Convergence in Panel Data:
Evidence from the Skipping Estimation
Etsuro ShiojiDepartment of Economics and Business
Universitat Pompeu Fabray
August 20, 1997
This paper demonstrates that, unlike what the conventional wisdom says,
measurement error biases in panel data estimation of convergence using OLS
with xed eects are huge, not trivial. It does so by way of the skipping
estimation: taking data from every m years of the sample (where m is an
integer greater than or equal to 2), as opposed to every single year. It is shownthat the estimated speed of convergence from the OLS with xed eects is biased
upwards by as much as 7 to 15%.

This research was partly nanced by the DGICYT grant PB94-1095. I would like tothank Fabio Canova and Xavier Sala-i-Martin for stimulating discussions.
yFrom October 1997, Department of Economics, Yokohama National University, 79-3
Tokiwadai, Hodogaya-ku, Yokohama 240, Japan. e-mail: shioji@econ.ynu.ac.jp.
Convergence in Panel Data:Evidence from the Skipping Estimation 1

1 Introduction
Estimating the speed of convergence has been one of the central issues ofrecent empirical literature of economic growth. At some point, it seemed likethere was a general consensus among economists on this issue: economies
do converge to their respective steady states, but at a very slow rate. However, many of the recent studies cast doubt on this view and conclude thatthe speed of convergence is actually very fast. It is important to resolve
this controversy because dierent speeds of convergence have very dierent
implications for an economy that deviates from its steady state. On one
hand, cross section estimation of speed of convergence generally yield theestimated speed of about two percent (Barro and Sala-i-Martin (1992), forexample). This speed means a half life of deviation from one's steady stateof about 34.3 years. On the other hand, panel data estimation of the speed,
the approach that has been employed by many recent authors, almost always
produce much higher estimates, ranging from 4 to 20% For example, De laFuente (1996) applies OLS with Fixed Eects to the Spanish regional dataand comes up with the estimated speed of about 12%. Canova and Marcet(1995) apply their Bayesian estimation method to the regional data from
Western Europe and countries in Europe, and come up with the estimatedspeed 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 distance estimation method to the Summers Heston data set and comes up with estimates of 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 andVillanueva (1993) apply dierent methods of panel data estimation to theSummers 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 themean of these estimates) would mean a half life of deviation from one's own
steady state of about 6.6 years. Dierent estimates have very dierent implications for growth theories as well. The neoclassical growth model of Solow(1956) and Swan (1956), when one takes a narrow view on capital stock andidenties it solely as physical capital, implies a speed of 5-6 % (Barro andSala-i-Martin (1995)). Hence, if the results from cross section regressions areto be trusted, the denition of capital stock in the model must be extended
to include human capital or some other types of capital. On the other hand,
if the mean of the panel data estimates of about 10% is to be trusted, there
is no way the neoclassical growth model alone can explain such a high speedof convergence, as the speed of 10% implies a negative share of capital if theConvergence in Panel Data:Evidence from the Skipping Estimation 2

neoclassical growth model is taken literally (De la Fuente (1996))1 .
This paper criticizes the panel data estimation of convergence with noninstrumental variables approach for being seriously biased. I will argue thatthe true speed of convergence is likely to be much closer to the ones foundin the cross section regressions. My investigation will concentrate on OLS
with Fixed Eects of De la Fuente (1996), as analytical results are easy toderive in this case. I plan to write a companion paper that deals with themethod of Canova and Marcet (1995), but there I will have to rely heavily
on Monte Carlo experiments. That these estimates are biased itself is wellknown. There are two types of biases. First, these estimates are subject tosmall sample biases. Second, they are subject to measurement error biases.
They both tend to bias the estimated speed of convergence upwards. The
extent of the latter type of biases depends on the standard deviation of themeasurement error relative to that of the true shocks. It has customarily beenargued that this type of bias cannot be too serious, as errors in measurementof income or output in developed countries cannot be so large compared totrue shocks to the economy (Canova and Marcet (1995)).
However, this argument is not sucient to reject the importance of themeasurement error biases when we are talking about estimating long run
tendencies of economies using annual panel data. A typical exercise in thisliterature is to estimate the following model:
yit = (1 ; )  yit; + i + uit(1)
where yit is output per capita of region i in year t relative to, say, the mean of
the whole country, is a parameter that is called the speed of convergence,that is common across regions and periods (j1 ; j < 1), i is the region
specic xed eect term, uit is shock to the true output per capita with mean
0 and variance u which is assumed to be serially and spatially independent.
The main point is that this is a growth model. In other words, this modelis meant to capture long run movements in output per capita. When Solow
wrote his paper on growth, for example, he did not mean his model to bea representation of year-to-year behavior in output. Hence, yit should beinterpreted as long run output per capita, net of short run movements suchas business cycle elements and temporary shocks (e.g. a bad harvest, a
temporary increase in oil prices, eects of having an expo in one year andthus getting millions of tourists and zillions of governmental subsidies, etc.).
The error term uit should also be interpreted as representing shocks to the1

