ECON 351* -- Introducton (Page 1) Econometrcs: What's It All About, Ale? Usng sample data on observable varables to learn about economc relatonshps, the unctonal relatonshps among economc varables. Econometrcs conssts manly o: estmatng economc relatonshps rom sample data testng hypotheses about how economc varables are related the exstence o relatonshps between varables the drecton o the relatonshps between one economc varable -- the dependent or outcome varable -- and ts hypotheszed observable determnants the magntude o the relatonshps between a dependent varable and the ndependent varables that are thought to determne t. Sample data consst o observatons on randomly selected members o populatons o economc agents (ndvdual persons, households or amles, rms) or other unts o observaton (ndustres, provnces or states, countres). Example 1 We wsh to nvestgate emprcally the determnants o households' ood expendtures, n partcular the relatonshp between households' ood expendtures and households' ncomes. Sample data consst o a random sample o 38 households rom the populaton o all households. For each household n the random sample, we have observatons on three observable varables: oodexp = annual ood expendture o household, thousands o dollars per year ncome = annual ncome o household, thousands o dollars per year hhsze = household sze, number o persons n household ECON 351* -- Introducton: Fled 351lec01.doc... Page 1 o 11
ECON 351* -- Introducton (Page 2). lst oodexp ncome hhsze oodexp ncome hhsze 1. 15.998 62.476 1 2. 16.652 82.304 5 3. 21.741 74.679 3 4. 7.431 39.151 3 5. 10.481 64.724 5 6. 13.548 36.786 3 7. 23.256 83.052 4 8. 17.976 86.935 1 9. 14.161 88.233 2 10. 8.825 38.695 2 11. 14.184 73.831 7 12. 19.604 77.122 3 13. 13.728 45.519 2 14. 21.141 82.251 2 15. 17.446 59.862 3 16. 9.629 26.563 3 17. 14.005 61.818 2 18. 9.16 29.682 1 19. 18.831 50.825 5 20. 7.641 71.062 4 21. 13.882 41.99 4 22. 9.67 37.324 3 23. 21.604 86.352 5 24. 10.866 45.506 2 25. 28.98 69.929 6 26. 10.882 61.041 2 27. 18.561 82.469 1 28. 11.629 44.208 2 29. 18.067 49.467 5 30. 14.539 25.905 5 31. 19.192 79.178 5 32. 25.918 75.811 3 33. 28.833 82.718 6 34. 15.869 48.311 4 35. 14.91 42.494 5 36. 9.55 40.573 4 37. 23.066 44.872 6 38. 14.751 27.167 7. descrbe Contans data rom oodexp.dta obs: 38 vars: 3 7 Sep 2000 23:30 sze: 608 (99.9% o memory ree) ------------------------------------------------------------------------------- 1. oodexp loat %9.0g ood expendture, thousands $ per yr 2. ncome loat %9.0g household ncome, thousands $ per yr 3. hhsze loat %9.0g household sze, persons per hh ------------------------------------------------------------------------------- Sorted by: Note: dataset has changed snce last saved ECON 351* -- Introducton: Fled 351lec01.doc... Page 2 o 11
ECON 351* -- Introducton (Page 3). summarze Varable Obs Mean Std. Dev. Mn Max ---------+----------------------------------------------------- oodexp 38 15.95282 5.624341 7.431 28.98 ncome 38 58.44434 19.93156 25.905 88.233 hhsze 38 3.578947 1.702646 1 7 Queston: What relatonshp generated these sample data? What s the data generatng process? Answer: We postulate that each populaton value o oodexp, denoted as oodexp, s generated by a relatonshp o the orm: where ( ncome, hhsze u ood exp = ) + the populaton regresson equaton ood exp = the dependent or outcome varable we are tryng to explan = the annual ood expendture o household (thousands o $ per year) ncome = on ndependent or explanatory varable that we thnk mght explan the dependent varable ood exp = the annual ncome o household (thousands o $ per year) hhsze = a second ndependent or explanatory varable that we thnk mght explan the dependent varable ood exp = household sze, measured by the number o persons n the household ( ncome, hhsze ) = a populaton regresson uncton representng the systematc relatonshp o ood exp to the ndependent or explanatory varables ncome and hhsze ; u = an unobservable random error term representng all unknown and unmeasured varables that determne the ndvdual populaton values o ood exp ECON 351* -- Introducton: Fled 351lec01.doc... Page 3 o 11
