Space Is more than Geography1
Nathaniel Beck and Kristian Gleditsch
Department of Political Science
University of California, San Diego
La Jolla, CA 92093
USA
Draft of March 22, 2003. Prepared for the Joint Meetings of the European
Consortium for Political Research, Edinburgh, Mar.-Apr., 2003
1 We
thank James Morrow and Randy Siverson for the use of their data. Kyle Beardsley was most
helpful in updating the data and otherwise making it useful for us. James P. LeSage and Kelly Pace
have generously made available their code for spatial analysis in Matlab and were extraordinarily
helpful in answering our numerous questions. Finally, we thank Michael Ward for impressing on
us the importance of space.
Spatial econometric models have begun to make inroads into the study of political science,
and, in particular, the study of international relations.1 Spatial econometrics has its roots in
the study of geography, so, not unnaturally, these applications have typically used geographic
notions in their spatial econometrics. But there is no inherent reason for this. In this paper,
we discuss two very different applications to IR data that, while using spatial econometrics,
do not use geographic notions of space.
Our first application is technical. Much IR analysis is based on the dyad year design.
Much attention has been paid to the interdependence of the dyads, with most attention
having been paid to whether the successive annual observations on the same dyad are independent (Beck and Katz, 1996). More generally, time-series–cross-section analysts have
worried about whether observations on different units at the same time point are independent (Beck and Katz, 1995). But while the latter endeavor proved somewhat deleterious,
the notion that different observations are dependent cannot be ignored. Here we consider
one type of interdependence that arises in dyadic data; two dyads that contain a common
member are unlikely to be independent. Perhaps even more seriously, analysts often study
the directed dyad AB and the directed dyad BA; these two dyads are particularly unlikely
to be independent. We use some spatial econometrics to help with this issue. But rather
than use a geographic notion of closeness, we posit that two dyads are close if they share a
common member, and are especially close if they are the reverse of each other. We apply
this approach to the study of politics and trade of Morrow, Siverson and Tabaras (1999).
A more interesting substantive issue is that spatial econometric methods allow us to use
Deutsch’s (1954) notion of community based on flows of people, communication, trade in our
models. Most econometric models assume that each unit (a nation or dyad) is independent of
all others (with all dependencies being in the same unit at different points in time). Spatial
econometricians have begun to break this down, allowing for attributes such as democracy or
war to be a function of those attributes in neighboring units, where neighboring is used in the
usual geographic sense. This alone was a major breakthrough, but we can do better, allowing
for units to be near each other insofar as they are members of a Deutschian community.
We can also use other political notions of neighborliness. Thus two nations may be
neighbors if they share a common IGO membership or are in a joint alliance. Thus, for
example, nations may become more democratic if they are in common market or tariff
agreement with other democratic nations; this is one of the arguments as to why it was
important that China join the WTO. (Or nations may become more democratic in an attempt
to join a common market of democratic states, as the example of Turkey may perhaps show.)
This also can be modeled with spatial methods.
In the next section of the paper we lay out the basic spatial econometric model. Since the
model is well described elsewhere (Anselin, 1988), and we use only well known estimation
techniques, we do not discuss the estimation of spatial models here. But in the following
sections, we show how spatial methods, with non-geographic notions of “distance.” can be
used in several models of interest to IR scholars.
1
See, for example, the various articles in the special issue of Political Analysis (10:3), as well as Gleditsch
(2002) and Gleditsch and Ward (2000).
1
1
The spatial econometric model/
While much of our interest is in time-series—cross-section (TSCS) data, where units (nations
or dyads) are observed annually over a long time period, we begin with a simple crosssectional setup.
Let yi represent some dependent variable of interest, and, as usual, we assume it is a
linear function of covariates, xi and some unmeasured variables, “the error,” εi , so
yi = xi β + εi .
(1)
Linearity here is purely for ease of notation, and any non-linear extensions available in typical
econometric models are available in the spatial framework.
The most critical assumption, other than that the specification is correct, is that the
covariates are independent of the error process. We begin with that assumption also. However, we weaken the assumption that the error process (and hence the yi |xi ) are independent
across observations. The basic spatial insight is that “errors” (best though of as omitted or
unmeasured variables) in unit i are related to the “errors” in nearby units. While nearby
is typically used in a geographic sense, there is no reason why we cannot use any notion of
nearness that we prefer, so long as this is specified by the analysts.
