JEL Classification: C51, O51, R12 Key words: Convergence, Time Series Analysis, Regional Growth

Transcription

1 Regonal and Sectoral Economc Studes. AEEADE. Vol. 4-2 (2004) A IME SERIES ES OF REGIONAL CONVERGENCE IN HE USA WIH DYNAMIC PANEL MODELS, SEDGLEY Norman and ELMSLIE Bruce * Abstract A good deal of controversy surrounds the emprcal regularty of convergence. If captal s share s taken to be 1/3, as n natonal accounts, then convergence should occur at a much faster rate than observed. Problems are worse f the economy s open. Wth perfect captal moblty convergence should occur at an nfnte rate. Convergence estmates appear to be as slow for state economes as for natonal economes, even though the assumpton of perfect captal moblty s a closer approxmaton of realty for these economes. Some argue that other varables, most promnently human captal, must be ncluded n any cross sectonal estmaton of convergence. Supposedly, ths addton of varables can brng the mpled rate of convergence n lne wth emprcal estmates by controllng for dfferences n the steady state level of per capta ncome. hs paper extends the analyss of Islam (1995) to US states by estmatng dynamc panel data models. hs s a more approprate method of allowng for dfferent steady states. We fnd that the data suggests states converge very quckly, mplyng a hgh degree of captal moblty, f each state economy s allowed to have ts own steady state captured through ts own fxed effect. hese results demonstrate the ptfalls of applyng closed economy models to study growth n very open economes and the dangers of addng varables to the estmaton whch have, at best, only a weak relatonshp to dfferental steady states. JEL Classfcaton: C51, O51, R12 Key words: Convergence, me Seres Analyss, Regonal Growth * Norman Sedgley, Department of Economcs, Sellnger School of Busness and Management Loyola College n Maryland. Baltmore, MD Emal: nsedgley@loyola.edubruce and Elmsle Whttemore School of Busness and EconomcsDepartment of EconomcsUnversty of New HampshreMcConnell HallDurham, NH

2 Regonal and Sectoral Economc Studes. AEEADE. Vol. 4-2 (2004) 1. Introducton he growth model developed by Solow (1956) has domnated most economsts understandng of the growth process for more than four decades. Unfortunatly, the emprcal evdence does not appear to favor ths neoclasscal approach n one mportant way. here exsts a sgnfcant dscrepancy between the actual factor shares n natonal ncome accounts and mpled factor shares as calculated from the estmate of the convergence coeffcent. Gven the standard Cobb-Douglas producton functon t s straght forward to calculate the rate of convergence to the steady state, β. ypcally, β s found to be n the neghborhood of 2% to 2.5% n studes nvolvng countres, U.S. States, regons of Western Europe, Canadan provnces, and Japanese prefectures, (Barro, 1997). Gven reasonable estmates of populaton growth, n, technologcal change, x, and deprcaton of captal, δ, of 2%, 2%, and 5% respectvely (see Barro and Sala --Martn 1995 and Romer, 1996), the slow rate of convergence mples that captal s share, α, s n the neghborhood of 72% of value added. hs s approxmatly double the value obtaned from the actual natonal accounts, whch suggests captal s share s close to 33%. he most popular explanaton of ths slow rate of convergence consdered n the lterature s the lack of accountng for human captal as a factor of producton. Mankw, Romer, and Wel (M-R-W) (1992) attempt to patch up the model and brng emprcal estmates of β n lne wth obseved factor shares by augmentng the neoclasscal producton functon wth human captal. Holtz-Eakn (1993) extends the analyss to states. Whle these approaches represent mportant contrbutons, the addton of these varables s not the best way of approachng the problem f an estmate of the rate of convergence s desred. hs paper argues that usng a panel data approach s the preferred method of estmaton. 6