ECON 351* -- Introducton (Page 4) Queston: What mathematcal orm does the populaton regresson uncton ( ncome, hhsze ) take? Answer: We hypothesze that the populaton regresson uncton -- or PRF -- s a lnear uncton: ( ncome, hhsze ) = β0 + β1ncome + β2hhsze Implcaton: The populaton regresson equaton -- the PRE -- s thereore ood exp = = β ( ncome, hhsze ) 0 + β ncome 1 + β 2 + u hhsze + u Observable Varables: oodexp the value o the dependent varable oodexp or the -th household ncome the value o the ndependent varable ncome or the -th household hhsze the value o the ndependent varable hhsze or the -th household Unobservable Varable: u the value o the random error term or the -th household n the populaton Unknown Parameters: the regresson coecents β 0, β 1 and β 2 β 0 = the ntercept coecent β 1 = the slope coecent on ncome β 2 = the slope coecent on hhsze The populaton values o the regresson coecents β 0, β 1 and β 2 are unknown. ECON 351* -- Introducton: Fled 351lec01.doc... Page 4 o 11
ECON 351* -- Introducton (Page 5) Example 2 We wsh to nvestgate emprcally the determnants o pad workers' wage rates. In partcular, we want to nvestgate whether male and emale workers wth the same characterstcs on average earn the same wage rate. Sample data consst o a random sample o 526 pad workers rom the 1976 US populaton o all pad workers n the employed labour orce. For each pad worker n the random sample, we have observatons on sx observable varables: wage ed exp ten emale = average hourly earnngs o pad worker, dollars per hour = years o educaton completed by pad worker, years = years o potental work experence o pad worker, years = tenure, or years wth current employer, o pad worker, years = 1 pad worker s emale, = 0 otherwse marred = 1 pad worker s marred, = 0 otherwse. descrbe Contans data rom wage1.dta obs: 526 vars: 6 16 Apr 2000 16:18 sze: 94,680 (90.7% o memory ree) ------------------------------------------------------------------------------- 1. wage loat %9.0g average hourly earnngs, $/hour 2. ed loat %9.0g years o educaton 3. exp loat %9.0g years o potental work experence 4. ten loat %9.0g tenure = years wth current employer 5. emale loat %9.0g =1 emale, =0 otherwse 6. marred loat %9.0g =1 marred, =0 otherwse ECON 351* -- Introducton: Fled 351lec01.doc... Page 5 o 11
ECON 351* -- Introducton (Page 6). lst wage ed exp ten emale marred wage ed exp ten emale marred 1. 21.86 12 24 16 0 1 2. 5.5 12 18 3 0 0 3. 3.75 2 39 13 0 1 4. 10 12 31 2 0 1 5. 3.5 13 1 0 0 0 6. 6.67 12 35 10 0 0 7. 3.88 12 12 3 0 1 8. 5.91 12 14 6 0 1 9. 5.9 12 14 7 0 1 10. 10 17 5 3 0 1 11. 4.55 16 34 2 0 1 12. 10 8 9 0 0 1 13. 6 13 8 0 0 1 14. 5 9 31 9 0 1 15. 4.5 12 13 0 0 1 16. 5.43 14 10 3 0 1 17. 2.83 10 1 0 0 0 18. 6.8 12 14 10 0 1 19. 6.76 12 19 3 0 1 20. 4.51 12 5 2 0 0 (output omtted) 507. 6.15 12 35 12 1 0 508. 11.1 15 1 4 1 0 509. 3.35 15 3 1 1 1 510. 5 12 7 3 1 0 511. 3.35 7 35 0 1 0 512. 6.25 12 13 0 1 1 513. 3.06 12 14 10 1 0 514. 5.9 12 9 7 1 1 515. 8.1 12 38 3 1 1 516. 14.58 18 13 7 1 0 517. 9.42 14 23 0 1 1 518. 9.68 13 16 16 1 0 519. 8.6 16 3 2 1 0 520. 3 12 38 0 1 1 521. 3.33 12 45 4 1 1 522. 4 12 22 11 1 1 523. 2.75 13 1 2 1 0 524. 3 16 19 10 1 1 525. 2.9 8 49 6 1 0 526. 3.18 12 5 0 1 1. summarze wage ed exp ten emale marred Varable Obs Mean Std. Dev. Mn Max ---------+----------------------------------------------------- wage 526 5.896103 3.693086.53 24.98 ed 526 12.56274 2.769022 0 18 exp 526 17.01711 13.57216 1 51 ten 526 5.104563 7.224462 0 44 emale 526.4790875.500038 0 1 marred 526.608365.4885804 0 1 ECON 351* -- Introducton: Fled 351lec01.doc... Page 6 o 11
ECON 351* -- Introducton (Page 7) Queston: What relatonshp generated these sample data? What s the data generatng process? Answer: We postulate that each populaton value o wage, denoted as wage, s generated by a populaton regresson equaton o the orm: ( ed, exp, ten, emale, marred ) u wage = + where: wage = the dependent or outcome varable we are tryng to explan = the average hourly earnngs o pad worker (dollars per hour) ed = one ndependent or explanatory varable that we thnk mght explan the dependent varable wage = the years o educaton completed by pad worker (years) exp = a second ndependent or explanatory varable that mght explan wage = the potental work experence accumulated by pad worker (years) ten = a thrd ndependent or explanatory varable that mght explan = tenure, years wth current employer, o pad worker (years) emale = a ourth ndependent or explanatory varable that mght aect = 1 pad worker s emale, = 0 otherwse marred = a th ndependent or explanatory varable that we thnk mght explan the dependent varable wage = 1 pad worker s marred, = 0 otherwse ( ed, exp, ten, emale, marred ) wage wage = a populaton regresson uncton representng the systematc relatonshp o wage to the ndependent varables ed, exp, ten, emale and marred u = an unobservable random error term representng all unknown varables and unmeasured that determne the ndvdual populaton values o wage ECON 351* -- Introducton: Fled 351lec01.doc... Page 7 o 11
