Stochastic estimation of firm inefficiency using distance functions.

1. Introduction

Both the cost function and the distance function are valid representations of multioutput technologies. The arguments of the cost function are output quantities and input prices, while the arguments of the distance function are input and output quantities. The input distance function measures the extent to which the firm is input efficient in producing a given set of outputs, while the cost function assumes cost minimization by a firm which is input efficient. The shadow cost function is a generalization of the cost function and depends on output quantities and shadow (internal to the firm) input prices rather than actual (market) input prices. Thus, the shadow cost function assumes shadow cost minimization but not actual cost minimization. This function has frequently been employed in fixed-effects estimation of allocative efficiency (AE) and technical efficiency (TE). An observed input vector is technically efficient if it is on the isoquant of the observed output vector. An input vector is allocatively eff icient if the radial contraction of the input vector to the isoquant is cost minimizing.

Identification of parameters that measure AE can be achieved by the estimation of a shadow cost system comprising the actual cost equation and the share or input demand equations, which are expressed in terms of shadow prices and output quantities. Shadow cost systems have been estimated by Atkinson and Halvorsen (1984), Sickles, Good, and Johnson (1986), Eakin and Kniesner (1988), Kumbhakar (1992), and Atkinson and Cornwell (1994), among others. Given panel data, the deviation of shadow from actual input prices can be measured by price-specific inefficiency parameters that may vary across firms and over time. However, the estimation of the effect of allocative inefficiency on input usage is indirect. One must compare the fitted input demands obtained from the estimated share or input demand equations with and without AE imposed.

The distance function has served mainly as a theoretical device. An important exception has occurred with efficiency measurement, for which the reciprocal of the input distance function, termed the Farrell input measure of TE, has been widely computed nonstochastically using the linear programming techniques discussed in Diewert and Parkan (1983) and Fare, Grosskopf, and Lovell (1985). Stochastic estimation of distance functions has been carried out by Grosskopf, Hayes, and Hirschberg (1995), Coelli and Perelman (1996), Grosskopf et al. (1997), and Atkinson, Cornwell, and Honerkamp (2002). However, these studies estimate a distance equation without the price equations necessary to identify AE, thereby failing to provide a parametric method for the direct estimation of AE.

As an alternative to both the standard distance function approach and the shadow cost system approach, in this paper we derive and estimate a shadow input distance system using the generalized method of moments (GMM) procedure. This system is a shadow input distance function, which is a function of input shadow quantities and output quantities, and the first-order conditions from the dual shadow cost minimization problem. Since we estimate these equations jointly, we impose the assumption of shadow cost minimization on our estimation problem. Unlike the shadow cost system, however, the shadow input distance system allows one to directly estimate the effects of allocative inefficiency, since the shadow input distance function is defined in terms of output quantities and shadow input quantities. The latter quantities indicate the cost-minimizing ratios of inputs the firm wishes to utilize but cannot because of some constraint. After estimating the shadow distance function, for which shadow input quantities are specified as parametric functions of actual input quantities, we decompose TE from noise. Finally, we compute returns to scale and the cost savings obtained by attaining AE and TE. We illustrate these procedures using panel data on U.S. railroads, an approach that allows us to identify input-specific, firm-specific, and time-varying AE parameters. (1)

The remainder of this paper is organized as follows. In section 2, we develop the cost-distance function duality assuming AE. In section 3, we add the parametric specification of AE to this dual relationship. In section 4, we discuss issues regarding stochastic estimation of the input distance function, AE, and TE. An application to U.S. railroads is presented in section 5, and conclusions follow in section 6.

2. The Cost-Distance Duality: Assuming Allocative Efficiency

In this section we assume AE, which implies that shadow prices equal actual prices and that shadow quantities equal actual quantities. Let x = ([x.sub.1], ..., [x.sub.N])' [member of] [R.sup.N.sub.+] denote a vector of N nonnegative inputs and let y = ([y.sub.1], ..., [y.sub.M])' [member of] [R.sup.M.sub.+] denote a vector of M nonnegative outputs. We define the technology set, [L.sup.t], as the set of all feasible input-output vectors in period t, i.e.,

[L.sup.t] = {(x, y) [member of] [R.sup.N.sub.+] X [R.sup.M.sub.+] : x can produce y in period t} (2.1)

where t = 1, ..., T. Following Shephard (1970), we denote input requirements sets by

L[(y).sup.t] = {x [member of] [R.sup.N.sub.+] : (x, y) [member of] [L.sup.t]}, y [member of] [R.sup.M.sub.+]. (2.2)

We assume that inputs are freely disposable, i.e., if x [member of] L[(y).sup.t]. We also assume that the input requirement set, L[(y).sup.t], is a convex set for all y [member of] [R.sup.M.sub.+].

The input distance function is defined as

[D.sub.i](y,x,t) = [[lambda].sup.*] = max{[lambda]: (x/[lambda]) [member of] L[(y).sup.t]} (2.3)

where [lambda] is a scalar such that 1 [less than or equal to] [lambda]. The input distance function indicates the radial contraction of inputs required to move the input vector x to the isoquant for y. See Figure 1.

Free disposability of inputs guarantees that x [member of] L[(y).sup.t] if and only if [D.sub.i](y,x) [greater than or equal to] 1. Also, given our assumption of free disposability and convexity, the input distance function is homogeneous of degree one, nondecreasing, and concave in x (see Shephard 1970). The cost function corresponding to the input distance function is defined by

C(y,p,t) = [min.sub.x] {px : x [member of] L[(y).sup.t]}, (2.4)

where p ([p.sub.1],..., [p.sub.N]) [member of] [R.sup.N.sub.+] denotes the vector of N nonnegative input prices. An equivalent definition of the cost function is

C(y,p,t) = [min.sub.x] {px : [D.sub.i](y,x,t,) [greater than or equal to]1}, (2.5)

since x [member of] L[(y).sup.t] if and only if [D.sub.i](y,x,t) [greater than or equal to] 1. Thus, the input distance function indicates the radial reduction in inputs required to move the firm to the frontier but not necessarily to the cost-minimizing position.

