It's Still 2%:
Evidence on Convergence from 116 Years of the
US States Panel Data
Etsuro ShiojiDepartment of Economics and Business
Universitat Pompeu Fabray
September 21, 1997
Abstract

A new debate over the speed of convergence in per capita income acrosseconomies is going on. Cross sectional estimates support the idea of slow
convergence of about two percent per year. Panel data estimates support
the idea of fast convergence of ve, ten or even twenty percent per year.
This paper shows that, if you do it right, even the panel data estimation
method yields the result of slow convergence of about two percent per year.

 This research was partly nanced by the DGICYT grant PB94-1095. I would like to thankFabio Canova and Xavier Sala-i-Martin for stimulating discussions.
y From October 1997, Department of Economics, Yokohama National University, 79-3 Tokiwadai, Hodogaya-ku, Yokohama 240 Japan. E-mail: shioji@econ.ynu.ac.jp.

1. Introduction
There has been a debate on what the right speed of convergence in per capita
incomes across economies is. On one hand, economists who use cross sectional
approach tend to nd a low estimated speed of convergence of about two to three
percent per year. On the other hand, recent studies that take advantage of thepanel dimension of the data using xed eect approach tend to nd much faster
speed of convergence, ranging from 5% to as large as 20%. This dierence isimportant not only because the two sharply dierent conclusions have very different welfare implications but also because they have very dierent implications
for model building in growth theory.
A problem with the existing panel data studies is that the sample periods are
too short. As some of the panel data techniques, most notably OLS with xedeects, are plagued by small sample biases, it would be ideal to use a longer timeseries.. The US states personal income data oers an ideal ground for this kind of
empirical studies in this respect. Not only has the BEA collected the data since
1929 till 1996 (so far), earlier estimates for the years 1880, 1900 and 1920 are
available from Easterlin (1960). All together, we currently have the richness of
116 years of data on growth, although data is missing for some earlier years. OLS
with xed eects cannot easily deal with this kind of data with missing points,
but I show that, with maximum likelihood approach, this situation can be dealt
with in a straightforward manner.
What I nd is the following. If I use the data from after 1929, estimatedspeed of convergence is much faster than two percent. However, once I take into
account the period prior to 1929, the estimated speed becomes much closer to two
percent. One may argue that this is because there was an important structural
break around 1929, and that we could not put the periods before and after the
break together to estimate the speed of convergence. The core part of the paperargues that this is not the case. Indeed, this dierence across the periods is exactly
what one would expect from econometric theory, even if there was no structural
break, given the nature of the data. In fact it will be argued that the true speedof convergence must be much closer to the estimate that includes periods both
before and after 1929. Hence, it is concluded that, even when one uses panel dataapproach, if the data is right, the method uncovers the true speed of convergence which is far from 10% or 20%, as some authors who took this approach argue
for, but is close to 2%, as researchers who use the cross sectional approach claim.
The rest of the paper is organized as follows. The next section oers a quick
overview of the literature. Section 3 explains the econometric approach. Section2

4 demonstrates results from estimation. It is shown that estimated speed ofconvergence is fairly dierent, depending on whether the period prior to 1929 isincluded or not. Section 5 explains why this does not have to mean that there
was an important structural break and that we could not put the two periods
together into a single sample. This section further argues that the estimates fromthe sample that includes years prior to 1929 are much more likely to be close to
the true speed of convergence. Section 6 concludes by saying that the true speedof convergence seems to be close to 2%, rather than 10% or 20%.

