a note on the use of aggregate data in individual choice models: discrete consumer choice among...

Journal of Econometrics 18 (1982) 313-335. North-Holland Publishing Company

A NOTE ON THE USE OF AGGREGATE DATA IN INDIVIDUAL CHOICE MODELS

Discrete Consumer Choice Among Alternative Fuels for Residential Appliances

Raymond S. HARTMAN* Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Boston University, Boston, MA 02215, USA

Received February 1981, final version received October 1981

In order to exploit the usefulness of qualitative models of individual choice and the availability of aggregate energy demand data, this paper develops a maximum likelihood technique for incorporating aggregate data into individual choice models. The paper treats the use of aggregate data as a measurement error problem; it develops consistent estimators of individual taste by correcting for the implied measurement error. Using the maximum likelihood technique, a number of individual choice models are estimated for residential demand for space heaters, water heaters, ranges and clothes dryers. The extent of asymptotic bias generated by the uncorrected use of aggregate data is documented.

1. Introduction and overview

The qualitative choice models used in many microeconomic analyses incorporate the preferences of a ‘representative’ individual; as a result, these models should be estimated utilizing data on the actual choices of individuals.’ However, data experiments focusing on individual choices are expensive to gather and have not been readily available.’ Furthermore, those experiments that do gather individual data are usually geographically (and sometimes socioeconomically) stratified; at times, they omit important explanatory variables. As a result, variation in tastes and socioeconomic/demographic characteristics across U.S. regions and groups of individuals may not be fully captured.

*The author gratefully acknowledges the comments and computer assistance of Jerry Hausman, Paul Ruud, William Wallace, Knut Mork, and a very helpful anonymous referee. The contributions of Jerry Hausman, Paul Ruud, and William Wallace cannot be over-emphasized.

‘As in Domencich and McFadden (1976), Hausman (1979) Hausman and Wise (1978) McFadden (1975), McFadden and Train (1978), and McFadden, Tye and Train (1976).

*The Midwest Research Institute (1979) data is one extensively-used example. As the importance of micro data has become more apparent, a greater effort has been directed toward improving and expanding these data sources.

0304-4076/82/0000-0000/$02.75 0 1982 North-Holland

314 R.S. Hartman, Aggregate data in individual choice models

On the other hand, compilations of aggregate data are fairly accessible. For example, a body of aggregate data that is readily available and used extensively for analysis of fuel choice3 consists of pooled time-series of state cross-sections. However, the use of such aggregate data for estimating individual choice models can be defended by its success as an empirical tool rather than its appropriateness for the underlying behavioral model: McFadden and Reid (1975) demonstrate that the use of such aggregate data for parameter estimation and prediction with disaggregated choice models leads to serious inconsistencies unless rather restrictive conditions are met.4

This paper attempts to bridge the gap between models of individual choice and aggregate data. The reasons for bridging the gap are immediate. The use of individual choice models has proved very useful in demand analysis, in particular for energy demand. Furthermore, the aggregate time-series/cross- sectional data available for many demand analyses (again, particularly energy demand) offer a rich resource to be exploited. The capability to combine such choice models and such aggregate data would be extremely useful. The aggregate data could then be used to estimate the individual choice models directly or to supplement micro data sources for estimation.

The paper proceeds by formally treating the use of aggregate data as a measurement error problem. Section 2 analyzes the measurement error generated when using aggregate data to estimate individual choice models. Only aggregate data is discussed; however, the results are easily generalized to the case of mixed aggregate and micro data. A standard choice model is introduced and a method is developed for correcting for the measurement error and obtaining consistent maximum likelihood estimates. In order to apply these results, section 3 briefly discusses the issues involved in residential appliance choice. Given a simple version of the resulting model of fuel choice and using aggregate data, section 4 presents estimates of the individual parameters of taste and indicates the extent of asymptotic bias generated when aggregate data are used without correction. Systematic parameter bias toward zero is demonstrated when such measurement error is present. Consistent estimates are presented when the extent of actual measurement error is incorporated and corrected for.

Finally,‘section 5 extends the empirical results of section 4. The simple model of appliance choice used in section 4 to demonstrate measurement bias makes choice a function of fuel price and appliance cost. Such a simple model is useful for demonstrating the effect of measurement error inherent in using aggregate data. However, there exist two additional potential sources of parameter inconsistency: misspecification and simultaneity.

%ee Anderson (1972, 1973, 1974), Baughman and Joskow (1974a, b, 1976), Hartman (1979b,d), Hartman and Hollyer (1977), Joskow and Baughman (1976), and Lin et al. (1976).

4Their results are developed for a binary probit model estimated by regression techniques.