Assuming that observed cost equals minimum cost and letting [[nabla].sub.y][D.sub.i](y,x) = [[partial][D.sub.i](y,x)/[partial][y.sub.1],...,[partial][D.sub.i](y, x)/[partial][y.sub.M] and [[nabla].sub.y]C(y,p) = [[partial]C(y,p)/[partial][y.sub.1],...,[partial]C(y,p)/[partial][y.s ub.M], Fare and Primont (1995) show that [[eta].sup.C][y,p) = [[eta].sup.[D.sub.i](y,x) wherer

[[eta].sup.C](y,p,) = C(y,p,t)/[[nabla].sub.y]C(y,p,t)y

and, in terms of the distance function,

[[eta].sup.[D.sub.i]](y, x) = -[D.sub.i](y,x)/[[nabla].sub.y][D.sub.i](y,x)y (2.6)

are measures of returns to scale using the cost and distance functions, respectively. We report scale elasticities for our data set in section 5.

3. The Cost-Distance Duality: Allowing Allocative Inefficiency

We now wish to introduce parameters to explicitly measure allocative inefficiency. Assuming shadow cost minimization, Atkinson and Cornwell (1994) employ a shadow cost function corresponding to the actual cost function in Equation 2.3,

C(y,[p.sup.*],t) = [min.sub.x]{[p.sup.*]x : x [member of] [L(y).sup.t]}, (3.1)

where [p.sup.*] =[[p.sup.*.sub.1],...,[p.sup.*.sub.N]] =[[k.sub.1][p.sub.1],...,[k.sub.N][p.sub.N]] is a (1 X N) vector of shadow prices. That is, [p.sup.*] is the price that makes the optimal input vector, h(y, [p.sup.*], t), equal to the actual input vector, x. The [k.sub.n] parameters, n = 1,..., N, measure the divergence of actual prices from shadow prices for the firm. Atkinson and Comwell (1994) employ panel data to measure both AE and TE assuming fixed effects. Using Equation 3.1, these authors derive and estimate a shadow cost system, consisting of actual costs and N - 1 actual shares, expressed in terms of shadow input prices and output quantities. With panel data, the [k.sub.n] parameters can theoretically be made time- and firm-specific. The extent of parameterization depends on the independent variation in the data.

Estimation of the shadow cost system yields direct measurement of the divergence between shadow prices and actual prices. However, often the researcher is more concerned with the effect of allocative inefficiency on input utilization. That is, one typically wishes to know the degree to which one input is over- or underutilized relative to another input rather than to know the ratio of their relative shadow prices. Using the shadow cost system, this calculation is indirect. One must first compute the fitted input quantities from the input share or demand equations assuming AE. These estimates are then compared with the fitted input quantities obtained after allowing for allocative inefficiency by estimating the AE parameters. However, by computing the shadow distance system, one directly obtains estimates of the effect of failing to attain AE on input utilization from the estimated AE parameters.

To derive the shadow input distance function in terms of shadow input quantities, we reverse the roles of shadow input prices and input quantities in the shadow cost minimization problem. Now one assumes shadow cost minimization in terms of shadow input quantities and actual input prices. From Equation 2.4 the dual cost function becomes

C(y,p,t) = [min.sub.[x.sup.*]]{[px.sup.*] :[D.sub.i](y,[x.sup.*],t) = 1}, (3.2)

where [x.sup.*] = [[k.sub.1][x.sub.1], ..., [k.sub.N][x.sub.N]] is a (1 X N) vector of shadow input quantities that solves the minimization problem in Equation 3.2. Note that in contrast to Equation 3.1, we assume that actual input prices equal shadow input prices so that the shadow costs are now defined as the sum of shadow input quantities times actual input prices rather than as the sum of actual input quantities times shadow input prices. If the shadow quantity for a given input differs from the actual quantity, the firm is using a nonoptimal amount of this input. This could result from (i) satisficing behavior, (ii) actions by labor unions to limit labor input, (iii) shortages of other inputs, (iv) production delays, (v) regulated production, or (vi) production quotas or target levels.

In the shadow distance model, actual costs are equivalent to shadow costs. By contrast, in the shadow cost model, actual (social) and shadow costs to the firm typically diverge. All costs are now assumed to be borne by the firm, whether or not it achieves AE. Hence, the notation for the shadow cost function is identical to that for the cost function, namely, C(y, p, t).

The first-order condition corresponding to Equation 3.2 is

[P.sub.n] = [mu][partial][D.sub.i](y, [x.sup.*],t)/[partial][x.sub.n], n = 1,...,N, (3.3)

where [mu] is the Lagrangian multiplier and [[partial][D.sub.i](y,[x.sup.*],t]/[partial][x.sub.n] is the partial derivative of [D.sub.i](y, [x.sup.*], t) with respect to [x.sub.n], evaluated at [x.sup.*.sub.n]. Multiplying both sides by [x.sup.*.sub.n] and summing, we obtain

[summation over(l)][p.sub.l][x.sup.*.sub.l] = [mu] [summation over (l)] [partial][D.sub.i](y, [x.sup.*], t)/[partial][x.sub.l] [x.sup.*.sub.l]. (3.4)

Since [D.sub.i](y, [x.sup.*], t) is linearly homogeneous in [x.sub.*], via Euler's theorem,

