Why not use standard anel unit root test for testing PPP Johan Lyhagen Deartment of Information Science, Usala University Abstract In this aer we show the consequences of alying a anel unit root test that assumes indeendence between the cross-sections when testing for a urchasing ower arity relationshi The distribution of the tests investigated, including the IPS test of Im et al (00), are influenced by a common stochastic trend which is usually not accounted for The result is that the emirical size tends to one with the number of cross-sections Hence, it is of crucial imortance to account for this cross-sectional deendency Financial suort from the Jan Wallander and Tom Hedelius Foundation, research grant number P005-0:, is gratefully acknowledged I thank olf Larsson and anonymous referees for valuable comments Citation: Lyhagen, Johan, (008) "Why not use standard anel unit root test for testing PPP" Economics Bulletin, Vol, o - Submitted: Setember 9, 00 Acceted: May, 008 UL: htt://economicsbulletinvanderbiltedu/008/volume/eb-0c0098adf
Introduction There is a large amount of literature on testing urchasing ower arity (PPP) due to the economic imortance of the relationshi PPP states that in the long run the exchange rate adjusted rice levels in two countries should be the same Otherwise it is ro table to exort/imort goods The most common way to test the PPP relationshi is to aly the standard augmented Dickey-Fuller unit root test (ADF) to the real exchange rate " t = ln (P it ) ln (P jt )+ln ( ijt ) where P it (P jt ) is the rice level in country i (j) and ijt is the exchange rate between country i and country j; all indexed for time eriod t: Shiller and Perron (985) show that the ower of the ADF test is very low for the number of observations encountered in real world data sets Hence, PPP is often rejected, subsequent studies use a anel version with the urose to increase the ower The most commonly used anel unit root tests are the ones of Levin and Lin (99, 99), LL, (later ublished as Levin, Lin and Chu, 00) and Im, Pesaran and Shin (00), IPS Other tests have been roosed, see eg the surveys of Froot and ogo (995), Banerjee (999), Baltagi and Kao (000) and Breitung and Pesaran (00) Therefore testing PPP using anel unit root tests are widely used, however, this aer shows that the inference used in these alications are likely to be wrong, ie the actual size may be very far from the nominal one The base currency used introduces a common stochastic trend which is not accounted for in the distribution of the test statistics Our aer analytically derives some useful exressions which hel to understand the consequences of the common stochastic trend Eg the LL test is shown to diverge with the number of crosssections A Monte Carlo simulation is carried out to analyze the consequences for two anel unit root tests The tests investigated in the Monte Carlo are the IPS and Hadri (000), H, which is the anel unit roots version of the anel cointegration test of McCoskey and Kao (998) The IPS test has a null hyothesis of unit root therefore we also have a look at the H test which has stationarity as null hyothesis Strictly, the H test is a test of stationarity but, for simlicity, through out the aer the term testing for unit roots is used for all tests The result is that for very small anels, =, the size is aroximately correct but for larger anels, 0; the size can be signi cantly distorted, ie the emirical size is much too large It should be noted that nowadays there are aers dealing with cross-sectional deendencies through the error term, see eg Banerjee, Marcellino and Osbat (00, 005) and through common factors, see eg Moon and Perron (00) and Bai and g (00) Further, O Connell (998) recognise the fact that PPP data would be cross-sectional deendent but fail to realize the true nature of this deendency (ie PPP imlies a common stochastic trend and not only cross-sectional deendencies through the error term) The aer is organized as follows: The next section introduces the statistical PPP model and this PPP seci cation is used throughout the aer Section analyzes the consequences for some anel unit root tests The Monte Carlo simulation in Section is used to show how large the consequences can
