TESTING FOR UNIT ROOTS IN PANELS IN THE PRESENCE OF STRUCTURAL CHANGE WITH AN APPLICATION TO OECD UNEMPLOYMENT

Transcription

1 ESING FOR UNI ROOS IN PANELS IN HE PRESENCE OF SRUCURAL CHANGE WIH AN APPLICAION O OECD UNEMPLOYMEN Christian J. Murray a and David H. Papell b a Department of Economics, University of Houston, Houston, X , USA el: (73) , Fax: (73) , cjmurray@uh.edu b Department of Economics, University of Houston, Houston, X , USA el: (73) , Fax: (73) , dpapell@uh.edu Abbreviated itle: esting for Unit Roots in Panels in the Presence of Structural Change

2 ABSRAC here has been extensive research on testing for unit roots in the presence of structural change and on testing for unit roots in panels. his paper takes a small step towards combining the two research agendas. We propose a unit root test for non-trending data in the presence of a one-time change in the mean for a heterogeneous panel. he date of the break is determined endogenously. We perform simulations to investigate the power of the test, and apply the test to a data set of annual unemployment rates for 7 OECD countries from 955 to 990.

3 I. Introduction he work of Perron (989) has inspired extensive research on testing for unit roots in the presence of structural change. Banerjee, Lumsdaine, and Stock (992), Zivot and Andrews (992), and Perron (997), among many others, develop tests which allow the break to be determined endogenously and Lumsdaine and Papell (997) extend the tests to allow for two breaks. Starting with Levin and Lin (992), much work has also been done on testing for unit roots in panels, including papers by Im, Peseran, and Shin (997), Maddala and Wu (999), and Bowman (999). his paper takes a small step towards combining the two research agendas. We propose a unit root test for non-trending data in the presence of a one-time change in the mean for a heterogeneous panel. he date of the break, which is common across the countries of the panel, is determined endogenously and, in the additive outlier framework, is assumed to occur instantaneously. he speed of mean reversion is also common across countries. he intercepts, coefficients on the break dummy variable, and serial correlation structure, however, are country specific. In the context of testing for a unit root in the presence of structural change, our test is most closely related to the work of Perron and Vogelsang (992). hey develop a test for a unit root in non-trending data in the presence of a one-time change in the mean of a single series, with the date of the change determined endogenously. In the panel unit root context, the most closely related work is Papell (997), who utilizes a feasible generalized least squares (SUR) method which allows for both contemporaneous and heterogeneous serial correlation.

4 Levin and Lin (992) and Bowman (999) show that, in the absence of structural change, panel unit root tests have good power in moderately sized samples of 0 or more countries, even with fairly long persistence. We conduct two power experiments, both involving panels of non-trending, stationary series with a one-time change in the mean. First, using conventional panel unit root tests, we find very low power to reject the unit root null. Second, using tests that incorporate structural change, the power is much improved. We apply the test to a data set of annual unemployment rates for 7 OECD countries from 955 to 990. Using the panel tests in the presence of structural change, we find much stronger rejections of unit roots than can be found with univariate tests that do not incorporate structural change, panel tests that do not incorporate structural change, or univariate tests that do incorporate structural change. II. Panel Unit Root ests in the Presence of Structural Change In this section, we develop panel unit root tests in the presence of structural change. We first discuss conventional Augmented Dickey-Fuller (ADF) unit root tests, panel unit root tests which do not incorporate structural change, and single-equation unit root tests with structural change, and then describe how to combine elements from the latter two tests to construct a panel unit root test with structural change. While our tests are for non-trending data, an extension to trending data would be straightforward. he most common tests for unit roots are Augmented Dickey-Fuller tests. ADF tests for non-trending data involve running the following regression: u t k t + ci ut i + i= = µ + αu ε, () t

