1 Introduction The time series properties of economic series (orders of integration and cointegration) are often of considerable interest. In micro pa

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1 Unit Roots and Identification in Autoregressive Panel Data Models: A Comparison of Alternative Tests Stephen Bond Institute for Fiscal Studies and Nuffield College, Oxford Céline Nauges LEERNA-INRA Toulouse and University College London Frank Windmeijer Institute for Fiscal Studies, 7 Ridgmount Street, London WC1E 7AE May 003 Abstract We compare the finite sample behaviour of various unit root tests for micro panels where the number of individuals is typically large, but the number of time periods is often very small. As in this case some econometric estimators do not identify the parameters of interest when the processes are random walks, it is important to test for unit roots/identification. We find that a t-test based on OLS estimation results provides a simple robust test with hight power for cases when the variance of the unobserved heterogeneity is relatively small. Its behaviour is similar to the underidentification test as proposed by Arellano, Hansen and Sentana (1999) for the GMM estimator on a first-differenced model. JEL Classification: C1, C3 Key Words: Generalised Method of Moments, Identification, Unit Root Tests 1

2 1 Introduction The time series properties of economic series (orders of integration and cointegration) are often of considerable interest. In micro panels, where the number of individuals is typically large but the number of time periods is often very small, these properties can also be crucial for identification of econometric models. Where differencing transformations are used to eliminate unobserved individual-specific effects, identification requires the existence of instrumental variables that are correlated with first-differences of the series. In the extreme case of a pure random walk, lagged values of the series are uncorrelated with first-differences, and the widely used first-differenced instrumental variables estimators will provide no information on the parameter of interest. It is therefore important toassess the time series properties of the series under consideration. Various test procedures have been proposed in the literature recently. In this paper we will develop these tests, paying particular attention to those that are appropriate for micro panels, and relating unit root tests to the issue of identification. To focus our discussion, we will concentrate on testing the value of ff in the first-order autoregressive model with unobserved individual-specific effects i and serially uncorrelated disturbances v it y it = ffy it 1 +(1 ff) i + v it ;jffj»1 for i =1; :::; N and t =; :::; T. This model reduces to arandomwalk when ff =1. Properties of estimators of ff in this setting are well known. For ff < 1, ordinary least squares gives an upward biased and inconsistent estimate, due to the correlation between the lagged dependent variable and the omitted

3 individual-specific effect. However for ff = 1, ordinary least squares is consistent. Regardless of the true value of ff, within groups (or least squares dummy variables) gives a downward biased estimate, which is inconsistent as N becomes large for fixed T (Nickell, 1981). A widely used estimator in this context eliminates the individual-specific effects by first-differencing or related transformations, and then uses suitably lagged levels of the series as instruments for the equations in first differences. In the AR(1) example, lagged levels of the dependent variable dated t and earlier are available as instruments. These orthogonality conditions can be exploited efficiently in a Generalised Method of Moments (GMM) estimator, which was developed in Holtz-Eakin, Newey and Rosen (1988), and Arellano and Bond (1991). However for the parameter to be identified using this first-differenced GMM estimator it is important that the instruments are correlated with the endogenous variable in the first-differenced equations. More generally, if the instruments are weak in the sense of Staiger and Stock (1997), simulation studies have shown that the finite sample properties of the first-differenced GMM estimator can be very poor. This happens when the series are highly persistent. In the extreme case when the series are a random walk y it = y it 1 + v it there is no correlation between y it 1 = v it 1 and lagged levels of the series dated t and earlier. In this case the first-differenced GMM estimator does not identify ff, and will not provide any information about this parameter even in very large samples. Blundell and Bond (1998) proposed an alternative GMM estimator that imposes a restriction on the initial conditions (y i1). They only considered 3

4 stationary models with ff < 1. However, following Binder, Hsiao and Pesaran (000) we can extend this to include the unit root case. The required properties of the initial conditions are E [(y i1 i ) i ] = 0 ifff < 1 (1) Var(y i1) < 1 ifff =1 () The former restriction ensures that y it is uncorrelated with i for stationary processes, and the latter restriction ensures that y it is correlated with y it 1 for non-stationary processes. Then regardless of its true value, the parameter ff is identified using lagged differences y is dated t 1 and earlier as instrumental variables for the levels equations. For ff<1 there is additional information in the first-differenced equations, and both sets of moment conditions are combined in the 'system' GMM estimator developed in Blundell and Bond (1998). Testing for a unit roots in micro panels is thus important in order to assess whether the first-differenced GMM estimator is identified or whether other estimators need to be considered. The time series properties of particular series may be of independent interest, and when applied to residuals from regression models, these procedures will also form the basis for cointegration tests. Most of the recent literature on testing for unit roots in panel data has focused on panels with a small number of cross-sectional units, N, observed over many time periods, T, typical of panel data for countries or industries. 1 A much smaller literature exists on testing for unit roots when the crosssectional dimension of the panel is large, but the time dimension is small and treated as fixed, the type of panel data often encountered when individuals, households or firms are surveyed. We will focus on the latter case. 1 For a recent survey, see Baltagi and Kao (000). 4

