Modeling Consumer Decision Making and Discrete Choice Behavior

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Econometrics in Health Economics Discrete
Choice ModelingandFrontier Modeling and
Efficiency EstimationProfessor William
GreeneStern School of BusinessNew York
UniversitySeptember 2-4, 2007
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Frontier and Efficiency Estimation

• Session 5
• Efficiency Analysis
• Stochastic Frontier Model
• Efficiency Estimation
• Session 6
• Panel Data Models and Heterogeneity
• Fixed and Random Effects
• Bayesian and Classical Estimation
• Session 7
• Efficiency Models
• Stochastic Frontier and Data Envelopment Analysis
• Student Presentation Silvio Daidone and
  Francesco DAmico
• Session 8 Computer Exercises and Applications

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The Production Function

• A single output technology is commonly described
  by means of a production function f(z) that gives
  the maximum amount q of output that can be
  produced using input amounts (z1,,zL-1) gt 0.
• Microeconomic Theory, Mas-Colell, Whinston,
  Green Oxford, 1995, p. 129. See also Samuelson
  (1938) and Shephard (1953).

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Thoughts on Inefficiency

• Failure to achieve the theoretical maximum
• Hicks (ca. 1935) on the benefits of monopoly
• Leibenstein (ca. 1966) X inefficiency
• Debreu, Farrell (1950s) on management
  inefficiency
• All related to firm behavior in the absence of
  market restraint the exercise of market power.

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A History of Empirical Investigation

• Cobb-Douglas (1927)
• Arrow, Chenery, Minhas, Solow (1963)
• Joel Dean (1940s, 1950s)
• Johnston (1950s)
• Nerlove (1960)
• Christensen et al. (1972)

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Inefficiency in the Real World

• Measurement of inefficiency in markets
  heterogeneous production outcomes
• Aigner and Chu (1968)
• Timmer (1971)
• Aigner, Lovell, Schmidt (1977)
• Meeusen, van den Broeck (1977)

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Production Functions

• Production is a process of transformation of a
  set of inputs, denoted x ? into a set of
  outputs, y ?
• Transformation of inputs to outputs is via the
  transformation function T(y,x) 0.

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Defining the Production Set

• Level set
• The Production function is defined by the
  isoquant
• The efficient subset is defined in terms of the
  level sets

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Isoquants and Level Sets
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The Distance Function
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Inefficiency
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Production Function Model with Inefficiency
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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 must be
  greater than g(y,w). Implies the idea of a cost
  frontier.
• lnC lng(y,w) u, u gt 0.

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Specification
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Corrected Ordinary Least Squares
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Modified OLS

• An alternative approach that requires a
  parametric model of the distribution of ui is
  modified OLS (MOLS). The OLS residuals, save for
  the constant displacement, are pointwise
  consistent estimates of their population
  counterparts, - ui. suppose that ui has an
  exponential distribution with mean ?. Then, the
  variance of ui is ?2, so the standard deviation
  of the OLS residuals is a consistent estimator of
  Eui ?. Since this is a one parameter
  distribution, the entire model for ui can be
  characterized by this parameter and functions of
  it. The estimated frontier function can now be
  displaced upward by this estimate of Eui.

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COLS and MOLS
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Deterministic Frontier Programming Estimators
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Estimating Inefficiency
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Statistical Problems with Programming Estimators

• They do correspond to MLEs.
• The likelihood functions are irregular
• There are no known statistical properties no
  estimable covariance matrix for estimates.
• They might be robust, like LAD. Noone knows
  for sure. Never demonstrated.

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A Model with a Statistical Basis
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Extensions

• Cost frontiers, based on duality results
• ln y f(x) u ?? ln C g(y,w) u
• u gt 0. u gt 0. Economies of scale and
• allocative inefficiency blur the relationship.
• Corrected and modified least squares estimators
  based on the deterministic frontiers are easily
  constructed.

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Data Envelopment Analysis
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Methodological Problems

• Measurement error
• Outliers
• Specification errors
• The overall problem with the deterministic
  frontier approach

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Stochastic Frontier Models

• Motivation
• Factors not under control of the firm
• Measurement error
• Differential rates of adoption of technology
• frontier is randomly placed by the whole
  collection of stochastic elements which might
  enter the model outside the control of the firm.
• Aigner, Lovell, Schmidt (1977), Meeusen, van den
  Broeck (1977)

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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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Least Squares Estimation

• Average inefficiency is embodied in the third
  moment of the disturbance ei vi - ui.
• So long as Evi - ui is constant, the OLS
  estimates of the slope parameters of the frontier
  function are unbiased and consistent. (The
  constant term estimates a-Eui. The average
  inefficiency present in the distribution is
  reflected in the asymmetry of the distribution,
  which can be estimated using the OLS residuals

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Application to Spanish Dairy Farms
N 247 farms, T 6 years (1993-1998)
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Example Dairy Farms
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The Normal-Half Normal Model
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Normal-Half Normal Variable
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Decomposition
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Standard Form
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Estimation Least Squares/MoM

• OLS estimator of ß is consistent
• Eui (2/p)1/2su, so OLS constant estimates a
  (2/p)1/2su
• Second and third moments of OLS residuals estimate

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A Problem with Method of Moments

• Estimator of su is m3/-.218011/3
• Theoretical m3 is lt 0
• Sample m3 may be gt 0. If so, no solution for su
  . (Negative to 1/3 power.)

