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Nonstationary Time Series Data and Cointegration ECON 6002 Econometrics Memorial University of Newfoundland Adapted from Vera Tabakova’s notes

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12.1 Stationary and Nonstationary Variables 12.2 Spurious Regressions 12.3 Unit Root Tests for Stationarity 12.4 Cointegration 12.5 Regression When There is No Cointegration

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Figure 12.1(a) US economic time series Yt-Y t-1 On the right hand side “Differenced series” Fluctuates about a rising trend Fluctuates about a zero mean

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Figure 12.1(b) US economic time series Yt-Y t-1 On the right hand side “Differenced series”

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Stationary if:

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Each realization of the process has a proportion rho of the previous one plus an error drawn from a distribution with mean zero and variance sigma squared It can be generalised to a higher autocorrelation order We just show AR(1)

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We can show that the constant mean of this series is zero

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We can also allow for a non-zero mean, by replacing yt with yt-mu Which boils down to (using alpha = mu(1-rho))

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Or we can allow for a AR(1) with a fluctuation around a linear trend (mu+delta times t) The “de-trended” model, which is now stationary, behaves like an autoregressive model: With alpha =(mu(1-rho)+rho times delta) And lambda = delta(1-rho)

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Figure 12.2 (a) Time Series Models

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Figure 12.2 (b) Time Series Models

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Figure 12.2 (c) Time Series Models

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The first component is usually just zero, since it is so far in the past that it has a negligible effect now The second one is the stochastic trend

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A random walk is non-stationary, although the mean is constant:

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A random walk with a drift both wanders and trends:

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Both independent and artificially generated, but…

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Figure 12.3 (a) Time Series of Two Random Walk Variables

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Figure 12.3 (b) Scatter Plot of Two Random Walk Variables

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Dickey-Fuller Test 1 (no constant and no trend)

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Easier way to test the hypothesis about rho Remember that the null is a unit root: nonstationarity!

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Dickey-Fuller Test 2 (with constant but no trend)

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Dickey-Fuller Test 3 (with constant and with trend)

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First step: plot the time series of the original observations on the variable. If the series appears to be wandering or fluctuating around a sample average of zero, use Version 1 If the series appears to be wandering or fluctuating around a sample average which is non-zero, use Version 2 If the series appears to be wandering or fluctuating around a linear trend, use Version 3

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An important extension of the Dickey-Fuller test allows for the possibility that the error term is autocorrelated. The unit root tests based on (12.6) and its variants (intercept excluded or trend included) are referred to as augmented Dickey-Fuller tests.

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F = US Federal funds interest rate B = 3-year bonds interest rate

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In STATA: use usa, clear gen date = q(1985q1) + _n - 1 format %tq date tsset date TESTING UNIT ROOTS “BY HAND”: * Augmented Dickey Fuller Regressions regress D.F L1.F L1.D.F regress D.B L1.B L1.D.B

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In STATA: TESTING UNIT ROOTS “BY HAND”: * Augmented Dickey Fuller Regressions regress D.F L1.F L1.D.F regress D.B L1.B L1.D.B

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In STATA: Augmented Dickey Fuller Regressions with built in functions dfuller F, regress lags(1) dfuller B, regress lags(1) Choice of lags if we want to allow For more than a AR(1) order

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In STATA: Augmented Dickey Fuller Regressions with built in functions dfuller F, regress lags(1)

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In STATA: Augmented Dickey Fuller Regressions with built in functions dfuller F, regress lags(1) Alternative: pperron uses Newey-West standard errors to account for serial correlation, whereas the augmented Dickey-Fuller test implemented in dfuller uses additional lags of the first-difference variable. Also consider now using DFGLS (Elliot Rothenberg and Stock, 1996) to counteract problems of lack of power in small samples. It also has in STATA a lag selection procedure based on a sequential t test suggested by Ng and Perron (1995)

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In STATA: Augmented Dickey Fuller Regressions with built in functions dfuller F, regress lags(1) Alternatives: use tests with stationarity as the null KPSS (Kwiatowski, Phillips, Schmidt and Shin. 1992) which also has an automatic bandwidth selection tool or the Leybourne & McCabe test.