5 The tests can be divided into two basic types: those based on estimates of ff that are consistent both under the unit root null and under the alternative; and those based on estimators which are only consistent (or have biases that can be characterised) under the null. Estimators that are consistent As the OLS estimator is consistent under the null of a random walk, a simple test is a t-test on the estimated OLS parameter. The asymptotic distribution of this t-test is standard in the case of micro panels, where we rely on N! 1 with T fixed to derive asymptotic approximations. The power of this test will however be influenced by the variance of the unobserved heterogeneity in the model under the alternative that ff < 1. As the OLS estimator is biased upward under the alternative, the power to reject ff =1 will be low when Var( i ) =V ar (v it ) is large. An alternative to OLS is to use atwo-step GMM estimator, using lagged levels y is dated t 1 and earlier as instruments for the levels equations. This will be more efficient than OLS in the presence of heteroskedasticity, and can be expected to increase the power of the t-test in this case. The GMM test for overidentifying restrictions also provides additional information on the appropriateness of the random walk model specification. Motivated by concern over the power of the simple OLS test when Var( i ) =V ar (v it ) is high, Breitung and Meyer (1994) proposed an alternative test based on the OLS estimator in a long-differenced model. In this model the OLS estimator has an asymptotic bias that is independent of the variance of the individual effects, and can be calculated under specific assumptions. Similarly Harris and Tzavalis (1999) derived the asymptotic bias and variance of the within groups estimator for the random walk model, and proposed a test for the null of a unit root based on the bias corrected estimates. These corrections are 5

6 derived for a model with normal, homoskedastic errors, and the tests may not be robust to deviations from these assumptions. In many applications it is likely that there is conditional heteroskedasticity of complex forms that are difficult to characterise. The simpler tests based on OLS or GMM estimates in the untransformed model will be asymptotically robust to this. Moreover there is likely to be a power trade-off between tests based on estimators whose bias under the alternative does not depend on the unobserved heterogeneity, but which are less efficient under the null. A related test is developed by Arellano, Hansen and Sentana (1999). They propose a general test for the identification of the parameters in models estimated by GMM. Applied to the first-differenced GMM estimator for the AR(1) model in equation (1), this becomes a test for unit roots. Whilst their test will be useful in a much wider range of models, it will be interesting to compare its performance in this setting, where a range of alternative tests of the same hypothesis are available. We find that a t-test based on OLS estimation results provides a simple robust test with high power for cases when the variance of the unobserved heterogeneity is relatively small. Its behaviour is similar to the underidentification test as proposed by Arellano, Hansen and Sentana (1999) for the first-differenced GMM estimator. The test proposed by Breitung and Meyer (1994) does not loose power with increasing variance, but its power can be quite low. The test by Harris and Tzavalis (1999) is shown to very sensitive to features of the underlying data. Especially when there is heteroscedasticity over time, its size properties are severely distorted. The next section introduces the model and GMM estimators. Section 3 describes the underidentification tests and Section 4 the alternative unit root testing procedures. Section 5 presents some Monte Carlo results and Section 6

7 6 concludes. Model and GMM Estimation Consider the simple dynamic AR(1) panel data model y it = ffy it 1 + u it (3) u it = (1 ff) i + v it ; for i =1; :::; N and t =; :::; T ; N is large and T is fixed. Note that there are no individual effects when ff = 1. The observations are independent across individuals and the error term satisfies E ( i )=0; E (v it )=0 fori =1;:::;Nandt=:::; T and E (v it v is )=0 fori =1; :::; Nandt 6= s: If it is only assumed that the v it are uncorrelated with y i1: E (y i1v it )=0 fori =1; :::; Nand t =:::; T; then there are the following (T 1) (T ) = linear moment conditions available for the estimation of ff by GMM E (y is u it )=0 fort =3;:::T;s=1;:::;t ; (4) where u it = u it u it 1 = y it ff y it 1, see for example Arellano-Bond (1991). Specifying the instrument set as Z D i = 6 4 y i1 0 0 ::: 0 ::: 0 0 y i1 y i ::: 0 ::: ::: ::: y i1 ::: y i(t ) : 7

