Stata 12/13 Tutorial 4

Transcription

1 Stata /3 Tutoral 4 TOPIC: Condtonal and Margnal Effects of Contnuous Explanatory Varables n Lnear Regresson Models DATA: auto.dta (a Stata-format dataset frst created n Stata /3 Tutoral ) TASKS: Stata /3 Tutoral 4 deals wth computng the condtonal and margnal effects of ndvdual contnuous explanatory varables on the dependent varable n lnear regresson models. It also demonstrates how to create a lne graph of the margnal effect of a contnuous explanatory varable and how to save the lne graph n several dfferent fle formats for future use. The condtonal effect of an explanatory varable on the dependent varable s the ceters parbus relatonshp between the values of that explanatory varable and the condtonal mean of the dependent varable for fxed values of all other explanatory varables n the regresson functon. The margnal effect of an explanatory varable on the dependent varable s the ceters parbus (or partal) effect of a unt ncrease n that explanatory varable on the condtonal mean of the dependent varable. The Stata commands that consttute the prmary subject of ths tutoral are: regress scalar lncom margns margnsplot graph graph save Used to perform OLS estmaton of multple lnear regresson models. Used to save as scalars elements of the coeffcent vector and the coeffcent covarance matrx after OLS estmaton. Used after estmaton to compute lnear combnatons of coeffcent estmates and assocated statstcs. Used after OLS estmaton to compute estmates of the margnal effects of contnuous explanatory varables. Used to draw lne graphs of the margnal effects of contnuous explanatory varables n lnear regresson models. Used to graph or plot the condtonal and margnal effects of contnuous explanatory varables n lnear regresson models Saves the graph currently dsplayed n the Graph wndow to a dsk fle n the current Stata workng drectory. Page of 33 pages

2 graph export Exports the graph currently dsplayed n the Graph wndow to a fle n the current Stata workng drectory. graph use Loads a graph on dsk nto memory and dsplays t n the Graph wndow. NOTE: Stata commands are case senstve. All Stata command names must be typed n the Command wndow n lower case letters. HELP: Stata has an extensve on-lne Help faclty that provdes farly detaled nformaton (ncludng examples) on all Stata commands. Students should become famlar wth the Stata on-lne Help system. In the course of dong ths tutoral, take the tme to browse the Help nformaton on some of the above Stata commands. To access the on-lne Help for any Stata command: choose (clck on) Help from the Stata man menu bar clck on Stata Command n the Help drop down menu type the full name of the Stata command n the Stata command dalog box and clck OK Page of 33 pages

3 Preparng for Your Stata Sesson Before begnnng your Stata sesson, use Wndows Explorer to copy the Stata-format dataset auto.dta to the Stata workng drectory on the C:-drve or D:-drve of the computer at whch you are workng. If you dd not save the Stata-format dataset auto.dta you created durng Stata /3 Tutoral and brng t wth you on a portable storage devce such as a flash memory stck, you wll have to repeat the relevant secton of Stata /3 Tutoral before proceedng wth ths tutoral. On the computers n Dunnng 35, the default Stata workng drectory s usually C:\data. Start Your Stata Sesson To start your Stata sesson, double-clck on the Stata or Stata 3 con on the Wndows desktop. After you double-clck the Stata or Stata 3 con, you wll see the famlar screen of fve Stata wndows. Record Your Stata Sesson -- log usng To record your Stata sesson, ncludng all the Stata commands you enter and the results (output) produced by these commands, make a text-format.log fle named 45tutoral4.log. To open (begn) the log fle 45tutoral4.log, enter n the Command wndow: log usng 45tutoral4.log Ths command opens a text-format (ASCII) fle called 45tutoral4.log n the current Stata workng drectory. Note: It s mportant to nclude the.log fle extenson when openng a log fle; f you do not, your log fle wll be n smcl format, a format that only Stata can read. Page 3 of 33 pages

4 Record Only Your Stata Commands -- cmdlog usng To record only the Stata commands you type durng your Stata sesson, you can use the Stata cmdlog usng command. To start (open) the command log fle 45tutoral4.txt, enter n the Command wndow: cmdlog usng 45tutoral4 Ths command opens a plan text-format (ASCII) fle called 45tutoral4.txt n the current Stata workng drectory. All commands you enter durng your Stata sesson are recorded n ths fle. At the end of ths tutoral, you wll close your command log fle n the same way you close your regular log fle. Loadng a Stata-Format Dataset nto Stata use To check that the Stata-format dataset auto.dta s n the current Stata workng drectory of the computer at whch you are workng, type n the Command wndow: dr auto.* You should see n the Stata Results wndow the flename auto.dta. To load, or read, nto memory the Stata-format dataset auto.dta, type n the Command wndow: use auto Ths command loads nto memory the Stata-format dataset auto.dta. To summarze the contents of the current dataset, use the descrbe command. Recall from Stata /3 Tutoral that the descrbe command dsplays a summary of the contents of the current dataset n memory, whch n ths case s the Stataformat data fle auto.dta. Type n the Command wndow: descrbe Page 4 of 33 pages

