Panel unit root and cointegration methods

Transcription

1 University of Vienna, Dept. of Economics Master in Economics Vienna 2010

2 Outline of the talk (1) Unit root, cointegration and estimation in time series. 1a) Unit Root tests (Dickey-Fuller Test, 1979); 2a) Cointegration tests: single equation method (Engle-Granger, 1987); 3a) Estimators: OLS, DOLS (Saikkonen, 1991), FMOLS (Phillips and Hansen, 1990).

3 Outline of the talk (2) Unit root, cointegration and estimation in panel data. 1b) Limits of time series approach; 2b) Advantages and Disadvantages of the nonstationary panel methods. 3b) First and second generation of panel unit root tests, cointegration and estimation methods.

4 Time series unit root tests: regression equation (1) y t = ρy t 1 + ε t (1) y t = βy t 1 + ε t (2) where β = (ρ 1) and ε t is white noise process (E(ε) = 0, E(ε 2 t ) = σ 2 <, E(e t e j ) = 0, for t j). The test of H 0 : β = 0 (ρ = 1) has a non-standard distribution (Brownian or Wiener process).

5 Asymptotic distribution of Dickey-Fuller tests (2) Given t observation, the OLS estimator of ρ in (1) is: ( T ˆρ = t=1 y 2 t 1 ) 1 T y t y t 1 (3) The limiting distribution of the OLS estimator ˆρ when ρ = 1 is: T (ˆρ 1) 1 0 W (r)dw (r) 1 0 W (r)2 dr t=1 = (1/2){[W (1)]2 1} 1 0 [W (R)]2 dr The previous distribution can be used for testing the unit root null hypothesis H 0 : ρ = 1, that s K = T(ˆρ 1) or we can normalize it with the standard of the OLS estimator and construct the t-statistics. (4)

6 Asymptotic distribution of Dickey-Fuller tests (3) The test statistics is: tˆρ = (ˆρ 1) ˆσ ρ = (ˆρ 1) {s 2 T t=1 y, (5) t 1 2 }1/2 where ˆσˆρ is the usual OLS standard error for the estimated coefficient and s 2 denotes the OLS estimate of the residual variance: as T, tˆρ T s 2 t=1 = (y t ˆρy t 1 ) 2 T 1 (6) 1 0 W (r)dw (r) [ 1 0 W (r)2 dr] = 1/2{[W (1)2 1]} 1/2 {[ 1 0 W, (7) (r)2 dr]} 1/2

7 Cointegration: concept (1) An important property of I(1) variables is that a linear combination of these two variables that is I(0) may exist. If this is the case, these variables are said to be cointegrated. The concept of cointegration was introduced by Granger (1981). Consider two variables y t and x t that are I(1). Then y t and x t are said to be cointegrated if there exist a β such that y t βx t is I(0). We denoted this as CI (1, 1). More generally, if y t is I(d) and and x t is I(d), then y t and x t are CI (d, b) if y t βx t is I (d b) with b > 0.

8 Cointegration: concept (2) What the previous concept means is that the regression equation: y t = βx t + µ t (8) makes sense since y t and x t do not drift too far apart each other over time. Thus, there is a long run equilibrium relationship between them (see the geometric interpretation of cointegration below). If y t and x t are not cointegrated, that is y t βx t = µ t is also I(1), then y t and x t would drift apart from each other over time. In this case, the relationship between y t and x t that we obtain by regressing y t on x t would be spurious.

9 Cointegration: graphics (1) Nominal interest rate on a 20 year US saving and loan credit instrument (R20) and AAA Moodys bond rate (R30)

10 Cointegration: geometric interpretation (4) Suppose p it > p jt, demand will go to location j: i.) Shocks to the economy make us to move out of the equilibrium; ii.) The adjustment does not have to be instantaneous but eventually; iii.) Long run equilibrium p it = p jt, this us a linear attractor p jt 3 ( p p ) i3 j3 4 pit p jt 5 2 ( p p ) i1 j1 1 ( pi 1 p j1) 45 p it

11 Cointegration Tests: Engel and Granger model (1) Consider two series y 1t I(1), y 2t I(1) and a simple two-equation model: y 1t = βy 2t + u 1t, u 1t = u 1t 1 + ε 1t, (9) y 1t = αy 2t + u 2t, u 2t = ρu 2t 1 + ε 2t, ρ < 1 (10) The second equation describes a particular combination of the series which is stationary. Hence y 1t and y 2t are C(1,1). The null hypothesis is taken to be no co-integration or ρ = 1 in (10).

12 Cointegration Tests: Engel and Granger model (2) 1. The cointegrating equation (10) is estimated by OLS and the residuals are saved. 2. Several tests on the residual are provided (i.e. Durbin-Watson, Dickey-Fuller and Augmented Dickey-Fuller). If the residuals are nonstationary, the series are no cointegrated. Otherwise, the series are cointegrated. For examples, if the residuals are nonstationary, the DW test will approach to zero and thus the test rejects no co-integration hypothesis if DW is too big.

13 Cointegration Tests: Engel and Granger model (3) Consider again y 1t and y 2t that are both I(1). Suppose there is cointegration, that s u t in (10) is I(0) and α is the cointegrating vector (for the case of two variables, scalar). If there is cointegration, we can show that α is unique. Because, if we have y 1t = γy 2t + v 2t where v 2t is also I(0), by substraction we have (α γ)y 2t + u 2t v 2t is I(0). But u 2t v 2t is I(0) which means (α γ)y 2t is I(0). This is not possible since y 2t is I(1).

14 Cointegration Tests: Engel and Granger model (4) The equation-system (9-10), can be re-written in reduced as follows: y 1t = α α β u 1t β α β u 2t (11) y 2t = 1 α β u 1t 1 α β u 2t (12) These equation show that both y 1t and y 2t are driven by a common I(1) variable. This is known as the common trend representation of the cointegrated system.

15 OLS estimator (1) If the variables in (10) are not cointegrated, ρ = 1, then the OLS is quite likely to produce spurious results (high R 2, t-statistics that appear to be significant), but the results are without economic results (if the residuals have a stochastic trend, any error in period t never decays, so that the deviation from the model is permanent. It s hard to imagine attaching any importance to an economic model having permanent errors).