Spatially lagged errors
Letting wi be a vector for how “close” the other observations are to unit i, and setting
wi = 0, and letting ε be the vector of all errors, we get the “spatially lagged error” model
yi = xi β + εi + λwi0 ε.
(2)
If λ = 0, this reduces to the standard non-spatial model.
If λ 6= 0, OLS is still consistent, but the reported standard errors will be wrong and the
estimated β̂ will be inefficient. This can be fixed by typical GLS reasoning, though complications require a full maximum likelihood estimation.2 Our interest here is on alternative
weighting schemes that are more relevant to political scientists than the typical geographic
schemes used by geographers (where the elements of W are either zero or one based on
contiguity, or a measure of the physical distance between unit i and the various other units).
The spatially lagged error model corresponds to the model in time series analysis where
the errors show some temporal correlation process. The analogy to the time series serially
correlated error model is useful. This analogy tells us that the only way that observations are
interdependent is through unmeasured variables that are correlated, in this case across space.
The model is odd, in that space matters in the “error process” but not in the substantive
portion of the model. Moreover, if we add a new variable to the model, so we move it from
the “error” to the substantive portion of the model, the spatially lagged error model assumes
that this variable no longer has a spatial impact of nearby observations. This assumption
2
For this and all other econometric details, the reader should consult Anselin’s (1988) text.
2
seems to us hard to defend. It is the case, however, that the spatially lagged error model
does seem appropriate for our problem of dealing with interconnected dyads. We return to
this issue in the next section.
If we have TSCS data, with observations indexed both by unit i, and time, t, (and
assuming without loss of generality a rectangular data structure), the spatially lagged error
model becomes
yi,t = xi,t β + εi,t + λwi0 εt
(3)
where εt is the vector of errors for all units i at time t. Implicit in this notation is that
all spatial impacts occur instantaneously, so that only contemporaneous errors show spatial autocorrelation. We will typically deal with temporal dynamics by including a lagged
dependent variable in the model, yielding
yi,t = φyi,t−1 + xi,t β + εi,t + λwi0 εt .
(4)
This equation is only easy to estimate if the error process shows no temporal correlations,
so the lagged y is independent of the error process. This assumption is often reasonable in
practice, and can be tested via a Lagrange multiplier test (Beck and Katz, 1996).
Spatial Lags
The spatially lagged error model corresponds to the time-series serially correlated errors
model. The “spatial lag” model corresponds to the time-series lagged dependent model. In
this model, the dependent variable is impacted by the values of the dependent variable in
nearby units, with nearby suitably defined. It differs from the spatially lagged errors model,
in that both the error term and the covariates in nearby units impact the current unit. Thus,
for example, let the dependent variable be the level of democracy in a country. Is it likely
that this is partly a function of democracy in nearby countries, rather than just being related
to common unmeasured variables in nearby units?
Again, we start with the simple cross-sectional model. Using the same notation as above,
and letting y be the vector of values for y, the spatial lag model has
yi = xi β + κwi0 y + εi .
(5)
This is a difficult model to estimate, but it can be done by complicated maximum likelihood.
As above, we are interested in weighting vectors that are more politically inspired than the
distance or proximity weights used by geographers.
This model can be generalized for TSCS data by
yi,t = xi,t β + κwi0 yt + εi,t
(6)
where the notation implies that the weighting vector is time invariant (although it is easy to
allow it to vary over time, so long as this is specified ex ante). Here yt refers to the vector
of observations on the dependent variable for all units at time t.
3
Since TSCS models normally show temporal dynamics, we can add a temporal lag of y
to the model, yielding
yi,t = xi,t β + φyi,t−1 + κwi0 yt + εi,t .
(7)
As above, estimation is much simplified if the remaining errors are temporally independent,
which can be assessed via a Lagrange multiplier test.
TSCS data allow us for an interesting alternative to the spatial lagged model. Suppose
that we continue to maintain that yi,t is related to the neighboring y’s, but we believe that
this impact occurs with a one period lag. This yields the model
yi,t = xi,t β + φyi,t−1 + κwi0 yt−1 + εi,t .
(8)
If the errors are temporally independent, this model is easy to estimate via OLS. In our
applications in Section 3 we compare the results of the spatial lag model with the simpler to
estimate temporally lagged spatial lag model.
2
Allowing for dyadic interdependence
Much recent work in TSCS analysis has attempted to allow for interdependent observations.
But one form of interdependence that has not been well studied is that dyads which contain
a common member may be interdependent, and, in data sets with directed dyads, any dyad
and its reverse should be strongly related.3 .
