Top

Empirical Economics

Published in:

01-12-2014

Imposing curvature conditions on flexible functional forms for GNP functions

Authors: Guy Chapda Nana, Bruno Larue

Published in: Empirical Economics | Issue 4/2014

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

This paper empirically investigates the implications of the choice of dates when imposing local restrictions to maintain convexity in output prices and concavity in factor endowments in the the estimation of a GNP function approximated by a translog (TL) function. Using macroeconomic data for Switzerland, we compare the TL to a symmetric normalized quadratic (SNQ) function on which global curvature restrictions can be imposed. When both functions are unrestricted for curvature, convexity in prices is violated more often than concavity in factor endowments. The number of points for which both curvature conditions hold is sensitive to the date at which local restrictions are imposed on the TL, but estimated TL elasticities are robust. Through searching for an appropriate date, the TL matched the SNQ’s ability to impose curvature conditions at all points. However, many TL and SNQ elasticities differ in sign and magnitude. The likelihood dominance criterion and in-sample forecasts comparisons favored the TL. Thus, choosing a functional form solely based on the possibility of imposing global curvature conditions is not advised.

previous article Fractional integration and cointegration in US financial time series data

next article Trade facilitation and country size

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

The gross national product (GNP) function has two sets of curvature restrictions as it is convex in product prices and concave in factor endowments. Other functions have one set of restrictions: the expenditure, cost, and input distance functions are respectively concave in product prices, input prices, and inputs, while the profit function is convex in product and input prices, and the indirect utility function is quasi-convex in product prices.

The Minflex Laurent proposed by Barnett (1983) is locally flexible, but not globally regular. The generalized Barnett is globally regular and locally quasiflexible (Diewert and Wales 1987) while the Fourier estimated by Gallant and Golub (1984) is globally flexible, but attaining regularity locally is difficult and the very large number of parameters may cause overfitting. The asymptotically ideal model (AIM) of Barnett and Yue (1998) is globally regular at all orders of approximation, but inference is problematic.

Diewert and Wales (1987), Fisher et al. (2001), Feng and Serletis (2008) and Serletis and Shahmoradi (2007) have relied on a classical estimation framework. Most functional form comparisons have

focused on consumer demand (e.g., Barnett and Serletis 2008) or cost functions (e.g., Griffiths et al. 2000; Feng and Serletis 2008). Work on profit functions is rarer, but Diewert and Ostensoe (1988) show that global curvature conditions can be imposed without destroying flexibility.

However, Tombazos (2003b) proposes a new methodology of enforcing curvature indirectly in the case of the translog variable profit function when curvature fails in the first round of estimations.

Our review of the literature shows that the TL and SNQ functional forms are by far the most common when it comes to the estimation of GNP functions. We conjecture that the small number of observations typically available for the estimation of GNP functions makes more complex functional forms less appealing. The degrees of freedom problem are most obvious at low levels of product aggregation. Kee et al. (2008, p. 670) invoke the short time span of data and the large number of explanatory variables to limit substitution patterns and the number of coefficients to estimate. In other areas, like efficiency frontier analysis, the Cobb–Douglas is common.

Switzerland is a small and stable open economy. This simplifies modeling because output prices can be treated as exogenous, and structural change is not a concern. This contrasts with Canada which embarked on a long transition when it began negotiating a free trade agreement with the United States (Chapda Nana et al. 2012).

Wong (1995)

Wiley et al. (1973) showed that a sufficient condition for matrix \(\mathbf{A}\) to be negative semi-definite is \(\mathbf{A}=\mathbf{-TT}^{\mathbf{{\prime }}},\) where T is a lower triangular matrix. Similarly, Diewert and Wales (1987) show that A is positive semi-definite if it can be written as \(\mathbf{A}=\mathbf{TT}^{{{\prime }}}.\)

With the condition \(\sum a_{{ik}} = 0\), the last element in each row of matrix A is a linear combination of the remaining elements in that row. Owing to the symmetry of A, we can obtain all its elements simply by estimating a (\(I-1\times I-1)\) matrix where we had deleted the last row and the last column. Finally, with linear algebra, we know that if A is semi-positive definite its leading principal sub-matrix is also semi-positive definite. The same reasoning is applied to B.

The negative sign associated with the import share is explained by the treatment of imports as intermediate inputs.

This estimator was also used by Kohli (1978, 1992), Kohli and Werner (1998), and Sharma (2002).

Holt (1998) compared the Berndt and Savin one-parameter specification to more flexible ones. Surprisingly, the one-parameter specification produced a higher likelihood than some alternatives, but the application of the LDC criterion confirms the most flexible specification as dominant even though differences in estimated demand coefficients are small.