5 where u t is the variable of interest. he null hypothesis of a unit root is rejected if the value of the t-statistic for α (in absolute value) is greater than the appropriate critical value. While the critical values are non-standard, they are readily available. here is substantial evidence that the lag truncation parameter k is best selected according to data-dependent methods rather than choosing a fixed k a priori. We follow the method suggested by Campbell and Perron (99), Hall (994), and Ng and Perron (995). Start with an upper bound k max on k. If the t-statistic on the coefficient of the last lag is significant, (using the 0% value of the asymptotic distribution of.645), then k max = k. If it is not significant, then k is lowered by one. his procedure is repeated until the last lag becomes significant. If no lag is significant, then k is chosen to equal zero. Panel unit root tests in the ADF framework for non-trending data with heterogeneous intercepts, which are equivalent to including country-specific dummy variables, involve estimating the following regressions: u jt k j = µ + αu. (2) j jt + c ji u jt i + ε jt i= he subscript j =,...,N indexes the elements of the panel which, for convenience of exposition, we will call countries. While Levin and Lin (992) show that imposing homogeneous intercepts results in substantial increases in power, there is rarely any support for such a restriction in practice. We estimate Equation (2) by feasible generalized least squares (SUR), with the coefficient α equated across countries and the lag length k j set equal to the value chosen by the single equation models described in Equation (). 2 his method accounts for

6 contemporaneous and serial correlation, both of which are often important in practice. 3 In Papell (997), this method is used to investigate purchasing power parity. he critical values for panel unit root tests computed by Levin and Lin (992) do not incorporate serial correlation in the disturbances. While, if the number of observations is large enough, the panel ADF statistic converges to the asymptotic distribution of the panel Dickey-Fuller statistic with no serial correlation, this is a serious problem in samples of the size normally used, especially when the recursive t-statistic method is used to select the lag length. Using Monte Carlo methods, we compute finite sample critical values for our test statistics which account for both serial correlation and cross correlation in the residuals. First, we generate unit root series for panels of 5, 0, 5, and 20 countries with 50, 00, and 200 observations. We then fit autoregressive (AR) models to the first differences of each series, using the Schwarz criterion to choose the optimal model, and then treat the optimal estimated AR models as the true data generating process for the errors of each of the series. For each panel, we construct pseudo samples using the optimal AR models with iid N (0, σ ) where 2 2 σ is the estimated innovation variance of the optimal AR model. 4 We then integrate the AR models to get the data in levels. Our test statistic is the t-statistic on α in Equation (2), with the lag length k j for each series chosen by univariate methods as described above. he critical values for the finite sample distributions, obtained from 0,000 replications, are reported in able. We now discuss univariate tests for a unit root in the presence of structural change for non-trending data, using the methods of Perron and Vogelsang (992). Additive

7 Outlier (AO) models, where the structural change occurs instantaneously, are estimated by the following two equations: 5 u t = µ + δdu + ρ, (3) t t and k k ωidbt i + αρt + ci ρt i + i= 0 i= ρ = ε, (4) t t where ρ t is the estimated residual from Equation (3). 6 B is the break date, DB t = if t = B +, 0 otherwise, and DU t = if t > B, 0 otherwise. 7 Equations (3) and (4) are estimated sequentially for each break year B = k+2,...,-, where is the number of observations. he break date is chosen to minimize the t-statistic for α, and datadependent methods are used to select the lag length k. he null hypothesis of a unit root is rejected if the t-statistic on α is sufficiently large (in absolute value). he finite sample critical values of Perron and Vogelsang (992) can be used to assess the significance of the unit root statistic. We proceed to construct a test for unit roots in panel data in the presence of structural change. With heterogeneous intercepts, the panel AO model is estimated by the following two equations: u jt = µ + δdu + ρ, (5) j jt jt and ρ jt = k kj ω ji DB jt i + αρ jt + c ji ρ jt i + i= 0 i= ε jt, (6)