3 Sedgley, N. and Elmsle, B. A me Seres est Of Regonal Convergence n the USA 2. Related Lterature he standard neoclasscal growth model consders a closed economy. Begn wth a Cobb-Douglas producton functon wth two factors of producton, Labor (L) and Captal (K). echnology s assumed to augment labor and s measured by the parameter A. he growth rate of ths parameter s exogenous and equal to x, whle labor grows at a constant rate equal to n. α 1 α Y = K (AL) (1) Wth a constant savngs rate, s, captal accumulates accordng to the smple dentty: K = sy δk (2) Equaton 1 s expressed n "ntensve" form by dvdng both sdes by α AL. Equaton 1 becomes y = k. he lower case ndcates a per capta measure and the "hat" mples that the varable s measured n terms of effectve labor, AL. From ths framework the well known emprcal growth framework s derved (see appendx 1). ln( y ( )) / y (0)) = a (1 e β ) ln( y (0)) + ε (3) β (1 e ) α a = x + [ ln s * ln( n + x + δ )] (4) 1 α β = ( 1 λ)( x + n + δ ) (5) * ndexes economes, and s s the steady state savngs rate. As t stands, equaton 3 performs very poorly n emprcal work unless the researcher s careful to select a sample of smlar countres or regons such as OECD natons, ol producng countres, US states etc (Barro and Sala--Martn, 1991, 1995; Mankw Romer and Wel, 1992; De 7

4 Regonal and Sectoral Economc Studes. AEEADE. Vol. 4-2 (2004) Long, 1998). he obvous problem wth drectly estmatng equaton 3 s that t mplctly assumes a = a j for all j. In other words each economy s convergng to the same steady state. hs noton of absolute convergence, of course, s not what neoclasscal growth theory predcts. Not properly accountng for these dfferences n steady states bases the estmate of convergence downward f steady state per capta ncome s postvely correlated wth the ntal level of per capta ncome. If the omtted varables are related to the ntal level of productvty such that a = b + b ln( y (0)) + ε o 1 (6) then the estmaton of the coeffcent on ln( y ( 0)) n equaton 3 captures two effects. Frst, t captures the drect effect of the ntal ncome on the subsequent growth rate, the convergence effect. It also captures an ndrect effect due to ts relaton to the omtted varable. If b 1 s postve, and hgher steady states are postvely related to hgher levels of ntal productvty, then the coeffcent from whch the convergence estmate s calculated s based upward, causng a downward bas n the estmaton of the convergence coeffcent, β. Addng varables such as nvestment as a percentage of GDP, the populaton growth rates, schoolng, and others often leads to a sgnfcant estmate of β. Once varables are added β can be brought nto the range of 2% to 3%, dependng on the sample perod and group of economes studed. Problems reman wth ths estmate of condtonal convergence, however, snce t s too low. Accordng to equaton 5 an estmate of β of 2.5% and a reasonable estmate of ( n + x + δ ) of 9% mples that a captal's share, α, s equal to 73%. Estmates of captal's share from natonal ncome accounts suggest a more accurate measure of captal's share s 33%. 8

5 Sedgley, N. and Elmsle, B. A me Seres est Of Regonal Convergence n the USA he most notable attempt to account for the slow rate of convergence and provde emprcal support for the quanttatve mplcatons of the neoclasscal growth model s Mankw et al.(1992). hey argue that, n order to test the neoclasscal model, the concept of captal must be expanded to nclude human captal, as n the followng producton functon: α λ α λ Y = K H ( AL) 1 (7) where H represents human captal. After augmentng the producton functon, t s straghtforward to show that the convergence coeffcent s β = ( 1 α λ)( n + x + δ ). hus, accountng for human captal suggests a slower rate of convergence by the value of λ. Mankw et al.(1992) clam that for a group of 98 countres, after controllng for human captal, the rate of convergence across countres should be around 2.5%, as estmated by most studes. hus, by addng human captal to the producton functon, the authors appear to brng the slow rate of convergence n lne wth a rate necessary to brng predcted factor shares n lne wth the actual fgures. Further dffcultes arse, however, when the model s augmented to nclude the porton of human captal accumulated through learnng by dong. Persson and Malmberg (1996) drectly extend the model by ncludng varables to control for the demographc structure of the populaton, argung that growth should be postvely related to the proporton of the populaton who are of workng age. hey test the mplcatons of ths model usng data for US states, and fnd that growth s postvley related to the percentage of the populaton aged years old and years old ndcatng an mportant role for the learnng by dong component of overall human captal. If the producton functon s ammended to nclude human captal, t s mportant to nclude both schoolng and tranng n the estmaton. 9