In Shazam, the form of the \(R\) matrix can be specified by changing the option of the command nl for “auto” for a diagonal matrix with the same coefficient or for “dhro” for a diagonal matrix with different terms or for “across” for a full rank matrix.

The system \(R^{2}\) is computed as in McElroy (1977).

The model was estimated 38 times, corresponding to the local imposition of properties at each date of our sample. Results reported in Table 2 are those obtained for 2000, one of the best dates of imposition.

We chose to present the results for the period 1970–1980 because the results for 1981–2007 provide similar insights. Naturally, they are available upon request.

We do not present the analysis for the case where theoretical properties are imposed at different dates because the results are qualitatively the same.

A more complete and supplementary discussion of elasticities is proposed in the next subsection when comparing elasticities for the TL and SNQ.

We conducted a matched pairs test of the mean difference for each elasticity with curvature restrictions imposed locally at 1978 and 2000.

We also looked at the evolution of the elasticities over time and generally, the estimated elasticities are significant, have the expected sign and tend to be larger in absolute value near the end of the sample.

The LDC as a mean to rank functional forms was brought to our attention by a referee.

In demand system estimation for example, MSE and MAE criteria have been used by Kastens and Brester (1996) for forecasting performance comparison while Wang and Bessler (2003) used Diebold–Mariano test for predictive accuracy assessment.

The main advantage of this test is that it can be applied with many loss functions, and few hypotheses about the asymptotic distribution of forecast errors are needed.

Barnett WA (1983) New Indices of money supply and the flexible laurent demand system. J Bus Econ Stat 1(1):7–23

Barnett WA, Geweke J, Yue YP (1991) Seminonparametric bayesian estimation of the asymptotically ideal model: the AIM demand system. In: Barnett W A, Powell J, Tauchen G (eds) Nonparametric and semiparametric methods, 127-174. Cambridge University Press, Cambridge

Barnett WA, Yue P (1998) Semiparametric estimation of the asymptotically ideal model: the AIM demand system. In: Rhodes G, Formby T (eds) Nonparametric and robust inference, advanced in econometrics 7. JAI, Greenwich

Barnett WA, Serletis A (2008) Consumer preferences and demand systems. J Econom 147(2):210–224CrossRef

Berndt ER (1991) The practice of econometrics: classic and contemporary. Addison Wesley, Reading

Berndt ER, Savin NE (1975) Estimation and hypothesis testing in singular equation system with autoregressive disturbances. Econometrica 43(5–6):937–957CrossRef

Chalfant J, Gray R, White K (1991) Evaluating prior beliefs in a demand system: the case of meat demand in Canada. Am J Agric Econom 73(2):476–490CrossRef

Chapda Nana G, Larue B, Gervais J-P (2012) Regional integration and dynamic adjustments: evidence from gross national product functions for Canada and the United States. N Am J Econom Finance 23(2):246–264CrossRef

Christensen LR, Jorgenson DW, Lau LJ (1970) Conjugate duality and the transcendental logarithmic production function. Unpublished paper presented at the Second World Congress of the Econometric Society, Cambridge, England

Deaton AS, Muellbauer JN (1980) An almost ideal demand system. Am Econom Rev 70(3):312–326

Diebold FX, Mariano RS (1995) Comparing predictive accuracy. J Bus Econom Stat 13(3):253–263

Diewert WE (1974) Applications of duality theory. In: Intriligator M, Kendrick D (eds) Frontiers in quantitative economics. North-Holland, Amsterdam

Diewert WE, Wales TJ (1987) Flexible functional forms and global curvature conditions. Econometrica 55(1):43–46CrossRef

Diewert WE, Ostensoe L (1988) Flexible functional forms for profit functions and global curvature conditions. In: Barnett W, Berndt ER, White H (eds) Dynamic econometric modeling. Cambridge University Press, Cambridge

Feng G, Serletis A (2008) Productivity trends in US manufacturing: evidence from the NQ and AIM cost function. J Econom 142(1):281–311CrossRef

Fisher D, Fleissig AR, Serletis A (2001) An empirical comparison of flexible demand system functional forms. J Appl Econom 16(1):59–80CrossRef

Gallant AR, Golub GH (1984) Imposing curvature restrictions on flexible functional forms. J Econom 26(3):295–321CrossRef

Griffiths WE, O’Donnell CJ, Cruz AT (2000) Imposing regularity conditions on a system of cost and factor share equations. Aust J Agric Resour Econom 44(1):107–127CrossRef

Harvey D, Leybourne S, Newbold P (1997) Testing the equality of prediction mean squared errors. Int J Forecast 13(2):281–291CrossRef

Heady, EO and Dillon, JL (1961) Agricultural production function. Iowa State University Press, Ames

Holt MT (1998) Autocorrelation specification in singular equation systems: a further look. Econom Lett 58(2):135–141CrossRef