R.S. Hartman, Aggregate data in individual choice models 315

Misspecification bias will occur if important explanatory variables have been excluded. Simultaneity bias will occur if there are supply relationships that are not incorporated into the analysis and estimation. For completeness, both of these possibilities are explored in section 5. Specifically, the choice models are estimated correcting for measurement error while incorporating additional effects on the fuel choice of a household. These additional effects include fuel price expectations, the availability of a natural gas hook-up for the household, and the fuel historically used by a household for an appliance being replaced (an inertia effect). Secondly, to correct for potential simultaneity bias due to residential appliance supply, instrumental variable estimates of appliance prices are used in estimating selected demand models.

2. Individual choice models and aggregate data

2.1. The standard model with individual data

Before exploring the applied problem and examining the empirical results, it is necessary to develop the formal estimation techniques. To that end, the standard utility formulation for individual choice models is that the utility Uij of alternative j to individual i is

U, = Oij(Xj, ai) + qj

= 2, p+ Fij, (lb)

where Xj is a vector of the observable attributes and levels of consumer services offered by alternative j, ai is a vector of observable characteristics of the individual i and the envirpnment in which individual i makes his/her choice, uij is the ‘average’ utility of a ‘representative’ individual, and cij is a random error term induced by purely random behavior, measurement error and/or unobserved characteristics of the individual and/or alternative.

Letting 2, represent combinations of Xi and ai, then oi,(Xj, ai) = Zij /f?, where p is assumed constant over the entire population (i.e., homogeneous tastes).”

Given utility function (1) the probability that an individual i chooses alternative k is

5This is the utility formulation utilized by most authors. Hausman and Wise (1978) utilize this model for heterogeneous tastes within a probit specification by decomposing the elements of p into both a constant (mean) and a random taste parameter. McFadden, Tye and Train (1976) also develop a heterogeneous preference model. Heterogeneous tastes could be introduced here; however, for simplicity I focus only on issues of aggregation.


ZUZ Pr(Z, p+ Eik > Zij /7+ Eij)

for all ,j # k, = Pr(sij-Eik <(Zi, -Zij)/l)

where (2)

Once the form of the utility function and the distribution F(E) of the sij are specified, unknown parameters (i.e., a) of i7,, and the parameters of F(E) can be estimated using the likelihood function

(3)

where the likelihood function L(0) is defined over all individuals i = 1,. ., m and alternatives k = 1,. . ., n, yi, = 1 if individual i chooses alternative k and = 0 otherwise, 1: = 1 Pi, = 1 for any individual i.

As is well known, the assumptions regarding the distribution F(E) determine the functional form of the Pi, as follows:

l &ij distributed as Weibull

This assumption is the basis for logit analyses. In this case qjk =sij-sik is distributed logistically [see Domencich and McFadden (1976)] and

P, = Pr (qjk < U, - Uij) for all j # k,

(4)

l ~~~ distributed as normal

In this case, qjk is also distributed normally and eq. (2) becomes

W’k 1

pi/c= j _..-I[... wp4(r;0,0)dr, ,..., drj ,..., dr, for jjk, (5) -m

where wzj= Bi,- Oij and c$(r.; 0,!2) is the multivariate normal frequency with


0 mean and covariance matrix Sz evaluated at r. If COV(&ij,Eik) is assumed zero

for all k#j, ‘independent’ probit results;6 otherwise, covariance probit results

[Hausman and Wise (197811. Substituting (4) or (5) into (3), using data on individuals for their choices

and a, and maximizing L with respect to 0 will yield estimates of the parameters of Oij (i.e. fl and F(E).

2.2. The model using aggregate data

Suppose as before we want to estimate the taste parameters ,L? in eq. (1 b) but hope to rely on the pooled time-series/cross-sectional data rather than data on individuals.’ For purposes of discussion, let eq. (1) become

uij = xj Bl + Ui pz + Eij,

and assume that only three choices are available (j = 1,2,3). Assume further

that our data source consists of pooled annual time-series of state cross- sections. In this case, we observe collections of individuals i (market shares) who made choices among j= 1,2, and 3 in a given cross-sectional unit (state) and time period (year) when their utility functions are assumed to be (6a).

Furthermore, we observe only average state estimates of Xj and ai rather than the actual individual values. For example, rather than observing the ‘true’ price of alternative j (X,), we observe xj= X,-vj (Xj observed with

error uj) where zj is the average price of alternative j in a state; and rather than observing the true income of individual i (a,), we observe &=ai--vi (ai observed with error vi), where Gi is the average income in a state. Using such aggregate data for the characteristics of individual i (a,) and the characteristics of the choices facing individual i (Xi), eq. (6a) becomes

Uij = Xjpl + Uip2 + Eij

= (2j + Vj)fll + (hi + VJaz + Eij

= r?jpl + ;ip* + (Eij + VjB~ + Vi&)

where the new error fij is induced by cij, in addition to the measurement

6With parameter estimates and predictive performance similar to that of logit when micro data is used.

70r on mixed aggregate/disaggregate data.


errors of Xj and a, (UjP1 +vip2) generated by using the aggregate data rather than the individual data.’