8 where ρ jt are the residuals from (5), DB jt = if t = B +, 0 otherwise, DU jt = if t > B, 0 otherwise, and j =,...,N indexes the countries. Using the Monte Carlo methods described above, with 2500 replications, we compute finite sample critical values for our test statistic, the t-statistic on α in equation (6). 8 III. Power of Panel Unit Root ests Finite sample critical values for panel unit root tests, which incorporate lag selection, are presented in able. Critical values for panel unit root tests with structural change are presented in able 2. As mentioned earlier, we allow for panels 5, 0, 5, and 20 countries (N), with 50, 00, and 200 observations (). In selecting the lag length, k max is set to 4, 8, and 2 for = 50, 00, and 200 respectively. ables and 2 reveal three properties of panel unit root statistics. An increase in leads to a decrease in the absolute value of the critical value of the unit root statistic, whereas an increase in N increases its absolute value. Also, allowing for structural change increases the absolute value of the panel unit root statistic. We now focus on the power of the t-statistic on α in Equations (3) and (4) and Equations (5) and (6). he range of α (the sum of the AR coefficients) we consider is 0.95, 0.90, and We consider mean shifts, δ, of 0.5 and.0. In the following empirical application, these values correspond to a one-half and full percentage point increase in the unemployment rate. We set the break date in the middle of the sample, i.e. B=/2. 9 ables 3 and 4 present the finite sample power of panel unit root tests without and with structural change, respectively. he AR length is again chosen by the Schwarz criterion. he number of repetitions used for able 3 is 2500, while 000 repetitions are

9 used for able 4. he upper bound on the standard error of rejection frequencies in able 4 is able 3 documents the generally poor power of panel unit root tests which fail to allow for a shift in mean which is indeed present. For the alternative closest to the null, α=0.95 and δ=0.5, power is essentially zero. Holding δ constant, power monotonically increases as α is lowered to 0.90 and 0.80, but it is only for the latter case where we begin to see decent power for a reasonable amount of data. Holding α constant, increasing δ monotonically reduces power. his is consistent with Perron s (989) finding that for a stationary time series, a larger mean shift increases the probability of spuriously finding a unit root. his is problematic in the context of our following empirical example. A value of δ= corresponds to a small (%), permanent change in the mean unemployment rate. Our results suggest that if α is close to but less than one, it is probable that panel unit root tests will incorrectly find that unemployment is integrated, rather than stationary around a one time shift in mean. able 4 demonstrates that allowing for a mean shift greatly increases power relative to able 3. For all values of α and δ considered, the power is at least 50%, and often times 00%, for a panel of at least 0 countries with at least 00 observations. Indeed, for =00, there are only two instances in which the power is less that 50%, and those occur for the smallest panel considered, N=5, and the most persistent value of α, IV. Empirical Example: Unit Roots in Unemployment We use annual series of unemployment for 7 OECD countries from 955 to 990. he source of the data is Layard, Nickell, and Jackman (99). We do not update

10 the data past 990. Unemployment rates rose sharply, especially in Europe, during the early 990s. In Papell, Murray, and Ghiblawi (2000), the single equation methods of Bai and Perron (998) detect considerable evidence of multiple structural changes with unemployment data extended through 997. esting for unit roots in panels with multiple structural changes, however, is well beyond the scope of this paper. Our empirical results, therefore, should be interpreted as an illustration of the techniques rather than as an economic analysis of postwar unemployment. he first step in our investigation is to test for unit roots using methods that do not account for structural change. he objective of this exercise is to provide a benchmark for our later results. We run Augmented Dickey-Fuller (ADF) tests, as in Equation (), for each of the 7 countries in the sample. he results of the ADF tests are reported in able 5. We set k max to 4. Using critical values from MacKinnon (99), we find that the null of a unit root cannot be rejected for any of the series at the 0% level. One possible reason for the failure of the ADF tests to reject the unit root hypothesis is the relatively short (36 years) time span of the data. 0 We investigate this possibility by conducting panel unit root tests, described by Equation (2), to exploit crosssection variability among the 7 unemployment rates. he results of the panel unit root tests are reported in able 6. he null hypothesis of a unit root cannot be rejected, at even the 0% level, either for the OECD countries as a whole or for smaller panels consisting of European (3), European Community (EC) (9), European Free rade Area (EFA) (4), Non-European (4), or Non-EC (EFA plus Non-Europe) (8) countries. 2 he results for the univariate AO model of Equations (3) and (4) are reported in able 7. he null hypothesis of a unit root is rejected for Finland, Ireland and Spain at