2

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Convergence in Panel Data:Evidence from the Skipping Estimation 3

long run level of output. It is true that, by taking deviations from the country
means, this approach reduces the business cycle elements in regional output
per capita to some extent, but, as regional business cycles are not perfectly
synchronized, and as some regions tend to have more share of sectors thatare sensitive to business cycles than others, it would not completely take out
business cycle elements. From this viewpoint of growth theory, all these short
run components in output per capita should be considered as measurement
errors in a broad sense of the word. Once we take this broad view ofmeasurement errors, it is not clear anymore if these errors can be consideredsmall compared to true shocks.
The question is, then, how to know the extent of biases due to measurement errors in the broad sense. This paper proposes a simple approachbased on the skipping estimation. There is nothing fancy about this. All
it means is that, instead of using the whole yearly data in output, we use
data from every m years, where m is equal to or greater than two. It will
be shown that, if the measurement error biases are trivial, this skipping
should push the estimated speed of convergence upwards compared to thenon-skipping estimate. This is because skipping worsens the small sample
bias. Only when the measurement error biases are serious, skipping pushes
the estimated speed of convergence downwards. This is because skipping
lessens the measurement error bias. Hence, by looking at eects of skippingon the estimated speed of convergence, we can infer the extent of biases due
to measurement errors (in the broad sense).
I present evidence from the US states, the Japanese prefectures, andOECD countries. In all the cases, it is shown that skipping lowers the estimated speed of convergence or at least has no sizable eect, for m not toolarge. Hence, these results suggest that the measurement error biases areserious, not trivial. It is shown that, in some cases, the measurement errors
may be biasing the estimates upwards by as much as 20%.
Before I proceed to the main text, let me clarify what this paper does
NOT say. The purpose of this paper is not to promote panel data estimationof convergence WITH instrumental variables. It is true that the instrumental variable estimator of Anderson and Hsiao (1981), which was employed
by Evans, yields consistent estimates even in the presence of measurement
errors if one takes enough number of lags in picking up instruments. However, I found that this method is way too inecient to draw any meaningfulconclusion: when I applied this method to the US states, the point estimate
for the speed of convergence varied between -4.6% and 8.4%, its standard

Convergence in Panel Data:Evidence from the Skipping Estimation 4errors being between 13% and 19%. In other words, in none of the cases
could I reject the null hypothesis of the true speed being zero. On the other
hand, the GMM approach of Arellano and Bond (1991), the technique usedby Caselli, Esquivel and Lefort (1996), yields a much more ecient estimate.
The problem with this method is that the number of instruments needed increases very quickly as the sample size increases in the time dimension. This
makes use of yearly data impossible in cases where data spans for more than
15 years or so, due to constraints on computer ability. One has to take datafrom every few years in order to avoid crushing a computer. In other words,
the very nature of the method forces one to do the skipping.
The rest of the paper is organized as follows. Section 2 explains how theextent of the biases depends on various parameters. Section 3 explains how
skipping aects these biases. Section 4 presents empirical results and argues
that OLS with Fixed Eects produces serious biases. Section 5 concludes.

2 Biases in the OLS with Fixed Eects
It has been established that, even in the absence of measurement errors, thatis, even if the model in equation (1) is a correct one, the OLS with xedeect estimator for , which will be denoted by ^ here, is subject to small
sample bias. Let's say there are N regions (i = 1; 2; :::; N ) and the sample
last for T periods (t = 1; 2; :::; T ). Then, as N goes to innity with T xed,
^ does not converge to the true . Nerlove (1971) shows this using Monte
Carlo experiments. Nickell (1981) derives the precise analytical expression
for the bias:^ ; = B =B
whereT
B = T1  (T ; 1) ; T  (1 ; ) + (1 ; )
1

1

2

2

2

(
T)
1
1
2

(1
;
)
(
T
;
1)
;
T

(1
;
)
+
(1
;
)
B = 1 ; (1 ; )  1 ; T ;