be Section 5 concludes the aer ull hyothesis of unit roots and the PPP Testing the null hyothesis of a unit root in a univariate series is often based on the Dickey-Fuller tye of equation (or the augmented tye): x t = x t + e t () which under the null hyothesis of a unit root ( = 0) becomes x t = e t () A anel version may be based on, as LL and IPS, x t x t x t x t x t 5 = x t 5 + e t e t e t 5 () where is the number of cross-sections, is the rst di erence oerator and e it are disturbances with nite variance and indeendent and identically distributed over time ote that i might or might not be equal to j ; i = j The anel null hyothesis is = = ::: = = 0: Under the alternative hyothesis, deending on which test used, some or all i are less than zero From () it can be shown that there are random walks in the system under the null hyothesis Let ~ i denote the log of the rice level in country i and r i+ the log of the exchange rate between country i and + : The PPP relationshi states that ~ it +t + r i+t = " it should be a cointegrating relationshi To make the notation simler we let it = ln (P it i+t ) ; ie the rice level in country i is in the currency of country + Further, assume that all i and + are nonstationary There are numerous emirical evidences that rices are nonstationary, see eg Culver and Paell (99) We can also justify it from the same kind of argument that claims that stock rices are nonstationary If there are cointegration between it and +;t the same stochastic trend drives the two variables As a consequence, jt ; i = j; share the same trend, ie all rices are driven by the same stochastic trend If there is no cointegration there are two stochastic trends, one for each rice level In a anel setu the economic model is t t t +t +t +t 5 = " t " t " t 5 ()
where we test simultaneously if " it has a non-stationary behavior through a anel unit root test As we do not estimate the cointegrating relationshi testing for cointegration coincide with testing for unit roots Under the null of no cointegration the rst rices are generated by t e t t e t t 5 = e t and the rice level for country + by +;t = e+;t : () Hence, the variables that are used in the anel unit root test are generated according to " t " t t t +;t +;t e t e t e +;t e +;t " t with covariance matrix 5 = t +;t 5 5 = e t e +;t 5 (5) () E e t e t e +;t e +;t 5 e t e t e +;t e +;t 0 = ( 0 + 0 ) + 0 = : (8) 5 e t e +;t e t e +;t where is the covariance matrix for the rst rice levels and is the variance for the + and is an vector of ones The exectation E [e t ; : : : ; e t ] 0 e +;t is It is imortant to note that each equation of () contains one common tochastic trend besides the not common one Further, for simlicity, we assume that the long run variance is the same Under the alternative of cointegration the data generating rocess is " t t +;t " t 5 = t +;t (9) 5 " t t +;t where +;t is generated according to () ote here that there is only one stochastic trend driving all rices Some notation: The Brownian motion generated by " it = it +;t is denoted B i () = W i () W () + and when B i () is normalized to have unit covariance matrix B i = W i W + : Further,! denotes the limit when T! :
Consequences for some tests of omitted crosssectional deendencies LL The LL test in the version of Levin and Lin (99) is based on the regression, i = ; :::; ; " it = " it + e it (0) where e it is indeendent across i and t; and are identically distributed with mean 0 and variance The null hyothesis is = : The anel estimator roosed for this simle model is P P T t= ^ = " it" it P P T t= " it () with t statistic t = ^ P h P h T P T t= " it T P T t= " it e it i i = () They showed that the t statistic converges to a standard normal distribution under the assumtion of indeendent random walks To see the consequences in the PPP case, the three arts of the t statistic are analyzed First it is obvious that ^ = P " i =T is a consistent estimator of, hence ^=! with T The inner art of the denominator tends, with T; to T TX t= " it = T TX ( it +t ) t= = TX T t=! Wi dr it it +t + +t W i W + dr + W +dr () where W i and W + are Brownian motions with unit variance Here, we have assumed that the long run variance is the same Considering the asymtotics, the limits are X Wi dr! 0:5 () X W i W + dr! 0 (5)