6 Regonal and Sectoral Economc Studes. AEEADE. Vol. 4-2 (2004) hs addton ncreases the rate of convergence to about 5.8%. hs somewhat faster rate of convergence s no longer consstent wth the neoclasscal model augmented for human captal. A convergence coeffcent of 5.8% suggests that human captal s share, gven the parameter values outlned n Mankw et al. (1992), s equal to -30%! In ther calculaton of factor shares, (n + x + δ) s taken to be.06, whle α s.33. Gven the parameter estmates used n the current paper of (n + x + δ) and α equallng 9% and 33% respectvly, human captal s share s 2.6%. Ether estmate s far from the value between 33% and 50% Mankw et al. suggest. Fnally, the whole debate concernng the speed of convergence becomes more complcated f an open economy s specfed. he rate of convergence ncreases to nfnty as captal becomes perfectly moble n a small open economy. hs s easy to see, snce a small economy's nterest rate s determned by ts margnal product of captal. If r s an (exogenous) world nterest rate and = f ' δ then free captal flows ensure that r = r and the captal to labor rato converges nstantly to the world captal to labor rato. If the captal to labor rato s too low r wll be greater than r and the dfference wll spark large captal flows untl captal to labor ratos are equated. hus, even rates of convergence n lne wth factor shares after accountng for human captal suffer from the same crtcsms once an open economy s specfed. here are many reasons why captal s not perfectly mobl, and they undoubtedly apply to states as well as natons, albet to a lesser degree. Cohen and Sachs (1986) show that convergence s not nfnte f there s the potental for one economy to default on what t owes to another. Captal moblty can be added n a straghtforward way for a small economy. Equaton 2, expressed n ntensve form, s augmented wth captal nflows ( f f ' δ > r ) and outflows (f f ' δ < r ). ˆα 1 = sk ( n + x + δ ) + ψ ( α 10 α 1 δ r r), 0 ψ (8)

7 Sedgley, N. and Elmsle, B. A me Seres est Of Regonal Convergence n the USA Expressng equaton 8 n terms of ln( k ˆ ), takng a taylor seres expanson around the steady state, and notng that ˆ*) ˆ*) (1 α) ln( k (1 α) ln( k s* e + ψαe = ( n + x + δ) + ψ ( r + δ ) leads drectly to the followng modfcaton of the convergence coeffcent. β = ( 1 α )( n + x + (1 + ψ ) δ + ψr) (9) ψ s a measure of captal moblty. If ψ = 0 then the economy s closed (only ntertemporal trade s allowed n a one sector model). In ths case equaton 9 s equvalent to equaton 5. If ψ = then β = as well and convergence s nstantaneous. here lkely exsts varyng degrees of captal moblty across dfferent samples of economes. hs presents yet another problem for slow convergence estmates across US states. he assumpton of perfect captal moblty should be a better approxmaton of captal moblty across states than across natons, but convergence appears to occur at the standard 2.5% rate n most condtonal convergence studes across states. 3. Emprcal Results he rate of convergence s estmated usng data collected for the 48 contnental US States and the Dstrct of Columba (DC). Alaska and Hawa are excluded to make the results of the analyss comparable wth other studes of growth across US states. (Barro and Sala -- Martn, 1991, 1992, 1995; Persson and Malmberg, 1996). he data spans the years 1972 to hese years are chosen because relable Gross State Product measures do not date earler than represents the most recent year that allows for our dvson of the data. he data s dvded nto three sub-perods each of an equal number of years ( , , ). here s nothng specal about the perods chosen. he productvty measure s Gross State Product per worker. A schoolng varable s ncluded, the percentage of the state's populaton over age 25 wth a college degree, 11