8 the GMM estimator minimises ψ 1 NX N i=1! 0 ψ 1 Z D0 i u i W N N NX i=1 Z D0 i u i! where u i = [ u i3; u i4;:::; u it ] 0, and W N is a positive semi-definite weight matrix that converges to a positive definite matrix W as N! 1. Under general conditions, an optimal two-step estimator is based on the weight matrix ψ 1 NX W N = N i=1 Z D0 0 i d u i d u i ZD i! 1 ; where d u i are the residuals based on an initial consistent estimator for ff. If further an error components structure on the error term and mean stationarity on the process are imposed, implying that E ( i v it )=0 fori =1; :::; Nand t =:::; T y i1 = i + " i fori =1; :::; N and E (" i )=E( i " i )=0 fori =1; :::; N; see for example Ahn-Schmidt (1995), Arellano-Bover (1995) and Blundell- Bond (1998), there are the following extra (T ) linear moment conditions available: E (u it y it 1 )=0 fort =3; :::; T: (5) The so-called system GMM estimator for ff is obtained by stacking the residuals from the differenced and level equations, and extending the instrument matrix to Zi S = Zi D y i : 0 0 y i(t 1) 8

9 The system GMM estimator combines the moment conditions for the model in first-differences with those for the model in levels. A simpler (and less efficient) GMM levels estimator, that is based on the (T 1) (T ) = moment conditions E (u it y i;t s )=0;fort =3; :::; T and1» s» t ; (6) relates only to the equations in levels. These can be expressed as E Z L0 i u i =0; where Z L i is the (T ) (T 1) (T ) = matrix given by Z L i = 6 4 y i 0 0 ::: 0 ::: 0 0 y i y i3 ::: 0 ::: 0 : : : ::: : ::: : ::: y i ::: y it ; and u i is the (T ) vector (u i3;u i4; :::; u it ) 0. 3 Unit Roots and Identification 3.1 First-Differenced GMM For the first-differenced GMM estimator that utilises moment conditions (4), the endogenous lagged differences y i;t 1 are instrumented by lagged levels y i1; :::; y it. Clearly, when ff = 1, the rank condition is not satisfied as the instruments are uncorrelated with the endogenous variable, and therefore ff is not identified in this case. Arellano, Hansen and Sentana (1999), henceforth AHS, propose a general test of the identification of the parameters in models estimated by GMM. For the simple AR(1) panel data model their test for underidentification is a test for the validity of the moment conditions E y t 1 i y it =0; (7) 9

10 where y t 1 i = (y i1; :::; y it 1 ) 0. The Sargan test statistic for overidentifying restrictions has an asymptotic χ distribution with T (T 1) = degrees of freedom when the model is underidentified and the moment conditions (7) are valid. When the Sargan test rejects, the model is not underidentified. For this model it is clear that a test for identification is equivalent to a test for a unit root, H0 : ff = 1, and we will compare the performance of the AHS test for underidentification to various tests for a unit root, as described in the next section. The AHS test is equivalent to the Anderson-Rubin test for H0 : ff = 1 in the first differenced GMM model. 3. System GMM For the system estimator the T extra moment conditions (5) remain valid when ff =1, even though the process is clearly not mean-stationary in this case. Consider the first stage regression for the levels equation, when T = 3, y i = ß y i + r i : If ff = 1, it follows that ß = 1 and r i = y i1. Denote TP the number of periods the process has been in existence before the sample is drawn. Then, for any fixed TP, plim N!1 bß OLS = 1, and the model is (asymptotically) identified. If TP goes to infinity at a faster rate than N, then plim N!1 bß OLS does not exist, and the model is not identified, although bß OLS is centered around 1. For any given sample, the ratio of N to TP determines how well the distribution of the level or system GMM estimator is approximated by its asymptotic distribution. Even if the model is not identified, the level and system estimator will be centered around 1, as the SLS estimator will be centered around the mean of the OLS estimator in the levels equation. This is no longer true if there are individual specific drifts. 10

