Testing For A Structural Break In Dynamic Panel Data Models With Common .

2.1 Stochastic Framework We study a linear dynamic panel data model with regressors and a multi-factor error structure. Our aim is to detect a possible break in the structural parameters of the model, i.e. the autoregressive and/or slope coefficients. Consider the following model: ρ0 yi,t 1 x0 β 0 λ0 f 0 εit t τ; it i t yit η0y 0 0 0 0 t τ, τ i,t 1 xit δτ λi ft εit (1) where both λi and ft0 are r 1 vectors, while (ητ0 , δτ0 ) replaces (ρ0 , β 0 ) from the break at time τ , τ 2; t 1, . . . , T . Notice that since T is held fixed in our study, ft0 is treated as a parameter vector to be estimated together with the structural parameters of the model. Our testing problem can be studied more formally by defining the following hypotheses: H0 : There are no structural breaks H(τ ) : There is a structural break at time τ , where τ is known H1 : There is a structural break at time τ , where τ is unknown. If H1 is true then we shall denote the true point in time where the break occurs, by τ0 . The model above can be expressed in vector form as (b) (a) (b) (a) yi ρ0 yi, 1 ητ0 yi, 1 Xi β 0 Xi δτ0 (IT λ0i )f 0 εi , (b) (2) (a) where yi (yi1 , yi2 , . . . , yiT )0T 1 , yi, 1 (yi0 , yi1 , . . . , yi,τ 2 , 0, . . . , 0)0T 1 , yi, 1 (0, . . . , 0, (b) (a) yi,τ 1 , . . . , yi,T 1 )0T 1 , Xi (xi1 , xi2 , . . . , xi,τ 1 , 0k 1 , . . . , 0k 1 )0T k , Xi (0k 1 , . . . , 0k 1 , 0 xiτ , . . . , xiT )0T k , f 0 vec (F 0 ) , F 0 (f10 , f20 , . . . , fT0 )0T r and εi (εi1 , εi2 , . . . , εiT )0T 1 ; the value ‘ 1’ in the suffix indicates that the corresponding values have been lagged by one time period; the superscripts ‘(b)’ and ‘(a)’ indicate that the vectors correspond to the periods before τ and from τ onwards respectively, regardless of whether or not a break has occurred. The following assumption is employed throughout the paper. Assumption 1: (i) (xit , λi , εit , yi0 ) are independently and identically distributed for i 1, . . . , N , with each component having finite fourth moment. (ii)There exists a (0, ) such that ρ0 a, ητ0 a, kβ 0 k a, and kδτ0 k a. (iii) f 0 is non-stochastic and there exists b (0, ) such that kf 0 k b. (iv) E (εit yi0 , . . . , yit 1 , λi , xi1 , . . . , xih ) 0, 5

for t 1, . . . , T , i 1, . . . , N , and some positive integer h. The value of h depends on whether regressor xit is strictly (weakly) exogenous, or endogenous. Assumption 1 is fairly standard in this literature, for example, it is in line with Assumption 2 in Robertson and Sarafidis (2015), and Assumptions BA.1, BA.3, and BA.4 in Ahn et al. (2013). The independence assumption over i 1, . . . , N, in the first part of Assumption 1, can be relaxed so long as the moment functions that will be defined below converge to their respective expectation in probability. The requirement of identical distribution in the same assumption, can also be relaxed. For example, εit could be heterogeneously distributed across both i and t, while conditional moments of λi could also depend on i (see e.g. Juodis and Sarafidis 2015). We do not consider such generalizations in this paper in order to avoid unnecessary notational complexity. Assumption 1 (iv) implies that the idiosyncratic errors are conditionally serially uncorrelated. This can be relaxed in a straightforward way and allow for serial correlation of (a) a moving average form by carefully selecting the moment conditions, or (b) an autoregressive form by including further lags of y and x into the model. In addition, this assumption implies that the idiosyncratic error is conditionally uncorrelated with the factor loadings. This is standard in the dynamic panel data literature, and allows for lagged values of y in levels to be used as instruments. Moreover, the value of h in Assumption 1 (iv) characterises the exogeneity properties of the covariates. For example, for h T (respectively, h t) the covariates are strictly (respectively, weakly) exogenous, or otherwise they would be endogenous (see Arellano 2003, pg. Section 8.1). Our methodology is valid regardless of the value of h mutatis mutandis. 2.2 Moment Conditions Our approach is based on a class of Method of Moments estimators known as Factor Instrumental Variables [FIV] estimators, for consistent inference in model (1); see Robertson and Sarafidis (2015). This approach involves building moment functions that contain parameters to capture the unobserved, unrestricted covariances between the instruments employed and the unobserved factor component. In particular, notice that under Assumption 1(iv) there exists a d 1 vector wi that contains potential “instruments”, i.e. variables within the model that are potentially orthogonal to the purely idiosyncratic error component; the precise set of variables that satisfies this orthogonality property may depend on period t. Thus, let St be a ζt d selector matrix of 0’s and 1’s that picks up from wi variables at period t that are uncorrelated with εit . Thus, in each period t, 6