16 OLS estimator (2) Stock (1987) show that OLS estimator is superconsistent: ˆα converge to its true value at the rate T(superconsistency) instead of the usual rate T (consistency). Although ˆα is superconsistent, Banerjee et al. (1986) and Banerjee et al. (1993) show that through Monte Carlo studies that there can be substantial sample biases (the dynamic is missed). This missing information also causes the DF test in the Engle-Granger approach to be less powerful than the cointegration test based on the t-statistics in the Error Correction model (ECM)

17 OLS estimator and the dynamic model (ECM)(1) Consider the following ADL(1,1) model: y t = α 0 + α 1 y t 1 + β 0 z t + β 1 z t 1 + ε t (13) where ε t iid(0, σ 2 ) and α 1 < 1. In a statistic equilibrium (all changes has ceased), we have E(y t ) = E(y t 1 ) =... = y and E(z t ) = E(z t 1 ) =... = z.

18 OLS estimator and the dynamic model (ECM)(2) By getting the expectation of (13), we have: y = α 0 + α 1 y + β 0 z t + β 1 z (14) and then or y = α 0 + (β 0 + β 1 )z 1 α 1 k 0 + k 1 z (15) E(y t ) = k 0 + k 1 E(z t ) (16) where k 1 is the long-run multiplier of y with respect to z.

19 OLS estimator and the dynamic model (ECM)(3) Now subtract y t 1 from both side of (13) and then add and subtract β 0 z t 1 on the right-hand side to get: y t = α 0 + (α 1 1)y t 1 + β 0 z t + (β 0 + β 1 )z t 1 + ε t (17) and finally add and subtract (α 1 1)z t 1 on the right side, yielding y t = α 0 +(α 1 1)(y t 1 z t 1 )+β 0 z t +(β 0 +β 1 +α 1 1)z t 1 +ε t (18)

20 OLS estimator and the dynamic model (ECM)(4) Alternatively, we could have added and subtracted (β 0 + β 1 )z t 1 on the right side, to get y t = α 0 + (α 1 1)(y t 1 k 1 z t 1 ) + β 0 z t + ε t (19) where (α 1 1) represents the short-run adjustment to a discrepancy (a measure of the speed of adjustment of y to a discrepancy between y and z in the previous period).

21 Dynamic model and cointegration test (1) Write in a different form (19) with no constant term: y t = a z t + b(y z) t 1 + ε t (20) The parameter b is the error correction coefficient. For y t = lny and z = lnz, a denotes the short run-elasticity of Y with respect to Z. Without loss of generality, the cointegrating vector for (y t, z t ) is (1,-1) if y t and z t are cointegrated. We assume, for simplicity, that the cointegrating vector is known.

22 Dynamic model and cointegration test (2) The variable y t and z t are cointegrated, or not, depending on whether b < 0 and b = 0. Thus, tests of cointegration rely on upon some estimate of b. In ECM approach, equation (20) is estimated by OLS y t = â z t + ˆbw t 1 + ε t (21) where the disequilibrium is: w t = y t z t (22) The t-statistics based on ˆb is the ECM statistics, t ECM. It is used to test the null hypothesis that b = 0, i.e, that y t and z t are not cointegrated with cointegrating vector [1,-1].

23 Dynamic model and cointegration test (3) The DF statistics derives form a different regression, so it s helpful to establish the relationship between the DF regression equation and the ECM in (20). Subtract z t from both side of (20) and re-arrange: (y z) t = b(y z) t + [(a 1) z t + ε t ] (23)

24 Dynamic model and cointegration test (4) It should be noted that (22) and (23) can be rewritten as: where the disturbance e t is w t = bw t + e t (24) e t = (a 1) z t + ε t ] (25)

25 Dynamic model and cointegration test (5) OLS estimation of (24) generates: w t = bw t + ẽ t (26) The t-statistics based on b is the DF statistics, T DF. This statistics is also used for testing whether y t and z t are cointegrated.

26 Dynamic model and cointegration test (6) In contrast to the estimated ECM in (21), the estimated DF equation (26) ignores potential information contained in z t

27 OLS estimator, endogeneity and serial correlation In addition, the asymptotic distribution of the OLS estimator depends on nuisance parameters arising from endogenity of the regressors and serial correlation in the errors. To solve these problems, two estimators are proposed: FMOLS (fully modified OLS) and DOLS (dynamic OLS)

28 FMOLS estimator (1) Consider the following model: y t = µ + β x t + u 1t = θ z t + u 1t, (27) x t = u 2t, (28) for t = 1,..., T, θ = (µ, β ), z t = (1, x t). For u t = [u 1t, u 2t ], we assume that the functional central limit theorem (FCLT) can be applied as follows: [Tr] 1 T t=1 W (r) = W 1(r) (29) W 2 (r) for 0 r 1, where W (r) is a Brownian motion on [0, 1] with a variance-covariance matrix Ω (W ( ) BM(Ω)).

29 FMOLS estimator (2) Note that the long-run variance of u t and its one-sided version can be expressed as Ω = Σ u + Π + Π Λ = Σ u + Π, with Σ u = lim T 1 T t=1 E(u tu t) and T Π = lim T 1 T 1 T j j=1 t=1 E(u tu t+j ). T

30 FMOLS estimator (3) Ω and Λ can be conformably partioned with u t as: Ω = ω 11 ω 12 ω 21 Ω 22 Λ = λ 11 λ 12 λ 21 Λ 22

31 FMOLS estimator (4) It is known that the OLS estimator of θ, denoted by ˆθ, is consistent but inefficient in general. The centered OLS estimator with a normalizing matrix D T = diag{ T, TI n } weakly converges to ( 1 D T (ˆθ θ) 0 ) 1 ( 1 ) W 2 (r)w 2(r)dr W 2 (r)dw 1 (r) + λ 21 0 (30) and we can observe that this limiting distribution contains the second-order bias from the correlation between W 1 ( ) and W 2 ( ) and the non-centrality parameter λ 21.

32 FMOLS estimator (5) As explained in Phillips and Hansen (1990) and Phillips (1995), the former bias arises from the endogeneity of the I(1) regressor x t while the non-centrality bias comes from the fact that the regression errors are serially correlated. Phillips and Hansen (1990) argue that the second-order biases have no effect on the consistency of the estimators, but result in asymptotic distributions of scaled estimators, such as T ( ˆβ β) in (27), having non-zero means.

33 FMOLS estimator (6) In order to eliminate the second-order bias, Phillips and Hansen (1990) proposes correcting the single-equation estimates non-parametrically in order to obtain meadian-unbiased and asymptotically normal estimates.

