Lecture 10 Splines Introduction to Splines

Transcription

1 ES8 Economerics. Inroducion o Splines. Esiming Spline Mehod One. Esiming Spline Mehod To.4 Cuic Splines Lecure Splines. Inroducion o Splines Firs pplied Poirier & Grer (974) in sud of profi res in hree disinc periods. The reed ime mens of liner spline: Period Period Period u u u Figure. We migh perform hree sepre regressions on his d nd his resul ould look like Figure.. There is nohing in he unresriced esimion process o ensure h he funcions mee he join poins (or knos) nd. Figure. illusres liner spline, or pieceise liner funcion, hich elimines insnneous jumps or disconinuiies in he funcion he join, or kno. Figure. Figure. Here he is is ime. I need no e nd one cn fi spline o, e.g., ges regressed on ers eperience [The Mincer curve]. The liner spline m e fied in o s, nd hese re oulined in he folloing o secions.

2 . Esiming Spline Mehod One The funcion is given s: Figure. The funcion is hen reprmeised: u Figure. Compring Figure. nd. e ge: Figure. Fiing Figure. OLS is OK. Tesing he signicnce of is sking heher here is rend in he firs period, heher he slope in he second period is signicnl dferen from in he firs, nd heher he slope in he hird period is signicnl dferen from in he second.

3 . Esiming Spline Mehod To An lernive mehod, hich is he primr focus of he lecure, is o use resriced les squres. Looking Figure. he resricions implied he join poins re: Figure. These m e se up s: Figure. Figure. in mri form is: R r Figure. No le us define he resriced les squres esimor s, This mus sisf our resricions.: R r Figure.4 The ssumed model is u X. The ojecive is o choose so s o minimize he sum of squred residuls sujec o sisfing his consrin. Therefore e se up Lgrngin, minimising he sum of squres sujec o Figure.4.

4 X X R r X X X R r R Figure.5 The premulipling λ is for simplici in he dfereniion. λ is vecor of Lgrngen mulipliers one for ech consrin. The firs dfereniion is discussed in greer deil in ls erms lecure in deriving he OLS esimor. Rerrnging he second equion in Figure.5 nd premulipling R X X gives: Figure.6 R X X X RX X R R Using he hird equion in Figure.5, nd he OLS formul =(X X) X Y, e cn ge: Figure.7 r R R X X R R X X R r R Reurning o he second equion in Figure.5 e cn rerrnge his o ge: X X R X X X R X Figure.8 Insering Figure.7 ino Figure.8 e ge: R r R R X X R r R X X X X X R RX X X X R Figure.9 Here is he unresriced OLS esimor. Figure.9 defines he resriced les squres esimor for n se of resricions (no jus he ones e hve se up reling o splines) emodied in R r. 4

5 This emple used ime, u orks equll ell ih oher coninuous vriles such s eperience in he Mincer equion..4 Cuic Splines One disdvnge ih he mehod presened so fr is h he firs derivive is no coninuous. Thus he join poins here cn e shrp chnge in he slope, hich is prol unrelisic. To overcome his e cn use cuic or qudric splines. We ill illusre ih cuic spline. Suppose e hve ovrile relionship ih o knon knos or splines nd. No insed of liner relionship, e ill use hirddegree polnomil in. i.e.: i i i i u i,, Figure 4. Where he suses re defined : i i i Figure 4. The resricions implied coninui he knos re hen: Figure 4. We furher impose coninui of he firs derivives ih respec o of he cuic spline funcion he o join poins, hich implies: Figure 4.4 Coninui of he second derivive implies: Figure 4.5 5

6 Thus he cuic spline merel llos disconinuiies in he hird derivives he join poins. The cuic spline m e esimed fiing Figure 4. nd esiming he elve prmeers sujec o he [] resricions se ou s in Figure.9. Think h he resricion mri in. ould look like ih hese er resricions dded. Useful references: Hudson, J. (996) Trends in MuliAuhored Ppers in Economics, Journl of Economic Perspecives Vol., No., pp. 558 Johnson, J. & DiNrdo, J., 997. Economeric Mehods. Fourh Ediion. McGrHill, Ne York, NY. Buse, A. & Lim, L., 977. Cuic Splines s Specil Cse of Resriced Les Squres. Journl of he Americn Sisicl Associion, Vol. 7, No. 57 (Mr 977), pp Poirier, D.J. & Grer, S.G., 974. The Deerminns of Aerospce Profi Res Souhern Economic Journl, Vol. 4, No. (Oc 974), pp.88. 6