[summation over (l)] [partial][D.sub.i](y, [x.sup.*], t)/[partial][x.sub.l] [x.sup.*.sub.l] = [D.sub.i](y, [x.sup.*], t). (3.5)

Since by assumption

1 = [D.sub.i](y, [x.sup.*], t), (3.6)

We can write Equation 3.4 as

[mu] = [summation over (l)] [p.sub.l][x.sup.*.sub.l] = C(y, p, t). (3.7)

This allows us to express the nth equation in Equation 3.3 as

[p.sub.n] = ([summation over (l)][p.sub.l][x.sup.*.sub.l]) [partial][D.sub.i](y, [x.sup.*], t)/[partial][x.sub.n]. (3.8)

4. Econometric Estimation

A Generalized Translog Distance Function

Since we cannot expect Equation 3.6 to hold with equality for all observations, we introduce a composed error term. Given panel data on F firms in T time periods, Equation 3.6 becomes

1 = [D.sub.i]([y.sub.ft], [x.sup.*.sub.ft] t)h([[epsilon].sub.ft]), (4.1)

where [x.sup.*.sub.ft] = [[k.sub.l][x.sub.l,ft],..., [k.sub.N][x.sub.N,ft]] and

h([[epsilon].sub.ft]) = exp([v.sub.ft] - [u.sub.ft]) (4.2)

is a two-component error term f = 1,..., F; t = 1,..., T. It comprises a one-sided iid error, [u.sub.ft] > 0, and statistical noise, [v.sub.ft], assumed to be iid with zero mean. The one-sided component allows us to measure firm TE. As a flexible approximation to the underlying true distance function, the translog input distance function, with error term, is written as

0 = ln[[D.sub.i]([y.sub.ft], [x.sup.*.sub.ft], t)h([[epsilon].sub.ft])] = ln[D.sub.i]([y.sub.ft], [x.sup.*.sub.ft], t) + ln h([[epsilon].sub.ft])

= [[gamma].sub.0] + [summation over (m)] [[gamma].sub.m] ln [y.sub.mft] + (1/2) [summation over (m)] [summation over (u)] [[gamma].sub.mu] ln [y.sub.mft] ln [y.sub.uft] + [summation over (m)] [summation over (n)] [[gamma].sub.mn] ln [y.sub.mft] ln [x.sup.*.sub.nft]

+ [summation over (n)] [[gamma].sub.n] ln [x.sup.*.sub.nft] + (1/2) [summation over (n)] [summation over (l)] [[gamma].sub.nl] ln [x.sup.*.sub.nft] ln [x.sup.*.sub.lft] + [summation over (n)] [[gamma].sub.nt] ln [x.sup.*.sub.nft]t

+ [summation over (m)] [[gamma].sub.mt] ln [y.sub.mft]t + [[gamma].sub.t]t + [[gamma].sub.lt][t.sup.2] + ln h([[epsilon].sub.ft]). (4.3)

The [u.sub.ft] can be modeled as a fixed or random effect. We adopt the fixed-effects approach for time-varying inefficiency proposed by Cornwell, Schmidt, and Sickles (1990):

[u.sub.ft] = [[beta].sub.f0][d.sub.f] + [[beta].sub.f1][d.sub.f]t + [[beta].sub.f2][d.sub.f][t.sup.2], (4.4)

where [d.sub.f] is a dummy variable for firm f and [[beta].sub.f0], [[beta].sub.f1], and [[beta].sub.f2] are parameters to be estimated for this firm. [[beta].sub.f0] captures time-invariant, firm-specific differences in the technology, whereas [[beta].sub.f1] and [[beta].sub.f2] capture time-varying, firm-specific differences in technology. Equation 4.4 is employed in a second-stage regression and is not part of the initially estimated shadow distance system, as explained below. The fixed-effects approach avoids having to make assumptions required by the random-effects approach. Using the latter approach, one must specify the distribution of the errors and assume that the explanatory variables are exogenous with respect to the [u.sub.ft] in particular. Both assumptions may be tenuous and should always be tested. (2)

In order to control for time-invariant, firm-specific effects in our estimated model, we reexpress Education 4.3 as

0 = ln D([y.sub.ft], [x.sup.*.sub.ft], t) - [[beta].sub.f0][d.sub.f] + [v.sub.ft] - [u.sup.*.sub.ft], (4.5)

where [u.sup.*.sub.ft] = [u.sub.ft] - [[beta].sub.f0][d.sub.f]. This equation and N price equations derived from Equation 4.3 using Equation 3.8 comprise our translog shadow distance system. Before estimation, we impose symmetry by requiring [[gamma].sub.mu] = [[gamma].sub.um] [for all] m, u, m [not equal to] u and [[gamma].sub.nl] = [[gamma].sub.ln], [for all] n, l, n [not equal to] l. Since [D.sub.i](y, [x.sup.*], t) is linearly homogeneous in [x.sup.*], the following restrictions are also imposed:

[summation over (n)] [[gamma].sub.n] = 1, [summation over (n)] [[gamma].sub.nl] = [summation over (l)] [[gamma].sub.nl] = [summation over (n)] [summation over (l)] [[gamma].sub.nl] = 0, [summation over (n)] [[gamma].sub.mn] = 0, [for all] m. (4.6)

We impose the restriction [[gamma].sub.0] = 0 in order to achieve the identification of all of the firm dummy variable parameters. The results for the AE parameters and the structural parameters do not vary with the choice of this normalization.

