Yarine Fawaz ECONOMETRICS I

Transcription

1 Yarne Fawaz ECONOMETRICS I

2 Organzaton of the course Classes: Classes to 0: -Introducton -Bvarate model -Multvarate model -Inference and tests -Advanced topcs: Lmted dependent varables; Interactons; etc. -Applcatons. Evaluaton: Man task: a PAPER, answerng your own queston, usng data of your choce. Group work: 3 or 4 max. Due by January, 4 th (30/00). One ndvdual home task: due by December 5 th (20/00). Indvdual or group work. One fnal exam: late January (50/00), on Tuesday, 4 th of January.

3 Other logstcs Concepts you should be famlar wth: Key concepts of math and statstcs as revsed n Appendx A, B, and C of ntroductory Wooldrdge. Webpage of the class: Materal for homework Help for your paper Readngs Resources to help you learn and use Stata: Datasets used n the class: Wooldrdge datasets Textbooks for Econometrcs I: Wooldrdge J., Introductory Econometrcs: A Modern Approach, 4e, Cncnnat, OH: Southwestern, Angrst J. & Pschke J.-S., Mostly Harmless Econometrcs: An Emprcst's Companon, Prnceton Unversty Press.

4 Want to talk to me? Ask questons n class! Come see me durng the break Come see me n my offce Thursday -3; or by appontment Send me an emal: yarne.fawaz@yahoo.fr

5 Outlne for ths class Motvaton How does econometrcs dffer from statstcs? Some early alerts! From an economc to an econometrc model Data Bvarate Regresson Model

6 What s econometrcs? Use of statstcal methods for: Estmatng economc relatonshp E.g. Each addtonal year of educaton ncreases average hourly wage by 54 cents n the 70s n the US. E.g. 50% of agrcultural output declne durng early transton n Eastern Europe can be attrbuted to prce declnes. Testng economc theores E.g. Is there a causal relatonshp between nsttutons/governance and growth? E.g. Is there a causal relatonshp between the educaton of the mother and educaton of her chldren? Impact analyss E.g. What s the mpact of an agrcultural extenson program on yelds n Indonesa? E.g. What s the mpact of condtonal cash transfers on school enrollment of chldren n Mexco?

7 Why? Forecastng Predctng of behavor economc agents advce on future polces Evaluatng exstng/past polces

8 How? Infer a causal effect: correlaton does not mply causalty. Noton of Ceters Parbus: other (relevant) factors beng equal. When carefully appled, econometrcs methods can smulate a ceters parbus experment.

9 How does econometrcs dffer from statstcs? Natural scences: often expermental data =>Causal nference automatc. Socal scences: Sometmes experments =>Random assgnment of nterventons Often can t do experments => Need for a more careful use of the data (polcy evaluaton) Need for new tools to deal wth the complextes of economc data and to test the predctons of economc theores.

10 Example of ssues wth the data: Impossble expermentaton and spurous correlaton What are the returns to educaton?

11 Spurous correlaton may be a concdence Y-axs: Percentage of a state's populaton that s Obese X- axs: Number of Placemarks wth Keyword Chrstanty / Total number of Placemarks Bvarate correlaton: 0.729

12 Possble reverse causalty? Det soda and obesty: postve correlaton Sports and obesty: negatve correlaton Alcohol and earnngs: Bethany L. Peters & Edward Strngham : "No Booze? You May Lose: Why Drnkers Earn More Money Than nondrnkers, 2006, Journal of Labor Research. Great paper to dscuss for all sorts of reasons ncludng measurement error, response bas, causalty, etc.

13 Econometrc Model () Example : relatonshp between wage and educaton Economc model: wage = f (educ) Econometrc model: Wage 0 * educ u Wage: Dependent (or left-hand sde) varable Educ: Independent (or rght-hand sde) varable 0, : Parameters: to be estmated! u: Error term: Captures the unobserved part of the relatonshp We wll need to make assumptons about these

14 Econometrc Model (2) Example 2: relatonshp between wage and educaton depends on other varables as well. Economc model: wage = f (educ,exper,tran) Econometrc model: Y 0 educ 2exper 3 tran u Interpretaton of parameters: Ceters parbus.e. keepng everythng else constant

15 Data Cross-secton: sample of many enttes (ndvduals, frms,, countres) at one pont n tme. Man advantage: random samplng most of the tme. ex: affars.dta ; wage.dta. Tme seres: Observatons on one or several varables at many ponts n tme. Careful to: - observatons cannot be assumed to be ndependent over tme - data frequency. ex: consump.dta Pooled cross-sectons: both CS and TS. Allows to ncrease sample sze and see how a relatonshp changes over tme. ex: cps78_85.dta Panel: Observatons on several enttes at several ponts n tme. Same cross-sectonal unts are followed over tme. Man advantage: allows to control for unobserved characterstcs of ndvduals, frms, etc=>easer to nfer causalty; allows to see lag effects. ex: wagepan.dta