and the trivial X The limit of the numerator is W+dr = W +dr: () T TX t= " it e it = T TX ( it + ) (e it e yt ) t= = TX ( it e it it e yt +t e it + +t e yt ) T t=! W i dw i W i dw + W + dw i + W + dw + It is well known that P Wi dw i! (0; 0:5) : The quantities P Wi dw + () and P W+ dw i both have mean 0 and variance 05 but with more kurtosis than what would be imlied by a normal distribution The art that in uences the statistic the most is the last in equation (), X W + dw + = W + dw + (8) Because W + dw + is either ositive of negative (or zero with robability zero) it tends to or with : This imlies that the t statistic tend to or with This result is consitent with Lemma in Moon and Perron (00) De ning the new test statistic ~t = t (9) would have the asymtotic distribution W+ dw + ~t! 0:5 + W + dr : (0) = IPS With no lags, the asymtotic version of the IPS test is ( t E [t i j t = i = ]) V ar [ t j i = ] When the t statistics are indeendent V ar [t j i = ] = P V ar [t ij i = ] : We know that the exectation of the mean remains the same if the t statistics
are deendent as in the case of PPP A closer look at the individual t reveals statistics t i = ^ P T T t= " it e it = () P T T t= " it! Wi dw i Wi dw + W+ dw i + W + dw + W i dr W i W + dr + W + dr = () The variance of t would be comlicated and we have not found an analytical exression but it is easily seen that the mean is ositive The H test The anel cointegration test of McCoskey and Kao (998) is easily modi ed to become a anel test for unit roots as done in Hadri (000) The LM test statistic is LM = T P P T t= S it s () where S it is the artial sum of the ith variable, S it = tx e ij () and s is a consistent estimator (assuming iid errors) of the long run variance, s = T j= X t= The anel test is the standardized LM test statistic, TX e it (5) (LM LM ) s LM () where LM and s LM are the exectation and the variance of the LM test statistic It can be shown, see eg Shin (99), that the LM test statistic is distributed as the quantity W : As seen from equation () ; in the PPP case the asymtotic distribution would be P B () i LM! s () P W () i dr W () i W () + dr + W () + dr = s (8)
so variance would be a ected If we assume that all rices have the same variance LM! X Wi dr W i W + dr + A Monte Carlo simulation W +dr : (9) To evaluate the consequences of the resence of cross-sectional deendencies of the PPP tye a small Monte Carlo simulation is carried out For simlicity we assume that all variables have the same variance but we change the correlation between the base rice and the other rices, = 0:5; 0:5; 0; 0:5; 0:5 We choose = ; 5; 0; 50; 00; 00 and 00: This will allow us to observe what the sizes converge to and how fast The length of the random walks aroximating the Brownian motions is 800 and the number of relicates is 00000 A 5% nominal size is used throughout The results of the Monte Carlo simulation is comuted in Tables () and () for IPS and the H test resectively The tables show the sizes of the tests when the original test rocedure are used For the IPS test the mean and variance used are asymtotic versions of those resented in Im, Pesaran and Shin (00) After standardization, the IPS test statistic is comared to the Gaussian distribution The test of Levin and Lin (99) is not simulated as the distribution is shown to be divergent Tables () () in here The results from the Monte Carlo simulation are that the emirical size for low values of is hardly a ected however for higher values the emirical size seems much distorted The emirical size is much bigger than the nominal size When the correlation decreases from 05 the e ect on the emirical size becomes further distorted The IPS test erforms badly with a emirical size of over 50% for = 00 and correlation less than 05 For both tests the emirical size seems to slowly tend to one with the number of cross-sections 5 Conclusion In this aer we have shown the consequences to the distribution of some anel unit root test statistics when testing the PPP theory All the tests investigated are in uenced by a large extent In most cases the emirical size becomes much too large, rarely it is not in uenced at all The emirical size usually increases with the number of cross-sections in the anel and when the correlation between the base rice level and the other rice levels decreases in absolute value The simulation shows that the emirical size tends to one for the three test statistics although slowly For ractical uroses the results of this aer have two major imlications when testing the PPP hyothesis Firstly, the emirical size of a anel unit root