2. Overview
Estimating the speed of convergence has been one of the central issues of recent
empirical literature of economic growth. At some point, the general consensus
among economists on this issue was that economies converge to their respective
steady states at a very slow rate of two percent per year. This speed means a halflife of deviation from one's steady state of about 34.3 years. However, many of
the recent studies cast doubt on this view using panel data estimation with xedeects, which almost always produces much higher estimates for the speed ofconvergence that ranges from 4 to 20% For example, De la Fuente (1996) applies
OLS with Fixed Eects to the Spanish regional data and comes up with theestimated speed of about 12%. Canova and Marcet (1995) apply their Bayesianestimation method to the regional data from Western Europe and countries in
Europe, and come up with the estimated speed of about 20% for the former and
10% for the latter (though the values vary depending on the precise specication).
Islam (1995) applies OLS with Fixed Eects as well as the minimum distanceestimation method to the Summers Heston data set and comes up with estimatesof between 4% and 9.3 % depending on the method and countries included in the
regression. Also, Casselli, Esquivel and Lefort (1996), Evans (1995), and Knight,
Laoyza and Villanueva (1993) apply dierent methods of panel data estimation to
the Summers Heston data set and come up with estimated speed of convergencemuch higher than 2%. A speed of 10% (which seems to be more or less the meanof these estimates) would mean a half life of deviation from one's own steady stateof about 6.6 years. Dierent estimates have very dierent implications for growththeories as well. The neoclassical growth model of Solow (1956) and Swan (1956),
when one takes a narrow view on capital stock and identies it solely as physicalcapital, implies a speed of 5-6 % (Barro and Sala-i-Martin (1995)). Hence, if theresults from cross section regressions are to be trusted, the denition of capitalstock in the model must be extended to include human capital or some other typesof capital. On the other hand, if the mean of the panel data estimates of about3

10% is to be trusted, there is no way the neoclassical growth model alone can
explain such a high speed of convergence, as the speed of 10% implies a negative
share of capital if the neoclassical growth model is taken literally (De la Fuente(1996))1 .
A problem with the past studies on convergence that used panel data is thatthe sample period tends to be very short. In an extreme case, the European regions
sample in Canova and Marcet (1995) spans a period of about ten years only. The
Summers and Heston data set runs for about 40 years. As our focus is to uncoverlong run relationship among per capita regional (national) per capita income,there is a worry that these are not suciently long. In fact, it is known thatOLS with Fixed Eect estimation of convergence is subject to important small
sample biases (Nerlove (1971), Nickell (1981), Hsiao (1986), Shioji (1997)), when
the sample is small in the time dimension. In a sample of 40 years, assuming thatthe true speed of convergence is under 10%, the estimated speed of convergenceis asymptotically (in the sense of the number of regions or countries approachinginnity) biased upwards by about 5%. In that sense, the US state personal income
series seems to be the ideal data set to examine the robustness of the results of
the recent studies, as the available data spans the period of 1880 to 1996, or 116
years.

3. Maximum Likelihood Estimation
It is known that, when the data exists for every year between the starting year
and the end year, OLS with Fixed Eects estimation coincides with maximum
likelihood estimation (Hsiao (1986)). However, in my sample, data from earlier
years estimated by Easterlin is available for years 1880, 1900 and 1920 only, andthe BEA data set starts from 1929. This situation is tricky to handle by OLS
etc., but it can be handled in a straightforward manner by maximum likelihood.
Assume that the model is

yit ; yi =   (yit; ; yi) + uit

(3.1)

1

where yit is output per capita of region i in year t relative to, say, the mean of thewhole country, yi is its steady state value which is assumed to be time-invariant
but could vary across the states,  1 ;  is the speed of convergence, thatis common across regions and periods (jj < 1), i is the region specic xedeect term, uit is shock to the true output per capita which follows N (0; u) and2

On the other hand, if one takes the lower bound estimate of 4%, this value is consistent
with the neoclassical growth model with the narrow denition of capital stock.
1

4

is assumed to be serially and spatially independent. Suppose that the data isavailable for N regions (i = 1; 2;   N ) for a set of periods denoted by T which isT  ftj gzj> tj for all j . Equation (3.1) implies, for each j=0where t = 0; tz = T and tj
between 1 and z,
yit ; yi = t ;t ;1  (yit ;1 ; yi) + eit
whereeit = uit +   uit ; +   uit ; +    + t ;t ;1  uit ;1 :Note thatt ;t ;1 u :V AR(eit ) = 1 ;1; 
0

