A Computationally Simple Cointegration Vector Estimator ... - CiteSeerX

A Computationally Simple Cointegration Vector Estimator for Panel Data

Nelson C. Mark and Donggyu Sul1

October 1999

Abstract We study a panel version of the dynamic OLS estimator of a cointegration vector. The estimator is asymptotically normally distributed, can be made fully parametric, is computationally simple, and can achieve substantial improvements in precision over the single equation estimator. In a series of Monte Carlo experiments, we find that the asymptotic distribution theory provides a reasonably close approximation to the exact finite sample distribution. As an empirical application, we use panel dynamic OLS to estimate coefficients of the long-run money demand function in a panel data set of 19 countries with annual observations from 1957 to 1996. We estimate the income elasticity to be 1.08 (asymptotic s.e.=0.17) and the interest rate semi-elasticity to be -0.02 (asymptotic s.e.=0.003). Keywords:Nonstationary Panel Data, Cointegration Vector Estimation, Money Demand

1

Department of Economics, The Ohio State University. We thank Paul Evans, Masao Ogaki, and seminar participants at Georgetown University and OSU for useful comments. Contact information [email protected], http://www.econ.ohio-state.edu/Mark/nmark.htm for Mark, and [email protected], http://arps.sbs.ohio-state.edu/sul/home.htm for Sul.

Introduction In this paper, we extend the single equation dynamic OLS technique for estimating and testing hypotheses about a cointegration vector suggested by Saikkonen (1991) and Stock and Watson (1993) to a panel setting. The environment that we study imposes homogeneity on the cointegrating vector across individuals but allows for heterogeneous short run dynamics as well as for individual-specific (fixed) effects.1 The panel dynamic OLS estimator that we study is asymptotically normal and Wald statistics formed to test linear restrictions on the cointegration vector are asymptotically chi-square. The estimator is fully parametric and offers a computationally convenient alternative to the ‘fully modified’ panel cointegration vector estimator suggested by Pedroni (1997) and Phillips and Moon (1998). We employ panel dynamic OLS to estimate the long-run money demand function using an annual panel data for 19 countries with observations from 1957 to 1996. Since 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 only that convergence towards the asymptotic distribution takes place at rate T —where T is the length of the time series—but it says nothing about the sampling variability of a particular estimator for any finite 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 series typically available to researchers [e.g., Inder (1993), Stock and Watson (1993)]. Even the limited amount of heterogeneity that we allow into the short-run dynamics across individuals is sufficient to generate considerable diversity in single-equation DOLS estimates of the true homogeneous cointegration vector. Combining cross-sectional and time-series information in the form of a panel can, in these situations, provide much more precise point estimates of the cointegration vector and more accurate 1 This is the same type of heterogeneity considered by Pedroni (1997). Phillips and Moon (1998) also allow for heterogeneous cointegrating vectors across individuals.

1

asymptotic approximations to the exact sampling distribution. As in Pedroni (1997) and Phillips and Moon (1998), the panel dynamic OLS estimator is asymptotically normally distributed which contrasts sharply to single equation estimators which converge to random variables expressed as functionals of Brownian motion.2 We obtain the asymptotic properties of panel dynamic OLS by first letting the number of time series observations T go to infinity then secondly letting the number of cross sectional units N go to infinity. Under regularity conditions laid out by Phillips and Moon (1998), the sequential T → ∞, N → ∞ asymptotic behavior of the estimator is identical to the limiting asymptotic behavior when N and T grow large simultaneously. Next, we conduct a series of Monte-Carlo experiments to examine the small sample properties of the panel dynamic OLS estimator. For comparison, we also study the behavior of a weighted version of the panel estimator, naive OLS with fixed effects (the least squares dummy variable [LSDV] method) run on pooled data, and univariate dynamic OLS. We find that panel dynamic OLS generally performs well under the relatively simple short-run dynamic designs that we consider. Unweighted panel dynamic OLS performs much better than the weighted version of the estimator. The unweighted estimator also attains a striking improvement in estimation precision over that of single equation dynamic OLS even a modest number of cross-sectional units. Finally, we apply panel dynamic OLS to estimate the long-run M1 demand function with a panel of annual observations extending from 1957 to 1996 for 19 countries.3 The countries in our analysis 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. Our money demand study builds on recent time-series contributions by Baba, Hendry and 2

Kao and Chiang (1998) discuss a weighted version of the panel dynamic OLS estimator. Evans (1997) also used the panel DOLS technique in his study of government consumption and economic growth. 3

2

Starr (1992), Ball (1998), Hoffman, Rasche, and Tieslan (1995), Lucas (1988) and Stock and Watson (1993), and the cross-sectional studies by Mulligan and Sala-iMartin (1992), and Mulligan (1997), most of which has focused on U.S. data.4 In time-series studies of long-run money demand, it is common to find that the point estimates exhibit substantial dependence on the particular sample period being studied. For example, using annual U.S. data spanning from 1903 to 1987, Stock and Watson’s (1993) dynamic OLS estimate of the income elasticity of long-run M1 demand is 0.97. When the sample spans from 1903 to 1945, their estimate is 0.89 but drops to 0.27 when the data span from 1946 to 1987. Ball (1998) extends these data and obtains an estimate 0.42 when the observations span from 1946 to 1996. A second source of tension arises from large differences between the estimates from time-series studies and those from cross-section studies. Time-series income elasticity estimates from post WWII U.S.data are typically well below 1, which implies that economies of scale exist in money management. On the other hand, Mulligan and Sala-i-Martin’s cross-sectional estimates range between 0.82 to 1.37 whereas Mulligan (1997) obtains an estimate of 0.83. Another dimension of uncertainty over single-equation time-series estimates can be seen by looking at dynamic OLS estimates of long-run money demand across the individual countries in our sample. Using real GDP as a scale variable and a short term interest rate to measure the opportunity cost of holding money and including a linear trend, we obtain such disparate income elasticity estimates as -1.23 for New Zealand and 2.42 for Canada. When linear trends are not included in the regression the income elasticity estimates range from 0.45 for the U.S. to 2.64 for Norway. Similarly, single-equation dynamic OLS estimates of the interest semi-elasticity range from 0.02 for Ireland (which also happens to have the wrong sign) to -0.09 for the UK with trend, and from 0.02 for Ireland to -0.16 for Norway without trend. We argue that with only 40 annual observations, the observed point estimate heterogeneity 4

Less recent cross-sectional studies include Meltzer (1963) and Gandolfi and Lothian (1976).

3

could plausibly be generated from an underlying process characterized by a homogeneous cointegration vectors and heterogeneous short run dynamics. When we include heterogeneous linear trends and estimate the cointegrating vector by panel dynamic OLS, our point estimate of the income elasticity is 1.08 and the point estimate of the interest semi-elasticity is -0.02. Moreover, these estimates, which are more in line with those from cross-sectional studies on U.S. data, are very stable as the span of the time-series dimension is varied and whether or not heterogeneous linear trends are included in the regression. The remainder of the paper is organized as follows. The next section presents the panel dynamic OLS estimator, and its asymptotic properties. In section 2, we conduct a Monte Carlo experiment to examine the small sample performance of the panel dynamic OLS estimator and the accuracy of the asymptotic approximations. Even with our relatively simple controlled dynamics, we find that single-equation DOLS can have very poor sampling properties when the error dynamics display persistence. In sharp contrast, the asymptotic approximations work quite well for the panel dynamic OLS estimator. In section 3 we apply the estimation technique to data and estimate long-run money demand elasticities from a panel of 19 countries. Section 4 concludes the paper.

1

Cointegration Vector Estimation with Panel Data We use the following notation. Underlined variables denote vectors, bold faced

variables denote matrices, but scalars have no special notation. We consider balanced panels of individuals indexed by i = 1, . . . , N tracked over time periods t = 1, . . . , T . W (r) is a vector standard Brownian motion for 0 ≤ r ≤ 1, and [T r] denotes the largest integer value of T r for 0 ≤ r ≤ 1. We drop the notational dependence on r, so integrals such as R

R1 0

W (r)dr are written as

R

W and

R1 0

W (r)dW (r)0 are written as

W dW 0. Scaled vector Brownian motions are denoted by B = ΛW where Λ is a

4

scaling matrix. The model we study allows for individual specific effects so perhaps a more accurate description of our estimator would be to call it the dynamic LSDV estimator. However, in the interests of simplicity, we refer to our estimator as the panel dynamic OLS.

