ECONOMICS 351* -- Stata 10 Tutorial 7. Stata 10 Tutorial 7. TOPIC: Estimation and Hypothesis Testing in Multiple Linear Regression Models

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1 ECONOMICS 5* -- Stata 0 Tutoral 7 Stata 0 Tutoral 7 TOPIC: Estmaton and Hypothess Testng n Multple Lnear Regresson Models DATA: autodta (a Stata-format data fle you created n Stata Tutoral ) TASKS: Stata Tutoral 7 ntroduces you to OLS estmaton of multple lnear regresson models contanng two or more regressors, and demonstrates how to perform varous common types of hypothess tests n such models It s ntended to gve you some practce n formulatng multple lnear regresson models and n computng F-tests and t-tests of lnear coeffcent restrctons on such models It ntroduces you to the specfcaton and nterpretaton of multple lnear regresson models that nclude as regressors polynomal terms n explanatory varables and nteracton terms between contnuous explanatory varables It also ntroduces the Stata lncom command, a post-estmaton command that computes lnear combnatons of coeffcent estmates, specfcally pont estmates, standard errors, t- statstcs, two-tal p-values, and two-sded confdence ntervals for specfed lnear combnatons of coeffcent estmates The Stata commands that consttute the prmary subject of ths tutoral are: regress test lncom Used to perform OLS estmaton of multple lnear regresson models Used to compute F-tests of lnear coeffcent equalty restrctons after OLS estmaton Used after estmaton to compute lnear combnatons of coeffcent estmates and assocated statstcs The Stata statstcal functons ntroduced n ths tutoral are: ttal(df, t 0 ) nvttal(df, p) Ftal(df, df, F 0 ) nvftal(df, df, α) Computes p-values of calculated t-statstcs Computes rght-tal crtcal values of t-dstrbutons Computes p-values of calculated F-statstcs Computes rght-tal crtcal values of F-dstrbutons NOTE: Stata commands are case senstve All Stata command names must be typed n the Command wndow n lower case letters ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page of 9 pages

2 ECONOMICS 5* -- Stata 0 Tutoral 7 Overvew of Regresson Models Used n ths Tutoral The followng multple lnear regresson models are estmated and tested n ths Stata tutoral prce β + β weght + β mpg + u () 0 0 ln( prce ) α + α ln(weght ) + α ln(mpg ) + u () prce β + β weght + β mpg + β weght + u (7) 0 0 prce β + β weght + β mpg + β weght + β mpg + u () prce β0 + βweght + βmpg + βweght + β4mpg + β5weghtmpg + u () 4 Preparng for Your Stata Sesson Before begnnng your Stata sesson, use Wndows Explorer to copy the Stataformat dataset autodta to the Stata workng drectory on the C:-drve or D:-drve of the computer at whch you are workng On the computers n Dunnng 50, the default Stata workng drectory s usually C:\data On the computers n MC B, the default Stata workng drectory s usually D:\courses Start Your Stata Sesson To start your Stata sesson, double-clck on the Stata 0 con n the Wndows desktop After you double-clck the Stata 0 con, you wll see the now famlar screen of four Stata wndows ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page of 9 pages

3 ECONOMICS 5* -- Stata 0 Tutoral 7 Record Your Stata Sesson and Stata Commands -- log usng, cmdlog usng To record your Stata sesson, ncludng all the Stata commands you enter and the results (output) produced by these commands, make a log fle named 5tutoral7log To open (begn) the log fle 5tutoral7log, enter n the Command wndow: log usng 5tutoral7log Ths command opens a fle called 5tutoral7log n the current Stata workng drectory Remember that once you have opened the 5tutoral7log fle, a copy of all the commands you enter durng your Stata sesson and of all the results they produce s recorded n that 5tutoral7log fle An alternatve way to open the log fle 5tutoral7log s to clck on the Log button; clck on Save as type: and select Log (*log); clck on the Fle name: box and type the fle name 5tutoral7; and clck on the Save button To record only the Stata commands you type durng your Stata sesson, use the Stata cmdlog usng command To start (open) the command log fle 5tutoral7txt, enter n the Command wndow: cmdlog usng 5tutoral7 Ths command opens a plan text-format (ASCII) fle called 5tutoral7txt n the current Stata workng drectory All commands you enter durng your Stata sesson are recorded n ths fle Loadng a Stata-Format Dataset nto Stata -- use Load, or read, nto memory the dataset you are usng To load the Stata-format data fle autodta nto memory, enter n the Command wndow: use auto Ths command loads nto memory the Stata-format dataset autodta ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page of 9 pages

4 ECONOMICS 5* -- Stata 0 Tutoral 7 Famlarze Yourself wth the Current Dataset To famlarze (or re-famlarze) yourself wth the contents of the current dataset, type n the Command wndow the followng commands: descrbe, short descrbe summarze Computng lnear combnatons of coeffcent estmates -- lncom Basc Syntax: lncom exp [, level(#)] where exp s a user-specfed lnear combnaton of coeffcent estmates The lncom command computes pont estmates, standard errors, t-statstcs, p- values, and two-sded confdence ntervals for a specfed lnear combnaton of coeffcent estmates from the most recently executed model estmaton command, such as the regress command Note that the lnear combnaton specfed n exp cannot contan an equalty sgn () The level(#) opton on the lncom command specfes the confdence level to be used n computng the two-sded confdence nterval for the specfed lnear combnaton of regresson coeffcents The value of # specfes the desred confdence level n percentage ponts; the default confdence level s 95 percent (e, α 095) Examples: The lncom command s most useful n multple regresson models contanng two or more explanatory varables Several examples of ts use are gven n subsequent sectons of ths tutoral Estmaton and Inference n Multple Lnear Regresson Models: Introducton Ths secton ntroduces you to OLS estmaton of multple lnear regresson models contanng two or more regressors It also demonstrates how to perform varous common types of hypothess tests n such models ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 4 of 9 pages

5 ECONOMICS 5* -- Stata 0 Tutoral 7 Consder the followng multple lnear regresson model for prce : prce β + β weght + β mpg + u () 0 where prce the prce of the -th car (n US dollars); weght the weght of the -th car (n pounds); and mpg the mles per gallon for the -th car (n mles per gallon) Note: For PRE (), the number of regresson coeffcents s K Regresson equaton () contans two explanatory varables, weght and mpg The margnal effect of weght n equaton () s: prce weght β a constant The margnal effect of mpg n equaton () s: prce mpg β a constant To estmate model () by OLS for the full sample of N 74 sample observatons n dataset autodta, enter n the Command wndow: regress prce weght mpg Testng the ndvdual sgnfcance of the slope coeffcent estmates Indvdual sgnfcance tests are two-tal tests of the followng statstcal hypothess: H 0 : β j 0 versus H : β j 0 ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 5 of 9 pages