We can easily define a weighting vector that says that one dyad is near another if both
share a common member; if the two dyads are the reverse of each other, the weight is some
value v > 1; for all other dyads, the relevant weighting entry is zero. While in principal one
could estimate the value of v, here we just try several values and choose the one yielding the
best fit.
Because y must be continuous,4 we chose a data set dealing with the political determinants
of trade (viewed as exports directed from A to B). While in principal we can deal with very
large data sets, in practice we have not been able to analyze our largest data set, which
consists of yearly dyadic trade for all dyads in the political system from 1950–2000. We
therefore decided to reanalyze the basic model of Morrow, Siverson and Tabaras (1999),
which analyzes the political determinants of trade amongst the seven major powers for
the period 1905–90. This study uses the exports of one country to another (in constant
dollars), yielding 42 directed dyads. Their estimation method assumed that all 42 dyads
were independent of each other. Here we compare the spatially lagged error model (Eq. 4 to
a more typical OLS estimation.
3
The only research on this that we know of is by Mansfield and Bronson (1997) which adjoins to each
model two dummy variables, one to represent each dyad. With much IR data, these fixed effects are not
ideal (Beck and Katz, 2001).
4
If y is discrete, the spatial setup is much more difficult and we do not pursue it here. See citeward:gled:2002 for one approach to this problem.
4
While the reader should see the original article for full details, Morrow et al. use both
political and non-political determinants of trade in a gravity model. Thus they regress the
log of exports on the GDP and population of both exporter and importer and the distance
between the two (all values logged).5 ’ Their political variable are whether the dyad is in
a Militarized Interstate Dispute (MIDL), the similarity of their alliance portfolios (TAUL),
whether the dyad is democracies (DEMDL), and whether the two nations are allied multiplied
by a dummy to mark the multipolar era (pre-1948) or the bipolar era (post-1948). Data from
both World Wars I and II were omitted.6
Morrow et al. use a complicated serially corrected error scheme to deal with temporal dependence, and spent much attention on how to model this given the gaps in the observations.
We choose the econometrically simpler, and theoretically more compelling, error correction
model. This model, due originally to Davidson, Hendry, Srba and Yeo (1978), assumes that
trade and the independent variables are in a long run equilibrium relationship; when the
system is out of equilibrium, it adjust back to the long run equilibrium, with the speed of
re-equilibration empirically determined. The model also allows for short run effects. Thus
the first difference of (logged) trade is a function of the first difference of the independent
variables, and the lags of all variables (including the lagged level of (logged) trade). The lag
of the trade variable gives the speed of re-equilibration; the coefficients on the other lagged
variables yield the long run equilibrium relationship (up to a constant). We drop the first
difference of variables such as distance, which are not meaningful (thus, distance only affects
the long run equilibrium relationship).
Observations with missing lagged values are simply dropped.7 This ensures that there is
no mistakes in using a model with one year dynamics. Since our methods are different from
those of Morrow et al., we present only our own results. (To make the output easier to read,
both trade and lagged trade were multiplied by 100; this of course has no impact other than
making the coefficients reported 100 times larger, except for lagged trade.)
The first columns in Table 1 are the results assuming dyadic interdependence (both with
common OLS standard errors and “panel correct standard errors” or PCSE’s.) In this case,
the PCSE’s are similar to the OLS standard errors.8
OLS with a lagged dependent variable is only consistent if the remaining errors are
serially independent. A Lagrange multiplier test for this unfortunately allows us to reject
the null hypothesis of no serial correlation of the remaining errors. Thus our estimates are
inconsistent. However, the estimated serial correlation of the errors is -.10 (with a standard
error of .02), indicating that the errors show little remaining serial correlation. The harm
5
All values were logged because of the nature of the gravity model of trade used by Morrow, et al. This
leads to non-standard coding for dummy variables and such, but since our interest is only on the changes in
coefficients and standard errors, we need not go into this here.
6
Some other missing data was also omitted. Thus measurements on Germany do not start until 1950,
and so there are no German dyadic measures for 1948 or 1949.
7
By using other data sources and filling in some missing data by interpolation where that made sense, we
were able to limit the number of dropped observations to those immediately following the two World Wars.
8
This means that PCSE’s alone are not capable of correcting for the dyadic dependence we find. This is
not surprising, since PCSE’s were not meant to be used in this way.
5
Table 1: The Political Determinants of Exports Amongst the Major Powers 1907–1990
OLS
Dyadic Interdependence
Variable
Coefficient Std. Err. PCSE Coefficient Std. Err.