:
T
2

2

2

2

These expressions are shown in Hsiao (1986). The bias is positive unless T is
too small. Hence, the estimated speed is biased upwards. For a reasonably
large value of T , the above bias is approximately^ ;  (2 ; )=(T ; 1)

Convergence in Panel Data:Evidence from the Skipping Estimation 5(Nickell (1981)). Note that, even if we are fortunate enough to have T as
large as 100 (which is almost unheard of in the literature of convergence
regressions), for a reasonable value of (say between 0:001 and 0:1), theupward bias is about two percent2. If what we are interested in were thevalue of the AR(1) coecient, 1 ; , this bias would not look that big. But
here what we are truly interested in is the value of and thus the bias oftwo percent could have a signicant consequence. For example, if the true
speed of convergence is two percent as many economists who work on crosssection regressions claim, then, the bias of two percent would mean that theestimated speed of convergence would on average be double the true speed.
The two percent convergence implies a half life of deviation from steadystate of 34.3 years. The four percent convergence implies a half life of 17.0
years. Thus the economic interpretation of the result would be substantiallydierent. As T goes to innity, this bias disappears.
When the measurement error is present, the model has to be rewritten
in the following way. Let yit be the observed output per capita of region i inperiod t, relative to the country mean in that period. Assume that yit canbe described by the following model:yit = (1 ; )  yit ; + i + uit
1

andyit = yit + vit
where yit is the true long run output per capita and vit is the measurement
error with mean 0 and variance v which is also assumed to be serially andspatially independent. Distributions of both uit and vit are assumed to be
space- and time-invariant. The model in equation (1) is equivalent to theabove model with vit = 0 for all i and t. If the measurement errors arepresent, it can be shown that the above expression for the asymptotic bias
for N = +1 is modied to:^ ; = (B + B )=(B + B )
where

  
1
T
;
1
B = (1 ; ) ; (1 ; )  T + T  v
u
2

1

3

2

4

2

3

2

A beauty of the small sample bias is that it is not very sensitive to the true value of .
Therefore, as long as one can assume that the true falls in some reasonable range, alikely range of values for this bias can be narrowed down fairly well. This, unfortunately,
is not the case for the measurement error bias.
2

Convergence in Panel Data:Evidence from the Skipping Estimation 6

and

  
1
B = 1 +  v :


2

T
u
Note that B is positive for any T > 1, and B is always positive. Hence,
the whole bias is still positive. It now depends on three things: the true ,T , and the ratio between the standard deviation of measurement errors andthat of the true shock, v =u . To measure the contribution of measurement
errors to the whole bias, note that, as stated before,
B =B  (2 ; )=(T ; 1)
while
B =B  1 ;hence, for a reasonably large T , B =B < B =B . It follows that the presenceof measurement errors tends to worsen the upward bias in the OLS with xedeects estimator for the speed of convergence.
The question is how important the worsening of the bias is. Table 1 shows
the asymptotic bias for dierent values of T and v =u. Note that these arebiases for the speed of convergence, , rather than the AR(1) coecient 1; .
Hence, a positive value means that the estimated speed is biased upwards,
and therefore that 1 ; is biased downwards. Moreover, the numbers are inpercentages. Table 1A corresponds to the case where the true speed is two
percent and in Table 1B the true speed is set to 10%. First, look at therst column in each of the tables. In this column, the standard deviation of
measurement errors is set to zero. Hence, in this case, only the small sample
bias exists. It can be seen that the bias is severe even for high values of T suchas 60 or 120. Next, look at the last row in each table. Here, T is set to +1.
Hence, in this case, there is no small sample bias and only the measurement
error bias exists. Note that the measurement error bias really kicks in only
when the ratio v =u is fairly large. For = 2%, the bias becomes large
only for the ratio greater than 0.5. For = 10%, the bias becomes large forthe ratio much greater than 0.2. As pure measurement errors are expected
to be much smaller than the true shock term, if we take the narrow view ofmeasurement errors, the bias cannot be that important. This is the reason
why the measurement error bias has not been considered as a big problem inthe literature. But if we take the broad view of measurement errors, values
such as v =u = 0:5 or even 1 are quite possible. And in such a case, bias
due to these errors could be serious. Moving to the other rows of the table,
it can be seen that the measurement error bias is even more serious when
4

3

1

2

3

4

4

1

2

3

4

Convergence in Panel Data:Evidence from the Skipping Estimation 7

the sample size is nite. For example, compare the last row of Table 1A(T = +1) with the fourth row (T = 60). In the last row, in the absence ofthe small sample bias, increasing the ratio v =u from 0 to 0.5 increases the
upward bias in the speed of convergence by less than 1 percent. However, inthe fourth row, even if T is as large as 60, increasing the ratio v =u from0 to 0.5 increases the upward bias in the speed of convergence by almost 3
percent. Hence, there is no simple relationship between the small sample
bias and the measurement error bias such as (total bias) = (small samplebias) + (measurement error bias). On the contrary, when the sample sizeis nite, the two types of bias reinforce each other to often create ahuge bias. For example, for the case where the true speed is 2%, T = 60
and v =u = 0:50, the total bias is as large as 7.40%, so the estimated speedof convergence will be on average 9.40%. Hence it is important to know theextent of measurement errors, to pin down the likely amount of bias.