11 the % level, Belgium, France, Italy and Norway at the 5% level, and Austria, Canada, Denmark, and the United Kingdom at the 0% level. he structural breaks are all positive, reflecting the general rise in unemployment among the OECD countries. he structural break occurs between 974 and 976 for nine out of eleven countries for which the unit root null can be rejected. he results of the panel unit root tests from Equations (5) and (6) that account for structural change, along with the associated critical values, are reported in able 8. 3 he unit root hypothesis is strongly (at the % level) rejected in favor of stationarity with a one-time break in 975 for the OECD, European, and EC countries and a break in 973 for the non-ec and EFA countries. For the non-europe countries, the unit root null could not be rejected at the 0% level. his panel, however, consists of only four countries. V. Conclusions he purpose of this paper was to develop and implement panel unit root tests in the presence of structural change. o that end, we combine methods from two previously disjoint literatures: testing for a unit root in panels and testing for a unit root in the presence of structural change. he resultant test allows for both serial and contemporaneous correlation, both of which are often found to be important in the panel unit root context. he motivation for the test comes from the hypothesis that conventional panel unit root tests, those that do not incorporate structural change, will have low power if the data are stationary with structural change. While this is well established in the univariate literature, it is only a conjecture in the panel context. We investigate this conjecture by

12 conducting power experiments for panels of non-trending, stationary series with a onetime change in the mean, and find that conventional panel unit root tests generally have very low power. We then conduct the same experiments using methods that test for a unit root in the presence of structural change, and find that the power of the tests is much improved. We apply our test to a data set of annual unemployment rates for 7 OECD countries from 955 to 990. For these countries, unit root tests that do not incorporate structural change, whether univariate or panel, provide no evidence against the unit root null. While univariate tests that incorporate structural change do provide some evidence against unit roots, the short span of the data suggests that power may be problematic. Using our panel test with a one-time structural change, we find very strong evidence of regime-wise stationarity. his evidence is both for the full panel and for a number of smaller sub-panels. Our work could be extended in a number of directions. While the test incorporates a one-time break in non-trending data, extensions to multiple breaks and/or trending data would be straightforward. Once variety in the number of breaks, type of breaks, number of countries, and number of observations are allowed for, the number of possibilities increases rapidly. With the availability of programs for calculating critical values, we suspect that it will be more fruitful to develop tests on a case-by-case basis rather than attempt to achieve generality. 4

13 REFERENCES Abuaf, N. and P. Jorion (990), Purchasing Power Parity in the Long Run. he Journal of Finance, 45, Bai, J., and P. Perron (998), Estimating and esting Linear Models with Multiple Structural Changes. Econometrica, 66, Banerjee, A., R.L. Lumsdaine, and J.H. Stock (992), Recursive and Sequential ests of the Unit Root and rend-break Hypotheses: heory and International Evidence. Journal of Business and Economic Statistics 0, Bowman, D. (999), Efficients ests for Autoregressive Unit Roots in Panel Data. IFDP #646, Board of Governors of the Federal Reserve System. Breuer, J., R. McNown, and M. Wallace (2000), he Quest for Purchasing Power Parity with a Series-Specific est using Panel Data. working paper, University of South Carolina. Campbell, J.Y. and P. Perron (99), Pitfalls and Opportunities: What Macroeconomists Should Know About Unit Roots. In O.J. Blanchard and S. Fischer (eds.), NBER Macroeconomic Annual, 4-20, MI Press: Cambridge. Froot, K.A., and K. Rogoff (995), Perspectives on PPP and Long-Run Real Exchange Rates. In G. Grossman and K. Rogoff (eds.), Handbook of International Economics, Vol. 3, Amsterdam: North Holland, Hall, A.R. (994), esting for a Unit Root in ime Series with Pretest Data-Based Model Selection. Journal of Business and Economic Statistics, 2, Im, S., H. Pesaran, and Y. Shin (997), esting for Unit Roots in Heterogenous Panels. working paper, University of Cambridge. Layard, R., S. Nickell, and R. Jackman (99), Unemployment: Macroeconomic Performance and he Labour Market. Oxford University Press: Oxford. Levin, A. and C.-F. Lin (992), Unit Root ests in Panel Data: Asymptotic and Finite-Sample Properties. University of California-San Diego Discussion Paper Lumsdaine, R.L. and D.H. Papell (997), Multiple rend Breaks and the Unit Root Hypothesis. he Review of Economics and Statistics, 79,