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The first difference of the random walk is stationary It is an example of a I(1) series (“integrated of order 1” First-differencing it would turn it into I(0) (stationary) In general, the order of integration is the minimum number of times a series must be differenced to make it stationarity

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So now we reject the Unit root after differencing once: We have a I(1) series

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In STATA: ADF on differences dfuller D.F, noconstant lags(0) dfuller D.B, noconstant lags(0)

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If you have unit roots in the time series in your model, you risk the problem of spurious regressions However, spuriousness will not arise if those series are cointegrated, so that determining whether cointegration exists is also key The series are cointegrated if they follow the same stochastic trend or share an underlying common factor In that case you can find a linear combination of your nonstationary variables that is itself stationary

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You must make sure that you have a balanced (potentially) cointegrating regression, so you want to find out the level of integration of your series (usually they are all I(1)) The coefficients in that linear combination form the cointegrating vector, which should have one of its elements normalized to one, because the cointegrating vector is only defined up to a factor of proportionality The cointegrating vector may include a constant, in order to allow for unequal means of the two series

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The estimator from a cointegrating regression is superconsistent

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Two main approaches can be used to check if there is cointegration: The residual approach The error correction approach

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Two main approaches can be used to check if there is cointegration: The residual approach. The classic Engle-Granger approach, based on testing whether the error of the (potentially) cointengrating regression is itself stationary The error correction approach, which test whether the error correction term is significant

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Not the same as for dfuller, since the residuals are estimated errors no actual Ones (also no constant!) Note: unfortunately STATA dfuller as such will not notice and give you erroneous critical values! They would lead to an overoptimistic conclusion

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Check: These are wrong!

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Now these are right! Using egranger with option regress

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The null and alternative hypotheses in the test for cointegration are:

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Let us consider the simple form of a dynamic model: Here the SR and LR effects are measured respectively by: Rearranging terms, we obtain the usual ECM:

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Where the LR effect will be given by: And is a partial correction term for the extent to which Yt-1 deviated from its Equilibrium value associated with Xt-1

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This representation assumes that any short-run shock to Y that pushes it off the long-run equilibrium growth rate will gradually be corrected, and an equilibrium rate will be restored is the residual of the long-run equilibrium relationship between X and Y and its Coefficient can be seen as the “speed of adjustment”

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Tthis representation assumes that any short-run shock to Y that pushes it off the long-run equilibrium growth rate will gradually be corrected, and an equilibrium rate will be restored Usually So the SR effect is weaker than the LR effect

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If you have cointegration, you can run an Error Correction Model, so you can estimate both the long run and the short run relationship between the relevant variables The integration of the variables suggests that we should not use them in a regression, but rather only their differences. We may obtain inconsistent estimates (the spurious regression problem) However, the fact that they are cointegrated (a weighted average of the variables is stationary, I(0)) means that you can include linear combinations of the variables in regressions of their differences in and Error Correction Model (ECM)

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By having already concluding that the variables are cointegrated, we have implicitly decided that there is a long-run causal relation between them. Then the causality being tested for in a VECM is sometimes called “short-run Granger causality”

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The ECM analysis can show (by the magnitude and significance of the EC terms) that when values of the relevant variables move away from the equilibrium relationship implied by the contegrating vector, there was a strong tendency for the variable(s) to change so that the equilibrium would be restored The ECM analysis under cointegration allows us not to throw away the information on the LR effect behind the relationship

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The EC term will be significant if there is a cointegrating relationship Therefore, you can test the existence of cointegration by looking at the significance of that coefficient

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First Difference Stationary The variable y t is said to be a first difference stationary series. Then we revert to the techniques we saw in Ch. 9

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Manipulating this one you can construct and Error Correction Model to investigate the SR dynamics of the relationship between y and x

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where and

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To summarize: If variables are stationary, or I(1) and cointegrated, we can estimate a regression relationship between the levels of those variables without fear of encountering a spurious regression. Then we can use the lagged residuals from the cointegrating regression in an ECM model This is the best case scenario, since if we had to first-differentiate the variables, we would be throwing away the long-run variation Additionally, the cointegrated regression yields a “superconsistent” estimator in large samples

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To summarize: If the variables are I(1) and not cointegrated, we need to estimate a relationship in first differences, with or without the constant term. If they are trend stationary, we can either de-trend the series first and then perform regression analysis with the stationary (de-trended) variables or, alternatively, estimate a regression relationship that includes a trend variable. The latter alternative is typically applied.