Table 1: Biases in Estimated Speed of Convergence: ^ ;in percentage
A: When the True Speed is 2 %Tn 
0 0.1 0.2 0.5
12v
u

10 26.65 27.07 28.30 35.69 51.77
20 13.74 14.01 14.81 19.99 33.96
40 6.83 6.99 7.47 10.70 20.52
60 4.47 4.59 4.95 7.40 15.16
120 2.11 2.19 2.42 4.03 9.34
1000 0.21 0.25 0.38 1.26 4.30
+1 0.00 0.04 0.15 0.96 3.73
B: When the True Speed is 10 %Tn uv
0 0.1 0.2 0.5
1
10 24.32 24.76 26.03 33.58 49.29
20 11.96 12.29 13.27 19.42 34.86
40 5.63 5.88 6.63 11.49 24.94
60 3.60 3.82 4.49 8.86 21.34
120 1.70 1.90 2.48 6.35 17.75
1000 0.19 0.37 0.88 4.34 14.76+1 0.00 0.17 0.68 4.08 14.37

71.89
59.60
44.27
36.16
25.66
14.76
13.40
2
67.69
59.45
50.98
47.24
43.14
39.38
38.86

Convergence in Panel Data:Evidence from the Skipping Estimation 8

3 the Skipping Estimation
This paper tries to investigate the extent of the measurement error bias
using the skipping estimation. The idea is to use the data from every m
years, instead of every single years. Hence, the estimated equation ishiyit = (1 ; )m  yit;m + 1 + (1 ; ) +    + (1 ; )m;  i + emit
whereemit = uit + (1 ; )  uit; +    + (1 ; )m;  uit;m :This model can be estimated with standard OLS with xed eects:
yit = m  yit;m + im + emit(2)
where m is an empirical counterpart of (1 ; )m. Let its estimated value be^ m. Then I recover estimated by the formula^ m = 1 ; ^m=m:
The bias of this estimator can be derived analytically. Appendix A presents
this derivation. The bias is computed numerically under various dierent
assumptions in Table 2 and Table 3. Table 2 deals with the case where
T = +1, or the case when there is no small sample bias. Table 2A shows
the case in which the true speed of convergence is 2%, and Table 2B shows
the case in which the speed is 10%. Dierent rows correspond to dierent
values of m. The values of m is chosen for the sake of comparison with thelater results. It is clear that, in this case, the bias decreases with m. In otherwords, skipping reduces the measurement error bias.
1

1

1

+1

1

Table 2: Biases in the Skipping Estimation
When T is innite(in percentage)
A: When the True Speed is 2 %mn uv
0 0.1 0.2 0.5
12
1 0.00 0.04 0.15 0.96 3.73 13.40
2 0.00 0.04 0.15 0.94 3.66 13.13
3 0.00 0.04 0.15 0.92 3.59 12.87
4 0.00 0.04 0.15 0.90 3.51 12.61
5 0.00 0.04 0.14 0.89 3.44 12.36
6 0.00 0.04 0.14 0.87 3.37 12.11
10 0.00 0.03 0.13 0.80 3.11 11.17
12 0.00 0.03 0.12 0.77 2.99 10.73
15 0.00 0.03 0.12 0.72 2.81 10.10
20 0.00 0.03 0.11 0.65 2.54 9.13

Convergence in Panel Data:Evidence from the Skipping Estimation 9

B: When the True Speed
0 0.1 0.2 0.5
1 0.00 0.17 0.68 4.08
2 0.00 0.15 0.61 3.67
3 0.00 0.14 0.55 3.31
4 0.00 0.12 0.49 2.98
5 0.00 0.11 0.45 2.68
6 0.00 0.10 0.40 2.41
10 0.00 0.07 0.26 1.58
12 0.00 0.05 0.21 1.28
15 0.00 0.04 0.16 0.93
20 0.00 0.02 0.09 0.55

mn uv

is 10 %
1
2
14.37 38.86
12.93 34.98
11.64 31.48
10.48 28.33
9.43 25.50
8.49 22.95
5.57 15.06
4.51 12.20
3.29 8.89
1.94 5.25

