An Ordered Tobit Model of Market Participation

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An Ordered Tobit

Model of Market Participation:

Evidence from Kenya and Ethiopia∗

Marc F. Bellemare†

Christopher B. Barrett‡

September 20, 2004

Abstract

This paper develops a two-stage econometric model, which we label“ordered tobit”, to differentiate between two competing hypothesesregarding household-level market behavior: whether households makemarket participation and volume (i.e. amount of participation) deci-sions simultaneously or sequentially. The first stage models the house-hold’s choice of whether to be a net buyer, autarkic, or a net seller inthe market. The second stage models the quantity bought (sold) for

net buyers (sellers) based on observable household characteristics. Us-ing household data from Kenya and Ethiopia, we find evidence in favorof sequential decision-making. Further, we find that fixed transactionscosts significantly affect both the participation and the volume deci-sions, offering additional evidence in favor of a well-known behavioralanomaly.

∗ROUGH DRAFT – NOT FOR CITATION. COMMENTS GREATLY APPRECI-ATED.

†Corresponding Author and Ph.D. Candidate, Department of Applied Economics andManagement, Cornell University, Ithaca, NY, 14853-7801, [email protected]. We thankJ.S. Butler, David Just, John McPeak and Christian Vossler for their useful comments;Erin Lentz, Sharon Osterloh and Amare Yirbecho for data assistance; the Pastoral RiskManagement (PARIMA) project of the USAID Global Livestock Collaborative ResearchSupport Program (GL CRSP) for financial support of the data collection; as well as theLITEK and BASIS Projects of the USAID and the SAGA cooperative agreement forsupporting data analysis. The usual disclaimer applies.

‡Professor, Department of Applied Economics and Management, Cornell University,Ithaca, NY, 14853-7801, [email protected].

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1 Introduction

Increased reliance over the past decade or so on markets as the foundation fordevelopment strategies puts a premium on understanding household marketparticipation. If many households do not participate actively in markets ordo not respond to market signals, market-based development strategies mayfail to facilitate wealth creation and poverty reduction. Although in simpletextbook models households almost surely participate in all markets, in ruralareas of the developing world significant market frictions commonly impedemarket participation, dampening households’ capacity to take advantageof market opportunities and governments’ capacity to influence microeco-nomic behavior through changing market incentives. Development agencies

and policymakers worldwide have therefore made increasing market partic-ipation a priority.

Yet there has been scant research on market participation, especially in de-veloping country settings where significant frictions make this question mostsalient. Goetz (1992) studied the participation of Senegalese agriculturalhouseholds in the coarse grains market, using a probit model of households’decision to participate in the market (whether as buyers or sellers, withoutdistinction) followed by a second-stage switching regression model of the ex-tent of market participation. Key et al. (2000) developed a structural modelto estimate structural supply functions and production thresholds for Mexi-

can farmers’ participation in the maize market, based on a censoring modelwith an unobserved censoring threshold. Their model usefully differentiatesbetween the effects of fixed transactions costs (FTCs) and of proportionaltransactions costs (PTCs). Holloway et al. (2001) used a Bayesian double-hurdle model to study participation of Ethiopian dairy farmers in the milkmarket when non-negligible fixed costs lead to non-zero censoring, as in Keyet al.

These papers begin, however, from fundamentally different assumptions onthe nature of households’ market participation choices. Goetz and Hollowayet al. explicitly assume sequential choice: households initially decide whether

or not to participate in the market, then decide on the volume purchasedor sold conditional on having chosen market participation. Key et al., bycontrast, implicitly model the household as making the discrete market par-ticipation choice simultaneously with the continuous decision as to volumespurchased or sold.

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Our contribution in this paper is threefold. First, and of most general value,

we introduce a method that nests within it both the simultaneous and thesequential formulations of household marketing behavior, allowing for directtesting of which approach the data most support. The estimation methodwe introduce has the added virtue of being simpler to implement than themore elaborate structural model developed by Key et al. It can be appliedto a relatively broad range of problems, as we briefly discuss in the con-cluding section. Second, we add new empirical results to the thin literatureon market participation, in our case looking at pastoralists’ participation inlivestock markets in southern Ethiopia and northern Kenya. This new appli-cation adds insights from markets for durable assets — livestock — missingin the extant studies on grain and milk. Finally, our data also permit us to

address some interesting empirical questions related to possible behavioralanomalies in household marketing behavior.

The rest of the paper is structured as follows. In section 2, we lay out asimple theoretical model of household marketing behavior, highlighting theimplications of different assumptions about whether households make (dis-crete) participation and (continuous) volume decisions. Then, in section 3,we present the ordered tobit estimator, a two-stage econometric model thattreats both sales and purchases as censored dependent variables, but mod-els the actual participation decision as an ordered decision by partitioningthe real line into three mutually exclusive and collectively exhaustive posi-

tions vis-a-vis the market: net buyer, autarkic and net seller. After brieflydescribing the data in section 4, section 5 then reports the estimation re-sults from applying this novel method to study livestock marketing behavioramong a population of poor herders in east Africa. The concluding sectionfocuses on both the practical policy implications of our empirical findingsand prospective other uses of the ordered tobit model.

2 A Theoretical Model of Market

Participation

Pastoralist households in the drylands of East Africa routinely make deci-sions as to whether to buy or sell livestock, the principal form of wealth in theregion. Under the maintained hypothesis that market behavior is driven by ahousehold’s objective of maximizing the discounted stream of consumptionit enjoys, one can usefully focus attention on the choice problem that relatesoptimal (non-negative) quantities bought and sold, Qb∗ and Qs∗, respec-

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tively, to household attributes and the environmental factors that condition

consumption and market behaviors. For a representative household, let C trepresent discretionary consumption over period t. The household posessesa vector of assets at the beginning of period t. Let W t be liquid but non-productive household wealth, H t reflect the size of a household’s herd, andAt equal the amount of cultivable land it operates. The productive assets— herd and land size — generate income over period t according to themapping Y t = y(H t, At).1 The household may also incur obligatory, norms-driven ceremonial expenses, X t, associated with births and deaths, whichwe treat as exogenously determined.