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.

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Slide Principles of Econometrics, 3rd Edition Augmented Dickey-Fuller test Autoregressive process Cointegration Dickey-Fuller tests Mean reversion Order of integration Random walk process Random walk with drift Spurious regressions Stationary and nonstationary Stochastic process Stochastic trend Tau statistic Trend and difference stationary Unit root tests

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Slide Principles of Econometrics, 3rd Edition Kit Baum has really good notes on these topics that can be used to learn also about extra STATA commands to handle the analysis: For example, some of you should look at (quarterly) seasonal unit root analysis (command hegy4 in STATA implements the test suggested by Hylleberg et al. 1990) Panel unit roots would be here Further issues

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Slide Principles of Econometrics, 3rd Edition A host of new tests have been developed to try and overcome the shortcomings of the first Dickey-Fuller ones Alternative: pperron uses Newey-West standard errors to account for serial correlation, whereas the augmented Dickey-Fuller test implemented in dfuller uses additional lags of the first-difference variable. Also consider now using DF-GLS (Elliot Rothenberg and Stock, 1996) to counteract problems of lack of power in small samples. It also has in STATA a lag selection procedure based on a sequential t test suggested by Ng and Perron (1995) that uses a Modified AIC (command dfgls) Further issues: more powerful tests

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Slide Principles of Econometrics, 3rd Edition Some tests use stationarity as the null hypothesis: See kpss which implements the test suggested by Kwiatowski etal. (1992) Kwiatowski, D., Phillips, P. C. B., Schmidt, P. and Shin, Y. (1992). `Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root', Journal of Econometrics, 54, See also Leybourne, S.J. and B.P.M. McCabe. A consistent test for a unit root. Journal of Business and Economic Statistics, 12, 1994, This type of test is complementary to the dfuller-type ones, so if they do not give you consistent results, you might be facing fractional unit roots or long range dependence [See lomodrs in Stata to learn more and Baum et al. 1999] Further issues: reverting the null

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In STATA:

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Slide Principles of Econometrics, 3rd Edition You may want to consider unit root tests that allow for structural Breaks, otherwise with a more basic test you might think you are detecting a unit root, while all you have is a structural break See Perron, Pierre The Great Crash, The Oil Price Shock and the Unit Root Hypothesis. Econometrica, 57, 1361–1401. Perron, P Testing for a unit root in a time series with a changing mean, Journal of Business and Economic Statistics, 8:2, Perron, Pierre Further Evidence on Breaking Trend Functions in Macroeconomic Variables. Journal of Econometrics, 80, 355–385. Perron, P. and T. Vogelsang Nonstationarity and level shifts with an application to purchasing power parity, Journal of Business and Economic Statistics, 10:3, You can also take a look at the literature review in this working paper: Further issues: unit root tests and structural breaks

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Slide Principles of Econometrics, 3rd Edition You may want to consider unit root tests that allow for structural Breaks: Stata has zandrews Zivot, E. & Andrews, W. K. Further Evidence on the Great Crash, the Oil Price Shock, and the Unit-Root Hypothesis Journal of Business and Economic Statistics, 1992, 10, And Cleamo1 Cleamao2 Clemio1 Clemio2 Clemente, J., Montañes, A. and M. Reyes Testing for a unit root in variables with a double change in the mean, Economics Letters, 59, Further issues: unit root tests and structural breaks

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Slide Principles of Econometrics, 3rd Edition Apart from the fact that in your cointegration relationship you must choose one variable to be the regressand (giving it a coefficient of one) When you deal with more than 2 regressors you should consider the Johansen’s method to examine the cointegration relationships This is because when there are more than 2 variables involved, there can be multiple cointegrating relationships!!! In this case, you we exploit the notion of Vector Autoregression (VAR) Models that involve a structural view of the dynamics of several variables The generalization of these VAR techniques in this case resulted in the Vector Error Correction Models (VECM) Further issues