1.1

The Econometric Model

Let {(yit , x0it )0 } be a (k + 1) dimensional vector of observations where yit is a scalar and xit is a k−dimensional vector. Observations on each individual i obey the triangular representation, yit = αi + γ 0xit + uit ∆xit = ζ i + vit ,

(1) (2)

where, αi is an individual specific effect, ζ i is a vector of drift parameters and {wit } = {(uit , vit0 )0 } is a covariance stationary vector process. Upon normalization, (1, −γ 0 ) is a cointegration vector between yit and xit . For concreteness, the cointegration vector is assumed to be unique. We assume that the cointegration vector (1, −γ 0 ) is identical across individuals. The homogeneity of γ is an identifying assumption that can often be justified by economic theory. Any heterogeneity across individuals stems from disparate short run dynamics of the {wit } across individuals. Our Monte Carlo study in section 3 illustrates that substantial improvements in estimation precision can accrue to panel dynamic OLS over single-equation dynamic OLS when the dynamics of the equilibrium error are persistent. The underlying data generating process is given by Assumption 1. (Data Generation.) {wit } is independent across i = 1, . . . , N ,

5

and has the moving average representation, wit = Ψi (L)²it

(3)

where wi0 = 0, ²it ∼ i.i.d. with E²it = 0, E(²it ²it0 ) = I, Ψi (L) =

P∞

j=0

Ψij Lj is a

(k + 1) × (k + 1) dimensional matrix lag polynomial in the lag operator L, where P∞

r=0

mn mn r|ψir | < ∞, and ψir is the m, n−th element of the matrix Ψir .

As in Phillips and Moon (1998), Kao (1999), Levin and Lin (1993), and Pedroni (1997), the wit are assumed to be independent across individuals, (E[w itw jt−k ] = 0, i 6= j, −∞ ≤ k ≤ ∞). Ψi (L) reflects an orthonormal transformation of the underlying Wold vector moving average representation by the lower-triangular Cholesky decomposition. We follow Phillips and Moon (1998) and Pedroni (1997) by letting the coefficients in the Ψi (L) polynomial vary stochastically but independently across individuals i. The variation in Ψi (L) is the source of heterogeneity across individuals. Let W i be a k + 1-dimensional standard Brownian motion. Under assumption 1 it follows that for each i = 1, . . . , N , w it obeys the functional central limit theorem, r] X 1 [T D √ w it → Ψi (1)W i (r) ≡ B i (r), T t=1

(4)

as T → ∞, where B i (r) = (B1i (r), B 02i (r))0 is a randomly scaled mixed Brownian motion with conditional (on individual i) covariance matrix, 

Ωi =  

Ω11,i

Ω021,i

Ω21,i Ω22,i



∞

X  0 (Γj,i + Γ0j,i )  = Ψi (1)Ψi (1) = Γ0,i + 

where Γj,i

(5)

j=1



0





0 uit   uit−j   Γ11,j,i Γ21,j,i  = E(wit w0it−j ) = E     =  vit vit−j Γ21,j,i Γ22,j,i

6

Let Ei (·) denote the expectation conditioned on individual i.5 Then for each i,

R

Ei ( B i B 0i ) = 12 Ωi . We impose regularity across cross-sectional units in

Assumption 2. Ωi is independent and identically distributed across individuals i and satisfies the law of large numbers,

1 N

PN

i=1

p

Ωi → E(Ωi ) = Ω.

Assumption 2 is a rather crude way to control the behavior of the observations across individuals. Alternatively, we can adopt assumptions 4 through 7 of Phillips and Moon (1998). They provide a more refined set of regularity conditions under which assumption 2 is satisfied and their conditions also ensure that the sequential asymptotic results we establish by first by letting T → ∞ then letting N → ∞ coincide with the limiting behavior when T and N are simultaneously driven to infinity. Since these conditions are rather lengthy, they are listed in the appendix. Before studying the panel dynamic OLS estimator, we first discuss pitfalls associated with running OLS on the pooled data.

1.2

Pooled OLS What are the consequences of simply running OLS on the pooled time-series

cross-section data?6 As shown by Phillips and Moon (1998) and Pedroni (1997), OLS is consistent and can be used as a first pass to get point estimates. The pooled OLS estimator of γ is, γˆ pool =

"N T XX

xitx0it

i=1 t=1

#−1 " N T XX

#

xit yit .

(6)

i=1 t=1

5

That is, conditional on Ψi (L). In empirical work, researchers would typically need to control for individual–specific effects. The point about running OLS on the pooled data applies to that case as well. 6

7

The residual asymptotic bias however, is increasing in the number of cross-sectional √ units, N. For fixed N, NT (ˆ γ pool − γ) converges to a non-standard distribution as T → ∞, but it will diverge when N → ∞. Divergence means that standard hypothesis tests become useless because the test statistics will also diverge. To see why, let s2 =

1 PN i=1 NT 2

PT

t=1 (yit

0 − γˆ pool xit )2 . Since γˆ pool is consistent, s2 → σ 2 , as p

T → ∞ where σ is a constant. Now

1 NT2

PN PT i=1

t=1

p

xit xit0 → 12 Ω22 , and the Wald

statistic formed to test the null hyptothesis Rˆ γ pool = r is, 

Jpool

−1

ÃN T !−1 i0 XX 1 h 0 = 2 R(ˆ γ pool − γ) R xit xit R0  s i=1 t=1 

Ã

h

N X T i0 1 h √ 1 X = 2 R N T (ˆ γ pool − γ) R x x0 s NT 2 i=1 t=1 it it

i

R(ˆ γ pool − γ)

!−1

−1

R0 

h √

i

R NT (ˆ γ pool − γ)

√ The weighting matrix converges but the random vectors R N T (ˆ γ pool − γ) do not so the Jpool statistic will also diverge.

1.3

Panel Dynamic OLS Estimator with Individual-Specific Effects We now turn our attention to the panel dynamic OLS estimator. Following

Saikkonen (1991) and Stock and Watson (1993), we assume that uit is correlated with at most pi leads and lags of v it = ∆xit. To control for this correlation, project uit onto these pi leads and lags, uit =

pi X

0 δ (i) r v it−r

+

r=−pi 0

u†it

=

pi X

0

δ r(i) ∆xit−r + u†it = δ i0 z it + u†it

(7)

r=−pi 0

0

(i) (i) 0 where δ i = (δ(i) −pi , . . . , δ 0 . . . , δ pi ) is a (2pi + 1)k−dimensional vector of projection 0 coefficients and z it = (∆xit−p , . . . , ∆x0it , . . . ∆x0it+pi )0 is a (2pi + 1)k− dimensional i

vector of leads and lags of the first differences of the variables xit . The projection 8

error uit† is by construction, orthogonal to all leads and lags of vit . Substituting the projection representation of uit into eq.(1), we obtain yit = αi + γ 0 xit + δi0 z it + uit† ,

(8)

where the newly defined vector process w †it = (u†it , v 0it )0 is itself a covariance stationary vector process that obeys the functional central limit theorem, 

u†it





† B1i (r)



1 X  D   † √   → B i (r) =  , † T t=1 vit B 2i (r) [T r]

(9)

as T → ∞, where B i†(r) is a scaled vector Brownian motion with conditional (upon i) covariance matrix, 

Ω†i =  

Ω†2 11,i

0

0

Ω22,i

0





 † † 0  = Ψi (1)Ψi (1) ,

and Ψ†i (1) =  

Ω†11,i

0

0

P22,i

0

  ,

where Ω22,i = P22,i P022,i is the lower triangular Cholesky decomposition. We note that † the properties of v it have not been changed and that B1i and B †2i are independent.

Since our primary interest lies in estimating and testing hypotheses about γ, the αi are viewed as nuisance parameters. To control for these individual specific effects, let an upper bar over a variable denote its time-series average and let a ‘tilde’ denote the deviation of an observation from its time series average. That is, y˜it = yit − y¯i , x ˜ it = xit − x ¯ i , u˜†it = u†it − u¯†i , and ˜z it = z it − ¯zi , where y¯i = u¯†i =

1 T

PT

t=1

u†it , and ¯z i =

1 T

PT

t=1

1 T

PT

t=1

yit , x ¯i =

1 T

PT

t=1

xit ,

z it .

Now taking the time-series average of the observations and using eq.(8) we have, y¯i = αi + γ 0 x ¯ i + δ0i z¯i + u¯†i ,

9

(10)

and subtracting eq.(10) from eq.(8) yields, y˜it = γ 0x˜it + δ 0i ˜z it + u˜it† ,

(11)

which eliminates the individual specific effects. To conduct inference, we require a consistent estimate of Ω†11,i . A potential complication is that eq.(11) provides us with u˜it† for which to estimate Ω†2 11,i , the long-run variance of u†it . However, lemma 1 tells us that we can consistently estimate Ω†11,i from u˜it.

Lemma 1: limT →∞ Var

³

√1 T

´

PT

† ˜1it = Var t=1 u

³

√1 T

† †2 t=1 u1it = Ω11,i .

Now to set up estimation, define ˜qit to be the 2k(1 + whose first k elements are x˜it , elements k(1+

´

PT

PN

Pi−1

i=1

j=1 (2pj +1))+1

pi ) dimensional vector to k(1+

Pi

j=1 (2pj +1))

are z˜it with 0s elsewhere. That is, let ˜q 1t

=

q˜2t .. .

=

0 (˜ x01t , ˜z 1t , 00, . . . , 00 )0

(˜ x02t , 00, ˜z 02t, . . . , 00 )0 . .. .