Under the assumption that the household makes its market participation and

marketed amount2 decisions simultaneously, household livestock marketingbehavior can thus be described by

maxC t,Q

jt

E ∞t=0

δ tU (C t) ∀ j ∈ {b, s} (1)

s.t. C t ≤ y(H t, At) + W t − X t (2)

H t+1 = H t + g(H t, et) + Qbt − Qs

t ≥ 0 (3)

W t+1 = W t − X t − C t + y(H t, At) + p∗st Qst − p∗bt Qbt ≥ 0 (4)

where E is the expectation operator, δ is the household’s discount rate andg(H t, et) represents the biological recruitment (growth) rate of the herd asa function of beginning period herd size and current local environmentalconditions, et. This model is essentially the dynamic generalization of thestructural model presented in Key et al. (2000). The p∗ j are the shadowprices for purchases ( j = b) and sales ( j = s). The shadow prices reflect theboundaries of the “price band” that defines household endogenous valuation

1We use uppercase letters to reflect household attributes and lowercase letters to rep-resent community-level conditions or functional relationships.

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We use the terms “amount”, “extent” and “volume” interchangeably to represent thenonnegative continuous variable reflecting net sales or net purchases. We also abstract fromthe possibility that households could be both buyers and sellers in the same period. In thedata set we use, there were no such observations. Further, in places where transactionscosts drive a significant wedge between buyer and seller shadow prices, there should notbe observations of both purchases and sales within the same (sufficiently disaggregated)period.

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of a resource that may or may not be traded (de Janvry et al., 1991). At the

upper boundary of the price band, households buy, paying a shadow pricethat adds the fixed and variable transactions costs of market participationto the underlying market price, pm

t . At the lower boundary, households sell,receiving net unit value equal to pm

t less the fixed and variable transactionscosts of market participation. Thus,

p∗bt = (1 + vct) pmt + f ct

p∗st = (1 − vct) pmt − f ct

where vc represents the (proportional) variable costs, vct ∈ [0, ∞), suchas market taxes and transport fees per unit sold, and f c summarizes non-

negative fixed costs, including the cost of the person’s transport to and frommarket, search, screening and negotiation costs, etc. Controlling for seasonalvariation — described by the function zk(ζ t) for k ∈ { p, f c, vc} — futuremarket prices, fixed costs and variable costs follow a random walk:

pmt+1 = pmt + z p(ζ t)f ct+1 = f ct + zfc(ζ t)vct+1 = vct + zvc(ζ t)

Rewriting this dynamic optimization problem as a Bellman equation (notshown) one can derive the household’s optimum marketing decisions as

Qb∗t = q b(At, H t, W t, X t, et, f ct, pmt , vct, δ ,ζ t), and (5)

Qs∗t = q s(At, H t, W t, X t, et, f ct, pmt , vct, δ ,ζ t). (6)

The theoretical predictions of this model are several. We would expect thatQb∗

t (Qs∗t ) is decreasing (increasing) in At because if a household cultivates,

its mobility is restricted, thereby limiting the size of the herd it can man-age sustainably, given local forage and water resources. Sales (purchases)should be decreasing (increasing) in income and wealth as households uselivestock sales (purchases) to smooth consumption. Sales should be increas-ing in household demographic shocks that necessitate ceremonial expendi-

tures, X . Both sales and purchases should be decreasing in fixed and variablecosts, while sales (purchases) should be increasing (decreasing) in the mar-ket price. This system of reduced form equations is estimable as a bivariatecensored regression model.

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However, the preceding specification relies on the earlier assumption that

the discrete household choice to participate in the market is made simulta-neously with the continuous choice as to the number of animals to buy orsell conditional on having chosen to go to market. If, however, participationand volume choices are made sequentially, as other papers in this literatureassume (e.g. Goetz, 1992; Holloway et al., 2001), then the preceding modelwill be misspecified. If households make decisions sequentially, we need tobreak each period down into sub-periods.

In the interests of parsimony, we break each period t into only two sub-periods: r = 0 when the household makes the discrete participation decision,and r = 1 when those households that have chosen to participate in the

market as either net buyers or net sellers make their continuous decision asto net sales or purchase volume. This changes the household’s optimizationproblem to

maxC rt,I

jrt,Q

jrt

E 1

r=0

∞t=0

δ tU (C rt) ∀ j ∈ {b, s} (7)

s.t. C rt ≤ y(H rt, Art) + W rt − X rt (8)

H 1t = H 0t + g(H 0t, e1t) ≥ 0 (9)

H 0t+1 = H 1t + g(H 1t, e0t) + I b1tQb1t − I s1tQ

s1t ≥ 0 (10)

W 1t = W 0t + X 0t − C 0t + y(H 0t, A0t) + pstI s1tQst − p∗bt I b1tQ

bt ≥ 0 (11)

W 0t+1 = W 1t − X 1t − C 1t + y(H 1t, A1t) − (I s1tst + I b1t)f ct ≥ 0 (12)

where the indicator variable I brt = 1 if the household chooses to be a netbuyer (I brt = 0 otherwise) and I srt = 1 if it chooses to be a net seller (I srt = 0

otherwise), with a complementary slackness condition that I brt · I

srt = 0. This

implies that I b0t = I b1t and I s0t = I s1t, with Qb0t = Qs

0t = 0. In this formulation,the boundary shadow prices no longer include the fixed costs of marketparticipation, since those are paid in subperiod 0 when the household makesthe discrete market participation choice. So the relevant marginal cost orrevenue per animal bought or sold, respectively, is

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pbt = (1 + vct) p

mt pst = (1 − vct) pmt

The household’s optimum continuous marketing decisions under the assump-tion of sequential decision making therefore does not include the fixed costsalready incurred:

I b∗rt = ib(At, H t, W t, X t, et, pmt , f ct, vct, ζ t) (13)

I s∗rt = is(At, H t, W t, X t, et, pmt , f ct, vct, ζ t) (14)

Qb∗rt = q b(At, H t, I brt, W t, X t, et, pmt , vct, ζ t) (15)

Qs∗rt = q s(At, H t, I srt, W t, X t, et, pmt , vct, ζ t) (16)

The relationship between the purchase or sales quantities and the discretemarket participation choice is a form of selectivity correction akin to thaton which Goetz (1992) focused. Here, however, we distinguish between netbuyers and net sellers. Because net buyers and net sellers can be strictly or-dered along the real line describing net sales (S t ≡ Qs∗

t − Qb∗t ) positions, we

can treat the {I b∗rt , I s∗rt }pair as an ordinal variable: {I b∗rt = 1, I b∗rt = 0; I b∗rt =0, I s∗rt = 0; and I b∗rt = 0, I s∗rt = 1}, a point we exploit in developing the es-

timator in the next section. In the next section we also introduce a moreformal test for whether household choice appears sequential or simultaneous,based on the correlation between the sub-period 0 choice over {I b∗rt , I s∗rt }andthe sub-period 1 choices of Qb∗

rt and Qs∗rt .