POLITICAL VARIABLES
FIRST DIFFERENCES
DEMOC
26.14
12.21
8.58
16.36
13.99
MIDL
−14.28
5.22
6.52
−14.02
5.53
TAUL
−11.54
13.65
12.78
−12.50
15.62
ALLIANCE (pre-1940)
−24.92
8.44
7.82
−16.92
9.13
ALLIANCE (post-1940)
−.61
10.68
8.79
−2.59
11.77
LEVEL (LAGGED)
DEMOC
8.80
3.10
2.86
8.84
2.71
MIDL
−42.14
7.00
8.82
−41.83
7.35
TAUL
11.42
7.79
7.06
9.85
8.38
ALLIANCE (pre-1940)
−26.34
5.43
5.10
−18.42
5.93
ALLIANCE (post-1940)
−2.59
4.86
4.39
−1.38
5.27
CONTROLS
FIRST DIFFERENCES
GNP Exporter
POP Exporter
GNP Importer
POP Importer
LEVELS (LAGGED)
GNP Exporter
POL Exporter
GNP Importer
POP Importer
DISTANCE
Trade
Intercept
λ
R2
N=2446
34.72
5.37
13.61
9.20
3.71
5.15
3.66
5.81
4.13
4.96
4.82
4.53
33.99
25.58
12.00
54.32
4.24
54.67
3.70
55.55
3.98
1.48
4.19
0.82
−3.47
−.08
−33.94
1.10
2.19
1.05
2.19
0.90
0.01
11.67
1.22
2.07
1.28
2.44
0.80
.01
14.23
4.73
1.55
4.66
0.91
−4.20
−0.09
−37.35
.38
0.67
0.46
0.65
0.39
0.96
0.01
14.50
0.01
.13
6
.19
induced by the small amount of serial correlation in the errors is small, and the known
corrections for this harm are difficult to use, impose their own harm, and the degree of this
harm is not proportion to the harm caused by the small amount of serial correlation of the
errors. Thus we work with the OLS results using the lagged dependent variable, simply
noting they are not as perfect as we would prefer.
Since we have an error correction model, we must interpret both the long run equilibrium
values and the short run features of the model. First, we note that trade is very slow to
return to equilibrium, with only about 8% of the disequilibrium value offset in any given
year. Trade moves very slowly (or appears very smooth). However, eventually trade will
return to its equilibrium value.
Starting with the political variables we see that democratic dyads have a higher equilibrium level of trade, and that dyads which are frequently engaged in a militarized dispute
have a lower equilibrium level of trade. While dyads with similar alliance portfolios tend
to trade more, this result does not attain standard levels of significance. Dyads which were
allied in the multipolar (pre-World War II) period tend to trade less (in equilibrium); in the
post World War II bipolar period there is essentially no relationship between being allied
and the level of trade. While this finding puzzles us, Morrow, Siverson and Tabares (1990)
have similar findings, and make a security argument for the reasonableness of this finding.
Looking at first differences, the outbreak of a MID depresses trade. While changes in
alliance patterns are rare, entering an alliance in the multipolar period depresses trade;
similarly, change2 in dyadic democracy scores are rare, but when a dyad does become more
democratic its trade does appear to increase.
The economic control variables behave as expected, though neither the population of the
exporter or importer is statistically significant. There is a strong equilibrium relationship
between trade and the distance of the sender and receiver and the GNPs of both parties;
these are not surprising. Increases in GNP also increase trade in the short run.
The OLS model was re-estimated allowing for spatially lagged errors. The weighting
matrix used had an entry of one for dyads that contain a common dyadic partner (e.g. AB
and AC), three for the dyad and its reverse (AB and BA) and zero otherwise. The value
three was chosen for reverse dyads to note that they are more related than dyads which
simply contain a common member. We estimated models allowing the weights for the dyad
and its reverse to be every integer from two to ten, with three yielding the best fit based
statistics. (It should be noted that the results were almost identical for all weight values
under five, and one could have chosen any of these.) This model is presented in the two right
hand columns of Table 1.
We had expected the introduction of spatially lagged errors, which takes account of
the interconnectedness of the dyads, to have more effect than it actually did. While the
coefficient on the spatial error lag, λ, is large (.38), with a tiny standard error, the substantive
coefficients are little affected by the introduction of spatially lagged errors.