34 FMOLS estimator (6) The Fully modified OLS is: ( T ) 1 ( T ) θ + = z t z t z t y t + T Ĵ+ t=1 t=1 (31) where the transformations: y + t = y t ˆω 12 ˆΩ 1 22 u 2t (32) and J + 0 = ˆλ 2 ˆ ˆ Λ 22 Ω 1 22 ωˆ 21 allows for correcting for the endogeneity bias and the non-centrality bias. (33)

35 DOLS Estimator (1) Contrary to nonparametric approach provided by Phillips and Hansen, the DOLS method proposed by Saikkonen (1991) is based on parametric regressions. Saikkonen proposes to augment the leads and lags of the first difference of y 2t as regressors and to estimate K y t = θ z t + π j x t j + u 1t, (34) j= K

36 DOLS Estimator (2) The DOLS estimator is defined as the OLS of θ for (84) θ = ( T K t=k+1 z t z t ) 1 ( T K t=k+1 z t ỹ 1t ) (35) where z t and ỹ 1t are regression residuals of z t and y t on w t = (u 2,t+K,..., u 2,t K ), respectively.

37 DOLS Estimator (3) The regression form (84) is based on the fact that under some regularity conditions, the regression errors u 1t in (27) can be expressed as u 1t = j= π ju 2t j + v t (36) where j= π j <, with being the standard Euclidian norm; further, v t is uncorrelated with u 2t j for all j.

38 DOLS Estimator (4) From (36), we observe that u 1t = π ju 2t j + v t (37) j >K The uncorrelatedness of v t with all the leads and lags of u 2t is an important property to prove that the DOLS method successfully eliminates the second-order bias of the OLS Estimator.

39 Some limits of time series approach 1. In time series analysis with unit root processes, many of the estimators and statistics of interest have been shown to have limiting distributions which are complicated functionals of Wiener processes. 2. The power deficiencies of pure time series-based tests for unit roots and cointegration.

40 Low power of unit root tests (1) Finite sample properties Monte Carlo simulations have shown that the power of the various Dickey-Fuller and Phillips-Perron tests is very low; unit root tests do not have power to distinguish between a unit root and near unit root process (see Dickey and Fuller, 1997). Thus, these test will too often indicate that a series contains a unit root. Moreover, they have a little power to distinguish between trend stationary and drifting processes. In finite sample, any trend stationary process can be arbitrarily well approximated by a unit root process, and a unit root process ca be arbitrarily well approximated by a trend stationary process.

41 Low power of unit root tests (2) Finite sample properties Consider the following random walk plus noise model: y t = µ t + η t (38) µ t = µ t 1 + ε t (39) where η t and ε t are both independent white-noise process with variance of ση 2 and σ 2, respectively. Suppose that we can observe the {y t } sequence, but cannot directly observe the separate shocks affecting y t.

42 Low power of unit root tests (3) Finite sample properties If σ 2 0, {y t } is the unit root process: T y t = µ 0 + ε t + η t (40) t=1 If σ 2 = 0, then all values of {ε t } are constant, that s: ε t = ε t1 =... = ε 0. Now, define this initial values of ε 0 as a 0. It follows that µ t = µ 0 + a 0 t and {y t } is trend stationary: y t = µ 0 + a 0 t + η t (41)

43 Low power of unit root tests (4) Finite sample properties The difference between the difference stationary process (40) and trend process (41) concerns the variance of ε t. Since we observe the composite effect of the two shocks, but not the individual components η t and ε t, we can see that there is no simple way to determine whether σ 2 is exactly equal to zero, in particular when the Data Generating Process (DGP) is such that ση 2 is large relative to σ 2. In a finite sample, arbitrarily increase ση 2 will make it virtually impossible to distinguish a TS and DS series.

44 Low power of unit root tests (5) Finite sample properties In addition, it also follows that a trend stationary process can be arbitrarily well approximate a unit root process. If the stochastic portion of the the trend stationary process has sufficient variance, it will be not possible to distinguish between the unit root and the trend stationary hypothesis.

45 Low power of unit root tests (6) For example, the random walk plus drift model: y t = a 0 + y t 1 + ε t, can be arbitrarily well represented by the model y t = a 0 + ρy t 1 + ε t by increasing σ 2 and allowing ρ to be close to unity. Both these models can be approximated by (41).

46 Low power of unit root tests (7) For applications It turns out that for tests of unit root hypothesis versus stationary alternatives the power depends very little on the number of observations per se but is rather influenced in an important way by the span of the data. For a given number of observations, the power is largest when the span is longest. For a given span, additional observation obtained using data sampled more frequently lead only to a marginal increase in power, the increase becoming negligible as the sampling interval is decreased (Perron, 1990, JBES).

47 Low power of unit root tests (8) In most applications os interest, a data set containing fewer annual data over a long time period will lead to tests having higher power than if use was made of a data set containing more observations over a short time period. These results show that, whenever possible, tests of unit root hypothesis should be performed using annual data over a long time period.

48 Nonstationary Panel data With the growing use of cross-country data over time to study purchasing power parity, growth convergence and international R&D spillovers, the focus of panel data econometrics has shifted towards studying the asymptotics of macro panels with large N(number of countries) and large T (length of the time series) rather than the usual asymptotics of micro panels with large N and small T. A strand of literature applied time series procedures to panels, worrying about nonstationarity, spurious regression and cointegration.

49 Why nonstationary panel data? Advantages The use of data from countries for which the span of time series data is insufficient and would in this way preclude the analysis of many economic hypothesis of interest; 2. The benefits coming from better power properties of the testing procedure with respect to standard time series technique; 3. The fact that many issue of economic interest, such as convergence or purchasing power parity lend themselves naturally to being analyzed in a panel framework; 4. Unit root and cointegration tests have Normal standard asymptotic distribution.

50 ...Disavantages (1) 1. Panel data generally introduce a substantial amount of unobserved heterogeneity, rendering the parameters of the model cross section specific; 2. The panel test outcomes are often difficult to interpret if the null of the unit root or cointegration is rejected. The best that can be concluded is that a significant fraction of the cross section units is stationary or cointegrated. The panel tests do not provide explicit guidance as to the size of this fraction or the identity of the cross section units that are stationary or cointegrated;

51 ...Disavantages (2) 3. With unobserved I(1) common factors affecting some or all the variables in the panel, it is also necessary to consider the possibility of cointegration between the variables across the groups (cross section cointegration) as well as within group cointegration;

52 ...Disavantages (3) 4. Economic applications. For example, panel unit root tests are not able to rescue purchasing power parity (PPP). The results on PPP with panels are mixed depending ont the group of countries studied, the period of study and the type of unit root test used. In addition, for PPP, series. The null hypothesis of a single time series is different from the null hypothesis of panel data, so the panel data tests are the wrong answer to low power of unit root tests in single time series.