(12)

˜q Nt = (˜ x0Nt , 00, . . . , 00, z˜0N t )0 This allows us to rewrite the regression in the compact notation, y˜it = β 0 ˜qit + uit† − u¯†i

(13)

where β = (γ 0, δ 10 , . . . , δ0N )0 . The panel dynamic OLS estimator for the model with individual specific fixed effects is,

10

"

βˆ pdols = "

=

N X T X i=1 t=1 N X T X i=1 t=1

= β+

#−1 "

˜q it˜q 0it #−1 "

˜q it˜q 0it

"N T XX i=1 t=1

˜qit˜q0it

N X T X i=1 t=1 N X T X

#

˜qit y˜it ³

˜qit q˜0it β +

i=1 t=1 #−1 " N T XX i=1 t=1

where the last equality follows from the fact that

u†it

−

u¯†i

# ´

(14)

#

˜q it u†it

PT t=1

˜q it [uit† − u¯†i ] =

PT

qit u†it . t=1 ˜

We

can now state the following result for the asymptotic properties of panel dynamic OLS.

Proposition 1. (Asymptotic Properties of the Panel Dynamic OLS Estimator) (i) For fixed N, as T → ∞,  √



       

       D  →      

NT (γ pdols − γ) √ T (δˆ1 − δ 1) ... √ T (δˆN − δ N )

 ³

1 N

PN R ˜ ˜ 0 ´−1 i=1 B 2i B 2i

√1 N

f ˜ Q−1h

1

1

... f ˜ Q−1h N

 PN R ˜ i=1 B 2i dW1i        

(15)

N

˜ i is a Gaussian random vector, and where for each i = 1, . . . , N , h f Qi = E(z it z 0it ) is a symmetric positive definite matrix of constants.

(ii) As T → ∞, N → ∞, √ h ³

f=2 E Ω e where V 22,i

´i−1

f NT (ˆ γ pdols − γ) → N(0, V), D

h

ih ³

†2 e e E Ω11,i Ω22,i E Ω 22,i

11

´i−1

.

(16)

Estimating the Asymptotic Covariance Matrix. The asymptotic covariance matrix of γˆpdols can be consistently estimated as follows. First, for each i estimate the long-run †2 variance, Ω11,i = limT →∞ Var

³

√1 T

PT

t=1

´

u†it . Here are two options for estimating Ω†2 11,i .

The first is a parametric approach based on an autoregressive approximation described ˆ˜†it be the residuals from the panel dynamic OLS in Hamilton (1994) (p.610). Let u

i ˆ˜†it , u ˆ˜†it = Pgj=1 ˆ˜†it−j + ηit .7 The estimate of regression. For each i, fit an AR(gi ) to u φij u P P b† = σ the long-run variance is, Ω ˆ1i /(1 − gi φˆij ), where σ ˆ 2 = (T − gi )−1 T ηˆit .

11,i

j=1

1i

t=gi +1

A second option is to do the nonparametric Newey–West (1987) procedure where Pm PT b †2 = ω ˆ˜it† u ˆ˜†it−j . Ω ˆ i,0 + 2 j=1 [1 − j/(m + 1)]ˆ ωi,j , where ω ˆ i,j = (1/T ) t=j+1 u 11,i b †2 , let Having obtained Ω 11,i b

e Ω 22 = c f = D

N X T 2 X x˜it x˜0it 2 NT i=1 t=1

Ã

(17)

N T 2 X 1 X †2 b Ω11,i x˜it x˜0it 2 N i=1 T t=1

!

(18)

c p b †2 ˜ e e f p ˜ Then Ω 22 → E(Ω22,i ) = Ω22 , and D → E(Ω11,i Ω22,i ), as T → ∞, N → ∞. The

asymptotic covariance matrix of γˆ pdols is consistently estimated by c

b −1 c b −1

f = 2Ω e f e V 22 DΩ22

(19)

From proposition 1, it follows that a Wald test of linear restrictions on the cointegration vector can easily be constructed. Suppose we wish to test the hypothesis, c

f as T → ∞, N → ∞, Rˆ γ pdols = r where R is an m×k restriction matrix. Since f V→V 7

p

The lag length gi can be determined by Campbell and Perron’s (1991) top-down t-test approach. That is start with some maximal lag order ` and estimate the autoregression on ˆ²it . If the absolute value of the t-ratio for φˆi` is less than some appropriate critical value, c∗ , reset gi to ` − 1 and reestimate. Repeat the process until the t-ratio of the estimated coefficient with the longest lag exceeds the critical value c∗ .

12

under the null hypothesis, the Wald statistic, " 0

Jpdols = [Rˆ γ pdols − r] R

µ·

¸

1 f c V 2 NT

¶−1

0

R

#−1

[Rˆ γ pdols − r]

√ √ c D f 0 ]−1 [R NT (ˆ = [R NT (ˆ γ pdols − γ)]0 [RVR γ pdols − γ)] → χ2m

(20)

as T → ∞, N → ∞.

1.4

Weighted Panel Dynamic Least Squares

An asymptotically equivalent estimator can be constructed by scaling the observations for individual i by Ω†11,i . Doing so results in the weighted least squares version of the panel estimator, which was discussed in Kao (1999). Denote the scaled observations with an asterisk where y˜it∗ = (1/Ω†11,i )˜ yit , x ˜ ∗it = (1/Ω†11,i )˜ xit , z˜∗it = (1/Ω†11,i )˜ z it , † and u˜it∗ = (1/Ω11,i )˜ u†it and write the compact representation of the regression equation

with the transformed observations as, y˜it∗ = β 0 ˜q∗it + u∗it − u¯∗i

(21)

where β = (γ 0 , δ10 , . . . , δ 0N )0 , and qit∗ is defined analogously to eq.(12). Since the estimators of δTj and γ are asymptotically independent, we address our attention to the cointegration vector estimator whose properties are summarized in

˜∗ = Corollary 1. (Asymptotic Properties of weighted panel dynamic OLS). Let Ω 22 † −1 f∗ = 2Ω e . As T → ∞, N → ∞, ˜ 22,i /Ω11,i ). Let V E(Ω 22 ³ ´ √ f∗ . NT (ˆ γ ∗wpdols − γ) → N 0, V

13

(22)

Again, tests of linear restrictions on the parameter vector γ can be formed in the standard fashion. It is straightforward to see that the Wald statistic formed to test m ≤ k linear restrictions Rˆ γ ∗wpdols = r where R is an (m × k) matrix of constants f∗ = and r is an m × 1 vector of constants. Let V

1 NT 2

PN PT i=1

0

t=1

x ˜it∗ x ˜∗it . Then as

T → ∞, N → ∞, ∗ Jwpdols

2

=

h

∗ R(ˆ γ wpdols

i0

− γ)

"

∗−1 c f RV R0

#−1

h

i

2 R(ˆ γ ∗wpdols − γ) → χm . D

(23)

Performance in Monte Carlo Experiments In this section, we investigate the small sample properties of panel dynamic OLS,

weighted panel dynamic OLS and their associated test statistics in a series of of Monte Carlo experiments. We examine the panel cointegrating regression with a single regressor and also with two regressors. The models that we study are nested within the following specification. yit = αi + γ1 x1it + γ2 x2it + uit where









 uit   ψ11,i ψ12,i ψ13,i     v    1it  =  ψ21,i ψ22,i ψ23,i 



v2it



ψ31,i ψ32,i ψ33,i



(24) 



  uit−1   ²1it     v     1it−1  +  ²2it 





v2it−1

  ,  

(25)

²3it

iid

(∆x1it, ∆x2it ) = (v1it , v2it ) and ²t = (²1it , ²2it , ²3it )0 ∼ N(0, Σi ). The true values are γ1 = γ2 = 1, and the elements of the covariance matrix Σi of the innovations and the autoregressive matrix ψkj,i are randomized across individuals. In each of the experiments, we generate 5,000 random samples of T observations for N individuals. For each sample the elements of Σi and the coefficients Ψrs,i , (r, s = 1, 2, 3), i = 1, . . . , N are independent draws from U[a,b] , the uniform distribution with support [a, b]. We vary the support of these uniform distributions across experiments 14

to modulate the degree of persistence in the short-run dynamics. We set the diagonal elements of Σi to 1 and draw the covariances from the uniform distribution. In the one-regressor case, low persistence of the equilibrium error is achieved by drawing the coefficient on the ‘own’ lag from a distribution with a low upper bound (say ψ11,i ∼ U[0,0.5] ) whereas high persistence is achieved by shifting up the support (say ψ11,i ∼ U[0.8,0.95] ). We proceed in an analogous fashion in the two-regressor case. We examine the performance of the panel dynamic OLS estimator, the weighted panel dynamic OLS estimator, single-equation dynamic OLS, and naive LSDV (panel OLS with fixed effects) applied to the pooled data. For each estimator, we report 5 and 95 percentiles of their asymptotic t-ratios. We also report the median asymptotic standard error for the panel dynamic OLS estimator. We report dynamic OLS run on the first individual of the panel. In our random coefficients DGP, the DOLS estimator is identically distributed across individuals so the decision to use the first individual is innocuous. 2 leads and lags of ∆xit are included in each equation to control for the asymptotic bias. Ω†11i is estimated by the parametric method using second-ordered autoregressions. The DGPs underlying tables 1 through 3 correspond approximately to those specified by Stock and Watson for their table 1. It should be noted that our single-equation dynamic OLS results are not directly comparable to Stock and Watson’s because our random coefficient specification induces a level of sampling variability into the Monte Carlo distribution that is different from what they considered. In tables 1–3, there are no individual-specific effects in the DGP but we include a constant in the singleequation dynamic OLS regressions and we subtract off the time-series mean of the observations (as if) controlling for individual-specific effects in the panel regressions. The DGP used to generate results in Table 1, corresponds to Stock and Watson’s specification for row 1 of their table 1B. The DGP of Table 2, covers rows 1-11 of Stock and Watson’s table 1B. Here, the equilibrium error displays very little persistence and none of the estimators exhibit much small sample bias in tables 1 and 2.

