Testing neoclassical convergence in regional incomes and earnings

ELSEVIER Regional Science and Urban Economics 26 (1995) 565-590

ECONOMICS

Testing neoclassical convergence in regional incomes and earnings Gerald A. Carlino a'*, L e o n a r d Mills b aResearch Department, Federal Reserve Bank of Philadelphia, Philadelphia, PA 19106, USA bFederal National Mortgage Association, 4000 Wisconsin Avenue, NW, Washington, DC 20016, USA

Received 3 September 1993; revised 13 October 1995

Abstract The time-series properties of per capita income and per capita earnings in the regions of the United States are tested for consistency with the neoclassical growth model's prediction of convergence. We find evidence for per capita income convergence for U.S. regions during the 1929-90 period after allowing for a trend break in 1946. These findings support the neoclassical model's prediction of convergence. The evidence for per capita earnings convergence is, however, less conclusive. Shocks to per capita earnings are found to be more persistent than shocks to per capita income. This implies that the regional distribution of transfer payments tends to smooth the effects of deviation on relative regional per capita earnings and reinforce trends in per capita income convergence. Keywords: Income convergence; Regional economic growth; Time series models JEL classification: C32; El0; 040; R00; R l l ; R23

1. Introduction A m a i n t e n e t o f S o l o w ' s (1956) n e o c l a s s i c a l g r o w t h m o d e l , with d i m i n i s h ing r e t u r n s to c a p i t a l , is t h a t t h e g r o w t h r a t e o f a c o u n t r y ' s p e r c a p i t a * Corresponding author. [email protected].

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G.A. Carlino, L. Mills / Reg. Sci. Urban Econ. 26 (1996) 565-590

income should be inversely related to its relative starting level of income per capita. That is, poor countries should tend to catch up to rich ones through time, implying that per capita incomes converge in levels. R o m e r (1989), Lucas (1988), and others have noted that the Solow model fails to reconcile observed international differences in per capita incomes. This failure has spawned a new and growing literature on endogenous growth theory. The essential feature of these models is that growth is endogenous because it occurs through increasing returns to physical capital (Romer, 1986) or externalities associated with human capital complimenters (Lucas, 1988) without exogenous increases in technical progress, as in the neoclassical growth model. The presence of these non-convexities may generate multiple equilibria in the steady-state growth paths across economies. Such heterogeneity in the growth experience implies that economies need not converge. The empirical evidence on the convergence hypothesis is the subject of continuing debate. Baumol (1986) suggests a cross-sectional test for convergence that relates an economy's growth rate of income per capita during the period under analysis to that economy's initial level of per capita income. For convergence, a negative relationship between growth and initial conditions must be found. Barro (1991) finds evidence of convergence when the cross-country (98 countries) regression of per capita output growth rates on initial per capita levels includes proxies for human capital development. Barro and Sala-i-Martin (1992) find evidence of convergence for the U.S. states, and Barro and Sala-i-Martin (1991) find evidence of convergence across 73 regions of Western Europe. Barro and Sala-i-Martin (1991) report that the rate of convergence is similar at around 2% per year for both the U.S. states and E u r o p e a n regions. In sum, the cross-sectional evidence supports the neoclassical growth model's prediction of convergence. O t h e r researchers have tested a stochastic definition of convergence. T o achieve convergence in these models, per capita income disparities between economies must follow a zero mean stationary process. Bernard (1991) and Bernard and Durlauf (1995a) find little evidence of stochastic convergence in cross-country per capita output for either bivariate or multivariate data samples during the 1900-87 period. Quah (1990) tests for convergence in levels and growth rates in cross-country incomes during the postwar period. Quah finds evidence of convergence in growth rates but not in levels. In a related paper, Brown et al. (1990) find little evidence of stationarity for the U.S. regions. Thus, unlike the cross-sectional studies, the bulk of the time-series studies find little evidence of convergence. Unlike previous tests for convergence that rely on either cross-section or time-series methods, Carlino and Mills (1993) argue that both time-series and cross-sectional tests are necessary for convergence. Carlino and Mills argue that two conditions must be met for convergence to hold: (i) shocks to relative regional per capita income should be temporary (stochastic convergence), and (ii) regions having per capita incomes initially above their

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567

compensating differential should exhibit slower growth than those regions having per capita earnings initially below their compensating differential (cross-sectional convergence). They show that if one allows for a break in trend growth rates of regional per capita income in 1946, then the timeseries evidence is not inconsistent with the cross-sectional finding of convergence for the U.S. regions) O n e limitation of the Carlino and Mills (1993) paper, and of the convergence literature in general, is that it focuses on per capita income. T h e neoclassical model relies on factor migration to lead to convergence of wages and earnings and not necessarily per capita income. To the extent that localized externalities in physical and h u m a n capital are internalized at the regional level, per capita incomes need not converge. M o r e important, per capita income includes transfer payments that may reinforce or counter any convergence trend in per capita earnings. This p a p e r extends Carlino and Mills (1993) by analyzing the time-series properties of both per capita income and per capita earnings in the m a j o r regions of the United States. Specifically, per capita incomes and per capita earnings are tested for consistency with the neoclassical growth model's prediction of conditional convergence, z For per capita income, we find evidence for stochastic convergence for U.S. regions during the 1929-90 period, but only after allowing for a b r e a k in the rate at which the various regions were converging in 1946. The per capita income results also suggest that the U.S regions have achieved cross-sectional convergence and that most of this convergence took place prior to 1946. We take these findings to support the view of conditional convergence emanating from the neoclassical growth model. The findings are less substantial for the theoretically more appealing per capita earnings data. Shocks to per capita earnings are found to be m o r e persistent than shocks to per capita income. This implies that the regional distribution of transfer payments tends to smooth the effects of deviation of regional per capita earnings and to reinforce per capita income convergence.

2. The convergence hypothesis As Fig. 1 shows, the log of relative per capita earnings differed widely across regions in 1929. In the New England, Mideast, G r e a t Lakes, and Far West regions, per capita earnings were well above the national average. Per

Bernard and Jones (1995a) take a cross-section, time-series approach similar to Carlino and Mills (1993) to study convergence of productivity across U.S. states and regions. Bernard and Jones (1995b) apply the model to study productivity convergence across 14 OECD countries. 2The current paper also differs from Carlino and Mills (1993) in that it provides alternative tests for convergence, as well as a critical discussion of these alternative measures. We also elaborate on the alternative concepts of convergence found in the literature.

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G.A. Carlino, L. Mills / Reg. Sci. Urban Econ. 26 (1996) 565-590

Percent of U.S. Average

0 0.6

4

- -

0.2 °l

l

~

,..~. ...............,.,...

.............