5 To compute summary statstcs for the varables n the current dataset, use the summarze command. Recall from Stata /3 Tutoral that the summarze command computes descrptve summary statstcs for all numerc varables n the current dataset n memory. Type n the Command wndow: summarze The Condtonal Effects of Indvdual Explanatory Varables: Defnton Nature: The condtonal effect of an explanatory varable on the dependent varable s the ceters parbus relatonshp between the values of that explanatory varable and the condtonal mean of the dependent varable for fxed values of all other explanatory varables n the regresson functon. Consder the followng lnear regresson model for car prces (whch s Model of Stata /3 Tutoral ): prce = β + u. () The populaton regresson functon for regresson equaton () s wrtten n general as: E(prce, ) = β () There are two explanatory varables n model () and. The condtonal effect of each s obtaned by settng the other explanatory varable equal to some specfed (or selected) value n the populaton regresson equaton n other words, by holdng constant the other explanatory varable. Condtonal effect of on prce The condtonal effect of on prce s obtaned by settng the other explanatory varable n regresson functon () namely equal to some specfed value such as =. The condtonal effect of on prce for = s thus: Page 5 of 33 pages

6 E(prce, ) = β = β = ( β ) + ( β ) (3) As the above equaton ndcates, the condtonal effect of on prce s a quadratc functon of once the value of s fxed at the specfed value. Interpretaton: The condtonal effect of on prce represents the ceters parbus, or "other thngs equal," relatonshp between mean prce and the explanatory varable. In other words, t solates the relatonshp between and prce from the relatonshps of the other explanatory varable(s) to prce. But t controls for, or adjusts for, the effect(s) on prce of the other explanatory varable(s) n ths case the explanatory varable. It s therefore often referred to as the regresson-adjusted relatonshp of to prce. Estmaton: The condtonal effect of on prce for = s estmated by replacng the populaton regresson coeffcents β j (j =,,, 5) n expresson (3) for E(prce, ) wth ther OLS estmates ˆβ j (j =,,, 5): Ê(prce, ) = (ˆ β ˆ ˆ 4 ) + (ˆ β ˆ 5 ) ˆ 3 (4) Condtonal effect of on prce The condtonal effect of on prce s obtaned by settng the other explanatory varable n regresson functon () namely equal to some specfed value such as =. The condtonal effect of on prce for = s thus: Page 6 of 33 pages

7 E(prce, ) = β = β = ( β ) + ( β ) (5) The above equaton ndcates that the condtonal effect of on prce s a quadratc functon of once the value of s fxed at the specfed value. Estmaton: The condtonal effect of on prce for = s estmated by replacng the populaton regresson coeffcents β j (j =,,, 5) n expresson (5) for E(prce, ) wth ther OLS estmates ˆβ j (j =,,, 5): Ê(prce, ) = (ˆ β ˆ ˆ 3 ) + (ˆ β ˆ 5 ) ˆ 4 (6) The Margnal Effects of Indvdual Explanatory Varables: Defnton The margnal effect of an ndvdual explanatory varable s a functon that gves the change n the mean value of the dependent varable n response to a one-unt ncrease n the value of the explanatory varable, holdng constant the effects of all other explanatory varables n the model. Consder agan Model for car prces, whch s gven by the populaton regresson equaton (): prce = β + u. () Re-wrte the populaton regresson functon (or PRF) for Model as: E(prce, ) = β () Page 7 of 33 pages

8 Margnal effect of on prce The margnal effect of on prce s obtaned by partally dfferentatng regresson equaton (), or equvalently the populaton regresson functon (), wth respect to : prce = E ( prce, ) = 3 5 β (7) Estmates of the margnal effect of on prce are obtaned by substtutng n (7) the OLS coeffcent estmates ˆβ j for the populaton regresson coeffcents β j : estmate of prce = Ê ( prce, ) β ˆ ˆ ˆ (8) = 3 5 The estmated margnal effect of on prce can be evaluated at any chosen values of and. Margnal effect of on prce The margnal effect of on prce s obtaned by partally dfferentatng regresson equaton (), or equvalently the populaton regresson functon (), wth respect to : prce = E ( prce, ) = 4 5 β (9) Estmates of the margnal effect of on prce are obtaned by substtutng n (9) the OLS coeffcent estmates ˆβ j for the populaton regresson coeffcents β j : estmate of prce = Ê ( prce, ) β ˆ ˆ ˆ () = 4 5 Page 8 of 33 pages

9 The estmated margnal effect of on prce can be evaluated at any chosen values of and. Computng "Typcal" Values of Varables as Scalars Computng both condtonal and margnal effects requres the selecton of certan fxed values of the explanatory varables and. In the foregong expressons for the condtonal and margnal effects of and, these selected fxed values are denoted as and. But what specfc values mght one select for and? Common choces for the values of and are sample means and sample percentles such as the medan (5-th percentle), the frst quartle (5-th percentle), and the thrd quartle (75-th percentle). In ths secton, we use the summarze and scalar commands to create these values as scalars. To create as scalars the sample mean and the 5-th, 5-th and 75-th sample percentles of the varable, enter the followng commands: summarze, detal return lst scalar bar = r(mean) scalar 5 = r(p5) scalar 5 = r(p5) scalar 75 = r(p75) scalar lst bar To create as scalars the sample mean and the 5-th, 5-th and 75-th sample percentles of the varable, enter the followng commands: summarze, detal return lst scalar bar = r(mean) scalar 5 = r(p5) scalar 5 = r(p5) scalar 75 = r(p75) scalar lst bar Page 9 of 33 pages