8 Regonal and Sectoral Economc Studes. AEEADE. Vol. 4-2 (2004) to llustrate mpact of allowng for state fxed effects. o avod endogenety problems ths varable s measured n 1970, 1980, and Consderng equaton 3, t s clear that a method of estmaton n a cross secton must, n some way, account for the dfferng values of a across economes. Examnaton of equaton 4 shows that, whle a can vary across from one economy to another; t s theoretcally constant for a partcular economy. All parameters ncluded n equaton 4 are theoretcally measured at the steady state where, by defnton, output per unt of effectve worker s constant. he most common method of estmatng condtonal convergence proceeds by addng varables to the estmaton of equaton 3. Supposedly these varables are correlated wth the long run level of an economy's producton functon. hs s the methodology advocated by M-R-W (1992) and extended n many growth studes. here are two serous problems n proceedng n ths manner. o llustrate arguments consder the Solow model augmented wth human captal va equaton 7. In ths case α * λ * α + λ ln( y*) = ln s K + ln s h ln( n + x + δ ) 1 α λ 1 α λ 1 α λ * * where s h and sk are the steady state measures of the percentage of output used to produce physcal captal and human captal respectvely. An updated verson of equaton 3 s: 12

9 Sedgley, N. and Elmsle, B. A me Seres est Of Regonal Convergence n the USA β β ln( y( ))/ y(0)) (1 e ) α * (1 e ) λ * = x + ln s, K + ln s, H (1 e β 1 α λ ) α + λ (1 e ln( n+ x + δ) 1 α λ (10) β ) ln( y (0)) + ε 1 α λ In an attempt to control for steady state dfferences across the economes, varables to control for dfferences n steady state rates of populaton growth, steady state dfferences n the percentage of output devoted to equpment nvestment, steady state dfferences n the percentage of output devoted to human captal nvestment, as well as a ktchen snk full of other varables, are often ncluded. he frst problem wth ths approach s obvous. Current measures of schoolng and nvestment may be very poor ndcators of the values these varables eventually take n the long run. If the Solow model s extended to a framework that ncludes optmzaton (Ramsey, 1928) t s clear that savngs rates can rse or fall as the economy transtons to the steady state. It s unclear exactly what role measures of current savngs rates play n the estmaton of equaton 10 snce no a pror relatonshp can be establshed between current savngs rates, s, and * * * steady state savngs rates, s, j (.e. s, j > s, j or s, j < s, j ). Any varable added to the regresson potentally suffers from ths crtcsm. hs s partcularly true of human captal measures snce they are, at best, poor ad-hoc proxes for the steady state percentage of output devoted to human captal formaton. he second major problem relates to the restrctve assumptons of the model. he mplcatons of openng up the closed economy model are dscussed above. If the data used n estmatng equaton 10 are generated from open economes then measures of nvestment may have lttle relatonshp to the parameters n equaton 10. Consder ncludng a measure of the percentage of the current populaton wth a college degree (School) to the regresson n an attempt to control for 13

10 Regonal and Sectoral Economc Studes. AEEADE. Vol. 4-2 (2004) * dfferences n s H, the steady state proporton of output nvested n human captal. If human captal mgrates then the School varable tells lttle about any partcular economy's domestc nvestment n human captal. hs common approach to modelng s, of course, necessary f an nvestgator's purpose s to understand why steady states dffer. hs s a worthy but dffcult research agenda. If, however, an answer to the queston of how fast economes converge s desred then an alternatve approach s preferred. Islam (1995) reports stronger evdence of convergence across countres when each s allowed to have ther own ntercept term n a panel regresson. Hs estmates of β ncrease from the standard 2.5% estmate to somewhere n the range of 4% to 6%. Augmentaton of Solow's model and the addton of ad-hoc varables are unnecessary. hese estmates are n accord wth factor shares f captal s hghly mmoble across natonal borders. Smlarly, Sedgley and Elmsle (2003) show evdence of absolute convergence across OECD economes when the steady state s allowed to shft over tme. Equaton 3 s estmated drectly usng the panel data set on state per worker Gross State Product. It s expected that the rate of convergence should ncrease and take a value greater than the 4% to 6% value reported by Islam f captal (fnancal and human) s more moble across states than across the natons he studes. o date the evdence suggests that states converge at the same standard 2.5% rate reported n most cross country studes, as outlned n the lterature revew. Equaton 3 s estmated usng both a one way and a two way fxed effects specfcaton. he results are reported n able 1. he top rows report results based on parttoned least squares. he bottom rows report the results usng two-stage parttoned least squares. he nstruments are ftted values of frst stage regressons. Ftted values of the frst stage regressons are based on regressons of each rght hand 14