_ '"=~:~:-:

I

-0.4 -0.6 1929 NENG

1939 MF~T

1949 GLAK

1959 PLNS

SEST

1969 SWST

1979 RKMT

1989 [:'WST

Fig. 1. Relative regional per capita earnings: percent of U.S. average. Note: The graph depicts regional per capita earnings relative to per capita earnings nationally. Data are in logs.

capita earnings in the Mideast region were almost 29% above the national average in 1929, and they were about 26% above in the Far West region. In the New England region, per capita earnings were more than 19% above the national average in 1929, and they were almost 17% above in the Great Lakes region. In contrast, per capita earnings in the Southeast region were almost 43% below the national average in 1929. In the remaining areas, however, they were at least 8% below average. Hoover and Giarrantani (1984) have identified several reasons for the vast inequality of regional per capita earnings in the years before World War II. One reason was the relatively low level of agricultural prices, which depressed earnings in this sector. Accordingly, in regions where agriculture was relatively important, per capita earnings lagged. In addition, national immigration policies virtually halted the influx of cheap labor after World War I, removing the constraints on wage increases in industrial regions that had been employing most of that labor, mainly the Mideast and Great Lakes. The period between 1929 and the late 1970s saw a pronounced trend toward convergence of regional per capita earnings. All of the low-income regions made substantial gains, while the high-income regions lost ground. According to Borts and Stein (1964), a main source of the convergence during the period was the shift of labor from low-wage agricultural employment to higher-paying non-agricultural jobs. This shift of employment largely occurred within regions. In the last 10 years, per capita earnings have tended to diverge across U.S. regions after decades of gradual convergence. In 1990, for example, earnings in the states of New England and the mid-Atlantic coast were well

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above the U.S. average• In 1978, however, earnings in both regions were close to the U.S. average• Meanwhile, in the Great Lakes and Southwest, per capita earnings, already low in 1978 relative to the rest of the country, had fallen even farther behind the U.S. average• In addition to short-run fluctuations in per capita earnings, differences in region-specific factors may result in differentials in real per capita earnings across regions in the long run. For example, regions may tend to specialize in the production of goods, and therefore worker characteristics (e.g. education, skill level, and participation rates) may differ across regions. Even if the value placed on these characteristics is the same across regions, earnings will differ because regional characteristics are different (Sahling and Smith, 1983)• In addition, wages may also differ in equilibrium to compensate workers for regional amenities, unemployment insurance, agglomeration economies, as well as for differences in the prices of localized goods (Roback, 1982).3 The possibility that economies may converge up to a constant differential is referred to as conditional convergence (Mankiw et al., 1992)• The widening gap in regional earnings after 1978 may be the result of changes in these regional-specific factors• If this is the case, then the widening gap would reflect a permanent adjustment toward a new long-run equilibrium• Alternatively, the widening gap may reflect the effects of powerful but temporary national economic shocks, such as energy and agricultural shocks, that affect regions differently• 2.1. Implications f o r convergence

In contrast to the foregoing discussion, the popular notion of convergence is a disequilibrium phenomenon. That is, the convergence hypothesis assumes that regions are initially out of equilibrium. Over time, however, factors will migrate across regions to achieve equlhbrmm. This equilibrium does not have relative regional earnings that are exactly equalized; instead, they are equalized up to the compensating differential (conditional convergence). Moreover, the compensating differentials equilibrium is usually taken to be time-invariant. •

•

•

4

2.2. Three concepts o f convergence

The convergence hypothesis has recently been the subject of growing research interest both across countries and across regions. There are three concepts of convergence. One concept, referred to as ~r-convergence by 3For a survey of the literature on equilibrium interregional wage differentials, see Dickie and Gerking (1989). 4Barro and Sala-i-Martin (1992) discuss the determinants of the transition dynamics of per capita earnings in a neoclassical growth model with Cobb-Douglas technology.

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Barro and Sala i-Martin (1991), considers the cross-sectional dispersion in per capita earnings, o--convergence occurs if the cross-sectional dispersion in per capita earnings, typically measured by the coefficient of variation, declines through time. Another approach, introduced by Baumol (1986) and referred to as [ 3 - c o n v e r g e n c e by Barro and Sala-i-Martin (1991), occurs when poor regions tend to grow faster than rich regions, such that the poor regions catch up to the rich ones in terms of the level of per capita earnings through time. As Barro and Sala-i-Martin point out, /3-convergence works toward o--convergence but may be offset by new disturbances that tend to increase crosssectional dispersion. Quah (1990) notes that the more relevant issue for an economy subject to random shocks through time is how persistent the effects of economic shocks are on regional per capita earnings. Thus, the final concept of convergence considered in the current paper is referred to as s t o c h a s t i c c o n v e r g e n c e . Bernard (1991), Bernard and Durlauf (1995a), and Quah (1990) develop a definition of convergence using the notions of unit roots and cointegration as described in Engle and Granger (1987). In these models, convergence in per capita earnings requires that permanent shocks to the national economy are associated with permanent shocks to regional economies. If some component of regional per capita earnings deviations is due to permanent regional-specific shocks, such as localized technology shocks, convergence may not be achieved. Thus, this definition of convergence requires that deviation in a region's per capita earnings relative to per capita earnings in the nation be characterized by a non-zero mean stationary stochastic process. But, as Carlino and Mills (1993) point out, time-series evidence alone may be insufficient to settle the convergence issue, even if we are able to reject the unit root null for regions within a country. Recent models of growth emphasize the role of dynamic externalities for regional growth. For example, Lucas (1988) suggests that regions with greater human capital could provide greater incentives for innovation and, therefore, relatively higher growth rates than regions with relatively less human capital. There is some evidence for divergence, since regions have differing levels of human capital, and agglomeration economies that tend to be positively associated with per capita income and earnings growth. If regions are characterized by endogenous growth mechanisms, /3-convergence is not guaranteed. Alternatively, if regional differences in human capital and agglomeration economies are time-invariant, these differences would be captured by the equalizing differentials, suggesting regions do converge conditional on these, and other, differentials. 3. Cross-sectional tests for convergence

o--convergence is perhaps the simplest, and most widely used, test for

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G.A. Carlino, L. Mills / Reg. Sci. Urban Econ. 26 (1996) 565-590

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

,

i

,LJ

, i l l ,

,,

i

,

rl

J

i~

ii

ii

,i,

,

1929 1939 1949 1959 1969 1979 1989 Fig. 2. Inequality of regional per capita earnings. Note: Earnings dispersion is measured by the unweighted cross-sectional standard deviation of the log of per capita earnings. convergence. Fig. 2 provides evidence of tr-convergence for the B E A (Bureau of E c o n o m i c Analysis) regions during the 1929-90 period by considering the unweighted cross-sectional standard deviation of the logarithm of relative per capita earnings over time, denoted o-,. In 1932, ~rt rose from 0.29 in 1929 to 0.33. The dispersion in relative regional per capita earnings fell dramatically between 1932 and 1949, declining to 0.17 in 1949. A f t e r 1949, ~rt declined slowly but steadily to 0.09 in 1982. Since 1982, tr, has risen to its late-1960s level of about 0.13, although it declined to 0.12 in 1990. 5 A n u m b e r of studies have documented this pattern of declining cross-sectional dispersion in per capita earnings measures until the late 1970s or early 1980s, followed by increasing dispersion thereafter. 6 B a u m o l (1986) develops a cross-sectional technique to examine fl-convergence in income across countries that involves estimating the following equation: gi = constant + 711,0 + e i ,

(1)

where g~. is the average growth in gross domestic product ( G D P ) per w o r k - h o u r in country i over the sample period, Yi0 is the initial level of G D P per work-hour, and e~ is the error term. /3-convergence implies that y < 0; countries that have a relatively high (low) level of G D P per hour initially grow slowly (quickly). In a cross-sectional regression covering 16 A similar pattern of it-convergence is found for regional per capita income. 6 See Barro and Sala-i-Martin (1991), Browne (1989), Coughlin and Mandelbaum (1988), Eberts (1989), and Rowley et al. (1991).