In the sequential choice model, several things change. Most importantly, notethat fixed costs should no longer have any effect on sales quantity decisions,only on the market participation choices, I b∗rt and I s∗rt . Conditional on findingthat the data support the sequential formulation of the household marketingchoice, tests of the exclusionary hypothesis that fixed costs are unrelated toquantities sold or purchased thus serve as tests of the prospective behavioral

anomaly that households take fixed costs into account when they reallyshould not. Moreover, because fixed costs are incurred before sales revenuesare generated — sometimes by quite a gap in time if there are paymentsdelays, as commonly takes place in rural markets in the developing world— the earlier negative relationship between income or liquid wealth andsales that one expects to find when choice is simultaneous breaks down.

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There need not be any relationship between sales and wealth, although there

should still be a positive relationship between wealth and purchases, due tothe budget constraint. The rest of the predicted relations between sales orpurchase quantities and the explanatory variables are as in the simultaneouschoice case. But the subtle distinction between whether a household makesits market participation and purchase or sales volume decisions sequentiallyor simultaneously thus has significant implications for several relationshipsof interest in market participation studies. We now present an estimator thatpermits estimation of the discrete choice over I jrt as well as the continuouschoice over Q j

rt and allows one to test whether the sequential or simultaneouschoice model fits the data better.

3 An Econometric Model of Market

Participation

This section develops the ordered tobit model we implement in the next sec-tion. The idea behind the model comes from the assumed sequence and jointestimation of the household’s marketing decisions, as just described. The keyinsight is that because a household’s net sales (sales minus purchases) volumespans the real line3, one can partition the continuous market participationoutcome into three distinct categories: net buyer (households whose net salesare strictly negative), autarkic (households whose net sales are equal to zero)

and net seller (households whose net sales are strictly positive) households.Because these categories are logically ordered, and since it is informative todistinguish between net buyers and net sellers rather than just lump themtogether as “market participants”, we can first estimate an ordered probitparticipation decision, then estimate a censored model of net sales or netpurchase volume. By testing the correlation between the first and secondstage regression residuals, we can then establish whether the participationand extent (or volume) decisions are made sequentially or simultaneously.

3.1 The Participation Decision

In the first stage, households decide whether they will be net buyers of livestock, autarkic, or net sellers of livestock. As described previously, if weconsider a household’s net sales S t ∈ R, i.e., its total sales of livestock minusits total purchases of livestock, we can partition R into three distinct parts,

3In the presence of non-zero censoring points, regions between zero and the censoringpoint(s) may have zero density.

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which correspond respectively to net buyer (S t < 0), autarkic (S t = 0) and

net seller (S t > 0) households. We can then let y1i represent the categoryto which household i belongs, since the net buyer, autarkic, and net sellerpartition leaves us with an ordered response. Let the ordered response y1 besuch that

y1i = 0 if S ti < 0

y1i = 1 if S ti = 0 (17)

y1i = 2 if S ti > 0

As is usually done in an ordered probit, we assume that

y∗1 = x1β 1 + 1 (18)

where y∗1 is a latent variable, i.e., the utility the household gets from partici-

pating in the market in this case, x1 is an (N ×K ) matrix of covariates whichdoes not contain the unit N -vector, β 1 is a (K × 1) vector of parametersto be estimated, and 1|x1 is distributed standard normal. In what follows,let J = 2, first because it does not entail any loss of generality, and secondbecause the categorical ordered response can only take three values in theapplication of section 5.

Let α1 < α2 be unknown threshold parameters, and define

y1i =

0 if y∗1i ≤ α1

1 if α1 < y∗1i ≤ α2

2 if y∗1i > α2

(19)

This is the set-up of a regular ordered probit (Wooldridge, 2002). The prob-ability of observing each of the ordered responses is such that:

P (y1 = 0|x1) = P (y∗1 ≤ α1|x1) = Φ(α1 − x1β 1)P (y1 = 1|x1) = P (α1 ≤ y∗

1 ≤ α2|x1) = Φ(α2 − x1β 1) − Φ(α1 − x1β 1)P (y1 = 2|x1) = P (y∗1 > α2|x1) = 1 − Φ(α2 − x1β 1)

Defining I

(y1i = 0), I

(y1i = 1) and I

(y1i = 2) as indicator variables, thelog-likelihood function of the ordered probit is thus:

(α, β ) = N

i=1{I(y1i = 0) ln[Φ(α1 − x1iβ 1)]+

I(y1i = 1)ln[Φ(α2 − x1iβ 1) − Φ(α1 − x1iβ 1)]+ (20)

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I(y1i = 2) ln[1 − Φ(α2 − x1iβ 1)]},

where Φ(·) denotes the standard normal cumulative density function (cdf).

3.2 The Volume Decision

Let us now turn to the second stage decision of how much net buyer house-holds buy and how much net seller households sell. Each decision could bestudied using a tobit model. Let the subscripts 2 and 3 denote net buyer andnet seller households, respectively. Recall that in a tobit with left-censoringat zero, y∗

2i = max{0, y2i} and that we assume that

y∗

2 = x2β 2 + 2 (21)where y∗

2 is a latent variable, i.e., the utility the household derives from buy-ing animals, in this case, x2 is an (N × L) matrix of covariates thought toaffect net purchases (or an (N × M ) matrix of covariates thought to affectnet sales for net seller households), β 2 is an (L × 1) (or an (M × 1)) vectorof parameters to be estimated, and 2|x2 is distributed normal with meanzero and variance σ 2

2.

Defining I(y2i = 0) and I(y2i > 0) as indicator variables, the log-likelihoodfunction of the censored tobit is thus:

(β 2, σ2) = N

i=1{I(y2i = 0) ln[1 − Φ((x2iβ 2/σ2)]}

+I(y2i > 0)

ln φ[(y2i − x2iβ 2)/σ2] − ln(σ22/2)

, (22)

where φ(·) denotes the standard normal probability density function (pdf).

If, however, households take these two decisions — whether to participateon the marker as net buyer, autarkic, or net seller households and how muchto buy or sell conditional upon being net buyers or net sellers, respectively— sequentially, we need to estimate these two decisions in order, for whichwe propose the following framework.

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3.3 The Ordered Tobit Model

Our “ordered tobit”4 specification allows us to study fixed and variabletransactions costs separately, as do Key et al. (2000), but using an esti-mator that we find converges more readily than does their somewhat morecumbersome likelihood function. This approach also allows for non-zero cen-soring points, as in Key et al. and Holloway et al. (2001).