10 Computng the Condtonal Effects of Indvdual Explanatory Varables Frst, f your current dataset does not already nclude them, you wll have to generate the regressors, and that enter Model but that are not ncluded n the current dataset. In that case, enter the commands: generate sq = ^ generate sq = ^ generate = * summarze prce sq sq Now estmate regresson equaton () by OLS. Enter the command regress prce sq sq Use the scalar command to save the OLS coeffcent estmates, and the scalar lst command to dsplay the saved scalars. Enter the commands: scalar b = _b[] scalar b = _b[] scalar b3 = _b[sq] scalar b4 = _b[sq] scalar b5 = _b[] scalar b = _b[_cons] scalar lst b b b3 b4 b5 b Condtonal Effect of The estmated condtonal effect of on prce for some gven value of s the followng quadratc functon of : Ê(prce, ) = (ˆ β ˆ ˆ 4 ) + (ˆ β ˆ 5 ) ˆ 3 (4) Create as a varable the condtonal effect of on prce for = the sample mean of. Enter on one lne the generate command: generate cebar = b + b*bar + b4*(bar^) + (b + b5*bar)* + b3*sq Page of 33 pages

11 Create as a varable the condtonal effect of on prce for = the 5-th sample percentle of. Enter on one lne the generate command: generate ce5 = b + b*5 + b4*(5^) + (b + b5*5)* + b3*sq Create as a varable the condtonal effect of on prce for = the 5-th sample percentle of. Enter on one lne the generate command: generate ce5 = b + b*5 + b4*(5^) + (b + b5*5)* + b3*sq Create as a varable the condtonal effect of on prce for = the 75-th sample percentle of. Enter on one lne the generate command: generate ce75 = b + b*75 + b4*(75^) + (b + b5*75)* + b3*sq Lst and compute summary statstcs for the values of the four condtonal effects of you have just created, after frst sortng the sample observatons n ascendng order of the sample values of. Enter the commands: sort lst cebar ce5 ce5 ce75 summarze cebar ce5 ce5 ce75 Condtonal Effect of The estmated condtonal effect of on prce for some gven value of s the followng quadratc functon of : Ê(prce, ) = (ˆ β ˆ ˆ 3 ) + (ˆ β ˆ 5 ) ˆ 4 (6) Create as a varable the condtonal effect of on prce for = the sample mean of. Enter on one lne the generate command: generate cebar = b + b*bar + b3*(bar^) + (b + b5*bar)* + b4*sq Page of 33 pages

12 Create as a varable the condtonal effect of on prce for = the 5-th sample percentle of. Enter on one lne the generate command: generate ce5 = b + b*5 + b3*(5^) + (b + b5*5)* + b4*sq Create as a varable the condtonal effect of on prce for = the 5-th sample percentle of. Enter on one lne the generate command: generate ce5 = b + b*5 + b3*(5^) + (b + b5*5)* + b4*sq Create as a varable the condtonal effect of on prce for = the 75-th sample percentle of. Enter on one lne the generate command: generate ce75 = b + b*75 + b3*(75^) + (b + b5*75)* + b4*sq Lst and compute summary statstcs for the four condtonal effects of you have just created, after frst sortng the sample observatons n ascendng order of the sample values of. Enter the commands: sort lst cebar ce5 ce5 ce75 summarze cebar ce5 ce5 ce75 Graphng the Condtonal Effects of Indvdual Explanatory Varables The estmated condtonal effect of on prce for any gven value of s a quadratc functon of. In order to better understand the nature of these condtonal effect functons, t s often nstructve to draw lne graphs of the estmated condtonal effect of on prce ; these plot the estmated values of the condtonal effect of aganst the sample values of sorted n ascendng order. Graphng the Condtonal Effect of Frst, to ensure that the sample observatons are sorted n ascendng order of the sample values of, enter the followng sort command: sort Page of 33 pages

13 Use the graph twoway lne command to draw a twoway lne graph of the estmated condtonal effect of on prce as a functon of whle holdng the value of the other contnuous varable constant at ts sample mean value. Enter the commands: graph twoway lne cebar lne cebar Note that the command name graph twoway lne can be abbrevated to smply lne. Use the yttle ( ) and xttle( ) optons to add your own ttles to the vertcal y-axs and horzontal x-axs of the above lne graph. Enter on one lne the command: lne cebar, yttle("condtonal Mean Car Prce ($)") xttle("car Weght (pounds)") Use the ttle( ) and subttle( ) optons to add ttles to the above fgure. Enter on one lne the command: lne cebar, yttle("condtonal Mean Car Prce ($)") xttle("car Weght (pounds)") ttle("condtonal Effect of Weght on Car Prce") subttle("for Sample Mean Value of Fuel Effcency ()") Use the graph twoway lne command to draw n the same dagram lne graphs of the estmated condtonal effects of on prce as functons of for the 5- th, 5-th and 75-th sample percentle values of. Enter on one lne the command: graph twoway lne ce5 ce5 ce75, yttle("condtonal Mean Car Prce ($)") xttle("car Weght (pounds)") ttle("condtonal Effect of Weght on Car Prce") subttle("for Selected Percentle Values of Fuel Effcency ()") Modfy the above dagram by usng the ylabel( ) and xlabel( ) optons to change the default labelng and tckng of values on the left vertcal and bottom horzontal axes of the dagram. Enter on one lne the command: Page 3 of 33 pages

