Representation Theorem for Convex Nonparametric Least Squares. Timo Kuosmanen

Transcription

1 Representaton Theorem or Convex Nonparametrc Least Squares Tmo Kuosmanen 4th Nordc Econometrc Meetng, Tartu, Estona, 4-6 May 007

2 Motvaton Inerences oten depend crtcally upon the algebrac orm chosen. It s not uncommon to nd that alternatve orms t the data almost equally well but have very derent mplcatons Ths problem would arse more requently n economc lterature t were gven attenton commensurate wth ts mportance. Many economsts uncrtcally accept the approprateness o a unctonal orm chosen largely on the bass o conventon or convenence; others try several orms or ther relatons but report only the one that n some sense looks best a posteror. Hldreth, C. (1954): Pont Estmates o Ordnates o Concave Functons, Journal o the Amercan Statstcal Assocaton 49(67),

3 Convex Nonparametrc Least Squares (CNLS) a.k.a. convex regresson or nonparametrc least squares subject to shape costrants Reles on the regularty propertes o the mcroeconomc theory (monotoncty, convexty + homogenety) Does not requre arbtrary assumptons about the unctonal orm or the smoothness o the regresson uncton

4 CNLS n Statstcs Hldreth, C. (1954): Pont Estmates o Ordnates o Concave Functons, Journal o the Amercan Statstcal Assocaton 49(67), Hanson, D.L., and G. Pledger (1976): Consstency n concave regresson. Annals o Statstcs 4(6), Dykstra, R.L. (1983): An algorthm or restrcted least squares regresson, Journal o the Amercan Statstcal Assocaton 78, Nemrovsk, A.S., B.T. Polyak, and A.B. Tsybakov (1985) Rates o Convergence o Nonparametrc Estmates o Maxmum Lkelhood Type, Problems o Inormaton Transmsson 1, Mammen, E. (1991): Nonparametrc regresson under qualtatve smoothness assumptons. Annals o Statststcs 19, Meyer, M.C. (1999) An Extenson o the Mxed Prmal-Dual Bases Algorthm to the Case o More Constrants than Dmensons, J. Statstcal Plannng and Inerence 81, Mammen, E., and C. Thomas-Agnan (1999): Smoothng splnes and shape restrctons, Scandnavan Journal o Statstcs 6, Groeneboom, P., G. Jongbloed, and J.A. Wellner (001): Estmaton o convex unctons: characterzatons and asymptotc theory, Annals o Statstcs 9, Meyer, M.C. (003) A Test or Lnear vs. Convex Regresson Functon usng Shape- Restrcted Regresson, Bometrka 90(1), 3 3. Meyer, M.C. (006) Consstency and Power n Tests wth Shape-Restrcted Alternatves, Journal o Statstcal Plannng and Inerence 136,

5 Advantages o CNLS Draws ts power rom the standard regularty condtons o the mcroeconomc theory Monotoncty, concavty/convexty, homogenety Conceptually and pedagogcally smple No unctonal orm or smoothness assumptons

6 Major barrers o applcaton lack o explct regresson uncton computatonal complexty dculty o statstcal nerence

7 Purpose o ths study Derve a contnuous representor uncton that helps us to lower each o these three barrers. Explct estmator o margnal propertes orecastng ex post economc modelng Computaton by quadratc programmng (QP) Operatonal multple regresson ormulaton Statstcal nerence by bootstrappng

8 CNLS model Regresson model: y = ( x ) + ε, = 1,..., n Assumptons: Functon belongs to the amly o monotonc ncreasng and globally concave unctons F. Errors ε are uncorrelated random varables wth E(ε)=0 and Var( ε ) = σ < = 1,..., n

9 CNLS problem n mn ( y ( x )) = 1 s. t. F

10 CNLS vs. OLS CNLS problem OLS as quadratc programmng mn ( y ( x )) s. t. n = 1 F mn ( y ( α + β x )) α, β n = 1

11 CNLS vs. OLS CNLS problem OLS as quadratc programmng mn ( y ( x )) s. t. n = 1 F mn ( y ( α + β x )) α, β n = 1 s. t. β 0 = 1,..., n; α + β x α + β x h, = 1,..., n h h

12 Representaton theorem Innte dmensonal problem s mn ( y ( x )) s. t. n = 1 F Quadratc programmng problem s g mn ( y ( α + β x )) α, β n = 1 s. t. β 0 = 1,..., n; α + β x α + β x h, = 1,..., n h h

13 Representaton theorem Innte dmensonal problem s mn ( y ( x )) s. t. n = 1 F Quadratc programmng problem s g mn ( y ( α + β x )) α, β n = 1 s. t. β 0 = 1,..., n; α + β x α + β x h, = 1,..., n h h s = s g

14 Representor uncton CNLS representor s always a pecewse lnear uncton o orm αˆ 1 + βˆ 1 x, αˆ + βˆ x,, gˆ( x) = mn... αˆ + ˆ n βn x, whereα^, β^ are estmated CNLS coecents. Functon g^ s one o the optmal solutons to the orgnal, nnte dmensonal CNLS problem

15 Margnal propertes substtuton rate between varables k and m : gˆ( x ) gˆ( x ) x x = ˆ β ˆ β k k m m Substtuton elastcty: e k, m ( x ) ˆ β = ˆ k β m x x m k

16 Smulated example True regresson uncton: (x) = ln(x) + 1 Random sample o 100 observatons o the x values rom Un[1,11], Observed y values perturbed by addng a random error term drawn ndependently rom N(0,0.6 ).

17 Smulated example y observatons true g representor o CNLS log-lnear OLS curve x

18 Smulated example y observatons true g representor o CNLS log-lnear OLS curve x

19 Monte Carlo smulaton Three derent speccatons or the true regresson uncton Cobb-Douglas Generalzed Leonte Pece-wse lnear Three derent levels o error varance Sample sze smulatons n each scenaro

20 Mean Squared Error (MSE) True uncton Std. dev. CNLS estm. Cobb-Douglas estm Translog estm. 1) Cobb-Douglas ) Gener. Leonte ) Pece-wse lnear

21 Applcatons o CNLS? Consumer demand analyss Combne the parametrc demand system estmaton technques wth the nonparametrc revealed preerence approach. Producton analyss Combne the parametrc stochastc ronter analyss (SFA) and the nonparametrc data envelopment analyss (DEA) nto a uned ramework o ronter estmaton. Envronmental economcs CNLS enables one to estmate benet, damage and abatement cost unctons n a nonparametrc ashon.

22 Thank you or your attenton! Workng paper avalable by request Questons / comments are welcome to E-mal: Tmo.Kuosmanen@mtt.