The specification of the ordered tobit model is as follows. Let y1i denote thecategory — net buyer, autarkic, or net seller — household i belongs to, asdefined in equation (1) above. The specification of the first-stage decisionremains exactly the same as in section (3.1), so the first-stage decision is

just a regular ordered probit. The innovation comes at the second stage.Let y2i > 0 be the total units of livestock purchased by household i andlet y3i < 0 be the total units of livestock sold by household i. Note thatthese two variables define clear, mutually exclusive subsets of the dataset: ahousehold cannot simultaneously be a net buyer and a net seller.

We could treat the full problem separately by estimating two type II tobits(or heckits, following Heckman, 1979), one for net buyer households, andone for net seller households or even estimating them jointly as a bivariatetobit. A better way to study the determinants of pastoralists’ market par-ticipation in northern Kenya and southern Ethiopia might be to estimate an

ordered probit in the first stage and then append two linear regressions tothe y1 = 0 and y1 = 2 categories: one for net buyers, and one for net sellers,respectively, and then test whether or not the ordered tobit specificationis supported by the data. Note that in the following analysis, x1 = x2 andx1 = x3 (but note that x2 could be equal to x3), where x1 is the vector of first-stage regressors, x2 is the vector of second-stage regressors thought toaffect the volume of purchases, and x3 is the vector of second-stage regres-sors thought to affect the volume of sales. Thus, the end result is an orderedprobit combined with two of what Amemiya (1985) refers to as Type II tobitmodels, i.e. tobit models in which the “if” binary decision is regressed ona set of covariates different from the set of covariates on which the “how

much” decision is regressed.5

4Klein and Sherman (1997) also combine the ordered probit and tobit estimators intowhat they term an “orbit” estimator, but in the reverse order. They first estimate acensored regression and then use the parameters from that first stage to fit an orderedresponse model. Our approach thus differs significantly from theirs.

5Groot and van den Brink (1999) use a similar estimator, but incorporating a Type I

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Dropping boldface notation for vectors, the log-likelihood for each observa-tion i is such that

i(α, β , σ) = {I(y1i = 0)

lnΦ(α1 − x1iβ 1) + ln φ

(y2i−x2iβ )σ2

− ln

σ22

2

+I(y1i = 1) {ln[Φ(α2 − x1iβ 1) − Φ(α1 − x1iβ 1)]}

+I(y1i = 2)

ln[1 − Φ(α2 − x1iβ 1)] + ln φ

(y3i−x3iβ )σ3

− ln

σ23

2

} ,

where α is a (2×1) vector of unknown threshold parameters, β = (β 1, β 2, β 3)is a ([K + L + M ] × 1) vector of parameters, and σ is (2 × 1) vector of vari-ance parameters, one for each linear component, i.e., net purchases and net

sales. Thus, the model will estimate K + L + M +4 parameters by maximumlikelihood.

Thus, the three error terms in the model, 1, 2 and 3 follow a trivariatenormal distribution, i.e., ∼ N (0, Σ), where 0 is a (3 × 1) vector of zerosand Σ is the (3 × 3) variance-covariance matrix between the equations of theordered tobit model:

Σ =

1 σ12 σ13

σ21 σ22 σ23

σ31 σ32 σ23

.

Also note that we can rewrite this matrix in terms of correlation coefficientsbetween the error terms,

R =

1 ρ12 ρ13

ρ12 1 ρ23

ρ13 ρ23 1

,

where ρij is the correlation coefficient between the error terms of equationsi and j, respectively. Thus, much like in a bivariate probit model (Mad-

dala, 1983; Gourieroux, 2000), a non-zero correlation coefficient betweentwo equations will be evidence in favor of the joint, sequential model. Thatis, rejecting the joint hypothesis that H 0 : (ρ12, ρ13) = 0 is, in effect, a

rather than a Type II tobit.

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rejection of separately estimating an ordered probit, then two linear regres-

sions in favor of estimating the ordered tobit model.6 The advantage of thismodel is that it sequentially estimates the participation and the amount-of-participation decisions while correcting for selection bias at the second stage.

The ordered tobit model has been the object of very little published work.Groot and Maassen van den Brink (1999) study overpayment and earningssatisfaction, developing a computationally similar but atheoretical model.Ranasinghe and Hartog’s (1997) unpublished working paper explores in-vestment in post-compulsory education in Sri Lanka. Yet, the prospectiveapplications of this model are many — as we discuss in the concluding sec-tion — and it is rather easy to estimate with any statistical package that

accommodates maximum likelihood.7

4 Data and Descriptive Statistics

We apply the ordered tobit estimator to study livestock market participationby pastoralists in a large, contiguous area of northern Kenya and southernEthiopia. Observers have long been puzzled by the limited use of livestockmarkets by east African pastoralists who hold most of their wealth in theform of livestock, who face considerable income variability, and who regu-larly confront climatic shocks that plunge them into massive herd die-offs

and loss of scarce wealth. It would seem that opportunistic use of marketswould permit herders to increase their wealth by buying when prices arelow and selling when prices are high and to smooth consumption throughconversion between livestock and cash useful for purchasing food. Yet suchbehavior seems relatively rare (Osterloh et al. 2003, Lybbert et al. 2004,McPeak 2004).

The data come from an ongoing study of risk management among eastAfrican pastoralists. The data consists of a panel of 337 pastoralist house-holds from eleven sites in the arid and semi-arid lands of northern Kenya andsouthern Ethiopia. Each household was observed quarterly between June

2000 and June 2002. We pool all nine time periods together and treat thedataset as a cross-section, first due to the inherent complexity that an ex-

6Note that we do not test any hypothesis on ρ23. Since every non-autarkic observationis either a net buyer or a net seller, ρ23 is not identified.

7For this paper, estimation was conducted using STATA’s ml set of commands andgenerally converged in under ten iterations using the Newton-Raphson algorithm.

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tension of the ordered tobit model to a panel setting would involve, and

second because of the highly unbalanced nature of our panel.8 The descrip-tive statistics presented here thus treat household i in period t and householdi in period s as two distinct observations for s = t. Further details on thesurveys, sites and instruments are available in Barrett et al. (2004).