Since the [k.sub.nft] values are derived assuming cost minimization, we can identify only N - 1 of them subject to a normalization of any arbitrarily chosen [k.sub.nft]. Let the normalization be imposed on input N so that [k.sub.Nft] is restricted to some positive constant, [for all]f, t. The values of the objective function, the values of the estimated efficiency parameters, and the values of all structural parameters are unaffected by the choice of the numeraire or its specific value. For the remaining inputs, we specify

[k.sub.nft] = [k.sub.nf] + [k.sub.n1]t + [k.sub.n2][t.sup.2], [for all] n = 1,...,N - 1. (4.7)

For details, see Atkinson and Cornwell (1994).

Estimating Allocative Inefficiency

The estimated [k.sub.nft] values tell us whether the firm has employed the efficient level of each input relative to the numeraire input. From Equation 3,8, the conditions for cost minimization are

Pnf/Plf = [partial] [D.sub.i]([y.sub.ft],[x.sup.*.sub.ft],t)/[partial][x.sup.*.sub.n]/[pa rtial][D.sub.i]([y.sub.ft],[x.sup.*.sub.ft],t)/[partial][x.sup.*.sub. l], [for all] n [not equal to] 1. (4.8)

For firm f at time t, we determine relative over- and underutilization of any pair of inputs [x.sub.nft] and [x.sub.lft] in comparison to the cost-minimizing ratio ([k.sub.nft][x.sub.nft])/([k.sub.lft][x.sub.lft]) by simply computing [k.sub.nft]/[k.sub.lft]. From Figure 1, the supporting hyperplane, labeled (p1, p2), yields C(y, p, t). This hyperplane is tangent to the input requirement set, L(y), at [x.sup.*], which is the cost-minimizing point consistent with Equation 4.8.

If we possessed a priori information about the absolute shadow price of one input, we could estimate the absolute shadow prices of the other inputs using the estimated ratios of price equations in Equation 4.8. However, we have no such a priori information, and we would surmise that such information would typically not be available. Thus, in most real-world data settings, absolute values of shadow prices cannot be estimated using our distance system. (3) Our price equations allow us to determine only the relative efficiency of input usage, estimated from ratios of fitted [k.sub.nft] [k.nft].

When a cost system comprising a cost function and its share equations is estimated, one share equation must be dropped from the estimated system because of the linear dependence of the error terms. Such is not the case with our input distance system, since the N price equations and the distance function do not constitute a linearly dependent system. Estimation of the distance function in Equation 4.5 and N - 1 price equations from Equation 3.8 would be sufficient to identify the parameters m Equation 4.7, up to a normalization on [k.sub.Nft]. This follows because we are estimating relative price efficiency, not absolute price efficiency. However, the estimation of all N price equations and the distance function in Equation 4.5 generates a net increase in our degrees of freedom. This occurs because we impose cross-equation restrictions (symmetry and linear homogeneity in Eqn. 4.6) and add no new parameters by including the additional equation that contains new data on the Nth input price. This should increase the small-sample efficiency of our parameter estimators.

Typically, the researcher is not interested in shadow prices per se and would use them only to compute the relative over- or underutilization of input quantities. For example, one may be interested in the effect of import quotas or job hiring quotas on input quantities. With rate-of-return regulation in a static environment, one may be concerned about the overuse of capital relative to other inputs. With the provision of public services by local municipalities--police protection, sewage and water treatment, and cultural activities--one may wish to focus on the extent to which public managers overuse capital inputs (e.g., an oversized fleet of police cars or an inefficiently large public water and sewage works or community performing arts center) relative to other inputs.

In this case, one advantage of the shadow distance system is that relative inefficiency in terms of the use of each input is computed directly using ratios of the estimated values of [k.sub.nft] and [k.sub.lft]. With a shadow cost system, relative inefficiencies for each input must be obtained indirectly by first estimating shadow prices from the cost function and share or input quantity equations. If share equations are employed, they must be solved for fitted input quantities. Otherwise, each fitted input demand equation must be computed. Then, for input n, one must compute the ratio of inefficient fitted demand (obtained using the estimated value of [k.sub.nft]) to efficient fitted demand (obtained using [k.sub.nft] = 1).

Alternative Specification of the Distance Function

An alternative specification of the distance function can be employed that yields a variable on the left-hand side of Equation 4.1, rather than a vector of ones. For simplicity, assume that [k.sub.n] = 1, [for all]n. Linear homogeneity can be imposed by normalizing all [x.sub.nft] and the left-hand side in Equation 4.1 by an arbitrarily chosen input, say, the Nth, in which case a conventional regression model can be obtained:

1/[x.sub.Nft] = D([y.sub.ft],[x.sub.ft], t)h([[epsilon].sub.ft]) (4.9)

where

[x.sub.ft] = ([x.sub.1ft]/[x.sub.Nft],...,[x.sub.N-1,ft]/[x.sub.Nft]).

This model is equivalent to Equation 4.1 after linear homogeneity is imposed via parametric restrictions (for the translog in Eqn. 4.3 by substituting Eqn. 4.6) and yields identical estimated coefficients and standard errors. However, direct estimation of Equation 4.1 with linear homogeneity imposed via parametric restrictions has the advantage of automatically generating the fitted distance function, and the partial derivative of its log with respect to time yields technical change. By contrast, Equation 4.9 would have to be rearranged to obtain Equation 4.1 following estimation before the derivative is taken to compute technical change.

Instrumental Variables

Consistent estimation of the N price Equations 3.8 and 4.5 using GMM requires that the model satisfy the moment conditions E([v.sub.ff]\[Z.sub.ft]) = 0, where Z is a vector of instruments, and that we are able to apply the uniform weak law of large numbers so that the difference between the average sample and population moments converges in probability to zero. The Hansen (1982) test of overidentifying restrictions is used to determine the validity of the instrument set. For our reported results, we employ the set of instruments that produces the smallest test statistic for the null hypothesis that the overidentifying restrictions are valid.