+1

j

j

j

j

j

j

j

2

1

j

j

j

2

2( j

j

j

)

j

j

+1

2

2

The maximum likelihood estimators for , u, and yi are given by maximizing
the following log likelihood function
L  ; Nz  ln 2 ; Nz  ln  + Nz  ln(1 ;  )
2

2

u

2

z 

X
; N  ln 1 ;  t ;t ;1
2( j

2

j =1

1

2

N X
z
X
;

;

2 2u i=1 j =1

j

2

2

)

h

1





i2

 ; t ;t ;  y
 :

y
;
y
;
y
it
it
i
;
i
t
;
t
;
1;
2( j

j

j 1)

j

j 1

j 1

The estimated variance-covariance matrix of the unknown parameter is given by
the negative of the inverse of the Hessian of the above function evaluated at the
maximum.

4. Empirical Evidence

4.1. Evidence from Non-Skipping DataFirst I present the result for the case when the whole available data is used. As
was stated previously, the data is from 1880, 1900, 1920 and every year from 1929
to 1996. Out of 50 US states, earlier data is not available for Alaska and Hawaii,
and the 1880 data is not available for Oklahoma. That leaves 47 states in mysample (N = 47). Table 1a (subtitled No-SKIP) demonstrates the result. To
save space, only estimated speed of convergence  1 ;  together with its standard
error and estimated u are reported. In the rst column I report the results when
the whole sample is used. In the second column I report the result when only thedata from 1880, 1900, 1920 and 1929 are used and in the third column I report5the result for the case when only the data from 1929-1996 is used (the method forthis case is OLS with xed eects which coincides with maximum likelihood).
Table 1a (No-SKIP)
Estimation result when the whole data is usedperiodall 1880,1900,20,29 1929-96
1;
3.91%
3.64%
8.6%
(std.) (0.30%)(0.74%) (0.67%)
u 0.042
0.032
0.042
First, note that the estimated speed of convergence for the case when all thedata is used is much lower than what is normally found in the literature of panel
data estimation of convergence. Second, the estimated speed is much lower in theearlier period than in the later period. Third, the estimated speed from the overall
period is much closer to that of the earlier period than that of the later period.
Hence, several questions arise. (1) Why is the estimated speed so much lower thanwhat is normally found using similar methods? (2) Why is the estimated speed
so much lower than the earlier period? Isn't that an indication that there was animportant structural break between the two periods? That is, isn't it unjustiable
to put the two periods together into one sample and estimate a single equation?
I investigate these questions in the rest of the paper.

4.2. Measurement Error?

One dierence between the earlier data and the later data is that the former is an
estimate done retrospectively after many years from whatever data was available,
whereas the latter is a series of estimates made contemporaneously using the datacollected with the specic purpose of estimating personal incomes. Hence, it isconceivable that the earlier data is subject to measurement errors of greater extent.
However, this does not explain the dierence in the estimated speed of convergence
between the two periods. As is shown in Shioji (1997), measurement errors tend
to bias the estimated speed of convergence upwards. That is, if anything, thetrue speed of convergence in the earlier period is likely to be even lower than theestimated one. Hence, it does not explain why the earlier estimate is so much
lower than the later one.