15

In table 3, in which our DGP corresponds to Stock and Watson’s used to generate their table 1C, the equilibrium error displays moderate persistence and each of the estimators display some small sample bias. We make five general observations about tables 1–3. First, for fixed T , the sampling distributions for panel dynamic OLS is not very sensitive to whether N = 10 or N = 20. This invariance is largely due to the iid assumption across individuals. Second, for T = 40, the panel dynamic OLS asymptotic standard errors are sometimes too small by a factor of 2 or 3 which is reflected in departures of the t-ratio distributions from the standard normal. For example, in table 1, for T = 40, N = 10, the difference between the 95 and 5 percentile of the γˆpdols distribution is 0.011. Dividing this number by 2 yields 0.006 but the median asymptotic standard error from covariance matrix (19) is 0.002. While there is some small sample distortion in the asymptotic t-ratio distribution for T = 40, the asymptotic approximation appears to work reasonably well for T = 300. Third, the performance of weighted panel dynamic OLS is inferior to panel dynamic OLS. The weighted estimator typically displays more severe small sample bias and the distribution of its asymptotic t-ratio displays larger departures from the standard normal. Although computationally the most convenient, the observations in the weighted estimator have a random variable (the estimated long run standard deviation) in the denominator, which evidently contributes to its poor small sample performance. Fourth, panel dynamic OLS typically displays less small sample bias and is much more precise than single-equation dynamic OLS. Of the experiments that we considered. Table 2 contains the only case in which the small sample bias of panel dynamic OLS exceeds that of single equation dynamic OLS, and this occurs only for T = 40. Even so, the 5 to 95 percentile interval of the single equation dynamic OLS distribution is 0.393 but is only 0.058 for panel dynamic OLS. so even though the asymptotic distribution for single equation dynamic OLS sometimes does a better job in approximating the Monte Carlo distributions, panel dynamic OLS is a much more precise estimator. Fifth, naive OLS with fixed effects

16

typically displays only moderate small sample bias (table 3 is the exception), but as can be seen, it’s asymptotic t-ratio is useless for drawing inference. In table 4, we allow for an individual specific effect with αi ∼ U[−1.0,1.0] , and an equilibrium error that displays a high degree of persistence. The distributions of panel dynamic OLS and weighted panel dynamic OLS are similar, but the asymptotic standard errors of the former are more accurate. Both of these panel estimators are substantially more precise than the single equation dynamic OLS estimator. For T = 40, N = 10, the difference between the 95-percentile and the 5 percentile of the panel dynamic OLS estimator distribution is 0.29 whereas it is 1.22 for dynamic OLS. Table 5 reports results from the analogous high persistence in the equilibrium error DGP for the case with two regressors. For this experiment, the panel dynamic OLS statistics are seen to provide more reliable inference than either the weighted panel dynamic OLS estimator or single equation dynamic OLS. Based on the available Monte Carlo evidence, our recommendation is to use the panel dynamic OLS estimator and not to use the weighted panel dynamic OLS estimator.

3

Long-Run Money Demand In this section we employ panel dynamic OLS to estimate coefficients of the long-

run M1 demand function. Economists have long been interested in obtaining precise estimates of money demand for at least two reasons. First, knowing the income elasticity of money demand tells us the rate of monetary expansion that is consistent with price level stability in the long run. Second, knowing the interest elasticity of money demand allows us to calculate the area under the demand curve and therefore to assess the welfare costs of long-run inflation [Baily (1956)]. In addition, because a stable money demand function is one of the building blocks of the IS-LM model, economists have historically been interested in knowing how well this particular aspect

17

of the model performed. While this motive has become less important in the era of dynamic general equilibrium models, Lucas (1988) shows that such a neoclassical model with a cash in advance constraint generates a standard money demand function. Although this is a heavily researched topic, there remains substantial uncertainty regarding long-run money demand elasticities. Time-series estimates using U.S. data are sensitive to the sample period used. The fragility of the estimates naturally leads to concern over the structural stability of the money demand function. For example, using annual observations from 1900 to 1958, Lucas (1988) obtains an M1 (permanent) income elasticity of 1.06 and an (short-term) interest rate semi-elasticity of -0.07. But when estimated over the period 1958–1985, the income elasticity drops to 0.21 and the interest semi–elasticity becomes -0.01.8 Cross-sectional estimates, on the other hand [Mulligan and Sala-i-Martin (1992) and Mulligan (1997)], find income-elasticities near 1. We follow Stock and Watson (1993), Ball (1999), and Hoffman et. al. (1995) in approaching long-run money demand as a cointegrating relationship. Our analysis suggests that the instability exhibited by the time-series estimates do not reflect underlying shifts in behavioral relationships but instead are indicative of inherent difficulties associated with estimation from relatively short sample spans in environments with persistent short run dynamics. Combining observations across countries allows us to obtain relatively sharp and stable estimates of money demand elasticities and the panel cointegration approach seems well suited to take up King’s (1988) suggestion to extend the money demand analysis beyond the United States. In his words, “the results of such investigations would provide us with sharper estimates of the long run values of Friedman’s (1956) ‘numerical constants of monetary behavior’ when we approach the difficult problem of the short run demand for money.” 8 Stock and Watson (1993), Hoffman et.al. (1995), and Ball (1999) examine the stability issue whereas Baba, Hendry and Starr (1992) attempt to model the underlying sources of instability.

18

The equation that we estimate is, ·

¸

Mit ln = αi + µt + γy ln Yit + γr Rit + uit Pit

(26)

for i = 1, . . . , 19, where Mit is an M1 measure of money, Pit is the price level, Yit is real GDP, and Rit is a nominal short term interest rate. Data definitions and sources are given in appendix B. In addition to individual specific effects, αi , we allow for possibly heterogeneous linear trends. These trends are intended to capture changes in the financial technology that affects money demand independently of income and the opportunity cost of holding money.9 The homogeneity restrictions we require for identification may seem at first blush to be unrealistic, because the countries under consideration differ in terms of industrial development, banking laws, and financial technology. We think, however, that the differences between two countries (the U.S. and New Zealand, say) relevant for long-run money demand at a point in time is less severe than the differences in a single country, say the U.S. between 1900 and 1995. Viewed in this light, cross-sectional homogeneity does not seem to be any more restrictive than imposing time homogeneity on long–spaning time-series data.

3.1

Comparison between Single-equation and panel dynamic OLS estimates

The econometric analysis of this section is predicated on the assumption that the equilibrium error is stationary—that is, that the observations cointegrate. While the question of whether a series is I(0) or I(1) is difficult to answer given a finite number 9

Panel dynamic OLS is robust to limited forms of cross-sectional dependence in which the individuals are bound together by a single-factor aggregate shock. In particular, suppose that instead of eqs.(1) and (2), the econometric model is given by yit = x0it γ + uit , where uit = θt + ²it , ∆xit = δθt + eit , where θt is a serially uncorrelated common time effect that is independent of eit and ²it . When uit is projected onto leads and lags of ∆xit , because θt is independent of eit , the projection error is orthogonal to both θt and eit .

19

of observations, we present some panel evidence on the cointegration by way of the Fisher test proposed by Maddala and Wu (1998). The Fisher test proceeds as follows. First, estimate eq.(26) by panel OLS with individual specific effects. For each individual, let uˆit be the estimated residual from this panel regression. Second, run the augmented Dickey–Fuller test for a unit root in uˆit , and get pi , p-value for the studentized coefficient. Maddala and Wu show that −2 ln

PN

i=1

2 ln(pi ) ∼ χ2N under the null hypothesis of no cointegration.10 When there

is not trend included in the panel regression, the test statistic is 45.2 (p-value=0.196), which isn’t exactly overwhelming evidence in favor of cointegration. Stronger evidence of cointegration is obtained when heterogeneous linear trends are included. Here, we obtain a test statistic of 58.2 (p-value=0.019). All of our money demand estimates are performed using 2 leads and 2 lags of ∆ ln Yit and ∆Rit in the regressions. All long run variances are estimated with the parametric method using an AR(2) specification. We begin with single equation dynamic OLS estimates. Figures 1 and 2 provide a graphical summary of the results where we have plotted the dynamic OLS point estimates of the income elasticity and the interest semi-elasticity for regressions run without and with trend. The figures also include the panel dynamic OLS estimates. The estimates and asymptotic standard errors are reported in table 6. As can be seen, the individual equation estimates display substantial cross-sectional variability which make them difficult to interpret. In the dynamic OLS regressions without trend, the income elasticities are all positive, ranging from 0.134 (Belgium) to a whopping 2.64 (Norway), but the interest semi-elasticity has the wrong sign for Belgium, France, Ireland, and Japan. When we include a trend in the regression, income elasticity estimates are negative for Finland, Iceland, Norway, and New Zealand, and 10 In the single equation analysis, the distribution of the test statistic depends on the number of regressors because of the sampling variability of the first-stage estimation. Pedroni (1997) shows that the first-stage sampling variability can be ignored in the panel setting and that the panel OLS estimates can be treated as fixed.