Computing Technical Efficiency

After the estimation of the distance system, we require a consistent estimator of [u.sub.ft] to compute [TE.sub.ft]. Extending Atkinson, Cornwell, and Honerkamp (2001) to the case of two-stage estimation, we proceed by first calculating the negative of the residuals from Equation 4.5 as [[beta].sub.f0][d.sub.f] - [v.sub.ft] + [u.sup.*.sub.ft] = [u.sub.ft] - [v.sub.ft], which is consistent for up [u.sub.ft] - [v.sub.ft], given that the coefficient estimators in the first stage are consistent, as just discussed. The estimator of [u.sub.ft] - [v.sub.ft] is then regressed on the right-hand side of Equation 4.4. The fitted values of this regression are consistent for [u.sub.ft].

While we have specified that [u.sub.ft] > 0, we have not imposed this restriction on [u.sub.ft]. Thus, we transform [u.sub.ft] by subtracting u = [min.sub.ft]([u.sub.ft]), which is the frontier intercept across all firms and over all time periods, to obtain [u.sup.*.sub.ft] = [u.sub.ft] - u [greater than or equal to] 0. With in [D.sub.i](y, x, t) representing the estimated translog portion of Equation 4.3 (i.e., those terms other than h([[epsilon].sub.ft])), adding and subtracting u yields

0 = ln [D.sub.i](y, x, t) + [v.sub.ft] - [u.sub.ft] + u - u = ln [D.sub.i](y, x, t) + [v.sub.ft] - [u.sub.ft], (4.10)

where ln [D.sub.i](y, x, t) = ln [D.sub.i](y, x, t) - u is the estimated frontier distance function in period t. Using [u.sub.ft], we estimate firm f's level of TE in period t as

[TE.sub.ft] = exp(-[u.sup.*.sub.ft]), (4.11)

where our normalization of [u.sup.*.sub.ft] guarantees that 0 < [TE.sub.ft] [less than or equal to] 1.

There are three basic methods that can be used to obtain standard errors and confidence intervals for estimated technical efficiency measures. The first is to employ the approach of Battese and Coelli (1988). However, this requires assumptions about the distribution of [v.sub.ft] and [u.sub.ft], which we avoid having to make during the estimation of our distance system by employing the GMM estimator. The second method is to employ the multiple comparison with the best technique, which was first applied to efficiency rankings by Horrace and Schmidt (2000). The application of this method would require that we supply estimated firm-specific parameters representing efficiencies and the estimated covariances of the estimators of these parameters. However, exp(-[u.sub.ft]) is a complex function of these parameters, structural parameters, and data, which makes the application of the Horrace and Schmidt method difficult. The third method is to employ the technique of Krinsky and Robb (1986), which directly simulates the standard error of exp(-[u.sub.ft]). One proceeds by using the fitted version of Equation 4.4 to draw randomly from a multivariate normal distribution with a mean equal to the estimated coefficients in Equation 4.4 and a covariance equal to their estimated covariance matrix. (4) Then, we compute the simulated values of the left-hand side of Equation 4.4, [u.sup.*.sub.ft], given these draws for our parameters and the original data in Equation 4.4. Finally, we compute our simulated technical efficiency scores as TE = exp(-[u.sup.*.sub.ft]). This process is repeated 5000 times, and then the sample standard deviation of the [TE.sub.ft], for each firm is computed for all of its time-series observations. (5)

Cost Savings from Eliminating Allocative and Technical Inefficiency

In order to compute the cost savings resulting from AE, TE, or both types of efficiency, we use Equation 3.7 to obtain

C(y, p, t) = [px.sup.*], (4.12)

where [x.sup.*] is the fitted [x.sup.*], based on the fitted values of [k.sub.nft]. To compute the percentage of cost savings due to the attainment of TE and AE, we compute efficient costs by multiplying [x.sup.*] by estimated [TE.sub.ft] so that the firm is operating on the efficient isoquant. To calculate the percentage of savings in cost due solely to the attainment of TE, in Equation 4.12 we replace [x.sup.*] with x, which is scaled by estimated

[TE.sub.ft]. Finally, to compute the cost savings from AE alone, we take the difference between the first and second set of cost estimates.

5. Data and Results

Our data set consists of annual observations from 1951 to 1975 (so that T = 25) for 12 (F = 12) class 1 U.S. railroads: the Atchison, Topeka, and Santa Fe; the Baltimore and Ohio; the Chesapeake and Ohio; the Denver, Rio Grande, and Western; the Louisville and Nashville; the Chicago, Milwaukee, and St. Paul; the Missouri-Pacific; the Norfolk and Western; the Reading; the Chicago, Rock Island, and Pacific; the Southern Pacific; and the Union Pacific. During the period studied, state commissions required that railroads provide passenger service even if unprofitable and established minimum rate regulation. After the early 1970s, railroads received relief from both types of regulation. Thus, limiting our analysis to this period assures a relatively homogeneous regulatory environment.

We use data on prices and quantities for four inputs and four outputs. The input quantities are ways and structures (WS), labor (L), fuel (E), and materials and equipment (ME). As output quantities we include measures of actual quantities transported--freight ton miles and passenger miles--as well as two output attributes, average passenger trip length in miles and average freight haul in miles. More details of the construction of quantities and prices are given in Caves, Christensen, and Swanson (1980) and references therein. We wish to thank Paul Wilson and Subal Kumbhakar for providing us with the data used in this paper. Table 1 presents the weighted averages of firm input cost shares over time; the weights are based on total revenues. The average cost shares are approximately 0.16, 0.48, 0.05, and 0.31 for WS, L, E, and ME, respectively. There was a steady decline in the cost shares of E and L from 1951 to 1970. At the same time, there was a generally steady increase in the cost share of ME.