53 : 1 generation The common feature of first generation of nonstationary methods is the restriction that all cross-sections are independent. Under this independence assumption the Lindberg-Levy central limit theorem or other central limit theorems can be applied to derive the asymptotic normality of panel test statistics.

54 : 2 generation The second generation panel methods relax the cross-sectional independence assumption. In this context, the first issue is to specify the cross-sectional dependencies, since as pointed out by Quah (1994). The second problem is that cross-sectional dependency is very hard to deal with in non-stationary panels. In this case the usual t-statistics unit root tests have limit distributions that are dependent in a very complicated way upon various nuisance parameters defining correlations across individual units. There does not exist any simple way to eliminate the nuisance parameters in such systems, and a lot of different testing procedures have been proposed.

55 First generation: Cross-section independent hypothesis 1. Unit root tests: Levin, lin and Chu (2002); Maddala and Wu (1999) and Im, Pesaran and Shin (1997, 2003) 2. Cointegration tests: residual based tests (Kao, 1999, JE). 3. Estimation and inference: OLS, DOLS and Fully Modified OLS (Kao and Chiang, 2000)

56 Panel Unit Root tests(1) Homogeneous alternative: Levin, lin and Chu (LLC) (2002). Model specifications: {y it } is generated by one of the following three models: y it = δ i y it 1 + ζ it (42) y it = α 0i + δ i y it 1 + ζ it (43) y it = α 0i + α 1i t + δ i y it 1 + ζ it, (44) where 2 δ 0 for, i=1,...,n; t=1,...,t, and the errors, ζ it are distributed independently across individuals and follow a stationary invertible ARMA process.

57 Panel Unit Root tests (2) In Model 1, the panel unit root test procedure evaluates the null hypothesis H 0 : δ = 0 against the alternative H 1 : δ < 0. The series y i t has an individual-specific mean in Model 2, but does not contain a time trend. In this case, the panel test procedure evaluates the null hypothesis that H 0 : δ = 0 and α 0i = 0, for all i, against H 1 : δ < 0 and α 0i R. Finally, under Model 3, the series y i t has an individual-specific mean and time trend. In this case, the panel test procedure evaluates the null hypothesis that H 0 : δ = 0 and α 1i = 0, for all i, against the alternative H 1 : δ < 0 and α 1i R. LLC (2002) formulate a panel unit root test procedure which consists of three steps.

58 Panel Unit Root tests (3) In the first step, the ADF regressions for each individual in the panel is carried out: P i y it = δ it y it 1 + θ il y it L + α mi d mt + ε it (45) L=1 where d mt denotes the vector of deterministic variables and α mi indicate the corresponding vector of coefficients for the specific model m (m {1, 2, 3}). 1 1 The models are identified as follows: m = 1 denotes an ADF with no constant and trend; m = 2 indicates an ADF with the constant term; m = 3 denotes an ADF with constant and trend.

59 Panel Unit Root tests (4) After having determined the order of the ADF regression, LLC run two auxiliary regressions of y it and y it 1 against y it L (with L = 1,..., p i ), and generate two orthogonolized residuals, ê it and ˆν it. To control for heterogeneity across individuals, LLC derive the normalized residuals ẽ it and ν it by dividing by the standard error form equation(6): ẽ it = êit ˆσ εi and ν it 1 = ˆν it 1 ˆσ εi.

60 Panel Unit Root tests (5) The second step requires estimating the ratio of the long run to short run innovation standard deviation, s i = σ yi σ εi, for each individual. Finally, the pooled t-statistic is computed: tρ = t ρ N T ŜN ˆσ 2 ˆε STD(ˆρ)µ m ˆT σ m ˆT where t ρ is the t-statistic in the regression (46) ẽ it = ρ ν it 1 + ε it, (47) Ŝ N is the estimated average standard deviation ratio,s N = 1 N N i=1 s i, T is the time dimension, STD(ˆρ) = ˆσ ε [ N T i=1 t=2+p i ν it 1 2 ] 1 2, µ m ˆT and σ are the mean m ˆT and the standard deviation adjustments.

61 Panel Unit Root tests (6) Using Lindberg-Levy central limit theorem and sequential limit theory (T followed by N ), LLC(2002) obtain the following limit distribution: model t ρ 1 t ρ N(0, 1) tρ N N(0, 1) ( 3 t ρ + ) 3.75N N(0, 1)

62 Panel Unit Root tests (7) Heterogeneuos alternative: Im, Pesaran and Shin (1997, 2003) IPS propose a test based on the average of the ADF statistics computed for each individual in the panel. The IPS test is based on y i,t = α i +β i y i,t 1 +Σ K j=1δ ij y i,t 1 +ξ i,t, i = 1, 2,..., N; t = 1, 2,..., T (48)

63 Panel Unit Root tests (8) The null hypothesis of a unit root can be now defined as H 0 : β i = 0 for all i against the alternatives H 1 : (β i < 0, i = 1, 2,..., N 1 < N and β i = 0, i = N 1 + 1, N 2,..., N). The alternative hypothesis β i may differ across cross-sectional units. Formally we assume that under the alternative hypothesis the fraction of the individual processes that are stationary is non-zero, namely if lim N (N 1 /N) = δ, 0 < δ 1. This condition is necessary for the consistency of the panel unit root tests.

64 Panel Unit Root tests (9) The IPS test simply uses the average of the N ADF individual t-statistics, t it : from which t NT = 1 N N t it (49) i=1 Z t = N 1 2 [ t NT E( t T )] [Var( t T )] 1 2 (50) where E( t T ) and Var( t T ) are respectively the theoretical mean and variance of t NT. The Z t statistic has an asymptotic standard normal distribution under the null of a unit root.

65 Panel Unit Root tests (10) Fisher s Test: Maddala and Wu (1999) MW (1999) proposed a new simple test based on Fisher s suggestion which consists in combining p-values from individual unit root test. Let the p-value of τ i be p i = Pr(τ τ i ) = τi f (x)dx (51) where f (x) is the probability density function of x. The density function of p i can be obtained by the method of transformation: g(p i ) = f (τ i ) J, where J = dτ i dp i is the Jacobian of the transformation and J is its absolute value.

66 Panel Unit Root tests (11) Since f (τ i ) = dp i dτ i, the Jacobian is 1 f (τ i ) and g(p i) = 1 for 0 p i 1. In other terms, p i is uniformly distributed on the interval [0, 1](p i U[0, 1]). Subsequently, we set y i = 2 ln(p i ).