16 THE SIMPLE (OR BIVARIATE) REGRESSION MODEL

17 Outlne Defnton and ntuton Estmaton of slope and ntercept Crtera for a good estmate? Interpretaton of estmated coeffcents How good s the estmaton at explanng the dependent varable?=> R-squared What about non-lnear relatonshps? More on nterpretaton Statstcal propertes of OLS 4 assumptons and unbasedness 5 th assumpton: homoscedastcty Precson of OLS estmates standard error

18 Intuton behnd OLS Explan y n terms of x: how y vares wth changes n x? Y: wage X: educaton. Plot Y aganst X and then fnd the lne that sums up ths relatonshp the best educ wage Ftted values

19 What s the equaton of ths lne? A smple equaton relatng y to x s: y 0 x u (2.) Where: y s the dependent/predcted/explaned/response varable x s the ndependent/predctor/explanatory/control varable u s the error/dsturbance term: factors other than x that affect y =>unobserved factors β 0 s the ntercept β s the slope.

20 How do we get the best estmates of β 0, β? We have a random sample of sze n wth observatons on ndvduals. We thnk there s a lnear relatonshp between y and x for each : y 0 x u Here u s the error term for observaton snce t contans all factors affectng y other than x. The goal s to obtan good estmates of β 0, β. ˆ s the estmated value of β 0 0. yˆ ˆ 0 ˆ x s the predcted value of y. Logcal crtera: mnmze ˆ y yˆ y ˆ ˆ x. u 0

21 Ex.: Returns to educaton n the US. scatter wage educ lft wage educ y ŷ û educ wage Ftted values Can you spot y, yˆ, uˆ?

22 Mnmze the sum of squared resduals We want small resduals=>mn Why squared? What do we obtan? Ordnary Least Squares estmates: Frst order condtons => (2.9) (2.7) We then form the OLS regresson lne (2.20) n n x y u ) ˆ ˆ ( ˆ n n x x y y x x 2 ) ( ) )( ( ˆ y x 0 ˆ ˆ y x 0 ˆ ˆ ˆ

23 How can we be sure of capturng a ceters parbus relatonshp between y and x? Two crucal assumptons on u: E(u) = 0 (2.5) Mean of u s 0. E(u x) =E(u)= 0 (2.6) Zero condtonal mean assumpton: u s unrelated (=>more than uncorrelated) to x. When s t a reasonable assumpton? These assumptons allow to predct y gven a certan x: E(y x)=β 0 +β x.

24 Ex.: Returns to educaton n the US (). use descrbe educ wage storage dsplay value varable name type format label varable label educ wage float %9.0g float %9.0g. sum educ wage Varable Obs Mean Std. Dev. Mn Max educ wage Descrpton of the data: Random sample of 526 ndvduals from the 976 Current Populaton Survey. Average number of years of educaton n the sample s 3, but educatonal attanment vares wdely (between 0 and 8 years). Smlarly, wages dffer wdely and range between a mnmum of 53 cent per hour to a maxmum of almost 25$ per hour. Average wage s about 6$ per hour.

25 Ex.: Returns to educaton n the US (2). regress wage educ Source SS df MS Number of obs = 526 F(, 524) = Model Prob > F = Resdual R-squared = Adj R-squared = Total Root MSE = wage Coef. Std. Err. t P> t [95% Conf. Interval] educ _cons : A -year ncrease n educaton s predcted to ncrease the hourly earnngs by 0.54 US $, ceters parbus. Therefore, four more years of educaton ncrease the predcted wage by 4(0.54)= 2.6 or $2.6 per hour. These are farly large effects. Because of the lnear nature of (2.27), another year of educaton ncreases the wage by the same amount, regardless of the ntal level of educaton. 0 : A person wth 0 years of educaton s predcted to have negatve hourly earnngs of -.90 dollars, ceters parbus. (ths s not necessarly very meanngful the low number of observatons wth years below 6 can explan ths predcton). ˆ ˆ

26 Ex.: Returns to educaton n the US (3). scatter wage educ lft wage educ y ŷ û educ wage Ftted values When s y under or overpredcted?

