Obtaining estimates for the standard errors of long-run parameters.
Gurney, Andrew
OBTAINING ESTIMATES FOR THE STANDARD ERRORS OF LONG-RUN PARAMETERS
This note provides a practical illustration of the reparameterisation
described by Wickens and Breusch (1988) which enables estimates of
long-run coefficients and their standard errors to be derived from an
autoregressive distributed lag equation. Equation A in table 1 shows one
such equation, which provides an estimate of the consumption function in
the United States. The t-statistics on the level of consumption, income
and wealth indicate that these items are not statistically different
from zero at conventional 5 per cent significance levels. But these
statistics are only relevant in assessing the significance with respect
to explaining the one period ahead level of consumption. The decision
to include or exclude these variables will however also have
implications for the long-run properties of the equation. The
Wickens-Breusch reparameterisation allows this information to be readily
extracted.
Point estimates of long-run effects can be easily extracted from
equation A. This is done by setting all difference terms to a constant
(zero here for simplicity) and by transferring the term in consumption
to the left-hand side of the equation: 0.0916 C = - 0.0112 + 0.0965 Y +
0.0110 W - 0.00101 R C = - 0.122 + 1.053 Y + 0.120 W - 0.0110 R
However obtaining estimates for the standard errors of these
long-run coefficients is not straightforward, and hence it is not easy
to assess their statistical significance.
The Wickens-Breusch solution to this problem is to reparameterise
equation A so that the levels term in C is taken to the left-hand side,
and the equation is renormalised using the coefficient estimate obtained
in equation A. This is done in equation B. Since we have changed the
dependent variable the coefficients on all the variables have also
changed, as have measures such as the equation standard-error and
R-squared. But there are a number of indicators that equations A and B
are equivalent. The first of these is that the implied long-run
coefficients calculated for equation A are now directly given by the
coefficients on the relevant levels variable. A second indicator is
provided by tests such as the Durbin-Watson statistic and the LM test
for serial corrleation. These tests are not dependent on the form of
the dependent variables, but only on the equations' residuals,
whose pattern is identical in the two cases (although their magnitude
will be different). As table 1 shows these tests give identical answers
for equations A and B. However the RESET test for functional form, which
regresses squared residuals on squared fitted values is different in the
two cases. This is because the form of the dependent variable, and
consequently its fitted values have changed.
The main interest of B is in the t-statistics on the long-run
coefficients. These show that the levels terms in Y, W and R are all
well-defined relative to the level of C, and hence that all these terms
should be retained in the regression. The intuition behind the apparent
differences in significance of, say, the term in Y-.sup.1. between
equations A and B, is that in equation A we are assessing the
significance of Y-.sup.1 only in relation to the immediate level of C.
Additional long-term effects are contained in the interaction of other
terms involving lagged C and lagged Y. The reparameterisation in B
allows for these by gathering together all the lagged terms in C,
enabling direct estimates of the long-run parameters to be revealed in
ordinary least squares estimation. The differences between the
equations show the importance of the Wickens-Breusch transformation (or
an equivalent technique) or assessing the long-run properties of an
equation where the dependent variable is specified as a single-period
change.
Equation C returns to the parameterisation used in equation A, but
with restriction that long-run consumption is proportional to long-run
income imposed. The restriction can be tested using the conventional
F-test, and is readily accepted. Equation B also suggests that the
restriction could be easily imposed. The coefficient on Y.sup.1 is
1.054 with a standard-error of 0.067, and hence a t-test indicates that
a unit coefficient is easily acceptable. Equation D is the
Wickens--Breusch reparameterisation of equation C. Note that the
reparameterisation differs from that in equation B in its treatment of
C.-.sup.1 as well as in including a term in Y-.sup.1. This reflects the
fact that the coefficients on C.sup.1 change between equations A and C.
Using ordinary least squares, it does not appear to be possible to
proceed directly from equation B to equation D.
