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.

]8

oo

o d-o

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quantity

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ooo

ooo

0"~oo0

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.

22

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.

25

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.