How Reliable is Duality Theory in Empirical Work?
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Transcript of How Reliable is Duality Theory in Empirical Work?
How Reliable is Duality Theory in Empirical
Work?
2016 AAEA Meetings, Boston MA
Francisco RosasUniversidad ORT Uruguay & Center for Economics Research-cinve
Sergio H. LenceIowa State University
August 1st, 2016
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 1 / 26
Background
Duality Theory
Neoclassical production theory establishes dual relationshipbetween profit/cost/revenue function and production function.
Given a profit function, its parameters appear (in a specific way)in the underlying or “consistent” production function.
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 2 / 26
Background
Duality Theory in Empirical WorkWe focus on the empirical applications of duality theory to estimateproduction parameters:
elasticities of substitutionprice elasticitieseconomies of scale/scope measures
It usually consists of:1 Profit/cost/revenue function approximation using parametric
functional form (NQ, TL, GL)
2 Derivation of input demand and output supply functions(Hotelling’s lemma)
3 Parameter estimation using netput prices & quantities data
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 3 / 26
Background
Objectives of our Research Agenda
Analyze whether theoretical duality relationships hold in practice
Show steps to construct a DGP by Monte Carlo simulation tomimic observed U.S. agriculture datasetsSome real-world features imply noise in data used for estimationWidely used datasets help calibration of noise with realisticlevels
Conclude about the extent to which duality theory recovers true(known) parameters of technology when typicaldata/econometric methods are applied
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 4 / 26
Background
Why is this important?
Price elasticities are widely used for decision-making
design of agricultural public policyfirm’s decision makingcomputing GDP and other macroeconomic aggregatesglobal agricultural models for projections (partial & generalequilibrium)
Up to our knowledge, this issue has not been explicitly addressedyet
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 5 / 26
Background
Outline
1 Background
2 Data Generating Process (DGP)ModelDGP StepsData
3 Econometric Estimation
4 Results
5 Conclusions
6 Fin
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 6 / 26
Data Generating Process (DGP) Model
ModelFirm’s problem:
max EU[W1] = max[yyy ] {EU[W0 + π]}= max[yyy ]
{EU[W0 + ppp′yyy + y0]
}= max[yyy ]
{EU[W0 + ppp′yyy − G (yyy ,KKK ;ααα)]
}Solution, expected quantities:
yyy ∗ = yyy(ppp,KKK ;βββ)
Restricted profit function:
πR = πR(ppp,KKK ;βββ)
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 7 / 26
Data Generating Process (DGP) Model
Model
Operationalize firm’s problem by using parametric functional forms:
Utility function U(W1): constant absolute risk aversion (CARA)
Production function G (yyy ,KKK ;α): quadratic in yyy and KKK
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 8 / 26
Data Generating Process (DGP) Model
Sources of Uncertainty
Firms face a probability distribution of quantities and prices duringproduction decision process.
Idiosyncratic output quantity shock: ψft = ψ(yft , vft)
Heteroskedastic: higher quantity → lower coefficient of variationψft between +/- 10% and 60% of output mean
Systematic output price shock: e
Deviation from firm-specific prices p∗ftLognormally distributed
Shocks calibrated to match features of real-world datasets
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 9 / 26
Data Generating Process (DGP) DGP Steps
Data Generation Process
Each dataset is a panel of 1.5 million “decision vectors”:F = 10,000 firms per region
R = 3 regions
T = 50 time periods (years)
Each decision vector [yyy ft |pppft ,KKK ft ,WWW 0,ft , λf ;aaaf ] composed of:8 variable netputs quantities and prices: yyy ft ,pppft
Set of production parameters aaaf and 1 quasifixed netput KKK ft
Initial wealth WWW 0,ft and risk aversion coefficient λf
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 10 / 26
Data Generating Process (DGP) DGP Steps
STEP 1: Unobserved Production Parameters: αααf
Randomly draw firm/region-specific production functionparameters
Calibrate unobserved firm heterogeneity
Moments of generated parameters heavily determine moments ofnetput quantities
Parameter size: impose correlation of parameters within the firmSkewness: non-symmetric Beta distribution (2007 U.S. Ag.Census data)Variation/heterogeneity across firms:
Yield dispersion not attributable to weather shocksFixed-effects regression (ARMS and PRISM panels)
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 11 / 26
Data Generating Process (DGP) DGP Steps
STEP 2.1: (Endogenous) Netput Prices ppp∗t
Endogenous prices for noisy dataset: time-specific “national”netput prices (ppp∗t )
Netput quantities at aggregate level affect price (endogeneity)
Farmers face an aggregate market for inputs/outputsppp∗t implicitly solves aggregate demand = aggregate supply
ΦΦΦtpppηt = FFFXXXpppt + FFFϕϕϕ
Shocks from market (ΦΦΦt) induce time variation of netput pricesSolution prices described by an AR(1) process (calibrated tomatch CME and Eldon Ball’s price datasets)
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 12 / 26
Data Generating Process (DGP) DGP Steps
