Regression and Inference Under Smoothness Restrictions Christopher F. Parmeter 1 Kai Sun 2 Daniel J. Henderson 3 Subal C. Kumbhakar 4 1 Department of Agricultural and Applied Economics Virginia Tech 2,3,4 Department of Economics Binghamton University The 44th. Annual Conference of the CEA
Introduction Research Question Motivation How to impose generalized economic constraints on parametric model for the estimation of a single/multiple-output technology? Most of the applied econometrics literature on the estimation of technology ignores the imposition of economic constraints. Negative marginal products or technical regress Various methods developed to impose constraints
Research Question Motivation Motivation Production Function Example Cobb-Douglas ln Y = α + β K ln K + β L ln L + u ln Y / ln K = β K 0 MPK 0 Translog ln Y =α + β K ln K + β L ln L + 0.5β KK (ln K) 2 + 0.5β LL (ln L) 2 + β KL ln K ln L + u ln Y / ln K = β K + β KK ln K + β KL ln L 0 MPK 0 ln Y / ln K is now observation-specific.
Motivation Introduction Research Question Motivation Recent work on constrained regression Hall and Huang (2001) O Donnell, Rambaldi and Doran (2001) Racine, Parmeter and Du (2009) Henderson and Parmeter (2009) What is new? Complement O Donnell et al. (2001) with classical approach Extend Racine et al. (2009) to any linear estimator
A Simple Simulation Graphical Illustration Data Generating Process: y = 10 + 3x + x 2 3x 3 + x 4 + u where x iidu[0, 2.5] u iidn(0, 0.01) Estimate ŷ = ˆβ 0 + ˆβ 1 x + ˆβ 2 x 2 + ˆβ 3 x 3 + ˆβ 4 x 4 Constraint: ŷ/ x 0.
Without Constraint Introduction A Simple Simulation Graphical Illustration
With Constraint Introduction A Simple Simulation Graphical Illustration
Constrained Regression: Estimation Constrained Regression: Estimation Constrained Regression: Inference Idea: Transform the response so that certain equality or inequality constraints are satisfied. Y = Xβ + u; Ŷ = Xb, where X = [1 x x 2 x 3 x 4 ] The unconstrained gradient: Ŷ(X)/ x = ( X/ x) b = [0 1 2x 3x 2 4x 3 ] (X X) 1 X Y n = x (X X) 1 X Y = A(x)Y = A i (x)y i The constrained gradient: i=1 Ŷ(X p)/ x = n i=1 A i(x)p i Y i = n i=1 A i(x)y i 0.
Constrained Regression: Estimation Constrained Regression: Inference Constrained Regression: Estimation Weight selection criterion p that minimizes D(p) = (p u p) (p u p) subject to Ŷ(X p)/ x 0 p: a weighting vector for the response [p 1, p 2,..., p n ] p u : a uniform weighting vector If p = p u, constraints will be non-binding.
Constrained Regression: Estimation Constrained Regression: Inference Constrained Regression: Inference Null hypothesis: Constraints are valid. A bootstrap approach: 1. Estimate the model under constraints, get D(ˆp), Ŷ(X ˆp), û; 2. Bootstrap û; obtain u ; 3. Create a new response, Y = Ŷ(X ˆp) + u ; 4. Estimate the model using the new sample under constraints, obtain D(ˆp ); 5. Repeat step 2 to 4 (B) times, calculate ˆP B = 1 B B b=1 I(D(ˆp ) D(ˆp)).
Econometric Model Data Results Future Work Multiple-output Technology Cannot use a production function Y: output vector (Q 1); X : input vector (K 1) Start with transformation function f (Y, X, t) = 1 or ln f (Y, X, t) = 0 ln f (Y, X, t) f (ln Y, ln X, t)
Econometric Model Data Results Future Work Multiple-output Technology Impose the restriction of homogeneous of degree 1 in inputs, and use X 1 (the first input) as the normalizing input: ln f (Y, X, t) ln X 1 f (ln Y, ln X, t) where ln X = ln(x /X 1 ) is a (K 1)-vector Estimate: ln X 1 = f (ln Y, ln X, t, id ) + u where id is a categorical variable for state effects f ( ): either a Translog (parametric) or an unknown smooth function (kernel-based nonparametric as in Li and Racine (2006)).
Econometric Model Data Results Future Work Multiple-output Technology Input distance function Shephard (1953) Färe and Primont (1995) Kumbhakar and Lovell (2000) Economic constraints: ln X 1 / ln X k 0 ( k = 2,..., K) ln X 1 / ln Y q 0 ( q = 1,..., Q) ln X 1 / t 0
Econometric Model Data Results Future Work Data: United States Department of Agriculture Table: Summary Statistics of the Variables Symbol Variable Name Mean Sd. Min. Max. X 1 Capital 638932 579797 7351 3330621 X 2 Land 681420 734737 4015 4659258 X 3 Labor 1616318 1409715 18189 8450988 X 4 Intermediate inputs 1915233 1753558 12917 9451845 Y 1 Livestock 1813097 1697585 9101 8497604 Y 2 Crop 2355130 2695164 25216 19386468 Y 3 Agricultural services 237087 309676 981 26603678 1. All of the variables are measured as real index numbers. 2. The sample consists of 1200 observations (for 48 states and 25 yearly data (1980 2004) for each state).
Econometric Model Data Results Future Work Estimation Results (Translog)
Econometric Model Data Results Future Work Estimation Results (Nonparametric)
Econometric Model Data Results Future Work Hypothesis Testing Seven joint monotonicity constraints Technical progress Constant returns to scale
Econometric Model Data Results Future Work Future Work Impose non-linear constraints on linear estimator Impose linear constraints on a seemingly-unrelated regression (SUR) system Extend the methodology to non-linear estimator