20

the interest semi-elasticity is positive for Finland, France, and Iceland. If we maintain an underlying belief that the financial systems and transactions technologies across modern economies are essentially similar, the cross-sectional variability in these estimates must reflect the inherent difficulty of obtaining good estimates rather than uncovering evidence of disparate economic behavior. Our panel dynamic OLS estimates are shown at the bottom of table 6 and also in Figures 1 and 2. When we run the panel regression with heterogeneous trends, we estimate the income elasticity to be 1.08 (asymptotic s.e.=0.17) and the interest semi-elasticity to be -0.02 (asymptotic s.e.=0.003). When we omit the linear trends, we estimate the income elasticity to be 0.86 (asymptotic s.e.=0.08) and the interest semi-elasticity to be -0.02 (asymptotic s.e.=0.01). To illustrate the problem of estimation instability in the time-series dimension, figures 3–6 display recursive single-equation dynamic OLS coefficient estimates for the US, UK, France, and Japan and panel dynamic OLS with all 19 countries. The recursive estimates are obtained by sequentially incrementing the endpoint of the sample from 1979 to 1995. In comparison, the recursive panel estimates are quite stable compared to the single-equation dynamic OLS estimates.11

4

Conclusions Heterogeneity and persistence in the short run dynamics are sufficient to create

substantial variability in single-equation cointegration vector point estimates. The result is that these estimators can be quite sensitive to the particular time span of the observations as well as to the particular individual being studied. This small sample fragility can be encountered in spite of the superconsistency of these estimators. In these environments, panel dynamic OLS delivers more precise estimates. It is 11 Plots of recursive estimates for all 19 countries along with 2 standard error bands are presented in appendix C.

21

asymptotically normally distributed and easy to compute. Relevant test statistics are also easy to compute and have standard asymptotic distributions. The asymptotic distributions were found to provide reasonably close approximations to the exact sampling distributions. We applied the panel dynamic OLS technique to estimate the long-run money demand function using a panel of 19 countries with annual data from 1957 to 1996. The estimates in which we have the most confidence are an income elasticity of 1.08 and an interest rate semi-elasticity of -0.02.

22

References Baba, Yoshihisa, David F. Hendry, and Ross M. Starr (1992). ‘The Demand for M1 in the U.S.A., 1960–1988,’ Review of Economic Studies, 59, pp. 25–61. Bailey, Martin J. (1956). ‘The Welfare Cost of Inflationary Finance,’ Journal of Political Economy, 64, (April) PP 93–110. Ball, Laurence (1998). ‘Another Look at Long-Run Money Demand,’ NBER Working Paper. No. W6597. Campbell, John Y. and Pierre Perron (1991). ‘Pitfalls and Opportunities: What Macroeconomists Should Know about Unit Roots,’ in O.J. Blanchard and S. Fisher (eds.) NBER Macroeconomic Annual, Vol. 6, MIT Press, Cambridge, Mass. Evans, Paul (1997). ‘Government Consumption and Growth,’ Economic Inquiry, 35(2), (April), pages 209-17. Friedman, Milton (1956). ‘The Quantity Theory of Money – A Restatement,’ in Milton Friedman (ed.), Studies in the Quantity Theory of Money. Chicago: University of Chicago Press. Hamilton, James D. (1994). Time Series Analysis, Princeton, Princeton University Press. Hoffman, Dennis L., Robert H. Rasche, and Margie A. Tieslau (1995). ‘The Stability of Long-Run Money Demand in Five Industrial Countries,’ Journal of Monetary Economics, 35, pp. 317–339. Hsiao, Cheng (1986). Analysis of Panel Data, New York, Cambridge University Press. Inder, Brett (1993). “Estimating Long-Run Relationships in Economics: A Comparison of Different Approaches, Journal of Econometrics, 57(1-3), May–June, pp. 53–68. Kao, Chihwa and Min-Hsien Chiang (1998). ‘On the Estimation and Inference of a Cointegrated Regression in Panel Data,’ mimeo, Syracuse University. Kao, Chihwa (1999). ‘Spurious Regression and Residual-Based Tests for Cointegration in Panel Data,’ Journal of Econometrics, 90, pp. 1-44. King, Robert G (1988). ‘Money Demand in the United States: A Quantitative Review: A Comment,’ Carnegie-Rochester Conference Series on Public Policy, 29, pp. 169–172. Levin, Andrew and Chien-Fu Lin (1993) ‘Unit Root Tests in Panel Data: New Results,’ mimeo, Board of Governors of the Federal Reserve of System. 23

Lucas, Robert E. Jr. (1988). ‘Money Demand in the United States: A Quantitative Review,’ Carnegie-Rochester Conference Series on Public Policy, 29, pp. 137–168. Meltzer, Allan H. (1963). ‘The Demand for Money: A Cross-Section Study of Business Firms,’ Quarterly Journal of Economics 77, pp. 405-422. Mulligan, Casey B. (1997). ‘Scale Economies, the Value of Time, and the Demand for Money: Longitudinal Evidence from Firms,’ Journal of Political Economy 105, pp. 1061–1079. Mulligan, Casey B., and Xavier Sala-i-Martin (1992). ‘U.S. Money Demand: Surprising Cross-Sectional Estimates,’ Brookings Papers on Economic Activity pp. 285– 329. Pedroni, Peter (1997). ‘Fully Modified OLS for Heterogeneous Cointegrated Panels and the Case of Purchasing Power Parity,’ mimeo, Department of Economics, Indiana University. Phillips, Peter C. B. and Hyungsik R. Moon (1998). ‘Linear Regression Limit Theory for Nonstationary Panel Data,’ Econometrica forthcoming. Saikkonen, Pentti (1991). ‘Asymptotically Efficient Estimation of Cointegration Regressions,’ Econometric Theory 7:1–21. Sims, Christopher A., James H. Stock, and Mark W. Watson (1990). ‘Inference in Linear Time Series Models with Some Unit Roots,’ Econometrica 58: 113-44. Stock, James H. and Mark W. Watson (1993). ‘A Simple Estimator of Cointegrating Vectors in Higher Order Integrated Systems,’ Econometrica 61:783–820.

24

0.05

Ireland Belgium

France

0 0

0.5

Iceland

Interest Semi-Elasticity

US Portugal -0.05

Japan Finalnd Austria 1 1.5 PANEL Spain

2

Germany

2.5

3

Denmark

Netherlands Canada Switzerland

Australia

New Zealand UK -0.1

-0.15 Norway

-0.2 Income Elasticity

Figure 1: Single equation DOLS and Panel DOLS estimates of long-run money demand, no trend

0.04

0.02 Ireland Finland France -1

Interest Semi-Elasticity

-1.5

-0.5

0 Iceland

0

0.5 Netherlands

1

1.5 PANEL

-0.02

2

2.5

3

Germany

Spain US

Denmark Belgium

-0.04

Austria Portugal

Australia Switzerland -0.06

Japan

Canada

-0.08 New Zealand UK

Norway -0.1 Income Elasticity

Figure 2: Single equation DOLS and Panel DOLS estimates of long-run money demand, with linear trend

25

Figure 3: Recursive Single-Equation DOLS Income Elasticity Estimates, No Trend

Figure 4: Recursive Single-Equation DOLS Interest Semi-Elasticity Estimates, No Trend

26

Figure 5: Recursive Single-Equation DOLS Income Elasticity Estimates, With Trend

Figure 6: Recursive Single-Equation DOLS Interest Semi-Elasticity Estimates, With Trend

27

Table 1: Single Regressor, No Individual Effects "

Σi =

T 40 100 300 40 100 300

T 40 100 300 40 100 300

N 10 10 10 20 20 20

N 10 10 10 20 20 20

5% 0.995 0.999 1.000 0.997 0.999 1.000

γˆpdols 50% 1.000 1.000 1.000 1.000 1.000 1.000

5% 0.992 0.997 0.999 0.994 0.998 0.999

γˆpool 50% 0.997 0.999 1.000 0.997 0.999 1.000

1 U[0.4,0.6] · 1

#

"

,

Ψi =

0 U[−0.1,0.1] 0 U[0.85,0.95]

#

A. Panel DOLS t-ratio asy. se γˆwpdols 95% 5% 95% 50% 5% 50% 1.006 -2.597 2.877 0.002 0.991 0.997 1.002 -1.943 2.079 0.001 0.997 0.999 1.000 -1.751 1.702 0.000 0.999 1.000 1.004 -2.356 3.076 0.001 0.993 0.997 1.001 -1.865 2.141 0.001 0.998 0.999 1.000 -1.830 1.788 0.000 0.999 1.000 B. Pooled OLS and Single-equation DOLS t-ratio γˆdols 95% 5% 95% 5% 50% 95% 1.001 -2.939 0.413 0.964 1.000 1.037 1.000 -3.165 0.366 0.992 1.000 1.008 1.000 -3.217 0.233 0.998 1.000 1.002 1.000 -3.384 0.025 0.965 1.000 1.036 1.000 -3.762 -0.152 0.992 1.000 1.008 1.000 -3.936 -0.306 0.998 1.000 1.002