Our estimated distance system comprises five equations: Equation 4.5, and one Equation 3.8 for each input. This system is estimated subject to symmetry plus the restrictions from Equation 4.6 that guarantee linear homogeneity in input quantities. Using a chi-square test at the 0.05 level, we test for and accept the additional restrictions that [[gamma].sub.nt] = [[gamma].sub.mt] = 0, [for all] m, n, which imply that time is separable from both input and output quantities. We impose the additional restriction that the two output attributes are separable from time and from both the input and the output quantities. Finally, we parameterize [k.sub.nft] in terms of Equation 4.7, where we set [k.sub.ME,ft] = 1, [for all]f, t, so that estimated [k.sub.nft], n [not equal to] ME are interpreted relative to this normalization.

We address the validity of the overidentifying restrictions using the Hansen (1982) J test based on estimation of our distance system via GMM. Allowing for heteroskedasticity and autocorrelation of unknown form, we used the Newey and West (1987) covariance matrix estimator with a lag of four periods. We examined a large number of potential instruments and selected the set on the basis of the moment conditions that generated the largest p value for the J test statistic. The data rejected any set of moment conditions that included either the outputs or the inputs as exogenous variables. We failed to reject the null hypothesis that the moment conditions are valid only when we employ firm dummies, time dummies, and an interaction between a time trend and firm dummies as instruments. The value of the I test statistic for this case was 64.79 with 154 degrees of freedom and a p value of 1.0. Table 2 presents the values of the Hansen J test for alternative sets of candidate instruments.

A Wald test rejected the null hypothesis of homotheticity of the technology at the 0.01 level. This implies that we also reject homogeneity and linear homogeneity. Using Equation 2.5 for only freight ton miles and passenger miles, we computed average returns to scale of 1.17 for our sample, estimating our shadow distance system assuming AE. Our results in terms of actual quantities are similar to those obtained by Caves, Christensen, and Swanson (1981), who estimated a cost function using a sample with more railroads but fewer years than employed here.

An input distance function should be monotonic nondecreasing and concave in inputs and monotonic nonincreasing in outputs. At the median observation, we find that the estimated distance function satisfies these properties.

The estimated shadow distance system fits the data quite well. The [R.sup.2] values are 0.31, 0.42, 0.15, 0.68, and 0.60 for the ME, L, E, WS, and distance equations, respectively. Although the estimated [R.sup.2] for Equation 4.3 is 0, we compute an [R.sup.2] from an equivalent normalized form of the shadow distance equation by first dividing the left-hand side of Equation 4.1 and all input quantities on the right-hand side by the quantity of any arbitrarily chosen input. Since this imposes linear homogeneity in input quantities, we do not need to impose the parametric adding-up restrictions from Equation 4.6, which also guarantee linear homogeneity in input quantities. However, all of the other restrictions imposed on the nonnormalized form of the shadow distance function need to be imposed on the normalized version. Since the use of this version in our shadow distance system yields estimated parameters that are identical to those obtained using the nonnormalized form, we simply evaluate the normalized vers ion of Equation 4.1 at the fitted values already obtained.

All but five of the structural parameters in Equation 4.3 were significantly different from zero at the 0.05 level with a two-tailed test. However, these coefficients were significant at no greater than the 0.19 level with a two-tailed test. Detailed results are available from the authors on request.

Fitted values for [k.sub.nft] are presented in Table 3. All but one of the AE parameters on the right-hand side of Equation 4.7 were significantly different from zero at the 0.05 level with a two-tailed test.

Table 4 reports allocative inefficiencies over time. Relative to materials and equipment, [k.sub.nft] < 1 implies that the cost-minimizing position of firm f would have been to use less of input n than it actually employed. Using the percentage of total revenues as weights, the weighted averages over all railroads for [k.sub.WS], [k.sub.L], and [k.sub.E] are 0.55, 1.19, and 0.19, respectively. Since [k.sub.ME, ft] = 1, these estimates indicate that the cost-minimizing solution for the railroad industry would have been to scale WS by 0.55, L by 1.19, and E by 0.19 relative to ME. In this relative sense, the industry was overcapitalized with WS relative to L and ME but undercapitalized with WS relative to E. Furthermore, L was substantially underused relative to E and slightly underused relative to ME. Finally, E was substantially overused relative to ME, which is consistent with a railroad running partially filled cars on routes that could not be abandoned because of regulation. As stated earlier, these result s do not depend on the choice of the numeraire input or its actual value. Over time we observe declines in both the overutilization of WS relative to L and the overutilization of E relative to WS. The underutilization of L relative to E also decreased. The overutilization of WS and E relative to ME increased over time, while the relative underutilization of L relative to ME before 1966 changed to relative overutilization after that date.

Table 5 reports estimated firm-specific parameters [[beta].sub.f0] and their estimated standard errors. All but one of these coefficients is significant at the 0.05 level with a two-tailed test.

The estimates of revenue-weighted average [TE.sub.f], [TE.sub.f], and their standard deviations for each firm are reported in Table 6. Since our normalization of [u.sup.*.sub.ft] is based on the minimum [u.sub.ft] over all f and t, the estimated efficiency scores are smaller than they would be if this normalization were based on the minimum [u.sub.ft] in each period. Average firm inefficiency over all periods is about 42%, with substantial variation from the least to the most efficient firm, which was the Denver, Rio Grande, and Western Railroad, with an average efficiency score of 0.95. Kumbhakar (1987) computed similar average efficiency scores from a frontier cost system. Estimated standard deviations are less than one tenth of their corresponding average efficiency scores.