6 ECONOMICS 5* -- Stata 0 Tutoral 7 Such ndvdual sgnfcance tests can be performed usng ether a two-tal t-test or an F-test The feasble test statstcs for such ndvdual sgnfcance tests are ether the t- statstc for ˆβ j gven by t(ˆ β j ) βˆ j β j sê(ˆ β ) j ~ t[n K] or the F-statstc for ˆβ j gven by ( βˆ β ) j j F(ˆ β j) ~ F[, N K] Vâr(ˆ β ) j Example : In regresson equaton (), perform a two-tal test of the hypothess H 0 : β 0 versus H : β 0 The null hypothess H 0 : β 0 mples that the margnal effect of the explanatory varable weght equals zero n PRE (): H 0 : β 0 prce β weght 0 Ths hypothess can be tested usng ether a two-tal t-test or an equvalent F-test A two-tal t-test: Under the null hypothess H 0 : β 0, the sample value of the t-statstc for β ˆ s t 0 βˆ ˆ ˆ β β 0 β (ˆ β ) sê(ˆ β ) sê(ˆ β ) sê(ˆ β ) ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 6 of 9 pages

7 ECONOMICS 5* -- Stata 0 Tutoral 7 The null dstrbuton of t β 0 (ˆ ) s: t 0 (ˆ β ) ~ t[n K] t[74 ] t[7] Compute the sample value of the t-statstc Enter the commands: scalar t0b _b[weght]/_se[weght] scalar lst t0b Compute the two-tal p-value of the calculated t-statstc Enter the command: dsplay *ttal(7, abs(t0b)) Another way to do what the precedng three commands have done s to use the Stata lncom command to compute both the sample value of the t-statstc and ts two-tal p-value Enter the command: lncom _b[weght] What nference would you draw from the outcome of ths two-tal t-test? Compute the two-tal 00α percent crtcal values t [7] α / and t [7 α / ] of the t[7]-dstrbuton Use the nvttal(df, p) functon, where p α/ At the % sgnfcance level for whch α 00, α/ 00/ 0005 Enter the commands: dsplay nvttal(7, 0005) (returns value of t [7] ) dsplay -nvttal(7, 0005) (returns value of t 005 [7] ) At the 5% sgnfcance level for whch α 005, α/ 005/ 005 Enter the commands: dsplay nvttal(7, 005) (returns value of t 0 05 [7] ) dsplay -nvttal(7, 005) (returns value of t 05 [7] ) 0 0 ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 7 of 9 pages

8 ECONOMICS 5* -- Stata 0 Tutoral 7 At the 0% sgnfcance level for whch α 00, α/ 00/ 005 Enter the commands: dsplay nvttal(7, 005) (returns value of t 0 05 [7] ) dsplay -nvttal(7, 005) (returns value of t 05 [7] ) 0 An F-test: Under the null hypothess H 0 : β 0, the sample value of the F- statstc for ˆβ s ( βˆ β ) ( βˆ 0) F 0(βˆ ) βˆ βˆ Vâr(βˆ ) Vâr(βˆ ) Vâr(βˆ ) sê(βˆ ) The null dstrbuton of F β 0 (ˆ ) s: F0 (ˆ β ) ~ F[, N K] F[, 74 ] F[, 7] Compute the sample value of the F-statstc Enter the commands: scalar F0b _b[weght]^/_se[weght]^ scalar lst F0b Compute the p-value of the calculated F-statstc Enter the command: dsplay Ftal(, 7, F0b) Another way to do what the precedng three commands have done s to use the Stata test command to compute both the sample value of the F-statstc and ts p-value Enter ether one of the followng test commands: test weght 0 test weght What nference would you draw from the outcome of ths F-test? Compute the 00α percent crtcal value F α [, 7] of the F[, 7]-dstrbuton ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 8 of 9 pages

9 ECONOMICS 5* -- Stata 0 Tutoral 7 Use the nvftal(df, df, α) statstcal functon to compute crtcal values of the F[df, df ]-dstrbuton, where df s the numerator degrees of freedom, df s the denomnator degrees of freedom, and α s the chosen sgnfcance level In the present case, df (the number of coeffcent restrctons specfed by the null hypothess H 0 ) and df 7 (the degrees of freedom for the RSS of the unrestrcted regresson equaton ()) The followng commands compute and dsplay the crtcal values of the F[,7] dstrbuton for sgnfcance levels α 00, 005, 00 At the % sgnfcance level, α 00 Enter the command: dsplay nvftal(,7, 00) (returns value of F 0 [, 7] ) At the 5% sgnfcance level, α 005 Enter the command: dsplay nvftal(,7, 005) (returns value of F 05 [, 7] ) At the 0% sgnfcance level, α 00 Enter the command: dsplay nvftal(,7, 00) (returns value of F 0 [, 7] ) These are the crtcal values of F α [, 7] for testng H 0 : β 0 aganst H : β 0 Would you reject the null hypothess β 0 n favor of the alternatve hypothess β 0 at the 0 percent sgnfcance level, at the 5 percent sgnfcance level, and/or at the percent sgnfcance level? Do the two tests the two-tal t-test and the F-test yeld the same nference? Example : In regresson equaton (), perform a two-tal test of the hypothess H 0 : β 0 versus H : β 0 The null hypothess H 0 : β 0 mples that the margnal effect of the explanatory varable mpg equals zero n PRE (): H 0 : β 0 prce β mpg 0 ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 9 of 9 pages