Table 3 deals with the case where T is nite. Table 3A and 3B correspondto the case when T is 60, and Table 3C and 3D correspond to the case when T
is 40. In Table 3A and 3C, the true speed of convergence is set to 2%, and in
Table 3B and 3D it is 10%. Note, rst, that, when there is no measurement
error (the rst column in each panel), the bias worsens with m. That is,skipping worsens the small sample bias. When both the small sample
bias and the measurement error bias are present, the result becomes a mixtureof the two extreme cases. When measurement errors are unimportant, saywhen v =u is 0.1 (the second column in each panel), skipping always worsens
the bias. The bias-correcting eect of skipping starts to appear as v =u
increases. With v =u = 0.2 (the third column), there is a minor reductionin the bias only when m is increased from 1 to 2. As m is increased further,the bias worsens. Note that, when the true is high (namely when it is10%) the bias worsens drastically for a large m. With v =u = 0:5 (thefourth column), there is a sizable improvement in the bias till m reaches 4
or 5 and then the bias starts to worsen. Again , the worsening is drastic
when is large. With v =u = 1.0 (the fth column), the bias reaches itstrough when m is much larger (10 in Table 3A, 6 in Table 3B, 8 in Table 3Cand 5 in Table 3D). The conclusion is that, if measurement errors arereally unimportant, skipping should worsen the bias, not improveit. Hence, by studying if skipping increases the estimated (a symptomof worsening bias) or decreases it (a symptom of improving bias), one could
know the extent of the measurement error bias.

Convergence in Panel Data:Evidence from the Skipping Estimation 10

Table 3: Biases in the Skipping Estimation(in percentage)
A: T = 60; = 2%
mn uv
0 0.1 0.2 0.5
12
1 4.47 4.59 4.95 7.40 15.16 36.16
2 4.56 4.62 4.80 6.03 10.04 21.88
3 4.66 4.70 4.82 5.64 8.33 16.46
4 4.77 4.80 4.89 5.50 7.51 13.64
5 4.89 4.91 4.98 5.47 7.06 11.94
6 5.01 5.03 5.09 5.49 6.81 10.81
10 5.67 5.68 5.71 5.93 6.64 8.72
B: T = 60; = 10%mn uv
0 0.1 0.2 0.5
12
1 3.60 3.82 4.49 8.86 21.34 47.24
2 3.86 3.97 4.30 6.53 13.21 29.26
3 4.16 4.24 4.46 5.96 10.53 21.97
4 4.52 4.58 4.75 5.88 9.37 18.25
5 4.96 5.00 5.14 6.06 8.89 16.19
6 5.49 5.52 5.64 6.42 8.84 15.09
10 9.83 9.85 9.94 10.55 12.45 17.97
C: T = 40; = 2%v
mn  u0 0.1 0.2 0.5
12
1 6.83 6.99 7.47 10.70 20.52 44.27
2 7.01 7.09 7.33 8.95 14.05 27.77
4 7.43 7.47 7.58 8.37 10.88 17.80
5 7.68 7.71 7.80 8.41 10.36 15.68
8 8.68 8.69 8.74 9.05 10.03 12.52
D: T = 40; = 10%mn uv
0 0.1 0.2 0.5
12
1 5.63 5.88 6.63 11.49 24.94 50.98
2 6.08 6.20 6.58 9.05 16.28 32.58
4 7.30 7.36 7.55 8.80 12.52 21.35
5 8.15 8.20 8.35 9.36 12.36 19.47
8 13.30 13.33 13.43 14.09 16.02 20.61

Convergence in Panel Data:Evidence from the Skipping Estimation 11

4 Evidence from the Skipping Estimation
Evidence from the US states

Skipping requires a reasonably long series. Perhaps the longest existingseries that has been used in the literature of convergence is the US state percapita personal income. BEA has collected data every year from 1929. The
latest year for which the data was available at the time of this research was
1996. This means that there is data for 68 years, for the 48 contiguous state.
Using the values for 1929 as the initial values (yi ), we are still left with
T = 67 years of data. Thus the US states data seems to be the most ideal
place to start with. Table 4 shows the estimated speed of convergence from
OLS with Fixed Eects estimation for dierent values of m. Unfortunately,
67 is not as nice a number as 60: it cannot be divided by too many integers.
Hence, the sample size is dierent for dierent values of m. Dene Tm as thelargest integer that is equal to or smaller than 67 and that can be divided bym. Then Tm is the sample size for that particular value of m. The second
column shows this sample size, and the third column shows the implied lastperiod of the sample. For example, for m = 2, the largest integer that issmaller than 67 that can be divided by m is 66, so the sample period starts
from 1929 (for t = 0) and ends in 1995 (t = 66). The fourth column isthe estimated speed of convergence in percentage, and the last column isthe standard error of the estimate, also in percentage (see Appendix B for
the formula). All the data sources used in this paper are summarized inAppendix C.
0