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Slide Principles of Econometrics, 3rd Edition Apart from the fact that in your cointegration relationship you must choose one variable to be the regressand (giving it a coefficient of one) When you deal with more than 2 regressors you should consider the Johansen’s method to examine the cointegration relationships You can use vecrank in Stata to run this test Johansen, S. and K. Juselius Maximum likelihood estimation and inference on cointegration with applications to the demand for money. Oxford Bulletin of Economics and Statistics, 522, 169–210. Further issues

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Slide Principles of Econometrics, 3rd Edition Since: many interesting relations involve relatively short time–series and unit root tests are infamous when applied to a single time series for their low power there may be hope from tests that can be used on short series but available across a cross–section of countries, regions, firms, or industries Further issues: unit root tests for panels

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Slide Principles of Econometrics, 3rd Edition We need to logically extend the unit roots testing machinery for univariate time series to the panel setting We can choose the null We need to consider how stationary (or nonstationary) a panel has to be for us to deem it all stationary (or nonstationary) We can use a logic of pooling the series and finding one indicator or averaging the indicators we find in each series instead We can use the residual approach or the ECM approach Further issues: unit root tests for panels

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Slide Principles of Econometrics, 3rd Edition One key issue with panel unit root tests is that they should try and consider cross-sectional dependence Only the second-generation tests can account for it, the first-generation tests assume cross-sectional independence Further issues: unit root tests for panels

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Slide Principles of Econometrics, 3rd Edition STATA offers: MADFULLER for MADF test, which is an extension of the ADF test (not good for longitudinal panels) The test's null hypothesis should be carefully considered will be violated if even only one of the series in the panel is stationary A rejection should thus not be taken to indicate that each of the series is stationary Sarno, L. and M. Taylor, Real exchange rates under the current float: Unequivocal evidence of mean reversion. Economics Letters 60, 131–137. Taylor, M. and L. Sarno, The behavior of real exchange rates during the post–Bretton Woods period. Journal of 9 International Economics, 46, 281–312. Further issues: unit root tests for panels

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Slide Principles of Econometrics, 3rd Edition STATA offers: Levin Lin Chu (old levinlin now xtunitroot llc) One of the first unit root tests for panel data, originally circulated in working paper form in 1992 and 1993, published, with Chu as a coauthor, in 2002 This model allows for two–way fixed effects and unit–specific time trends This test is a pooled Dickey–Fuller (or ADF) test, potentially with differing lag lengths across the units of the panel Unlike the MADF test, it works with short wide panels Assumes that the autoregressive parameter rho is identical for all cross section units (homogeneous alternatives) Further issues: unit root tests for panels

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Slide Principles of Econometrics, 3rd Edition STATA offers: ipshin now xtunitroot ips The Im–Pesaran–Shin test extends the LLC to allow for heterogeneity in the value of rho (heterogeneous alternatives) Under the null, all series nonstationary; under the alternative, a fraction of the series are assumed to be stationary in contrast to the LLC test, which presumes that all series are stationary under the alternative hypothesis IPS use a group–mean Lagrange multiplier statistic to test the null hypothesis. The ADF regressions (which may be of differing lag lengths) are calculated for each series, and a standardized statistic computed as the average of the LM tests for each equation Im, K., Pesaran, M., and Y. Shin, Testing for unit roots in heterogeneous panels. Mimeo, Department of Applied Economics, University of Cambridge. Further issues: unit root tests for panels

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Slide Principles of Econometrics, 3rd Edition STATA offers: hadrilm now xtunitroot hadri Hadri et al. LM test whose null hypothesis is that all series in the panel are stationary, just like the KPSS test differs from that of Dickey–Fuller style tests in assuming stationarity rather that nonstationarity Hadri, K., Testing for stationarity in heterogeneous panel data. Econometrics Journal, 3, 148–161. Further issues: unit root tests for panels