14 lne ce5 ce5 ce75, yttle("condtonal Mean Car Prce ($)") xttle("car Weght (pounds)") ttle("condtonal Effect of Weght on Car Prce") subttle("for Selected Percentle Values of Fuel Effcency ()") ylabel((5)) xlabel((5)5) Note the effects of the added optons on the appearance of the fgure, n partcular on the value labelng of the two axes. Note too that the command name has been abbrevated from graph twoway lne to smply lne. Fnally, use the legend ( ) opton to change the default legend of the above dagram. Enter on one lne the command: lne ce5 ce5 ce75, yttle("condtonal Mean Car Prce ($)") xttle("car Weght (pounds)") ttle("condtonal Effect of Weght on Car Prce") subttle("for Selected Percentle Values of Fuel Effcency ()") ylabel((5)) xlabel((5)5) legend(label( " = 5-th percentle") label( " = 5-th percentle") label(3 " = 75-th percentle")) Graphng the Condtonal Effect of Frst, sort the sample observatons n ascendng order of the sample values of. Enter the followng sort command: sort Use the graph twoway lne command to draw a lne graph of the estmated condtonal effect of on prce as a functon of, condtonal on the sample mean value of. Enter on one lne the command: graph twoway lne cebar, yttle("condtonal Mean Car Prce ($)") xttle("fuel Effcency ( = mles per gallon)") Use the graph twoway lne command to draw n the same dagram lne graphs of the estmated condtonal effect of on prce as a functon of for the 5- th, 5-th and 75-th sample percentle values of. Use approprate ttle( ), subttle( ), ylabel( ), and legend( ) optons to make your fgure more readable. Enter on one lne the command: Page 4 of 33 pages

15 lne ce5 ce5 ce75, yttle("condtonal Mean Car Prce ($)") xttle("fuel Effcency ( = mles per gallon)") ttle("condtonal Effect of Fuel Effcency on Car Prce") subttle("for Selected Percentle Values of Car Weght ()") ylabel((5)) legend(label( " = 5-th percentle") label( " = 5-th percentle") label(3 " = 75-th percentle")) Computng the Margnal Effects of Indvdual Explanatory Varables The lncom command s a convenent tool not only for evaluatng margnal effects for selected values of the explanatory varables, but also for performng two-tal tests of the null hypothess that the margnal effect s zero for the specfed values of the explanatory varables. Frst, re-estmate the model by OLS by enterng ether one of the followng commands: regress regress prce sq sq Margnal Effects of The estmated margnal effect of on prce for some gven values of and of s calculated by the followng lnear functon of and : estmate of prce = Ê ( prce, ) ˆ = β 3 5 ˆ ˆ Use the followng lncom command to evaluate the estmated margnal effect of on prce for the sample mean values of and : lncom _b[] + *_b[sq]*bar + _b[]*bar Note that the lncom command also computes () the estmated standard error of the estmated margnal effect of on prce, and () the t-rato for a two-tal test of the null hypothess that the margnal effect of on prce equals zero for the average car n the sample. Would you retan or reject the null hypothess at the 5 percent sgnfcance level, at the percent sgnfcance level? Page 5 of 33 pages

16 Use the followng lncom command to evaluate the estmated margnal effect of on prce for the 5-th sample percentles of and. Enter the commands: lncom _b[] + *_b[sq]*5 + _b[]*5 return lst Would you retan or reject the null hypothess that the margnal effect of on prce equals zero for a car whose weght and fuel effcency equal the 5-th sample percentle values? Use the followng lncom command to evaluate the estmated margnal effect of on prce for the 5-th sample percentles of and : lncom _b[] + *_b[sq]*5 + _b[]*5 Would you retan or reject the null hypothess that the margnal effect of on prce equals zero for the medan car n the sample? Use the followng lncom command to evaluate the estmated margnal effect of on prce for a car n the 5-th sample percentle of and the 75-th sample percentle n : lncom _b[] + *_b[sq]*5 + _b[]*75 Would you retan or reject the null hypothess that the margnal effect of on prce equals zero for a car n the 5-th sample percentle of and the 75-th sample percentle n? Fnally, you can evaluate the margnal effect of on prce for all 74 cars n the sample. Enter the followng commands: generate me = b + *b3* + b5* summarze me Use the graph twoway scatter command to plot the estmated margnal effect of on prce aganst for all cars n the sample, after frst sortng the sample observatons n ascendng order of the observed values of. Enter the commands: Page 6 of 33 pages