6

4.3. Evidence from the Skipping Estimation

Another dierence between the earlier data and the later one is that the former isthe skipped data, that is, it does not come from every single year, whereas thelatter does come from every single year. Shioji (1997) shows that, for the case of
OLS with Fixed Eects estimation, when the data is subject to large measurement
errors, skipping (taking data from every m years (m  2) as opposed to every
single year) tends to cut down the upward bias that exists due to these errors, atleast for m not too large. That is, skipping tends to lower the estimated speedof convergence (and brings it closer to the true speed). Hence, the presence ofskipping in the earlier sample and its absence in the later sample may explain why
the estimated speed is lower in the former. And Shioji (1997) also shows that,using the data from 1929 to 1996, skipping indeed lowers the estimated speed ofconvergence for the US states, which suggests that the data is in fact subject tolarge measurement errors. And he argues that, when we talk about the convergence regression, the word measurement errors must be dened in the broad
sense: whatever is a deviation from the growth model, such as business cycle components and temporary shocks to income, has to be considered as measurement
errors from the viewpoint of growth theory.
Now I investigate this possibility. Obviously we cannot solve this problem
by nding data for the earlier period, which does not exist. Instead I will see
what happens when I do skipping for the later sample. In Table 1b (labelled
SKIP-2), I use data from every two years for the period between 1930 and
1996. In Table 1c (labeled SKIP-5), I use data from every ve years for theperiod between 1930 and 1995. In Table 1d (labeled SKIP-10), I use the datafrom every 10 years for the period between 1930 and 1990. In each of these tables,in the rst column I report the estimated speed of convergence for the case where
the data from 1880, 1900, 1920 are combined with the data from the later period.
In the second column I use data from only after 1930. In Table 1e (labelled
SKIP-20), I use the data from every 20 years. In the rst column, I use thedata from 1880, 1900, 1920, 1940, 1960, 1980, and 1996. In the second column,
the data from the rst half, namely 1880, 1900, 1920 and 1940 is used. In thethird column the data from 1940, 1960, 1980 and 1996 is used.
Table 1b (SKIP-2)
For 1930 onwards, every two years
period 1880, 1900, 1920, 1930-96 1930-96
1;
3.07%
7.6%
(std.)(0.40%) (0.60%)
u0.036 0.034
7

Table 1c (SKIP-5)
For 1930 onwards, every ve years
period 1880, 1900, 1920, 1930-95 1930-95
1;
2.46%
6.2%
(std.)(0.28%) (0.52%)
u0.030 0.042
Table 1d (SKIP-10)
For 1930 onwards, every ten years
period 1880, 1900, 1920, 1930-90 1930-901;
2.44% 4.73%(std.)(0.29%) (0.50%)
u0.030 0.045
Table 1e (SKIP-20)
Every twenty years (except between 1980 and 1996)
period 1880-1996 1880-1940 1940-1996
1;
2.21%
3.48%
5.68%
(std) (0.23%) (0.51%) (0.66%)
u
0.028
0.029
0.017
A quick comparison from Table 1a through Table 1e reveals that the estimatedspeed of convergence goes down monotonically as the skipping becomes larger andlarger, when the whole period is used as a sample. When only the latter period
is used, the estimate goes down monotonically except between SKIP-10 andSKIP-20. These are evidence that the non-skipping estimates in Table 1a aresubject to large upward bias due to measurement errors. Thus the true speed ofconvergence must be close to about 2% in Table 1d or 1e than about 4% in Table
1a. Also, note that the dierence between the whole period estimate and the later
period estimate shrinks somewhat as the skipping becomes larger. This suggeststhat the huge dierence between these two in Table 1a was at least partially due
to the fact that there was a fair amount of skipping for the earlier period but there
was no skipping in the later period. However, the most important message from
Table 1d and 1e is that, even with a large skipping, a large dierence between
the whole period estimate and the later period estimate remains. Hence, it looks
like the doubt that there was an important structural change between the earlier
period and the later period is justied.
In the next section it is shown that this needs not be the case. This dierence
between the whole period (or the earlier period) estimate and the later periodcounterpart is exactly what is expected from econometric theory. Estimates fromthe whole sample in tables 1d and 1e are fairly reliable estimates of the true speedof convergence.
8