11 The AHS test for underidentification for the GMM levels model using moment conditions (6) is a test for the moment conditions E y it y t i =0; where y t i =( y i; :::; y it ) 0. 4 Tests for Unit Roots 4.1 OLS Under the null H0 : ff =1,the OLS estimator in model (3) is unbiased and consistent, and a simple t-test based on OLS results is given by where Var(bff OLS )= y 0 1 y 1 t OLS = bff OLS 1 q Var(bff OLS ) 1 NX i=1 1 yi; 1 0 e ie 0 i y i; 1 y y ; (8) with e i = y i y i; 1 bff OLS, y i = (y i;:::;y it ) 0, y i; 1 = (y i1; :::; y it 1 ) 0, and y 1 = (y 0 1 ;y0 ; :::; y0 N) 0. Under the null, t OLS has an asymptotic standard normal distribution. Under the alternative, ff < 1, the OLS estimator is biased upwards, more so when the variance of i is large, and the power of the test will therefore depend on the magnitude of ff. Under covariance stationarity the probability limit of the OLS estimator is given by 4. Differencing ff plim N!1 bff OLS = ff +(1 ff) ff + : ff v 1 ff In response to this sensitivity toff of the simple test based on the OLS estimator in the levels equation, Breitung and Meyer (1994) propose a modified 11

12 Dickey-Fuller statistic, based on the OLS estimator for ff in the transformed model y it y i1 = ff (y it 1 y i1)+" it ; t =3; :::; T; (9) where " it = v it (1 ff)(y i1 i ). Clearly, the OLS estimator in this model is unbiased when ff =1in which case p N (bffd1ols 1)! N 0;ff D1 whith ff D1 = P T 1 j= (T j) 1 when the vit are homoskedastic, and the simple t-test is valid under the null of a unit root. Again this test is robust to general forms of heteroskedasticity when using robust standard errors similar to (8). When ff<1 the OLS estimator is again upwards biased, however the asymptotic bias is not a function of ff when the process is mean stationary and the power of the test is therefore not affected by the magnitude of ff in that case. Under covariance stationarity the probability limit of the OLS estimator in (9) is given by plim N!1 bff D1OLS = ff +1 : Consider next the simple model in first differences y it y it 1 = ff (y it 1 y it )+(v it v it 1 ) ; t =3;:::;T; (10) Under the null of a random walk, the probability limit of the OLS estimator in model (10) is given by plim N!1 bff DOLS =1+ P Ni=1 P T t=3 v it 1 (v it v it 1 ) P Ni=1 P T t=3 v it 1 =0; irrespective of heteroskedasticity of the v it. Therefore, when ff =1, p N (bffdols )! N 0;ff D 1

13 with ff D = (T ) 1 if the v it are homoskedastic. In general, the variance can again be easily estimated allowing for general heteroskedasticity. When ff<1 the bias of the estimator is again independent of ff when the process is mean stationary. Under covariance stationarity the probability limit of the OLS estimator in (10) is given by plim N!1 bff DOLS = ff 1 : Therefore, the probability limit of the bias corrected" (under the nul) estimator bff DOLS + 1 is equal to (ff +1)=, i.e. the same as the probability limit of the OLS estimator in model (9). 4.3 Within Groups Harris and Tzavalis (1999) base a test for the unit root hypothesis on a bias correction of the within groups estimator under the null. Under the assumptions that v it ο iid N(0;ff v) and the y i1 are fixed observable constants, which implies that y i1 is uncorrelated with the sequence fv it g, Harris and Tzavalis (1999) show that, under the null of a unit root in model (3), p N (bffwg 1 B)! N (0;C) ; where bff WG is the within groups estimator of ff, and B and C are given by 3 B = 3 T ; C = 3 17 (T 1) 0 (T 1) + 17 A simple test then is (bff WG 1 B) = C=N, which has an asymptotic standard normal distribution under the null. 5T 3 (T ) q 3 Note that these expressions differ from those in Harris and Tzavalis (1999, p.07) due to the fact that our first observation is y i1, not y i0. Therefore, their panel length T is replaced by T 1 in our case. 13 :