17 sort lst me graph twoway scatter me Add approprate ylabel( ), xlabel( ), ttle( ), subttle( ), yttle( ) and xttle( ) optons to the above graph twoway scatter command to make the fgure more readable. Enter on one lne the command: graph twoway scatter me, ylabel(-(5)) xlabel((5)5) ttle("margnal Effect of Weght on Car Prce") subttle("as Functon of WGT, All Cars n Sample") yttle("margnal effect of on prce ($ per pound)") xttle("car weght n pounds ()") Use the graph twoway scatter command to plot the estmated margnal effect of on prce aganst for all cars n the sample, after frst sortng the sample observatons n ascendng order of the observed values of. Enter the commands: sort lst me graph twoway scatter me Add approprate ylabel( ), xlabel( ), yttle( ), xttle( ), ttle( ), and subttle( ) optons to the above graph twoway scatter command to make the fgure more readable. Enter on one lne the command: graph twoway scatter me, ylabel(-(5)) xlabel((5)4) yttle("margnal effect of on prce ($ per pound)") xttle("mles per gallon ()") ttle("estmated Margnal Effect of Wgt on Prce") subttle("as Functon of MPG, All Cars n Sample") Margnal Effects of The estmated margnal effect of on prce for some gven values of and of s calculated by the followng lnear functon of and : estmate of prce = Ê ( prce, ) ˆ = β 4 5 ˆ ˆ Page 7 of 33 pages

18 Use the followng lncom command to evaluate the estmated margnal effect of on prce for the sample mean values of and : lncom _b[] + *_b[sq]*bar + _b[]*bar Would you retan or reject the null hypothess that the margnal effect of on prce equals zero for the "average" car n the sample? Use the followng lncom command to evaluate the estmated margnal effect of on prce for the 5-th sample percentles of and : lncom _b[] + *_b[sq]*5 + _b[]*5 Would you retan or reject the null hypothess that the margnal effect of on prce equals zero for a car whose weght and fuel effcency equal the 5-th sample percentle values? Use the followng lncom command to evaluate the estmated margnal effect of on prce for the 5-th sample percentles of and : lncom _b[] + *_b[sq]*5 + _b[]*5 Would you retan or reject the null hypothess that the margnal effect of on prce equals zero for the medan car n the sample? Use the followng lncom command to evaluate the estmated margnal effect of on prce for a car n the 5-th sample percentle of and the 75-th sample percentle of : lncom _b[] + *_b[sq]*75 + _b[]*5 Would you retan or reject the null hypothess that the margnal effect of on prce equals zero for a car n the 5-th sample percentle of and the 75-th sample percentle of? Fnally, you can evaluate the margnal effect of on prce for all 74 cars n the sample. Enter the followng commands: generate me = b + *b4* + b5* summarze me Page 8 of 33 pages

19 Use the graph twoway scatter command to plot the estmated margnal effect of on prce aganst for all cars n the sample, after frst sortng the sample observatons n ascendng order of the observed values of. Enter the commands: sort lst me graph twoway scatter me Add approprate yttle( ), xttle( ), ttle( ), and subttle( ) optons to the above graph twoway scatter command to make the fgure more readable. Enter on one lne the command: graph twoway scatter me, ttle("margnal Effect of MPG on Car Prce") subttle("as Functon of MPG, All Cars n Sample") yttle("margnal effect of on prce ($ per )") xttle("mles per gallon ()") Use the graph twoway scatter command to plot the estmated margnal effect of on prce aganst for all cars n the sample, after frst sortng the sample observatons n ascendng order of the observed values of. Enter the commands: sort lst me graph twoway scatter me Add approprate yttle( ), xttle( ), ttle( ), and subttle( ) optons to the above graph twoway scatter command to make the fgure more readable. Enter on one lne the command: graph twoway scatter me, ttle("margnal Effect of MPG on Car Prce") subttle("as Functon of WGT, All Cars n Sample") yttle("margnal effect of on prce ($ per )") xttle("car weght ()") Computng Margnal Effects of Contnuous Explanatory Varables margns In the two precedng sectons, you have computed the margnal effect of and the margnal effect of for selected values of the explanatory varables usng the lncom command. In ths secton, you are ntroduced to the Stata margns command Page 9 of 33 pages

