Econometrics: What's It All About, Alfie?

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1 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

2 ECON 351* -- Introducton (Page 2). lst oodexp ncome hhsze oodexp ncome hhsze descrbe Contans data rom oodexp.dta obs: 38 vars: 3 7 Sep :30 sze: 608 (99.9% o memory ree) 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

3 ECON 351* -- Introducton (Page 3). summarze Varable Obs Mean Std. Dev. Mn Max oodexp ncome hhsze 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

4 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

5 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 :18 sze: 94,680 (90.7% o memory ree) 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

6 ECON 351* -- Introducton (Page 6). lst wage ed exp ten emale marred wage ed exp ten emale marred (output omtted) summarze wage ed exp ten emale marred Varable Obs Mean Std. Dev. Mn Max wage ed exp ten emale marred ECON 351* -- Introducton: Fled 351lec01.doc... Page 6 o 11

7 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

8 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

9 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

10 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

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 Example 2 -- a lnear regresson model or pad workers' wage rates: wage = β + β ed + β exp + β ten + β emale + β marred + u 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