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G.A. Carlino, L. Mills / Reg. Sci. Urban Econ. 26 (1996) 565-590

countries, Baumol finds evidence in favor of convergence from long-run data on a selected group of countries. Barro and Sala-i-Martin (1991, 1992) use this technique to examine B-convergence in state per capita income as well as gross state product for the United States. In a version of the Baumol equation that is augmented by including variables designed to capture differences in industrial structure across states, Barro and Sala-i-Martin (1991) find evidence for convergence in state-level data. In fact, they find evidence of a rate of convergence of around 2% per year for the U.S. states. We estimated Eq. (1) for the eight B E A regions shown in Fig. 1 during the 1929-90 period for both per capita income and per capita earnings:

Real per capita income gy, i = 3.3343 - 0.000301"/0 ,

(18.2)

R 2 =

0.8677. (2a)

(-6.85)

Real per capita earnings gr. i = 3.0958 - 0.00035Ri0, (16.8)

(-6.23)

R 2 = 0.8440. (2b)

In Eqs. (2a) and (2b) gy. i and gr. i represent compound average annual real growth rates from 1929 to 1990 in the levels of real per capita incomes and real per capita earnings, respectively, and Yi0 and Rio represent the levels of real per capita income and real per capita earnings in 1929, respectively. Figures in parentheses are t-statistics. While Eqs. (2a) and (2b) are subject to a n u m b e r of econometric problems, taken together with o--convergence found for per capita income and per capita earnings, these equations support cross-sectional convergence] The signs on Y~0 and Rio are negative and highly significant. The estimates indicate that for every $1000 that real per capita income (earnings) was above the national average in 1929, it lowered the real per capita income (earnings) growth rate by 0.30 (0.35) percentage points per year during the 1929-90 period. In addition to econometric shortcomings, the Baumol technique has a number of other limitations. This technique examines average growth cross-sectionally over a long period. Consequently, it cannot be used to address the question of how, and if, a shock to a specific region's relative income or relative earnings dissipates over time. This may be an important consideration because national and region-specific shocks may affect different regions in v In addition to well-known concerns raised by De Long (1988), Friedman (1992) and Q u a h (1993) have shown that such cross-sectional tests are biased toward a finding of convergence a m o n g economies.

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573

different ways (Sherwood-Call, 1988). Furthermore, a cross-sectional regression assumes that all regions have an identical rate of convergence. Finally, /3-convergence is not meaningful if shocks to relative regional per capita earnings are highly persistent.

4. Time-series tests for convergence

While the cross-sectional evidence supports the convergence hypothesis, it is possible that relative regional per capita earnings are separate random walks with drift. The reason that we find cross-sectional 'convergence' may be that cross-sectional convergence tests examine only the two end-points in the sample for each region. It is possible that the time series on relative regional per capita earnings is non-stationary, so that the appearance of convergence at the two end-points is random. 8 This suggests that time-series test for convergence is useful in examining the dynamic path of relative regional per capita earnings. We consider the time-series properties of per capita earnings in the U.S. regions relative to per capita earnings in the nation. Initially, we assume that there is a time-invariant compensating-differentials-equilibrium level of relative regional earnings toward which each region is moving over time. U n d e r this assumption, the log of relative per capita earnings in region i at time t, R I , , consists of two parts, the time-invariant equilibrium differential, R I ~ , and the deviations from this equilibrium, u , (for ease in exposition, the region i subscript is suppressed): RI, = RI e + u,.

(3)

It is important to note that we let R l e ~ O, thus allowing for conditional convergence. We model u t as consisting of a deterministic linear trend and a stochastic process: u, = Po + fit + u, ,

(4)

where v0 is the initial deviation from equilibrium and/3 is the deterministic rate of convergence. The process describing u, is a dynamic version of the Baumol (1986) hypothesis: /3-convergence requires that if a region is above its compensating differential initially, i.e. v0 > 0, then it should grow more slowly than the nation, i.e./3 < O. Similarly, if u0 < O, then/3 > O. Unlike the cross-sectional approach, where the rates of convergence are assumed to be Bernard and Durlauf (1995b) show that cross-sectional convergence tests have poor size properties and will frequently reject the no convergence null when the data are actually generated by an endogenous growth model with multiple equilibria.

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574

e q u a l across regions, the time-series e q u a t i o n , Eq. (4), allows the rate of c o n v e r g e n c e to differ across regions. Substituting (4) into (3) yields: where a =

RI, = a +/3t + ~ ,

(5)

R I e + vo .

E s t i m a t e s of a in Eq. (5) do not separately identify R I e and v0. N o n e t h e l e s s , a a n d / 3 should be negatively related u n d e r the cross-sectional notion o f / 3 - c o n v e r g e n c e . 9 T h e time-series notion of stochastic c o n v e r g e n c e is also c a p t u r e d by Eq. (5). T h a t is, for a specific region's earnings to /3-converge to the n a t i o n ' s in the deterministic m a n n e r described a b o v e , d e v i a t i o n s f r o m the relative trend growth, v,, must be t e m p o r a r y . T h e v, t e r m is m o d e l e d as an A R M A ( 2 , 0 ) process, r e p r e s e n t e d by (1 - p L ) ( 1 -

4,L)v,

= e,,

(6)

w h e r e L is the lag o p e r a t o r , p and 4, are the two roots, 14,1 < 1, and e, is the serially u n c o r r e l a t e d shock to v,. Shocks to relative regional earnings will be t e m p o r a r y if Ipl< 1. If lpl = 1, then v, is said to have a unit root, and shocks to relative regional earnings are p e r m a n e n t . ~° Substituting (6) into (5), eliminating the lag o p e r a t o r , and rearranging yields the D i c k e y - F u l l e r test for the p r e s e n c e of a unit root: ARI, = a + bt + c ARI,

~ + dRl,

(7)

i + ~, ,

where a = a ( 1 - p)(1 - 4,) +/3p(1 - 4,) +/34,,(1 - p ) , b =/3(1 - p)(1 - 6 ) , c = p4, and d = -(1-p)(1If d = O , i.e. permanent.~ L

p=l,

¢). then

shocks

to

relative

regional

earnings

are

It is possible for v0 to be large but opposite in sign from R1 °. In this case, a and/3 could be positively related under/3-convergence. The empirical results presented below suggest that this is counterfactual. m Incorporating moving-average terms does not substantially alter the results presented below. Because of space considerations, these results are not reported in this paper but are available upon request from the authors. ~t Solving c and d for p and 4, results in the pair of solutions to a quadratic equation: p, 4, = 0.5(1 + c + d ) - 0.5[(1 + c + d) ~ - 4d] ~'2 . Thus, with two autoregressive terms there is the possibility that p and ~b are a pair of complex roots.

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575

4.1. D a t a a n d e s t i m a t i o n

T h i s s u b s e c t i o n e x a m i n e s t h e t i m e - s e r i e s p r o p e r t i e s for b o t h r e l a t i v e regional per capita income and relative regional per capita earnings. The logarithm of annual data on personal income per capita and per capita e a r n i n g s f o r t h e B E A r e g i o n s d u r i n g t h e p e r i o d 1 9 2 9 - 9 0 a r e u s e d in this s t u d y . 12 C o l u m n t w o o f T a b l e 1 p a r t (a), p r e s e n t s t h e p o i n t e s t i m a t e s o f p a n d t h e r - s t a t i s t i c s in p a r e n t h e s e s for t h e null h y p o t h e s i s p = 1 (d = 0) f r o m e s t i m a t i n g E q . (7) using d a t a o n r e l a t i v e (to the n a t i o n ) r e g i o n a l p e r c a p i t a

Table 1 (a) Persistence in relative per capita income, 1929-1990 Region

p (unit root)

Persistence measures Parametric

Non-parametric

5 (S.E.)

