where
n = number of observations;
B2s = a vector of the 2SLS estimates of inter-
est;
Bols = a vector of OLS estimates of interest;
and
V(q) = the variance-covariance matrix of the
vector (B28-Bols),
represented by n(V2g - Vols) or n times
the differences between variance-co-
variance matrices of 2SLS and OLS es-
timates.
The statistic W has a χ2 with one degree of
freedom as its asymptotic distribution if the
null hypothesis is true.
The Wu test produced no evidence of differ-
ences between the 2SLS and OLS estimates of
the EC equation. (The W statistic calculated for
the soybean and soybean meal prices in the EC
equation was 1.31, which is not significant at the
five-percent level.) We concluded that EC im-
ports did not influence the U.S. soybean and
soybean meal prices and assumed the prices are
exogenous. We then assumed that the other
five markets, whose shares ranged from 25
percent down to 4 percent, were also price
takers and that the U.S. soybean and soybean
meal prices were exogenous to all six equations.
We applied the same test to a market-wide
equation using data aggregated across all 19
countries. The W statistic calculated for soy-
bean and soybean meal prices in the equation
was 0.67, which is not significant at the five-
percent level. Thus, the W statistics calculated
for both the EC and world (19-country) equa-
tions were insufficient at the five-percent level
to suggest that the coefficients are subject to
simultaneous equation bias. Consequently,
2SLS estimates are not appropriate. However,
2SLS estimation of the world equation is a
commonly accepted, if not recommended, pro-
cedure (in the absence of the Wu test). There-
fore, we present 2SLS estimation results to
compare with the OLS and SUR estimation
results.
Testing for Aggregation Bias
To test the null hypothesis that the param-
eters on the linearly aggregated variable (pork
production) were the same across all six
equations,
versus the alternative,
Ha: at least one b3i ≠ b3j, (i ≠ j),
where b3i is the parameter on pork production in
the ith equation, the six market specific equa-
tions were estimated first by SUR without any
restrictions. Then the equations were reesti-
mated with the restriction that the estimators
on the pork production variable were the same
across all six equations.
Testing the results of this restriction deter-
mines whether we consider the parameters on
the aggregated variable the same across the
individual markets. If the restriction on the
pork production estimator significantly alters
the variance-covariance matrix of errors be-
tween the six equations, we can reject H0 and
conclude that estimates from a single equation
would contain aggregation bias.
To determine if the restricted estimations
were significantly different from the unre-
stricted, the statistic g was used,
g = (r - RB)'(RCR')1(r - RB),
where r = RB represents a matrix of linear
restrictions on the coefficient vector B; C =
[X'(∑^1Φ I)X]' ; X represents the matrix of the
exogenous variables; ∑ is the variance-
covariance matrix of errors between equations;
I is an identity matrix; and ® denotes Kronecker
product (Judge et al., p. 28). The statistic g is χ2
distributed, with degrees of freedom equal to
the number of restrictions (five in our case). Jn
deriving and estimating g, ∑^1 is replaced by ∑^1
(see Judge et al., pp. 472-76). Our calculated g
statistic was 62, significant at the one percent
level, leading to rejection of the hypothesis that
the parameters on all the aggregated variables
are the same across country-specific markets.
Thus, one of the conditions for using aggregate
data to estimate a single equation is violated.
Weighted-Market-Share Estimation
From our aggregation-bias test, we concluded
that single equation estimation using this ex-
port demand data, aggregated across country-
specific markets, contains aggregation bias.
Comparing single-equation elasticity estimates
with trade-weighted elasticities from the six
equations may help reveal the extent of the
bias.
Parameter estimates and t-values from the
unrestricted SUR estimations of the six market-
specific equations are in Table 1. Results from a
total-export single OLS equation and a total-
export 2SLS system of equations are in Table 2.
Elasticities, calculated at the sample means
from each of the six SUR equations, were first
weighted by that market’s share of U.S. soybean
exports for 1983-85, and then added to obtain
aggregate U.S. elasticity estimates across all
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