While magnitudes of TE and AE are important for firm managers, their impact on costs may also be of concern. We computed the revenue-weighted average reductions in total costs if the firm were to achieve AE, TE, or both using the procedure discussed above. For all firms, the average cost savings resulting from TE, from AE, and from both TE and AE are approximately 63%, 12%, and 75%, respectively.

6. Summary and Conclusions

Frequently, fixed-effects measures of AE and TE have been estimated using a shadow cost function, whose arguments are output quantities and shadow input prices. However, this approach yields indirect measurement of relative input distortions. In this paper, we developed a procedure for the stochastic estimation of a shadow distance system that yields direct measurement of relative input distortions. Such estimation may be more useful than the indirect estimation of relative input misallocation that can be obtained from the estimation of a shadow cost function. Both procedures require the same assumption, namely, that the firm is cost minimizing. We jointly estimate the shadow distance function and the first-order conditions from a cost minimization problem in which the arguments of both are shadow input quantities and output quantities. Typically, interest centers on the efficiency of input utilization rather than on the divergence of shadow from actual prices per se. Although shadow input quantities and shad ow input prices provide dual measures of the relative efficiency of input utilization, only the former are direct measures. Since it is unclear a priori which of the arguments of the distance function are exogenous, we compute the GMM estimator of our shadow distance system using alternative sets of instrumental variables and test the validity of each set of overidentifying restrictions for a panel of U.S. railroads. We estimate TE, AE, returns to scale, and the cost reduction percentage if the industry were to become allocatively efficient, technically efficient, or both.

[FIGURE 1 OMITTED]

Table 1

Industry Average Factor Cost Shares over Time (Firm-Weighted Averages)

Period     WS      L       E       ME

1951     0.0830  0.5801  0.0784  0.2584
1952     0.0976  0.5733  0.0673  0.2618
1953     0.1258  0.5401  0.0602  0.2740
1954     0.1276  0.5342  0.0549  0.2833
1955     0.1252  0.5424  0.0550  0.2774
1956     0.1286  0.5530  0.0522  0.2662
1957     0.1544  0.5278  0.0481  0.2698
1958     0.1758  0.5068  0.0403  0.2770
1959     0.1687  0.5071  0.0411  0.2831
1960     0.1930  0.4825  0.0378  0.2866
1961     0.2241  0.4349  0.0360  0.3050
1962     0.2273  0.4445  0.0360  0.2922
1963     0.2272  0.4373  0.0367  0.2988
1964     0.2214  0.4332  0.0347  0.3106
1965     0.2170  0.4339  0.0358  0.3133
1966     0.2147  0.4238  0.0358  0.3257
1967     0.2150  0.4147  0.0352  0.3352
1968     0.2045  0.4163  0.0354  0.3438
1969     0.1954  0.4176  0.0356  0.3514
1970     0.1934  0.4177  0.0339  0.3550
1971     0.1633  0.4368  0.0351  0.3648
1972     0.1351  0.4863  0.0384  0.3402
1973     0.1213  0.5004  0.0433  0.3351
1974     0.0508  0.5123  0.0814  0.3555
1975     0.0632  0,5412  0.0886  0.3070
Average  0.1621  0.4839  0.0471  0.3068

Table 2

Sensitivity of Results to the Choice of Instruments

Instrument Set  [chi square]  p-Value

1                  463.12      0.00
2                  604.76      0.00
3                  565.23      0.00
4                   64.34      1.00

Each instrument set contains the linearly independent elements in A =
([d.sub.j], [d.sub.t], [d.sub.f]t], where t is a trend. The four cases
are distinguished as follows: (1) A and [In [p.sub.n], [(In
[p.sub.n]).sup.2], In [p.sub.n] and (In [p.sub.n])t]; (2) A and [[In
[y.sub.m]).sup.2], In [y.sub.m] In [y.sub.n]1, and (In [y.sub.m]t]; (3)
A and [In [x.sub.m] [In [x.sub.n]).sup.2], In [x.sub.n] In [x.sub.t],
and (In [X.sub.n])t]; (4) A only.

Table 3

Estimated Allocative Inefficiency Parameters

           [k.sub.WS]  [k.sub.L]   [k.sub.F]

t          -0.0013     -0.0743     -0.0029
           (0.0019)    (0.0040) *  (O.0004) *
[t.sup.2]  -0.0002      0.0016      0.0001
           (0.0001) *  (0.0001) *  (0.0000) *
ATS         0.3173      1.6211      0.3580
           (0.0182) *  (0.0490) *  (0.0149) *
B&O         1.0830      1.6406      0.1161
           (0.0412) *  (0.0869) *  (0.0139) *
C&O         1.0397      1.3316      0.1136
           (0.0448) *  (0.0632) *  (0.0134) *
DRGW        0.8760      1.0718      0.1438
           (0.0320) *  (0.0518) *  (0.0130) *
L&N         0.8593      1.4843      0.1445
           (0.0295) *  (0.0822) *  (0.0169) *
MILW        0.3926      2.1684      0.4278
           (0.0244) *  (0.0838) *  (0.0254) *
MP          1.1274      1.4729      0.1352
           (0.0656) *  (0.0793) *  (0.0380) *
NW          0.2090      0.9058      0.4018
           (0.0164) *  (0.0448) *  (0.0133) *
RDG         0.9937      1.9169      0.2082
           (0.0534) *  (0.1062) *  (0.0275) *
ROCK        0.7569      1.7583      0.3080
           (0.0367) *  (0.1048) *  (0.0286) *
SP          0.3898      3.2042      0.0386
           (0.0261) *  (0.1744) *  (0.0039) *
UP          0.4496      1.6440      0.2365
           (0.0201) *  (0.0429) *  (0.0089) *

Asymptotic standard errors are in parentheses. Railroad abbreviations
are as follows: ATS, Atchison, Topeka, and Santa Fe; B&O, Baltimore and
Ohio; C&O, Chesapeake and Ohio; DRGW, Denver, Rio Grande, and Western;
L&N, Louisville and Nashville; MILW, Chicago, Milwaukee, and St. Paul;
MP, Missouri-Pacific; NW, Norfolk and Western; RDG, Reading; ROCK,
Chicago, Rock Island, and Pacific; SP, Southern Pacific; UP, Union
Pacific.

