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OXFORD BULLETIN OF ECONOMICS AND STATISTICS,65,5(2003)0305-9049 Cointegration Vector Estimation by Panel
DOLS and Long-run Money Demand*
翻译在线翻译Nelson C.Mark ,à,§and Donggyu Sul–
Ohio State University(e-mail:mark.1@osu.edu)
woolen
àUniversity of Notre Dame
§NBER
少儿有声读物
–University of Auckland(e-mail:d.sul@)
Abstract
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We study the panel dynamic ordinary least square(DOLS)estimator of a homogeneous cointegration vector for a balanced panel of N individuals obrved over T time periods.Allowable heterogeneity across individuals include individual-specific time trends,individual-specificfixed effects and time-specifi
c effects.The estimator is fully parametric,computationally convenient,and more preci than the single equation estimator.Forfixed N as Tfi1,the estimator converges to a function of Brownian motions and the Wald statistic for testing a t of s linear constraints has a limiting v2(s) distribution.The estimator also has a Gaussian quential limit distribution that is obtainedfirst by letting Tfi1and then letting Nfi1.In a ries of Monte-Carlo experiments,wefind that the asymptotic distribution theory provides a reasonably clo approximation to the exactfinite sample distribution.We u panel DOLS to estimate coefficients of the long-run money demand function from a panel of19countries with annual obrvations that span from1957to1996.The estimated income elasticity is1.08(¼0.26)and the estimated interest rate mi-elasticity is)0.02(¼0.01).
*This paper was previously circulated under the title‘AComputationally Simple Cointegration Vector Estimator for Panel Data’.For valuable comments on earlier drafts,we thank Ronald Bewley, Roger Moon,Peter Phillips,minar participants at Georgetown University,Ohio State University, the2001New Zealand Econometric study group meeting,the University of California at Santa Barbara,the University of Southern California,and an anonymous referee.The usual disclaimer applies.
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ÓBlackwell Publishing Ltd,2003.Published by Blackwell Publishing Ltd,9600Garsington Road,Oxford OX42DQ,UK and350Main Street,Malden,MA02148,USA.
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I.Introduction
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This paper considers the extension of the single equation dynamic ordinary least squares(DOLS)method of Saikkonen(1991)and Stock and Watson (1993)for estimating and testing hypothes about a cointegrating vector to panel data.We call the estimator panel DOLS.We discuss its limit distribution and apply it to estimate the long-run money demand function using a panel data t of19countries with annual obrvations spanning from1957to1996.
Panel DOLS is fully parametric and offers a computationally convenient alternative to the panel‘fully modified’OLS estimator propod by Pedroni (1997)and Phillips and Moon(1999).Properties of panel DOLS,when there arefixed effects in the cointegrating regression,have been discusd by Kao and Chiang(2000).We take this to be the starting point for our analysis.In our environment,the cointegrati
ng vector is homogeneous across individuals but we allow for individual heterogeneity through disparate short-run dynamics, individual-specificfixed effects and individual-specific time trends.Moreover, we permit a limited degree of cross-ctional dependence(CSD)through the prence of time-specific effects.
