Chapter 12

Chapter 12

Nonstationary Time Series Data and Cointegration

ECON 6002Econometrics Memorial University of Newfoundland

Adapted from Vera Tabakova’s notes

Chapter 12: Nonstationary Time Series Data and Cointegration

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

Slide 12-2Principles of Econometrics, 3rd Edition

12.1 Stationary and Nonstationary Variables

Figure 12.1(a) US economic time series Slide 12-3Principles of Econometrics, 3rd Edition

Yt-Y t-1On the right hand side

“Differenced series”

Fluctuates about a rising trend

Fluctuates about a zero mean


Figure 12.1(b) US economic time series Slide 12-4Principles of Econometrics, 3rd Edition

Yt-Y t-1On the right hand side

“Differenced series”



(12.1a)

(12.1b)

(12.1c)

tE y

2var ty

cov , cov ,t t s t t s sy y y y

Stationary if:



12.1.1 The First-Order Autoregressive Model


(12.2a)1 , 1t t ty y v

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)



(12.2a)1 , 1t t ty y v

1 0 1

22 1 2 0 1 2 0 1 2

21 2 0

( )

..... tt t t t

y y v

y y v y v v y v v

y v v v y

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

21 2[ ] [ .....] 0t t t tE y E v v v



(12.2b)

1( ) ( )t t ty y v

1 , 1t t ty y v

We can also allow for a non-zero mean, by replacing yt with yt-mu

Which boils down to (using alpha = mu(1-rho))

( ) / (1 ) 1/ (1 0.7) 3.33tE y



(12.2c)

1( ) ( ( 1)) , 1 t t ty t y t v

1t t ty y t v

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)


Figure 12.2 (a) Time Series Models



Figure 12.2 (b) Time Series Models



Figure 12.2 (c) Time Series Models


12.1.2 Random Walk Models


(12.3a)1t t ty y v

1 0 1

2

2 1 2 0 1 2 01

1 01

( )

ss

t

t t t ss

y y v

y y v y v v y v

y y v y v

The first component is usually just zero, since it is so far in thepast that it has a negligible effect nowThe second one is the stochastictrend


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


0 1 2 0( ) ( ... )t tE y y E v v v y

21 2var( ) var( ... )t t vy v v v t


A random walk with a drift both wanders and trends:


1 0 1

2

2 1 2 0 1 2 01

1 01

( ) 2

ss

t

t t t ss

y y v

y y v y v v y v

y y v t y v

(12.3b)1t t ty y v



0 1 2 3 0( ) ( ... )t tE y t y E v v v v t y

21 2 3var( ) var( ... )t t vy v v v v t

12.2 Spurious Regressions


1 1 1

2 1 2

:

:

t t t

t t t

rw y y v

rw x x v

21 217.818 0.842 , .70

( ) (40.837)t trw rw R

t

Both independent and artificially generated, but…


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



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


12.3 Unit Root Test for Stationarity

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


(12.4)1t t ty y v

(12.5a)

1 1 1

1

1

1

t t t t t

t t t

t t

y y y y v

y y v

y v


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


0 0

1 1

: 1 : 0

: 1 : 0

H H

H H

Easier way to test the hypothesis about rho

Remember that the null is a unit root: nonstationarity!


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


(12.5b)1t t ty y v


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


(12.5c)1t t ty y t v

12.3.4 The Dickey-Fuller Testing Procedure

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 3Slide 12-25Principles of Econometrics, 3rd Edition




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.


(12.6)11

m

t t s t s ts

y y a y v

1 1 2 2 2 3, ,t t t t t ty y y y y y

12.3.5 The Dickey-Fuller Tests: An Example


1 1

1 1

0.178 0.037 0.672

( ) ( 2.090)

0.285 0.056 0.315

( ) ( 1.976)

t t t

t t t

F F F

tau

B B B

tau

F = US Federal funds interest rate

B = 3-year bonds interest rate



In STATA:

use usa, cleargen date = q(1985q1) + _n - 1format %tq datetsset date

TESTING UNIT ROOTS “BY HAND”:* Augmented Dickey Fuller Regressionsregress D.F L1.F L1.D.Fregress D.B L1.B L1.D.B



In STATA:

TESTING UNIT ROOTS “BY HAND”:* Augmented Dickey Fuller Regressionsregress D.F L1.F L1.D.Fregress D.B L1.B L1.D.B

_cons .1778617 .1007511 1.77 0.082 -.0228016 .378525 LD. .6724777 .0853664 7.88 0.000 .5024559 .8424996 L1. -.0370668 .0177327 -2.09 0.040 -.0723847 -.001749 F D.F Coef. Std. Err. t P>|t| [95% Conf. Interval]

Total 17.5433842 78 .224915182 Root MSE = .35436 Adj R-squared = 0.4417 Residual 9.54348876 76 .12557222 R-squared = 0.4560 Model 7.99989546 2 3.99994773 Prob > F = 0.0000 F( 2, 76) = 31.85 Source SS df MS Number of obs = 79