ζt 0 instruments are available, expressed in vector form as zit St wi , for which the orthogonality condition E(zit εit ) 0 holds true. The total number of moment conditions P is given by ζ Tt 1 ζt . Let S diag (S1 ; . . . ; ST ), and Zi0 S (IT wi ), while (b) (a) (b) (a) µτ,i (θτ ) Zi0 {yi ρyi, 1 ητ yi, 1 Xi β Xi δτ } S (IT G) f , (3) where G E (wi λ0i ) and θτ (g 0 , f 0 , ρ, β 0 , ητ , δτ0 )0 with g vec (G). G denotes the matrix that contains the unrestricted covariances between the matrix of instruments and the factor loadings. Remark 1. The above definition for µτ,i (θτ ) requires some explanation. To develop a Method of Moments estimator for inference based on the foregoing moment conditions, one typically considers terms of the form (b) (a) (b) (a) ei (θτ ) yi [ρyi, 1 ητ yi, 1 Xi β Xi δτ (IT λ0i )f ] with µτ (θτ ) E[Zi0 ei (θτ )]. Thus, µτ (θτ0 ) 0 is treated as a moment condition and µτ (θτ ) as the corresponding moment function. The method adopted in the literature for P 0 simple settings suggests to estimate µτ (θτ0 ) by N 1 N i 1 Zi ei (θτ ). However, in the present case λi is an unobserved random variable and so the foregoing suggestive method is not suitable. Therefore, we adopt a method which essentially integrates out the unobserved random variable. Taking expectations in the expression above yields the following vector-valued moment function: (b) (a) µτ (θτ ) E [µτ,i (θτ )] m ρm 1 ητ m 1 M (b) β M (a) δτ S (IT G) f , (4) (j) (j) (j) with m E[Zi0 yi ]ζ 1 , m 1 E[Zi0 yi, 1 ]ζ 1 , and M (j) E[Zi0 Xi ]ζ k , for j {a, b}. It follows that for the true parameter vector θτ0 , we have µτ (θτ0 ) 0. Remark 2. Observe that the last term of expression (4) can be written as S (IT G) f S (F Id ) g Svec(GF 0 ). But since Svec(GF 0 ) Svec(GU U 1 F 0 ) for any r r invertible matrix U , the parameters G and F are not identified per se without normalizing restrictions. This issue is well-known in factor models. Following standard practice, in what follows we assume that a set of normalizing restrictions is available to ensure identification. The actual choice is not important; see, for example, Robertson and Sarafidis (2015). Therefore, θτ0 corresponds hereafter to the true parameter vector containing the normalized values of f 0 and g 0 . 7

