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THE SENSITIVITY OF
CONSUMER SURPLUS ESTIMATION
TO FUNCTIONAL FORM SPECIFICATION
Duangkamon Chotikapanich and William E. Griffiths
No. 94- December 1996
ISSN 0 157 0188
ISBN 1 86389 399 7
THE SENSITIVITY OF CONSUMER SURPLUS ESTIMATION
TO FUNCTIONAL FORM SPECIFICATION
Duangkamon ChotikapanichDepartment of Economics
Curtin University of Technology
William E. GriffithsDepartment of EconometricsUniversity of New England
The Sensitivity of Consumer Surplus Estimation
to Functional Form Specification
Abstract
The travel cost method utilized by Beal (1995) for estimating consumer surplus isre-examined. An alternative more algebraically consistent estimation strategy is suggested.We demonstrate that estimation of consumer surplus is inherently unreliable when only asmall number of observations are available, and when the consumer surplus estimatedepends critically on extrapolation of the demand curve beyond the sample observations.
2
1. INTRODUCTION
In a recent paper in the Review of Marketing and Agricultural Economics, BeN (1995)
assesses the value of Carnarvon Gorge National Park for recreational use by estimating
the consumer surplus associated with selected demand equations. She was kind enough
to provide us with her data so that we could use her model as one of our examples in a
paper we are writing on using Bayesian techniques for estimating areas in economics.
One of the things that struck us when we looked more closely at her paper was how
sensitive consumer surplus estimates can be to functional form specification. In
addition, there were some methodological issues which we thought could benefit from
further discussion. These issues are (i) provision of standard errors for consumer
surplus estimates, (ii) choice of functional form, and (iii) prediction in log-log models.
To provide what we hope will be regarded as useful insights on these questions, we
have organized this paper as follows. In the next section we summarize the Beal
methodology for estimating consumer surplus associated with the use of the National
Park. We point out that the 2-step, 2-equation strategy used by Beal is internally
inconsistent and we estimate consumer surplus using an alternative more consistent
strategy. We also indicate how a standard error for the consumer surplus estimate can
be calculated. The large discrepancy between the consumer surplus estimates from the
two strategies made us wonder how sensitive such estimates are to the functional form
specification. So, in the section that follows, we estimate the consumer surplus implied
by each of the functional forms tested by Beal. We also compute standard errors of the
estimates and give the maximized log-likelihood values for each of the functions. These
values are likely to be a better guide to functional form selection than the R: s used by
Beal. In a final section, we comment on prediction in log-log models; our comments
help explain why Beal’s equation seemed to give a biased prediction of total visits for a
zero price.
2. AN ALTERNATIVE CONSUMER SURPLUS ESTIMATE
2.1 BeN’s Estimation Technique
BeN estimates two demand functions, a camping demand function and a
demand function for day visits. We restrict our discussion to the camping demand
curve. The points we make are equally valid for the demand for day visits.
Potential visitors to the park are divided into 12 geographical zones.
Associated with travel to the park from each zone is a travel and time cost variable
(TCi) that takes the place of price in a conventional demand equation. A demand
equation for camp use is estimated by relating the i-th zone’s average annual visitation
rate per thousand population (Vi) to travel cost, say,
Vi = f(TCi) i = 1,2 .....12 (2.1)
The relationship between aggregate demand Q and the travel costs for each zone is
given by12 12
Q = ~_~popi V,. = ~_~popi f(TCi ) (2.2)i=1 i=1
where pop~ is the population of the i-th zone. Because it represents just one point on
an aggregate demand curve, equation (2.2) by itself cannot be used to estimate
consumer surplus. It describes the existing demand that relates to the existing range of
zonal travel costs. The "travel cost method" overcomes this problem by assuming that
consumers of recreation react to changes in a hypothetical entry price (P) in the same
way as they would to changes in travel costs. It" such is the case, augmenting the travel
cost variable with an entry price variable yields the function12
Q= ~_,pop~ f(ZCi + P ) (2.3)i=1
which can be used to trace out a complete demand function where aggregate demand
Q is related to price P. The relevant consumer surplus measure in BeN’s study is the
area under this aggregate demand function.