10 (S.E.)

5

10

New England

0.78 -+ 0.15i (-2.30)

0.31 (0.06)

0.10 (0.03)

1.58

2.14

Mideast

0.91 ( - 1.56)

1.05 (0.27)

0.64 (0.38)

1.58

2.17

Great Lakes

0.76 (-2.60)

0.29 (0.18)

0.07 (0.10)

0.97

0.57

Plains

0.87 (-1.80)

0.35 (0.15)

0.18 (0.15)

0.47

0.43

Sout he ast

0.86 (-2.45)

0.50 (0.16)

0.23 (0.16)

1.06

0.91

Southwest

0.91 ( - 1.42)

0.72 (0.22)

0.46 (0.30)

1.15

1.51

Rocky Mountain

0.83 (-2.24)

0.37 (0.17)

0.14 (0.14)

0.72

0.66

Far West

0.72 (-2.57)

0.24 (0.14)

0.07 (0.08)

0.72

0.57

~zIdeally, we should deflate regional per capita incomes using regional price deflators. Since regional price indexes are not available, many authors simply use the national CPI. This deflation serves no purpose in our approach, since all of our income measures are already in relative (to the nation) terms.

576

G.A. Carlino, L. Mills / Reg. Sci. Urban Econ. 26 (1996) 565-590

Table 1 (continued) (b) Persistence in relative per capita earnings, 1929-1990 Region

p (unit root)

Persistence measures Parametric

Non-parametric

5 (S.E.)

10 (S.E.)

5

10

New England

0.77 ---0.64i (-2.17)

0.29 (0.06)

0.08 (0.03)

2.23

2.61

Mideast

0.88 ( - 1.62)

0.72 (0.26)

0.38 (0.31)

1.26

1.46

Great Lakes

0.76 (-2.64)

0.26 (0.16)

0.07 (0.09)

0.94

0.49

Plains

0.76 (-2.63)

0.16 (0.10)

0.04 (0.05)

0.38

0.29

Southeast

0.72 (-3.13)

0.26 (0.16)

0.05 (0.08)

0.91

0.57

Southwest

0.87 (-1.63)

0.58 (0.24)

0.29 (0.27)

1.09

1.29

Rocky Mountain

0.74 (-2.64)

0.20 (0.14)

0.44 (0.07)

0.64

0.50

Far West

0.83 (-2.20)

0.33 (0.16)

0.13 (0.13)

0.83

0.71

Note: The critical values for the unit root tests are generated by Monte Carlo simulations.

** Indicates significance at 5% (critical value of -3.49); * indicates significance at 10% (critical value of -3.18). i n c o m e s o v e r the e n t i r e sample. 13 U n d e r the null hypothesis of a u n i t root, t h e s e statistics are n o t t-distributed. H o w e v e r , we t a b u l a t e d the d i s t r i b u t i o n of the r - r a t i o s from M o n t e C a r l o s i m u l a t i o n s . T h e p o i n t estimates of p r a n g e in value from 0.7 to 0.9. T h e r-statistics r e p o r t e d in c o l u m n two of T a b l e 1, part (a), i n d i c a t e that the null h y p o t h e s i s of a u n i t root c a n n o t be r e j e c t e d for any of the eight regions. T h e finding t h a t p is n o t significantly different f r o m u n i t y implies that shocks to relative r e g i o n a l p e r capita i n c o m e s are p e r m a n e n t . T h a t is, o n c e s h o c k e d , relative r e g i o n a l per capita i n c o m e s do n o t r e t u r n to a d e t e r m i n i s t i c t r e n d . 13Given that the existing literature considers per capita income convergence, we first discuss our findings for relative regional per capita income and take up relative regional per capita earnings later.

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577

Tests for a unit root are often criticized because they have low power against alternatives close to one. Moreover, as pointed out by Cochrane (1988) and others, a permanent component may be present in a time series, but this component may not be responsible for a large proportion of the total variation in the series. In response to these criticisms, researchers have used parametric and non-parametric methods to examine the amount of persistence in a time series. Following this recent literature, we rely on persistence measures rather than unit root tests to gauge the persistence of shocks to relative regional per capita incomes and relative regional per capita earnings. 14 The parametric approach to measuring persistence was developed by Campbell and Mankiw (1989) and uses the moving-average representation implied by an A R I M A model. Persistence is then estimated by examining the effect of a unit innovation in each region in period t on the deviation of R I from trend in period t + N, denoted ~J,t+No In our case, N

~t+N = Z p N - i ~ t* i-O

(8)

N i W h e n p = 1, (8) becomes ~'£t+N ~- ~'i=0 (~ and the effect of a unit innovation approaches ( 1 - ~ b ) -1 as N - - - ~ . Thus if u~ is a random walk, and therefore R I , is a random walk, then ~b = 0, which implies that/x,+~ = 1. In this case, a unit deviation from trend today leads to a unit deviation indefinitely. If ~b < 0, then tz,+~ will be less than 1 and the unit root is 'less important' than in a random walk. Similarly, 4) > 0 implies/x,+~ > 1 and the unit root is 'more important' than in a random walk. The point estimate of the impulse response function, /2,+N, depends on the estimated values of p and 4) as well as the time horizon N. For a point estimate of p that is less than one, /2,+ N goes to zero as N goes to infinity. In this case, while shocks are not literally permanent, they may be highly persistent, and /2t+N is a measure of that persistence. Column three (four) of Table 1, part (a), presents the parametric persistence measures for N = 5 (10) years. For every region, at least 24% of today's shock remains 5 years out. 15 Less persistence is indicated 10 years out for all regions, but substantial persistence is nonetheless found for a n u m b e r of regions.

~4Levin and Lin (1992, 1993) have developed unit root tests in panel data that overcome some of the power problems of univariate estimators. Bernard and Jones (1995a) use these techniques to analyze the contribution of individual sectors to convergence of aggregate productivity for the U.S. states, and Bernard and Jones (1995b) apply these techniques to 14 OECD countries. ~5The numbers in parentheses in Table 1 are estimated standard errors, computed using a first-order Taylor series approximation.

578

G.A. Carlino, L. Mills / Reg. Sci. Urban Econ. 26 (1996) 565-590

O n e problem with this measure of persistence is that it may be sensitive to the parametric specification of the model. A n alternative, non-parametric measure is the variance-ratio statistic developed by Cochrane (1988): N

1 Var(RIt+ u - RI,) - 1+ 2 ~(1V ( N ) =- N Var(Rlt+ 1 - RI,) i=1

(]/N))",,

where n i is the ith autocorrelation of ARI,. The intuition for this nonparametric measure is similar to that for the parametric measure. For example, if R I t is a random walk, then ARIt is serially uncorrelated and n i = 0 for all i. Hence V ( N ) = 1 in this case. The variance-ratio will be greater than one if A R I t is predominantly positively autocorrelated, which is analogous to the case where th > 0. Similarly, the variance-ratio will be less than one if A R I t is predominantly negatively autocorrelated. The final two columns of Table 1, part (a), present the variance-ratio estimates for N = 5 and N = 10 years. For N = 5, the point estimates are close to or greater than one for the majority of regions. The standard errors for these statistics depend on the null hypothesis being tested. The usual case tested is a random walk, i.e. the null that V ( N ) = 1. In this case, the asymptotic standard error is 0.36 given the 61 observations and N = 5.16 On the basis of this standard error, the point estimates are within two standard deviations of unity. Persistence is still evident after 10 years. However, as with the parametric measures for N = 10, inferences are hard to draw because of the large estimated standard errors. For the random walk null, the estimated standard error is 0.49, and we cannot reject this null hypothesis.