Table 1 provides descriptive statistics for our survey dataset. Almost 70percent of the households are male-headed, with an average size of 7.4 peo-ple and a dependency ratio of nearly 0.5.9 Most households own livestock,with an average herd size of more than 20 tropical livestock units (TLU),a standard measure for aggregating across ruminant species such as camels,cattle, goats and sheep.10 Herds reproduce, on average, at a rate of about 6.1

percent annually (animal births/total herd size). Pastoralists have a strongpreference for holding cows for milk and calves, so herds are more than two-thirds female, on average. Property rights in livestock can be complex, withimplications for livestock marketing patterns. Households often give or lendanimals to one another without surrendering all rights in the animal. Forexample, it is common for a household to “own” an animal given to it by arelative, yet the household is not permitted to sell or slaughter the animalnor to give it to anyone outside of the clan or village. These encumbered orrestricted property rights affect less than ten percent of a household’s herd,on average, yet may matter to marketing decisions, especially with respectto purchasing cows (for which restricted gifts may be a substitute) or selling

bulls. Less than one-third of households own poultry and mean land hold-ings are small, at about 1.4 hectares, much of which goes uncultivated anygiven year due to insufficient rainfall. Other assets owned by the householdinclude bicycles, radios, wooden beds, tables and other furniture, watches,lanterns, ploughs, small shops or other businesses, non-local breed animals,vehicles and urban property, all valued in Kenyan shillings (Ksh).11 Thevalue of these assets amounts to a bit more than US$35 per capita, whilehousehold income (the sum of milk and crop production, sales of firewood,

8The number of observations per time period ranged from 233 to 255, reflecting amixture of attrition and interruption.

9A household’s dependency ratio is calculated by dividing the number of individuals

under 15 years of age plus the number of individuals over 64 years of age by the totalnumber of individuals in the household. Thus, a dependency ratio of zero indicates ahousehold whose resources are not strained at all, whereas a dependency ratio of oneindicates a household whose resources are extremely strained.

10One TLU equals 0.7 camel, 1 cattle, 10 goats or 11 sheep.11For Ethiopian households, we use 1 Ethiopian birr = 8.75 Ksh. Note that US$1.00 ∼=

Ksh75.

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charcoal, crafts and hides and skins, and wage and salary earnings) over

the preceding quarter averaged around $1.70 per day. Fixed and variablecost expenditures on market participation represent a perhaps surprisinglymodest share of price.12 Variable costs related to per animal transport costsand market fees add (for buyers, subtract for sellers) only about 12 per-cent to the small stock (goat or sheep) price and less than 2 percent to thelarge stock (camel and cattle) price. Fixed costs associated with transportand lodging expenditures of the individual who sells or buys animals andany market fees unrelated to volumes sold or purchased are about 40 per-cent larger than variable costs per TLU. We omit descriptive statistics forlocation and time variables for brevity.

5 Results and Analysis

This section first presents estimation results for the simple model of simulta-neous choice based on two tobit regressions — one for net buyer households,one for net seller households — consistent with the model outlined earlierwhen household market participation and volume choices are made simul-taneously. Then we present estimation results from the ordered tobit modelthat corresponds to the model of sequential household choice. Finally, wetest the null hypothesis that the simpler, simultaneous choice model sufficesin modeling household livestock marketing behavior in these data.

5.1 Two Tobit Models

As a first step in understanding how the households in our dataset behave,we estimate two tobit models — one for net buyer households, one for netseller households — using the classic type I tobit model. The estimated co-efficients are reported in Table 2 for both the net buyer and net seller tobitmodels.13

Household demographic characteristics affect livestock marketing patterns.Female-headed households buy and sell significantly fewer animals than their

12

In our analysis, fixed fees include accommodations, food and transportation for theherder as well as bribery, security expenditures and medications. Variable fees are feesper animal paid to county or municipal authorities as well as District Veterinary Officer(DVO) inspection and other veterinary fees.

13In reporting the estimated coefficients, we will omit reporting the results for the timeand location dummies, and in our estimation results, the superscripts ∗ , ∗∗ and ∗∗∗ indicatea coefficient significant at the 90, 95 and 99% levels of confidence, respectively.

15



male-headed counterparts. Livestock sales increase with household size and

with (human) births into the household, while a household’s dependencyratio decreases the number of TLUs bought. The greater effect of birthson sales than of household size likely reflects cultural ceremonial expensesassociated with a new birth.

A household’s wealth and income also appear to affect livestock marketingpatterns. Household land and other assets increase the number of animalssold, thus indicating a certain complementarity between various forms of wealth. Land also decreases the number of animals bought, likely due to thefact that households who possess more land cultivate more, which in turnsrestricts their ability to undertake long-distance migrations in search of for-

age and water, and thus the herd size they can manage within a confinedspace. Livestock sales increase with household income.14

Finally, we turn to the effect of the main variables of interest: prices andtransaction costs. Variable costs appear to increase the number of animalssold, a puzzling results since one would expect the volume of trade to bestrictly decreasing in variable costs. Finally, household behavior with respectto the price shows downward-sloping demand for large stock (camels andcattle) but no significant supply price response for either small stock orlarge stock and no demand response to small stock price. We discuss suchresults in greater detail in the next subsection.

5.2 The Ordered Tobit

We now explore whether these estimation results change if we instead modelhousehold marketing as a sequence of choices and employ the ordered to-bit model. In essence, we first estimate an ordered probit, then two linearregressions — one for the amount of participation conditional on being anet buyer, and one for the amount of participation conditional on being anet seller — in the second stage. Since both linear components include aselection term, the usual inverse Mills ratio (IMR), our model is actually anordered type II tobit. Following Heckman (1979), we correct the variance-

14Household income is likely endogenous. We therefore instrumented for income usingthe household head’s education – a series of dummy variables for whether or not the headof the household had no education, between 1 and 12 years of education, 13 or more yearsof education, or had completed an adult literacy class – time-and-location interactionterms, beginning period herd size and land assets. Since the instrumenting equation forincome had an adjusted R2 of 0.2665, we can rule out overfitting.

16



covariance matrices for each of the second-stage regressions (see Appendix A

for a derivation of the inverted Mills ratios as well as the Heckman-correctedvariance matrix in the case of the ordered type II tobit). The only differencebetween our method and that of Heckman comes from the first-stage, andin that sense, our model offers a modest extension to Heckman (1979).

The ordered probit model of discrete market participation yields intuitiveresults (Table 3). The non-zero censoring points are of opposite signs, withthe lower censoring threshold at -1.59 TLU net sales and the upper thresholdat 0.95, each statistically significantly different from zero. These estimatessuggest that purchases or sales of less than one TLU are generally uneco-nomical, given the monetary and nonmonetary costs of market participation

in this region. People are more willing to enter the market for smaller vol-ume sales than purchases, likely reflecting the fact that sales of small stock(goats or sheep) are means by which households meet immediate cash needsrelated to payment of school fees, food purchases and ceremonial or emer-gency health expenses.

The basic household demography effects of the simpler tobit model carryover. Female-headed households are more likely to be autarkic than to benet sellers and are more likely to be net buyers than to be autarkic, ceteris

paribus . Human births positively affects market participation, i.e., it makesnet buyer households more likely to be autarkic and autarkic households

more likely to be net sellers. Animal births likewise exerts a positive ef-fect on the ordered market participation variable. The more animal births ahousehold herd enjoys in a period, the more likely it is to be autarkic insteadof being a net buyer and the more likely it is to be a net seller instead of being autarkic.