11 Sedgley, N. and Elmsle, B. A me Seres est Of Regonal Convergence n the USA sde varable on lagged values of all varables ncluded n the stage two regresson. he lagged value of per capta ncome s sgnfcant n each set of regresson results, mplyng catch up s mportant across state economes. he frst model, wthout fxed effects or perod effects, suggests a rate of convergence between 1.8% and 3.9%, slower than the 4% to 6% value found across natons n Islam's (1995) results. Whle t s encouragng to fnd evdence of convergence ths slow speed of convergence s nconsstent wth the ntuton that captal and labor are more moble across states than across natons. he half-lfe to convergence measures the amount of tme t takes an economy to close half of the gap between the current level of productvty and the steady state level of productvty. he half-lfe s calculated from ln( y) ( e Bt Bt = 1 )ln y * + e ln( y( 0 )), where y* s the steady state value of output per worker. At the tme t when y s Bt Bt halfway between y(0) and y* t must be true that ( 1 e ) = e. Solvng for t gves a formula for the half-lfe of t = ln( 2)/ β. he results of the frst model are consstent wth most estmates of convergence n the sense that convergence s sgnfcant but slow. he able reports that the half-lfe to convergence based on a specfcaton absent of fxed and/or perod effects s lkely to be from 20 to 40 years across US states. he F test n the last column tests the sgnfcance of the addtonal fxed or perod effects as they are added to the model. When parttoned least squares s used the fxed effects are statstcally sgnfcant f schoolng s ncluded, whle the perod effects are sgnfcant f schoolng s excluded. he schoolng varable s nsgnfcant n the two way model. When two stage least squares s used as the estmaton technque the results become more consstent. hs estmator s preferred snce t accounts for possble endogenety and measurement error. We focus, therefore, on the 2SLS estmates. 15

12 Regonal and Sectoral Economc Studes. AEEADE. Vol. 4-2 (2004) Allowng for fxed effects ncreases the estmate of convergence and lowers the half-lfe dramatcally. he model ncludng schoolng suggests that the half-lfe falls from 20 years to only 4 years as the rate of convergence jumps from 3.4% to 17.7%. he F test suggests the addton of the fxed effects sgnfcantly ncreases the R-squared. A study of why steady states mght shft s the doman of new growth theory. If a perod effect s ncluded these shfts can be tested for sgnfcance. he addton of perod effects s sgnfcant but has lttle mpact on the estmated rate of convergence to the steady state. Not surprsngly the ad-hoc measure of human captal, School, becomes nsgnfcant once equaton 3 s estmated drectly usng a panel data approach. When convergence s estmated wthout the addtonal unneeded School varable t s clear that the estmate of convergence s mpacted dramatcally by properly accountng for the fxed effects and perod effects across states. he rate of convergence ncreases from 1.8% to 16.2% whle the half-lfe falls from 39 years to just 4 years. It s very mportant to fnd evdence that states converge so much more quckly than natons gven smple ntuton concernng captal and labor moblty. he tme to close half the gap toward the steady state s a mere 1/3 of the fastest estmates across natons mpled by Islam's (1995) results, who uses the same methodology. 4. Concluson he neoclasscal growth framework, stll a workhorse of growth theory, s often crtczed because emprcal estmates of the convergence parameter are not n lne wth expectatons formed on the bass of captal s share n natonal ncome accounts. More specfcally, the rate of convergence s too slow to correspond wth the captal share of 33% typcally reported for a narrow concept of physcal captal. he most sgnfcant attempt to account for ths emprcal anomaly s provded by Mankw et al (1992) who take Solow serously and augment the neoclasscal model to nclude a broader concept of captal that encompasses human captal. hey then show that, n a cross 16