The table shows that, when m = 1, that is, when there is no skipping,
the estimated speed of convergence is 8.6%. This result of fast convergenceis comparable to the ndings of De la Fuente (1996) and Canova and Marcet(1995). The question is if this is because is truly large, or it simply reects
a bias that is inated by the presence of measurement errors. Going down therows, the table shows the general tendency for the estimated to decrease
with m for smaller values of m and then to rise with m for larger m's. Thatis, the bias improves as skipping becomes larger, at least for smaller m's.
This could not happen if measurement errors were truly negligible. In fact,
the estimate reaches its bottom at m = 11, which suggests that measurement
errors are actually quite large. In Table 3 in the previous section, Panel A(in which T is set to 60) shows that, if the true speed is 2%, the bias reaches
its minimum at m = 10 only when the ratio v =u is as large as 1.0. In
such a case, Table 3A indicates that the non-skipping estimate of the speed(the case where m = 1) is biased by as much as 15%. Even if we take aconservative stance and assume that v =u is 0.5, the non-skipping bias isConvergence in Panel Data:Evidence from the Skipping Estimation 12

still over 7%. Hence, the estimated speed of convergence is subject to aserious upward bias.

Table 4: Evidence from the US States
the estimated speed is in percentage
m Tm end year ^ m std.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

67
66
66
64
65
66
63
64
63
60
66
60
65
56
60
64

96
95
95
93
94
95
92
93
92
89
95
89
94
85
89
93

8.64
7.48
6.67
7.33
6.84
6.20
6.02
5.73
6.37
5.21
3.90
4.72
5.07
5.28
7.06
6.36

Evidence from the Japanese prefectures

0.67
0.63
0.59
0.64
0.61
0.57
0.58
0.53
0.61
0.55
0.44
0.52
0.52
0.58
0.68
0.58

Data for the Japanese per capita Prefectural Income is available for 19501990. As I was particularly concerned with the quality of the 1950 data (seeAppendix C), I use the 1951 values for initial values (t = 0). There are
currently 47 prefectures in Japan but data on Okinawa is not available forearlier years as it was under US occupation till 1972. Hence the sample size
is T = 39 and N = 46. The result is summarized in Table 5 in the same way
as in Table 4. First, note from the rst row that the non-skipping estimate
suggests a very high speed of convergence of 16:9% per year. However, going
down the rows, there is a broad tendency for the estimate to decrease withm (with a notable exception of the case m = 2) for smaller m and then toincrease with m. The bottom is reached at m = 6. Table 3C and 3D (inwhich T is set to 40) suggest that this can happen only if the ratio v =u
is greater than 0.5. Moreover, the reduction in the estimated speed due toskipping is very large: between m = 1 and m = 6, the estimated speed goes
down by more than 6%. A further analysis showed that such a large drop
can happen only when v =u is between 0:7 and 0:8. In such cases, the bias

Convergence in Panel Data:Evidence from the Skipping Estimation 13

bottoms at m = 5 (if the true is 2%) or at m = 4 (if the true is 10%
and v =u is 0:7), which is broadly consistent with the nding in Table 5.
Assuming that v =u is 0:7, the non-skipping estimate (the case in whichm = 1) is biased upwards by 14.11% (if the true is 2%) or by 16.38% (ifthe true is 10%). This can be seen from the rst rows of Table 3C and 3D.
Hence, in either case, it can be concluded that the non-skipping estimate isbiased upwards by almost 15%.