Looking at levels, for the political variables only the puzzling multipolar alliance coefficient was affected; while still significantly negative, it is reduced in magnitude by about one
third. On the first difference of the political variables, the impact of a change in whether
7
a dyad is democratic (again, a rare event) is decreased by about one third, and the impact
of becoming allied in the multipolar period declines similar (and is now marginally significant at best). The only changes in the control variables are in the impact of a change in
population, which is usually of little interest. While the coefficients on population change
dramatically, the standard errors signal to us that the model simply has trouble with the
changes in population.
While we think that the spatially lagged error results are more accurate, it is not clear
that it is worth the effort to produce the changes we have seen. This does not mean that in
other cases we would not see more dramatic changes, but, at least for this exercise, the gains
from introducing spatially lagged errors to deal with dyadic interdependence do not appear
great. While we should not sneer at even small gains, the problem with the use of spatial
econometrics is that it requires specialized software, which often means we cannot use many
techniques that are available for the non-spatial model. Thus, for example, we cannot use
any kind of semi-parametric method if we use a spatial model (this is purely a function of
software, but such a constraint is non-trivial). Thus we are left with a choice of what types
of mistakes we are willing to tolerate.
Why does taking account of dyadic interdependence seem to have such a small effect on
our results. One reason may be that with TSCS data, and a lagged dependent variable, many
of the dyadic interdependencies are already included in the lagged value of the dependent
variable. Thus the greatest gains from allowing for spatially correlated errors may be in
purely cross-sectional data. But at present this is purely speculation. We now move from
models with spatially correlated errors to models which include a spatial lag of the dependent
variable, where space is given a broad definition.
3
Spatial econometrics and Deutschian communities
Following Lipset’s social requisites hypothesis, an extensive literature has examined how
social and economic attributes influence the likelihood that countries will be democratic.
However, there are many reasons to suspects that the level of democracy in one country
could be influenced by the level of democracy in other states.9 Previous analyses have considered diffusion in the context of relations between countries that are geographic neighbors.
However, there is no particular reason why connections between states must be limited to
geographic distance per se.
Defining connectivity and space
We use three different plausible definitions of connectivity between states. Clearly, diffusion
between countries may be likely to occur between nearby countries in a geographical sense.
This is the basis for out first connectivity criterion, using the geographical distance between
9
See Gleditsch (2002) for a more extended discussion.
8
states. We use the minimum distance data from Gleditsch and Ward (2001) to define countries as connected if they are within 500 km of one another. This yields a binary connectivity
matrix where each entry wij is 1 if state i and state j are within 500 km from each other.
Each neighboring country is given equal weight in the row for country i. Here, and elsewhere
in this section, we normalize the matrix so that each row sums to 1.
In many cases, the primary reference countries are not necessarily the closest countries,
but countries that are seen as somehow “similar”. A second possible specification of closeness
could thus be connectivity defined by shared cultural attributes. Exactly what is meant by
shared culture can be somewhat difficult to pin down. Possible candidates include language,
religion, and broad cultural groupings akin to the “civilizations” suggested by Huntington
(1996). Each of these alternatives have their advantages and disadvantages. First, most
available cross-national language data, generally codes the primary languages as proper
names in a very restrictive manner. This could lead to overly narrow definitions of shared
culture that fail to take into account that many identified national languages such as Finnish
and Estonian may be closely related and mutually intelligible to their speakers.10 By contrast,
defining shared culture based on religion would lump together countries such as Switzerland,
Korea, and Paraguay, which though all predominantly Christian states, have few similarities
or connections.
In this application, we opt for a middle ground based on cultural clusters. We use a modified version of Henderson and Tucker’s (2001) operationalization of civilizations, and classify
countries as being either Buddhist, Caribbean, Islamic, Japanese, Jewish, Latin-American,
Orthodox, Pacific Islanders, Sinic or Western.11 Each country is coded as connected to other
countries in the same cultural cluster. This allows for very distant countries with high levels
of contact such as Great Britain and New Zealand to be considered cultural “neighbors”.
Our final specification of connectivity is based on the size of trade flows. A country is
considered connected to all other countries that it has some trade with. However, countries
tend be more dependent or influenced by its major trading partners, where the bilateral trade
flows are large relative to the country’s total trade. The trade connectivity matrix differs from
the previous distance and culture matrices in two notable ways. First, whereas the distance
and culture matrices assign equal weights to any geographical neighbor or country in the
cultural cluster, the trade matrix consists of weights where the importance of another state
is given by its proportion of a country’s total trade. This weights large trading partners
much more heavily than smaller trading partners. Moreover, in the distance and culture
matrix, any neighbor of A must always have A as a non-trivial neighbor. In the trade
matrix, however, it will often be the case that one country, say El Salvador, has another
country, say the United States, as its major trading partner, yet in tern is a relatively small
and trivial trading partner to the other country.