20 as a convenent means of computng estmates of the margnal effects of contnuous explanatory varables followng OLS estmaton of a lnear regresson model usng the regress command, and of performng two-tal tests of the null hypothess that the margnal effect s zero for the specfed values of the explanatory varables. For ths purpose, the margns command s often much smpler and faster to use than the lncom command. However, a requrement of the margns command s that the estmaton command t follows s formulated n what Stata calls factor-varable notaton, whch dentfes for Stata whether a partcular explanatory varable s a contnuous or a categorcal varable.. If an explanatory varable s contnuous, then ts name must be entered n the varable lst of the estmaton command wth the prefx c., whch dentfes the varable as a contnuous varable. For example, the varable must appear n the varable lst of the regress command as c.. Smlarly, the varable must appear n the varable lst of the regress command as c... If an explanatory varable s categorcal, then ts name must be entered n the varable lst of the estmaton command wth the prefx., whch dentfes the varable as a categorcal, or ndcator, varable that takes only non-negatve nteger values denotng the varous levels of the varable. For example, f the bnary categorcal varable sex takes only the two values for males and for females, then t must appear n the varable lst of the regress command as.sex. Smlarly, f the mult-level categorcal varable group takes the fve values,, 3, 4 and 5, then t must appear n the varable lst of the regress command as.group. You wll be ntroduced to the use of the margns command for categorcal varables n a later tutoral. Recall that Model for car prces s gven by the populaton regresson equaton (): prce = β + u. () Re-wrte the populaton regresson functon (or PRF) for Model as: E(prce, ) = β () Page of 33 pages

21 Frst, use the regress command to re-estmate by OLS regresson equaton () for prce usng the c. prefx to dentfy the contnuous explanatory varables. Enter the command: regress prce c. c. c.#c. c.#c. c.#c. Note the use of the factor-varable operator #, whch specfes nteractons or products of two varables. Thus, c.#c. s squared ( ), c.#c. s squared ( ), and c.#c. s the nteracton term. Computng Margnal Effects of The estmated margnal effect of on prce for some gven values of and of s calculated by the followng lnear functon of and : estmate of prce = Ê ( prce, ) ˆ = β 3 5 ˆ ˆ Use any of the followng margns commands to evaluate the estmated margnal effect of on prce at the sample mean values of and : margns, dydx(c.) margns, dydx() margns, dydx(c.) at((mean) ) Note that all three of these margns commands produce dentcal results. That s because the default values of the explanatory varables at whch the margns command computes estmates of margnal effects are the sample mean values. In addton to an estmate of the condtonal margnal effect of on prce, each of the above margns commands also computes () the estmated standard error of the estmated margnal effect of on prce, () a z-rato (or large sample t- rato) for performng a two-tal test of the null hypothess that the margnal effect of on prce equals zero for the average car n the sample, (3) a two-tal p- value for the z-rato, and (4) a two-sded 95% confdence nterval for the margnal Page of 33 pages

22 effect of on prce. Would you retan or reject the null hypothess at the 5 percent sgnfcance level, at the percent sgnfcance level? Use the at( ) opton on the margns command to estmate the margnal effect of on prce for the 5-th sample percentles of and. Enter the two margns commands: margns, dydx(c.) at((p5) (p5) ) margns, dydx(c.) at((p5) ) Note that these two margns commands produce dentcal results; the second just saves a few keystrokes. Would you retan or reject the null hypothess that the margnal effect of on prce equals zero for a car whose weght and fuel effcency equal the 5-th sample percentle values? Use any of the followng margns commands wth the at( ) opton to estmate the margnal effect of on prce for the 5-th sample percentles of and : margns, dydx(c.) at((p5) (p5) ) margns, dydx(c.) at((medan) (medan) ) margns, dydx(c.) at((medan) ) Would you retan or reject the null hypothess that the margnal effect of on prce equals zero for the medan car n the sample? Use the followng margns command to evaluate the estmated margnal effect of on prce of a car n the 5-th sample percentle of and the 75-th sample percentle n : margns, dydx(c.) at((p5) (p75) ) Would you retan or reject the null hypothess that the margnal effect of on prce equals zero for a car n the 5-th sample percentle of and the 75-th sample percentle n? Fnally, you can evaluate the margnal effect of on prce at several dfferent values of and a fxed value of to examne how the condtonal margnal effect of vares wth whle holdng the values of the other explanatory Page of 33 pages

23 varables constant at specfed fxed values. To llustrate, suppose we want to estmate the margnal effect of on prce at the sample medan value of and the followng values of :,,,5,,5,,75, 3,, 3,5, 3,5, 3,75, 4,, 4,5, 4,5. These values of approxmately span the range of sample values of n the sample data. Enter the followng margns command, notng n partcular the formulaton of the at( ) opton: margns, dydx(c.) at( = ((5)45) (medan) ) The above at( ) opton fxes the value of the contnuous explanatory varable at ts sample medan value and computes the condtonal margnal effect of on prce at dstnct values of, begnnng at, pounds and ncreasng n ncrements of 5 pounds to 4,5 pounds. Now we can use the margnsplot command to draw a lne graph of the estmated margnal effect of on prce aganst for cars wth the medan value of and the dstnct values of specfed n the precedng margns command. Enter the followng margnsplot command: margnsplot Note that ths margnsplot command dsplays the 95% confdence lmts for the margnal effect estmates at each of the data ponts. To suppress the dsplay of the confdence ntervals, add the noc opton to the above margnsplot command; noc s short for no confdence nterval. Enter the followng margnsplot command wth the noc opton: margnsplot, noc Add approprate ttles to the lne graph produced by the foregong margnsplot commands usng the ttle( ), subttle( ) and yttle( ) optons. The xttle( ), note( ) and capton( ) optons can also be used to further annotate the graph. Enter on one lne the followng margnsplot command wth ttle( ), subttle( ) and yttle( ) optons: margnsplot, ttle(condtonal Margnal Effect of WGT on Car Prce) subttle("at Sample Medan of MPG =, wth 95% Confdence Lmts") yttle(change n car prce (dollars)) Page 3 of 33 pages