The ftrst point we make in this paper is that BeN’s specification for the
aggregate demand function in (2.3) is algebraically inconsistent with her choice of the
functional form for f(rc~). In addition, specifying (2.3) in an algebraically consistent
4
way has a considerable effect on the estimate of consumer surplus. More explicitly,
after some preliminary testing, Beal chooses the log-log equation
ln(l~,. )= ~o + ~1 ln(TCi ) (2.4)
for the visitation rate demand curve. The estimates obtained were ~o = 9.8426 and
~1 = -1.8743. Equation (2.1) becomes, therefore,
~/ = exp{~o + ~1 ln(TCi )} = el~° (TCi (2.5)
and the corresponding aggregate demand curve, with the entry price variable P
included, becomes
O =eOO ~ poPi (rci + (2.6)i=1
If equation (2.5) is the legitimate function to use for Vi, then, presumably, the relevant
consumer surplus measure is the area under the aggregate demand function in equation
(2.6). However, this was not the function utilized by Beal. Instead, she used equation
(2.6) to predict values for Q for 7 selected values of P. An 8th value was the observed
value for Q when P=0. These values of Q and P were used to estimate a number of
alternative aggregate demand functions from which the linear function
~) = 15764.7 - 252.09 P (2.7)
was chosen. Based on this estimated demand curve, the consumer surplus, which is the
total area under the curve, was estimated as 492,931.
2.2 An Alternative Consumer Surplus Estimate
From section 2.1 it can be seen that Beal chose to approximate the aggregate
exponential demand function (2.6) with the linear function (2.7) and to use the latter
to calculate the consumer surplus. An alternative way which is internally more
consistent is to calculate the consumer surplus directly from the total area under the
exponential demand function (2.6). Figure 1 shows the two curves drawn from (2.6)
and (2.7). It can be seen that the linear function is only a reasonable approximation for
prices up to $20 and that, by using the linear function to approximate the exponential
one, we largely underestimate the total area under the curve.
To calculate the area under the exponential curve (2.6) we need to be aware
that this exponential curve is asymptotic to the price axis. That is, as P ~ ~,, Q --~ 0.
The area under the curve is given by
P* hl
~S = ego lim ~ 2 (P°Pi)(TCi + P) d PP % ,,~ 0 i
lim 2 (P°Pi) TCi + P* ) - TCii
gl+l
Since ~: islessthan-1, lim (TCi+P*) is zero and the ~S becomesp* _.~
gl+l
dS- 7~e ~.,(popiXTCi)~i+1 i
(2.8)
The consumer surplus calculated from this definition is 2,954,560. By using the linear
function to approximate the exponential function the consumer surplus has been
underestimated by as much as 6 times.
2.3 Standard Error for the Consumer Surplus
When estimating any quantity it is important to have some idea of the reliability
of that estimate. One indicator of reliability is provided by the standard error and a
consequent interval estimate. An approximate standard error for the consumer surplus
estimate can be obtained from the squared root of the asymptotic variance which is
given by (Judge et al 1988, p. 542)
(2.9)
For the consumer surplus defined in (2.8), the derivatives are
6
[~1+1.~.,(popi)(TC,)
3(CS) _ ef~° lh+: eI~°
~]~1 ([~1 +1)2 ~(pop,)(TC,), (]~1 +1) "~(pop,)(TC,), ln(TC, )
Using v&(]~o)= 1.1005, vS_r(]~:)= 0.0377, ctv(]~o,]~,)= -0.2016 and evaluating the
derivatives at ~o and ~: the standard error calculated for the consumer surplus (2.8) is
792,214. This value, along with the estimated consumer surplus gives a 95%
confidence interval as (1,189,507 , 4,719,613). This very wide interval shows there
is considerable uncertainty associated with our consumer surplus estimate.
3. SENSITIVITY OF CONSUMER SURPLUS ESTIMATION
The enormous difference between the estimate of consumer surplus from
Beal’s linear aggregate demand function and the estimate implied by her visitation rate
function raises questions about the sensitivity of consumer surplus estimation to
functional form specification. In this section, we look at the consumer surplus implied
by the other five functional forms of the demand function that were considered by
Beal. For each of the functional forms, we provide the point estimate, the standard
error and the 95% interval estimate of the consumer surplus. A comparison of these
results reveals the critical nature of functional form selection. Also, for a goodness-of-
fit comparison, we give values of the maximized log-likelihoods under an assumption
of normally distributed errors. Since the six functional forms do not have the same
dependent variables, the maximized log-likelihood provides a better basis for
comparison than the R2s used by Beal.