10 ECONOMICS 5* -- Stata 0 Tutoral 7 Ths hypothess can be tested usng ether a two-tal t-test or an equvalent F-test A two-tal t-test: Under the null hypothess H 0 : β 0, the sample value of the t-statstc for β ˆ s t 0 βˆ ˆ ˆ β β 0 β (ˆ β ) sê(ˆ β ) sê(ˆ β ) sê(ˆ β ) The null dstrbuton of t β 0 (ˆ ) s: t 0 (ˆ β ) ~ t[n K] t[74 ] t[7] Compute the sample value of the t-statstc Enter the commands: scalar t0b _b[mpg]/_se[mpg] scalar lst t0b Compute the two-tal p-value of the calculated t-statstc Enter the command: dsplay *ttal(7, abs(t0b)) Another way to do what the precedng three commands have done s to use the Stata lncom command to compute both the sample value of the t-statstc and ts two-tal p-value Enter the command: lncom _b[mpg] What nference would you draw from the outcome of ths two-tal t-test? Re-compute the two-tal 00α percent crtcal values t [7] α / and t [7 α / ] of the t[7]-dstrbuton at sgnfcance levels α 00, 005, 00 Use the nvttal(df, p) functon wth p α/ Enter the commands: dsplay nvttal(7, 0005) (returns value of t [7] ) dsplay -nvttal(7, 0005) (returns value of t 005 [7] ) 0 ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 0 of 9 pages

11 ECONOMICS 5* -- Stata 0 Tutoral 7 dsplay nvttal(7, 005) (returns value of t 0 05 [7] ) dsplay -nvttal(7, 005) (returns value of t 05 [7] ) dsplay nvttal(7, 005) (returns value of t 0 05 [7] ) dsplay -nvttal(7, 005) (returns value of t 05 [7] ) 0 0 An F-test: Under the null hypothess H 0 : β 0, the sample value of the F- statstc for ˆβ s ( βˆ β ) ( βˆ 0) F 0(βˆ ) βˆ βˆ Vâr(βˆ ) Vâr(βˆ ) Vâr(βˆ ) sê(βˆ ) The null dstrbuton of F β 0 (ˆ ) s: F0 (ˆ β ) ~ F[, N K] F[, 74 ] F[, 7] Compute the sample value of the F-statstc Enter the commands: scalar F0b _b[mpg]^/_se[mpg]^ scalar lst F0b Compute the p-value of the calculated F-statstc Enter the command: dsplay Ftal(, 7, F0b) Another way to do what the precedng three commands have done s to use the Stata test command to compute both the sample value of the F-statstc and ts p-value Enter ether one of the followng test commands: test mpg 0 test mpg What nference would you draw from the outcome of ths F-test? ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page of 9 pages

12 ECONOMICS 5* -- Stata 0 Tutoral 7 Re-compute the 00α percent crtcal values F α [, 7] of the F[, 7]- dstrbuton at sgnfcance levels α 00, 005, 00 At the % sgnfcance level, α 00 Enter the command: dsplay nvftal(,7, 00) (returns value of F 0 [, 7] ) At the 5% sgnfcance level, α 005 Enter the command: dsplay nvftal(,7, 005) (returns value of F 05 [, 7] ) At the 0% sgnfcance level, α 00 Enter the command: dsplay nvftal(,7, 00) (returns value of F 0 [, 7] ) These are the crtcal values of F α [, 7] for testng H 0 : β 0 aganst H : β 0 Would you reject the null hypothess β 0 n favor of the alternatve hypothess β 0 at the 0 percent sgnfcance level, at the 5 percent sgnfcance level, and/or at the percent sgnfcance level? Do the two tests the two-tal t-test and the F-test yeld the same nference? Testng the jont sgnfcance of all slope coeffcent estmates A jont sgnfcance test of all slope coeffcent estmates s a test the hypothess that the two slope coeffcents n PRE () are jontly equal to zero The null and alternatve hypotheses are: H 0 : β 0 j, β 0 and β 0 () j H : β 0 j, β 0 and/or β 0 j The followng sequence of test commands wth the accumulate opton wll perform a jont F-test of the null hypothess H 0 n () Enter the test commands: test weght 0 test mpg 0, accum ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page of 9 pages

13 ECONOMICS 5* -- Stata 0 Tutoral 7 Note that the accumulate opton can be abbrevated as accum In fact, t can even be abbrevated as a Ths seres of test commands computes the sample value of the F-statstc and ts p-value Alternatvely, a sngle test command can be used to compute ths jont F-test when the null hypothess specfes zero or excluson restrctons on two or more regresson coeffcents, as t does n the case of the null and alternatve hypotheses n () above Enter the command: test weght mpg Ths test command performs a jont F-test of the null hypothess that the regresson coeffcent of the regressor weght equals zero and the regresson coeffcent of the regressor mpg equals zero Use the nvftal(df, df, α) statstcal functon to compute crtcal values of the F[df, df ]-dstrbuton, where df s the numerator degrees of freedom, df s the denomnator degrees of freedom, and α s the chosen sgnfcance level In the present case, df (the number of coeffcent restrctons specfed by the null hypothess H 0 ) and df 7 (the degrees of freedom for the RSS of the unrestrcted regresson equaton ()) The followng commands compute and dsplay the crtcal values of the F[,7] dstrbuton for sgnfcance levels α 00, 005, 00 At the % sgnfcance level, α 00 Enter the command: dsplay nvftal(,7, 00) (returns value of F 0 [, 7] ) At the 5% sgnfcance level, α 005 Enter the command: dsplay nvftal(,7, 005) (returns value of F 05 [, 7] ) At the 0% sgnfcance level, α 00 Enter the command: dsplay nvftal(,7, 00) (returns value of F 0 [, 7] ) These are the crtcal values approprate for testng hypothess () above ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page of 9 pages

14 ECONOMICS 5* -- Stata 0 Tutoral 7 Would you reject the null hypothess n () at the 0 percent sgnfcance level, at the 5 percent sgnfcance level, and/or at the percent sgnfcance level? Testng Equalty of Slope Coeffcents n Multple Lnear Regresson Models: Introducton to the lncom Command Recall the multple lnear regresson model for prce consdered n the prevous secton: prce β + β weght + β mpg + u () 0 In ths regresson equaton, there s n general no reason to thnk that the slope coeffcents β and β wll be equal, because the explanatory varables weght and mpg wll usually have dfferent unts of measurement However, consder the followng log-log regresson model for car prces: ln( prce ) α + α ln(weght ) + α ln(mpg ) + u () 0 In the log-log regresson model (), the slope coeffcents α and α are partal elastcty coeffcents, and hence do not depend on the unts of measurement for prce, weght and mpg In fact, ln(prce ) α the partal elastcty of prce wrt weght ; ln(weght ) ln(prce ) α the partal elastcty of prce wrt mpg ln(mpg ) In log-log models such as that gven by populaton regresson equaton (), testng the equalty of pars of slope coeffcents makes perfect sense, snce these slope coeffcents are comparable n terms of ther magntude Before estmatng regresson equaton (), t s necessary to create the varables ln( prce ), ln( weght ) and ln( mpg ) Enter the commands: generate lnprce ln(prce) generate lnweght ln(weght) generate lnmpg ln(mpg) ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 4 of 9 pages