4.2. Per capita earnings

While the convergence literature has focused on relative per capita personal income, one problem is that this measure includes income earned outside the region, i.e. transfer payments, interest, dividends, and rent. The convergence hypothesis, however, depends on the migration of factors to eliminate differences in earnings across regions. Browne (1989) reports that earnings per worker accounted for most of the shifts observed in per capita income in the 1970s and 1980s, with property income reinforcing those in earnings. Transfer income played a smaller role in these two decades. To i~ Following Campbell and Mankiw (1989), we report the bias-corrected version of the variance-ratio statistic: [ T / ( T - N ) ] V ( N ) . This correction adjusts for the bias in the statistic due to drift. The asymptotic standard error of the estimated V(N) is the true V(N), under the null, divided by [3/4(T/(N + 1))] 1~2.

G.A. Carlino, L. Mills / Reg. Sci. Urban Econ. 26 (1996) 565-590

579

see what difference might result in our sample, we repeated the estimations using per capita earnings instead of per capita income. ~7 Column two of Table 1, part (b), presents the point estimates of p and the t-statistics for the unit root hypothesis. As with per capita income, the unit root hypothesis is not rejected at the 10% level for any region using data on per capita earnings. Columns three and four of Table 1, part (b), present the p a r a m e t r i c persistence measures for N = 5 and N = 10 years. The final two columns of Table 1, part (b), present the non-parametric measures of persistence for N = 5 and N = 10 years. The results for both the parametric and non-parametric measures of persistence presented in part (b) of Table 1 are quite similar to those presented in part (a), although they display s o m e w h a t less persistence. Nonetheless, the bulk of the evidence still seems to support the no convergence null. T o summarize, the findings suggest that p = 1 and, more generally, that a large degree of persistence is present in relative regional per capita income and per capita earnings. This finding is inconsistent with the stochastic notion of regional convergence toward some constant equilibrium.

5. Allowing for a trend break in 1946

P e r r o n (1989) and Campbell and Perron (1991) emphasize the importance of p r o p e r modeling of the deterministic c o m p o n e n t in time-series tests of persistence. T h a t is, by constraining the model to follow a single trend t h r o u g h o u t the entire sample, a one-time change in the deterministic path could mistakenly be interpreted as a persistent stochastic shock. In the present context of regional convergence, it is important to consider two types of shifts in the deterministic trend: (i) the compensating differentials equilibrium m a y have changed, a n d / o r (ii) the rate at which the various regions are converging may have changed. Accordingly, our time-series model can be modified to allow for a break at time t*. Let R I ~ represent the equilibrium and Ul, the r a n d o m error for 1 <~ t ~< t* - 1: R I 1 , = R I ~ + ul, ;

where Ult

=

Vl0 "~- [31t + Vt .

(9)

Similarly, for t/> t*: R I 2 , = R I 2 + u2, ,

where u2, = V2o + [32(t - t* + 1) + v,.

(10)

Combining (9) and (10) and modeling the error term as an A R M A ( 2 , 0 ) ~7Ideally, we should examine earnings per worker instead of earnings per capita. Data on regional employment are, however, available only after 1947, providing too few observations for time-series analysis. In addition, time series on employment at the regional level based on consistent sampling techniques are available for even shorter time periods.

G.A. Carlino, L. Mills / Reg. Sci. Urban Econ. 26 (1996) 565-590

580

process described by (6) yields the model that will be used to test for stationarity around a broken trend: A R I t = a i D ~ + a 2 D 2 + b l D t ~ + b z D t 2 + c ARIt_~ + d R I , _ 1 + et ,

(11)

where a I = ai(1 - p ) ( 1 - ~b) +/31p(1

--

q~) +/3~b(1 - ~b),

a 2 = %(1 - p)(1 - ~b) +/320(1 - ok) +/32¢b(1 - 4 , ) ,

b 1 =/3,(1 - p)(1 - ~b), b 2 =/32(1 - p)(1 - ~b), a I = R I ~ +/21o Ol2

=

,

R I ~ + V2o ,

D~=I

for t < t *

and

D~=0

D 2 =0

for t < t *

and

0

2 =

for t > 1 t * , 1

for t / > t * ,

DT l=t

for t < t *

and

DT~=O

DT 2=0

for t < t *

and

DT 2=t+t*

for t / > t * , fort/>t*.

Easterlin (1960) reports a b r e a k in the convergence trend during the World War II years in his per capita earnings study. Following the m e t h o d o l o g y of Carlino and Mills (1993) we impose an exogenously d e t e r m i n e d b r e a k in 1946 on both relative per capita income and relative per capita earnings. Loewy and Papell (1995), hereafter referred to as LP, and D e m e e s t e r and Van R o m p u y (1994), hereafter referred to as DV, have e x t e n d e d our basic model by allowing for endogenously determined b r e a k points--specifically they use an asymptotic approach suggested by Zivot and A n d r e w s (1992). After endogenously estimating the break year, these authors find somewhat m o r e evidence for rejection of the unit root null in relative regional per capita incomes than that reported in Carlino and Mills. H o w e v e r , as with the unit root test, the power of the LP and D V test to discriminate precisely enough between trend break dates that are close is p r o b a b l y low. In fact, the alternative estimates of the b r e a k dates from LP and D V are in the 1940-50 range, which is consistent with Carlino and Mill's 1946 date. So there seems to be both anecdotal and statistical evidence that s o m e w h e r e around World War II, the evolution of relative regional incomes changed. But as the papers by LP and D V m a k e clear, the choice of 1946 as the b r e a k year does not alter our findings in an appreciable way. In addition, our test for stochastic convergence primarily relies on the analysis of several persistence measures and not on unit root tests because of the p o o r power properties of unit root tests.

G.A. Carlino, L. Mills / Reg. Sci. Urban Econ. 26 (1996) 565-590

581

Table 2, part (a), presents the r-statistics for the null hypothesis p = 1 and the persistence measures for per capita income, allowing for a break in 1946. The results presented in Table 2, part (a), provide evidence supporting the stationarity of shocks to relative regional earnings. The point estimates of p are low, and the unit root hypothesis is rejected in three of the regions analyzed in this paper. The parametric persistence measures indicate little or no persistence 5 and 10 years out. The non-parametric persistence measures given in the final two columns of Table 2, part (a), support these conclusions. Table 2, part (b), presents the r-statistics for the null hypothesis p = 1 and the persistence measures for per capita earnings, allowing for a break in 1946. The findings for stationarity with a break in 1946 are not as strong for the per capita earnings version. Although the point estimates of p are low, the unit root hypothesis is rejected only for the Plains region. In addition, the parametric persistence measures tend to be higher 5 years out for per Table 2 (a) Persistence in relative per capita income, 1929-1990, break in 1946 Region

p (unit root)

Persistence measures Parametric

Non-parametric

5

10

(S.E.)

(S.E.)