3.1 Consumer Surplus Estimates and theft Standard Errors
The procedure for obtaining the consumer surplus estimated in the previoussection can be summarized as follows. It" the zonal demand is the following function
V~ = f (TC~) (3.1)
7
then the aggregate demand for the site is
Q = ~_~(popi)f (TCi + P ) (3.2)i
And consumer surplus is defined as
P
CS= f ~_,(popi)f (TCi+P)dP (3.3)0 i
where P* is that price for which Q = 0 in equation (3.2). If the demand function in
equation (3.2) is asymptotic to the price axis, then it is necessary to take limit of
equation (3.3) as P*~ ,,,, . The expression for the estimated variance of CS from
which we can obtain standard errors is given in equation (2.9). Specific expressions for
V, Q and CS for each of the functional forms considered by Beal are summarized in the
Appendix, together with the derivatives of CS with respect to 15o and I~1- These
derivatives are used in the calculation of the standard error of the estimated consumer
surplus.
Table 1 reports the values of the point estimates, their standard errors and
corresponding 95% interval estimates of consumer surplus obtained from the different
functional forms. The interval estimates are derived assuming, in each case, that ~S
has a limiting distribution that is normal. It should be noted that the demand curve for
equation 4 is asymptotic to the price axis, and, as price approaches infinity, the curve
does not approach the axis quickly enough for the area to be f’mite. The most striking
thing about the entries in Table 1 is that the estimated CS’s obtained from the different
functional forms vary considerably, from 442,324 to 4,105,566. Also, from the large
standard errors and the wide 95% interval estimates, we see that all consumer surplus
estimates are very unreliable, even il" the underlying functional form is known. This is
particularly the case for equations 1, 2 and 5 where the 95% interval estimates contain
meaningless negative ranges. In the sampling theory approach to inference (as distinct
from Bayesian inference), there does not appear to be any simple way of incorporate
prior information to ensure that interval estimates include only a positive range. Also,
it appears that a small degree of uncertainty or unreliability in the estimation of the
parameters I~ o and ~1 can lead to a large degree of uncertainty in the estimation of
CS. Very few of Beal’s estimates for I~0 and 13x are not significantly different from
zero, yet three out of five of the CS estimates derived from them have standard errors
that are bigger than the estimates themselves. It is clear that any valuation of the
Carnarvon Gorge National Park on the basis of consumer surplus estimation is
particularly tenuous.
TABLE 1Point and Interval Estimates of Consumer Surplus
for Different Functional Forms
Functional Form
(se( ~s ))95% interval estimate
equation 1 4,105,566V = 9o + ~1TC (5,933,944)
equation 2V= ~o +~llnTC
1,002,818(1,742,918)
(-7,524,965) - (15,736,100)
(-2,413,301) - (4,418,937)
equation 3 2,875,357 (948,632) - (4,802,082)In V = [3o + ~ITC (983,023)
equation 41/V = [~o +~1TC
equation 5V = ~3o +~31(1 / TC)
equation 6lnV = 13o +131 lnTC
442,324(1,442,770)
2,954,560(792,214)
(-2,385,506) - (3,270,153)
(1,401,821 ) - (4,507,299)
3.2 Comparing Functional Forms
As mentioned earlier, the use of R2 as a criterion for model selection is
questionable for models with different dependent variables. An alternative, which
makes comparisons in terms of comparable units of measurement, is the value of the
maximised log-likelihood function. We are not suggesting that choosing the model that
has the largest maximized log-likelihood is a foolproof model selection criterion. Such
a procedure is subject to sampling error and does not consider whether the data
"significantly" favour one model over another (see, for example, Box and Cox, 1964
9
and Fisher and McAleer, 1981). However, as a descriptive goodness-of-fit measure, it
is preferable to R2. To give an expression for the maximized log-likelihood consider the
equation:
gl (V) -" 13o .-b 131 g2 (TC) + e
where it is assumed that e is an independent normal random variable with mean zero
and variance ~z, and g! (V) and g2 (TC) are functions of V and TC, respectively. For
example, for equation 2, gl (V) = In V and g2 (TC) = TC. After substituting maximum
likelihood estimators for 13o, 131 and ~2into the log-likelihood function, this function
becomes
L = -~[ln(2rt) + l - ln(n)]-~ ln(SSE) + ~_,ln(
where n = total number of observations, SSE =
Og~(V)
sum of squarederrors, and
[Ogl (V)/OV] is the Jacobian of the transformation from gl (V) to V. It is the presence
of this last term that makes a ranking of models on the basis of L possibly different
from a ranking of models on the basis of R2.