Using eq. (6b), the probability that an individual i chooses alternative k is given as follows:

pik = W(Eij + UjDl + UiiJJ -t&k + ukBl + UiBJ < Gik - zij)fl

= Pr(,;,- E;:k < (Z, - Z,)8) for all j # k,

where = Pr(fjk < tzik - zij)EL

fjk = Cij - Fiik.

(7)

Once we have characterized yljk, the Pi, are substituted into a version of (3) that takes into account the use of the pooled data9 as follows:

where the pooled time-series runs from year t, to t,, covering 50 cross-

sectional units (states) for Mskt persons in state s in period t that choose alternative k.

As before, the assumptions regarding the distributions of Eij, Ui, vk, and vi will determine the functional form of the Pi,. Assuming joint normality’O of the random variables, we have fjk normal, where

(9)

‘This formulation is similar to that of Hausman and Wise (1978) except they attribute the additional randomness to tastes rather than measurement error. With individual data, randomness in tastes will probably be most important; when using aggregate data, randomness due to measurement error will probably be most important. Of course, both types of randomness could be parameterized and tested if the data existed. Such data would require observations on individuals with repetitions where the measurement errors were capable of characterization.

‘The use of aggregate data implies that Msltr will be determined by shares data; hence, it too is subject to measurement error. I do not examine the effects of that measurement error here. Formally, P, and P, in (8) should be differentiated by state and time (P$), given the variation of independent variables across s and t. I do not do so in an attempt to avoid too much notation.

“This is the only assumption that insures analytic tractability. For example, for rjjijt to be logistic, the 6 must be distributed as Weibull. However, random variables distributed as Weibull are closed under maximization [Domencich and McFadden (1976, pp. 61-62)]. Since the Cij consist of the sum of random variables aij. vi, and rj, the distribution of Iij is not easily characterized if sij, vi, and v, are Weibull. Likewise, if qj is distributed as Weibull and the vi, ulr, and vj are distributed normally, cj (hence rjjk) is the sum of both Weibull and normal random variables. The distribution of that sum is extremely complicated. Only when sij. vi, uI, and vj are distributed normally will cj and rjjk be distributed normally; in that case the P, will be analytically tractable.

R.S. Hartman. Aggregate data in individual choice models 319

In (9) it is evident that for personal attributes (a,) that enter utility additively, measurement errors will have no effect” on estimation - (vi*- z+) =O. If, as

assumed, measurement errors occur in the attributes of the alternatives (Xj) or in interaction terms of the Xj and ui, thenl’

Assuming oj-N(O, I$) and sijw N(0, CJ~‘) for all j where 5 is a diagonal covariance matrix with the variances of measurement errors of Xj along the diagonal, assuming all covariances among measurement errors (i.e., off- diagonal elements of 5) and between measurement errors and sij are zero, and assuming cov(sij, sik) = ajk for all j and k, we obtain

‘(fjk) = ‘((&ij - Eik) + (“j - uk)’ PI) = 0, Pb)

and

PC)

For further ease of exposition, assume j?, is a scaler, then

and for 11;r and f3r with bivariate normal density function g,(~2,,rj;,;QR,) and covariance matrix

we have from eq. (5)

where b, is a standard bivariate normal distribution with correlation

“If there are no measurement errors in the attributes of the alternatives (uj = ut = 0) and the attributes of the alternatives enter utility additively (i.e., not in interaction with personal attributes), then tjjk=~Eii-~ik and standard logit or probit analysis is possible given assumptions about eij and .sik (section 2.1).

‘ZAssuming fll includes parameters for attributes of the alternatives and all interaction terms.


coefficient p1.13 The probabilities of Pi, and Pi3 are calculated in the same

manner.

2.3. Model estimation Clearly, for the estimation of an individual choice model using aggregate

data, the full Pij [eq. (1 l)] and Qj [eq. (lo)] need to be incorporated into likelihood function (8) for j= 1,2,3. The Pij and 52, for j= 1,2,3 are formally developed in appendix A, in addition to the first-order conditions for estimation of p and ojk = COV(S,~, EJ.

Before proceeding to the applied problem and using these estimation results, it is useful to examine eqs. (10) and (11) in greater detail. If there were no measurement error in the problem, rrzl = &, = gf3 =O, and the likelihood function and estimation problem would reduce to the standard probit problem using micro data. In this case, the taste parameter pi would not appear in 52, and the bounds of integration.

If, however, aggregate data are used (cri, # 0, of2 # 0, crz3 # 0), and standard probit techniques are utilized, the standard likelihood function would ignore the presence of 8: and the measurement error variances in 52, and Pi,. The resulting estimation of pi would be inconsistent. If the oij are non-zero, their presence in Pi, and 52, must be explicitly treated for consistent estimation.14

3. Consumer choice among alternate fuels for residential energy using appliances

Having indicated the desired consistent estimator (section 2 and appendix A), let us proceed with the analysis of residential appliance and fuel choice, which offers an example where individual choice models and aggregate data can be usefully combined.