Table 5: Evidence from the Japanese Prefectures
the estimated speed is in percentage
m Tm end year ^ m std.
1
2
3
4
5
6
7
8
9

39
38
39
36
35
36
35
32
36

Evidence from OECD countries

90
89
90
87
86
87
86
83
87

16.88
18.65
12.00
13.69
12.84
10.27
11.02
12.59
12.56

1.18
1.34
1.00
1.13
1.14
1.02
1.09
1.27
1.29

Data on GDP per capita for 24 OECD countries is taken from the SummersHeston data set. The data is available for the period from 1950 to 1990.
Hence, using the values for 1950 as initial values, in this case, N = 24 andT = 40. The result is summarized in Table 6. There is no clear tendency
for the estimated speed to either go up or go down with m. In fact, given
the relatively large standard errors, it could reasonably concluded that theestimate is more or less invariant to m. Hence, the result is not as clear
cut as the previous two cases. But looking at Table 3C and 3D suggests
that, as long as v =u is small, say less than 0.2, one should nd a clear
pattern of the estimated speed increasing with m. Hence, the result suggeststhat v =u should be at least larger than 0.2 even in this case. Even if it
is 0.2, the non-skipping estimate would still be biased upwards by around
7% (the rst rows in Table 3C and 3D). Hence, the conclusion is that thenon-skipping estimate for OECD countries is likely to be biased upwards byat least around 7% even in this case.

Convergence in Panel Data:Evidence from the Skipping Estimation 14

Table 6: Evidence from OECD countries
the estimated speed is in percentage
m Tm end year ^ m std.
1
2
3
4
5
6
7
8
9
10

40
40
39
40
40
36
35
40
36
40

90
90
89
90
90
86
85
90
86
90

5.72
6.05
6.01
5.77
5.07
6.16
6.10
5.18
6.20
5.12

0.74
0.79
0.75
0.76
0.68
0.76
0.77
0.67
0.78
0.67

5 Conclusion
This paper has shown that the speed of convergence estimated by OLS with
Fixed Eects is subject to a large upward bias, not only because of the small
sample bias but also because of the measurement error bias.
Researchers who use the panel data approach to study convergence criticize the cross sectional approach for not using all the data available, andthus throwing away a lot of potentially important information that is there.
They argue that the panel data approach is superior because it makes useof all the information in the data. However, the convergence equation is a
growth theory equation. It is not supposed to explain year-to-year behaviors
in output per capita. The measurement error bias, when measurement errors are dened in the broad sense, is a bias that comes from using too richinformation for a simple model. Use of a rich data set would require a richmodelling of economies that goes beyond a simple convergence equation thatcan be derived from, for example, the Solow-Swan model.

Appendix
A Biases in the Skipping Estimation
To derive the bias in the estimated coecient m, the AR(1) coecient in
the skipping estimation, note rst that equation 2 is no dierent from theConvergence in Panel Data:Evidence from the Skipping Estimation 15

usual OLS with xed eects model except that the length of each period is minstead of 1. So, the formula for the biases in OLS with Fixed eect, which
is presented in Section 2 of the text, can be directly applied. All one needs
to do is to replace T  in the original formula by T=m. Also, 1 ; has tobe replaced by m, and u, the standard deviation of the true shock, has tobe replaced by em. Then one obtains^ m ; m = ;(Bm + Bm )=(Bm + Bm )where
1  (T=m ; 1) ; (T=m)  m + m T=m
Bm = (T=m)(1 ; m)
1

1

3

2

4

2

2

(
T=m )
1
1
2

(
T=m
;
1)
;
(
T=m
)

+
m
m
m
B = 1;
 1 ; T=m ; (1 ; ) 
(T=m)
m
m
! 

1
T=m
;
1

v
Bm = m ; m  T=m + (T=m)  
em
2

2

2

2

2

3

and

2

! 

1
Bm = 1 + T=m  v :
2

4

em

In the above, em is the variance of emit which is2em = 11;;(1(1;; ))  u:
m

2

2

2

2

As I estimate the AR(1) coecient by taking^ m = 1 ; ^m=m
1

and noting that

= 1 ; m=m;
1

the bias in the estimated is^ m ; = ; [(1 ; )m ; (Bm + Bm )=(Bm + Bm )] =m + (1 ; ):
1

3

2

4

1

Convergence in Panel Data:Evidence from the Skipping Estimation 16

B Standard Errors for the Skipping Estimator
The original (non-skipping) model is
yit =   yit; + i + uit
where  is dened as 1 ; . Dene y~i as the deviation of yit from its regional(country) mean. Then the above model can be rewritten as
y~it =   y~it; + uit:
(3)This model can be estimated by OLS, which is equivalent to maximum likelihood. The skipping model can be written as
y~it =  y~it;m + eit
(4)
where
= m
(5)
and
eit = uit +   uit; +   uit; +    + m;  uit;m :(6)Equation (4) can be estimated by OLS to yield ^ and ^ e , and they are
their respective maximum likelihood estimators. Note that maximizing the
likelihood with respect to and e is equivalent to maximizing it with respectto  and u . Hence, by using equations (5) and (6) on ^ and ^ e , we can derive
the maximum likelihood estimators for  and u which I will denote ^ and^ u :
1