15 ECONOMICS 5* -- Stata 0 Tutoral 7 Estmate by OLS regresson equaton () Enter the command: regress lnprce lnweght lnmpg Test : Use the results of OLS estmaton of the log-log model () to test the proposton that the partal elastcty of prce wth respect to weght s equal to the partal elastcty of prce wth respect to mpg The null and alternatve hypotheses are: H 0 : H : α α 0 (4) α α α α 0 α α Use a Stata lncom command to compute a two-tal t-test of the null hypothess (4) above Enter the command: lncom _b[lnweght] - _b[lnmpg] Carefully nspect the results of ths lncom command State the nference you would draw from ths t-test Now use a Stata test command to compute an F-test of the null hypothess (4) Enter ether of the followng commands: test lnweght - lnmpg 0 test lnweght lnmpg Carefully nspect the results of ths test command How do you know that ths F-test yelds the same nference as the two-tal t-test of the same null hypothess (4)? Test : Use the results of OLS estmaton of the log-log model () to test the proposton that the partal elastcty of prce wth respect to weght s equal n magntude but opposte n sgn to the partal elastcty of prce wth respect to mpg ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 5 of 9 pages

16 ECONOMICS 5* -- Stata 0 Tutoral 7 The null and alternatve hypotheses for ths test are: H 0 : H : α α + α 0 (5) α α + α 0 α α Use a Stata lncom command to compute a two-tal t-test of the null hypothess above Enter the command: lncom _b[lnweght] + _b[lnmpg] Carefully nspect the results of ths lncom command State the nference you would draw from ths t-test Use a Stata test command to compute an F-test of the null hypothess (5) Enter ether of the followng commands: test lnweght + lnmpg 0 test lnweght -lnmpg Carefully nspect the results of ths test command How do you know that ths F-test yelds the same nference as the two-tal t-test of the same null hypothess (5)? Interacton Varables: Polynomal Terms n Explanatory Varables Nature: Polynomal terms refer to nteger powers of explanatory varables For example, the second-order polynomal term n the varable X j s X j -squared, ; the thrd-order polynomal term n the varable X j s X j -cubed, X j Usage: Polynomal terms n explanatory varables are one way of allowng for nonlnear relatonshps between the dependent varable Y and ndvdual explanatory varables such as X j Consder the followng regresson equaton: X j Y β + β X + β X + β X + u (6) 0 ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 6 of 9 pages

17 ECONOMICS 5* -- Stata 0 Tutoral 7 Compare the margnal effects on Y of the two explanatory varables X and X n equaton (6) The margnal effect of X n equaton (6) s: Y X β + β X a lnear functon of X The margnal effect of X n equaton (6) s: Y X β a constant Example : a polynomal n one explanatory varable, weght Consder the followng multple regresson equaton: prce β + β weght + β mpg + β weght + u (7) 0 where prce the prce of the -th car (n US dollars); weght the weght of the -th car (n pounds); weght the square of weght ; and mpg the mles per gallon for the -th car (n mles per gallon) Note: For PRE (7), the number of regresson coeffcents s K 4 Regresson equaton (7) contans two explanatory varables, weght and mpg, but three regressors weght, weght, and mpg Margnal Effects of the Explanatory Varables The margnal effect of weght n equaton (7) s: ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 7 of 9 pages

18 ECONOMICS 5* -- Stata 0 Tutoral 7 prce weght β + β weght a lnear functon of weght The margnal effect of mpg n equaton (7) s: prce mpg β a constant Before estmatng regresson equaton (7), t s necessary to create the varable weght Enter the command: generate weghtsq weght^ Now estmate by OLS regresson equaton (7) Enter the command: regress prce weght mpg weghtsq Testng the margnal effects of ndvdual explanatory varables Test : Test the hypothess that the margnal effect of weght on prce s zero for all cars n regresson equaton (7) The margnal effect of weght on prce s obtaned by partally dfferentatng regresson equaton (7) wth respect to weght : prce weght β + β weght (,, N) Suffcent condtons for prce weght 0 for all are β 0 and β 0 We therefore want to test these two coeffcent excluson restrctons The null and alternatve hypotheses are: H 0 : β 0 j, β 0 and β 0 (8) j H : β 0 j, β 0 and/or β 0 j ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 8 of 9 pages

19 ECONOMICS 5* -- Stata 0 Tutoral 7 The followng test command computes an F-test of H 0 aganst H Enter the command: test weght weghtsq Inspect the results generated by ths command State the nference you would draw from ths test Note: The above test command s equvalent to the followng sequence of test commands wth the accumulate opton: test weght 0 test weghtsq 0, accum Use the followng commands to compute the crtcal values of the F[,70] dstrbuton for sgnfcance levels α 00, 005, 00: dsplay nvftal(,70, 00) (returns value of F 0 0 [, 70] ) dsplay nvftal(,70, 005) (returns value of F 0 05 [, 70] ) dsplay nvftal(,70, 00) (returns value of F 0 [, 70] ) These are the crtcal values of the F[, 70]-dstrbuton approprate for testng hypothess (8) above Would you reject the null hypothess n (8) at the 0 percent sgnfcance level, at the 5 percent sgnfcance level, and/or at the percent sgnfcance level? Test 4: Test the hypothess that the margnal effect of weght on prce s constant n regresson equaton (7) e, that t does not depend on weght The necessary and suffcent condton for prce weght n equaton (7) to be constant s β 0 We therefore want to test ths coeffcent restrcton The null and alternatve hypotheses are: H 0 : β 0 (9) H : β 0 0 ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 9 of 9 pages