5

10

New England

0.67 ± 0.35i (-3.74)

0.22 (0.06)

0.05 (0.02)

1.09

1.06

Mideast

0.56 ± 0.29i (-3.54)

0.10 (0.03)

0.01 (0.00)

1.09

1.06

Great Lakes

0.25 -+ 0.21i (-4.39)**

0.00 (0.00)

0.00 (0.00)

0.96

0.53

Plains

0.56 (-4.60)**

0.04

0.00

0.35

0.26

(0.05)

(0.00)

Southeast

0.56 (-2.91)

0.07 (0.10)

0.00 (0.01)

0.73

0.39

Southwest

0.69 (-2.58)

0.17 (0.15)

0.03 (0.05)

0.79

0.88

Rocky Mountain

0.26 ± 0.13i (-3.97)*

0.00

0.00

0.56

0.35

(0.00)

(0.00)

0.66 (-2.41)

0.09 (0.15)

0.01 (0.04)

0.31

0.19

Far West

582

G.A. Carlino, L. Mills / Reg. Sci. Urban Econ. 26 (1996) 565-590

Table 2 (continued) (b) Persistence in relative per capita earnings, 1929-1990, break in 1946 Region

O (unit root)

Persistence measures Parametric

Non-parametric

5 (S.E.)

10 (S.E.)

5

10

New England

0.71 -+0.22i (-2.86)

0.23 (0.06)

0.05 (0.02)

1.29

1.29

Mideast

0.69 (-2.40)

0.31 (0.25)

0.05 (0.12)

1.04

1.04

Great Lakes

0.61 (-3.35)

0.10 (0.11)

0.01 (0.02)

0.92

0.45

Plains

0.53 (-4.19)**

0.02 (0.03)

0.00 (0.00)

0.31

0.23

Southeast

0.48 (-3.35)

0.07 (0.13)

0.00 (0.01)

0.76

0.39

Southwest

0.78 (-2.17)

0.33 (0.21)

0.10 (0.14)

0.92

1.00

Rocky Mountain

0.64 (-3.12)

0.11

0.01

0.60

0.42

(0.11)

(0.03)

0.68 (-2.76)

0.13 (0.12)

0.02 (0.04)

0.33

0.22

Far West

Note: The critical values for the unit root tests are generated by Monte Carlo simulations. **Indicates significance at 5% (critical value of -4.16); *indicates significance at 10% (critical value of -3.83).

capita earnings, c o m p a r e d with per capita income. H o w e v e r , as with the per capita i n c o m e results with a b r e a k in 1946, the p a r a m e t r i c and nonp a r a m e t r i c persistence m e a s u r e s indicate little or no persistence 10 years out for the per capita earnings data. T a b l e 3, part (a) (part (b)), presents the c o m p l e t e regression results for p e r capita i n c o m e (earnings) with a break in 1946. T h e c o l u m n h e a d e d a 1 shows the estimated differentials in relative regional per capita i n c o m e s (earnings) in 1929, while the c o l u m n headed/31 gives the estimated average a n n u a l rate o f c o n v e r g e n c e by region for the 1929-45 period. T h e results r e p o r t e d in parts (a) and (b) of Table 3 s u p p o r t cross-sectional or r c o n v e r g e n c e . In every case, a negative relationship b e t w e e n &~ and /31 is

-0.041 (-1.83)

-0.33 (-9.66)**

-0.71 (-19.00)**

-0.60 (-5.99)**

-0.22 (-5.75)**

Great Lakes

Plains

Southeast

Southwest

Rocky Mountain

0.28 (8.02)**

0.41 (12.15)**

Mideast

Far West

0.42 (6.16)**

oq

New England

Region

0.14 (8.98)**

0.064 (2.82)*

0.21 (17.02)**

-0.015 (-0.86)

-0.16 (-4.90)**

-0.39 (-26.01)**

0.058 (-4.26)**

0.12 (12.30)**

a2

- 0.0013 (-0.44)

0.011 (3.50)**

0.024 (3.48)**

0.019 (6.29)**

0.013 (4.66)**

0.0043 (2.26)*

-0.013 (-4.73)**

- 0.019 (-3.39)**

B1

- 0.0029 (-6.31)**

-0.0019 (-2.98)*

0.0016 (1.38)

0.0065 (12.11)**

0.00032 (0.64)

-0.0031 (-8.55)**

0.0069 (-3.23)*

0.0012 (1.44)

/32

0.66 (3.19)

0.26 (3.94)

0.69 (4.40)

0.56 (2.40)

0.56 (3.56)

0.25 (3.69)

0.56 (9.01)

0.67 (11.61)

P

Table 3 (a) Estimates of deterministic trend: per capita income, 1929-1990; break at 1946; ARMA (2,0)

-0.09 (-0.41)

-+0.13i (0.27)

0.09 (0.39)

0.14 (0.53)

-0.22 (-1.27)

+0.22i (0.91)

-+0.29i (2.17)

-+0.31i (3.32)

4'

0.21

0.25

0.21

0.28

0.45

0.27

0.36

0.53

R2

0.82

0.84

0.03

0.55

0.39

0.08

0.83

0.83

Q(21)

~

-0.18 (-2.54)**

Rocky Mountain

0.15 (9.06)**

0.13 (4.82)**

0.015 (0.41)

0.18 (10.14)**

-0.031 (-0.09)

- 0.16 (-3.30)**

- 0.38 (-24.28)**

-0.061 (-3.58)**

a2

-0.0015 (-0.42)

0.0084 (1.59)

0.022 (2.16)*

0.014 (4.70)**

0.009 (2.60)**

- 0.0012 ( - 0.42)

-0.0086 (-1.95)*

-0.017 (-2.12)*

/31

-0.0020 (-3.23)**

-0.0010 (-0.96)

0.0023 (1.31)

0.058 ( 10.34)**

0.00043 (0.69)

- 0.0036 (-6.19)**

-0.0004 (-0.44)

0.0017 (1.05)

f12

0.68 (5.03)

0.64 (4.00)

0.78 (6.14)

0.47 (0.77)

0.53 (3.90)

0.61 (3.36)

0.68 (2.86)

0.71 (11.26)

p

-0.13 (-0.72)

-0.01 (-0.54)

0.10 (0.51)

0.32 (0.48)

-0.38 (-2.38)

0.096 (0.42)

0.35 (1.19)

-+0.24i (1.94)

~

R2

0.20

0.19

0.14

0.26

0.45

0.20

0.18

0.45

Q(21)

0.68

0.96

0.11

0.48

0.60

0.57

0.85

0.95

Critical values 5% 10%

aI -2.20 -1.70

a2 -3.25 -2.73

/3t -3.13 - 2.66

Monte Carlo simulations under a unit root null. /32 -2.17 - 1.67

Note: r-statistics in parenthesis. **indicates significance at 5%; *indicates significance at 10%. The following critical values are generated by

0.25 (5.08)**

-0.51 (-3.08)**

Southwest

FarWest

-0.50 ( - 13.28)**

Southeast

0.067 (1.75)

Great Lakes

-0.25 (-5.56)**

0.30 (4.90)**

Mideast

Plains

0.37 (3.33)**

aI

NewEngland

Region

Table 3 (continued) (b) Estimates of deterministic trend: per capita earnings, 1929-1990; break at 1946; A R M A (2,0)

'~

_~ -~ ~

"~

.~

~"