27 Ex.: Returns to educaton n the US (4) What s the predcted daly wage of somebody wth years of educaton? y ˆ * x => predcted wage s +/- 5 US$ per hour (=5.05) Or you can also ask stata to compute t: reg wage educ predct wageh tab wageh f educ==

28 Ex.: Returns to educaton n the US (5) What happens f wage s expressed n daly earnngs? =>wage =wage*8 wage= β 0 +β educ wage /8= β 0 +β educ wage =8β 0 +8β educ So the new equaton s wage = β 0 +β educ, wth β 0 =8β 0 and β =8β. Concluson: If the dependent varable s multpled by the constant c, then the OLS ntercept and slope estmates are also multpled by c.

29 Ex.: Returns to educaton n the US (6). ge dalywage=wage*8. regress dalywage educ Source SS df MS Number of obs = 526 F(, 524) = Model Prob > F = Resdual R-squared = Adj R-squared = Total Root MSE = dalywage Coef. Std. Err. t P> t [95% Conf. Interval] educ _cons Interpretaton: A -year ncrease n educaton s predcted to ncrease the daly earnngs by 4.33 US $, ceters parbus.

30 Ex.: Returns to educaton n the US (7) What happens f educ s expressed n months? =>educ_m=educ_y*2: to express educaton n months, we need to multply educaton n years by 2. wage= β 0 +β educ_m/2 wage= β 0 +β /2 educ_m So the new equaton s wage= β 0 +β educ_m, wth β =β /2. Concluson: f the ndependent varable s dvded or multpled by some nonzero constant, c, then the OLS slope coeffcent s also multpled or dvded by c respectvely.

31 Ex. 2: Relatonshp between sze of house and house prce () Source: dataset hprce.dta, Wooldrdge, collected from the real estate pages of the Boston Globe durng 990. These are home sellng n the Boston, MA area. Estmated model: prce 0 sqrft u. use des prce sqrft storage dsplay value varable name type format label varable label prce float %9.0g house prce, $000s sqrft nt %9.0g sze of house n square feet. sum prce sqrft Varable Obs Mean Std. Dev. Mn Max prce sqrft regress prce sqrft Source SS df MS Number of obs = 88 F(, 86) = Model Prob > F = Resdual R-squared = Adj R-squared = Total Root MSE = prce Coef. Std. Err. t P> t [95% Conf. Interval] sqrft _cons

32 Ex. 2: Relatonshp between sze of house and house prce (2). scatter prce sqrft lft prce sqrft sze of house n square feet house prce, $000s Ftted values Interpretaton: A square foot ncrease of the sze of the house s predcted to ncrease the prce of the house by 40 $, ceters parbus. A house wth 0 square feet has an estmated prce of 204$, ceters parbus. (obvously the predcted prce of a house wth 0 square feet s however not very meanngful).

33 Algebrac propertes of the OLS n ˆ u 0 n x uˆ 0 The pont lne. ( x, y) s always on the OLS regresson

34 How good s the estmaton at explanng the dependent varable? Measure of sample varaton: Total Sum of Squares: SST n ( y y) 2 Part that s explaned by x: Explaned Sum of Squares: SSE n ( yˆ y) 2 Part that s unexplaned by x: Resdual Sum of Squares: SSR n 2 uˆ SST=SSE+SSR

35 Goodness of ft: the R-squared R 2 SSE SST SSR SST R 2 s the rato of the explaned varaton compared to the total varaton=> fracton of the sample varaton n y that s explaned by x. R 2 les between 0 and. Hgher value ndcates a better ft. Importance depends on the objectve of the estmaton: understand a relatonshp between two varables vs predctng an outcome.

36 Meanng of lnear regresson Careful: the smple regresson model s lnear n parameters, not n the explanatory varable!!! Is y= β 0 + β x 2 +u a lnear equaton? What about consumpton= /(β 0 + β nc)+ u? The mechancs of smple regresson do not depend on how y and x are defned, but the nterpretaton of the coeffcents does depend on ther defntons.

37 Incorporatng non-lnearty: logarthmc form wage 0 educ u β s the change n wage gven addtonal year of educaton => constant returns to educaton. log( wage) 0 educ u 00* β % change n wage gven addtonal year of educaton => ncreasng returns to an addtonal year of educaton, by usng log of dependent varable.

38 Ex. b: Returns to educaton n the US Source: Wooldrdge, WAGE.dta (data from 976 Current Populaton Survey) Estmated model: log( wage) 0 educ u use reg lwage educ Source SS df MS Number of obs = 526 F(, 524) = 9.58 Model Prob > F = Resdual R-squared = Adj R-squared = Total Root MSE = lwage Coef. Std. Err. t P> t [95% Conf. Interval] educ _cons

39 Ex. b: Returns to educaton n the US(2) We obtan: lwâge= educ. R 2 = wage ncreases by 8.3 percent for every addtonal year of educaton =>constant percentage effect of educaton on wage. The ntercept s not very meanngful, as t gves the predcted log(wage) when educ=0. R-squared => educ explans about 8.6 percent of the varaton n log(wage) (not wage). Fnally, ths equaton mght not capture all of the nonlnearty n the relatonshp between wage and schoolng. Ex: dploma effects.