As developed elsewhere [Hartman (1979a, b, c)], any analysis of residential energy demand should deal with the fact that fuels and fuel-using appliances/equipment are combined in varying ways to produce a particular residential service. This demand behavior can be decomposed into decisions regarding equipment purchase and equipment use [Hartman (1979a)]. It is the equipment purchase decision that is usually analyzed using choice models.i5

Many of the efforts in the literature utilize aggregate data without correcting for it. As indicated in section 2, this will generate inconsistent

/ (o:+a~-2u~*+~:((r,2,+(Tt*~~~6:+6:-2~1~+~~~~L+~v2~~~. 14Notice in this case, the measurement error variances are treated as data to be estimated

elsewhere and input for the estimation of p and elk = cov (Eij, E&

‘5Anderson (1972 1973), Baugham and Joskow (1974a, b, 1976), Charles River Associates (1979), Hartman (1479a,b,c,d), Hartman and Hollyer (1977), Hausman (1975), Joskow and Baugham (1976), and Lin et al. (1976).


parameter estimates and inconsistent aggregate energy choice estimates

[McFadden and Reid (1975)]. To help overcome this and other difficulties,16 this section specifies a choice model for residential appliances which will be estimated using the techniques developed above.

The specification of random utility used here is a version of models found in the literature [see Hartman (1979c), Hausman (1979)]. In that development, the size or capacity of an appliance for household i will be determined by a vector of household characteristics a, where a, includes weather effects, climate effects, and socioeconomic factors such as the size and age composition of the family, the size of the residence, etc. The capacity of the equipment can then be parameterized as a function of (a,). Conditional on (a,), the cost of capital and the cost of fuels (Pj), rational consumers will identify within each fuel the type of appliance to be purchased (characterized by appliance price Cj, appliance efficiency EFFj, and all other appliance characteristics aj) and the desired level of utilization [Hartman (1979c)].

Finally, consumers will make their fuel/appliance choice based on comparing utility among the possible choices j, j= gas, oil, and electricity. Using an indirect utility formulation, we have therefore the utility of alternative j to household i given asl7

Uij = V( ~, Cj, PjIEFFj, ~j; ~i) = ~ij + &ij = Zij 8+ ~ij, (12)

where x is household income. By specifying the form of < the exact components of Rj and ai, and the stochastic nature of cij and all measurement errors, we can substitute (12) into (7) and (8) to estimate the individual parameters of taste. This is done in section 4.

4. Empirical results: The extent of measurement bias

My purpose in this section is to explore the estimation results generated when using aggregate data in individual choice models. To that end, I do not specify complicated versions of the indirect utility in eq. (12). Empirical results from richer formulations of utility are presented in section 5.

IsOther deficiencies include the following:

0 The analyses have applied the choice models to shares of the stock of home heating equipment. The discrete choice models are really appropriate only for changes (new and replacement purchases) in the stock of capital [see Hartman (1979a, c)].

0 The characteristics of the fuels and complementary fuel-burning equipment in the choice set have not been adequately specified. Most choice models have based choice only upon operating costs and have not included capital costs [see Hartman (1979 a, c)].

0 Most formulations to date have utilized conditional logit, thereby suffering from the ‘independence of irrelevant alternatives’ [see Hartman (1978, 1979a)]. All three of these deficiencies are corrected here.

“See Hartman (1979~); a direct utility specification is also developed therein.


Rather, a straightforward model of utility is posited that includes operating costs and capital costs for the alternative fuel/appliance options. The end-use equipment for which the choice model is estimated include space heating equipment, clothes dryers, water heaters, and kitchen ranges.

Two models are tested for space heating which incorporate variables for operating costs and the cost of capital services. More specifically, the variables’* include:

Space heating

Capital costs (a,)

Operating costs (LYJ

Model 1 Cost of space heating equipment (C,)

Fuel price (P,)

Model 2 Cost of space heating equipment/household income (Cl/Y)

Fuel price * heating degree days (P, * HDD)

Both models are estimated for choices across natural gas, oil and electricity. For clothes dryers. water heaters and kitchen ranges, the variableslg include:

Clothes drying, water heating, and cooking

Capital costs (G(J

Operating costs (Q)

Equipment cost/ household income Fuel price/household

(C,/Y) income (P//y)

The models of choice for these three end-uses are estimated for natural gas and electricity.