5. Initial Value Distribution and Biases in Estimated Speedof Convergence
5.1. Case of OLS (Analytical results)

In this section I argue that the dierence between the whole period estimate (or theearlier period estimate) and the later period estimate is exactly what is expectedfrom theory. I will rst discuss the case of OLS with Fixed Eects, as analyticalresults are easy to derive in this case. In the next subsection I will discuss thecase of Maximum Likelihood using Monte Carlo experiments. Intuition behind
the two results is the same.
I will consider the general case where measurement errors are present, as thisseems to be the case in the data I am dealing with. Suppose that the true model
is equation (3.1), but that the observed relative income per capita, denoted xit,
is subject to measurement errors:
xit = yit + vit
where vit is the measurement error with mean 0 and variance v which is assumed
to be serially and spatially independent. Distributions of both uit and vit are
assumed to be space- and time-invariant. Let's say an econometrician use thisobserved incubi per capita to estimate an equation
xit =   xit; + i + uit
using OLS with Fixed Eects. Nickell (1981) and Hsiao(1986) derive the asymptotic bias (as N goes to innity) in the estimate for the case without measurement
errors. Shioji (1997) generalizes their result to the case with measurement errors:^ ;  = ;(B + B )=(B + B )where
1 (T ; 1) ; T   + T
B =T 
(1 ; )
2

1

1

1

2

(

1

1

2



4

2

2
(1

2

2

  
T
;
1
B =  ;   T + T  v
u
3

and

2

)


(T ; 1) ; T   + T
B = 1 ;   1 ; T ; ; ) 
:
T
2

where

3

1

2

2


  
B = 1 + T1  v :
u
9
2

4

The bias is negative unless T is very small.
A potential problem with the derivation of the above expression is that it
assumes that the initial values, yi , are drawn from their stationary distribution.
That is, it is implicitly assumed that the economies start from the stationary
states. This assumption is not necessarily appropriate in many cases in practice,and is denitely unrealistic in the case of the US states. Shortly before the sample
starts there was the Civil War, which most likely sent states far out of theirstationary distributions. In fact, cross sectional dispersion of state per capita
incomes declined almost monotonically between 1880 and the late 1970s (Barroand Sala-i-Martin (1992)) which seems to indicate that the states were far awayfrom the stationary distribution in 1880 and spent nearly a century to get backto their stationary distribution. When the initial value distribution is allowed todier from the stationary distribution, the above expression undergoes a slight but
potentially important change.
The stationary distribution of yi has mean yi andpstandard deviation u = 1 ;  . Now assume that the initial value distribution
has the same mean but the standard deviation is  (> 0) times larger. Then theabove expression becomes:
002^ ;  = ;(B + B )=(B + B + B )
1

where

"

3

2

4

5

#

 T ; ; 1  (1 ; T )   ; 1  1 :
B = T  ;1 ;T (1 ; )
1;
5

1

1

2(

1)

2

2

2

2

2

2

The only change is the presence of B in the new expression. When  is greaterthan 1, that is, when the initial value distribution is more disperse than thestationary distribution (as seems to be the case), B is positive for a large T .
And if that is the case, the absolute value of the bias decreases with . That is, alarge dispersion in the initial value distribution cuts down the bias in theestimated speed of convergence. The intuition is the usual signal-noise argument.
When the economies start out far away from their respective non-stochastic steadystates, the data contains more information on how quickly (or slowly) they moveback toward their steady states. This fact could potentially explain why thewhole period (or the earlier period) estimated speed of convergence was much
lower than the later period estimate. As I argued, it is quite conceivable that thestates started far away from their stationary distribution in 1880. Hence, when
the sample that starts from this year is used, the bias in the estimated speed ofconvergence is likely to be small. However, between 1880 and 1930, the economies
moved closer to their respective stationary distribution. Hence, when the sample
that starts from 1930 (or 1940) is used, the upward bias in the estimated speedof convergence is likely to be large. This could explain why the estimated speed
5

5

10

turned out to be much larger for the later period in the previous section. And if
this is true, there is no need to presume that there was an important structuralchange. In fact, it could be concluded that the whole period estimate of about
2% is a fairly bias-free estimate.