Now, suppose that the null hypothesis is true. Hence model (1) reduces to yit ρ0 yi,t 1 x0it β 0 λ0i ft0 εit , (t 1, . . . , T ). (5) By arguments very similar to those leading to (4), we obtain the moment function µ1 (θτ ) m ρm 1 M β S (IT G) f , (b) (6) (a) where m 1 m 1 m 1 and M M (b) M (a) . Note that the foregoing moment function involves only the parameter θ1 : (g 0 , f 0 , ρ, β 0 )0 , which is a subvector of θτ . In what follows, we will slightly abuse notation and define µ1 (θ1 ) and µτ (θτ ) as having the same value when the null hypothesis is satisfied. Consequently, µτ (θτ ) is well defined for τ 1. More specifically, it is the moment function in (4) for the full model (1) if τ 2, and the moment function in (6) for the reduced model (5) if τ 1. Before we formalize the testing methodology proposed in our paper, it will be instructive to use a simple example in order to illustrate the moment functions described above, which are employed by our estimator. To this end, we consider for simplicity the case where T 3, r 1 and β 0. Example 1. Under the null hypothesis we have E(µ1,i (θ1 )) E(zi1 εi1 ) E(zi2 εi2 ) E(zi3 εi3 ) E(yi0 εi1 ) E(yi0 εi2 ) E(yi1 εi2 ) E(yi0 εi3 ) E(yi1 εi3 ) E(yi2 εi3 ) m ρm 1 Svec(GF 0 ), m01 m02 m12 m03 m13 m23 ρ m00 m01 m11 m02 m12 m22 g0 f1 g0 f2 g1 f2 g0 f3 g1 f3 g2 f3 (7) where ms,t E (yis yit ) and θ1 (g0 , g1 , g2 , f1 , f2 , f3 , ρ)0 . Observe that the moment conditions are ordered by the time-index t of the equations from which they are derived and then by the time-index s of the instruments. On the other hand, under H(τ ) with 8

τ 3 we have E(µ3,i (θ3 )) E(zi1 εi1 ) E(zi2 εi2 ) E(zi3 εi3 ) (b) E(yi0 εi1 ) E(yi0 εi2 ) E(yi1 εi2 ) E(yi0 εi3 ) E(yi1 εi3 ) E(yi2 εi3 ) m01 m00 m02 m01 m m12 ρ 11 0 m03 0 m13 0 m23 0 0 0 η3 m02 m12 m22 g0 f1 g0 f2 g1 f2 g0 f3 g1 f3 g2 f3 (a) m ρm 1 η3 m 1 Svec(GF 0 ), with θ3 (g0 , g1 , g2 , f1 , f2 , f3 , ρ, η3 )0 . (8) Remark 3. As it is typically the case with breakpoint detection in general, identification of the structural break requires certain restrictions on the date when the break occurs. For example, in a time series autoregressive model it is required that the break occurs at period τ p 1, where p denotes the order of the AR process. On the other hand, in a standard dynamic panel data model of first order, identification requires that the break takes place at period τ 2 because first-differencing of the model removes the first time period. In the present paper identification also depends on the number of factors, as well as on the properties of the regressors. For r 1, identification requires τ 3; to see this, notice that if the break occurs at period t 2, then ρ is not identified because it is only “observable” at period t 1, which contains a single estimating equation, given by m01 ρm00 g0 f1 0, and 2 parameters that do not appear elsewhere, namely ρ and f1 . On the other hand, it is worth mentioning that if instruments with respect to exogenous regressors or other exogenous variables are used, identification can be accomplished even with τ 2. Remark 4. The vector-valued moment function above can be simplified when ft 1 for all t. In this case, the factor component degenerates to a single individual-specific effect, and the last term in (say) µτ (θτ ) reduces to S(ιT Id )g, where ιT is a T 1 vector of ones. Therefore, our model incorporates the fixed effects panel data model as a special case. As it will be shown in the next section, our estimator of θτ0 is the minimiser of the c µ̂τ (θτ ), where W c is some positive definite following quadratic form: Q̂τ (θτ ) µ̂0τ (θτ )W P weighting matrix, and µ̂τ (θτ ) N 1 N i 1 µτ,i (θτ ). 9

