Urban Water Demand Estimates Under Increasing Block Rates Page: 6
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6 GROWTH AND CHANGE, WINTER 1988
It is generally accepted that OLS estimates of water demand functions
for consumers facing block rate schedules yield biased estimates of price
elasticity (for example, see Terza 1986), since the price variable is correlated
with the random error term. As described by Henson (1984), the coeffi-
cient on marginal price is expected to be biased toward zero and the coeffi-
cient on difference to be biased away from zero, under an increasing block
structure. A Hausman (1978) specification test was used to test for this possi-
ble simultaneity bias. The Hausman test utilizes the result that the OLS
estimator is consistent and asymptotically efficient under the null hypothesis
that the price variables are exogenous, but is inconsistent under the alter-
native hypothesis. The instrumental variable estimators, bi , are consistent
under both the null and alternative hypotheses, but not asymptotically eff-
icient under the null hypothesis. The test statistic is
m = b' (Viv - Vol)-lIb,
where b = biv - bol,, and Viv and V01o are variance - covariance
estimates under instrumental variable and OLS techniques, respectively.
This statistic has an asymptotic chi-square distribution with degrees of
freedom equal to the number of explanatory variables. The calculated
Hausman's test statistic is 887, which greatly exceeds the critical value of
x2 at the 1 percent level of significance of 18.475 with seven degrees of
freedom. OLS estimates are presented in this study for purposes of
comparison.
While several instrumental variables techniques have been suggested, we
focus on two commonly used ones. The first technique introduces a separate
price equation in a Two-Stage Least Squares (2SLS) procedure introduced
by Wilder and Willenborg (1975). Price is regressed against lawn size,
weather, income, age and the two marginal prices. The predicted price is
used in the second stage as a regressor. Another approach was suggested
by McFadden, Puig, and Kirschner (1977). McFadden et al. regress actual
household consumption on a set of variables which represent the typical
bill. However, they did not have access to the actual rate schedules for their
data. This caused two problems, as Terza notes:
They were not able to regress observed electricity consumption on the actual
rate schedule. Instead, they were forced to approximate the rate schedule by
typical electric bills at various levels of consumption. In the second stage, they
were unable to use the actual rate schedule to compute the values of the price
instrument (Terza 1986, p. 1137).
We employ an application of the IV method as used by Terza, which is
not subject to these problems. In the first stage, observed water demand
was regressed on actual marginal prices that the household would face at
two different levels of water demand (10 and 20 thousand gallons and the
exogenous variables). In the second stage, the actual rate schedule was used
to obtain predicted marginal prices. The difference variable is calculated
using these predicted marginal prices. Both of these adjustments to the IV
method are likely to improve the reliability of the estimates (Terza 1986).r.; ii }' M: r 9'M1 1""k1'r 'OT." "'k i :/' 1 Fr ;fr. : rc ... rfs :rr r T:'"7 f?:7::r:::: . :" :rtii"}ti
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Nieswiadomy, Michael L. & Molina, David J. Urban Water Demand Estimates Under Increasing Block Rates, article, January 1988; [Hoboken, New Jersey]. (https://digital.library.unt.edu/ark:/67531/metadc71792/m1/6/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.