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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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Alternative Model Exponential
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Normal-Exponential Likelihood
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Truncated Normal Model
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Normal-Truncated Normal
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Other Models

• Other Parametric Models (we will examine gamma
  later in the course)
• Semiparametric and nonparametric the recent
  outer reaches of the theoretical literature
• Other variations including heterogeneity in the
  frontier function and in the distribution of
  inefficiency

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Estimating ui

• No direct estimate of ui
• Data permit estimation of yi ßxi. Can this be
  used?
• ei yi ßxi vi ui
• Indirect estimate of ui, using Euivi ui
• vi ui is estimable with ei yi bxi.

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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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Other Distributions

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Efficiency
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Application Electricity Generation
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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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Confidence Region
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Application (Based on Costs)
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Multiple Output Frontier

• The formal theory of production departs from the
  transformation function that links the vector of
  outputs, y to the vector of inputs, x
• T(y,x) 0.
• As it stands, some further assumptions are
  obviously needed to produce the framework for an
  empirical model. By assuming homothetic
  separability, the function may be written in the
  form
• A(y) f(x).

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Multiple Output Production Function
Inefficiency in this setting reflects the failure
of the firm to achieve the maximum aggregate
output attainable. Note that the model does not
address the economic question of whether the
chosen output mix is optimal with respect to the
output prices and input costs. That would
require a profit function approach. Berger (1993)
and Adams et al. (1999) apply the method to a
panel of U.S. banks 798 banks, ten years.
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Duality Between Production and Cost
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Implied Cost Frontier Function
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Stochastic Cost Frontier
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Cobb-Douglas Cost Frontier
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Translog Cost Frontier
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Restricted Translog Cost Function
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Cost Application to CG Data
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Estimates of Economic Efficiency
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Duality Production vs. Cost
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Multiple Output Cost Frontier
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Allocative Inefficiency and Economic Inefficiency
Technical inefficiency Off the isoquant.
Allocative inefficiency Wrong input mix.
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Cost Structure Demand System
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Cost Frontier Model
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The Greene Problem

• Factor shares are derived from the cost function
  by differentiation.
• Where does ek come from?
• Any nonzero value of ek, which can be positive or
  negative, must translate into higher costs.
  Thus, u must be a function of e1,,eK such that
  ?u/?ek gt 0
• Noone had derived a complete, internally
  consistent equation system ?? the Greene problem.
• Solution Kumbhakar in several recent papers.
• Very complicated near to impractical
• Apparently not of interest to practitioners

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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 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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WHO Health Care Study
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Application WHO Data
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One vs. Two Step
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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 ApplicationBanking Costs
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Heteroscedasticity in v and/or u

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

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Application WHO Data
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A Scaling Model
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Model Extensions

• Simulation Based Estimators
• Normal-Gamma Frontier Model
• Bayesian Estimation of Stochastic Frontiers
• Similar Model Structures
• Similar Estimation Methodologies
• Similar Results

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Normal-Gamma
Very flexible model. VERY difficult log
likelihood function. Bayesians love it.
Conjugate functional forms for other model parts
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Normal-Gamma Model
z N-?i ?v2/?u, ?v2.
q(r,ei) is extremely difficult to compute
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Normal-Gamma
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Simulating the Likelihood
?i yi - ?Txi, ?i -?i - ?v2/?u, ? ?v, and
PL ?(-?i/?) and Fq is a draw from the
continuous uniform(0,1) distribution.
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Application to CG Data
This is the standard data set for developing and
testing Exponential, Gamma, and Bayesian
estimators.
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Application to CG Data
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Bayesian Estimation

• Short history first developed post 1995
• Range of applications
• Largely replicated existing classical methods
• Recent applications have extended received
  approaches
• Common features of the application

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Bayesian Formulation of SF Model
Normal Exponential Model
vi ui yi - ? - ?Txi. Estimation proceeds (in
principle) by specifying priors over ?
(?,?,?v,?u), then deriving inferences from the
joint posterior p(?data). In general, the joint
posterior for this model cannot be derived in
closed form, so direct analysis is not feasible.
Using Gibbs sampling, and known conditional
posteriors, it is possible use Markov Chain Monte
Carlo (MCMC) methods to sample from the marginal
posteriors and use that device to learn about the
parameters and inefficiencies. In particular,
for the model parameters, we are interested in
estimating E?data, Var?data and, perhaps
even more fully characterizing the density
f(?data).
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Estimating Inefficiency

• One might, ex post, estimate Euidata
  however, it is more natural in this setting to
  include (u1,...,uN) with ?, and estimate the
  conditional means with those of the other
  parameters. The method is known as data
  augmentation.

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Priors Over Parameters
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Priors for Inefficiencies
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Posterior
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Gibbs Sampling Conditional Posteriors
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Bayesian Normal-Gamma Model

• Tsionas (2002)
• Erlang form Integer P
• Random parameters
• Applied to CG
• River Huang (2004)
• Fully general
• Applied (as usual) to CG

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Bayesian and Classical Results
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Methodological Comparison

• Bayesian vs. Classical
• Interpretation
• Practical results Bernstein von Mises Theorem
  in the presence of diffuse priors
• Kim and Schmidt comparison (JPA, 2000)
• Important difference tight priors over ui in
  this context.
• Conclusions?