Wealth has no discernible effect on market participation. The fixed costs of market participation exert an increasing, concave effect on market partic-ipation up through almost the 75th percentile of the data, at which pointthe effect turns negative. This implies that over most of the range of fixedcosts observed in these data, the marginal effect is greatest with respect to

purchase decisions, moving households from net purchases to autarky. How-ever, when high fixed costs are extremely high — beyond about Ksh415 —this encourages households to move from net seller positions to autarky.

The second stage net purchase and net sales volume choices conditional onexpected market participation likewise make sense (Table 4). Pronounced

17



and intuitive life cycle effects emerge, as households buy more and sell less up

through about age 50 — roughly the mean in these data — and then switchto selling more and buying less. As one would expect, non-productive house-hold assets and household income are both positively associated with pur-chase volumes. Wealthier people buy in larger volumes than poorer house-holds with equal probability of market participation. Household livestocksales volumes are decreasing in income, likely reflecting households’ use of livestock as a form of savings for partial consumption smoothing. Whenincome is high, they sell fewer animals and when income is low, they sellmore, ceteris paribus . Household land holdings are positively related to salesbecause pastoralists who own land have effectively sedentarized themselves,reducing the herd sizes they can manage within a fixed space subject to

considerable intertemporal variability in forage and water availability.

Herd size matters to livestock marketing patterns. Households with largerherds sell slightly more animals, although this effect, while statistically sig-nificant, is small in magnitude, indicating that marketing is not used toregulate herd sizes (Lybbert, 2004). The data do not support our hypoth-esis that complex indigenous livestock gifting and loaning institutions thatencumber some animals in many households’ herds impede livestock mar-keting behavior. Nor does it appear that the gender mix of a household’sherd matters to market participation or transactions volumes.

The multifunctional nature of livestock holding in pastoralist regions be-comes evident when we consider the estimated effect of livestock prices onnet sales and purchases. Larger stock (camels and cattle) are productiveassets held for long-term equity growth. Net sales increase modestly withprice while net purchases decrease sharply as prices rise, with price elastici-ties of supply and demand of 0.1575 and −2.7059, respectively, at the samplemeans. As in the previous section, demand for and supply of small ruminants(goats and sheep), however, appear characterized by backward-bending sup-ply curves and downward-sloping demand. The former is consistent with theuse of small ruminants as a sort of walking bank. Herders tend to liquidategoats and sheep, as needed, to meet immediate cash needs (Osterloh et al.

2004), thus the number they sell falls as price increases.

The anomalous results in these models relate to transactions costs. Variablecosts appear to exert a small, significant negative effect on sales volumes,as one would expect, but an anomalous positive effect on purchase volumes.Meanwhile, fixed costs appear to affect purchase (sales) volumes negatively

18



(postively) and significantly when the theoretical model based on sequential

choice predicts that fixed costs should have no effect whatsoever on the con-tinuous volume decision, only on the discrete participation decision, whichwas indeed affected by fixed costs. This could indicate that fixed costs re-duce the cash available to liquidity-constrained households, thereby limitingthe number of heads they can purchase and inducing them to sell a fewmore animals, or that sellers wish to “recover” their sunk costs once theychoose to participate as net sellers. These results could also simply reflectthe well-known behavioral anomaly that people take sunk costs into con-sideration even when they should not were they trying simply to maximizeprofits or utility subject to an irreversible expense. We must defer more indepth investigation of these alternative explanations to future research.

The results of the ordered tobit differ substantially from those of the twotype I tobit models under the assumption of simultaneous choice. Many of the more intuitive results only emerge from the more general estimationmethod we introduce here. For example, life cycle effects, fixed costs of mar-keting and the responsiveness of livestock sales to prices and to herd sizeare statistically significant only in the more general, two-stage model. Andwithout the first stage ordered probit selection control, assets (income) havenegative and statistically insignificant effects on net purchases (net sales).All of these qualitative differences suggest that the estimator we introducemore accurately reflects livestock marketing behavior among these house-

holds.

5.3 Hypothesis Tests

We have established that the simultaneous and sequential model specifica-tions yield different results and that the ordered tobit results appear moreintuitively plausible. But which model better fits the data statistically? Onemethod is to check whether the IMR variables are statistically significantin either of the second-stage linear components of the ordered tobit model.The weakness of that approach is that it depends fundamentally on the

instruments used to identify the first-stage choice. This data set has fewgood instruments for that purpose. We used only the number of children inthe household and animal births to identify the first stage equation and thusthe resulting IMR. Weak identification will cause imprecise estimation of theeffect of the IMR on second-stage sales or purchase volumes. And indeed,the IMR coefficient estimates are not statistically significantly different from

19



zero in either regression reported in Table 4.

An alternative and arguably superior method is to check whether the resid-uals of the second-stage linear components are correlated with the residualsof the first-stage ordered probit component, thus testing the sequential spec-ification versus estimating the ordered probit and the simultaneous linearcomponents separately. The estimated matrix of correlation coefficients be-tween the residuals of all three components of the ordered tobit is such that:

R =

1 0.1069∗ 0.2859∗

0.1069∗ 1 0.2343∗

0.2859∗ 0.2343∗ 1

.

Recall that ρ12 and ρ13 are the degrees of correlation between the decisionto participate respectively as a net buyer or a net seller and the extent of participation conditional on having decided to participate in the market.Thus, if the residual vectors 2 and 3 are both correlated with 1, i.e., if both ρ12 and ρ13 are statistically significantly different from zero, then thesequential choice ordered tobit model is preferred to separately estimatingan ordered probit and a simultaneous equations model for net buyers andnet sellers. Given that ρ12 and ρ13 are both significant, we infer that ourdata supports the ordered tobit model.

6 Conclusion

In this paper, we highlighted the subtle but important differences in behav-ior depending on whether households make (discrete) market participationand (continuous) sales or purchase volumes choices sequentially or simulta-neously. We then developed a two-stage econometric model that permits di-rect testing between these competing ways of understanding household levelmarketing behavior. This method should be applicable to a range of othereconomic problems similarly characterized by an ordered first-stage process

and a continuous second stage process. Examples include financial invest-ments — e.g., ranking risk tolerance and share of alternative instruments ina portfolio — and market integration in domestic and international trade— establishing whether markets are segmented, competitively integrated ornon-competitively integrated and identifying trade volumes. For now, we

20



leave such topics to future research. One could likewise adapt this basic ap-

proach to cover multinomial or count data, instead of ordinal data, in thefirst stage estimation.