67 Panel Unit Root tests (12) By the method of transformation, the probability density function of y i is h(y i ) = g(p i ) dp i dy i. Since g(p i ) = 1 and dp i dy i = p i 2 = 1 y 2 e i 2, then we get h(y i ) = 1 y 2 e i 2 which is the density of a chi-square with two degrees of freedom. The joint test statistic, under the null and the additional hypothesis of cross-sectional independence of the errors terms ε it in the ADF equation, has a chi-square distribution with 2N degrees of freedom: λ = 2 N ln(p i ) χ 2 2N (52) i=1 where N is the number of separate samples.

68 Panel Unit Root tests (13) For the Fisher test, MW apply the ADF(p) test for each individual series. Two models are estimated p y i,t = α i + ρ i y i,t 1 + γ ij y i,t j + ε it, j=1 p y i,t = α i + δ i t + ρ i y i,t 1 + γ ij y i,t j + ε it. j=1

69 Panel Unit Root tests: theoretical considerations The IPS and Fisher tests relax the restrictive hypothesis assumption of the LLC test that the autoregressive parameter of y it 1 is the same under the alternative hypothesis; The Fisher test has the advantage over the IPS test in that it does not require a balanced panel; The Fisher test can use different lag lengths in the individual ADF regressions and can be applied to any other unit root test. However, the Fisher test has disadvantage that the p-values have to be derived by Monte Carlo simulations.

70 The size and the power of panel unit root tests: DGP (1) The Theory and Practice of the Econometrics of Non-Stationary Panels (Banerjee and Wagner, mimeo): DGP 1 y it = α i (1 ρ) + ρy it 1 + u it (53) u it = ε it + cε it 1, (54) with ε it N(0, 1). The parameters chosen in the simulations are α = [α 1,..., α N ], ρ and c. Note that the formulation of the intercepts as α i (1 ρ) ensures that in the unit root case (when ρ = 1) no drift appears.

71 The size and the power of panel unit root tests: DGP (2) Consequently, when ρ = 1 α is equal to zero in the simulations for computational efficiency. Otherwise, the coefficients α i are chosen uniformly distributed over the interval 0 to 4, i.e. α i U[0, 4].

72 The size and the power of the panel unit root tests: DGP (3) DGP 2 y it = α i + α i (1 ρ)t + ρy it 1 + u it (55) u it = ε it + cε it 1, (56) with ε it N(0, 1). This formulation allows for a linear trend in the absence of a unit root and for a drift in the presence of a unit root. The coefficients α i are, as for the previous case, U[0,4] distributed.

73 The size and the power of the panel unit root tests: DGP (4) NOTE: The careful reader will have observed that our simulated DGPs all have a cross sectionally identical coefficient ρ under both the null and the alternative. Thus, we are in effect in a situation where we generate data either under the null hypothesis or under the homogenous alternative. We do this, because only the more restrictive homogenous alternative can be used for all tests described in the previous section. This implies to a certain extent that we do not explore the additional degree of freedom that the tests against the heterogeneous alternative (IPS and MW) possess.

74 The size of panel unit root tests DGP 1. a) c=0, T=10,15,20. LLC and MW tests are increasingly oversized with N increasing. IPS test exhibit satisfactory size behaviour. For T=50,100. LLC and MW also exhibit satisfactory size behaviour. b) c=-0.99 (negative serial correlation). Size distortion for any given T. DGP 2. a) c=0. T=10,15. Size distortion for LLC and MW (lesser rate). T=25, only MW show size distortion. b) Serially correlated errors (c 0.2). The size is below 0.1 for LLC for any combination of N,T.

75 The power of the panel unit root tests DGP 1. For ρ 0.9, N=10 and T 100 all test have power equal to 1. For larger value of ρ ρ {0.95, 0.99}, N 50 is required to have power tending to 1 for T 100 (for both c=0 and c 0). DGP2. For ρ 0.9 and T 100, all test have power equal to 1 for all values of N.

76 Panel cointegration tests (1) Homogeneous hypothesis Kao (1999) proposes the Dickey-Fuller test and the Augmented Dickey-Fuller (ADF). Let ê it be the estimated residual from the following regression: y it = α i + βx it + e it (57)

77 Panel cointegration tests (2) The equation (57) is estimated using LSDV (least square dummy variable) estimator. The DF test is applied to the estimated residuals: ê it = γê it 1 + ˆν it (58) The null hypothesis of no cointegration, H 0 : γ = 1, is tested against the alternative of cointegration for all i=1,...n (Homogenous hypothesis).

78 Panel cointegration tests (3) Kao (1999) proposed four DF-types tests: DF γ = NT (ˆγ 1) 10.2 (59) DF γ = DF t = 1.25tγ 1.875N (60) NT (ˆγ 1) + ( ˆσ4 ν 5ˆσ 4 0ν 3 N ˆσ 2 ν ˆσ 2 0ν ) (61)

79 Panel cointegration tests (4) ) DF t = ( 6N t γ + ˆσν /2ˆσ 0ν ) ( ) (62) (ˆσ 0ν 2 /2ˆσ2 ν + 3ˆσ ν/10ˆσ 2 0ν 2 While DF γ and DF t are based on the assumption of strict exogeneity of the regressors with respect to the errors in the equation, DFγ and DFt are for cointegration with endogenous regressors.

80 Panel cointegration tests (5) The ADF regression estimated is: p ê it = γê it 1 + φ j ê it j + ν it (63) The ADF test is applied to the estimated residual: where p is chosen so that the residual ν i,tp are serially uncorrelated. The ADF test statistic is the usual t-statistic of the equation (63). j=1

81 Panel cointegration tests (6) With the null hypothesis of no cointegration, the ADF test statistics can be constructed as: ADF = t ADF + ( 6N ˆσν 2ˆσ 0ν ) (64) ( ˆσ2 0ν ) + (10ˆσ 2ˆσ 2 ν 2 0ν ) where ˆσ 2 ν = Σ µε Σ µε Σ 1 ε, ˆσ 2 0ν = Ω µε Ω µε Ω 1 ε, Ω is the long-run covariance matrix and t ADF is the t-statistic in the ADF regression. Kao shows that all DF and ADF test converges to a standard normal distribution N(0,1).

82 Panel estimation methods (1) homogeneity hypothesis (i.e. the variances are constant across the cross-section units.) Kao and Chiang (2000) analysed the asymptotic distributions for ordinary least square (OLS), fully modified OLS (FMOLS), and dynamic OLS (DOLS) estimators in cointegrated regression models in panel data. They shows that the OLS, FOMLS, and DOLS estimators are all asymptotically normally distributed.

