Continuous vs. Discrete Goods

Transcription

1 CE 651 Transportaton Economcs Charsma Choudhury Lecture 3-4 Analyss of Demand Contnuous vs. Dscrete Goods Contnuous Goods Dscrete Goods x auto 1 Indfference u curves 3 u u 1 x bus

2 Outlne Data Modelng Prncples Assumptons Estmates Statstcal Tests Potental data problems 3 Data Cross-secton Actvtes of ndvdual persons, frms or other unts at sngle tme One observaton/ ndvdual Tme seres Movement of a varable over tme Annual, quarterly, monthly, weekly observatons etc. Mostly used for natonal or regonal level aggregaton of the observatons Pooled/panel Combnaton of tme seres and cross-secton Behavor of ndvdual persons, frms or other unts over tme 4

3 Examples Cross-sectonal data ID VMT o of cars n HH HH Income o of HH members o of Chldren n HH TAZ >80k k k Vehcle Mles Travelled n Examples Tme seres data Vehcle Mles Travelled n between Year Avg Fuel Prce/L Average VMT/ person Average car ownershp

4 Examples Pooled/Panel data ID 1 Year Avg Fuel Prce/L VMT Vehcle Mles Travelled n between o of cars n HH HH Income o of HH members o of Chldren n HH >80k >80k >80k k k k TAZ 7 Examples Pseudo panel data ID Vehcle Mles Travelled n between Year o of Avg Fuel Prce/L VMT o of cars HH o of HH Chldren n HH Income members n HH TAZ >80k k >80k k k k

5 Modelng Prncples Hypothess Example VMT= f (fuel cost, no of cars, hh ncome, hh sze) Lnear relatonshp Example VMT= α + βcost * cost + βcar * caro + βnc * hhinc + βsze * hhsze on-lnear relatonshp βcos t * cost VMT = α + βcar * caro β * hhinc In ths course we wll deal wth lnear relatonshps only In Regresson analyss, we estmate α and β s that best ft the observed data usng estmators nc 9 Estmators Our nterest Populaton Avalable Sample/samples from populaton Sample nformaton to obtan best possble estmates Estmator Rule that gves a reasonable estmate for each and every possble sample Estmators are rules Estmates are numbers produced by the estmator Desrable propertes Unbasedness Effcency Consstency (only for large sample) 10

6 Desrable Propertes We want our estmators to be Unbased Expected value of estmator close to true mean Bas = E β * ( ) βpopulaton Effcent For a gven sample sze, varance s smaller than any other unbased estmator Hgher effcency ndcates hgher relance on results * Consstent As ncreases β β populaton Ths assumpton s requred when we do statstcal tests (e.g. t-test) 11 Examples of Estmators Least/Mnmum error Mn ( Y Y ) = 1 Least/Mnmum absolute error Mn ( Y Y ) = 1 Ordnary Least square (OLS) Mn ( Y Y ) = 1 Weghted least square (WLS) Mn w ( Y Y ) = 1 1

7 Two Varable Lnear Regresson Model Model Y = α + βx + ε X = non stochastc ε = stochastc random term ( often follows certan dstrbutons) 13 ε Error ( ) Varables cannot provde perfect explanatons Errors are thngs that nfluence Y other than X Reasons Smplfcaton of realty e.g. VMT=f (no of cars, hh ncome, hh sze, hh chldren, locaton) Omtted varables ndvdual tastes, educaton, lfestyle patterns and many more Measurement errors Prvacy ssues Poor record keepng etc. 14

8 Error ( ε ) Predcton Error ε = Y Y Y * * = Predcted dependent varables= α + β X Sum squared error (SSE) ε = ( Y Y ) * In OLS, we mnmze SSE 15 Two Varable Lnear Regresson Model Model Y = α + βx + ε X = non stochastc ε = stochastc random term ( often follows certan dstrbutons) Soluton β = α XY X Y = β X If Y vares a lot when X vares lttle, wll be bg. In other words, s the magntude of nfluence of x on y β β 16

9 Statstcal Sgnfcance How dependable are the estmates? How sgnfcant s X n explanng Y? * If there s a hgh probablty that β s not 0, then β s statstcally sgnfcant The smaller the standard errors (varances) are relatve to the coeffcents, the more confdence we have n the estmates How to test? Use t-stats/ t-test t-stat = β * β * std error of β Compare wth t crtcal (at 95% or 90% level of confdence) at (-k) dof (=Obs number, k= number of estmated parameters) > t crtcal statstcally sgnfcant 17 Goodness-of-Ft How well the model fts the data Measure ε R 1 Y = 18

10 Multvarate Lnear Regresson Model Y = α + β X + β X + β X ε In matrx notaton Y=βX X = [1 X X X...] OLS Soluton β = ( X ' X) ( X ' Y) 19 Goodness-of-Ft R always ncreases as we add new varables Measure R whch accounts for k (number of estmated parameters) Model wth hgher R-bar sqr. has better goodness-of-ft n absolute terms 0

11 Example Chcago Trp Generaton Dependent varable average trps per occuped dwellng unt Independent varables average car ownershp average household sze three zonal socal ndces 1 Assgnment Varatons you can try Add other varables Use nteracton terms Use log on varables Pecewse lnear formulaton Evaluaton crtera Correct sgns Improvement n goodness-of-ft t-test

12 Assumptons of Classcal LR Model 1. Relatonshp between X and Y lnear. X non-stochastc and no exact lnear relatonshp exsts between two or more ndependent varables 3. Error has zero expected value (cancel out) E( ε ) = 0 4. Error has constant varance for all observatons E( ε ) = σ 5. o correlaton among errors E( ε ε ) = 0, for all j j 3 Gauss Markov Theorem If 1-5 s fulflled OLS s BLUE Best Lnear Unbased Estmator 4

13 Volaton 1 Collnearty Types Perfect correlaton Other hgh nterdependence multcollnearty Examples e.g. GPA=f(X1,X,X3, X4,X5) X1= parents educaton level X= average hours of study / day X3= average hours of study/ week X4= parents ncome X5= school X and X3 perfectly collnear X1 and X4 can be multcollnear 5 Volaton 1Collnearty (cont) Effect Perfect Cannot be estmated Multcollnear Dffcult to nterpret Affects statstcal sgnfcance Soluton Drop one varable Cauton May result bas 6

14 Volaton Heteroscedastcty Homoscedastc= constant varance Heteroscedastc = varance not constant Example Large frm bgger errors Larger TAZ bgger errors Effect Estmators unbased but neffcent Soluton Weghted least square (WLS) 7 Volatons 3 Seral correlaton Both cross secton and tme seres Can be postve or negatve e.g. Postve error ncorrect mleage readng egatve error mleage data taken n Jan 009 nstead of Dec 008 ; overestmaton of 008 VMT, underestmaton of 009 VMT Effect Estmators unbased but neffcent Soluton Pras-Wnsten, Cochrane-Orcutt, Durbn s Method 8