13 Sedgley, N. and Elmsle, B. A me Seres est Of Regonal Convergence n the USA secton of countres, schoolng s sgnfcant and rate of convergence remans wthn the 2% to 3% range typcally reported. Problems reman, however, snce convergence across states appears to occur at the same lethargc 2.5% rate reported n most cross-country studes. hs does not match up wth ntuton snce the relatvely hgh degree of captal and labor moblty across states suggests a much hgher rate of convergence across states than across countres. We follow Islam (1995) n usng panel data methods to estmate the rate of convergence. he fxed effects allow each economy to have a unque steady state wthout dependng on ad-hoc proxes for eventual steady state values of mportant parameters. State economes may dffer n ther long run producton functons due to ndustry concentraton, natural resource endowments, hstorcal access to trade and waterways, and many other hstorcal factors. Allowng for fxed effects appears to solve an mportant anomaly; state economes appear to converge at a substantally faster rate than natonal economes. hs makes sense due to the very open nature of trade, labor moblty, and captal moblty across states. he data suggests that half the gap between the steady state level of productvty and the current level of productvty s closed n only four years. Earler studes suggest that t takes much longer, perhaps 30 years, to move half way to the steady state. Appendx 1. Frst note that the growth of output can be expressed as: yˆ = α k yˆ ˆ and accordng to equaton (2) ( α 1) ln( ) = se ( n + x + σ ) 17

14 Regonal and Sectoral Economc Studes. AEEADE. Vol. 4-2 (2004) a aylor seres expanson around the steady state (denoted wth an *) yelds: ( α 1) ln( *) = se k In the steady state wth ( α 1)[ln( ) ln( ˆ*)] = ( n + x + σ )( α 1)[ln( ) ln( *)] k ˆ * k ˆ equal to zero the above equaton s: * hs together wth yˆ = k yˆ α mples: ˆ yˆ = ( n + x + σ )( α 1)[ln( yˆ) ln( yˆ*)] yˆ hs s a dfferental equaton wth the followng soluton: β ln( y$) ( e t βt = 1 )ln( y$*) + e ln( y$( 0)) he followng steps then derve the emprcal growth framework. B B 1 e 1 ( 1 e ) ln( y$ / y$( 0)) = ln y$( 0) ln y$( 0) + ln y $ * B 1 ( 1 e ) ln( y / y( 0)) = x + ln( y $ * / y $( 0 )), where y$ = y / A, y$* = y * / A*, 18

15 Sedgley, N. and Elmsle, B. A me Seres est Of Regonal Convergence n the USA y$ ( 0) = y( 0) / A(0)., and normalzng A(0)=1. 1 ln( y / y(0)) = (1 e x + ) (1 e ln( yˆ*) β β ) ln( y(0)) Whch s dentcal to equaton (3). References Barro, R. (1997) Determnants of Economc Growth: A Cross Country Emprcal Study, he MI Press, Cambrdge. Barro, R.. and Sala -I-Martn X. (1991) Convergence Across States and Regons, Brookngs Papers on Economc Actvty, 1, Barro, R. and Sala-I-Martn X. (1992) Convergence, Journal of Poltcal Economy, 100, 2, Barro, R. and Sala-I-Martn X. (1995) Economc Growth, McGraw- Hll, New York. Cohen, D. and Sachs J. (1986) "Growth and External Debt under Rsk of Debt Repudaton," European Economc Revew, 30, 3, Holtz-Eakn, D. (1993) "Solow and the States: Captal Accumulaton, Productvty, and Economc Growth," Natonal ax Journal, 46, Islam, N.(1995) "Growth Emprcs: A Panel Data Approach," Quarterly Journal of Economcs, November, Mankw, N. G, Romer, D., and Davd Wel (1992) A Contrbuton to the Emprcs of Economc Growth, Quarterly Journal of Economcs, 107, 2, Persson, J. and Bo Malmberg (1996) Human Captal, Demographcs and Growth across the U.S. States , Unpublshed Manuscrpt. 19

16 Regonal and Sectoral Economc Studes. AEEADE. Vol. 4-2 (2004) Ramsey, Frank (1928) "A Mathmatcal heory of Savngs," Economc Journal, 38, Romer, Davd (1996) Advanced Macroeconomcs, McGraw-Hll, New York. Sedgley, Norman and Bruce Elmsle (2003) me Seres ests of Convergence and a Shftng Steady State, Unpublshed Manuscrpt. Solow, Robert (1956), A Contrbuton to the heory of Economc Growth, Quarterly Journal of Economcs, LXX, Revsta publcada por la Asocacón Euro-Amercana de Estudos de Desarrollo Económco 20