Our measure of democracy is taken from the Polity data. We use the full 21 point institu10
For many Estonians, Finnish television provided a window to the West during the Soviet era.
We modify many of the classifications suggested by Henderson and Tucker. We suggest three new
categories (Caribbean, Jewish, and Pacific Islanders), and reclassify all countries labeled as “other” in their
data.
11
9
tionalized democracy scale suggested by Jaggers and Gurr (1995). Given some specification
of connectivity or dependence we can define the “spatial lag ” or average of y in a state i0 s
connected entities:
(9)
yiw̄ = wi0 y.
Clearly, the bases for each of these three definitions of connectivity matrices are quite
different, and as a result, they may yield very different spatial lag measures for democracy.
However, since culture and trading patterns also are geographically clustered, it could also
be the case that the three different definitions of connectivity will link many of the same
countries. The bivariate correlations in Table 2 indicate that both of these conjectures to
some extent are true. It is clearly the case that all of these measures are positively correlated
with democracy and that they are positively correlated with each other. However, the spatial
lag defined from the trade matrix tends to have much smaller bivariate correlations with the
other variables than is the case for the bivariate correlation between the distance and culture
based spatial lags.
One interesting feature of the trade matrix is that most countries have the bulk of their
trade with large, wealthier countries. These countries more often tend to be democratic. As
a result, for many developing countries, the “spatial lag” or the average democracy in its
trading partners is generally much higher than the country’s own value on the democracy
scale. Whereas the spatial lag of democracy defined over distance and culture has a bimodal
density function, much like the density of the democracy variable itself, the spatial lag of
democracy defined over the trade flow matrix has a single peak, with a mean much higher
than the median (or even mean) of the democracy variable. As such, different measures
clearly stipulate quite different linkages between states.
Variable
Table 2: Bivariate correlations, spatial democracy lags
Democracy L.Democracy(distance) L.Democracy(culture)
L.Democracy(distance)
0.567
L.Democracy(culture)
0.550
0.684
L.Democracy(trade)
0.385
0.248
0.214
Cross-sectional analyses
To explore the importance of spatial linkages between observation in the distribution, we
start by a cross sectional analysis of the most recently available year. For the distance and
culture data, this is 1998. For the trade data measure, we lack data on trade beyond 1996.
The results are shown in Table 3. As can be seen, the OLS results in the second column
suggest a strong positive relationship between GDP per capita and level of democracy.
However, the OLS coefficient estimate of the log of GDP per capita might be biased if
the observations on democracy are spatially clustered. The third column of Table 3 contains
10
Table 3: Democracy and Social Requisites, 1998/1996
Variable
Constant
Ln(GDPPC)
κ (distance)
κ (culture)
κ (trade)
N
R2
OLS,1998
Spatial autoregressive estimates
-21.03 (3.84) -13.01 (3.59) -10.00 (3.35) -20.20 (3.75)
2.81 (0.45)
1.71 (0.43)
1.31 (0.40)
2.29 (0.45)
0.53 (0.08)
0.64 (0.07)
0.53 (0.13)
161
161
161
160
0.19
0.21
0.25
0.14
estimates of a spatially autoregressive model with a lagged term defined by distances.
The first thing to note is that the results in the second column of Table 3 display clear
evidence of spatial clustering. The estimate of κ indicates positive spatial clustering, and the
clustering is statistically significant. Moreover, these results also indicate that the coefficient
estimate and standard error of the OLS models might display substantial bias. The coefficient
estimate is reduced to about 60% of its original size.
The results in the third column of Table 3 pertaining to cultural connections between
states also display clear evidence of clustering. Adjusting for dependence between observations reduces the size of the coefficient estimates for logged GDP per capita to less than half
of its original size.
The results for trade connectivities in the fourth column of Table 3 look substantially
different from the previous two connectivity matrices. Although the estimated κ indicates
significant clustering, the coefficient estimate for ln GDP per capita changes much less relative to the OLS coefficient estimate than was the case for the previous two specifications. The
standard error for κ is much larger than in the previous two specifications. One possibility
might be that the information in the trade defined spatial lag measure is substantially different than the information in the distance/culture based measure. Distance/culture might
reflect the regional dynamics in regional waves of transitions. The trade based measure,
however, may conceivably pick up some of the transitions in developing states that have a
high amount of trade openness, primarily with large and wealthy democracies.