For all of these end-uses, unit fuel price and equipment costs are appropriate measures of operating costs and capital service costs if weather/climate is fairly constant or irrelevant to use, if equipment lifetimes are similar across fuel-specific equipment and if household discount rates are fairly constant. It is assumed that use of water heaters, kitchen ranges and

‘*Fuel prices are the average price of natural gas per cubic foot (Gas Facts), marginal price of electricity (developed by Data Resources, Inc. for the Electric Power Research Institute) and #2 fuel oil prices per gallon (AGA househeating survey). Heating degree days are developed by the National Oceanic and Atmospheric Administration. Household income is developed from Survey of Current Business data. Finally, detailed equipment costs have been developed for alternative space heating equipment. The data is summarized in MIT Residential Energy Demand Group (1979).

lgThe price and income variables are the same as those used for space heating. The appliance equipment costs are developed by Merchandising Week. Again for greater discussion, see MIT Residential Energy Demand Group (1979).


clothes dryers is invariant to weather. However, it is clear that space heating use and operating costs will be very weather-sensitive. Since weather and climate differ across state and furthermore, since empirical work suggests that discount rates vary inversely with income [Hausman (1979)], space heating model # 2 adds weather into operating costs (Ps * HDD) and attempts to proxy varying discount rates by dividing by household income (C,/y). The other end-uses exclude weather but divide both costs by income. As a result, we have from eq. (12):

Space heating model 1

Space heating model 2

Uij=(Cj/Y)Crl +(Pj*HDD)N, +Eij,

All other end-uses

uij=(cj/Ybl +tpj/Yk2 +&ij

Pooled annual state averages for P,, HDD, Cj, and y are utilized. The results from estimation of space heating models 1 and 2 are given in

tables la and lb. The other end uses are compiled in table 2. Logit results are given which are equivalent to independent probit results when there is no measurement error or errors in variables (EIV). For comparison sake in examining the effects of aggregate data, all results utilize independent probit2’

For model 1 in table la, measurement error has a substantial effect on the parameter estimates. If measurement error exists at 10% of the average data,21 consistent parameter estimates (at extremely high levels of significance) are 3&70x larger than the inconsistent estimates generated by assuming no measurement error. If the measurement error is 15% of the variable means, consistent parameter estimates are 7.27 and - 11.24 compared with 3.96 and -3.60 generated if no measurement error is assumed. For measurement error above 15%, the likelihood function becomes ill conditioned and estimation becomes impossible. From these results, it is clear that ignoring measurement error attendent with aggregate data will generate severely inconsistent parameter estimates of household tastes (a1 and ~1~).

The space heating model in table lb is better specified and more robust to measurement error. With measurement error at 75% of the variable means,22

“The full covariance probit could have also been developed. “That is, measurement error variances are 10% of the value of the composite variables

appearing in utility. 22That is, 75% of C,/y and P, * HDD, appropriately scaled. See notes to table 1.


highly significant convergent estimates obtain which differ significantly from those obtained by ignoring such measurement error. The inconsistency is only 20% for tlI and 50% for CQ.

Table 1

Logit and probit results for space heating.”

(a) Model 1 %(Cr) %(Pr) L

Logit

Independent probit, no EIV

Independent probit, 10% ED’

Independent probit, 15% EIV

Independent probit, EIV> 15%

5.0954 -4.5665 -46435 (94.01) (-57.61)

3.9622 - 3.6048 - 46629

(967) (- 548)

5.3741 - 6.4704 -46532

(590) (-441)

7.2749 - 11.2463 - 46479

(332) (-261)

Likelihood function becomes ill-conditioned

Model 2 %(C,lY) a,(PI* HDD) L

Logit







0.54217 -0.55782 -47210 (98.8) (- 34.5)

0.4229 -0.4513 -47370 (9’19.5) (- 323.4)

0.4332

(884)

- 0.4839 -47349 (- 329)

0.4397

(869)

-0.5039 -47339 (-332)

0.4528

(836)

- 0.5436 -47322 (-336)

0.4906

(748)

- 0.6849 - 47287 (- 342)

0.5366

(646)

-0.9112 -47257 (- 326)

“EIV=errors in variables at X% of means, L = value of likelihood function, C, = system costs in dollars, scaled by l/100,

Y = household income, undeflated, P, =fuel price in cents per therm, kWh and gallons,

HDD = heating degree days, scaled by l/10000,

t-statistics for H,: cq =0 in parentheses. See MIT Residential Energy Demand Group (1979).

R.S. Hartman, Aggregate data in indiuidual choice models 325

Table 2 presents results for clothes dryers, water heaters and kitchen

ranges. These results are much more sensitive to measurement error. For clothes dryers, measurement error at the 0.10% level would imply parameter

inconsistency at the level of 8% for a, and 9% for a2 if uncorrected aggregate data were used. With measurement error variances at 0.50% of the means,

the inconsistencies would be 41% and 44% respectively. For water heaters at 0.10% EIV, the inconsistency is 8% and 18x, while at 0.50% EIV the likelihood function becomes ill conditioned so that estimation is impossible.

Table 2 Logit and probit results for water heaters, kitchen ranges, and clothes dryers.”