test is likely to be far from the nominal One reason for using a anel test is to increase the ower of the test but the increased size makes it di cult to judge if a rejection of the null hyothesis deends on increased ower or to a to large emirical size Secondly, as the distribution heavily deends on the correlation between the stochastic trends, these estimates are not available Therefore in ractice it is very di cult to correct the emirical size when using tests that do not consider the cross-sectional deendencies imlied by PPP It is interesting to note that the results hold irresectively of tests with a null of unit root or for no unit root null eferences Bai, J and g, S (00) A PAIC attack on unit roots and cointegration, Econometrica,- Baltagi, B, and Kao, C (000) onstationary anels, cointegration in anels and dynamic anels: A survey, Advances in Econometrics, 5, -5 Banerjee, A (999) Panel data unit roots and cointegration: An overview, Oxford Bulletin of Economics and Statistics, 0-9 Banerjee, A, Marcellino, M and Osbat, C (00) Some cautions on the use of anel methods for integrated series of macro-economic data, Econometrics Journal,-0 Banerjee, A, Marcellino, M and Osbat, C (005) Testing for PPP: Should we use anel methods, Emirical Economics 0, -9 Breitung, J and Pesaran, MH (00) Unit oots and Cointegration in Panels, in L Matyas, and P Sevestre (Eds), The Econometrics of Panel Data (Third Edition), Kluwer Academic Publishers Culver, S and Paell, D (99) Is there a unit root in the in ation rate? Evidence from sequential break and anel data models, Journal of Alied Econometrics, 5- Froot, KA and ogo, K (995) Persectives on PPP and long-run exchange rates, in G Grossman and K ogo (Eds), Handbook of International Economics, Vol III, orth-holland, Amsterdam Hadri, K (000) Testing for stationarity in heterogeneous anel data, Econometrics Journal, 8- Im, KS, Pesaran, MH and Shin, Y (99) Testing for unit roots in heterogenous anels Journal of Econometrics 5, 5- Levin, A and Lin, CF (99) Unit root tests in anel data: Asymtotic and nite samle roerties, Deartment of Economics, University of California at San Diego, Discussion Paer o 9- Levin, A and Lin, CF (99) Unit root tests in anel data: ew results, Deartment of Economics, University of California at San Diego, Discussion Paer o 9-5 Levin, A Lin, CF and Chu, C-SJ (00) Unit root tests in anel data: Asymtotic and nite samle roerties, Journal of Econometrics08, -
McCoskey, S and Kao, C (998) A residual-based test for the null of cointegration in anel data Econometric eviews, 5-8 Moon, H and Perron, B (00) Testing for a unit root in anels with dynamic factors, Journal of Econometrics, 8- O Connell, P (998) The Overvaluation of Purchasing Power Parity, Journal of International Economics, -9 Phillis, PCB and Moon, H (999) Linear egression Limit Theory for onstationary Panel Data, Econometrica, 05- Shiller and Perron, P (985) Testing the random walk hyothesis: Power versus frequency of observation, Economics Letters 8, 8-8 Shin, Y (99) A residual-based test for cointegration, Econometric Theory 0, 9-5
Tables n 5 0 50 00 00 00 05 00 00 005 000 00899 095 0 05 000 008 005 0 05 09 0599 0 0050 005 0099 0 09 059 0555-05 000 009 089 009 08 059 055-05 00 0 0808 0508 088 05 058 Table : Emirical size of the IPS test when testing for PPP ominal size is 5% n 5 0 50 00 00 00 05 00 00 009 005 00 05 0 05 0050 00 008 0 000 09 05 0 005 00 000 0908 08 0 09-05 000 0088 095 008 0 005 080-05 00 05 058 05 0 080 09 Table : Emirical size of the Hadri (000) LM test when testing for PPP ominal size is 5%