*~

G . A . Carlino, L. Mills / Reg. Sci. Urban Econ. 26 (1996) 5 6 5 - 5 9 0

585

found. TM In general, the rate of per capita earnings convergence in the 1929-45 period is slower than the rate found for per capita income convergence. This suggests that non-labor income played an important role in the convergence of per capita income. The column headed ot2 in Table 3, part (a) (part (b)), shows the estimated differential in relative regional per capita incomes (earnings) that existed in 1946, while the column headed /32 gives the rate of per capita income (earnings) convergence by region for the period 1946-90. With the exception of the Far West region, a slowdown in the rate of convergence after 1946 is indicated for the per capita income data. H o w e v e r , there is little evidence for a slowdown in the rate of convergence after 1946 for the per capita earnings data. In fact, for three regions (Great Lakes, Southeast, and Far West) /32 is somewhat larger in the post-war period. T a b l e 4 presents formal tests to determine which of the two potential sources is responsible for the b r e a k in the deterministic trend in 1946: a change in the compensating differential or a change in the rate of convergence. If the underlying compensating differential remained unchanged, i.e. R I 1 = R I 2 , then the initial deviation in the second subperiod, v20, as described in Eq. (10), would equal the initial deviation in the first subperiod plus the amount of 'catch up', Vl0 +/3~t*. Using R I e = R I 1 = R I 2 and u20 = Ul0 +/31t* in Eqs. (4)-(6), we have: a I=RI

~+vlo

and

a2 "= R l e

+/PlO + fll t* ,

which implies a 2 - a I -tilt*

= O.

(12)

Eq. (12) is a test for an unchanging compensating differentials equilibrium. Column 1 of Table 4, part (a) (part (b)), shows that we cannot reject the hypothesis of an unchanging equilibrium for any region of the United States for per capita income (earnings); all of the r-statistics reported are well below the critical value, even at the 10% level of significance. It thus a p p e a r s that a single compensating differential equilibrium prevailed during the 1929-90 period for either per capita income or earnings. 18Because the level of initial income and earnings is not a regressor in our model, we are not subject to the measurement error identified by De Long (1988). However, if BEA measures of regional per capita incomes and earnings have improved over time for those regions initially far from the national average, then our estimates of a and/3 could be biased toward convergence. Fortunately, regional per capita income and earnings data are not constructed from samples. Instead, the BEA data are taken from the unemployment insurance program (earnings) and IRS records (non-wage income) and are therefore a comprehensive measure of a region's income and earnings. Consequently, we see no a priori reason for our estimates of a and/3 to be biased toward convergence.

586

G.A. Carlino, L. Mills / Reg. Sci. Urban Econ. 26 (1996) 565-590

A second possible source of the break is that the rate at which the various regions were converging changed in 1946. Column 2 of Table 4, part (a) (part (b)), presents a formal test of the hypothesis that/31 =/32 for per capita income (per capita earnings). Considering per capita income, we are able to reject the hypothesis that/31 =/32 for five of the eight regions at the 5% level of significance. In addition, we are able to reject the hypothesis that/31 =/32 at the 10% level of significance for the Plains region; and the Southeast region is marginally significant as well. Therefore, a one-time change in the rate of convergence in 1946 is responsible for the finding of persistence in the entire sample results reported in Table 1, part (a). Our findings suggest that 1946 is not an important break point for per capita earnings. We are not able to reject the hypothesis that/31 =/32 for any region for the per capita earnings version. Taken together, these findings suggest that transfer payments played an important role in per capita income convergence.

Table 4 (a) Test for the sources of trend break in per capita income Region

Unchanging equilibrium n0:42 = al +/3,t*

Unchanging rate of convergence /40:/31 = [3zt*

New England

-0.020 (-0.60)

-0.020 (-3.47)**

Mideast

-0.036 (-1.12)

-0.012 (-4.33)**

Great Lakes

0.0031 (0.12)

0.0074 (3.09)*

Plains

0.032 (1.92)

0.013 (4.62)

-0.026 (-0.97)

0.013 (2.82)

Southeast Southwest

0.00013 (0.00)

0.023 (3.54)**

Rocky Mountain

0.00018 (0.01)

0.013 (3.80)**

Far West

-0.046 (-0.93)

0.0015 (0.62)

G.A. Carlino, L. Mills / Reg. Sci. Urban Econ. 26 (1996) 565-590

587

Table 4 (continued) (b) Test for the sources of trend break in per capita earnings Region

Unchanging equilibrium H0: a 2 = oq + til t*

Unchanging rate of convergence H0:131 = tizt*

New England

-0.025 (-0.46)

-0.018 (-2.19)

Mideast

-0,027 (-0.65)

-0.0082 (-1.79)

Great Lakes

0.045 (0.95)

0.0024 (0.71)

Plains

0.030 (1.22)

0.0089 (2.52)

Southeast

-0.040 (-0.91)

0.0086 (1.97)

Southwest

0.0096 (-0.19)

0.020 (2.11)

Rocky Mountain

-0.019 (-0.01)

0.0094 (3.80)

Far West

-0.071 (-1.35)

0.0045 (0.14)

**indicates significance at 5%; *indicates significance at 10%. The following critical values are generated by Monte Carlo simulations under a unit root null. Critical values 5% 10%

Unchanging equilibrium 3.54 2.99

Unchanging rate of convergence 3.45 2.8800

6. C o n c l u s i o n

T h i s p a p e r e m p l o y s t i m e - s e r i e s t e c h n i q u e s to e x a m i n e w h e t h e r t h e p a t t e r n o f r e l a t i v e r e g i o n a l p e r c a p i t a i n c o m e a n d e a r n i n g s in t h e U n i t e d S t a t e s o v e r t h e p a s t 60 y e a r s is c o n s i s t e n t with t h e c o n v e r g e n c e h y p o t h e s i s . W e a r e g e n e r a l l y u n a b l e to r e j e c t t h e unit r o o t null for a n y o f t h e B E A r e g i o n s for e i t h e r p e r c a p i t a i n c o m e o r e a r n i n g s . M o r e o v e r , w e find t h a t a n u m b e r o f alternative persistence measures indicate that region-specific shocks have h i g h l y p e r s i s t e n t effects. This l a c k o f s t o c h a s t i c c o n v e r g e n c e for U . S . r e g i o n s is c o n s i s t e n t with t h e e v i d e n c e r e p o r t e d in t h e t i m e - s e r i e s s t u d y o f B r o w n et

588

G.A. Carlino, L. Mills / Reg. Sci. Urban Econ. 26 (1996) 565-590

al. (1990). H o w e v e r , we find t h a t allowing for a b r e a k in t h e c o n v e r g e n c e r a t e in 1946 allows us to c h a r a c t e r i z e s h o c k s to U . S . r e l a t i v e r e g i o n a l p e r c a p i t a i n c o m e as t e m p o r a r y , a finding c o n s i s t e n t with s t o c h a s t i c c o n v e r g e n c e . M o r e o v e r , o u r findings for p e r c a p i t a i n c o m e also s u p p o r t t h e c r o s s - s e c t i o n a l n o t i o n o f f l - c o n v e r g e n c e a f t e r a l l o w i n g for a c o m p e n s a t i n g d i f f e r e n t i a l a m o n g r e g i o n s . W e t a k e t h e s e findings as e v i d e n c e s u p p o r t i n g t h e n e o c l a s s i c a l g r o w t h m o d e l . T h e t i m e - s e r i e s findings a r e less c o n c l u s i v e f o r t h e t h e o r e t i c a l l y m o r e a p p e a l i n g p e r c a p i t a e a r n i n g s d a t a . S h o c k s to p e r c a p i t a e a r n i n g s a r e f o u n d to b e m o r e p e r s i s t e n t t h a n shocks to p e r c a p i t a i n c o m e . This i m p l i e s t h a t the r e g i o n a l d i s t r i b u t i o n o f t r a n s f e r p a y m e n t s t e n d s to s m o o t h t h e effects o f d e v i a t i o n o n r e l a t i v e r e g i o n a l p e r c a p i t a e a r n i n g s a n d to r e i n f o r c e t r e n d s in p e r c a p i t a i n c o m e c o n v e r g e n c e .