14 Maddala, G.S., and S. Wu (999), A Comparative Study of Unit Root ests with Panel Data and a New Simple est. Oxford Bulletin of Economics and Statistics, 6, MacKinnon, J.G. (99), Critical Values for Cointegration ests, In R.F. Engle and C.W.J. Granger (eds.), Readings in Cointegration, Oxford University Press: Oxford. Ng, S. and P. Perron (995), Unit Root est in ARMA Models with Data Dependent Methods for the Selection of the runcation Lag. Journal of the American Statistical Association, 90, O Connell, P.G.J. (998), he Overvaluation of Purchasing Power Parity. Journal of International Economics, 44, -20. Papell, D.H. (997), Searching for Stationarity: Purchasing Power Parity under the Current Float. Journal of International Economics, 43, Papell, D.H. (2000), he Great Appreciation, the Great Depreciation, and the Purchasing Power Parity Hypothesis. working paper, University of Houston. Papell, D.H., C.J. Murray, and H. Ghiblawi (2000), he Structure of Unemployment. he Review of Economics and Statistics, 82. Perron, P. (989), he Great Crash, the Oil Price Shock, and the Unit Root Hypothesis. Econometrica, 57, Perron, P. (997), Further Evidence on Breaking rend Functions in Macroeconomic Variables. Journal of Econometrics, 80, Perron, P., and.j. Vogelsang (992), Nonstationarity and Level Shifts with an Application to Purchasing Power Parity. Journal of Business and Economic Statistics, 0, Zivot, E., and D.W.K. Andrews (992), Further Evidence on the Great Crash, the Oil- Price Shock, and the Unit Root Hypothesis. Journal of Business and Economic Statistics, 0,

15 able. Finite Sample Critical Values for Panel Unit Root ests without Structural Change % N % N % N

16 able 2. Finite Sample Critical Values for Panel Unit Root ests with Structural Change % N % N % N

17 able 3. Power of Panel Unit Root ests without Structural Change α=0.95, δ=0.5 α=0.95, δ= N N α=0.90, δ=0.5 α=0.90, δ= N N α=0.80, δ=0.5 α=0.80, δ= N N

18 able 4. Power of Panel Unit Root ests with Structural Change α=0.95, δ=0.5 α=0.95, δ= N N α=0.90, δ=0.5 α=0.90, δ= N N α=0.80, δ=0.5 α=0.80, δ= N N

19 able 5. Augmented Dickey-Fuller ests Country µ α k Australia Austria Belgium Canada Denmark Finland France Germany Ireland Italy Japan Netherlands Norway Spain Sweden U.K. U.S.A (.60) 0.88 (.26) (.48) 0.89 (.6) (0.82) (.42) 0.76 (.38) (.9) (.36) (2.04) 0.20 (.9) (.2) (.0) (.85) 0.43 (.82) 0.39 (.38).389 (2.4) (-.5) 0.95 (-.28) (-.40) (-.46) ( -0.4) 0.92 ( -.26) (-0.54) (-.32) (-.28) (-2.08) (-2.04) (-0.96) (-0.84) (-2.25 ) (-.37) (-.4) (-2.6) Note: he critical values for the ADF test, calculated from MacKinnon (99) with 36 observations, are ( percent), (5 percent), and -2.6 (0 percent). Numbers in parentheses are t-statistics