Table 2 reports the values of the log-likelihood function, along with the R2s
from Beal’s paper. Based on these values of the maximized log-likelihood functions,
equation 6 is the best equation, a choice that is consistent with that of Beak However,
the ranking of the other models has changed.
TABLE 2
Values of R~ and Log-Likelihood Function for Different
Functional Forms
Functional Form R~ L
eqn 1: V = 13o + 131TC 0.23 -34.1261
eqn 2: V = 13o + 13~ in TC 0.56 -30.7955
eqn3: lnV=f3o+13~TC 0.62 -14.1678
eqn 4: 1IV = 13o +13~TC 0.86 -13.5402
eqn 5: V = 13o +13~(1/TC) 0.83 -25.0763
eqn 6: In V = 13o +131 In TC 0.90 -5.9777
10
4. PREDICTION IN LOG-LOG MODELS
One last point of interest is the characteristics of prediction in log-log models
and their implications for consumer surplus estimation. As mentioned in Section 2,
Beal uses a 2-stage procedure where, to begin the second stage, Oj is obtained using
the predictive model
O_.j = Zpopi ~.ii
with l)i2 based on equation 6 (the log-log model), and given by the expression
~, = exp{~o +~, ln(TC,. + P,)} (4.1)
She notes that the result obtained from this predictor for Pa = 0 is 14,843. This value
largely underestimates the true value of 17,000. A possible reason for this outcome is
that the predictor ~a- is not an unbiased predictor. Consider the zonal demand function
In Vi = [3o + ~1 In TCi + ei where we assume ei - N(O, ~2) and hence
In(V/)- N[(13o + [3, ln(TCi )), ~2]
Vi - LogN[([3o +~, ln(TCi)), (y2]
From the properties of lognormal distributions (Atchison and Brown ,1966), the mean
of V/ is
E(Vi) = exp{(13° +131 ln(TCi )) +l ~2 }2
Thus, an unbiased predictor for V/j is
Qiij = exp{(’O + ’l ln(ZCi + Pj ))+~(y2} (4.2)
When [3o, [31 and ~y2 are replaced by their estimators, qj no longer retains its
unbiasedness property, but it is still likely to be a better predictor than 1)/~. Table 3
shows the new values of Oj obtained using the feasible version of ~, alongside those
obtained by Bea!. Note that the actual number of visits demanded Q~ at zero
additional entry fee (Pa = 0) is 17,000. The new prediction for Qj is very accurate
11
with a value of 0j : 17,003. If a new linear function is estimated on the basis of those
new values of 0r, the estimated consumer surplus is 697,771. This value emphasizes
once more how sensitive consumer surplus estimation can be to choice of
methodology.
P
TABLE 3
Alternative Predictions from the Log-Log Model
Beal’s O
0 17,003 17,000"5 15,735 13,7368 15,068 13,15610 14,656 12,76912 14,267 12,45615 13,723 11,98118 13,219 11,54020 12,904 11,265
CS 697,771 492,931
* Observed value
4.1 Stochastic Considerations in Consumer Surplus Estimation
The possible use of properties of the log-normal distribution to predict quantity
for Beal’s second stage of estimation raises questions about whether such properties
should be used for the more algebraically consistent procedure that we proposed. The
underlying issue here is how to treat the stochastic error term e in the visitation rate
demand function In V~ = I~o + 1~1 In TCi + ei. One approach is to ignore it. This is the
approach taken by Beal and by us so far. Another approach is to carry it through to the
aggregate demand function and consider the estimation of consumer surplus on the
basis of the average or expected demand function. In this case the function becomes
Q= y_~popi exp{[~o +[~1 ln(TCi + P)+ei} (4.3)i
And, if we assume e~ - N(0, ¢r2 ), then the expected aggregate demand function is
E(Q)=’~popi exp{~o +~l ln(TC~ + p)+lcj2} (4.4)i 2
12
Consumer surplus is obtained by integrating E(Q) in (4.4) between 0 and P* and
taking the limit of the integral as P*
A third approach is to first obtain consumer surplus by integrating Q in (4.3)
p*between 0 and P* and taking the limit as ~ ~,. The resulting CS depends on the
error term ei. Taking the expected value of this result gives an expected or average
CS. The two approaches are conceptually different. One measures CS as the area under
the "average" demand curve. The other recognizes that different errors lead to
different demand curves that lead to different consumer surpluses, and then takes the
average CS. Despite this conceptual difference, for the model we are considering, both
approaches lead to the same result, namely
(60+2)
dS = -e131+1
To derive the standard error of a CS estimate based on this expression would involve
partial derivatives of CS with respect to 13o, [31 and (~2 as well as the variance of ~y2.