28

95% 1.001 1.000 1.000 1.000 1.000 1.000

t-ratio 5% 95% -4.803 0.777 -4.098 0.433 -3.955 0.221 -5.519 0.222 -4.919 -0.278 -4.832 -0.427

t-ratio 5% 95% -2.386 2.409 -1.901 1.812 -1.713 1.702 -2.344 2.417 -1.921 1.850 -1.742 1.661

Table 2: Single Regressor, No Individual Effects

"

Σi =

T 40 100 300 40 100 300

N 10 10 10 20 20 20

5% 0.972 0.992 0.998 0.981 0.995 0.998

γˆpdols 50% 1.001 1.000 1.000 1.001 1.000 1.000

T 40 100 300 40 100 300

N 10 10 10 20 20 20

5% 0.991 0.996 0.999 0.996 0.998 0.999

γˆpool 50% 1.022 1.008 1.003 1.019 1.007 1.002

1 U[0.4,0.6] · 1

#

"

,

Ψi =

0 U[0.0,0.8] 0 U[−0.9,0.9]

#

A. Panel DOLS t-ratio asy. se γˆwpdols 95% 5% 95% 50% 5% 50% 1.030 -2.458 2.832 0.011 0.993 1.025 1.008 -1.826 1.935 0.004 0.997 1.009 1.002 -1.682 1.677 0.001 0.999 1.003 1.021 -2.455 2.820 0.007 0.998 1.022 1.006 -1.853 1.940 0.003 1.000 1.008 1.002 -1.703 1.740 0.001 1.000 1.002 B. Pooled OLS and Single-equation DOLS t-ratio γˆdols 95% 5% 95% 5% 50% 95% 1.073 -0.955 4.099 0.807 1.000 1.200 1.029 -1.154 4.172 0.939 1.000 1.060 1.010 -1.117 4.104 0.983 1.000 1.018 1.052 -0.611 4.744 0.812 1.001 1.204 1.020 -0.959 4.919 0.941 1.000 1.057 1.007 -0.943 4.872 0.982 1.000 1.018

29

95% 1.079 1.030 1.010 1.058 1.021 1.007

t-ratio 5% 95% -1.071 6.373 -0.859 5.489 -1.161 5.350 -0.320 7.355 -0.196 6.369 -0.562 6.287

t-ratio 5% 95% -2.267 2.287 -1.835 1.800 -1.646 1.711 -2.268 2.298 -1.869 1.787 -1.707 1.633

Table 3: Single Regressor, No Individual Effects

"

Σi =

T 40 100 300 40 100 300

N 10 10 10 20 20 20

5% 0.942 0.976 0.990 0.969 0.989 0.994

γˆpdols 50% 1.052 1.020 1.005 1.050 1.020 1.005

T 40 100 300 40 100 300

N 10 10 10 20 20 20

5% 1.029 1.010 1.003 1.051 1.022 1.007

γˆpool 50% 1.123 1.053 1.018 1.119 1.053 1.019

1 U[0.4,0.6] · 1

#

"

,

Ψi =

U[0.44,0.84] U[−0.1,0.02] U[−0.08,0.02] U[−0.2,0.0]

A. Panel DOLS t-ratio asy. se γˆwpdols 95% 5% 95% 50% 5% 50% 1.183 -1.191 3.549 0.049 1.023 1.122 1.072 -1.027 2.745 0.024 1.008 1.049 1.022 -1.174 2.369 0.009 1.002 1.016 1.141 -0.843 3.847 0.036 1.045 1.116 1.056 -0.663 3.078 0.017 1.018 1.048 1.017 -0.918 2.585 0.006 1.006 1.016 B. Pooled OLS and Single-equation DOLS t-ratio γˆdols 95% 5% 95% 5% 50% 95% 1.233 1.107 8.150 0.605 1.057 1.627 1.109 0.962 8.556 0.863 1.022 1.251 1.040 0.893 8.870 0.951 1.005 1.083 1.196 2.720 10.034 0.605 1.052 1.620 1.090 2.816 10.538 0.856 1.021 1.254 1.032 2.820 10.791 0.951 1.006 1.080

30

#

95% 1.237 1.102 1.034 1.200 1.084 1.028

t-ratio 5% 95% 0.737 6.794 0.498 4.913 0.408 4.430 1.984 8.323 1.573 6.059 1.429 5.377

t-ratio 5% 95% -2.170 3.456 -1.622 2.313 -1.586 1.984 -2.400 3.446 -1.655 2.409 -1.611 1.949

Table 4: Single Regressor with Individual Effects, High Persistence

"

αi ∼ U[−1,1], Σi =

T 40 100 300 40 100 300

N 10 10 10 20 20 20

5% 0.858 0.938 0.979 0.898 0.958 0.985

γˆpdols 50% 0.999 1.000 1.000 1.000 1.000 1.000

T 40 100 300 40 100 300

N 10 10 10 20 20 20

5% 0.883 0.942 0.979 0.915 0.960 0.985

γˆpool 50% 0.999 0.999 1.000 1.000 1.000 1.000

1 U[−0.5,0.5] · 1

#

"

,

Ψi =

U[0.75,0.90] U[−0.05,0.05] U[−0.05,0.05] U[0.0,0.4]

A. Panel DOLS t-ratio asy. se γˆwpdols 95% 5% 95% 50% 5% 50% 1.144 -2.543 2.524 0.057 0.868 1.000 1.061 -1.973 1.949 0.030 0.941 1.000 1.020 -1.809 1.709 0.011 0.980 1.000 1.097 -2.436 2.367 0.042 0.906 1.000 1.043 -1.909 1.931 0.022 0.958 1.000 1.014 -1.790 1.747 0.008 0.986 1.000 B. Pooled OLS and Single-equation DOLS t-ratio γˆdols 95% 5% 95% 5% 50% 95% 1.118 -4.705 4.905 0.401 1.005 1.620 1.058 -5.428 5.435 0.747 0.998 1.251 1.020 -5.735 5.353 0.913 1.000 1.087 1.081 -4.947 4.766 0.396 1.003 1.619 1.040 -5.447 5.460 0.753 0.998 1.241 1.014 -5.626 5.596 0.912 1.000 1.089

31

95% 1.134 1.060 1.020 1.095 1.042 1.014

#

t-ratio 5% 95% -3.667 3.736 -2.473 2.474 -2.082 1.984 -3.775 3.833 -2.471 2.491 -2.005 1.965

t-ratio 5% 95% -3.682 3.788 -2.354 2.284 -1.878 1.914 -3.608 3.597 -2.257 2.262 -1.866 1.812

Table 5: Two Regressors with Individual Effects, High Persistence 



1 U[−0.5,0.5] U[−0.5,0.5]  Σi =  · 1 U[−0.5,0.5]  , · · 1

T 40 100 300 40 100 300

T 40 100 300 40 100 300

T 40 100 300 40 100 300

T 40 100 300 40 100 300

N 10 10 10 20 20 20

N 10 10 10 20 20 20

N 10 10 10 20 20 20

N 10 10 10 20 20 20





U[0.75,0.9] U[−0.05,0.05] U[−0.05,0.05]  Ψi =  U[−0.05,0.05] U[0.0,0.4] U[−0.05,0.05]   U[−0.05,0.05] U[−0.05,0.05] U[0.0,0.4]

A. Panel DOLS γˆpdols,1 t-ratio γˆpdols,2 50% 5% 95% 50% 0.999 -3.370 3.316 1.001 1.000 -2.107 2.123 1.001 1.000 -1.821 1.857 1.000 1.001 -3.058 3.113 0.999 1.000 -2.080 2.118 1.000 1.000 -1.834 1.832 1.000 B. Weighted Panel DOLS γˆwpdols,1 t-ratio γˆwpdols,2 50% 5% 95% 50% 1.000 -4.917 4.869 1.001 1.000 -2.784 2.895 1.001 1.000 -2.221 2.269 1.000 1.002 -4.888 5.002 1.000 1.000 -2.830 2.821 1.000 1.000 -2.196 2.290 1.000 C. Pooled OLS γˆpool,1 t-ratio γˆpool,2 50% 5% 95% 50% 0.999 -4.889 4.892 1.000 1.000 -5.345 5.479 1.001 1.000 -5.558 5.691 1.000 1.002 -4.845 4.837 1.001 1.000 -5.414 5.586 1.000 1.000 -5.597 5.767 1.000 D. DOLS γˆdols,1 t-ratio γˆdols,2 50% 5% 95% 50% 1.000 -5.086 4.623 1.001 0.999 -2.678 2.518 1.002 1.001 -1.964 2.032 1.000 32 1.010 -4.548 4.592 0.995 1.002 -2.477 2.574 1.002 0.999 -1.935 1.995 1.001

t-ratio 5% 95% -3.209 3.218 -2.042 2.128 -1.868 1.840 -3.089 3.004 -1.964 2.099 -1.872 1.818 t-ratio 5% 95% -4.855 4.882 -2.688 2.798 -2.245 2.205 -5.008 4.859 -2.712 2.850 -2.318 2.232 t-ratio 5% 95% -4.781 4.838 -5.195 5.463 -5.851 5.646 -4.902 4.938 -5.349 5.461 -5.827 5.758 t-ratio 5% 95% -4.821 4.822 -2.618 2.581 -1.999 2.063 -4.772 4.691 -2.569 2.650 -2.032 1.964

