	78.4
	Observations	2360	2360	2360	2360
	Notes: *

Table VII . Dynamic RE Probit estimations (Model 3)

 VII , **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. Standard errors in parentheses.

		Only_Prod	Only_Proc	Single	Complex
	Lagged Innovation
	Only_Prod(t-1)	0.326*
		(0.173)
	Only_Proc(t-1)		0.151
			(0.220)
	Single(t-1)			0.559***
				(0.096)
	Complex(t-1)				0.311*
					(0.187)
	Organizational Innovation
	IntOrg(t)	0.295***	0.0214	0.309***	0.411***
		(0.035)	(0.037)	(0.030)	(0.045)
	ρ	0.372	0.343	0.001	0.073
		(0.096)	(0.118)	(0.003)	(0.130)
	-2lnL	1072.03	697.00	864.65	510.34
	Percent correctly predicted	84.4	71.6	89.0	84.3
	Observations	2360	2360	2360	2360
	Notes: *

Table VIII . Dynamic RE Probit estimations (Model 4)

 VIII , **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. Standard errors in parentheses.

		Only_Prod	Only_Proc	Single	Complex
	Lagged Innovation
	Only_prod(t-1)	0.211
		(0.176)
	Only_proc(t-1)		0.150
			(0.221)
	Prod_ou_proc(t-1)			0.352*
				(0.197)
	Prod_et_proc(t-1)				0.280
					(0.211)
	Organizational Innovation
	IntOrg(t-1)	0.080**	-0.013	0.101***	0.159***
		(0.033)	(0.038)	(0.033)	(0.035)
	ρ	0.426	0.342	0.161	0.106
		(0.089)	(0.118)	(0.147)	(0.144)
	-2lnL	1112.45	697.55	914.36	567.98
	Percent correctly predicted	82.2	71.9	87.9	77.8
	Observations	2360	2360	2360	2360
	Notes: *

A methodological change between the CIS06 and CIS08 reintegrated 'knowledge management' back into 'new business practices for organizing procedures' for CIS08.

In this case, the firm does not introduce any organizational innovations.

We interpret IntOrg(.) as a measure of the intensity of organizational innovation. It should depict the diversity of new practices implemented by the firm.

The total amount of in-house R&D is given directly in CIS. The total amount of external R&D is a variable that we constructed from an average of three inputs: (1) the amount dedicated to the purchase of external R&D; (2) the acquisition of machinery, equipment, and software dedicated to R&D; and (3) the acquisition of external knowledge.

In a first step, we also estimated simple models, assuming the absence of individual effects and exogenous initial conditions. The persistence parameters were positive and highly significant for all innovator profiles. However, in these unrealistic conditions, overestimation of the dependent variable is likely, so the significance of the persistence parameters does not mean that true persistence exists. These results are available on request.

In addition, the individual average of firm size (Sizemean) is positive and significant for pure product innovators and negative and significant for pure process innovators, which indicates substantial correlations between these variables and unobserved individual heterogeneity.