5. CONCLUSIONS
It is evident from the results in this paper that estimating consumer surplus can
be tricky. The results can be very sensitive to demand function specification and to the
chosen estimation methodology. The purpose of this paper was to expose this
sensitivity as well as to suggest ways in which the methodology could be improved by
computing standard errors and developing consumer surplus estimates which are
internally consistent with visitation rate equations. There are two general observations
that we would also like to make.
One of likely reasons for the sensitivity of Beal’s results to model specification,
and the impreciseness of the consumer surplus estimates as reflected in the width of
interval estimates, is the small number of observations. With only 12 observations it is
difficult to estimate most things accurately. In Beal’s case, however, the 12
observations were obtained via an aggregation of individual responses. The effect of
such an aggregation may have been a loss of information. A possible direction for the
future that might yield better, more reliable estimates is through the development and
estimation of models that utilize single observations on individuals rather than
13
observations that are computed by aggregating information. Also, it would be good to
get consumer surplus estimates and reliability measures that are not dependent on
functional form selection. We are now to investigate a Bayesian approach that may
yield some success along this lines.
Another fundamental difficulty associated with estimating consumer surplus
from models of the type considered here is the need to extrapolate the demand function
beyond the region of the observed sample. The problem is well depicted in Figure 1.
While any number of functions may fit the data reasonably well in the region of the
observed data points, it is the behaviour of the function beyond these data points that is
critical for the estimation of consumer surplus. Since the data contain no information
on this behaviour, the most critical part of the estimation procedure is fraught with
uncertainty. While this uncertainty is reflected by the sensitivity of the consumer
surplus estimates to functional form specification, it is not reflected in the width of an
interval estimate for a given functional form. The fact that these interval estimates are
very wide means that estimation of consumer surplus is subject to a great deal of
uncertainty even if we have perfect knowledge of the correct functional form.
However, as mentioned in the previous paragraph, it may be possible to correct this
problem with a larger number of observations.
14
APPENDIX
The expressions for V, Q, P* and CS for each of the functional forms
considered by Beal are summarized below, together with the derivatives of CS with
respect to I~o and I~1. These derivatives are used in the calculation of the standard
error of the CS in equation (3.3). In some cases (equations 2 and 5) where it is not
possible to derive an analytical expression for P*, it’s value was obtained numerically.
For these cases, the derivatives of CS with respect to I~o and I~ldepend on the
derivatives of P’with respect to [3o and [31. These derivatives of P’were obtained
using the differentiation of implicit functions.
Equation 1
* 11~1(p’)2~S = ~oP Zpopi + ~aP*ZpopiTCi + -~ Zpopii i i
0~7S -~ ° ~" P°Pi__ i-- Z POpiTCi
~0 ~1 i
O~S ~ 2° Z popi
2
2Z poPii
Equation 2
q = ~JO + ~l ln TCi
d= 2(popi)[f3o q-~l ln(TCi + P)]i
15
P* = 90.8365 is obtained numerically as the solution to
i
0
p* p*
i i
3~° = P*~__~popii + ~o 3~° ..T.P°Pi+ ~.~P°piln(TCii + P "3[~o
3~7SP*)ln(TC, + ) - TC, ~ ~+ pop,p, p, ln(TCi )} + 3P* ,v,~, - Z, ~o~,{(~c, + - o
+ ~Zi popiln(TCi+ "3~
where
Op , _ Oh/O~ l _ -Zi poPiln(TCi + P * )
13~ 3h/OP" ~ Zi P°Pi TCi + P~
Equation 3
lnq = l~o+l~lTCi
0 = e~° Z (poPi)e~l(Tci+P)
i
As ~ --4 O, P --~ ~,. Integrating and taking limits yields
dS - ~3, ~" (P°pi) e~’rci
16
~dS e�~° ~lrci
~1 ~ 21 Zi popi e
Equation 4
1/r~= ~0+ ~lrCi
1~ = Z,. POpi ~0 + ~1 (ZCi q- P)
As P --4 ~,, Q ~ O. We find that
,~.I 1 ({~ + 0 )1dS=lim ~1 P°piln o ~,TCi+~P*}-P°piln{~ +~TCi}
Asp* {~ ~ ~ }-~ and ~S -->---~oo, In 0+ ~TCi+ 1P* oo, ~,.
