Efficiency Measurement

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Efficiency Measurement

• William Greene
• Stern School of Business
• New York University

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Session 5

• Modeling Heterogeneity

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Production Function Model with Inefficiency
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The Stochastic Frontier Model
ui gt 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 ??xivi
and, as before, ui represents the inefficiency.
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The Normal-Half Normal Model
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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.
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Normal-Truncated Normal
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Truncated Normal Model
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Fundamental Tool - JLMS
We can insert our maximum likelihood estimates of
all parameters. Note This estimates Euvi
ui, not ui.
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Estimated Translog Production Frontiers
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Inefficiency Estimates
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Estimated Inefficiency Distribution
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Cost Inefficiency

• y f(x) ?? C g(y,w)
• (Samuelson Shephard duality results)
• Cost inefficiency If y lt f(x), then C mustbe
  greater than g(y,w). Implies the idea of a cost
  frontier.
• lnC lng(y,w) u, u gt 0.

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Stochastic Cost Frontier
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Estimates of Economic Efficiency
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Duality Production vs. Cost
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Where to Next?

• Heterogeneity Where do we put the zs?
• Other variables that affect production and
  inefficiency
• Enter production frontier, inefficiency
  distribution, elsewhere?
• Heteroscedasticity
• Another form of heterogeneity
• Production risk
• Bayesian and simulation estimators
• The stochastic frontier model with gamma
  inefficiency
• Bayesian treatments of the stochastic frontier
  model
• Panel Data
• Heterogeneity vs. Inefficiency can we
  distinguish
• Model forms Is inefficiency persistent through
  time?
• Applications

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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?

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Shifting the Outcome Function
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 N0,?v2 ui Ui where Ui
N?i, ?u2, ?i ?0 ?1?zi,
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Heterogeneous Mean
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Estimated Economic Efficiency
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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.

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One vs. Two Step
0.8 0.9 1.0
Efficiency computed without load factor, stage
length and points served.
Efficiency computed with load factor, stage
length and points served.
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Application WHO Data
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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

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A Latent Class Model
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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.

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Latent Class Application
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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.

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Nursing Home Costs

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

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Estimated Cost Efficiency
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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.

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LogL 464 with a common frontier model, 527 with
two classes
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Heteroscedasticity in v and/or u

• Varvi hi ?v2gv(hi,?) ?vi2
• gv(hi,0) 1,
• gv(hi,?) exp(?hi)2
• VarUi hi ?u2gu(hi,?) ?ui2
• gu(hi,0) 1,
• gu(hi,?) exp(?hi)2

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Application WHO Data
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A Scaling Model
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Unobserved Endogenous Heterogeneity

• Cost C(p,y,Q), Q quality
• Quality is unobserved
• Quality is endogenous correlated with
  unobservables that influence cost
• Econometric Response There exists a proxy that
  is also endogenous
• Omit the variable?
• Include the proxy?
• Question Bias in estimated inefficiency (not
  interested in coefficients)

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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)

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A Simulation Experiment
Conclusion Omitted variable problem does not
make the bias worse.