14 As the bias correction and derived variance are valid only under homoscedasticity, it is likely that the test performance will be poor under certain forms of heteroscedasticity. Kruiniger and Tzavalis (00) extend the test to allow for general forms of heteroscedasticity and certain types of serial correlation. 4.4 Maximum Likelihood An estimator that is consistent both under the null of ff = 1 and the alternative, ff < 1, is the ML estimator of a model in first differences when the process is covariance stationary under the alternative. The likelihood of the first differenced model can be formulated in many different ways, see for example Arellano (003), Kruiniger (00a) and Hsiao, Pesaran and Tahmiscioglu (00) and is the likelihood of the original levels model conditional on the ML estimates of the fixed effects. In the following we adopt the parameterisation of Hsiao et al. (00). The log-likelihood for the model in first differences under normality isgiven by 4 ln L = NT ln (ß) N ln jωj 1 NX i=1 v Λ0 i Ω 1 v Λ i ; where v Λ i =[ y i; y i3 ff y i; :::; y it ff y i;t 1 ] and Ω=ff v 6 4 with! = Var( y i) =ff v.! 1 0 ::: = ff vω Λ Under covariance stationarity,! = = (1 + ff). When! is restricted in this way, the ML estimator also estimates ff = 1 4 We do not include a possible constant to be estimated for the initial condition, which is assumed to be zero throughout. Inclusion of the constant does not alter any results. In the Appendix a constant is added for completeness. 14

15 consistently, and a simple t-test is valid, even though the parameter is on the boundary of the parameter space (see Hsiao et al. (00) and Kruiniger 00a). When covariance stationarity is not imposed on the model, but mean stationarity ismaintained y i1 = i + " i ; with a general variance of " i, ff ", it then follows that Var( y i) = ff v + (1 ff) ff ", or! = 1+(1 ff) ff " =ff v 1. Hsiao et al. (00) do not impose this restriction and do not try to estimate (1 ff) ff " as a separate parameter but simple treat! as an unknown parameter to be estimated. For Ω to be positive definite,! has to be larger than (T 1) =T, which is satisfied by the ML estimator. So although ff =1result in! =1,which is on the boundary of the parameter space, this is not immediate from the ML estimation procedure. However, as shown in the Appendix, the information matrix is singular at ff = 1, and so this ML estimator will provide a poor basis for testing the null of a unit root. 5 Setting! = 1 will result in an ML estimator that is consistent under the null, but biased under the alternative. The ML estimator for ff is given by bff = ψ X N! 1 w 0 i (Ω Λ ) 1 X N w i i=1 w 0 i (Ω Λ ) 1 y i i=1 where w 0 i = 0 y 0 i; 1. It is easily seen that this estimator is numerically identical to the OLS estimator in the long differenced model (9). This follows 5 Equivalently, the information matrix of the conditonal ML estimator as proposed by Lancaster (00) based on an orthogonal transformation of teh fixed effects is also singular at ff = 1 in this case because there are no fixed effects, i.e. no individual drifts when ff =1. 15

16 because where 6 4 y i y i1 y i3 y i1. y it y i1 P = and P 0 P =(Ω Λ ) 1 when! = = P 5 Monte Carlo Results 6 4 y i y i1 y i3 y i. y it y it In this section we present the results of an extensive Monte Carlo study, investigating the properties of the various estimators and test statistics as described in the previous sections. The general data generating process is y it = ffy it 1 +(1 ff) i + v it ; with i ο N 0;ff, and, initially, vit ο N (0; 1). 1. To investigate the size behaviour of the various tests, and the behaviour of the various estimators, we consider unit root processes for which y i0 =0, i.e. there is no pre-sample history, and processes that started up 50 periods ago.. To both investigate the behaviour of the estimators under the alternative, and to determine the power of the various test statistics, we consider processes with ff < 1 with (covariance) stationary initial conditions, ff y i0 v ο N i ; 1 ff. We also study the case when the vit are heteroscedastic. In all cases, N = 00, T =6. 16

17 5.1 Unit Root Table 1 presents the estimation results for the various estimators when the process is a random walk, for the case of no pre-sample history, the column is labeled y0 = 0, and for the case of 50 pre-sample periods, the column is labeled y 50 = 0. The estimators considered are the simple OLS estimator in the levels equation (3), denoted OLS in the table; the conditional ML estimator under covariance stationarity, denoted MLCS; the OLS estimator in the transformed model (9) as proposed by Breitung and Meyer (1994), which is equivalent to an ML estimator that sets! = 1, denoted D1OLS; the OLS estimator in the first differenced model, denoted DOLS; the simple within groups estimator; and the two-step GMM DIF and SYS estimators. As expected, the first differenced and within groups estimators are downward biased. As the correction factors are 1 for the first differenced model and 0:5 for within groups in this case, the bias corrected versions of these estimators are virtually unbiased. The GMM DIF estimator is very poorly behaved, as this estimator is not identified. The OLS, D1OLS and SYS estimator are virtually unbiased, as expected. 17