67.3087 is obtained numerically as the solution to
[h(l~o,]~l,P*) ~ ~i (popi) ~0 q-~l (ZCi.t..¯ p, = 0
= p*dS ~ o P " Z P°Pi "}- ~1 Z popi [In (TOi "t- )-ln(TC,.)]
i i
17
1 ~P*O~S _ ~_~ P°Pi [ln(TCi + p.)_ In TCi ]+ ~~ Y~ popi (TCi + p. ) ~
where
~ popi~P * _ ~h/~ o _ ~
~i(
1~3o Oh/OP* ~3~ popi (TCi + p, )~
REFERENCES
Aitchison, J. and J.A.C. Brown (1966), The Lognormal Distribution, CambridgeUniversity Press, Cambridge
Beal, D.J. (1995), "A Travel Cost Analysis of the Value of Camarvon Gorge NationalPark for Recreational Use", Review of Marketing and Agricultural Economics,63,292-303.
Box, G.E.P. and D.R. Cox (1964), "An Analysis of Transformations", Journal of theRoyal Statistical Society, Series B, 26, 211-243.
Fisher,G.R. and M. McAleer (1981), "Alternative Procedures and Associated Testsof Significance for Non-Nested Hypotheses", Journal of Econometrics, 16,103-119.
Judge, G.G., R.C. Hill, W.E. Griffiths, H. Ltitkepohl and T.C. Lee (1988),Introduction to the Theory and Practice of Econometrics, John Wiley andSons, New York.
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19
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Estimation of Elasticities of Substitution for CES Production Functions Using A ggregativeData on Selected Manufacturing Industries in Pakistan. George E. Battese andSohail J. Malik, No. 26 - April 1986.
Estimation of Elasticities of Substitution for CES and VES Production Functions UsingFirm-Level Data for Food-Processing Industries in Pakistan. George E. Battese andSohail J. Malik, No. 27 - May 1986.
On the Prediction of Technical Efficiencies, Given the Specifications of a GeneralizedFrontier Production Function and Panel Data on Sample Firms. George E. Battese,No. 28 -June 1986.
21
A General Equilibrium Approach to the Construction of Multilateral Index Numbers.D.S. Prasada Rao and J. Salazar-Carrillo, No. 29 - August 1986.
Further Results on lnterval Estimation in an AR(1) Error Model. H.E. Doran,W.E. Gfiffiths and P.A. Beesley, No. 30 - August 1987.
Bayesian Econometrics and How to Get Rid of Those Wrong Signs. William E. Griffiths,No. 31 - November 1987.
Confidence Intervals for the Expected Average Marginal Products of Cobb-DouglasFactors With Applications of Estimating Shadow Prices and Testing for RiskAversion. Chris M. Alaouze, No. 32 - September 1988.
Estimation of Frontier Production Functions and the Efficiencies of Indian Farms UsingPanel Data from ICRISAT’s Village Level Studies. G.E Battese, T.J. Coelli andT.C. Colby, No. 33 -January 1989.
Estimation of Frontier Production Functions: A Guide to the Computer Program,FRONTIER. Tim J. Coelli, No. 34 - February 1989.
An Introduction to Australian Economy-Wide Modelling. Colin P. Hargreaves, No. 35 -February 1989.
Testing and Estimating Location Vectors Under Heteroskedasticity. William Griffiths andGeorge Judge, No. 36 - February 1989.
The Management of Irrigation Water During Drought. Chris M. Alaouze, No. 37 - April1989.
An Additive Property of the Inverse of the Survivor Function and the Inverse of theDistribution Function of a Strictly Positive Random Variable with Applications toWater Allocation Problems. Chris M. Alaouze, No. 38 - July 1989.
A Mixed Integer Linear Programming Evaluation of Salinity and Waterlogging ControlOptions in the Murray-Darling Basin of Australia. Chris M. Alaouze and CampbellR. Fitzpatrick, No. 39 - August 1989.