24 You may want to save a graph to dsk n a format such that you can use t, and perhaps modfy t, ether later n your current Stata sesson or durng some future Stata sesson. The graph save command can be used to do ths. To save the graph created by the precedng margnsplot command to a dsk fle n the current Stata workng drectory and gve that dsk fle the name graph_tutoral4_f3.gph, enter the followng graph save command: graph save graph_tutoral4_f3.gph Note that specfyng the fle extenson.gph n the name gven to the dsk fle s optonal; all graph save commands wll create dsk fles wth the fle extenson.gph. Computng Margnal Effects of The estmated margnal effect of on prce for some gven values of and of s calculated by the followng lnear functon of and : estmate of prce = Ê ( prce, ) ˆ = β 4 5 Use any of the followng margns commands to evaluate the estmated margnal effect of on prce at the sample mean values of and : margns, dydx(c.) margns, dydx() margns, dydx(c.) at((mean) (mean) ) margns, dydx(c.) at((mean) ) Note agan that all four of these margns commands produce dentcal results because the default values of the explanatory varables at whch the margns command computes estmates of margnal effects are the sample mean values. Would you retan or reject the null hypothess that the margnal effect of on prce equals zero for the average car n the sample at the 5 percent sgnfcance level, at the percent sgnfcance level? ˆ ˆ Page 4 of 33 pages

25 Use the at( ) opton on the margns command to estmate the margnal effect of on prce for the 5-th sample percentles of and. Enter the two margns commands, whch you wll see produce dentcal results: margns, dydx(c.) at((p5) (p5) ) margns, dydx(c.) at((p5) ) Would you retan or reject the null hypothess that the margnal effect of on prce equals zero for a car whose weght and fuel effcency equal the 5-th sample percentle values? Use any of the followng margns commands wth the at( ) opton to estmate the margnal effect of on prce for the 5-th sample percentles of and : margns, dydx(c.) at((p5) (p5) ) margns, dydx(c.) at((medan) (medan) ) margns, dydx(c.) at((medan) ) Would you retan or reject the null hypothess that the margnal effect of on prce equals zero for the medan car n the sample? Use the followng margns command to evaluate the estmated margnal effect of on prce for a car n the 5-th sample percentle of and the 75-th sample percentle n : margns, dydx(c.) at((p5) (p75) ) Would you retan or reject the null hypothess that the margnal effect of on prce equals zero for a car n the 5-th sample percentle of and the 75-th sample percentle n? Fnally, you can evaluate the margnal effect of on prce at several dfferent values of and a fxed value of to examne how the condtonal margnal effect of vares wth whle holdng the values of the other explanatory varables constant at specfed fxed values. To llustrate, suppose we want to estmate the margnal effect of on prce at the sample medan value of and the followng 5 values of :, 4, 6, 8,,, 4, 6, 8, 3, 3, 34, 36, 38, 4. These 5 values of approxmately span the range of sample Page 5 of 33 pages

26 values of n the sample data. Enter the followng margns command, notng n partcular the formulaton of the at( ) opton: margns, dydx(c.) at( = (()4) (medan) ) The above at( ) opton fxes the value of the contnuous explanatory varable at ts sample medan value and computes the condtonal margnal effect of on prce at 5 dstnct values of, begnnng at and ncreasng n ncrements of to 4. We can now use the margnsplot command to draw a lne graph of the estmated margnal effect of on prce aganst for cars wth the medan value of and the 5 dstnct values of specfed n the precedng margns command. Enter the followng margnsplot command: margnsplot Note that ths margnsplot command dsplays the 95% confdence lmts for the margnal effect estmates at each of the 5 data ponts. To suppress the dsplay of the confdence ntervals, add the noc opton to the above margnsplot command; noc s short for no confdence nterval. Enter the followng margnsplot command wth the noc opton: margnsplot, noc Add approprate ttles to the lne graph produced by the foregong margnsplot commands usng the ttle( ), subttle( ) and yttle( ) optons. The xttle( ), note( ) and capton( ) optons can also be used to further annotate the graph. Enter on one lne the followng margnsplot command wth ttle( ), subttle( ) and yttle( ) optons: margnsplot, ttle(condtonal Margnal Effect of MPG on Car Prce) subttle("at medan WGT (3,9 pounds), wth 95% confdence lmts") yttle(change n car prce (US dollars)) To save the graph created by the precedng margnsplot command to a dsk fle n the current Stata workng drectory and gve that dsk fle the name graph_tutoral4_f3.gph, enter the followng graph save command: Page 6 of 33 pages