(4 Clothes dryers %(C,/Y) %(P//Y) L

Logit


Independent probit, 0.10% EIV



Independent probit, O.SOu/, EIV

Independent probit, 0.20% /o. 10%

- 5.7468 - 71.909 (- 1958.5) (- 926.93)

- 3.434356 -42.31929 (-43.17) (- 18.38)

- 3.7089 (- 38.6)

-46.6127 (- 18.1)

-5.768181 (- 15.65)

- 75.745 163 (-11.03)

- 10.264 (4.75)

- 134.4774 (- 4.23)

-272.4153 (-0.009)

- 3526.2268 ( - 0.009)

-3.835122 (- 35.99)

-48.291519 (17.44)

-2131.6x 10“

-2133.11 x 10“

-2130.6 x lo4

-2132.9 x lo4

-2132.41 x lo4

-2131.9 x lo4

-2130x lo4

(b) Water heaters %(C,/Y) %(P,/Y) L

Logit




lndependent probit, 0.50% EIV

Independent probit, 0.200/,/0.10”/,

- 4.0667 - 139.05 -2784.7 x 10“ (-713.46) (- 3062.8)

- 2.480393 - 80.2258 -2798.13 x 10“ (-21.36) (- 77.20)

-2.6932 -97.9115 -2778.72 x lo4 (19.75) (-60.87)

- 5.1787 -258.9160 -2740.35 x lo4 (12.78) (- 12.82)

Likelihood function becomes ill-conditioned

-2.7785 - 98.6644 -2778.1 x lo4 ( - 20.03) (-61.39)


Table 2 (continued)

64 Kitchen ranges ar(Crly) W.dv) L

Logit


Independent probit, O.lOo/, EIV




Indepedent probit, 5.00% EIV

Independent probit, 0.20% /O.lOo/,

0.26202 -41.678 (573.93) (-1139.1)

0.162983 - 25.9708 (16.44) (-31.77)

0.165318 (16.33)

-26.3130 (-31.06)

0.176118 (15.87)

-27.9519 (-38.39)

0.189929 (15.03)

- 30.09 (- 24.92)

0.283482 (10.91)

-44.5183 (- 13.75)

12.0719 (0.007)

- 1894.74 ( - 0.007)

0.165333 (16.33)

-26.3152 (-31.06)

-4111.5 x 104

-4111.6 x 10“

-411156 x lo4

-4111.3 x lo4

-4111.06x LO4

-4110.3 x lo4

-4109.9 x lo4

-4111.6 x lo4

“EIV=errors in variables at X% of means, L = value of the likelihood function at maximum, C, = appliance price for fuel f in dollars, P, =fuel price in cents per therm and kWH, y = real household income,

t-statistics for HO: parameter =0 in parentheses.

Prices are deflated by the consumer price index regionalized using the state cross- sectional index of Kent Anderson. Number of purchases of each choice [i.e., Mslir in eq. (8)] scaled by 10v4. C,/y and P,/y scaled by 100.0 to speed convergence. Data are fully discussed in MIT Residential Energy Demand Group (1979).

Kitchen range choice estimation is somewhat more robust to error. For measurement error at the 0.10% level, the inconsistency in oi, is 1% and in 8, it is 4%. At the 0.50% EIV level, the inconsistency is 7% for oi, and oi,. At levels of measurement error of 1.00% the inconsistency is 14%.

Table 2 also presents estimates of c~i and a, for levels of measurement errors that differ across capital and operating costs - 0.20% for capital costs and 0.10% for fuel prices. For clothes dryers the resulting estimates of a, and ~1~ are -3.835 and -48.292. For water heaters the estimates are -2.78 and -98.66. For kitchen ranges the estimates are 0.1653 and -26.32. The extent of inconsistency generated by using uncorrected aggregate data (no EIV) can be easily estimated from the table.


Tables 1 and 2 indicate the standard result23 that measurement error systematically biases the parameter estimates toward zero. It is clear from the results that the greater the measurement error, the greater is the bias toward zero.

Table 3 presents results for a subset of appliances (water heater and kitchen range choice) for actual measurement error estimates. Because table 2 indicates that consistent estimates can be extremely sensitive to measurement error, it may be impossible to estimate individual taste parameters if the measurement error is too large. This fact should not be surprising. As a result, the contents of table 3 are interesting. The sources for the estimates of measurement error and the technique for estimating the composite variable variances are indicated in the note to table 3. The measurement errors for fuel prices were the smallest, varying from 0.0% of the state mean (for single utility states such as Montana) to 71% for New York. Income measurement variances varied from 150% of the state mean for homogeneous states such as New Hampshire to 863% of the state mean for Alaska. The variances for some appliances prices are quite large - a full 8800% of the mean for gas water heaters, for example.

In spite of these large variances, the variances of the composite variables were sufficiently small to permit estimation of individual taste parameters for

Table 3 Actual results for water heaters and kitchen ranges.”

