autoregressive models are estimated using Gauss 386i autoreg procedure. Since the autocorrelation parameter
estimates decrease in size and the second and higher-order autocorrelation coefficients are not statistically
different from zero at the 10% level, it is concluded that a first order autoregressive [AR(1)] error term
specification is sufficient to correct for autocorrelation.
The estimated AR(1) models are presented in Table 3. Note that, although the autocorrelation
coefficients are statistically significant at the 1% level in both the corn and soybean price models, the standard
error estimates for the estimators of the slope parameters βc1, βc2, βs1, and βs2 are larger under the AR(1) models.
This is a common occurrence in applied modeling work, not withstanding of the fact that, under non-
independently distributed errors, the AR(1) slope parameter estimators are theoretically more efficient than OLS.
The cause of this apparent contradiction is that, in the presence of autocorrelation, the OLS standard error
estimators are biased, generally downwards, i.e. the OLS standard error estimates tend to underestimate the
correct standard errors and, thus, are not reliable.
Theoretically, if corn and soybean prices are correlated through time, more efficient slope parameter
estimators can be obtained by estimating the two models jointly, using a seemingly unrelated regression (SUR)
procedure. This procedure is available in Gauss, SAS, and other statistical analysis software under the
assumption of independently distributed errors, but not under autocorrelated errors. However, the log-likelihood
function used to estimate such a model (available from the authors upon request) is a straightforward bivariate
extension of equation (4). In the case of the soybean price model, the estimated standard errors for the estimators
of the regression slope parameters are moderately lower under this SUR-AR(1) procedure, but they are slightly
higher in the corn price model. The overall efficiency gains from SUR estimation are likely limited by the low
corn-soybean price correlation coefficient estimate under the assumption of error term normality. The SUR-
AR(1) model is then expanded to evaluate if the variances of the price distributions have been changing through
time. The two most common time-dependent heteroskedastic specifications are evaluated:
(8) σj = σjo + σj1t, and σj2 = σjo + σj1t, t = 1,...,50, j = c and s;
which make the standard deviation or the variance of the corn and soybean price distributions linear
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