Cross-sectional time-series results
Previous research has shown that factors that seem to be highly associated with differences
democracy in a cross-sectional comparison are not necessarily strong predictors of differences
over time. Political institutions are highly persistent and display little change over short time
periods. In this section, we estimate a model for democracy with pooled annual observations
for the period 1950–1998.
Pooling observations raises questions about whether observations can be assumed to
be independent over time and are likely to display serially correlated residuals. Unlike in
the previous section, where we used the error correction model, here we simply adjoin the
11
lagged dependent variable to the specification. The basic idea is that lagged values of the
dependent variable will incorporate prior influences on the dependent variable and usually
eliminate serial correlation in the residuals. Including the first lag of the dependent variable
also changes the interpretation of the model. In particular, β is the short run impact of a
change in an independent variable on y; βφ is the long run impact, where φ is the coefficient
of the lagged dependent variable. Since φ > .9 is common, this means that the long run
impact may be ten times the short run impact.
The second column of Table 4 contains the results of a standard OLS regression of
democracy on the log of lagged GDP per capita plus its own lag. As can be seen, the
autoregressive parameter is φ = 0.94, which is very close to 1. While this indicates that we
might have to worry about non-stationarity, the small standard error on φ̂ allows us to reject
the hypothesis φ = 1.12
As expected, the estimate of the short run impact of logged GDP per capita on democracy
is greatly reduced from the simple cross-sectional estimate. One way to think about this
is that much of the seeming ability of the log of lagged GDP per capita to account for
differences in democracy between countries is incorporated in the past values of democracy
or the autoregressive term. Alternatively, we note that the long run impact of a change in
logged GDP per capita is almost five, which is larger than the cross-sectional estimate.
Table 4: Democracy
Variable
OLS
Constant
-1.91 (0.27)
Democracyt−1 0.94 (0.00)
Ln(GDPPC)
0.24 (0.03)
κ (distance)
κ (culture)
κ (trade)
N
6242
2
R
0.92
and Social Requisites, 1950–98
Spatial autoregressive estimates
-1.26 (0.28) -1.07 (0.29) -2.11 (0.29)
0.93 (0.00) 0.93 (0.00) 0.94 (0.01)
0.16 (0.03) 0.14 (0.04) 0.23 (0.03)
0.06 (0.01)
0.05 (0.01)
0.05 (0.02)
6242
6242
5873
0.92
0.92
0.92
Columns three to five of Table 4 confirm that the two general results with respect to
spatial associations that we have previously observed also apply for the spatial autoregressive
term in the TSCS model with a lagged dependent variable. The first thing to note is that
the estimate of κ remains statistically significant. Comparing columns three to five with
the OLS estimates in column two reveal that accounting for the spatial connections between
12
The standard error is about .004, so a Dickey-Fuller statistic on the null of non-stationarity is about
25. While this statistic has fatter tails than the usual normal distribution, 25 is well above any critical
value for rejecting the null of stationarity at any level of significance that is as close to zero as one might
like. We also must ensure that the remaining errors show no serial correlation. A Lagrange multiplier test
showed an estimated serial correlation of the errors of -.10 (with a standard error of .01) indicating little
albeit significant remaining serial correlation. Again, we conclude that the the degree of this harm is not
proportion to the harm caused by the small amount of serial correlation of the errors.
12
observations yield a large decrease in the coefficient estimate for the log of lagged GDP per
capita; this decrease is comparable to what we saw in the simple cross-sectional analysis of
Table 3.
Our conjecture is that κ is much smaller for the TSCS data with a lagged dependent
variable than for the simple cross-sectional model because the lagged dependent variable
incorporates all the connectivities in the model, and thus reflects the spatial as well as the
temporal interconnectivities in the data. However, unlike the previous section, accounting
for even the small degree of remaining spatial connectivity in the data yields large changes
in the coefficient estimates.
We have previously noted the technical difficulties in estimating a model with the simultaneous spatial lag. The problem is due to the expected value of the product of the spatial
lag term and the error term being non-zero.13 The consistency of OLS depends on this, and
all simple GLS fix ups require us to start with a consistent estimator.