Our empirical results shed some light on the contemporary puzzle of whypastoralist households in the arid and semi-arid lands of east Africa makerelatively little use of livestock markets. Households follow strong life cyclesof accumulation, steadily building their herds over most of their adult lives.Fixed costs of market participation impede market participation. Householdskeep small stock as a sort of walking bank, adjusting goat and sheep salesvolumes to price changes in a manner suggesting that they are used to meetimmediate cash needs. Camel and cattle marketing, by contrast, appears

responsive to large stock prices, especially on the demand side. Perhapsmost revealingly, although livestock sales increase as a household’s herd sizegrows, the marginal increase is quite small, reflecting households’ preferencefor maximizing herd size, rather than liquidating animals and holding orinvesting cash in activities or assets other than livestock. While increasinglocal cattle prices through, for example, enhanced promotion of east Africanbreeds in international markets may elicit a supply increase, it appears thateast African pastoralists are less drawn to the commercialization of livestockthan to accumulating substantial herds.

21



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[1] Amemiya, T. (1985), Advanced Econometrics , Cambridge, MA: HarvardUniversity Press.

[2] Barrett, C.B., G. Gebru, J.G. McPeak, A.G. Mude, J. Vanderpuye-Orgle and A.T. Yirbecho (2004), “Codebook for Data Collected underthe USAID GL-CRSP PARIMA Project”, Cornell University.

[3] de Janvry, A., M. Fafchamps and E. Sadoulet (1991), “Peasant House-hold Behavior with Missing Markets: Some Paradoxes Explained”, Eco-

nomic Journal 101: 1400-1417.

[4] Goetz, S.J. (1992), “A Selectivity Model of Household Food Market-ing Behavior in Sub-Saharan Africa”, American Journal of Agricultural

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[5] Gourieroux, C. (2000), Econometrics of Qualitative Dependent Vari-

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[6] Greene, W. (2003), Econometric Analysis , Fifth Edition, New York,NY: Prentice-Hall.

[7] Heckman, J.J. (1979), “Sample Selection Bias as a Specification Error”,Econometrica 47: 931-959.

[8] Holloway, G., C.B. Barrett and S. Ehui (2001), “The Double HurdleModel in the Presence of Fixed Costs”, Working Paper, InternationalLivestock Research Institute, University of Reading and Cornell Uni-versity.

[9] Key, N., E. Sadoulet and A. de Janvry (2000), “Transactions Costsand Agricultural Household Supply Response”, American Journal of

Agricultural Economics 82: 245-259.

[10] Klein, R. and R. Sherman (1997), “Estimating New Product Demandfrom Biased Survey Data”, Journal of Econometrics 76: 53-76.

[11] Lybbert, T.J., C.B. Barrett, S. Desta and D. L. Coppock (2004),“Stochastic Wealth Dynamics and Risk Management Among A PoorPopulation”, Economic Journal , in press.

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Econometrics , Cambridge, UK: Cambridge University Press.

22



[13] McPeak, J.G. (2004),“Contrasting Income Shocks With Asset Shocks:

Livestock Sales in Northern Kenya”, Oxford Economic Papers 56: 263-284.

[14] Osterloh, S.M., J.G. McPeak, H. Mahmoud, W.K. Luseno, P.D. Lit-tle, G. Gebru and C.B. Barrett (2003), “Pastoralist Livestock Market-ing Behavior in Northern Kenya and Southern Ethiopia: An Analysisof Constraints Limiting Off-Take Rates”, Final Report on PARIMA-LEWS Collaborative Study under the USAID Global Livestock CRSP.

[15] Ranasinghe, A. and J. Hartog (1997), “Investment in Post-CompulsoryEducation in Sri Lanka”, Working Paper, Tinbergen Institute and Uni-versity of Amsterdam.

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[17] Wooldridge, J.M. (2002), Econometric Analysis of Cross-Section and

Panel Data , Cambridge, MA: MIT Press.

23



Table 1 – Descriptive Statistics.

Variable Mean Std. Dev.

Household Head Gender (1=Female) .3112314 .4631017

Household Head Age (Years) 49.19578 14.6925

Children (Under 15) 3.360288 2.359304

Household Size (Persons) 7.386343 3.873955

Land Assets (Hectares Owned) 1.414645 2.535116

Assets (Ksh) 19816.18 189207.2

Births (Persons) .0633139 .2435813

Deaths (Persons) .0166143 .1313166

Income (Ksh, Including Food Aid) 11645.82 21894.24

Herd Size (TLUs) 20.38237 30.06591

Percentage of Female TLUs .6800255 .2422851

Encumbered Male TLUs .4728469 2.823439

Encumbered Female TLUs 1.011073 4.466281

Average Price of Large Stock (Ksh) 5529.666 2655.46

Average Price of Small Stock (Ksh) 787.6219 425.2299

Animal Births (TLUs) 1.236916 3.348557

Net Buyer (1=Yes) .0377189 .1905584

Autarkic (1=Yes) .6834306 .4652423

Net Seller (1=Yes) .2788505 .4485345

Dependency Ratio .4799896 .1987313

Net Sales (TLUs) .2346206 1.59179

Fixed Costs (Ksh) 138.6515 251.3097

Variable Costs (Ksh/TLU) 98.52199 132.7719

Sales (Net Sellers) 1.038969 2.157152

Purchases (Net Buyers) 1.460714 4.943286

24



Table 2 – Estimation Results for the Two Tobits.

Quantity Bought Quantity Sold

Variable Coefficient Standard Error Coefficient Standard Error

Household Head Gender −3.436327∗∗∗ 1.022038 −.4782938∗∗∗ .1320222

Household Head Age .0465924 .1463452 .0026337 .023254

Household Head Age Squared −.0007354 .0013565 −.000067 .0002139

Individuals in Household .1014839 .1295241 .0600956∗∗∗ .0194627

Dependency Ratio −5.230665∗∗∗ 1.846062 .2419318 .3005247

Births −.0819576 1.341713 .6186621∗∗∗ .2076999

Deaths −1.605708 2.825491 .5551639 .3700035

Household Assets −4.57e − 07 2.19e − 06 3.14e − 06∗∗∗ 2.35e-07

Land −.328638∗ .1846291 .045275∗ .0258425

Income .000031 .0000684 .000019∗∗ 9.28e-06

Herd Size −.006388 .0278305 .002784 .0033879

% Female TLUs −.6688664 1.479763 .2102953 .2601015

Encumbered Males −.1415813 .2931889 .0220244 .0215265

Encumbered Females .1340441 .1462505 −.0046377 .0136714

Fixed Costs .0015043 .0022892 .000031 .0002743

Variable Costs .0032402 .0050344 .0019912∗∗∗ .0006443

Avg. Price Large Stock −.0003469∗∗ .0001726 .000013 .0000241

Avg. Price Small Stock .0017703 .0012699 −.0003029 .0001868

Constant −4.550694 3.875828 −1.029673 .6313306

ρ(b, s) −0.0813∗∗∗

25



Table 3 – Estimation Results for the First Stage of the Ordered Tobit.