* Significant at the 0.05 level with a two-tailed test.

Table 4

Allocative Inefficiencies over Time (Firm-Weighted Averages)

Period   [k.sub.WS]  [k.sub.L]  [k.sub.E]

1951       0.6505     1.7256     0.2091
1952       0.6426     1.6729     0.2061
1953       0.6403     1.6042     0.2032
1954       0.6300     1.5516     0.2011
1955       0.6306     1.4770     0.1983
1956       0.6422     1.4005     0.1945
1957       0.6363     1.3357     0.1943
1958       0.6190     1.3063     0.1937
1959       0.6093     1.2692     0.1916
1960       0.6011     1.2129     0.1911
1961       0.5837     1.1872     0.1891
1962       0.5727     1.1911     0.1811
1963       0.5671     1.1466     0.1797
1964       0.5521     1.1015     0.1814
1965       0.5222     1.0335     0.1890
1966       0.5097     1.0097     0.1896
1967       0.5036     0.9868     0.1881
1968       0.4926     0.9693     0.1873
1969       0.4808     0.9582     0.1864
1970       0.4719     0.9303     0.1873
1971       0.4643     0.9388     0.1839
1972       0.4448     0.9366     0.1849
1973       0.4350     0.9277     0.1860
1974       0.4272     0.9138     0.1862
1975       0.4117     0.9045     0.1876
Average    0.5497     1.1877     0.1908

Table 5

Estimated Firm-Specific Parameters

Firm  Parameter

ATS     0.2598
       (0.0391) *
B&O     0.3162
       (0.0388) *
C&O    -0.1207
       (0.0333) *
DRGW   -0.8798
       (0.0420) *
L&N    -0.1808
       (0.0367) *
MILW    0.3504
       (0.0378) *
MP      0.0023
       (0.0264)
NW     -0.7166
       (0.0317) *
RDG     0.2573
       (0.1273) *
ROCK    0.1488
       (0.0435) *
SP      0.8452
       (0.0554) *
UP      0.2166
       (0.0435) *

Asymptotic standard errors are in parentheses. Railroad abbreviations
are as follows: ATS, Atchison, Topeka, and Santa Fe; B&O, Baltimore and
Ohio; C&O, Chesapeake and Ohio; DRGW, Denver, Rio Grande, and Western;
L&N, Louisville and Nashville; MILW, Chicago, Milwaukee, and St. Paul;
MP, Missouri-Pacific; NW, Norfolk and Western; RDG, Reading; ROCK,
Chicago, Rock Island, and Pacific; SP, Southern Pacific; UP, Union
Pacific.

* Significant at the 0.05 level with a two-tailed test.

Table 6

Average Firm Technical Inefficiencies

         Efficiency      Estimated
Period     Score     Standard Deviation

ATS        0.2969          0.0161
B&O        0.2831          0.0125
C&O        0.4379          0.0208
DRGW       0.9471          0.0519
L&N        0.4696          0.0307
MILW       0.2710          0.0129
MP         0.3886          0.0165
NW         0.7955          0.0626
RDG        0.2952          0.0163
ROCK       0.3339          0.0161
SP         0.1656          0.0085
UP         0.3101          0.0182
Average    0.4162

Railroad abbreviations are as follows: ATS, Atchison, Topeka, and Santa
Fe; B&O, Baltimore and Ohio; C&O, Chesapeake and Ohio; DRGW, Denver, Rio
Grande, and Western; L&N, Louisville and Nashville; MILW, Chicago,
Milwaukee, and St. Paul; MP, Missouri-Pacific; NW, Norfolk and Western;
RDG, Reading; ROCK, Chicago, Rock Island, and Pacific; SP, Southern
Pacific; UP, Union Pacific.

Received November 1999; accepted February 2002.

(1.) Future research could compare the shadow input distance system with a shadow cost system.

(2.) One convenient way to test endogeneity and the validity of one's instrument set is demonstrated below using the GMM estimator and the Hansen (1982) J test. Testing distributional assumptions requires the use of maximum likelihood techniques. If competing densities are nested, standard nested hypothesis testing may be performed. Alternatively, one may carry out nonnested tests using methods such as the artifical nesting models proposed by Davidson and MacKinnon (1981), the generalized Wald test, or the generalized score test. See Gourieroux and Monfort (1995) for an excellent survey of these methods.

(3.) One could estimate absolute values of shadow prices from a shadow profit system. See, for example, Atkinson and Halvorsen (1980).

(4.) The assumption of normality of the estimators in Equation 4.4 in order to generate random values of estimated parameters is clearly justified on asymptotic grounds, assuming that the error term is iid.

(5.) These estimates are computed assuming that the estimated coefficients from the first-stage regression are known, rather than treating them as random variables to reflect their uncertainty. However, this is also true of the Battese and Coelli (1988) method.

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Scott E. Atkinson, *

Rolf Fare, +

Daniel Primont ++

* Department of Economics, University of Georgia, Athens, GA 30602, USA; E-mail [e-mail]; corresponding author.

+ Department of Economics, Oregon State University, Corvallis, OR 97331, USA.

++ Department of Economics, Southern Illinois University, Carbondale, IL 62901, USA.

We wish to thank Chris Cornwell for numerous comments on earlier drafts of this paper.