Table 6: Single-equation and Panel dynamic OLS estimates of long run money demand

Country Austria Belgium Denmark Finland France Germany Iceland Ireland Netherlands Norway Portugal Spain Switzerland UK Japan Australia New Zealand Canada US Panel

γˆy 0.901 0.134 1.460 1.019 0.677 1.548 0.594 0.507 1.112 2.641 0.517 1.203 1.020 1.738 0.889 0.926 1.349 1.245 0.445 0.860

No Trend (s.e.) γˆR (0.144) -0.009 (0.158) 0.009 (0.112) -0.043 (0.211) -0.006 (0.085) 0.010 (0.041) -0.019 (0.096) -0.010 (0.071) 0.022 (0.094) -0.045 (0.260) -0.160 (0.096) -0.037 (0.071) -0.030 (0.137) -0.062 (0.085) -0.089 (0.483) 0.009 (0.073) -0.043 (0.364) -0.076 (0.162) -0.057 (0.053) -0.029 (0.082) -0.020

(s.e.) (0.030) (0.028) (0.006) (0.011) (0.008) (0.010) (0.003) (0.009) (0.017) (0.026) (0.012) (0.006) (0.014) (0.007) (0.099) (0.006) (0.018) (0.017) (0.004) (0.006)

33

γˆy 1.552 1.183 0.684 -0.740 0.842 1.691 -0.451 1.670 0.309 -0.676 1.624 1.203 1.447 2.128 1.798 0.068 -1.233 2.420 1.169 1.079

With Trend (s.e.) γˆR (s.e.) (0.359) -0.037 (0.027) (0.318) -0.033 (0.019) (0.213) -0.036 (0.004) (0.708) 0.009 (0.009) (0.289) 0.004 (0.013) (0.223) -0.023 (0.010) (0.743) -0.004 (0.005) (1.004) 0.015 (0.010) (0.435) -0.011 (0.024) (1.307) -0.092 (0.036) (0.204) -0.043 (0.006) (0.149) -0.030 (0.007) (0.351) -0.053 (0.015) (0.639) -0.089 (0.007) (0.286) -0.076 (0.042) (0.199) -0.048 (0.004) (0.633) -0.084 (0.010) (0.689) -0.078 (0.019) (0.264) -0.033 (0.004) (0.170) -0.022 (0.003)

Trend -0.018 -0.029 0.021 0.069 -0.005 -0.004 0.047 -0.047 0.024 0.106 -0.047 0.000 -0.011 -0.009 -0.064 0.033 0.067 -0.048 -0.019 —

A.

Appendix

A1.

Phillips and Moon Regularity Conditions

Phillips and Moon (1998) show that the limiting T → ∞, N → ∞ behavior of a PN PT double-indexed random variable, XN,T = N1a i=1 YiT , where Yit = T1b t=1 yit, for a, b > 0 coincides with its limiting T, N → ∞ behavior if XNT converges uniformly in N as T → ∞. For our application, the regularity conditions laid out in assumptions 4 through 7 of Phillips and Moon are sufficient for the sequential asymptotic to imply the identical limiting behavior when both T, N grow large simultaneously. For convenience, we restate those regularity conditions using our notation. They are, i. {Ψij }N i=1 is iid across i for all j. 2 ii. EΨ16 ait < ∞ for all a = 1, 2, . . . , k where Ψait is the a-th element of vec(Ψit ).

iii.

P∞

iv.

P∞

v. vi.

t=0

t=0

P∞

t=0

t2 EΨ2ait < ∞ t4 [EΨ4ait ]1/4 < ∞

t2[EΨ8ait]1/8 < ∞

P∞

16 1/16 t=0 [EΨait ]

0 as tr(Ω) → ∞, and f(Ω) = O(|Ω|b ) for some b > 7 as |Ω| → 0. (Phillips and Moon show that the Wishart distribution has this property.)

A2.

Properties of OLS run on pooled data

The essential properties of pooled OLS are easily seen by considering the model with no individual specific effects so we set αi = 0 for each i. We begin with asymptotic divergence. First rewrite (6) as, √

"

N T 1 X 1 X NT (ˆ γ − γ) = x x0 N i=1 T 2 t=1 it it

34

#−1 "

N T 1 X 1X √ xit uit N i=1 T t=1

#

(A.1)

P

R

0 T For each i, T12 t=1 xitRxit0 → B 2i B 2i , as T → ∞ which is independent across individuals with mean Ei ( B 2i B 02i ) = 12 Ei (Ω22,i ) = 12 Ω22 , where Ω22 is a k × k symmetric R P p 0 positive definite matrix of constants. By the law of large numbers, N1 N i=1 B 2i B 2i → 1 Ω as N → ∞. Furthermore, from Hamilton’s (1994) proposition 18.1 for each i, 2 22 P D R 1 PT x u → B 2i dB 1i + Λ21,i as T → ∞, where Λ21,i = ∞ it it t=1 j=0 E(vit−j uit ) = T P ∞ E(∆x u ) is a k−dimensional random (across i) vector. By the law of large it−j it j=0 p 1 PN numbers, N i=1 Λ21,i → E(Λ21,i ) = Λ21 where Λ21 is a k−dimensional vector of constants. Thus for fixed N , as T → ∞, D

N T N Z N 1 X 1X 1 X 1 X D √ xit uit → √ B 2i dB 1i + √ Λ21,i N i=1 T t=1 N i=1 N i=1 √ N N Z 1 X NX = √ B 2i dB 1i + Λ21,i N i=1 N i=1

R

P

As N → ∞, √1N N i=1 B 2i dB 1i converges to a normal random variate but clearly diverges. To establish consistency, note that "

N T 1 X 1 X γˆpool − γ = x N i=1 T 2 t=1 it

P

P

T 1 1 0 For fixed N, N1 N i=1 T 2 t=1 xit xit → N 1 Ω as N → ∞. Also, for fixed N, 2 22 p follows that γˆpool → γ. D

#−1 "

√ N N

PN i=1

Λ21,i

#

N T 1 X 1 X x uit . N i=1 T 2 t=1 it

PN R

0 i=1 B 2i B 2i as T → ∞ P P p N T 1 1 i=1 T 2 t=1 xit uit → N

P

R

p

N 0 and N1 i=1 B 2i B 2i → 0 as T → ∞. So it

Proof of Lemma 1. We begin with Ã

!

¶ T TX −1 µ 1 X T −j Var √ u˜†it = E(˜ u†2 ) + 2 E(˜ u†it u˜†it−j ), it | {z } T T t=1 j=1 |

(a)

{z

(A.2)

}

(b) † 2 Expansion of the first term (a) yields, E(˜ u†2 u†it )2 − 2E(¯ u†i u†it). The it ) = E(uit ) + E(¯ second term in the expansion is

Ã

E(¯ u†i )2

T X 1 =E √ √ u†it T T t=1

!2

35

Ã

T 1 1 X = Var √ u†it T T t=1

!

Since

Ã

!

T 1 X †2 √ lim Var u†it = Ω11,i T →∞ T t=1

is finite, E(¯ u†i )2 → 0 as T → ∞. † Let Γ11,j = E(u†it u†it−j ). The last term in the expansion is, 

E(u†it u¯†i ) = Eu†it  

=

T 1X

T

u†ij 

j=1





t−1 X



T −t X



1 E(u†it )2 + E u†it u†it−j  + E u†it u†it+j  T j=1 j=1 

=



T −t X

t−1 X



1 † Γ11,0 + Γ†11,j + Γ†11,j  T j=1 j=1

Since the term in square brackets is finite for all T , E(u†itu¯†i ) → 0 as T → ∞. It follows that the term (a) in eq.(A.2) E(˜ u†it )2 → E(u†it )2 as T → ∞. Expanding a typical term in expression (b) of eq.(A.2) we have, E(˜ uit† u˜†it−j ) = E(uit uit−j ) − E(uit−j u¯†i ) − E(uit u¯†i ) + E(¯ u†i )2 The last two terms of this expansion werre shown to have limiting values of zero as T → ∞. Moreover, since E(u†itu¯†i ) → 0 as T → ∞ for arbitrary t, the second † term of the expansion, E(uit−j u¯†i ) also has a limiting value of 0. It follows that E(uit† u†it−j ) − E(uit uit−j ) → 0 as T → ∞, and therefore, Ã

!

Ã

!