83 Panel estimation methods (2) Kao and Chiang consider the following fixed-effect panel regression: y it = α i + x itβ + u it i = 1,..., N, t = 1,..., T, (65) where {y it } are 1 1, β is a k 1 vector of the slope parameters, {α i } are the intercepts, and {u it } are the stationary disturbance terms.

84 Panel estimation methods (3) Kao and Chiang also assumed that {x it } are K 1 integrated processes of order one for all i, where x it = x it 1 + ε it. Under these specifications, the previous equation defines a system of cointegrated regressions, i.e. is cointegrated under the hypothesis that {y it } and {x it } are independent across cross-sectional units.

85 Panel estimation methods (4) The innovation vector is w it = (u it, ε it ). The long-run covariance matrix, Ω, of w it, can be written as: Ω = E(w i,j w i,0 ) (66) J= = Σ + Γ + Γ (67) = Ω u Ω εu Ω uε Ω ε. (68)

86 Panel estimation methods (5) where and Γ = E(w ij w i0 ) = Γ u J= Σ = E(w ij w i0 ) = Σ u Σ εu Γ εu Σ uε Σ ε Γ uε Γ ε (69) (70) are partitioned conformably with w it.

87 Panel estimation methods (6) The-sided long-run covariance is defined as: = Σ + Γ (71) = E(w i,j w i,0 ) (72) J= = Ω u Ω εu Ω uε Ω ε. (73)

88 Panel estimation methods (7) Kao and Chiang derived limiting distributions for the OLS, FMOLS and DOLS estimators in a cointegrated regression. The OLS estimator of is β is N T N T ˆβ OLS = [ (x it x i )(x it x i ) ] 1 [ (x it x i )(y it ȳ i )] i=1 t=1 i=1 t=1 where x i = 1 T T i=1 x it and ȳ i = 1 T T i=1 y it represent the individuals means. The FMOLS estimator is derived by making corrections for endogeneity and serial correlations to the OLS estimator ˆβ OLS. (74)

89 Panel estimation methods (8) Let u + it = u it Ω uε Ω 1 ε ε it (75) û + it = u it ˆΩ uε ˆΩ 1 ε ε it (76) and y + it = y it Ω uε Ω 1 ε it (77) ŷ + it = y it ˆΩ uε ˆΩ 1 ε it (78)

90 Panel estimation methods (9) The endogeneity correction is achieved by modifying the variable y it in (65), with the transformation: ŷ + it = y it ˆΩ εu ˆΩ 1 ε x it (79) = α i + x itβ ˆΩ εu ˆΩ 1 ε x it (80) where ˆΩ εu and ˆΩ ε are consistent estimates of Ω εu and Ω ε.

91 Panel estimation methods (9) The serial correlation correction term takes the form: ( ) 1 ˆ + εu = ( ˆ εu ˆ 1 ε ) ˆΩ εu ˆΩ ε (81) = ˆ εu ˆ ε ˆΩ 1 ε ˆ εu (82) where ˆ εu and ˆ ε are kernel estimates of εu and ε.

92 Panel estimation methods (10) The FMOLS estimator is: N T N ( T ˆβ FMOLS = [ (x it x i )(x it x i ) ] 1 [ (x it x i )ŷ + it T ˆ + εu)] i=1 t=1 i=1 t=1 (83)

93 Panel estimation methods (11) The DOLS estimator can be obtained by running the following regression: y it = α i + x itβ q 2 j= q 1 c it x it+j + ν it (84) Kao and Chiang (2000) showed that the asymptotic distributions of the OLS, FMOLS and DOLS estimators are normal standard.

94 Cross-section dependence: Introduction(1) The cross-sectional independence assumption is quite restrictive in many empirical applications. More generally, this assumption raises the issue of the validity of the panel approach in macroeconomic, finance or international finance. For instance, the issue is to know if it is useful to test the non-stationarity of the GDP of a particular country, which is notably linked to the persistence of international shocks, without considering the relationships between this GDP and the GDP of the others countries which belong to the same economic area. Since co-movements in national business cycles are often observed (Backus and Kehoe, 1992), this issue is far from being only a technical problem of power and size distortion.

95 ... How does the independent tests work under a simple form of cross-section dependence? (1) Consider the following DGP: y it = φ i µ i + φ i y it 1 + u it, (85) The error term u it contains a time-specific effect θ t and a specific component ε it : u it = θ t + ε it, where ε it = λ i ε it 1 + e it. 2 2 The inclusion of time dummies (common time effect) appears to be a poor control for cross-sectional dependence, for example, in testing for purchasing power parity

96 ... How does the independent tests work under a simple form of cross-section dependence? (2) We assume e it to be jointly normal distributed with: E(e it ) = 0, (86) and σ ij for t=s E(e it, e js ) = 0 for t s. (87)

97 ... How does the independent tests work under a simple form of cross-section dependence? (3) If we let Σ denote (σ ij ) N i,j=1 then non-zero terms on the off-diagonal terms in Σ represents the existence of cross-correlations.

98 ... How does the independent tests work under a simple form of cross-section dependence? (3) Example: Σ, N = 2 and T = 3. i, t 1,1 1, 2 1, 3 2, 1 2, 2 2, 3 1,1 σ σ ,2 0 σ σ ,3 0 0 σ σ 12 2,1 σ σ ,2 0 σ σ ,3 0 0 σ σ2 2

99 ... How does the independent tests work under a simple form of cross-section dependence? (4) In general, when there is no cross-sectional correlation in the errors, the IPS test is slightly more powerful than the Fisher test, in the sense that the IPS test has higher power when the two have the same size. Both tests are more powerful than the LL test. When the errors in the different samples (or cross-section units) are cross correlated (as would often be the case in empirical work) none of the tests can handle this problem well.

100 ... How does the independent tests work under a simple form of cross-section dependence? (5) However, the Monte Carlo evidence suggests that this problem is less severe with the Fisher test than with the LL or the IPS test. More specifically, when T is large but N is not very large, the size distortion with the Fisher test is small. But for medium values of T and large N, the size distortion of the Fisher test is of the same level as that of the IPS test.

101 Cross-section dependent panels: GLS, cross-sectional demean and bootstrap method. (1) 1. O Connel (1998) The distribution of ˆρ in LLC (2002) is derived under the assumption that the variance-covariance matrix is diagonal (no correlation).