Imagine instead, however, that we use the spatial lag of the one year lag of the dependent
variable wi0 yt−1 (where yt−1 is the vector of all observations at time t − 1).14 This would
be associated with the errors at t − 1, but not the contemporaneous errors at t. Hence, we
can use OLS to estimate the model if the errors are serially uncorrelated. We report the
results of these estimates in Table 5. Interestingly, these results are relatively similar to
the considerably more complicated spatial autoregressive model estimates in Table 4. These
results suggest that serial correlation in a time-series cross-section can easily be dealt with
if the spatial correlation is not contemporaneous. Little seems to be lost in this application.
This is perhaps not too surprising, given the sticky nature of institutions and the lack of
change in dependent variable. Obviously this is only one example, but we plan to continue
to explore using the temporal lag of the spatial lag in place of the contemporaneous spatial
lag. If this is a good strategy, it allows for much simpler estimation (and hence the ability
to use other tools that are easily used with OLS).
4
Conclusions
Spatial econometric techniques are now starting to be used by political scientists. Because
of the geographic heritage of these models, their primary application has been to incorporate
physical notions of space (distance) into political models, and, particularly, to argue that
13
0 The consistency of OLS depends on E (Wyt ) ε = 0. However,
0 −1
plimN −1 (Wyt ) ε = plimN −1 ε0 W (I − κW) ε.
(10)
(W is the matrix built up from the spatial weighting vectors, wi .) Unless κ = 0, this probability limit will
not equal zero.
14
This is not simply done for econometric simplicity. It seems likely that while democracy in “nearby”
units may have an impact, this is likely to take place with some lag. Political economists, such as Garrett
(1998) and Franzese (2002) have informally used these ideas, including the lagged trade weighted average
of growth in one’s trading partners in a model of economic growth, noting that growth can be imported in
open economics.
13
Table 5: Democracy and Social Requisites, 1951–98,
Variable
OLS estimates
Constant
-1.32 (0.29) -1.03 (0.30) -2.12 (0.29)
Democracyt−1 0.93 (0.005) 0.93 (0.005) 0.94 (0.00)
Ln(GDPPC)
0.17 (0.03) 0.13 (0.04) 0.24 (0.03)
κ (distance)t−1 0.05 (0.01)
κ (culture)t−1
0.06 (0.01)
κ (trade)t−1
0.04 (0.01)
N
6077
6077
5709
2
R
0.92
0.92
0.91
All spatial lags entered with one year temporal lag
geographically nearby units are linked together (or, less usefully, that their error terms are
linked together).
While this approach is highly promising, we think it can be made much more fruitful
if we allow for interconnectivities that go beyond geography. Here we have shown that
spatial econometrics can help deal with dyadic interdependent observations and with notions
of community that flow from the work of Karl Deutsch and related scholars. These are
technically feasible and capable of enriching our econometric models, which all too often
assume that all units are independent of each other.
The results on dyadic interdependence are somewhat negative, in that, at least in our
example, taking dyadic interdependence into account does not have much of an effect on
substantive results. It is, however, useful to know this, since we might have suspected that
the results based on assuming dyadic independence are wrong.
We get larger effects in the Deutschian analysis of democracy; the spatial models used
there also seem more theoretically compelling. Thus at least a third of the effect of being rich
on being democratic appears to be a spatial effect, that is, it is an effect of one’s Deutschian
“neighbors” being democratic.
We are particularly interested in time-series—cross-section data, since so many IR data
sets are of that form. Here, the use of a lagged dependent variable may pick up many of
the features of spatial interdependence, since the lagged dependent variable already contains
such interdependence. But, in our model of democracy, we found that explicitly taking
“space” into account still mattered, even in the presence of a lagged dependent variable.
Finally, we note that for time-series–cross-sectional data, it may make sense to use the
temporal lag of the spatial lag, instead of the contemporaneous spatial lag in the specification.
This makes theoretical sense; nearby units may have an effect, but it is not likely to be
instantaneous. This makes estimation of models with the spatial lag easy, since OLS is
appropriative for such model if the errors are serially uncorrelated. This is easy to test for,
and in practice the errors will be nearly serially uncorrelated if there is a lagged dependent
variable in the model. The advantage of such an approach is that analysts can do many things
in an OLS context that are hard to do with the specialized software of spatial econometrics.
14
But in the end, our message is theoretical, not technical. In international relations and
comparative politics, we would expect units to be affected by the what takes place in other
units. We would expect the connectivity of units to be a function of political and social, as
well as geographic, variables. While the heritage of spatial econometrics is geographic, there
is no reason to limit spatial econometric models to geographic modes of thinking.
15
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