20 ECONOMICS 5* -- Stata 0 Tutoral 7 One way to test ths hypothess s to use the t-rato for ˆβ that s prnted by the regress command used to estmate equaton (7) The t-rato for ˆβ s gven n general by the expresson t βˆ (ˆ β ) ~ t[n K] t[74 4] t[70] under H 0 : β 0 sê(ˆ β ) 0 Alternatvely, you can calculate the above t-rato for by enterng the followng lncom command: lncom _b[weghtsq] ˆβ and ts two-tal p-value From the results of OLS estmaton of equaton (7), state the nference you would draw from ths t-test A second way to test ths hypothess s to compute an F-test of H 0 : β 0 aganst H : β 0 The sample value of the F-statstc for ths test s gven by the expresson ˆ β F0 (ˆ β ) ~ F[, N K] F[, 74 4] F[, 70] under H 0 : β 0 Vâr(ˆ β ) Enter ether of the followng test commands: test weghtsq test weghtsq 0 State the nference you would draw from ths F-test Compare the results of ths F-test wth those of the t-test performed above Are these two tests equvalent? Why or why not? Test 5: Test the hypothess that the margnal effect of mpg on prce s zero n regresson equaton (7), e, that prce does not depend on mpg The margnal effect of mpg on prce s obtaned by partally dfferentatng regresson equaton (7) wth respect to mpg : ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 0 of 9 pages

21 ECONOMICS 5* -- Stata 0 Tutoral 7 prce mpg β a constant (,, N) The necessary and suffcent condton for prce mpg n equaton (7) to equal zero s β 0 We therefore want to test ths coeffcent restrcton The null and alternatve hypotheses are: H 0 : β 0 (0) H : β 0 One way to test hypothess (0) s to use the t-rato for ˆβ that s prnted by the regress command used to estmate equaton (7) The t-rato for ˆβ s gven n general by the expresson t βˆ (ˆ β ) ~ t[n K] t[74 4] t[70] under H 0 : β 0 sê(ˆ β ) 0 Alternatvely, you can calculate the above t-rato for command: lncom _b[mpg] ˆβ by enterng the lncom From the results of OLS estmaton of equaton (7), state the nference you would draw from ths t-test A second way to test hypothess (0) s to perform an F-test of the sngle coeffcent restrcton specfed by the null hypothess H 0 : β 0 n (0) Enter ether of the followng test commands: test mpg test mpg 0 ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page of 9 pages

22 ECONOMICS 5* -- Stata 0 Tutoral 7 Note: The sample value of the F-statstc computed by the above test command equals the square of the sample value of the t-statstc for the null hypothess H 0 n (0): ( βˆ b ) ( βˆ 0) ( βˆ ) βˆ [ ] 0 ) t 0 ) F (βˆ Vâr(βˆ ) Vâr(βˆ ) Vâr(βˆ ) sê(βˆ ) Use the followng commands to verfy these relatonshps between t (ˆ β 0 ) and (ˆ ): F β 0 scalar t0b _b[mpg]/_se[mpg] scalar varb _se[mpg]^ scalar F0b _b[mpg]^/varb dsplay t0b dsplay sqrt(f0b) dsplay t0b^ dsplay F0b Use the followng commands to verfy that the two-tal p-value of the calculated t-statstc t (ˆ β 0 ) equals the p-value of the calculated F-statstc (ˆ ): F β 0 dsplay *ttal(70, abs(t0b)) dsplay Ftal(, 70, F0b) Fnally, calculate and compare the two-tal % crtcal value of the t[70]- dstrbuton and the % crtcal value of the F[, 70]-dstrbuton Enter the dsplay commands: dsplay nvttal(70, 0005)^ dsplay nvftal(, 70, 00) The results of these commands can be used to verfy that the α-level crtcal value of the F[, 70]-dstrbuton equals the square of the α/ crtcal value of the t[70] dstrbuton, e, that 0 0[, 70] t 0005[70] ( F ) or n general that F [, N K] t [N K α (βˆ ( ) α / ] ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page of 9 pages

23 ECONOMICS 5* -- Stata 0 Tutoral 7 Example : polynomals n two explanatory varables, weght and mpg Consder the followng multple regresson equaton: prce β + β weght + β mpg + β weght + β mpg + u () 0 4 where mpg the square of mpg Note: For PRE (), the number of regresson coeffcents s K 5 Regresson equaton () contans two explanatory varables, weght and mpg, but four regressors weght, weght, mpg, and mpg Margnal Effects of the Explanatory Varables The margnal effect of weght n equaton () s: prce weght β + β weght a lnear functon of weght The margnal effect of mpg n equaton () s: prce mpg β + β 4 mpg a lnear functon of mpg Before estmatng regresson equaton (), t s necessary to create the varable Enter the command: mpg gen mpgsq mpg^ Now estmate by OLS regresson equaton () Enter the command: regress prce weght mpg weghtsq mpgsq ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page of 9 pages

24 ECONOMICS 5* -- Stata 0 Tutoral 7 Test 6: Test the hypothess that the margnal effect of weght on prce s zero n regresson equaton () The margnal effect of weght on prce s obtaned by partally dfferentatng regresson equaton () wth respect to weght : prce weght β + β weght Suffcent condtons for prce weght 0 for all are β 0 and β 0 We therefore want to test these two coeffcent excluson restrctons on regresson equaton () The null and alternatve hypotheses are: H 0 : β 0 and β 0 H : β 0 and/or β 0 The followng test command computes an F-test of H 0 aganst H Enter the command: test weght weghtsq Inspect the results generated by ths command State the nference you would draw from ths test Test 7: Use a lncom command to compute the estmated value of the margnal effect of weght n equaton () for a car that weghs,000 pounds, and to perform a two-tal t-test of the null hypothess that the margnal effect of weght n equaton () for a car that weghs,000 pounds equals zero Recall that the expresson for prce weght n equaton () s: prce weght β + β weght Evaluated at weght 000, the estmated margnal effect of weght on prce n equaton () s: ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 4 of 9 pages

25 ECONOMICS 5* -- Stata 0 Tutoral 7 prce estmate of βˆ + ˆ β(000) weght The null and alternatve hypotheses are: H 0 : β + β (000) 0 H : β + β (000) 0 Enter ether of the followng two lncom commands: lncom _b[weght] + *_b[weghtsq]*000 lncom weght + *weghtsq*000 Note that n the second of the above lncom commands varable names are used to represent ther correspondng coeffcent estmates But f you fnd ths notaton confusng, just use the frst form of the above lncom commands Inspect the output generated by the above lncom commands Would you nfer that the margnal effect of weght on prce s dfferent from zero for a car that weghs,000 pounds? Test 8: Test the hypothess that the margnal effect of mpg on prce s zero n regresson equaton () The margnal effect of mpg on prce s obtaned by partally dfferentatng regresson equaton () wth respect to mpg prce mpg β + β 4 mpg Suffcent condtons for prce mpg 0 for all are β 0 and β 4 0 We therefore want to test these two coeffcent excluson restrctons The null and alternatve hypotheses are: ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 5 of 9 pages