102 Cross-section dependent panels: GLS, cross-sectional demean and bootstrap method. (2) Now, suppose that the correlation matrix taken the following form: 1 ω... ω ω 1... ω Ω = (88) ω ω... 1

103 Cross-section dependent panels: GLS (O Connel, 1998), cross-sectional demean and bootstrap method. (3) To define the GLS estimator of δ i in (42), let Y be the following matrix T N y 11 y y N1 y 12 y y N2 Y T N = y 1T y 2T... y NT Similarly, let X be the T N matrix of lagged y it.

104 Cross-section dependent panels: GLS (O Connel, 1998), cross-sectional demean and bootstrap method. (4) The GLS estimated of δ i is given by: ˆδ i GLS = tr(x Y Ω 1 ) tr(x X Ω 1 ) (89) The GLS estimator possesses an appealing feature that aids in the interpretation of cross-country estimates of δ i.

105 Cross-section dependent panels: GLS (O Connel, 1998), cross-sectional demean and bootstrap method. (5) Consider the Purchasing power parity. A basic intuition is that a a set of real exchange rates generated by different choices of numeraire are linear combinations of the one another. Thus changing the numeraire does not change the information that is used in the estimator, only its configuration (i.e its interdependence). By nature, GLS controls for interdependence.

106 Cross-section dependent panels: GLS (O Connel, 1998), cross-sectional demean and bootstrap method. (6) As a results, the GLS estimator is invariant to the linear combination of the real exchange rates that is used as numeraire. Its not necessary for Ω to be known for this to hold: invariance carries over to the feasible GLS estimator ˆδ i GLS = tr(x Y ˆΩ 1 ) tr(x X ˆΩ 1 ) (90) where ˆΩ is some consistent estimates of Ω.

107 Cross-section dependent panels: GLS (O Connel, 1998), cross-sectional demean and bootstrap method. (7) 2. Cross-sectional demean. Let y it be a balanced panel generated by (42) with ζ it = α i + θ t + ε t, where θ t is a single common time effect. You can control for the common time effect θ t. If you do, you subtract off the cross-sectional mean and the basic unit of analysis is ỹ it = y it 1 N N y jt (91) j=1

108 Cross-section dependent panels: GLS (O Connel, 1998), cross-sectional demean and bootstrap method. (6) Potential pitfalls of including common-time effect. Doing so however involves a potential pitfall. θ t, as part of the error-components model, is assumed to be iid. The problem is that there is no way to impose independence. Specifically, if it is the case that each y it is driven in part by common unit root factor, θ is a unit root process. Then y it = y it 1 1 N y jt N will be stationary. The transformation renders all the deviations from the cross-sectional mean stationary. j=1

109 Cross-section dependent panels: GLS (O Connel, 1998), cross-sectional demean and bootstrap method. (7) This might cause you to reject the unit root hypothesis when it is true. Subtracting off the cross-sectional average is not necessarily a fatal flaw in the procedure, however, because you are subtracting off only one potential unit root from each of the N time-series. It is possible that the N individuals are driven by N distinct and independent unit roots. The adjustment will cause all originally nonstationary observations to be stationary only if all N individuals are driven by the same unit root.

110 Cross-section dependent panels: GLS (O Connel, 1998), cross-sectional demean and bootstrap method. (8) 3. bootstrap method The method discussed here is called the residual bootstrap because we resampling from the residuals. The DGP under the null hypothesis is: K i y it = µ i + φ ij y it j + ε it (92) j=1 Since y t is a unit root process, its firs difference follows an autoregression. The individual equations of the DGP can be fitted by Least Squares. If a linear trend is included in the test equation a constant must be included in (92).

111 Cross-section dependent panels: GLS (O Connel, 1998), cross-sectional demean and bootstrap method. (9) To account for dependence across cross-sectional units, estimate the joint error covariance matrix Σ = E(ε t ε t) by ˆΣ = 1 T T t=1 (ˆε t ˆε t) where ˆε t = ( εˆ 1t,..., ε NT ˆ ) is the vector of OLS residuals.

112 Cross-section dependent panels: GLS (O Connel, 1998), cross-sectional demean and bootstrap method. (10) The parametric bootstrap distribution for τ (see (46)) is built as follows. 1. Draw a sequence of length T + R innovation vectors from ε N(0, ˆΣ). 2. Recursively build up pseudo-observations y it, i = 1,..., N, t = 1,..., T + R according to (92) with the ε t and estimated values of the coefficients µ i and ˆφ ij.

113 Cross-section dependent panels: GLS (O Connel, 1998), cross-sectional demean and bootstrap method. (11) 3. Drop the first R pseudo-observations, then run the LLC test on the pseudo-data. Do not transform the data by subtracting off the cross-sectional mean. This yields a realization of τ ρ in (46) generated in the presence of cross- sectional dependent errors. 4. Repeat a large number (2000 or 5000) times and the collection of τ ρ statistics form the bootstrap distribution of these statistics under the null hypothesis.

114 Cross-section dependent panel unit root tests: factor models and cross-section averages methods 1. Test based on factors models: Bai and NG, For these tests, the idea is to shift data into two unobserved components: one with the characteristic that is strongly cross-sectionally correlated (common factor) and one with the characteristic that is largely unit specific (idiosyncratic component). 2. Test based on cross-section averages: Pesaran (2007). Instead of basing the unit root tests on deviations from the estimated common factors, he augments the standard Dickey Fuller or Augmented Dickey Fuller regressions with the cross section average of lagged levels and first-differences of the individual series.

115 Panel unit root tests: factor models (1) 1. Bai and Ng propose to test the common factors and the idiosyncratic components separately. So, it is possible to know if the non-stationarity comes from a pervasive or an idiosyncratic source. Bai and Ng (2004) consider the following model Y i,t = D i,t + λ if t + e i,t, (93) where D i,t is a polynomial trend function, F t is an r 1 vector of common factors, and λ i is a vector of factor loading. The process Y i,t may be non-stationary if one or more of the common factors are non-stationary, or the idiosyncratic error is non-stationary, or both.

116 Panel unit root tests: factor models (2) In order to test the non-stationarity of the common factors, Bai and Ng (2004) distinguish two cases: only one common factor among the N variables (r = 1) and more than one common factor (r > 1).

117 Panel unit root tests: factor models (3) Among the r common factors, we allow r0 and r 1 to be stochastic common trends with r 0 + r 1 = r. The corresponding model in first difference is: y it = λ i + z it (94) where z it = e it and f = F it with E(f t ) = 0. Applying the principal-components approach to y it yields r estimated factors ft, the associated loadings λ t, and the estimated residuals, z it = y it ˆλ ˆf i t.