1

2

1

1

2

+1

2

2

2

2

2

2

An asymptotic variance for ^ is given by
"
#;
@
ln
L
AV AR (^) = ; @ () ;2e  2e
1

2

2

=^

=^

where ln L is the log likelihood function of the model (4). The question iswhat the derivative inside the bracket is. Applying the chain rule for thesecond derivative,
2

!2

3

@ ln L = 4 @ ln L  @ + @ ln L  @ 5@ ()@ ( ) @
@ @ ()
2
3
!
@
ln
L
@
(

)
@
ln
L
@
(

)
+ 4 @ ( )  @e + @ ( )  @ ()e 5
e
e
2

2

2

2

2

2

2

2

2 2

2

2

2

2

2

Convergence in Panel Data:Evidence from the Skipping Estimation 17

When evaluated at the maximum of the likelihood function, the second term
inside each of the squared brackets is equal to zero, so the above expression
simplies into!
!
@ ln L = @ ln L  @ + @ ln L  @ (e ) :
(7)
@ ()
@ ( ) @@@ (e )
I will rst deal with derivatives of and e with respect to . From equation(5), the rst one is given by@ = m  m; :@Next, from equation (6),
2

2

2

2

2

2

2

2 2

2

2

1

h

i

 e = 1 +  +  +  +    +   m;   u :Hence," ;#@ (e ) = 1  mX
i@
 i 2  i    u
2

2

4

2(

1

2

= 1


2

=1

"m;1
X

i=1

2

1)

2

2

3;1
# 2m
X
 i  2i  4 2(j;1)5  2e :j =1

For the second derivatives of the log likelihood function with respect to andu in equation (7), evaluated at the maximum of the likelihood function, I
approximate them by the negative of the inverse of their estimated variance:
2

# 1

" 2

;
^
= AV AR (^) V AR (^); @@ (ln)L ;2e 2e
2

and

" 2

=^

=^

# 1

;
 
^  
@
ln
L
; @ ( )= AV AR ^ e V AR ^ e = N2  ^Tem :e  ;2e 2e
Hence,
2

2 2

=^

=^

h

i2

^

AV AR (^); = m  ^m;  V AR (^);
1

4

2

1

8
;1
< "mX
N  Tm  >>  i=1
:

+ 2

1
^

2

1

2
3;1 9
# 2m
>
=
X
 i  2i  4 2(j;1)5 >; :
j =1

^

^

Convergence in Panel Data:Evidence from the Skipping Estimation 18

C Data Sources
US data

The original data is all 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
1992-1996, I downloaded the data from the home page of BEA. I linked thisseries with the previous series using the growth rate.

Japanese Data

The original data is almost entirely from the Economic Planning Agency(EPA) of Japan, Prefectural Economic Accounts. However, between 1950 and
1955, values for some prefectures are missing from the ocial data. Miyohei
Shinohara et. al., Chiiki Kozo no Keiryo Bunseki (Econometric Analysis
of Regional Structure, in Japanese) estimates these missing values. I use
these estimated numbers whenever the ocial numbers were not available. I
linked slightly dierent series using the growth rates at: 1965, 1975 and 1980.
Note on the data for 1950: The 1950 value for Tokyo, the wealthiest of theprefectures, appeared to be problematic. In 1950, its total prefectural income,according to the data, was 565.5 (in billions of yen). In 1951, it went down to513.2, while the rest of Japan grew at the rate of 26.3% (nominal). Between
1951 and 1952 no such obvious anomaly was found. Hence, I concluded thatthe data for 1950 was distorted, perhaps due to a typing mistake, and decided
not to use it. Hence, the whole analysis for Japan starts in 1951.

OECD data

All the data is taken from the Summers and Heston data set, available
from the home page of the University of Toronto. I used the series coded
RGDPL, real GDP per capita. I excluded member countries that participatedOECD recently, for the sake of comparison with other research and also forthe sake of data availability. The 24 countries in my sample are: Canada,
USA, Austria, Belgium, Denmark, France, Germany, Greece, Iceland, Ireland, Italy, Luxembourg, the Netherlands, Norway, Portugal, Spain, Sweden,
Switzerland, Turkey, UK, Japan, Finland, Australia, and New Zealand.

Convergence in Panel Data:Evidence from the Skipping Estimation 19

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