2.3 Main Results Our test statistic builds upon the moment conditions introduced in the previous section. Therefore, our approach retains the traditional attractive feature of Method of Moments estimators in that it exploits only the orthogonality conditions implied by the model and does not require subsidiary assumptions such as homoskedasticity or other distributional properties of the error process. In what follows, we list the remaining assumptions required to establish the main asymptotic properties of our method. Let Θ denote the parameter space that is obtained by a particular set of normalizing restrictions on (G, F ). Assumption 2. The parameter space Θ is compact and contains the true value θτ0 in its interior. For τ 1, the population moment function vector µτ (θτ ) is equal to 0 if and only if θτ θτ0 . Assumption 3. The variance-covariance matrix of the moment functions evaluated at θτ0 , which is defined as Φτ (θτ0 ) E[µτ,i (θτ0 )µ0τ,i (θτ0 )], and the derivative matrix of moment functions Γτ (θτ0 ) E[( / θτ0 )µτ,i (θτ0 )], both exist and have full rank (τ 1). The aforementioned assumptions provide the main conditions to ensure consistency and asymptotic normality of the estimator proposed in this paper. In particular, let c µ̂τ (θτ ), and define the Factor Instrumental Variable [FIV] estimator Q̂τ (θτ ) µ̂0 (θτ )W τ θ̂τ of θτ0 as follows: θ̂τ arg min Q̂τ (θτ ). (9) θτ Θ We will show that supθτ Θ Q̂τ (θτ ) Qτ (θτ ) converges to zero in probability, where Qτ (θτ ) µ0τ (θτ )W µτ (θτ ) and W is a positive definite weighting matrix. The consistency of θ̂τ follows from this result. c Φ̂ 1 (θ̂τ ) in The optimal choice of the weighting matrix is obtained by setting W τ (9) (see Hansen 1982). Since this requires an initial consistent estimate of θτ0 , effcient c Iζ , which estimation can be implemented in two stages: in the first stage one sets W (1) provides a first-step consistent estimate θ̂τ of θτ0 ; this can be used in the second stage to obtain a consistent estimate of the inverse of the variance-covariance matrix of the c Φ̂ 1 (θ̂τ(1) ) if Φ̂ 1 (θ̂τ(1) ) is non-singular, moment conditions and then rely on (9) with W τ τ c [Φ̂τ (θ̂τ(1) ) N 1 I] 1 . otherwise W Below we establish an important lemma for this paper; the proof is given in the Appendix. 10

Lemma 1. Suppose that Assumptions 1-3 are satisfied. Let Φτ Φτ (θτ0 ) and Γτ Γτ (θτ0 ). p 1 0 1 Then, as N , we have (i) θ̂τ θτ0 , (ii) N (θ̂τ θτ0 ) (Γ0τ Φ 1 N µ̂τ (θτ0 ) τ Γτ ) Γτ Φτ d d 1 op (1), (iii) N µ̂τ (θτ0 ) N (0, Φτ ), and (iv) N (θ̂τ θτ0 ) N (0, (Γ0τ Φ 1 τ Γτ ) ). In studies involving structural breaks, the case when the break point τ is given, and the case when it is unknown are both of interest. Hence, we will study both cases. Since θ̂1 and θ̂τ are estimators of the true value of the parameter, under the null H0 and under the alternative H(τ ) respectively, a suitable statistic for testing H0 vs H(τ ) is the following difference in the squared lengths of the sample moment functions: ψτ N [Q̂1 (θ̂1 ) Q̂τ (θ̂τ )]. (10) We now state the main result about the asymptotic null distribution of ψτ . The proof is given in the Appendix. Theorem 1. Suppose that Assumptions 1-3 hold, and τ 2. Then, under the null hypothesis H0 , the test statistic ψτ N [Q̂1 (θ̂1 ) Q̂τ (θ̂τ )] is asymptotically distributed as chi-squared with degrees of freedom equal to dim(ητ ) dim(δτ ) 1 k. This result is sufficient for us to test H0 vs H(τ ) . Now, we use this to develop a test of H0 against the more general alternative H1 wherein the breakpoint τ0 is unknown. To this end, we consider the foregoing test for each possible value of τ0 and then combine them. Recall that, we are interested to test H0 against the alternative that a structural break occurs at some time τ0 in {τ1 , . . . , τL }. Let ψ [ψτ1 , . . . , ψτL ]0 , (11) where ψτ N [Q̂1 (θ̂1 ) Q̂τ (θ̂τ )] has been proposed for testing H0 against H(τ ) . Clearly, if H0 is true then Q̂1 (θ̂1 ) and {Q̂τ1 (θ̂τ1 ), . . . , Q̂τL (θ̂τL )} are all estimators of the same quantity and hence max{ψτ1 , . . . , ψτL } is expected to be small. On the other hand, if H1 is true, then Q̂τ0 (θ̂τ0 ) is expected to be smaller than Q̂1 (θ̂1 ) and hence max{ψτ1 , . . . , ψτL } is expected to be large. Therefore, in order to test H0 vs H1 we propose using the statistic ψmax : max {ψτ }, 1,.,L (12) and reject the null for large enough values of ψmax . The unknown break point τ0 can be estimated by τs , where ψτs ψmax . 11