118 Panel unit root tests: factor models (4) Define for t = 2..., T ê it = t s=2 ẑit (i=1,...n) ˆF t = t s=2 ẑit, an r 1 vector. 1. If r = 1, let ADF F ê be the t statistics for testing δ i0 in the univariate augmented autoregression (with an intercept): ˆF it = c + δ 0 ˆF t 1 + δ 1 ê t 1 + δ p ˆF it p + error (95)

119 Panel unit root tests: factor models (5) 2. If r > 1, demean ˆF t and denote ˆF c t = ˆF t ˆF t, where ˆF t = (T 1) 1 T t=2 ˆF t. Start with m = r: A: ˆβ denotes the m eigenvectors associated with the m largest eigenvalues of T 2 T t=2 ˆF c t considered: c ˆF t. Two different statistics may be B.I: Let K(j) = 1 j (j+1), j = 0, 1,...J i) Let ˆξ t c be the residuals from estimating a first-order VAR in Ŷ c In addition, let ˆ c 1 = J j=1 K(j)(T 1 T ˆξ t=2 t j c c ˆξ t ) t.

120 Panel unit root tests: factor models (6) ii) Let v M c be the smallest eigenvalue of: T T Φ c c(m) = 0.5[ (Ŷt c Ŷ t 1 c + Ŷt 1Ŷ c t c ) T (ˆΣ c 1 + ˆΣ c 1 )]( Ŷt c Ŷ t 1) c 1 t=2 iii) Define MQ c c (m) = T [ˆν c c (m) 1]. t=2 (96)

121 Panel unit root tests: factor models (7) B.II: For p fixed that does not depend on N and T i) Estimate a VAR of order p in Ŷ t c to get ˆ (L) = Im ˆ 1 L... ˆ p Lp and filter Ŷ c t by ˆ (L), we have: ŷt c = ˆ (L)Ŷ t c ii) Let ˆν f c (m) be the smallest eigenvalue of: T T Φ f c(m) = 0.5[ (ŷt c ŷt 1 c + ŷt 1ŷ c t c )]( ŷt c ŷt 1) c 1 (97) t=2 t=2 iii) Define the statistics MQf c(m) = T [ˆνc f (m) 1]. C: If H 0 : r 1 = m is rejected, set m = m 1 and return to step A. Otherwise, ˆr 1 = m and stop.

122 Panel unit root tests: factor models (8) To test the stationarity of the idiosyncratic component, Bai and Ng (2004) propose to pool individual Augmented Dickey-Fuller (ADF ) t-statistics with de-factored estimated components ê it in the model with no deterministic trend p e i,t = δ i,0 ê i,t 1 + δ i,j ê i,t j + µ i,t. (98) j=1 Let ADFê c (i) be the ADF t-statistic for the i-th cross-section unit. The asymptotic distribution of the ADFê c (i) coincides with the Dickey-Fuller distribution for the case of no constant. However, these individual time series tests have the same low power as those based on the initial series.

123 Panel unit root tests: factor models (9) Bai and Ng (2004) propose pooled tests based on Fisher type statistics defined as in Choi (2001) and Maddala and Wu (1999). Let Pê c (i) be the p-value of the the ADF t-statistics for the i-th cross-section unit, ADFê c (i), then the standardized Choi s type statistics is: Z c ê = 2 N (i) 2N. (99) 4N i=1 log Pc ê The statistics (99) converge for (N, T ) to a standard normal distribution.

124 Panel unit root tests: cross-section averages methods, Pesaran (2007) (1) Pesaran proposed to augment the standard DF (or ADF) regression with the cross section averages of lagged levels and first-differences of the individual series. If residuals are not serially correlated, the regression used for the ith country is defined as: y it = α i + ρ i y i,t 1 + c i ȳ t 1 + d i ȳ t + e it. (100) where ȳ t 1 = (1/N) N i=1 y it 1 and ȳ t = (1/N) N i=1 y it

125 Panel unit root tests: cross-section averages methods (2) Let us denote t i (N, T ) the t-statistic of the OLS estimate of ρ i. The Pesaran s test is based on these individual cross-sectionally augmented ADF statistics, denoted CADF. A truncated version, denoted CADF, is also considered to avoid undue influence of extreme outcomes that could arise for small T samples.

126 Panel unit root tests: cross-section averages methods (3) In both cases, the idea is to build a modified version of IPS t-bar test based on the average of individual CADF or CADF statistics (respectively denoted CIPS and CIPS): CIPS(N, T ) = N 1 CIPS (N, T ) = N 1 N i=1 N i=1 t i (N, T ) (101) t i (N, T ) (102)

127 Panel unit root tests: cross-section averages methods (4) where the truncated CADF statistics is defined: K 1 if t i (N, T ) K 1 ti (N, T ) = t i (N, T ) if K 1 < t i (N, T ) < K 2 K 2 if t i (N, T ) K 2 (103) The constants K 1 and K 2 are fixed such that the probability that t i (N, T ) belongs to [K 1, K 2 ] is near to one. In a model with intercept only, the corresponding simulated values are respectively and 2.61

128 Panel unit root tests: cross-section averages methods (5) Pesaran consider also the case of serially correlated residuals. For an AR(p) error specification, the relevant individual CADF statistics are computed from a p th order cross-section/time series augmented regression: p p y it = α i + ρ i y i,t 1 + c i ȳ t 1 + d i,j ȳ t + β i,j y it j + µ it. j=0 j=0 (104)

129 Cointegration tests: cross-section dependence(1) Gengenbach, Palm and Urbain (2006) propose the following testing procedure which consist of two steps: 1. A preliminary PANIC analysis on each variable X i,t and Y i,t to extract common factors is conducted. Tests for unit roots are performed on both the common factors and the idiosyncratic components using by Bai and Ng (2004) procedure.

130 Cointegration tests: cross-section dependence (2) 2. a) If I(1) common factors and I(0) idiosyncratic components are detected, then a situation of cross-member cointegration is found and consequently the non-stationarity in the panel is entirely due to a reduced number of common stochastic trends. Cointegration between Y i,t and X i,t can only occur if the common factors for Y i,t cointegrate with those of X i,t. b) If I(1) common factors and I(1) idiosyncratic components are detected, then defactored series are used. In particular, Y i,t and X i,t are defactored separately. Testing for no-cointegration between the defactored data can be conducted using standard panel tests for no cointegration such as those of Pedroni s (1999, 2004) unit root tests.