Water heaters

Kitchen ranges

a,(C,/y)

-2.91303 (-23.86)

0.17343 (15.15)

%(PflY) L

-374.781 - 2703.09 (-28.70)

-28.11302 -4111.63 (-27.51)

“Using a h&-order Taylor approximation and assuming independence of the measurment errors,

YC,/Y)=(l/Yz) VC,) + q/Y’ V(Y) 3 ~(PJ/Y)~(llYZ) VpJ)+p:/Y” V(Y).

V(P,) is estimated from state utility data. The estimate for each state was chosen from the year between 196551975 for which the greatest number of utilities reported. The appropriate block rate was chosen for an assumed average usage; for example, 20&250 kWh for electricity. The variance was assumed to hold for all sample years. For the data source, see Data Resources, Inc. (1977). The raw data tiles are needed.

V(Y) is estimated by state for 1975 by the Bureau of the Census (1978). It is assumed to be constant across sample years.

V(C,) is estimated from a national sample of appliance distributors

and assumed to hold for each state. For example, 125 gas water heaters and 80 electric water heaters were sampled.

‘%ee, for example, Malinvaud (1970, pp. 144147).

JE-B

328 RX Hartman, Aggregate data in individual choice models

those appliances in table 2 that showed the greatest sensitivity to measurement error. The results indicate in table 3 that severe bias in CQ for

water heaters will occur if uncorrected aggregate data is used. Likewise, for kitchen ranges c(~ is more severely biased than tli.

5. Empirical extensions

In addition to the observation that aggregate data can generate severely inconsistent parameter estimates, a second observation is warranted for the results in section 4. For all estimates in that section, the operating cost effect is negative. Ceteris paribus, we would also expect the capital cost effect to be negative. This is true for water heaters and clothes dryers; however, the capital cost effect for both space heating equipment and kitchen ranges is positive. It would appear that this positive capital cost effect reflects either excluded equipment characteristics (such as quality, brand effects and optional specialty features), excluded household characteristics, or a simultaneity bias caused by not explicitly modeling the supply of appliances. The presence of excluded specialty appliance features and fuel-specific household preferences would exist more convincingly for space heating equipment and kitchen ranges. Clothes dryers and particularly water heaters

offer a smaller array of specialty options; hence, the choice in these cases is more truly ceteris paribus and the capital cost effect would be expected to be negative.

These considerations are explored more fully in table 4. The potential presence of misspecification (excluded variables) is assessed for the four appliances by better specifying operating costsz4 and expanding the set of exogenous variables affecting fuel choice (table 4a). The potential presence of simultaneity bias is assessed for dryers by generating an instrumental variable estimate for the appliance price (table 4b). The expanded variable set

and the instruments are defined formally in the note to table 4. The results are interesting. The price effects (equipment price and fuel

price) are negative for all appliance choice models in table 4a. Thus, the inclusion of a more complete set of exogenous variables eliminates apparent specification bias in the equipment price coefficient for ranges and space heaters found in section 4. These additional terms include the previous fuel used for an appliance that is being physically retired (a,), the household expectations regarding future fuel price increases (cc4) and the access to either fuel (~1~).

The signs of all the variables in table 4a are as expected. All own price effects are negative (a,,~,) although the appliance price effects for water heaters and ranges are not significantly different from zero. The effect of previous fuel (aJ ownership is positive, as expected. Thus, if an appliance is

24Effciency-corrected, Btu-corrected operating costs are used.


Table 4 Extended estimates.=

Single-equation estimates Dryers

Water heaters Ranges

Space heaters

Appliance price (a,)

Fuel price (a*)

Previous fuel (a3)

Fuel price expectations (a,)

Access to fuel (as)

Likelihood

-1.114528 (- 5.69)

- 8.947798 (- 8.45)

2.54273 (19.03)

1.112151 (9.77)

- 1799.69

- 0.053384 (-0.163)

-3.651624 ( - 10.05)

1.66774 (48.20)

0.583956 (7.68)

- 2038.49 _

- 0.003546 -0.01926 (-0.174) (-22.51)

-0.152731 - 1.447623 (-1.45) (148.12)

1.059886 1.414031 (40.56) (308.99)

-0.706592 (-1.56)

0.444922 (8.26)

3298.80 - 17300.06

(b) Instrumental-variable estimates Dryers

Appliance price (a,)

Fuel price (az)

Previous fuel (a3)

Fuel price expectations (a4)

Access to fuel (as)

Likelihood

- 1.382 (-5.53)

- 10.534 (7.51)

2.69 (16.37)

1.193 (9.32)

- 1799.93

V-statistics for H,: parameter = 0 in parentheses. Previous fuel variable indicates the probability that a household is retiring an appliance of a particular fuel. It is measured for gas ranges, for example, as (number of gas ranges retired in year t/total sales of ranges in t). Fuel price expectations variable in year t is measured as (Price,- Price,_,)/Price,_,. Access to fuel variable indicates the probability that a household has a hookup for a particular fuel. For electricity this variable equals one; for gas it is measured as (number of households with gas hookups/total households). The instrumental variables used for appliance price were time and the national CPI. Fuel price and appiiance price variables measured as in section 4. All variables come from the same data sources.