Acknowledgements W e t h a n k Y a n n i s I o n n i d e s a n d two a n o n y m o u s r e f e r e e s for t h e i r v a l u a b l e c o m m e n t s . W e also t h a n k Sally B u r k e for helpful e d i t o r i a l c o m m e n t s , a n d C h r i s t o p h e r D a w e s for r e s e a r c h assistance. T h e o p i n i o n s e x p r e s s e d h e r e a r e solely those of the authors and do not necessarily represent those of the F e d e r a l R e s e r v e B a n k o f P h i l a d e l p h i a , the F e d e r a l R e s e r v e S y s t e m , o r t h e Federal National Mortgage Association.

References Barro, R., 1991, Economic growth in a cross section of countries, Quarterly Journal of Economics 151,407-443. Barro, R. and X. Sala-i-Martin, 1991, Convergence across states and regions, Brookings Papers on Economic Activity 1, 107-158. Barro, R. and X. Sala-i-Martin, 1992, Convergence, Journal of Political Economy 100, 223-251. Baumol, W., 1986, Productivity growth, convergence and welfare, American Economic Review 76, 1072-1085. Bernard, A.B., 1991, Empirical implications of the convergence hypothesis, Center for Economic Policy Research, Stanford University, 239. Bernard, A.B. and S.N. Durlauf, 1995a, Convergence of international output, Journal of Applied Econometrics 10, 97-108. Bernard, A.B. and S.N. Durlauf, 1995b, Interpreting tests of the convergence hypothesis, Journal of Econometrics, in press. Bernard, A.B. and C.I. Jones, 1995a, Productivity and convergence across U.S. states and industries, Empirical Economics, in press. Bernard, A.B. and C.I. Jones, 1995b, Productivity across industries and countries: Time series theory and evidence, Review of Economics and Statistics, in press. Borts, G.H. and J.L. Stein, 1964, Economic growth in a free market (Columbia University Press).

G.A. Carlino, L. Mills / Reg. Sci. Urban Econ. 26 (1996) 565-590

589

Brown, S.J., N.E. Coulson and R.F. Engle, 1990, Non-cointegration and econometric evaluation of models of regional shift and share, NBER Working Paper 3291. Browne, L.E., 1989, Shifting regional fortunes: The wheel turns, New England Economic Review (May/June), 13-26. Campbell, J.Y. and G.N. Mankiw, 1989, International evidence on the persistence of economic fluctuations, Journal of Monetary Economics 23, 319-333. Campbell, J.Y. and P. Perron, 1991, Pitfalls and opportunities: What macroeconomists should know about unit roots, NBER Macroeconomics Annual (MIT Press, Cambridge, MA). Carlino, G.A. and L.O. Mills, 1993, Are U.S. regional incomes converging? A time series analysis, Journal of Monetary Economics 32, 335-346. Cochrane, J.H., 1988, How big is the random walk in GNP?, Journal of Political Economy 96, 893-920. Coughlin, C.C. and T.B. Mandelbaum, 1988, Why have state per capita incomes diverged recently?, The Federal Reserve Bank of St. Louis Review (September/October), 24-36. Demeester, N. and P. Van Rompuy, 1994, Are U.S. regional incomes converging? Comment, Catholic University of Leuven, mimeo. De Long, J.B., 1988, Productivity growth, convergence, and welfare: Comment, American Economic Review 78, 1138-1154. Dickie, M. and S. Gerking, 1989, Interregional wage differentials in the United States: A survey, in: J. Van Dijk, H. Folmer, H.W. Herzog, Jr. and A.M. Schlottmann, eds., Migration and labor market adjustment (Kluwer, Boston) 111-146. Easterlin, R.A., 1960, Interregional differences in per capita income, population, and total income, 1840-1950, in: Conference on Research in Income and Wealth, Trends in the American Economy in the Nineteenth Century, voi. 24, 73-140. Eberts, R.W., 1989, Accounting for the recent divergence in regional wage differentials, The Federal Reserve Bank of Cleveland Economic Review 25, 14-26. Engle, R.F. and C.W.J. Granger, 1987, Cointegration and error-correction: Representation, estimation and testing, Econometrics 55, 251-276. Friedman, M., 1992, Do old fallacies ever die?, Journal of Economic Literature 30, 2129-2132. Hoover, E. and F. Giarrantani, 1984, An introduction to regional economics (Alfred A. Knopf, New York). Levin, A. and C. Lin, 1992, Unit root in panel data: Asymptotic and finite-sample properties, University of California, San Diego Working Paper 92-23. Levin, A. and C. Lin, 1993, Unit root in panel data: New results, University of California, San Diego Working Paper 93-56. Loewy, M.B. and D.H. Papell, 1995, Are U.S. regional incomes converging? Some further evidence, Journal of Monetary Economics, in press. Lucas, R.E., 1988, On the mechanics of economic development, Journal of Monetary Economics 22, 3-42. Mankiw, N.G., D. Romer and D.N. Weil, 1992, A contribution to the empirics of economic growth, Quarterly Journal of Economics 429, 407-437. Perron, P., 1989, The great crash, the oil price shock, and the unit root hypothesis, Econometrica 57, 1361-1401. Quah, D., 1990, International patterns of growth: I. Persistence in cross-country disparities, mimeo. Quah, D., 1993, Galton's fallacy and tests of the convergence hypothesis, Scandinavian Journal of Economics 95, 427-443. Roback, J., 1982, Wages, rents, and the quality of life, Journal of Political Economy 96, 1257-1278. Romer, P., 1986, Increasing returns and long run growth, Journal of Political Economy 94, 1002-1037.

590

G.A. Carlino, L. Mills / Reg. Sci. Urban Econ. 26 (1996) 565-590

Romer, P., 1989, Capital accumulation in the theory of long run growth, in: R. Barro, ed., Modern business cycle theory (Harvard University Press, Cambridge, MA). Rowley, T.D., J.M. Redman and J. Angel, 1991, The rapid rise in state per capita income inequality in the 1980's: Sources and prospects, Economic Research Service, U.S. Department of Agriculture, Staff Report AGES 9104. Sahling, L.G. and S.E Smith, 1983, Regional wage differentials: Has the south risen again?, Review of Economics and Statistics 65, 131-135. Sherwood-Call, C., 1988, Exploring the relationships between national and regional economic fluctuations, Federal Reserve Bank of San Francisco, Economic Review, 15-25. Solow, R.M., 1956, A contribution to the theory of economic growth, Quarterly Journal of Economics 70, 65-94. Zivot, E. and D.W.K. Andrews, 1992, Further evidence on the great crash, the oil price shock, and the unit root hypothesis, Journal of Business and Economic Statistics 10, 251-270.