26 ECONOMICS 5* -- Stata 0 Tutoral 7 H 0 : β 0 and β 4 0 H : β 0 and/or β 4 0 The followng test command computes an F-test of H 0 aganst H Enter the command: test mpg mpgsq Inspect the results generated by ths command State the nference you would draw from ths test Test 9: Use a lncom command to compute the estmated value of the margnal effect of mpg n equaton () for a car that gets 5 mles to the gallon, and to compute a two-tal t-test of the null hypothess that the margnal effect of mpg n equaton () for a car that gets 5 mles to the gallon equals zero Recall that the expresson for prce mpg n equaton () s: prce mpg β + β 4 mpg Evaluated at mpg 5, the estmated margnal effect of mpg on prce s: prce estmate of βˆ + ˆ β4(5) mpg The null and alternatve hypotheses are: H 0 : β + β 4 (5) 0 H : β + β 4 (5) 0 Enter ether (or both) of the followng lncom commands: lncom _b[mpg] + *_b[mpgsq]*5 lncom mpg + *mpgsq*5 ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 6 of 9 pages

27 ECONOMICS 5* -- Stata 0 Tutoral 7 What s the estmated value of the margnal effect of mpg on prce for a car that has fuel effcency equal to 5 mles per gallon? Would you nfer that the margnal effect of mpg on prce s dfferent from zero for a car that has fuel effcency equal to 5 mles per gallon? Interacton Varables: Interactons Between Two Contnuous Varables Nature: Interactons between two contnuous varables refer to products of explanatory varables For example, f X j and X k are two contnuous explanatory varables, the nteracton term between them s the product X j X k The product X j X k s ncluded as a regressor n the populaton regresson functon Usage: An nteracton term between two contnuous varables allows the margnal effect of one explanatory varable to be a lnear functon of the other explanatory varable Consder the followng regresson equaton: Y β + β X + β X + β X + β X + β X X + u () Compute the margnal effects on Y of the two explanatory varables X and X n equaton () The margnal effect of X n equaton () s: Y X β + β X + β 5 X a lnear functon of both X and X The margnal effect of X n equaton () s: Y X β + β 4 X + β 5 X a lnear functon of both X and X ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 7 of 9 pages

28 ECONOMICS 5* -- Stata 0 Tutoral 7 Example : polynomals and an nteracton term n two explanatory varables prce β0 + βweght + βmpg + βweght + β4mpg + β5weghtmpg + u () Note: For PRE (), the number of regresson coeffcents s K 6 Regresson equaton () contans two explanatory varables, weght and mpg, but fve regressors weght, weght, mpg, mpg, and weght mpg Margnal Effects of the Explanatory Varables The margnal effect of weght n equaton () s: prce weght β + β weght + β 5 mpg a lnear functon of both weght and mpg The margnal effect of mpg n equaton () s: prce mpg β + β 4 mpg + β 5 weght a lnear functon of both weght and mpg Before estmatng regresson equaton (), t s necessary to create the nteracton term weght mpg ) Enter the command: ( generate wgtmpg weght*mpg Now estmate by OLS regresson equaton () Enter the command: regress prce weght weghtsq mpg mpgsq wgtmpg ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 8 of 9 pages

29 ECONOMICS 5* -- Stata 0 Tutoral 7 Test 0: Test the hypothess that the margnal effect of weght on prce s zero n regresson equaton () The margnal effect of weght on prce n equaton () s prce weght β + β weght + β 5 mpg Suffcent condtons for prce weght 0 for all are β 0 and β 0 and β 5 0 We therefore want to test these three coeffcent excluson restrctons The null and alternatve hypotheses are: H 0 : β 0 and β 0 and β 5 0 (4) H : β 0 and/or β 0 and/or β 5 0 The followng test command computes an F-test of H 0 aganst H n (4) Enter the command: test weght weghtsq wgtmpg Inspect the results generated by ths command State the nference you would draw from ths test Test : Use a lncom command to compute the estmated value of the margnal effect of weght n equaton () for a car that weghs,000 pounds and gets 0 mles to the gallon, and to perform a two-tal t-test of the null hypothess that ths margnal effect of weght n equaton () s equal to zero Recall that the expresson for prce weght n equaton () s: prce weght β + β weght + β 5 mpg Evaluated at weght 000 and mpg 0, the estmated margnal effect of weght on prce s: ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 9 of 9 pages

30 ECONOMICS 5* -- Stata 0 Tutoral 7 prce estmate of βˆ ˆ (000) ˆ + β + β5(0) weght The null and alternatve hypotheses are: H 0 : β + β (000) + β (0) 0 5 H : β + β (000) + β (0) 0 5 Enter ether (or both) of the followng lncom commands: lncom _b[weght] + *_b[weghtsq]*000 + _b[wgtmpg]*0 lncom weght + *weghtsq*000 + wgtmpg*0 Inspect the output generated by the above lncom commands Would you nfer that the margnal effect of weght on prce s dfferent from zero for a car that weghs,000 pounds and has fuel effcency equal to 0 mles per gallon? Test : Test the hypothess that the margnal effect of mpg on prce s zero n regresson equaton () The margnal effect of mpg on prce s obtaned by partally dfferentatng regresson equaton () wth respect to mpg prce mpg β + β 4 mpg + β 5 weght Suffcent condtons for prce mpg 0 for all are β 0 and β 4 0 and β 5 0 We therefore want to test these three coeffcent restrctons The null and alternatve hypotheses are: H 0 : β 0 and β 4 0 and β 5 0 (5) H : β 0 and/or β 4 0 and/or β 5 0 The followng test command computes an F-test of H 0 aganst H n (5) Enter the command: ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 0 of 9 pages