5.2. Case of ML (Numerical results)

As analytical results are hard to derive in the case of Maximum Likelihood, I relyon the Monte Carlo experiment. The experiment is in line with the empirical
exercise in Section 3. Number of regions, N , is set to 47. Total number ofyears, T , is 116. For each trial, I estimate the speed of convergence for theearlier period: t = 0 (for the year 1880); 20 (for 1900); 40 (for 1920) and 49 (for1929) and for the later period: t = 50 to 116 (for the years 1930 to 1996), andalso for the combined period, using maximum likelihood. I set the underlying
parameter values as follows. I set  to be 0.98, which means the true speed ofconvergence of 2%. I set v , the standard deviation of the measurement error
term, to be half of u . This follows the result of Shioji (1997). If we denemeasurement errors as literally errors in measuring incomes, this is probably toolarge. However, as has already been argued, when talking about growth theory,
any short run movements in incomes that are not captured by growth theory
should be considered as measurement errors. From this viewpoint the above
choice of value is by no means extreme. The question is how to choose , theratio between the standard deviation of the initial value distribution and thatof the stationary distribution. The latter can in principle be obtained from theestimates in section 3. However, I discarded this approach as the standard error
around its estimate was huge. Instead, in Table 2, I compute standard deviationin the dierence between yit and the estimated y across the states. The estimatesare based on the case of Table 1e (Skip-20), and I only report the average acrossyears for the later period. Results are similar when other estimates from section
3 are used.

11

Table 2

std. of the deviation from steady state
1880 0.494
1900 0.387
1920 0.223
1929-39 0.234
1940-49 0.166
1950-59 0.116
1960-69 0.080
1970-79 0.092
1980-89 0.084
1990-96 0.079
One gets an impression that, starting from a dispersion far larger than that of
the stationary state, around 0.5, the states more or less converged to the stationary distribution since around 1950, with the standard deviation oating aroundsomewhat below 0.1. Hence, the standard deviation of the initial distribution (ofyit ; yi) seems to be 4-5 times larger than that of the stationary distribution..
From this exercise, I set  = 4. I also set u = 0:019, so that the standard
deviation in the steady state distribution is 0.1. Finally, for each trial, I draw
non-stochastic steady state values yi from a uniform distribution with mean 0
and standard deviation = 0.5.
I run the experiment 100 times and simply take the average. The result isshown in Table 3.

Table 3: Monte Carlo Experiments

whole 0,20,40,49 50-116
average 3.56%
4.11% 7.40%
std. 0.44%
0.94% 0.95%
The rst column refers to the case when the whole period, namely t = 0; 20; 40; 49,
and 50 ; 116 is all used for estimation. The second column reports the case when
only t = 0; 20; 40 and 49 are used for estimation. In the third column I uset = 50 ; 116 only and use OLS for the estimation. The rst row reports theaverage of the point estimate over the 100 trials, and the second row reports thestandard deviation of the point estimate over these trials. Note that, despite thatthe true parameter is the same across periods, the average estimates for the wholesample and for the earlier period are much lower than that for the later period.
And the former two are much closer to the true value than the latter one. Hence,
it can be concluded that the dierences in the estimated speed of convergence12

across periods in section 4 do not necessarily imply the presence of a structural
break between the periods. In fact, the presence of these dierences is exactly
what you would expect from econometric theory, once you take into account thatthe initial distribution of the US states incomes was probably far more dispersethan the stationary distribution. Note how close the experiment results in Table
3 are to the empirical estimates in Table 1a. And the true speed of convergencehas to be much close to the whole period estimate (or the earlier period estimate)than the latter period estimate, that is, it is likely to be much closer to 2% than
8.6% in the third column of Table 1a. Denitely not 10% or 20%.

6. Conclusion
It is always a truth in econometrics that a good data set is a data set that contains
large variations. The US states data set is a good data set to study convergence not
only because a long time series is available but also because it shows a huge initial
distribution due to the Civil War and it records how the states came back toward
their respective steady states from this distribution. That is why it eloquently
tells how the process of convergence looks like. Using this data set, I have shownthat even the panel data estimation method yields the result of slow convergenceof about 2% per year.

Appendix

A. Data Sources
The original data for 1929-1996 is from Bureau of Economic Analysis (BEA).
For 1929-1992, I downloaded the data from Xavier Sala-i-Martin's home page. I
combined the two series that appear in his le using the growth rate. For 19921996, I downloaded the data from the home page of BEA. I linked this series withthe previous series using the growth rate.

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13

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