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Slide Principles of Econometrics, 3rd Edition STATA offers: hadrilm now xtunitroot hadri Hadri et al. LM test whose null hypothesis is that all series in the panel are stationary, just like the KPSS test differs from that of Dickey–Fuller style tests in assuming stationarity rather that nonstationarity Hadri, K., Testing for stationarity in heterogeneous panel data. Econometrics Journal, 3, 148–161. Further issues: unit root tests for panels

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Slide Principles of Econometrics, 3rd Edition STATA offers: nharvey The Nyblom–Harvey Test of Common Stochastic Trends Nyblom, J. and A. Harvey. Tests of common stochastic trends. Econometric Theory, 16, 2000, Nyblom, J. and A. Harvey. Testing against smooth stochastic trends. Journal of Applied Econometrics, 16, 415–429. Nyblom, J. and T. Makelainen, Comparison of tests for the presence of random walk components in a simple linear model. Journal of the American Statistical Association, 78, 856–864. Further issues: unit root tests for panels

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Slide Principles of Econometrics, 3rd Edition STATA offers: Breitung test Breitung, J The local power of some unit root tests for panel data. In Advances in Econometrics, Volume 15: Nonstationary Panels, Panel Cointegration, and Dynamic Panels, ed. B. H. Baltagi, Amsterdam: JAI Press. Breitung, J., and S. Das Panel unit root tests under cross-sectional dependence. Statistica Neerlandica 59: Harris-Tzavalis test Harris, D. and Inder, B. (1994). `ATest of the Null Hypothesis of Cointegration', in Non-Stationary Time Series Analysis and Cointegration, ed. C. Hargreaves, Oxford University Press, New York. Harris, R. D. F. and Tzavalis, E. (1999). `Inference for Unit Roots in Dynamic Panels where the Time Dimension is Fixed', Journal of Econometrics, 91, Fisher-type tests (combining p-values) Maddala, G.S. andWu, S. (1999), A Comparative Study of Unit Root Tests with Panel Data and a new simple test, Oxford Bulletin of Economics and Statistics, 61, Further issues: unit root tests for panels

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Slide Principles of Econometrics, 3rd Edition Other tests Pedroni, P.L., Critical values for cointegration tests in heterogeneous panels with multiple regressors. Oxford Bulletin of Economics and Statistics 61 (4), 653–670. Pedroni, P.L., Panel cointegration; asymptotic and finite sample properties of pooled time series tests with an application to the purchasing power parity hypothesis. Econometric Theory 20 (3), 597–625. Further issues: unit root tests for panels

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Slide Principles of Econometrics, 3rd Edition STATA offers: Pesaran’s pescadf (Lewandowski, 2007) and multipurt -- Running 1st and 2nd generation panel unit root tests for multiple variables and lags To detect cross-sectional dependence: xtcds and xtcd Pesaran, M. Hashem (2004) General Diagnostic Tests for Cross Section Dependence in Panels' IZA Discussion Paper No Sarafidis, V. & De Hoyos, R. E. On Testing for Cross Sectional Dependence in Panel Data Models The Stata Journal, 2006, 6, StataCorp LP, vol. 6(4), pages Further issues: unit root tests for panels

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Slide Principles of Econometrics, 3rd Edition We also need tests for cointegration in panels Both residual-based and ECM-based (see Breitung and Pesaran, 2005, for a review) Most residual-based cointegration tests, both in time series and in panels, require that the long-run parameters for the variables in their levels are equal to the short-run parameters for the variables in their differences The failure to meet this common-factor restriction can lead to a significant loss of power for residual-based cointegration tests Further issues: cointegration tests for panels

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Slide Principles of Econometrics, 3rd Edition STATA offers: xtwest a ECM-based cointegration test Persyn and Westerlund (2008) Westerlund, J Testing for error correction in panel data. Oxford Bulletin of Economics and Statistics 69: 709–748. Further issues: unit root tests for panels, another issue

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Slide Principles of Econometrics, 3rd Edition We need to worry now also about the heterogeneity of panels (e.g. see xtpmg ) Blackburne, E. F. & Frank, M. W. Estimation of Nonstationary Heterogeneous Panels The Stata Journal, 2001, 7, Also, it is also now necessary to consider the possibility of cointegration between the variables across the groups (cross section cointegration) as well as within group cointegration Further issues: unit root tests for panels, more issues

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