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Estimation and inference under economic restrictions


Estimation of economic relationships often requires imposition of constraints such as positivity or monotonicity on each observation. Methods to impose such constraints, however, vary depending upon the estimation technique employed. We describe a general methodology to impose (observation-specific) constraints for the class of linear regression estimators using a method known as constraint weighted bootstrapping. While this method has received attention in the nonparametric regression literature, we show how it can be applied for both parametric and nonparametric estimators. A benefit of this method is that imposing numerous constraints simultaneously can be performed seamlessly. We apply this method to Norwegian dairy farm data to estimate both unconstrained and constrained parametric and nonparametric models.

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  1. 1.

    See Ajgaonkar (1965) for a formal definition of linear estimators.

  2. 2.

    See Henderson and Parmeter (2009) for a survey of various methods (including this one) to impose constraints on nonparametrically estimated regression surfaces.

  3. 3.

    The recent work of Sauer (2006) also advocated for checking and testing for theoretical consistency, albeit in a different setting.

  4. 4.

    This is not always a bad strategy as Ryan and Wales (2000) have shown that imposing a constraint at a single point may result in that constraint being satisfied at many points.

  5. 5.

    It is feasible (but difficult computationally) to impose concavity in a linear in p setup using the ‘Afriat’ conditions (Matzkin 1994); see Parmeter and Racine (2012) for details.

  6. 6.

    To appreciate why it may prove necessary to allow for negative weighting, suppose we simply wished to constrain a surface that has both positive and negative regions to be uniformly positive. This could be accomplished by allowing some of the weights to be negative. However, probability weights would fail to produce a feasible solution as they are all non-negative.

  7. 7.

    A referee has correctly noted the tight link between CWB and empirical likelihood. However, in that setup one minimizes \(\sum\nolimits^n_{i=1}\log\left(p_i/p_u\right)\) subject to a set of moment conditions. CWB as proposed here with a quadratic norm makes solving the constrained problem easier to implement and offers (potentially) extra flexibility to satisfy the constraints by allowing p i  < 0.

  8. 8.

    Other useful economic constraints that satisfy (in)equalities that are linear in p include additive separability, homogeneity, diminishing marginal returns/products, and bounding of derivatives of any order.

  9. 9.

    Code in the R language is available from the authors upon request.

  10. 10.

    This methodology also works well with the general order local polynomial estimator as well as with spline regression. We leave a full treatment of this for future research.

  11. 11.

    Both Hall and Huang (2001) and Henderson and Parmeter (2009) argue that for the most part, the choice of distance metric has relatively little influence on the weights and the resulting estimates.

  12. 12.

    Note that X and Y here does not denote regressor and regressand, respectively.

  13. 13.

    See Färe and Primont (1995) and Kumbhakar and Lovell (2000) for more details on the properties of distance functions.

  14. 14.

    Nomenclature of Territorial Units for Statistics (NUTS) is a geocode standard, developed and regulated by the European Union, for referencing the subdivisions of countries for statistical purposes.

  15. 15.

    See “Appendix” for the estimates of the 65 parameters from our translog model.

  16. 16.

    Our unbalanced panel includes n ≡ ∑ N i=1 T i  = 4,333 total observations and we want to impose nine inequality constraints. In cases such as this, a relatively powerful computer in terms of speed and memory space is needed. For example, with a single constraint, the linear regression estimator is expressed as a n × n dimensional matrix. Our problem required three gigabytes of memory for the parametric model and four gigabytes for the nonparametric model.

  17. 17.

    We use the Silverman reflection method to construct the densities for the constrained gradients.

  18. 18.

    We will consider more formal comparisons in Sect. 5.3.

  19. 19.

    We are also interested in testing the null that the constrained translog model is correctly specified. Unfortunately, it is unclear how to perform the HLR test in this scenario. We expect that this model will also be rejected, as the deviations from the unconstrained model are minimal, but we do not make a formal claim at this time.


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The idea of using constraint weighted bootstrapping for parametric models came directly from Jeffrey S. Racine and we acknowledge his insights here.

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Correspondence to Subal C. Kumbhakar.

Appendix: Translog parameter estimates

Appendix: Translog parameter estimates

The parameter estimates for the unconstrained translog model with fixed effects are given in Table 5. Excluding the farm effects, we first note that 33 out of the possible 65 parameter estimates (excluding intercepts) are significant at the 10 % level. We also note that the fit of the model is very high with a squared correlation coefficient between the actual and fitted values equal to 0.9845.

Table 5 Translog with state effects: unrestricted versus restricted

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Parmeter, C.F., Sun, K., Henderson, D.J. et al. Estimation and inference under economic restrictions. J Prod Anal 41, 111–129 (2014).

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  • Restrictions
  • Equality
  • Inequality
  • Constraint weighted bootstrapping
  • Linear regression estimators

JEL Classification

  • C12
  • C13
  • C14