We prent two limit distributions for panel DOLS.Thefirst limit distribution is obtained for afixed number of cross-ctional units N,letting Tfi1.In this ca,panel DOLS converges in distribution to a function of Brownian motions and the Wald statistic for testing a t of s linear constraints has a limiting v2(s)distribution.This limit theory ems well suited for many applied macroeconomic or international problems.Here,rearchers often have available panel data ts of moderate N but much larger T.With the passage of time,the data ts will gain time-ries obrvations but they are unlikely to acquire many more cross-ctional units.1We also obtain the quential limit distribution byfirst letting Tfi1forfixed N,and then letting Nfi1as propod by Phillips and Moon(1999).Here,panel DOLS has a limiting Gaussian distribution and as in thefixed N ca,the Wald statistic has a limiting chi-square distribution.In the abnce of linear trends in the cointegrating regression,the quential limiting normality of the estimator is theoretically interesting but has less practical import becau the limit distribution of the test statistics is identical to the Tfi1distribution wit
h fixed N.However,when linear trends are prent,the quential limit theory produces considerable simplifications.Here,the estimator of the cointegration 1For example,if the obrvational unit is a national economy,the total number of countries may fluctuate over time,but is unlikely to go to infinity.While the break-up of the Soviet Union created veral new economies,the opposite trend is at work in Europe where the EMU may eventually combine to form a single economic unit.But beyond this,rearchers typically choo to group countries into class that share common characteristics such as income levels,stages of development or geography which often result in panels with5–20individuals.英语六级阅读技巧
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vector and the time-trend slope coefficients remain correlated for fixed N as T fi1but are asymptotically uncorrelated when T fi1then N fi1.As single equation cointegration vector estimators are super consistent,it is natural to ask what is to be gained by using the panel estimator.The answer is that super consistency means that convergence towards the asymptotic distribution occurs at rate T but it says nothing about the sampling variability of the estimator for a fixed value of T .In fact,the statistical properties of single-equation cointegration-vector estimators can be quite poor when applied to sample sizes associated with macroeconomic time ries typically available to rearcher
s (e.g.Inder,1993;Stock and Watson,1993).Moreover,even limited amounts of heterogeneity in the short-run dynamics across individuals can generate considerable disparities in single-equation DOLS estimates of the true homogeneous cointegration vector.In the situations,combining cross-ctional and time-ries information in the form of a panel can provide much more preci point estimates of the cointegration vector with reasonably accurate asymptotic approximations to the exact sampling distribution.In a ries of Monte-Carlo experiments,we study the small sample performance of panel DOLS and compare it with single-equation DOLS.Panel DOLS generally performs well under the short-run dynamic designs that we consider and can attain a striking improvement in estimation precision over that of single-equation DOLS with even a modest number of cross-ctional units.
We then apply panel DOLS to estimate the long-run demand for M1money.The countries in our study are Austria,Australia,Belgium,Canada,Denmark,France,Finland,Germany,Iceland,Ireland,Japan,Norway,New Zealand,the Netherlands,Portugal,Spain,Switzerland,the United Kingdom,and the United States.Here,we build on the time-ries contributions by Baba,Hendry and Starr (1992),Ball (1998),Hoffman,Rasche,and Tieslau (1995),Lucas (1988)and Stock and Watson (1993),and the cros
s-ctional studies by Mulligan and Sala-i-Martin (1992),and Mulligan (1997),most of which has focud on US data.2
杂乱无章The studies cited above report conflicting results along three dimensions.First,point estimates from time-ries studies exhibit substantial dependence on the sample period of the data.Income elasticity estimates from post-WWII US data typically lie well below 1–which implies the existence of economies of scale in money management –whereas estimates obtained from pre-WWII obrvations or tho that combine pre-and postwar obrvations are generally clo to 1.Using annual US data spanning from 1903to 1987,Stock and Watson’s (1993)DOLS estimate of the income elasticity is 0.97.When the sample spans from 1903to 1945,their estimate is 0.89but drops to 0.27when 2Less recent cross-ctional studies include Meltzer (1963)and Gandolfiand Lothian (1976).
657Cointegration vector estimation by panel DOLS and long-run money demand ÓBlackwell Publishing Ltd 2003
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the data span from1946to1987.Ball(2001)extends the data and obtains an estimate of0.42when th
e obrvations span from1946to1996.Using annual obrvations from1900to1958,Lucas’s(1988)estimate of the M1(permanent) income elasticity is1.06and his estimate of the(short-term)interest rate mi-elasticity is)0.07.Using data spanning from1958to1985,his income elasticity estimate drops to0.21and his interest mi-elasticity estimate is)0.01. Secondly,there is tension generated by the large difference between the estimates from time-ries studies and tho from postwar cross-ction studies. Mulligan and Sala-i-Martin’s(1992)estimates from a1989cross-ctional data t from the Survey of Consumer Finances range between0.82and1.37. Mulligan(1997)runs OLS on for a panel of12,000firms obrved from1961to 1992and obtains an income-elasticity estimate of0.83.Thirdly,there is substantial cross-country variation even amongst economies of similar income levels andfinancial market development.In our analysis,single-equation DOLS with trend gives such disparate income elasticity estimates as)1.23for New Zealand and2.42for Canada.The corresponding interest rate mi-elasticity estimates range from0.02for Ireland(which has the wrong sign)to)0.09for the UK.When trends are omitted,the income elasticity estimates range from0.13 for Belgium to2.64for Norway and the interest mi-elasticity estimates range from0.02for Ireland to)0.16for Norway.