Kastens TL, Brester GW (1996) Model selection and forecasting ability of theory-constrained food demand systems. Am J Agric Econom 78(2):301–312CrossRef

Kee HL, Nicita A, Olarreaga M (2008) Import demand elasticities and trade distortions. Rev Econom Stat 90(4):666–682CrossRef

Kohli U (1978) A gross national production function and the derived demand for imports and supply of exports. Can J Econom 11(2):167–182CrossRef

Kohli U (1993) A symmetric normalized quadratic GNP function and the US demand for imports and supply of exports. Int Econom Rev 34(1):243–255CrossRef

Kohli U (1992) Production, foreign trade, and global curvature conditions : Switzerland, 1948–1988. Swiss J Econom Stat 128(1):3–20

Kohli U (1991) Technology, duality, and foreign trade: the GNP function approach to modeling imports and exports. University of Michigan Press, Ann Arbor

Kohli U, Werner A (1998) Accounting for South Korean GDP growth: index number and econometrics estimates. Pac Econom Rev 3(2):133–152CrossRef

Larivière E, Larue B, Chalfant J (2000) Modeling the demand for alcoholic beverages and advertising specifications. Agric Econom 22(2):147–162CrossRef

Lau LJ (1978) Testing and imposing monotonicity, convexity and quasiconvexity constraints. In: Fuss M, McFadden D (eds) Production economics: a dual approach to theory and applications. North Holland, Amsterdam, pp 409–453

Lau LJ (1986) Functional forms in econometric model building. In: Griliches Z, Intriligator M (eds) Handbook of Econometrics vol 3. North Holland, Amsterdam, pp 1515–1566

McElroy MB (1977) Goodness of fit for seemingly unrelated regressions: glahn’s R2y. x and Hooper’s \(\bar{{\rm r}}2\). J Econom 6(3):381–387

Morey ER (1986) An introduction to checking, testing, and imposing curvature properties: the true function and the estimated function. Can J Econom 19(2):207–235CrossRef

Moschini G (1999) Imposing local curvature conditions in flexible demand systems. J Bus Econom Stat 17(4):487–490

Pollak RA, Wales TJ (1991) The likelihood dominance criterion: a new approach to model selection. J Econom 47(2–3):227–242CrossRef

Ryan DL, Wales TJ (2000) Imposing local concavity in the translog and generalized Leontief cost function. Econom Lett 67(3):253–260CrossRef

Ryan DL, Wales TJ (1998) A simple method for imposing local curvature in some fexible consumer-demand systems. J Bus Econom Stat 16(3):331–338

Serletis A, Shahmoradi A (2007) Flexible functional forms, curvature conditions, and the demand for assets. Macroeconom Dyn 11(4):455–486CrossRef

Sharma SC (2002) The Morishima elasticity of substitution for the variable profit function and the demand for imports in the United States. Int Econom Rev 43(1):115–135CrossRef

Terrell D (1996) Incorporating monotonicity and concavity conditions in flexible functional forms. J Appl Econom 11(2):179–194CrossRef

Tombazos CG (2003a) A production theory approach to the imports and wage inequality nexus. Econom Inq 41(1):42–61CrossRef

Tombazos CG (2003b) Curvature enforcement and the tenuous flexibility of the transcendental logarithmic variable profit function. Discussion Papers, Department of Economics, Monash University

Wang Z, Bessler DA (2003) Forecast evaluations in meat demand analysis. Agribusiness 19(4):505–523CrossRef

Wong KKG (2004) A new approach to modeling GNP functions: an application of two-stage technologies and modifying function approach. Aust Econom Pap 43(3):315–339CrossRef

Wong KY (1995) International trade in goods and factor mobility. MIT Press, Cambridge

Wiley DE, Schmidt WH, Bramble WJ (1973) Studies of a class of covariance structures models. J Am Stat Assoc 68(342):317–323CrossRef

Zellner A (1962) An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. J Am Stat Assoc 57(298):348–368CrossRef

Title: Imposing curvature conditions on flexible functional forms for GNP functions
Authors: Guy Chapda Nana
Bruno Larue
Publication date: 01-12-2014
Publisher: Springer Berlin Heidelberg
Published in: Empirical Economics / Issue 4/2014
Print ISSN: 0377-7332
Electronic ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-013-0784-4

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft"

Springer Professional "Wirtschaft+Technik"

Other articles of this Issue 4/2014

Succeeding in innovation: key insights on the role of R&D and technological acquisition drawn from company data

Trade facilitation and country size

Export and import market-specific characteristics

Forecasting spatially dependent origin and destination commodity flows

On estimation of almost ideal demand system using moving blocks bootstrap and pairs bootstrap methods

Bank competition in transition countries: Are those markets really in equilibrium?

Premium Partner