. regress D.F L1.F L1.D.F



In STATA:

Augmented Dickey Fuller Regressions with built in functionsdfuller F, regress lags(1)dfuller B, regress lags(1)

Choice of lags if we want to allow For more than a AR(1) order



In STATA:

Augmented Dickey Fuller Regressions with built in functionsdfuller F, regress lags(1)


MacKinnon approximate p-value for Z(t) = 0.2484 Z(t) -2.090 -3.539 -2.907 -2.588 Statistic Value Value Value Test 1% Critical 5% Critical 10% Critical Interpolated Dickey-Fuller

Augmented Dickey-Fuller test for unit root Number of obs = 79

. dfuller F, regress lags(1)







In STATA:


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)



In STATA:


Alternatives: use tests with stationarity as the nullKPSS (Kwiatowski, Phillips, Schmidt and Shin. 1992) which also has an automatic bandwidth selection tool or the Leybourne & McCabe test .

12.3.6 Order of Integration


1t t t ty y y v

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

12.3.6 Order of Integration


1t t t ty y y v

10.340

( ) ( 4.007)

t tF F

tau

10.679

( ) ( 6.415)

t tB B

tau

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


Slide 12-37

In STATA:

ADF on differencesdfuller D.F, noconstant lags(0)dfuller D.B, noconstant lags(0)





Z(t) -4.007 -2.608 -1.950 -1.610 Statistic Value Value Value Test 1% Critical 5% Critical 10% Critical Interpolated Dickey-Fuller

Dickey-Fuller test for unit root Number of obs = 79

. dfuller D.F, noconstant lags(0)

12.4 Cointegration


(12.7)1ˆ ˆt t te e v

(12.8c)

(12.8b)

(12.8a)ˆ1: t t tCase e y bx

2 1ˆ2 : t t tCase e y b x b

2 1ˆˆ3: t t tCase e y b x b t

12.4 Cointegration

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


12.4 Cointegration

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


12.4 Cointegration

The estimator from a cointegrating regression is superconsistent


12.4 Cointegration

Two main approaches can be used to check if there is cointegration:

The residual approach

The error correction approach


12.4 Cointegration

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


12.4 Cointegration


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

12.4.1 An Example of a Cointegration Test

Slide 12-45

(12.9)2ˆ 1.644 0.832 , 0.881

( ) (8.437) (24.147)t tB F R

t

1 1ˆ ˆ ˆ0.314 0.315( ) ( 4.543)

t t te e etau

Z(t) -4.543 -2.608 -1.950 -1.610 Statistic Value Value Value Test 1% Critical 5% Critical 10% Critical Interpolated Dickey-Fuller


. dfuller ehat, noconstant lags(1)

Check: These are wrong!


Slide 12-46

1 1ˆ ˆ ˆ0.314 0.315( ) ( 4.543)

t t te e etau

Now these are right!

Critical values from MacKinnon (1990, 2010)

Z(t) -4.543 -3.515 -2.898 -2.586 Statistic Value Value Value Test 1% Critical 5% Critical 10% Critical Number of lags = 1 N (test) = 79Augmented Engle-Granger test for cointegration N (1st step) = 81

Replacing variable _egresid.... egranger B F, regress lags(1)

Critical values from MacKinnon (1990, 2010)

Z(t) -4.543 -3.515 -2.898 -2.586 Statistic Value Value Value Test 1% Critical 5% Critical 10% Critical Number of lags = 1 N (test) = 79Augmented Engle-Granger test for cointegration N (1st step) = 81

LD. .3147476 .1021565 3.08 0.003 .1113281 .5181671 L1. -.3143204 .0691906 -4.54 0.000 -.4520965 -.1765443 _egresid D._egresid Coef. Std. Err. t P>|t| [95% Conf. Interval] Engle-Granger test regression

Using egranger with option regress


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


0

1

: the series are not cointegrated residuals are nonstationary

: the series are cointegrated residuals are stationary

H

H

12 Error Correction Models

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:

Slide 12-48


Slide 12-49

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


Slide 12-50

This representation assumes that any short-run shock to Y that pushes it off the long-run equilibrium growth rate will gradually be corrected, andan 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”


Slide 12-51

Tthis representation assumes that any short-run shock to Y that pushes it off the long-run equilibrium growth rate will gradually be corrected, andan equilibrium rate will be restored

Usually

So the SR effect is weaker than the LR effect


Slide 12-52

_cons -.0315012 .0477661 -0.66 0.512 -.1266156 .0636133 D1. .7529687 .1080218 6.97 0.000 .5378699 .9680675 F L1. -.2357887 .0723679 -3.26 0.002 -.3798917 -.0916858 _egresid D.B Coef. Std. Err. t P>|t| [95% Conf. Interval] Engle-Granger 2-step ECM

N (2nd step) = 80Engle-Granger 2-step ECM estimation N (1st step) = 81

Replacing variable _egresid.... egranger B F, ecm


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)Slide 12-53


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”

Slide 12-54


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

Slide 12-55


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

Slide 12-56

12.5 Regression When There Is No Cointegration 12.5.1 First Difference Stationary

The variable yt is said to be a first difference stationary series.