Variable Coefficient Standard Error

Household Head Gender −.1199407∗∗∗ .0641554

Household Head Age .0071154 .0115469

Household Head Age Squared −.0000848 .0001079

Individuals in Household .0180623 .0213252

Children .0091998 .0395143

Dependency Ratio .1057024 .2600769

Births .2543608∗∗ .1138981

Deaths .1213393 .2086002

Household Assets −3.91e − 08 1.98e − 07

Land .010639 .0141761

Income 2.95e − 06 4.71e − 06

Herd Size .0028195 .0018374

% Female TLUs .1717806 .1297437

Encumbered Males .0161459 .0144535

Encumbered Females −.0059672 .0083052

Fixed Costs .0005797∗∗ .0002375

Fixed Costs Squared −2.23e − 07∗ 1.34e − 07

Animal Births .0268813∗∗ .0121053

α1 −1.586686∗∗∗ .331772

α2 .951289∗∗∗ .3310051

26



Table 4 – Estimation Results for the Second Stage of the Ordered Tobit.

Quantity Bought Quantity Sold

Variable Coefficient Standard Error Coefficient Standard Error

Household Head Gender .7158475 133.4344 −.0084077 .52197595

Household Head Age .4730303 2.4767117 −.0637312∗∗∗ .01000649

Household Head Age Squared −.0044802∗∗∗ .00021205 .00061∗∗∗ 8.593e − 07

Individuals in Household −.6311107 4.8351958 −.0092062 .01833041

Dependency Ratio −16.29315 515.76254 .1582091 2.0633256

Births −4.036646 582.89715 −.050381 2.2515476

Deaths 3.279318 1067.772 .6814582 4.3248424

Household Assets .0000263∗∗∗ 1.471e − 09 3.56e − 06∗∗∗ 6.196e − 12

Land −.4244275 4.8468381 .0597178∗∗∗ .02002678

Income .000223∗∗∗ 5.178e − 07 −7.50e − 06∗∗∗ 2.147e − 09

Herd Size −.0884315 .13286771 .0016347∗∗∗ .00044273

% Female TLUs 3.885264 433.53984 −.0899614 1.7359938

Encumbered Males .4374481 8.5898427 −.0020435 .03373999

Encumbered Females −.282414 2.265696 −.0027516 .00897036

Fixed Costs −.0202091∗∗∗ .0007079 .0002797∗∗∗ 2.434e − 06

Variable Costs .0347987∗∗∗ .00021017 −.000108∗∗∗ 9.017e − 07

Avg. Price Large Stock −.0007148∗∗∗ 4.916e − 08 .0000296∗∗∗ 2.459e − 10

Avg. Price Small Stock .0010069∗∗∗ 6.612e − 06 −.0000728∗∗∗ 2.980e − 08

Inverted Mills Ratio 17.19722 5487.4599 −1.094857 27.982761

Constant -39.10176 22591.808 4.043882 65.490076

27



A Heckman-Corrected Variance Matrix

In this section, we present the Heckman correction used to get the rightstandard errors at the second stage of the ordered tobit model. The firststep is to consider the ordered probit model presented in equations (17) to(20) in section 3 above. From this first stage, we derive that

y1i =

0 if x1β 1 + 1 ≤ α1

1 if α1 < x1β 1 + 1 ≤ α2

2 if x1β 1 + 1 > α2

(23)

so that

y1i =

0 if 1 ∈ (−∞, α1 − x1β 1]1 if 1 ∈ [α1 − x1β 1, α2 − x1β 1)2 if 1 ∈ (α2 − x1β 1, ∞)

(24)

Thus, by symmetry of φ(·), the standard normal pdf, we have that P (y =0) = Φ(α1 − x1β 1) and P (y = 2) = Φ(x1β 1 − α2). We can then obtainthe inverted Mills ratios (IMRs) for net buyer and net seller households,respectively:

λb = φ(α1 − x1β 1)

Φ(α1 − x1β 1), (25)

and

λs = φ(x1β 1 − α2)

Φ(x1β 1 − α2). (26)

Using these, we can fully describe the two-step estimator used in section5. Our description closely follows that of Heckman’s two-step estimator byGreene (2003). The first step is to estimate the first-stage ordered probit bymaximum likelihood in order to obtain estimates for (α, β 1). Then, for each

observation i ∈ {1,...,N }, the IMRs λbi and λsi must be computed, but onemust also compute:

δ bi = λbi( λbi − α1 − x1i β 1) (27)

and

δ si = λsi( λsi − α2 − x1i β 1). (28)

The second step is to estimate (β 2, β bλ) and (β 3, β sλ) by a regression of netpurchases (net sales) on the set of covariates thought to affect net purchases

28



(net sales) and on

λb (

λs).

Letting j ∈ {b, s} denote net buyer and net seller households, respectively,the estimated residual variance of each second-stage linear component issuch that

σ2j =

1

n j j + δ j β 2 jλ, (29)

where

δ j = plim1

n

ni=1

δ ji , (30)

and where β 2 jλ is the square of the estimated coefficient for the j th IMR.

Moreover, we have that

ρ2 j =

β 2 jλ σ2j

. (31)

Once we have obtained the above values, we can finally compute the Heckman-corrected variance-covariance matrices for our ordered (type II) tobit model,which are equal to:

Var( β j , β jλ) = σ2j [X X ]−1[X (I − ρ2

j ∆ j)X + Q j ][X X ]−1, (32)

where the Q j matrices are such that

Q j = ρ2 j (X

∆ jX 1)Var( β 1)(X

1 ∆ jX ), (33)

and X is the data matrix of the second stage, which is identical for netbuyer and net seller households, i.e. X ≡ X b = X s, I − ρ2

j ∆ j is a diagonal

matrix whose diagonal terms are equal to 1 − ρ2 j δ ji , X 1 is the data matrix

of the first-stage ordered probit, and Var( β 1) is the variance matrix of thefirst-stage coefficients. Performing these computations thus yields efficient

estimates for the second-stage parameters of our ordered (type II) tobit andoffers an “ordered”, modest extension to Heckman’s (1979) seminal paper.

29

Documents

An Ordered Tobit Model of Market Participation