Efficiency Measurement William Greene Stern School of Business New York University Session 5 Modeling Heterogeneity

Production Function Model with Inefficiency The Stochastic Frontier Model = f (xi )TE i e vi ln y i = + xi + vi ui yi = + xi + i .

ui > 0, but vi may take any value. A symmetric distribution, such as the normal distribution, is usually assumed for vi. Thus, the stochastic frontier is +xxi+vi and, as before, ui represents the inefficiency. The Normal-Half Normal Model

Log Likelihood Function Waldman (1982) result on skewness of OLS residuals: If the OLS residuals are positively skewed, rather than negative, then OLS maximizes the log likelihood, and there is no evidence of inefficiency in the data. Normal-Truncated Normal

Truncated Normal Model Fundamental Tool - JLMS ( it ) it E[uit | it ]

, it 2 it ( it ) 1 We can insert our maximum likelihood estimates of all parameters. Note: This estimates E[u|vi ui], not ui.

Estimated Translog Production Frontiers Inefficiency Estimates Estimated Inefficiency Distribution Cost Inefficiency

y* = f(x) C* = g(y*,w) (Samuelson Shephard duality results) Cost inefficiency: If y < f(x), then C must be greater than g(y,w). Implies the idea of a cost frontier. lnC = lng(y,w) + u, u > 0. Stochastic Cost Frontier

Estimates of Economic Efficiency Duality Production vs. Cost Where to Next? Heterogeneity: Where do we put the zs?

Heteroscedasticity

The stochastic frontier model with gamma inefficiency Bayesian treatments of the stochastic frontier model Panel Data

Another form of heterogeneity Production risk Bayesian and simulation estimators

Other variables that affect production and inefficiency Enter production frontier, inefficiency distribution, elsewhere? Heterogeneity vs. Inefficiency can we distinguish Model forms: Is inefficiency persistent through time?

Applications Observable Heterogeneity As opposed to unobservable heterogeneity

Observe: Y or C (outcome) and X or w (inputs or input prices) Firm characteristics z. Not production or cost, characterize the production process.

Enter the production or cost function? Enter the inefficiency distribution? How? Shifting the Outcome Function ln y it f(xit , ) g (zit , ) h(t ) v it uit Firm specific heterogeneity can also be incorporated into the inefficiency model as follows: This modifies the mean of the truncated

normal distribution yi = xi + vi - ui vi ~ N[0,v2] ui = |Ui| where Ui ~ N[i, u2], i = 0 + 1zi, Heterogeneous Mean Estimated Economic Efficiency

One Step or Two Step 2 Step: Fit Half or truncated normal model, compute JLMS ui, regress ui on zi Airline EXAMPLE: Fit model without POINTS, LOADFACTOR, STAGE 1 Step: Include zi in the model, compute ui including zi Airline example: Include 3 variables

Methodological issue: Left out variables in two step approach. One vs. Two Step 0.8 Efficiency computed without load factor, stage length and points

served. Efficiency computed with load factor, stage length and points served. 0.9 1.0

Application: WHO Data Unobservable Heterogeneity Parameters vary across firms

Random variation (heterogeneity, not Bayesian) Variation partially explained by observable indicators Continuous variation random parameter models: Considered with panel data

models Latent class discrete parameter variation A Latent Class Model Latent Class Efficiency Studies Battese and Coelli growing in weather regimes for Indonesian rice farmers

Kumbhakar and Orea cost structures for U.S. Banks Greene (Health Economics, 2005) revisits WHO Year 2000 World Health Report Kumbhakar, Parmeter, Tsionas (JE, 2013) U.S. Banks. Latent Class Application

Inefficiency? Not all agree with the presence (or identifiability) of inefficiency in market outcomes data. Variation around the common production structure may all be nonsystematic and not controlled by management Implication, no inefficiency: u = 0.

Nursing Home Costs 44 Swiss nursing homes, 13 years Cost, Pk, Pl, output, two environmental variables Estimate cost function Estimate inefficiency

Estimated Cost Efficiency A Two Class Model Class 1: With Inefficiency

logC = f(output, input prices, environment) + vv + uu Class 2: Without Inefficiency logC = f(output, input prices, environment) + vv

u = 0 Implement with a single zero restriction in a constrained (same cost function) two class model Parameterization: = u /v = 0 in class 2.

LogL= 464 with a common frontier model, 527 with two classes Heteroscedasticity in v and/or u Var[vi | hi] = v2gv(hi,) = vi2 gv(hi,0) = 1, gv(hi,) = [exp(xhi)]2

Var[Ui | hi] = u2gu(hi,)= ui2 gu(hi,0) = 1, gu(hi,) = [exp(xhi)]2 Application: WHO Data A Scaling Model u i h(z i , u i * where f(u i *) does not involve z i Scales both mean and variance of u i

N Ln L(,, , , 0 ) = -(N/2) ln 2 - i 1 ln i + ln ( i / ui ) + 1 2 i i i i i i 1 2 ln i

i i i i exp(z i ), ui u exp(z i ), N

vi v exp( z i ), i ui / vi , i vi2 ui2 Unobserved Endogenous Heterogeneity Cost = C(p,y,Q), Q = quality

Econometric Response: There exists a proxy that is also endogenous

Quality is unobserved Quality is endogenous correlated with unobservables that influence cost Omit the variable? Include the proxy?

Question: Bias in estimated inefficiency (not interested in coefficients) Simulation Experiment Mutter, et al. (AHRQ), 2011 Analysis of California nursing home data Estimate model with a simulated data set Compare biases in sample average inefficiency compared to the exogenous

case Endogeneity is quantified in terms of correlation of Q(i) with u(i) A Simulation Experiment Mean Ineffi ciency vs. Gamma

.4 8 4 Ineffi c iency .4 1 7 .3 5 0

.2 8 3 .2 1 6 .1 5 0 .0 0 .2 0

.4 0 .6 0 .8 0 1 .0 0 GAM M A

QS _ INCL QS _ E XCL Conclusion: Omitted variable problem does not make the bias worse.