31 ECONOMICS 5* -- Stata 0 Tutoral 7 test mpg mpgsq wgtmpg Inspect the results generated by ths command State the nference you would draw from ths test Test : Use a lncom command to compute the estmated value of the margnal effect of mpg n equaton () for a car that weghs,000 pounds and gets 0 mles to the gallon, and to perform a two-tal t-test of the null hypothess that the margnal effect of mpg n equaton () for a car that weghs,000 pounds and gets 0 mles to the gallon equals zero Recall that the expresson for prce mpg n equaton () s: prce mpg β + β 4 mpg + β 5 weght Evaluated at weght 000 and mpg 0, the estmated margnal effect of mpg on prce s: prce estmate of βˆ ˆ (0) ˆ + β4 + β5(000) mpg The null and alternatve hypotheses are: H 0 : β + β (0) + β (000) H : β + β (0) + β (000) Enter ether (or both) of the followng lncom commands: lncom _b[mpg] + *_b[mpgsq]*0 + _b[wgtmpg]*000 lncom mpg + *mpgsq*0 + wgtmpg*000 Would you nfer that the margnal effect of mpg on prce s dfferent from zero for a car that weghs,000 pounds and has fuel effcency equal to 0 mles per gallon? What s the estmated value of ths margnal effect of mpg on prce? ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page of 9 pages

32 ECONOMICS 5* -- Stata 0 Tutoral 7 Testng the Jont Sgnfcance of All Slope Coeffcent Estmates -- Alternatve Methods A basc test of any estmated sample regresson equaton s a test of the jont sgnfcance of all ts slope coeffcent estmates The reason for the mportance of ths partcular hypothess test s that addresses an mportant queston about the emprcal usefulness and relevance of the postulated model That queston s: Are the regressors n the regresson functon jontly relevant n explanng the varaton of the dependent varable? Do all the regressors together actually help account for the varaton n the dependent varable? Ths secton demonstrates the equvalence of three alternatve methods for computng an F-test of the jont sgnfcance of all the slope coeffcent estmates n a multple lnear regresson model Snce a multple regresson model contans at least two non-constant regressors, the null hypothess for such a jont sgnfcance test specfes two or more coeffcent excluson restrctons An F-test s therefore approprate The Model To llustrate the alternatve methods of performng an F-test of the jont sgnfcance of all slope coeffcent estmates, use the multple lnear regresson model for car prces gven by equaton (): prce β0 + βweght + βmpg + βweght + β4mpg + β5weghtmpg + u () For regresson model (), the total number of regresson coeffcents s K 6, and the number of slope coeffcents s k K 6 5 Re-estmate model () by enterng the followng command: regress prce weght mpg weghtsq mpgsq wgtmpg Alternatvely, you can redsplay the results of the most recently ssued regress command by smply enterng the command: ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page of 9 pages

33 ECONOMICS 5* -- Stata 0 Tutoral 7 regress The Null and Alternatve Hypotheses H 0 : β j 0 for all j,, 5 β 0 and β 0 and β 5 0; (6) H : β j 0 for j,, 5 β 0 and/or β 0 and/or β 5 0 Method : the ANOVA F-statstc Method uses the ANOVA F-statstc derved from the Analyss-of-Varance table for the OLS sample regresson equaton obtaned by OLS estmaton of equaton () The ANOVA F-statstc can be computed usng two alternatve but equvalent formulas: ESS dfess ESS (K ) R (K ) (7) RSS df RSS (N K) ( R ) (N K) FANOVA RSS where: ESS the Explaned Sum of Squares for the unrestrcted OLS-SRE; RSS the Resdual Sum of Squares for the unrestrcted OLS-SRE; df ESS K the degrees of freedom for ESS; df RSS N K the degrees of freedom for RSS; K the total number of regresson coeffcents n the unrestrcted model; N the number of sample observatons n the estmaton sample R ESS RSS the R-squared for the unrestrcted OLS-SRE TSS TSS The null dstrbuton of the ANOVA F-statstc -- that s, the dstrbuton of F ANOVA under the null hypothess that all K slope coeffcents jontly equal zero s the F[K, N K] F[df ESS, df RSS ] dstrbuton: F ~ F[K, N K] F[df, df ] under H 0 : β j 0 j,, k ANOVA ESS RSS ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page of 9 pages

34 ECONOMICS 5* -- Stata 0 Tutoral 7 Recall from Stata Tutoral that selected results of the regress command are temporarly stored n e( ) functons To compute the ANOVA F-statstc, you wll want to save the values of the terms that enter the formulas for F ANOVA To do ths, enter the followng scalar commands: scalar N e(n) scalar ESS e(mss) scalar dfess e(df_m) scalar RSS e(rss) scalar dfrss e(df_r) scalar TSS ESS + RSS scalar dftss N - scalar lst N ESS dfess RSS dfrss TSS dftss Calculate the sample value of the ANOVA F-statstc usng the formula F ANOVA ESS dfess ESS (K ) (7a) RSS df RSS (N K) RSS Enter the commands: scalar Fanova (ESS/dfESS)/(RSS/dfRSS) scalar lst Fanova ESS RSS dfess dfrss Calculate and dsplay the p-value for the ANOVA F-statstc F ANOVA Enter the command: dsplay Ftal(dfESS, dfrss, Fanova) Calculate and dsplay the 5% and % crtcal values of the F[K, N K] F[df ESS, df RSS ] F[5, 68] dstrbuton Enter the commands: dsplay nvftal(dfess, dfrss, 005) dsplay nvftal(dfess, dfrss, 00) Now calculate the sample value of the ANOVA F-statstc usng the alternatve formula R dfess R (K ) (7b) ( R ) df ( R ) (N K) FANOVA RSS ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 4 of 9 pages

35 ECONOMICS 5* -- Stata 0 Tutoral 7 You wll frst want to generate the value of R, the R-squared for the estmated sample regresson equaton Enter the commands: scalar Rsq ESS/TSS scalar Fanova (Rsq/dfESS)/((-Rsq)/dfRSS) scalar lst Fanova Rsq dfess dfrss Calculate and dsplay the p-value for the ANOVA F-statstc F ANOVA Enter the command: dsplay Ftal(dfESS, dfrss, Fanova) Fnally, compare the sample values of the two ANOVA F-statstcs F ANOVA and F ANOVA Enter the command: scalar lst Fanova Fanova As expected, F ANOVA F ANOVA Method : the General Form F-statstc A second method of computng an F-test of hypothess (6) employs the followng general form of the F-statstc (see Note 7): F 0 (RSS RSS 0 0 (8) RSS ) df (df df ) where: RSS 0 the resdual sum of squares for the restrcted OLS-SRE; RSS the resdual sum of squares for the unrestrcted OLS-SRE; df 0 N K 0 the degrees of freedom for RSS 0, the restrcted RSS; df N K the degrees of freedom for RSS, the unrestrcted RSS; K 0 the number of free regresson coeffcents n the restrcted model; K the number of free regresson coeffcents n the unrestrcted model; df 0 df N K 0 (N K) N K 0 N + K K K 0 ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 5 of 9 pages