40 Another use of the log Use log of dependent and ndependent varables: Ex.: log( prce) 0 log( sze) u β s the % change n prce gven a % ncrease n sze = elastcty of prce wth respect to sze. => constant elastcty model.

41 Example 2b: Relatonshp between sze of house and house prce Estmated model: log( prce) 0 log( sqrft) u. use des prce sqrft lprce lsqrft storage dsplay value varable name type format label varable label prce float %9.0g house prce, $000s sqrft nt %9.0g sze of house n square feet lprce float %9.0g log(prce) lsqrft float %9.0g log(sqrft). reg lprce lsqrft Source SS df MS Number of obs = 88 F(, 86) = Model Prob > F = Resdual R-squared = Adj R-squared = Total Root MSE =.2044 lprce Coef. Std. Err. t P> t [95% Conf. Interval] lsqrft _cons

42 Summary of functonal forms nvolvng logarthms

43 Populaton parameters vs. estmated parameters Populaton model: y= β 0 + β x+u Use random sample of observatons {(x,y ): =,,n} to obtan an OLS estmate. β 0 and β are estmators of the true values, and dfferent random samples wll result n dfferent and ˆ. =>we never know the true ˆ 0 and ˆ, but we can establsh the statstcal propertes of ther dstrbutons over dfferent random samples. ˆ 0

44 Unbasedness of OLS Remember assumptons: Lnearty (n parameters!): Random samplng: Zero condtonal mean: E(u x)=0. Sample varaton: ( x ) 2 x 0. y n 0 x u, =,2,,n. Usng all these assumptons we can prove the frst mportant statstcal property of OLS: unbasedness ( ˆ 0) and E( ˆ). y 0 x E 0 u

45 What f the thrd assumpton fals? Usng data from 988 for houses sold n Andover, MA, from Kel and McClan(995), the followng equaton relates housng prce (prce) to the dstance from a recently bult garbage ncnerator (dst): log(prîce)= log(dst) n =35, R 2 =0.62 =>Interpret β, does t have the expected sgn? Is t an unbased estmator?

46 Homoskedastcty ˆ s centered about β, but how far s t from β on average? =>crteron to choose the best estmator: mnmze varance, or standard devaton. Assumpton 5: Homoskedastcty (constant varance) 2 Var ( u x) ˆ ˆ =>smplfy computaton of the varances of 0 and. Consequences: 2 Var ( y x) 2 Var ( u) so σ 2 s the error varance or dsturbance varance. σ s the standard devaton of the error.

47 Heteroskedastcty When s homoskedastcty not a realstc assumpton? Example: y: wage; x: educaton. Is V(wage educaton) constant across all educaton levels? Whenever V(y x) s a functon of x, there s heteroskedastcty => Need to thnk about t as an emprcal ssue.

48 Varances of OLS estmators Assumptons to 5 allow dervng a formula for the precson of the coeffcent estmates: Var( ˆ ) n More varaton n the unobservables =>more dffcult to get precse estmates. More varaton n the ndependent varables => easer to get precse estmates. 2 ( x x) 2

49 Errors vs. Resduals Populaton model: error for obs. y 0 x u, where u s the Estmated model: y the resdual for obs. ˆ yˆ uˆ 0 x ˆ uˆ, where û s u û : u u ( ˆ ) ( ˆ ) x so that E(u - û )=0 ˆ 0 0 σ 2 =E(u 2 ) => how to estmate σ 2?

50 Estmatng the error varance σ 2 =E(u 2 ) => a good guess s. BUT: The errors are never observable, whle the resduals are computed from the data. A realstc one: uˆ 2 n SSR n u 2. BUT: based. 2 An unbased computable one:. 2 Under assumptons to 5: E( ˆ 2 ). ˆ n 2 uˆ n 2 SSR n 2

51 Standard error of the regresson Also called standard error of the estmate and the root mean squared error ( root MSE n stata ): ˆ ˆ 2 It s an estmate of the standard devaton n the unobservables affectng y. It s an estmate of the standard devaton n y after the effect of x has been taken out.

52 Standard error of We can use to estmate the standard devaton of, whch s called standard error of. Plug nto. We get: Precson of estmate depends on : the resduals, the number of obs, the varaton of the ndep. var. 2 ˆ ˆ ˆ n x x Var 2 2 ) ( ) ˆ ( 2 ˆ 2 / / 2 ) ( 2) ( ˆ ) ( ˆ ) ˆ ( n n n x x n u x x se ˆ ˆ