With only40annual obrvations,the heterogeneity that we obrve in the point estimates may plausi
bly have been generated from an underlying data generating process with a homogeneous cointegration vector and heterogeneous short-run dynamics.When we include heterogeneous linear trends and estimate the cointegrating vector by panel DOLS,we obtain a point estimate of the income elasticity of1.08(¼0.26)and a point estimate of the interest mi-elasticity of)0.02(¼0.01).Moreover,the estimates,which are more in line with tho from cross-ctional studies on US data,are stable as the span of the time-ries dimension is varied and are reasonably robust to the inclusion or omission of heterogeneous linear trends.
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The remainder of the paper is organized as follows.The next ction des-cribes the reprentation of the non-stationary panel data and regularity con-ditions assumed in the paper.Section III describes the panel DOLS estimator and discuss its asymptotic properties.Section IV reports the results of a Monte-Carlo experiment to examine the small sample performance of the panel DOLS estimator and the accuracy of the asymptotic approximations.In ction V we prent our long-run money demand study and ction VI concludes the paper.Proofs of propositions and supplementary results from the money demand study are given in an appendix which is available upon request from the authors.3 3Alternatively,the appendix can be downloaded from the www.wcon.ohio-state.edu/Mark/ nmark.htm.
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II.Reprentation ofcointegrated obrvations in pane l
data Consider a balanced panel of individuals indexed by i ¼1,…,N obrved over time periods t ¼1,…,T .Vectors are underlined and matrices appear in bold face.W (r )is a vector standard Brownian motion for 0£r £1,and [Tr ]denotes the largest integer value of Tr for 0£r £1.As is standard in the literature,we will drop the notational dependence on r and write R 10W ðr Þd r as R W and R 10W ðr Þd W ðr Þ0as R W d W 0.Scaled vector Brownian motions are denoted by B ¼K W where K is a scaling matrix.For any matrix A ,||A ||denotes the Euclidian norm,[Tr(A 0A )]1/2.
We will be working with double indexed sums.In some instances –to deal with individual-specific fixed effects or common-time effects –the sums will involve transformations of the original obrvations.To handle such situations,we generically denote the sample cross-moment matrix averaged over N and T 2as M NT and let the preci definition depend upon the particular model under consideration.Also,we generically denote the limit of the moment matrix as T fi1for any given N by M N .A s N fi1,M N need not converge to a constant and we denote this limit as M N .Similarly,our generic notation for the sample cross-product vector between the regressors and the equilibrium error is m NT ,and the limit for fixed N as T fi1is m N .
As the model we study allows for individual specific effects,perhaps it would be more accurate to call the estimator dynamic least squares dummy variable (LSDV).However,in the interests of simplicity,we will refer to the estimator as panel DOLS.(i)Triangular reprentation
Let fðy it ;x 0it Þ0g be a (k +1)dimensional vector of obrvations where y it is a scalar and x it is a k -dimensional vector.Obrvations on each individual i obey the triangular reprentation
y it ¼a i þk i t þh t þc 0x it þu y it ;
ð1ÞD x it ¼v it ;ð2Þ
where (1,)c 0)is a cointegrating vector between y it and x it that is identical across individuals.The composite equilibrium error y it )c 0x it is potentially comprid of an individual-specific effect a i ,an individual-specific linear trend k i t ,and a common time-specific factor h t .The remaining idiosyncratic error u y it is independent across i but possibly dependent across t .An alternative reprentation for equation (2)allows x it to have an individual-specific vector of drift terms and for the trend in equation (1)to be induced by this drift.With
659Cointegration vector estimation by panel DOLS and long-run money demand ÓBlackwell Publishing Ltd 2003berth