Scaling. of price variables in these runs as follows:

Dryers, water Appliance cost * 100 Space Appliance cost/income / 100 heaters, ranges Fuel price * 1000 heating Fuel price * heating degree days / 10000

Fuel price increase /lO Fuel price increase / 10

being purchased to replace an appliance using a particular fuel, it is more likely the same fuel will be used for the replacement appliance. Fuel price


expectations (ad) are negative; if the price of a fuel has been rising and is expected to continue to rise, that fuel will be less likely chosen. Finally, access (CQ) to a given fuel (gas and electric hook up) will increase the probability of purchase.

Finally, table 4b presents estimates of the dryer choice equation using an instrumental variable estimate of appliance price. The results do not differ considerably from those in table 4a, suggesting that incomplete specification rather than simultaneity was the probable cause of inappropriate signs in section 4.

Appendix A: Full model statement and estimation

For estimation we need the P, and sZj for j= 1,2,3 in addition to the likelihood function (8). For the three alternatives we have the following covariance matrices for an individual in state s and period t where there are R variables measured with error:


where p in (lb) has R elements &,25 oiIr, &,, r&, are the measurement error variances for attributes r of alternatives 1, 2, 3 respectively ((rzr,, &, CT&, are

assumed to be unique for state s and year t - hence the subscript st for (cJ,~,, + &)J, ~j”, ajk characterize the multivariate residual variance in (sil, si2, si3) which will most likely pick up unobserved factors across states and time.26

Generalizing from (11) we have for a given individual with k = 1

64.2)

where wT2 = (2, - 2,)’ BIJG ~7s = (2, - Z,)‘p/JG b is the standard

bivariate normal with p1 =wr, ,,/JG Similar derivations hold for

P, and P,. Using the mathematics developed in appendix B, the first-order conditions

are now apparent. For the log of (8) we have for element Bi of 6’

y=g&~~> I

where 8i can be an element of p, gij or cii for i and j = 1,2,3.

For any j7g we have for k= 1

(A.3)

apt + b(w:,> w:,, PI)%, (A.4)

while for any crij

+ b(wT2, w apI

i-3, Pl)I,.

lJ (A.5)

Eq. (A.4) indicates how the choice model estimated from aggregate data differs from that estimated from individual data. For example, if individual

“Again, R could be a subset of all independent variables assuming a mix of aggregate and micro data.

26This randomness could be incorporated through state and time specific variances; this is not done here.


data is used, i3pl/d~g=0, and for aw:,/ag, and &@,/a& do,,ij/a~,=O for i and j= 1,2. Hence, the first-order condition (A.4) for 8, reduces to2’

where

(2, - z3)g = a(z, - z,)mag,

(‘4.6)

Similar derivations are possible for the remaining first-order conditions. The method used to maximize the likelihood function is that developed by

Berndt et al. (1974). 28 The resulting maximum likelihood estimates will be consistent and asymptotically normal. The asymptotic covariance matrix of the maximum likelihood estimates is equal to the inverse of the covariance matrix of the gradient of the likelihood function evaluated at the maximum e* = (8, gij)*.

Appendix B

For the likelihood function the probabilities are in the form

,JloI) w(4)

F=f’,= j j W,,X,,p(v))dX,dXz, -cc -m

(B.1)

where we arbitrarily work with P, and parameterize a,, a2 and p as functions of q. b is the standard bivariate normal density function. We require dF/iYy, which by the fundamental theorem of calculus is

aF

[

aF aa, aF aa, aF ap -_= --+--+--~ all aa, all da, atj ap all 1

03.2)

We can calculate the components of (B.2) as follows. Rewrite F in two equivalent formulations by reversing the order of integration:

%(q)-_p(‘l)X,

al(q) (1 -PklSP F= j .f 4(X2 1 XA dG,)dX, dX,,

--cc --CC U3.3)

27See Hausman and Wise (1978). 28The computer program is adapted from that developed by Hausman and Wise (1978).

R.S. Hartman, Aggregate data in individual choice models

and

Then

and

= d&(4) @ [

al(v9-drb2h) (1 _ p(r)2)+

1 .

333

(B.4)

VW

Finally, using the relation that the derivative of a multivariate normal distribution with respect to an element of the correlation matrix equals the second cross partial derivative with respect to the corresponding ordinates,29 we have

aF a2F -=--=&l, a2, P) = Wal(rl), a2(d dvl)). ap au, au,

Combining (BS), (B.6) and (B.7) into (B.2), we obtain

[

44 - PW2(d + 4(a2(q)) @ (1 _ p(r)2)+ 1 aa,(d

aq

(B.7)

“See Hausman and Wise (1978).

334 RX Hartman, Aggregate data in indioidual choice models

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