Now, to state the main result about the distribution of the test statistic, ψmax , let us introduce the following notation: For a full column rank matrix, B, let MB and PB be two projection matrices defined by PB B(B 0 B) 1 B 0 and MB I PB respectively. 1/2 Suppose that the null hypothesis holds true. Let Φτ root of Φ 1 τ so that Φ 1 τ 1/2 1/2 Φτ Φτ . denote the symmetric square Let Vτ MΦ 1/2 Γ1 MΦ 1/2 , where all the Γτ τ 1 quantities are evaluated at the true value of the unknown parameter. The next theorem provides the essential result for applying ψmax for testing H0 against H1 . Theorem 2. Suppose that Assumptions 1-3 and the null hypothesis H0 are satisfied. Let z N (0, Iζ ) where ζ is the number of moment conditions, and let Vτ1 , . . . , VτL be evaluated at the true value of the parameter specified by the null hypothesis. Then, d ψ {z 0 [Vτ1 , . . . , VτL ](IL z)}0 . Consequently, the test statistic ψmax is asymptotically distributed as Ξ max{z 0 Vτ1 z, . . . , z 0 VτL z}. Since the asymptotic null distribution of ψmax depends on the nuisance parameter θ10 through Vτ (θ10 ), we propose to approximate the distribution of ψmax (θ10 ) by that of ψmax (θ̂1 ) conditional on θ̂1 . Therefore, critical values can be estimated by simulation. Details are provided in Section 3 below. 3 3.1 Monte Carlo Simulations Simulation Design In this section we investigate the finite-sample properties of the test introduced previously. Our focus is on the impact of sample size, as well as on the location and magnitude of the break on the performance of our statistic. We study the pure AR(1) model, the choice of which is mainly motivated by our application that follows. The DGP is given by ρ0 yi,t 1 λi f 0 εit ; t τ t yit η0y λ f0 ε ; t τ, τ i,t 1 i t (13) it for i 1, . . . , N , t 1, . . . , T , where εit N (0,σε2 ), λi N (0,σλ2 ), ft0 N (0,σf2 ), and σε2 σf2 1. We set the factor number as r 1, noting that we have also examined a two-factor model; the results are very similar and therefore we are not including them here to save space. Define π σλ2 /(1 σλ2 ), which is the variance of the factor component λi ft0 as a proportion of the variance of the total error term λi ft0 εit . 12

The initial observation is generated as yi0 λi /(1 ρ0 ) N (0, 1). The vector of possible instruments is given by wi (yi0 , . . . , yi,T 1 )0 . For each t, we choose zit (yi0 , . . . , yi,t 1 )0 . We set N {60, 120, 300, 600, 1200}, T {6, 8}, τ0 {4, 6}, and ω 0 ητ0 ρ0 { 0.05, 0, 0.05, 0.10, 0.15}. We fix ρ0 0.5 and π 0.5. All the simulations are conducted using 10, 000 replications. The estimation algorithm that we employ involves an iterative procedure. In particular, notice that if either f or g is held fixed then equations (4) and (6) become linear in the remaining parameters. This feature can simplify the computations; specifically, to compute the global minimum of the objective function Q̂τ (θτ ) one can proceed as follows: (a) for a given fixed value of f , minimize Q̂τ (θτ ) with respect to the remaining parameters, (b) hold the value of g obtained i

the model. Our testing problem can be studied more formally by de ning the following hypotheses: H 0: There are no structural breaks H ( ): There is a structural break at time , where is known H 1: There is a structural break at time , where is unknown: If H 1 is true then we shall denote the true point in time where the break occurs .