20 able 6. Panel Unit Root ests Group Ν α t α OECD EUROPE EC NON-EC EFA NON-EUROPE Critical Values Group % 5% 0% OECD EUROPE EC NON-EC EFA NON-EUROPE

21 able 7. he Additive Outlier Model Country Break Year µ δ α K Australia Austria Belgium Canada Denmark Finland France Germany Ireland Italy Japan Netherlands Norway Spain Sweden U.K. U.S.A (6.99).704 (3.55) 2.77 (8.70) 5.45 (7.95) (8.29).95 (8.6) (6.35).47 (3.63) (0.4) (6.43).653 (2.9).945 (4.94) (6.96) (2.57).470 (0.40) 2.75 (6.4) (9.2) (0.6).460 (6.42) (3.99) (8.7) (.93) (8.65) 5.94 (.8) 3.37 (6.0) (8.9).907 (4.20) (2.38) (0.55).78 (4.9).463 (8.20) (2.0) (8.82) 2.4 (5.67) (-3.99) (-4.33) c (-4.96) b (-4.33) c 0.53 (-4.34) c (-6.64) a (-4.95) b (-3.63) (-7.58) a (-4.75) b (-3.53) (-4.06) (-4.78) b (-7.6) a (-3.87) (-4.60) c 0.25 (-4.0) Note: he critical values for the AO model, reported in Perron and Vogelsang (992), are ( percent), (5 percent), and (0 percent). Numbers in parentheses are t-statistics. Superscripts a, b, and c denote rejection of the unit root null at the %, 5%, and 0% significance levels respectively

22 able 8. Panel Unit Root ests with Structural Change Group N Break Year α t α OECD a EUROPE a EC a NON-EC a EFA a NON-EUROPE Critical Values Group % 5% 0% OECD EUROPE EC NON-EC EFA NON-EUROPE Note: Superscripts a, b, and c denote rejection of the unit root null at the %, 5%, and 0% significance levels respectively.

23 MacKinnon (99) shows how to calculate critical values for ADF tests for any sample size. 2 If the coefficient α is not equated across countries, as in Breuer, McNown, and Wallace (2000), the gains in power over univariate methods are much smaller. Im, Peseran, and Shin (997) report higher power without equating α across countries, but their alternative hypothesis is that one member of the panel, rather than all members, are stationary. 3 If there is no serial correlation (k = 0), or if the k s and c s are constrained to be equal across countries, as in O Connell (998), the FGLS estimator can be iterated to achieve maximum likelihood. hese restrictions, however, rarely (if ever) hold in practice. 4 For all of the critical value calculations, we generate 50 more observations than are reported, and then discard the first 50 observations. 5 Innovational outlier models, where the structural change occurs gradually, can also be estimated. 6 As explained by Perron and Vogelsang (992), the dummy variables DB t-i are included to ensure that the t-statistic on α in Equation (4) has the same asymptotic distribution as in the IO model and is invariant to the value of k. 7 he dummy variable DB t is included to allow for a change in the mean under the null. 8 Abuaf and Jorion (990) conduct panel unit root tests which allow for structural change, but the time of the break is assumed to be known a priori. 9 he results in ables 3 and 4 are qualitatively unchanged for B=/4 or 3/4. 0 Froot and Rogoff (995) show that, if a variable follows a stationary AR() process with a half life of three years, it would take 72 years of annual data to reject the unit root null using the 5% Dickey-Fuller critical value. he critical values, also reported in able 6, are calculated for the exact number of countries and observations in each of the panels, using the Monte Carlo methods described above. 2 he members of the EC (included in our data) are Belgium, Denmark, France, Germany, Ireland, Italy, Netherlands, Spain, and the United Kingdom. he EFA countries ate Austria, Finland, Norway, and Sweden. 3 he critical values are calculated for the exact number of countries and observations in each of the panels, using the Monte Carlo methods described above. 4 An example is Papell (2000), who develops a panel unit root test in the presence of three breaks in the slope, but none in the intercept, of the trend function, with further restrictions imposed for consistency with purchasing power parity.