Then we revert to the techniques we saw in Ch. 9


1t t ty y v

1t t t ty y y v

12.5.1 First Difference Stationary


(12.10a)1 0 1 1t t t t ty y x x e

1t t ty y v

t ty v

(12.10b)1 0 1 1t t t t ty y x x e

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

12.5.2 Trend Stationary

where

and


(12.11)

t ty t v

t ty t v

1 0 1 1t t t t ty y x x e

1 0 1 1t t t t ty t y x x e

1 1 2 0 1 1 1 1 2(1 ) ( )

1 1 2 0 1(1 ) ( )


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



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.


12 Summary

.


Keywords


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


Kit Baum has really good notes on these topics that can be used to learn also about extra STATA commands to handle the analysis:

http://fmwww.bc.edu/EC-C/S2003/821/EC821.sect05.nn1.pdf


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


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


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, 91-115.

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, 157-166.

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



In STATA:


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. 1989. The Great Crash, The Oil Price Shock and the Unit Root Hypothesis. Econometrica, 57,1361–1401.Perron, P. 1990. Testing for a unit root in a time series with a changing mean, Journal of Business and Economic Statistics, 8:2, 153-162.Perron, Pierre. 1997. Further Evidence on Breaking Trend Functions in Macroeconomic Variables. Journal ofEconometrics, 80, 355–385.Perron, P. and T. Vogelsang. 1992. Nonstationarity and level shifts with an application to purchasing power parity, Journal of Business and Economic Statistics, 10:3, 301-320.

You can also take a look at the literature review in this working paper:

http://ideas.repec.org/p/wpa/wuwpot/0410002.html

Further issues: unit root tests and structural breaks




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 HypothesisJournal of Business and Economic Statistics, 1992, 10, 251-270

And

Cleamo1 Cleamao2 Clemio1 Clemio2

Clemente, J., Montañes, A. and M. Reyes. 1998. Testing for a unit root in variables with a double change in the mean, Economics Letters, 59, 175-182

Further issues: unit root tests and structural breaks


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


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. 1990. 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


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


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



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



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, 1998. Real exchange rates under the current float: Unequivocal evidence of mean reversion.Economics Letters 60, 131–137.Taylor, M. and L. Sarno, 1998. The behavior of real exchange rates during the post–Bretton Woods period. Journal of9 International Economics, 46, 281–312.



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)



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 laglengths) are calculated for each series, and a standardized statisticcomputed as the average of the LM tests for each equation

Im, K., Pesaran, M., and Y. Shin, 1997. Testing for unit roots in heterogeneous panels. Mimeo, Department of Applied Economics, University of Cambridge.



STATA offers: hadrilm now xtunitroot hadri

Hadri et al. LM test whose null hypothesis is that all series in the panelare stationary, just like the KPSS test differs from that of Dickey–Fuller style tests in assuming stationarity rather that nonstationarity

Hadri, K., 2000. Testing for stationarity in heterogeneous panel data. Econometrics Journal, 3, 148–161.



STATA offers: hadrilm now xtunitroot hadri

Hadri et al. LM test whose null hypothesis is that all series in the panelare stationary, just like the KPSS test differs from that of Dickey–Fuller style tests in assuming stationarity rather that nonstationarity

Hadri, K., 2000. Testing for stationarity in heterogeneous panel data. Econometrics Journal, 3, 148–161.



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, 176-199.Nyblom, J. and A. Harvey. Testing against smooth stochastic trends. Journal of Applied Econometrics, 16, 415–429.Nyblom, J. and T. Makelainen, 1983. Comparison of tests for the presence of random walk components in a simplelinear model. Journal of the American Statistical Association, 78, 856–864.



STATA offers:

Breitung test

Breitung, J. 2000. 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, 161-178. Amsterdam: JAI Press.

Breitung, J., and S. Das. 2005. Panel unit root tests under cross-sectional dependence. Statistica Neerlandica 59: 414-433.

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, 201-226

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, 631-652.



Other tests

Pedroni, P.L., 1999. Critical values for cointegration tests in heterogeneous panels with multiple regressors. Oxford Bulletin of Economics and Statistics 61 (4), 653–670. Pedroni, P.L., 2004.

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.



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. 1240.

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 482-496



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


STATA offers:

xtwest a ECM-based cointegration test Persyn and Westerlund (2008)

Westerlund, J. 2007. 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


• 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 PanelsThe Stata Journal, 2001, 7, 197-208

• 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

Chapter 12

Documents

principles of econometrics

cointegration slide

nonstationary time series

b time series modelsslide

c time series modelsslide

c stationary

b us economic time series

nonstationary variablesfigure