STEP 2.2: Firm-specific Prices: ppp∗ft , ppp∗∗ft
Heterogenous firms face different prices
Deviations from average prices
Randomly draw F × R firm prices, such that:
Mean preserving spread from ppptBeta distribution (symmetric)Independent draws to favor identification (prices not correlatedwith firm size)Result: More variation than ARMS data and “lessconcentrated”
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 13 / 26
Data Generating Process (DGP) Data
Simulated Dataset, IDataset features real-world characteristics of information available topractitioners.Sources of noise calibrated realistically and favoring recoveryGenerated with 6 sources of noise:
Source 1: Solve firm’s expected utility max problem in each time t
Given:
ppp∗ft , KKK∗ft , W0,ft , aaaf ,
quadratic production function,coefficient of relative risk aversion λf ∼ U[2, 4], anddistribution of price and quantity shocks faced by firms
Method: Gaussian quadratures
Solution, expected quantities: yyy ∗ftResulting dataset → [yyy ∗ft ,ppp
∗ft ,KKK
∗ft ]
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 14 / 26
Data Generating Process (DGP) Data
Simulated Dataset, II
Source 2: Realized shocks of production and prices
Draw from ψft and e distributions and apply to Step 1 result
Source 3: Measurement error in variables
Calibrated as deviation from “true” value of price and quantity
Standard deviation from literature
Source 4: Omitted variables
Delete one output and one input
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 15 / 26
Data Generating Process (DGP) Data
Simulated Dataset, III
Source 5: Aggregated inputs and outputs
Aggregate two outputs into one
Aggregate two inputs into one
Revenue weighting average for price and quantity aggregation
Source 6: Firm aggregation
Aggregate across heterogeneous firms
Consistent with objective of analyzing duality theory intime-series estimation
Steps 1-6 result in dataset [yyy t ,pppt ,KKK t ]
Proceed to estimation
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 16 / 26
Econometric Estimation
Estimation I
Objective: estimate profit function parameters & computeelasticities
Approximate a Normalized Quadratic profit function
Derive supplies and demands system (Hotelling’s Lemma)
Estimation using dataset of only one region (R=1)
Noisy data: 100 samples of 6,000 firms each
Using the corresponding dataset, estimate parameters byiterated seemingly unrelated regression (SUR)
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 17 / 26
Econometric Estimation
Estimation II
Sources of noise treated in estimation:
Serial autocorrelation of errors: series in first differencesOmitted variables: IV approachPrice endogeneity: IV approach
Solution: profit function Hessian
Noisy data: [BBB]
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 18 / 26
Econometric Estimation
True versus Estimated Elasticities
Objective: compare estimated elasticities with true values
[EEE ]: Supply & demand elasticities with respect to (own- and cross-) pricesand quasi-fixed netputs
True elasticities [EEE ]f : firm-specific matrix of elasticities; i.e. a distribution
Estimated elasticities [EEE ] or [EEE ]: a single matrix
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 19 / 26
Results
Results - Simulated-Data EstimationEntries of the 4x4 price elasticities matrix (true vs. estimated)
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 20 / 26
Results
Results - Simulated-Data EstimationEntries of the 4x4 price elasticities matrix (true vs. estimated)
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 21 / 26
Results
Sensitivity Analysis - Simulated-Data Estimation, I
Summary of elasticity matrixNetput elasticities wrt. prices & quasi-fixed netputs (true vs.estimated)
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 22 / 26
Results
Sensitivity Analysis - Simulated-Data Estimation, II
Tradeoff: Increase sample size vs. increase firm heterogeneity
Estimation data: regions 1, 2, and 35 samples of 2,000 firms each, in each regionAggregate across heterogeneous firms of 3 regionsPool data: 750 observations (vs. 50 observations)Regional dummy variables
Qualitatively similar results
Estimated elasticities: 53% deviated from true values
Range: range 11% and 209%
Higher t-statistics
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 23 / 26
Conclusions
Conclusions
Showed steps to generate dataset that mimics key features ofreal-world data available to researchers
Evaluated duality theory econometrics that aims to recoverproduction parameters
Application: price elasticities using U.S. agricultural time-seriesdata
Concluded that for most elasticities duality approach yieldsbiased results
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 24 / 26
Conclusions
Future Research
Make simulated dataset publicly available
Evaluate other applications of duality theory with the simulatedpanel dataset:
Dig deeper into the contribution to estimation bias of eachsource of noise, to guide identification alternativesFocus on estimating the representative technology employingdifferent aggregation methods of technologically heterogeneousfarmersEmpirical performance of Duality with cross-sectional data
Each may be regarded as a stand alone paper
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 25 / 26
Fin
Rosas (ORT & cinve) Duality Theory Econometrics August 1st, 2016 26 / 26