27 graph save graph_tutoral4_f3.gph Note that specfyng the fle extenson.gph n the name gven to the dsk fle s optonal; all graph save commands wll create dsk fles wth the fle extenson.gph. Exportng a graph to dsk graph export You may also want to export a graph created by Stata to a fle on dsk so that you can nsert that graph nto a document created by a word processor such as MS Word. The graph export command can be used to do ths. Basc Syntax graph export newflename.suffx [, optons ] Ths command exports to a dsk fle n the current Stata workng drectory the graph currently dsplayed n the Graph wndow. The output format of ths fle s determned by the suffx of newflename.suffx. An alternatve way to specfy the output format of the exported fle s to use the as(fleformat) opton, where fleformat specfes the desred format of the exported fle. The avalable output formats that can be specfed wth the graph export command are lsted n the followng table. suffx mpled opton output format.ps as(ps) PostScrpt.eps as(eps) EPS (Encapsulated PostScrpt).wmf as(wmf) Wndows Metafle.emf as(emf) Wndows Enhanced Metafle.pct as(pct) Mackntosh PICT format.pdf as(pdf) PDF.png as(png) PNG (Portable Network Graphcs).tf as(tf) TIFF Notes: ps and eps are avalable wth all versons of Stata; png and tf are avalable for all versons of Stata except Stata(console) for Unx; wmf and emf are avalable only wth Stata for Wndows; and pct and pdf are avalable only wth Stata for Mackntosh. Page 7 of 33 pages

28 Source: StataCorp 7. Stata Statstcal Software Release : Graphcs. College Staton, TX: StataCorp LP, p. 4. To llustrate how to use the graph export command, frst use the Stata graph use command to read nto memory and dsplay n the Graph wndow the frst graph you saved to dsk durng ths tutoral, specfcally the graph named graph_tutoral4_f3. Enter the followng command: graph use graph_tutoral4_f3 To export and save ths lne graph n EPS (Encapsulated PostScrpt) format to a fle named graph_tutoral4_f3.eps n the current Stata workng drectory, enter the graph export command: graph export graph_tutoral4_f3.eps To export and save the same lne graph n Wndows Metafle format to a fle named graph_tutoral4_f3.wmf n the current Stata workng drectory, enter the graph export command: graph export graph_tutoral4_f3.wmf To export and save the same lne graph n Wndows Enhanced Metafle format to a fle named graph_tutoral4_f3.emf n the current Stata workng drectory, enter the graph export command: graph export graph_tutoral4_f3.emf Fnally, to export and save the same lne graph n PNG (Portable Network Graphcs) format to a fle named graph_tutoral4_f3.png n the current Stata workng drectory, enter the graph export command: graph export graph_tutoral4_f3.png To lst the several graph fles you have just saved to fles n the current Stata workng drectory, enter the followng dr command: dr graph_tutoral4_f3.* Page 8 of 33 pages

29 Insertng exported graph fles nto an MS Word document In ths secton, we llustrate how exported Stata graph fles n varous output formats appear when they are nserted nto the current document, whch s a Mcrosoft Word document. The exported fles we nsert are the followng: the fle graph_tutoral4_f3.eps, whch s n EPS (Encapsulated PostScrpt) format; the fle graph_tutoral4_f3.wmf, whch s n Wndows Metafle format; the fle graph_tutoral4_f3.emf, whch s n Wndows Enhanced Metafle format; and the fle graph_tutoral4_f3.png, whch s n PNG (Portable Network Graphcs) format. In MS Word, each of these Stata graph fles can be nserted nto a document by selectng Insert > Pcture and then choosng the flename of the graph you want from the lst of all pctures. The fle graph_tutoral4_f3.eps n EPS (Encapsulated PostScrpt) format: Page 9 of 33 pages

30 The fle graph_tutoral4_f3.wmf n Wndows Metafle format: Condtonal Margnal Effect of WGT on Car Prce at Sample Medan of MPG =, wth 95% Confdence Lmts Change n car prce (dollars) Weght (n pounds) Page 3 of 33 pages

31 The fle graph_tutoral4_f3.emf n Wndows Enhanced Metafle format: Condtonal Margnal Effect of WGT on Car Prce at Sample Medan of MPG =, wth 95% Confdence Lmts Change n car prce (dollars) Weght (n pounds) Page 3 of 33 pages

32 The fle graph_tutoral4_f3.png n PNG (Portable Network Graphcs) format: Page 3 of 33 pages

33 Preparng to End Your Stata Sesson Before you end your Stata sesson, you should do three thngs. Frst, you wll want to save the current data set. Enter ether of the followng save commands to save the current data set as Stata-format data set auto4.dta: save auto4 or save auto4, replace Second, close the log fle you have been recordng. Enter the command: log close Thrd, close the command log fle you have been recordng. Enter the command: cmdlog 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 portable electronc storage devce such as a flash memory stck. These fles wll be found n the Stata workng drectory, whch s usually C:\data on the computers n Dunnng 35. There are three fles you wll want to be sure you have: the Stata log fle 45tutoral4.log; the Stata command log fle 45tutoral4.txt; and the saved Stata-format dataset auto4.dta. Fnally, you wll probably want to take wth you the several Stata graph fles you saved to dsk durng ths tutoral. Use the Wndows copy command to copy any fles you want to keep to your own portable electronc storage devce (e.g., a flash memory stck). 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. Page 33 of 33 pages