18 Table 1. Estimation Results, ff = 1 y0 =0 y 50 =0 OLS Mean St Dev MLCS Mean St Dev D1OLS Mean St Dev DOLS Mean St Dev Within-Groups Mean St Dev GMM DIF Mean St Dev GMM SYS Mean St Dev Notes: based on 10,000 replications. N = 00, T = 6. Table presents the size properties for the various tests for unit roots and the overidentification test of Arellano, Hansen and Sentana (1999). For the two-step GMM estimators the estimated variances have been corrected using the small sample adjustment as developed in Windmeijer (000). The tests based on OLS, MLCS, D1OLS and HT tests have the correct size of 0:05 for both processes. The t-test based on the GMM DIF estimation results has of course very poor size properties. When y0 = 0 the GMM SYS t-test is slightly undersized, which gets worse when there are 50 pre-sample periods, as the parameter gets less well identified. The underidentification test statistic, denoted UI - DIF and UI - LEV, clearly shows that the DIF 18

19 estimator is not identified. It also confirms that the LEV (and therefore SYS) estimator is identified when y0 = 0 whereas it is less so when there are 50 pre-sample periods. Table. Test Results, H0 : ff =1 H1 : ff<1, size =0:05 y0 =0 y 50 =0 OLS MLCS D1OLS DOLS WG HT WG KT GMM SYS UI -DIF UI - LEV Notes: based on 10,000 replications, N = 00, T =6 All tests are one-sided tests at the 5% level. All tests based on GMM estimates use two-step estimates and corrected two-step standard errors. In UI tests, the null is underidentification. 5. Stationarity, ff < 1 Tables 3 and 4 present estimation and test results respectively for the alternative of a covariance stationary process. Values for ff considered are 0:90, 0:95 and 0:98. The two values of ff that are considered are 1 and 19

20 100 respectively. It is clear that the OLS and BM estimators are upward biased, whereas the Within Groups and GMM DIF estimators are downward biased.when ff = 1 the upward bias of the OLS estimator is small. The GMM SYS estimator displays a small bias, which is negative when ff =1. Table 3. Estimation Results, covariance stationary initial conditions ff =0:90 ff =0:95 ff =0:98 ff =1 ff =10 ff =1 ff =10 ff =1 ff =10 OLS Mean St Dev MLCS Mean St Dev D1OLS Mean St Dev DOLS Mean St Dev WG Mean St Dev GMM DIF Mean St Dev GMM SYS Mean St Dev Notes: based on 10,000 replications. N = 00, T = 6. Table 4 shows the rejection frequencies for the tests of a unit root. As expected, the simple OLS t-test performs very well when ff =1. Its power is considerably smaller when ff = 10. It is clear that the power of the 0

21 MLCS, D1OLS, DOLS and HT/KT tests are not affected by increases in ff. However, the power of these tests are quite low for higher values of ff, and the value of ff has to be very large for the standard OLS t-test to have lower power. 6 The power of the MLCS and D1OLS tests are almost identical, the bias of the D1OLS estimator offset by a smaller variance of the estimator as compared to MLCS. The t-test based on DOLS estimation results has smaller power than the D1OLS (and MLCS) tests, due to a larger variance. The power of the HG test is inbetween those of the D1OLS/MLCS and DOLS tests. Allowing for general heteroskedasticity, the KT test, reduces the power considerably. The GMM SYS t-test also has very low power at high values of ff. The UI-DIF finds that the model is identified less frequently than the OLS t-test rejects a unit root, especially when ff = 10. Also in that case, the underidentification test for the levels moment conditions almost never rejects the null of underidentification. 6 For example, for ff =0:95 and ff = 0 the rejection frequency for the OLS one-sided t-test is 0.086, for ff =0:98 and ff = 50 it is

22 Table 4. Test Results, stationary model, H0 : ff =1,H1 : ff<1 ff =0:90 ff =0:95 ff =0:98 ff =1 ff =10 ff =1 ff =10 ff =1 ff =10 OLS MLCS D1OLS DOLS WG HT WG KT GMM SYS UI - DIF UI - LEV Notes: based on 10,000 replications. N = 00, T = 6. All tests are one-sided tests at the 5% level. All tests based on GMM estimates use two-step estimates and corrected two-step standard errors. In UI tests, the null is underidentification. 5.3 Heteroskedasticity The DGP for the heteroscedasticity over time case is given by y it = ffy it 1 +(1 ff) i + v it ; t =; ::; T ff vt = (0:5+t=T ) and y i1 = i + " i ; ff " =1; ifff < 1 y i0 = 0;ifff=1:

23 Tables 5 and 6 present estimation and test results for the unit root process with heteroskedastic errors, whereas Tables 7 and 8 present the results for stationary processes. It is clear that the presence of heteroscedastitcity does not affect the estimation and most of the test results when there is a unit root. The only test for which the size is now very poor when there is heteroskedasticity over time is the HT test. It is clear that the KT test corrects this and has a good size performance. Table 5. Estimation Results, ff = 1, heteroskedastic errors y0 =0 OLS Mean St Dev 0.04 MLCS Mean St Dev D1OLS Mean St Dev DOLS Mean St Dev WG Mean St Dev GMM DIF Mean St Dev GMM SYS Mean St Dev Notes: based on 10,000 replications. N = 00, T = 6. 3

24 Table 6. Test Results, ff =1, heteroskedastic errors y0 =0 OLS MLCS D1OLS DOLS WG HT WG KT GMM SYS UI - DIF UI - LEV Notes: based on 5,000 replications. N = 00, T = 6. 4

25 Table 7. Estimation Results, heteroskedastic errors ff =0:90 ff =0:95 ff =0:98 ff =1 ff =10 ff =1 ff =10 ff =1 ff =10 OLS Mean St Dev MLCS Mean St Dev D1OLS Mean St Dev DOLS Mean St Dev WG Mean St Dev GMM DIF Mean St Dev GMM SYS Mean St Dev Notes: based on 10,000 replications. N = 00, T = 6. 5

26 Table 8. Test Results, stationary model, H0 : ff =1,H1 : ff<1 ff =0:90 ff =0:95 ff =0:98 ff =1 ff =10 ff =1 ff =10 ff =1 ff =10 OLS MLCS D1OLS DOLS WG KT GMM SYS UI - DIF UI - LEV Notes: based on 10,000 replications. N = 00, T = 6. All tests are one-sided tests at the 5% level. All tests based on GMM estimates use two-step estimates and corrected two-step standard errors. In UI tests, the null is underidentification. 6 Conclusions We have compared the finite sample behaviour of various unit root tests for micro panels where the number of individuals is typically large, but the number of time periods is often very small. As in this case some econometric estimators do not identify the parameters of interest when the processes are random walks, it is important to test for unit roots/identification. We find that a t-test based on OLS estimation results provides a simple robust test with high power for cases when the variance of the unobserved heterogeneity is relatively small. 7 7 See also Hall and Mairesse (001). The underidentification test as proposed by Arellano, 6

27 Hansen and Sentana (1999) for the GMM-DIF model, which isequivalent to a unit root test using the Anderson-Rubin statistic performs also quite well in thses cases. The test proposed by Breitung and Meyer (1994) does not loose power with increasing variance, but its power can be quite low when the process is covariance stationary under the alternative. The properties of this test are very similar to that of the t-test based on the conditional maximum likelihood estimation results under covariance stationarity. The test by Harris and Tzavalis (1999) is shown to be very sensitive to the underlying data features. When there is heteroskedasticity over time, its size properties are severely distorted. The Kruiniger-Tzavalis method of dealing with heteroskedasticity for the bias corrected within-groups estimator improves the size properties considerably at the cost of having considerably less power. The model is y i1 = c + i + " i y it = ffy i;t 1 +(1 ff) i + v it For the ML estimator as presented by Pesaran et al. (00) the loglikelihood for the model in differences is given by ln L = NT ln (ß) N ln jωj 1 NX i=1 v Λ0 i Ω 1 v Λ i ; where v Λ i = [ y i c Λ ; y i3 ff y i; :::; y it ff y i;t 1 ], c Λ = (ff 1) c, and Ω=ff v 6 4! 1 0 ::: = ff vω Λ with! = Var( y i) =ff v. The first-order derivatives of the likelihood func- 7