ECON 351* -- Introducton (Page 8) Queston: What mathematcal orm does the populaton regresson uncton, or PRF, ( ed, L, marred ) take? Answer: We hypothesze that the populaton regresson uncton -- or PRF -- s a lnear uncton. ( ed, L, marred ) = β0 + β1ed + β2 exp + β3ten + β4emale + β5marred Implcaton: The populaton regresson equaton -- the PRE -- s thereore wage = = β ( ed, exp, ten, emale, marred ) 0 + β ed 1 + β 2 exp + β 3 ten + β 4 + emale u + β 5 marred + u Observable Varables: wage the value o the dependent varable wage or the -th employee ed the value o the ndependent varable ed or the -th employee exp the value o the ndependent varable exp or the -th employee ten the value o the ndependent varable ten or the -th employee emale the value o the ndependent varable emale or the -th employee marred the value o the ndependent varable marred or the -th employee Unobservable Varable: u the value o the random error term or the -th pad worker n the populaton Unknown Parameters: the regresson coecents β 0, β 1, β 2, β 3, β 4 and β 5 β 0 = the ntercept coecent β 1 = the slope coecent on ed β 2 = the slope coecent on exp β 3 = the slope coecent on ten β 4 = the slope coecent on emale β 5 = the slope coecent on marred Our task: To learn how to compute rom sample data relable estmates o the regresson coecents β 0, β 1, β 2, β 3, β 4 and β 5. ECON 351* -- Introducton: Fled 351lec01.doc... Page 8 o 11
ECON 351* -- Introducton (Page 9) The Four Elements o Econometrcs Data Collectng and codng the sample data, the raw materal o econometrcs. Most economc data s observatonal, or non-expermental, data (as dstnct rom expermental data generated under controlled expermental condtons). Speccaton Speccaton o the econometrc model that we thnk (hope) generated the sample data -- that s, speccaton o the data generatng process (or DGP). An econometrc model conssts o two components: 1. An economc model: speces the dependent or outcome varable to be explaned and the ndependent or explanatory varables that we thnk are related to the dependent varable o nterest. Oten suggested or derved rom economc theory. Sometmes obtaned rom normal ntuton and observaton. 2. A statstcal model: speces the statstcal elements o the relatonshp under nvestgaton, n partcular the statstcal propertes o the random varables n the relatonshp. Estmaton Conssts o usng the assembled sample data on the observable varables n the model to compute estmates o the numercal values o all the unknown parameters n the model. Inerence Conssts o usng the parameter estmates computed rom sample data to test hypotheses about the numercal values o the unknown populaton parameters that descrbe the behavour o the populaton rom whch the sample was selected. ECON 351* -- Introducton: Fled 351lec01.doc... Page 9 o 11
ECON 351* -- Introducton (Page 10) Scentc Method The collecton o prncples and processes necessary or scentc nvestgaton, ncludng: 1. rules or concept ormaton 2. rules or conductng observatons and experments 3. rules or valdatng hypotheses by observatons or experments Econometrcs s that branch o economcs -- the dsmal scence -- whch s concerned wth tems 2 and 3 n the above lst. ECON 351* -- Introducton: Fled 351lec01.doc... Page 10 o 11
ECON 351* -- Introducton (Page 11) Recap We have consdered two examples o what are genercally called lnear regresson equatons or lnear regresson models. Example 1 -- a lnear regresson model or household ood expendture: ood exp = β + β ncome + β hhsze + u 0 1 2 Example 2 -- a lnear regresson model or pad workers' wage rates: wage = β + β ed + β exp + β ten + β emale + β marred + u 0 1 2 3 4 5 Regresson analyss has two undamental tasks: 1. Estmaton: computng rom sample data relable estmates o the numercal values o the regresson coecents β j (j = 0, 1,, K), and hence o the populaton regresson uncton. 2. Inerence: usng sample estmates o the regresson coecents β j (j = 0, 1,, K) to test hypotheses about the populaton values o the unknown regresson coecents --.e., to ner rom sample estmates the true populaton values o the regresson coecents wthn speced margns o statstcal error. ECON 351* -- Introducton: Fled 351lec01.doc... Page 11 o 11