Estimation of Risk Effects with Seemingly Unrelated Regressions and Panel Data.Guang H. Wan, William E. Griffiths and Jock R. Anderson, No. 40 - September1989.
The Optimality of Capacity Sharing in Stochastic Dynamic Programming Problems ofShared Reservoir Operation. Chris M. Alaouze, No. 41 - November 1989.
Confidence Intervals for Impulse Responses from VAR Models: A Comparison ofAsymptotic Theory and Simulation Approaches. William Griffiths and HelmutL0tkepohl, No. 42 - March 1990.
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A Geometrical Expository Note on Hausman’s Specification Test. Howard E. Doran,No. 43 - March 1990.
Using The Kalman Filter to Estimate Sub-Populations. Howard E. Doran, No. 44 - March1990.
Constraining Kalman Filter and Smoothing Estimates to Satisfy Time-VaryingRestrictions. Howard Doran, No. 45 - May 1990.
Multiple Minima in the Estimation of Models with Autoregressive Disturbances.Howard Doran and Jan Kmenta, No. 46 - May 1990.
A Method for the Computation of Standard Errors for Geary-Khamis Parities andInternational Prices. D.S. Prasada Rao and E.A. Selvanathan, No. 47 - September1990.
Prediction of the Probability of Successful First-Year University Studies in Terms of HighSchool Background: With Application to the Faculty of Economic Studies at theUniversity of New England. D.M. Dancer and H.E. Doran, No. 48 - September 1990.
A Generalized Theil-Tornqvist Index for Multilateral Comparisons. D.S. Prasada Rao andE.A. Selvanathan, No. 49- November 1990.
Frontier Production Functions and Technical Efficiency: A Survey of EmpiricalApplications in Agricultural Economics. George E. Battese, No. 50 - May i991.
Consistent OLS Covariance Estimator and Misspecification Test for Models withStationary Errors of Unspecified Form. Howard E. Doran, No. 51 - May 1991.
Testing Non-NestedModels. Howard E. Doran, No. 52 - May 1991.
Estimation of Australian Wool and Lamb Production Technologies: An Error ComponentsApproach. C.J. O~Donnell and A.D. Woodland, No. 53 -October 1991.
Competitiveness Indices and the Trade Performance of the Australian ManufacturingSector. C. Hargreaves, J. Harfington and A.M. Siriwardarna, No. 54 - October 1991.
Modelling Money Demand in Australian Economy-Wide Models: Some PreliminaryAnalyses. Colin Hargreaves, No. 55 - October 1991.
Frontier Production Functions, Technical Efficiency and Panel Data: With Application toPaddy Farmers in India. G.E. Battese and T.J. Coelli, No. 56 - November 1991.
Maximum Likelihood Estimation of Stochastic Frontier Production Functions with Time-Varying Technical Efficiency using the Computer Program, FRONTIER Version 2.0.T.J. Coelli, No. 57 - October 1991.
Securities and Risk Reduction in Venture Capital Investment Agreements. BarbaraCornelius and Colin Hargreaves, No. 58 - October 1991.
23
The Role of Covenants in Venture Capital Investment Agreements. Barbara Cornelius andColin Hargreaves, No. 59 - October 1991.
A Comparison of Alternative Functional Forms for the Lorenz Curve. DuangkamonChotikapanich, No. 60 - October 1991.
A Disequilibrium Model of the Australian Manufacturing Sector. Colin Hargreaves andMelissa Hope, No. 61 - October 1991.
Overnight Money-Market Interest Rates, The Term Structure and The TransmissionMechanism. Colin Hargreaves, No. 62 - November 1991.
A Study of the Income Distribution Underlying the Rasche, Gaffney, Koo and Obst LorenzCurve. Duangkamon Chotikapanich, No. 63 - May 1992.
Estimation of Stochastic Frontier Production Functions with Time-Varying Parametersand Technical Efficiencies Using Panel Data from Indian Villages. G.E. Battese andG.A. Tessema, No. 64 - May 1992.
The Demand for Australian Wool: A Simultaneous Equations Model Which PermitsEndogenous Switching. C.J. O’Donnell, No. 65 -June 1992.
A Stochastic Frontier Production Function Incorporating Flexible Risk Properties.Guang H. Wan and George E. Battese, No. 66 - June 1992.
Income InequaBty in Asia, 1960-1985: A Decomposition Analysis. Ma. Rebecca J.Valenzuela, No. 67 - April 1993.