T T 1 X 1 X lim Var √ u†it = lim Var √ uit = Ω†11,i T →∞ T →∞ T t=1 T t=1

Proof of Proposition 1. We know from single equation dynamic OLS that the estimators for γ and δi converge at different rates. To allow for these different convergence rates, we follow

36

Hamilton (1994) and Sims, Stock, and Watson (1990) by defining the scaling matrix  √  NT Ik √ 00 ··· 00     0 T Ip 1 · · · 00   0   0 0 ··· 0 GN T =  .   .. .. .. ..   . . . √ .   0 0 ··· T Ip N Now √ √ √ 0 0 0 GNT (βˆ − β) = ( NT (ˆ γ pdols − γ 0 ), T (δˆ1 − δ10 ), . . . , T (δˆ1 − δ 0N )0 "

=

G−1 NT

ÃN T XX i=1 t=1

!

#−1 "

−1 q˜itq˜0it GNT

GNT

ÃN T XX i=1 t=1

(A.3)

!#

q˜it u†it

The scaled moment matrix is,  PN PT

G−1 NT

ÃN T XX i=1 t=1

!

q˜it q˜0 it G−1 NT

     =      

PT

x ˜ x ˜0 t=1 it it NT2 0 T z˜ x ˜ t=1 1t 1t √ N T 3/2 T z˜ x ˜0 t=1 2t 2t √ N T 3/2

P P

t=1 √

x ˜ z˜0 t=1 2t 2t √ N T 3/2

PT

0 .. .

.. .

PT

PT

x ˜ z˜0 t=1 1t 1t √ 3/2 NT PT 0 z ˜ z˜ t=1 1t 1t T

i=1

z˜0 N T x ˜0 N T

PT

0

x ˜ z˜ t=1 N t Nt √ NT 3/2

···

00

.. .

··· .. .

00 .. .

···

0

z˜2t z˜0 2t T

t=1

0

N T 3/2

00

···

PT t=1

z˜N T z˜0 N T T

(A.4) To establish the limiting behavior of this matrix, begin with the off-diagonal blocks. First, PT

x˜it z˜0 it T

t=1

=

T 1X x˜ [v0 it−pi , · · · , v 0 it+pi ] T t=1 it



D ˜i + → A

pi X

˜0 , A ˜i + Γ 22,j

j=1

pX i −1

(A.5)

˜0 , · · · A ˜i + Γ 22,j

j=1

k−1 X



˜0 ,  Γ 22,j

(A.6)

j=1

˜ 22,i = E(˜vit ˜v0 ) and A ˜i = R B ˜ 2i dB ˜ + P∞ Γ ˜0 as T → ∞ where Γ it−j 2i j=0 22,j . Thus for each i, PT 0 ˜it z˜it p t=1 x →0 (A.7) T 3/2 0

so each of the off-diagonal blocks of G−1 NT

³P N PT

37

i=1

0

´

−1 ˜it q˜it GN t=1 q T converge in proba-

            

bility to 0. Now in the first diagonal block of the matrix in (A.4) we have for each i, 0 0 D R ˜ ˜ 1 PT x ˜ x ˜ → B B , as T → ∞, which is independent across i. By the law of 2 it 2i t=1 it 2i T 0 0 R p 1 PN R ˜ ˜ ˜ 22,i ) = 1 Ω ˜ . ˜ 2i B ˜ ) = 1 Ei (Ω large numbers, N i=1 B 2i B 2i → Ei ( B 2i 2 2 22 The remaining diagonal blocks converge in probability to symmetric positive defP 0 p ˜ i . We thus have for the scaled moment matrix, inite matrices, T1 Tt=1 z˜it z˜it → Q  "

G−1 NT

N X T X i=1 t=1

0

−1 q˜it q˜it GN T

#−1

  →    p

˜ −1 2Ω 00 · · · 22 ˜ −1 0 Q ··· 1 .. . ···

0

0

00 0 .. . ˜ Q−1

      

(A.8)

N

as T → ∞ and N → ∞. γˆ pdols and δˆi , 1, . . . , N are therefore asymptotically uncorrelated. Next,  PN PT  † G−1 NT

x ˜ u √ t=1 it 1t N T PT † z˜ u t=1 1t 2t T

i=1

  N X T  X q˜it u†it =     i=1 t=1 

PT t=1

.. .

z˜N t u†N t T

       

(A.9)

Since z˜it u†it is a k−dimensional random vector of stationary zero mean variates, PT D ˜ i is a zero mean Gaussian vector. √1 ˜itu†it → ˜hi as T → ∞, where h t=1 z T R P D R ˜ † ˜ ˜ ˜ We now concentrate on γˆpdols . For each i, T1 Tt=1 x˜it u†it → B 2i dB1i = Ω11,i B 2i dW1i , as T → ∞, which has conditional (on i) covariance matrix †2 Ω11,i

Z

Z

0 ˜ 2i B ˜ 2i ˜ 2i W ˜ 02i )P022,i . B = Ω†2 W 11,i P22,i (

By the law of large numbers as N → ∞, · ¸ Z Z N 1 X 0 0 p †2 †2 0 0 ˜ ˜ ˜ ˜ Ω P22,i ( W 2i W 2i )P22,i → E Ω11,i P22,i ( W 2i W 2i )P22,i N i=1 11,i ·

= E =

38

Ω†2 11,i Ei

µ

Z

P22,i

1 h †2 ˜ i E Ω11,i Ω22,i 2

˜ 2i W ˜ 02i P22,i W

¶¸

By the Lindeberg–Levy central limit theorem, as N → ∞ Z N h i 1 X D †2 ˜ ˜ 2i dW ˜ 1i → √ Ω†11,i B N 0, (1/2)E(Ω11,i Ω22,i ) . N i=1 √ f as T → ∞, N → ∞, where V f is given It follows that NT (ˆ γ pdols − γ) ⇒ N (0, V) in the text.

B.

The Data

Our data consists of annual time series observations from 1957 through 1996 for the following 19 countries: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Great Britain, Iceland, Ireland, Japan, Netherlands, New Zealand, Norway, Portugal, Spain, Switzerland, and The Unite States. The composition of the sample was determined by data availability. Our measure of money is from the IFS (line code 34) for all countries except for Great Britain where we used M0. The definition of money from the IFS is the sum of transferable deposits and currency outside banks. Price levels for all countries are measured using the CPI from the IFS (line code 64). IFS annual real GDP data (line code 99B) are used for all countries with the following exceptions: The June 1998 IFS CD–ROM reports only nominal GDP for Austria from 1957–1963, for Finland and Iceland from 1957–1959, and Portugal from 1957–1964. For these countries we generated our own measure of real GDP for the early part of the sample by deflating nominal GDP with the CPI. The June 1998 IFS CD–ROM reports real GDP for Germany only from 1979–1996. To obtain a complete series, we spliced this series to real GDP from 1960 to 1978 reported in the 1992 OECD Main Economic Indicators. For the period 1957 to 1959, we deflated nominal GDP by the CPI. The availability of short term interest rates is given in the table A.7. Interest rates are available over the entire 1957–1996 period only for the U.S., U.K., Belgium, Canada, and Japan. For the other countries, we estimated interest rates in the early part of the sample by covered interest parity with the U.S. for those countries with forward exchange rates, and uncovered interest parity for those that did not have forward currency trading. Since most of the missing data occurs during the Bretton Woods system of fixed exchange rates, the difference between uncovered interest parity and covered interest parity is pretty small.

39

Table A.7: Interest Rate Availability Country Interest Rate Source Dates Australia 13 Week T-Bill IFS 1969–1996 Austria Call Money IFS 1967–1996 Belgium Call Money IFS 1957–1996 Canada 30 day Prime Corp. Paper IFS 1957–1996 Denmark Call Money IFS 1972–1996 Finland Call Money OECD 1976–1996 France Call Money IFS 1964–1996 Germany Call Money IFS 1960–1996 Iceland 3 Month Time Deposit IFS 1976–1996 Ireland Deposit Rate IFS 1962–1996 Japan Call Money IFS 1957–1996 Netherlands Call Money IFS 1960–1996 New Zealand Call Money OECD 1973–1996 Norway Call Money IFS 1971–1996 Portugal Call Money IFS 1983–1996 Spain Call Money IFS 1974–1996 Switzerland Call Money OECD 1975–1996 U.S. 3 Month T-Bill NBER 1957–1996 U.K. 3 Month T-Bill IFS 1957–1996

40

Figure 7: Recursive Single-Equation DOLS Income Elasticity Estimates with 2 standard error bands, No Trend

C.

Recursive dynamic OLS estimates of long-run money demand.

Here, we present plots of recursive single-equation dynamic OLS estimates of income elasticities and interest semi elasticities for each of the 19 countries in the sample. Recursive estimates are obtained by sequentially incrementing the endpoint of the sample from 1979 to 1995.

41

Figure 8: Recursive Single-Equation DOLS Interest Semi-Elasticity Estimates with 2 standard error bands, No Trend

42

Figure 9: Recursive Single-Equation DOLS Income Elasticity Estimates with 2 standard error bands. Trend included.

43

Figure 10: Recursive Single-Equation DOLS Interest Semi-Elasticity Estimates with 2 standard error bands. Trend included.

44