36 ECONOMICS 5* -- Stata 0 Tutoral 7 Ths general form F-statstc nvolves the resdual sum of squares for both the unrestrcted OLS sample regresson equaton correspondng to the alternatve hypothess H and the restrcted OLS sample regresson equaton correspondng to the null hypothess H 0 To compute ths general form F-statstc, t s therefore necessary to estmate by OLS both the unrestrcted model mpled the alternatve hypothess H and the restrcted model mpled by the null hypothess H 0 The unrestrcted model mpled the alternatve hypothess H n (6) s smply regresson equaton (): prce β0 + βweght + βmpg + βweght + β4mpg + β5weghtmpg + u () The restrcted model mpled the null hypothess H 0 n (6) s obtaned by settng β 0 and β 0 and β 0 and β 4 0 and β 5 0 n the unrestrcted regresson equaton (): prce β + u 0 (9) The followng seres of commands llustrates the computaton of the general form F-statstc (8) Estmate by OLS the unrestrcted PRE () and retreve the unrestrcted RSS (RSS ) and ts degrees-of-freedom (df ) Enter the commands: regress prce weght mpg weghtsq mpgsq wgtmpg scalar RSS e(rss) scalar df e(df_r) scalar lst RSS df Save for later use the sample value of the F-statstc generated by the above regress command Enter the commands: scalar Fstateq0 e(f) scalar lst Fstateq0 dfess dfrss Estmate by OLS the restrcted PRE (9) and retreve the restrcted RSS (RSS 0 ) and ts degrees-of-freedom (df 0 ) Enter the commands: regress prce ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 6 of 9 pages

37 ECONOMICS 5* -- Stata 0 Tutoral 7 scalar RSS0 e(rss) scalar df0 e(df_r) scalar lst RSS0 df0 To see how to nterpret the OLS estmate of the ntercept coeffcent β 0 n the restrcted regresson equaton (9), compute the sample mean of the dependent varable prce Enter the command: summarze prce ~ Compare the results of ths command wth the restrcted OLS estmate β0 the ntercept coeffcent β 0 n restrcted regresson equaton (9): you wll observe that β ~ 0 prce, the sample mean of the dependent varable prce of Calculate and dsplay the sample value of the F-statstc Enter the commands: scalar F0 ((RSS0 - RSS)/(df0 - df))/(rss/df) scalar lst F0 RSS0 RSS df0 df Calculate the p-value of the calculated F-statstc F 0 Frst create scalars representng the numerator degrees of freedom dfnum df0 df K K0 and the denomnator degrees of freedom dfden df N K for the null dstrbuton of F 0 Enter the commands: scalar dfnum df0 - df scalar dfden df scalar lst dfnum dfden dsplay Ftal(dfnum, dfden, F0) Use the followng commands to compute the crtcal values of the F[dfnum, dfden] F[5, 68]-dstrbuton for sgnfcance levels α 00, 005, 00: dsplay nvftal(5, 68, 00) dsplay nvftal(5, 68, 005) dsplay nvftal(5, 68, 00) Fnally, compare the sample values of the two ANOVA F-statstcs F ANOVA and F ANOVA computed by Method wth the sample value of the F-statstc F 0 computed by Method and the sample value of the F-statstc that s outputted by the regress command for the unrestrcted equaton () Enter the command: ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 7 of 9 pages

38 ECONOMICS 5* -- Stata 0 Tutoral 7 scalar lst Fanova Fanova F0 Fstateq0 As expected, these four F-statstcs all have the same sample value What nference would you draw from these F-statstcs respectng the jont sgnfcance of the slope coeffcent estmates n equaton ()? Method : the test command As you have learned, a sngle test command can be used to perform an F-test of the jont sgnfcance of all slope coeffcent estmates n a lnear regresson model such as equaton () Ths test command produces the same sample value of the F-statstc as do Methods and above, but t uses yet another form of the F- statstc The followng seres of commands llustrates the equvalence of the test command and the other methods Re-estmate regresson equaton () Enter agan the followng regress command: regress prce weght mpg weghtsq mpgsq wgtmpg Now enter the followng test command: test weght mpg weghtsq mpgsq wgtmpg Ths test command performs a jont F-test of the null hypothess H 0 n (6) that all fve slope coeffcents n equaton () are jontly equal to zero aganst the alternatve hypothess H that at least one of these slope coeffcents s not equal to zero Compare the sample value and p-value of the F-statstc computed by ths test command wth those of the F-statstcs computed by Methods and above Enter the commands: scalar lst Fanova Fanova F0 Fstateq0 dsplay Ftal(dfnum, dfden, F0) dsplay Ftal(dfESS, dfrss, Fanova) dsplay Ftal(dfESS, dfrss, Fstateq0) ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 8 of 9 pages

39 ECONOMICS 5* -- Stata 0 Tutoral 7 Preparng to End Your Stata Sesson Before you end your Stata sesson, you should do three thngs Frst, you may want to save the current dataset (although you wll not need t for future tutorals) Enter the followng save command to save the current dataset as Stata-format dataset auto7dta: save auto7 Second, close the command log fle you have been recordng Enter the command: cmdlog close Thrd, close the log fle you have been recordng Enter the command: log close End Your Stata Sesson -- ext To end your Stata sesson, use the ext command Enter the command: ext or ext, clear Cleanng Up and Clearng Out After returnng to Wndows, you should copy all the fles you have used and created durng your Stata sesson to your own dskette These fles wll be found n the Stata workng drectory, whch s usually C:\data on the computers n Dunnng 50, and D:\courses on the computers n MC B There are two fles you wll want to be sure you have: the Stata log fle 5tutoral7log, and the command log fle 5tutoral7txt If you saved the Stata-format data set auto7dta, you wll probably want to take t wth you as well Use the Wndows copy command to copy any fles you want to keep to your own portable electronc storage devce (eg, flash memory stck) n the E:-drve (or to a dskette n the A:- drve) Fnally, as a courtesy to other users of the computng classroom, please delete all the fles you have used or created from the Stata workng drectory ECON 5* -- Wnter 008: Stata 0 Tutoral 7 Page 9 of 9 pages