A MIMIC Approach to the Estimation of the Supply and Demand for ConstructionMaterials in the U.S. Alicia N. Rambaldi, R. Carter Hill and Stephen Farber, No. 68 -August 1993.
A Stochastic Frontier Production Function Incorporating A Model For TechnicalInefficiency Effects. G.E. Battese and T.J. Coelli, No. 69 - October 1993.
Finite Sample Properties of Stochastic Frontier Estimators and Associated Test Statistics.Tim Coelli, No. 70 - November 1993.
Measurement of Total Factor Productivity Growth and Biases in Technological Change inWestern Australian Agriculture. Tim J. Coelli, No. 71 -December 1993.
An Investigation of Stochastic Frontier Production Functions Involving FarmerCharacteristics Using ICRISA T Data From Three Indian Villages. G.E. Battese andM. Bernabe, No. 72 - December 1993.
A Monte Carlo Analysis of Alternative Estimators of the Tobit Model. Getachew AsgedomTessema, No. 73 - April 1994.
24
A Bayesian Estimator of the Linear Regression Model with an Uncertain InequalityConstraint. W.E Gfiffiths and A.T.K Wan, No. 74 - May 1994.
Predicting the Severity of Motor Vehicle Accident lnjuries Using Models of OrderedMultiple Choice. C.J. O~onnell and D.H. Connor, No. 75 - September 1994.
Identification of Factors which Influence the Technical Inefficiency of Indian Farmers.T.J. Coelli and G.E. Battese, No. 76 - September 1994.
Bayesian Predictors for an AR(1) Error Model. William E. Griffiths, No. 77 - September1994.
Small Sample Performance of Non-Causality Tests in Cointegrated Systems. Hector O.Zapata and Alicia N. Rambaldi, No. 78 - December 1994.
Engel Scales for Australia, the Philippines and Thailand: A Comparative Analysis.Ma. Rebecca J. Valenzuela, No. 79 - August 1995.
Maximum Likelihood Estimation of HousehoM Equivalence Scales from an ExtendedLinear Expenditure System: Application to the 1988 Australian HousehoMExpenditure Survey. William E. Griffiths and Ma. Rebecca J. Valenzuela, No. 80 -November 1995.
Bayesian Estimation of the Linear Regression Model with anConstraint on Coefficients. Alan T.K. Wan and William E.November 1995.
Uncertain IntervalGriffiths, No. 81
Unemployment, GDP, and Crime Rate: The Short- and Long-run Relationship for theAustralian Case. Alicia N. Rambaldi, Tony Auld and Jonathan Baldry, No. 82 -November 1995.
Applying Linear Time-varying Constraints to Econometric Models: An Application of theKalman Filter. Howard E. Doran and Alicia N. Rambaldi, No. 83 - November 1995.
An Improved Heckman Estimator for the Tobit Model. Getachew Asgedom Tessema,Howard Doran and William Griffiths, No. 84 - March 1996.
New Guinea GoM or Bust: Detection of Trends in the Quality of Coffee Exports in PapuaNew Guinea. Chai McConnell, Alicia Rambaldi and Euan Fleming, No. 85 - March1996.
On the Estimation of Production Functions Involving Explanatory Variables Which HaveZero Values. George E. Battese, No. 86 - May 1996.
Bayesian Estimation of Some Australian ELES-based Equivalence Scales. WilliamGriffiths and Rebecca Valenzuela, No. 87 - May 1996.
Testing for Granger Non-causality in Cointegrated Systems Made Easy. Alicia N.Rambaldi and Howard E. Doran, No. 88 - August 1996.
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Inefficiency, Uncertainty and the Structure of Cost, Cost-share and Input-demandFunctions. C.J. OT)onnell, No. 89 - August 1996.
A Probit Analysis of the Incidence of the Cotton Leaf Curl Virus in Punjab, Pakistan.Munir Ahmad and George E. Battese, No. 90 - September 1996.
An Analysis of Attendance at Voluntary Residential Schools. Bernard Conlon, No. 91 -December 1996.
Estimation of Risk Response by Australian Wheat Producers. Alicia N. Rambaldi andPhil Simmons, No. 92 - December 1996.
Bayesian Methodology for Imposing Inequality Constraints on a Linear ExpenditureSystem with Demographic Factors. William E. Griffiths and DuangkamonChotikapanich, No. 93 - December 1996.