28 tions result in ψ bc Λ b = bff where and bff v = 1 NT b! = Ψ= 6 4! = ψ N X i=1 W 0 i bω Λ 1 Wi! 1 N X i=1 NX yi W i b 0 bω Λ 1 yi W i b i=1 (T 1) + T W 0 bω Λ 1 i yi 1 NX» yi bff v NT W i b 0 Ψ yi W i b ; i=1 3 W i = y i1 0 y i.. 0 y i;t 1 T T (T 1) T (T ) ::: T T (T 1) (T 1) (T 1) (T ) ::: (T 1)... :::. T (T 1) (T ) ::: 1 The second-derivatives of the log-likelihood function are given 0 = 1 ff @ ln (ffv) = NT ff 4 v NX i=1 NX W 0 i (Ω Λ ) 1 W i = 1 ff 4 v i=1 1 NX = ff v [1 + T (! 1)] i=1 1 ff 6 v W 0 i (Ω Λ ) 1 ( y i W i ) ln = 1 ff 4 v [1 + T (! ln = 7 5 W 0 i Ψ( y i W i ) ( y i W i ) 0 (Ω Λ ) 1 ( y i W i ) i=1 NX NT [1+T (! 1)] 1 ff v [1 + T (! 1)] : ( y i W i ) 0 Ψ( y i W i ) i=1 NX ( y i W i ) 0 Ψ( y i W i ) i=1 When ff =1,and therefore! =1and c Λ =0,the information matrix is singular. As in this case 8

29 " T ff v E ln L 0 = T(T 1) 0 E " # ln L = 0 v E " # ln L 0 = T(T E ln L (ffv) = T ff 4 v E ln L = T ff v E 1 ln = T # the information matrix is given by I ; ff v ;! = 6 4 which is clearly singular. References T=ff v T (T 1) = 0 T (T 1) = 0 0 T=ff 4 v T=ff v 0 T (T 1) = T=ff v T = [1] Arellano, M., 003, Panel Data Econometrics, Oxford University Press. [] Arellano, M. and S. Bond, 1991, Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations, Review of Economic Studies 58, [3] Arellano, M., L. Hansen and E. Sentana, 1999, Underidentification?, mimeo, CEMFI, Madrid. [4] Baltagi, B.H. and C. Kao, 000, Nonstationary Panels, Cointegration in Panels and Dynamic Panels: A Survey, in B.H. Baltagi (ed.) Nonsta

30 tionary Panels, Panel Cointegration, and Dynamic Panels, Advances in Econometrics 15, JAI Press, Elsevier Science, Amsterdam. [5] Binder, M., C.Hsiao and M.H. Pesaran, 000, Estimation and Inference in Short Panel Vector Autoregressions with Unit Roots and Cointegration, University of Cambridge DAE Working Paper No.0003 [6] Blundell, R. and S. Bond, 1998, Initial Conditions and Moment Restrictions in Dynamic Panel Data Models, Journal of Econometrics, 87, [7] Breitung, J. and W. Meyer, 1994, Testing for Unit Roots in Panel Data: Are Wages on Different Bargaining Levels Cointegrated? Applied Economics, 6, [8] Hall, B.H. and J. Mairesse, 001, Testing for Unit Roots in Panel Data: An Exploration Using Real and Simulated Data, mimeo, UC Berkeley. [9] Hansen, L.P., 198, Large Sample Properties of Generalized Method of Moments Estimators, Econometrica, 50, [10] Harris, R.D.F. and E. Tzavalis, 1999, Inference for Unit Roots in Dynamic Panels where the Time Dimension is Fixed, Journal of Econometrics, 91, [11] Holtz-Eakin, D., W.K. Newey and H. Rosen, 1988, Estimating Vector Auto-regressions with Panel Data, Econometrica, 56, [1] Kruiniger, H., 00a, Maximum Likelihood Estimation of Dynamic Linear Panel Data Models with Fixed Effects, Working Paper, Queen Mary, University of London. 30

31 [13] Kruiniger, H., 00b, On the Estimation of Panel Regression Models with Fixed Effects, Working Paper, Queen Mary, University of London. [14] Kruiniger, H. and E. Tzavalis, 00, Testing for Unit Roots in Short Dynamic Panels with Serially Correlated and heteroskedastic Disturbance Terms, Working Paper, Queen Mary, University of London. [15] Nickell, S.J., 1981, Biases in Dynamic Models with Fixed Effects, Econometrica, 49, [16] Hsiao, C., M.H. Pesaran and A.K. Tahmiscioglu, 00, Maximum Likelihood Estimation of Fixed Effects Dynamic Panel Data Models Covering Short Time Periods, Journal of Econometrics 109, [17] Staiger, D. and J.H. Stock, 1997, Instrumental Variables Regression with Weak Instruments, Econometrica, 65, [18] Windmeijer, F., (000), A Finite Sample Correction for the Variance of Linear Two-Step GMM Estimators, IFS working paper no. W00/19. 31