ECONOMICS 351* -- Stata 10 Tutorial 6. Stata 10 Tutorial 6. TOPICS: Functional Form and Variable Re-scaling in Simple Linear Regression Models

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1 ECONOMICS 35* -- Stata Tutoral 6 Stata Tutoral 6 TOPICS: Functonal Form and Varable Re-scalng n Smple Lnear Regresson Models DATA: auto.dta (a Stata-format data fle) TASKS: Stata Tutoral 6 has two prmary purposes: () to ntroduce you to some of the alternatve functonal forms commonly used n lnear-n-coeffcents regresson models; and () to nvestgate how varable re-scalng that s, changng the unts of measurement for Y and/or X affects OLS estmates of the slope coeffcent β and the ntercept coeffcent β n a smple lnear regresson equaton. The Stata commands that consttute the prmary subject of ths tutoral are: regress Used to perform OLS estmaton of smple lnear regresson models. predct Computes estmated Y -values and OLS resduals. graph twoway Draws scatterplots of sample data ponts and lne graphs of OLS sample regresson functons. NOTE: Stata commands are case senstve. All Stata command names must be typed n the Command wndow n lower case letters. LEARNING FROM THIS TUTORIAL: Stata Tutoral 6 contans some mportant analytcal results. You should make sure you understand them. HELP: Stata has an extensve on-lne Help faclty that provdes farly detaled nformaton (ncludng examples) on all Stata commands. 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 ECON 35* -- Wnter 8: Stata Tutoral 6 Page of 3 pages

2 ECONOMICS 35* -- Stata Tutoral 6 Preparng for Your Stata Sesson Before begnnng your Stata sesson, use Wndows Explorer to copy the Stataformat dataset auto.dta 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 35, the default Stata workng drectory s usually C:\data. On the computers n MC B9/B, the default Stata workng drectory s usually D:\courses. Start Your Stata Sesson To start your Stata sesson, double-clck on the Stata con n the Wndows desktop. After you double-clck the Stata con, you wll see the now famlar screen of four 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.log fle named 35tutoral6.log. To open (begn) the.log fle 35tutoral6.log, enter n the Command wndow: log usng 35tutoral6.log Ths command opens a fle called 35tutoral6.log n the current Stata workng drectory. Remember that once you have opened the 35tutoral6.log fle, a copy of all the commands you enter durng your Stata sesson and of all the results they produce s recorded n that 35tutoral6.log fle. An alternatve way to open the.log fle 35tutoral6.log 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 35tutoral6; and clck on the Save button. ECON 35* -- Wnter 8: Stata Tutoral 6 Page of 3 pages

3 ECONOMICS 35* -- Stata Tutoral 6 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 auto.dta nto memory, enter n the Command wndow: use auto Ths command loads nto memory the Stata-format dataset auto.dta. 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 summarze Alternatve Functonal Forms for the Smple Lnear Regresson Model Ths secton demonstrates () how to estmate by OLS dfferent functonal forms for the smple lnear regresson model relatng car prce (prce ) to car weght (weght ), () how to use the predct command to compute estmated or predcted values of the regressand ( $Y -values) for the sample observatons, and (3) how to use the graph twoway command to dsplay the OLS sample regresson functon correspondng to the observed sample values of the regressor weght.. The LIN-LIN (Lnear) Model: Ths model take the general form Y β + β X + u (a) Settng Y prce and X weght, PRE (a) takes the specfc form prce β + β weght + u (b) To estmate ths model (agan!) by OLS for the full sample of observatons n dataset auto.dta, and to calculate the estmated (or predcted) values of prce ECON 35* -- Wnter 8: Stata Tutoral 6 Page 3 of 3 pages

4 ECONOMICS 35* -- Stata Tutoral 6 for the sample observatons, enter n the Command wndow the followng commands: regress prce weght predct yhat The yhat varable created by the predct command takes the form Ŷ prîce βˆ + βˆ X βˆ + βˆ weght (,..., N) (c) where ˆβ and are the OLS coeffcent estmates for the LIN-LIN model. $ β To make a scatterplot of the sample data ( Y, X ) (prce, weght ) and a lne graph of the OLS sample regresson functon (c), frst sort the sample data by weght and then use the followng graph twoway command: sort weght graph twoway scatter prce weght lne yhat weght, yttle("car prce (U.S. dollars)," "observed and estmated") xttle("car weght (pounds)") ttle("lin-lin Model of Car Prce on Car Weght") subttle("ols Regresson and Scatterplot of Sample Data") legend(label( "Sample data ponts") label( "Sample regresson lne")) Ths command nstructs Stata to draw on the same set of coordnate axes both () a scatterplot of the sample data ponts (Y, X ) (prce, weght ) and () a lne graph of the estmated values of prce (.e., yhat Ŷ prî ) aganst the ce sample values of weght,.e, of the ponts ( Y $, X). Note that weght s the varable measured on the horzontal X-axs, and both prce and yhat are measured on the vertcal Y-axs.. The LOG-LOG (Double-Log) Model: Ths model takes the general form ln Y α + α ln X + u (a) where lny s the natural logarthm of Y and lnx s the natural logarthm of X. ECON 35* -- Wnter 8: Stata Tutoral 6 Page 4 of 3 pages

5 ECONOMICS 35* -- Stata Tutoral 6 Settng ln Y ln( prce) and lnx ln( weght ), PRE (a) takes the specfc form ln( prce ) α + α ln(weght ) + u, (b) where ln( prce ) the natural logarthm of the varable prce ; ln( weght ) the natural logarthm of the varable weght. Note: The natural logarthm s defned only for varables that take only postve values. Ths s the case for both prce and weght n the dataset auto.dta. Before estmatng the LOG-LOG model (), you must generate the natural logarthms of the varables prce and weght. Use the followng Stata generate commands to do ths. generate lnprce ln(prce) generate lnweght ln(weght) summarze lnprce lnweght To estmate the LOG-LOG model by OLS for the full sample of observatons and to calculate the estmated (or predcted) values of ln(prce ) for the sample observatons, enter n the Command wndow: regress lnprce lnweght predct lnyhatdl The lnyhatdl varable created by the predct command takes the form lnˆ Y ln(prîce ) αˆ + αˆ ln X αˆ + αˆ ln(weght ) (,..., N) (c) where ˆα and $α are the OLS coeffcent estmates for the LOG-LOG model and l$ny ln( $ prce) denotes the predcted values of lny. ECON 35* -- Wnter 8: Stata Tutoral 6 Page 5 of 3 pages

6 ECONOMICS 35* -- Stata Tutoral 6 To make a scatterplot of the sample data (lny, lnx ) and a lne graph of the OLS sample regresson functon (c), use the followng graph twoway command: graph twoway scatter lnprce lnweght lne lnyhatdl lnweght, yttle("ln(prce)," "observed and estmated") xttle("ln(weght)") ttle("log-log Model of Car Prce on Car Weght") subttle("ols Regresson and Scatterplot of Sample Data") legend(label( "Sample data ponts") label( "Sample regresson lne")) 3. The LOG-LIN (Sem-Log) Model: Ths model takes the general form ln Y γ + γ X + u (3a) Settng ln Y ln( prce ) and X weght, PRE (3a) takes the specfc form ln( prce ) γ + γ weght + u (3b) To estmate the LOG-LIN model by OLS for the full sample of observatons and to calculate the estmated (or predcted) values of ln(prce ) for the sample observatons, type n the Command wndow: regress lnprce weght predct lnyhatsl The lnyhatsl varable created by the predct command takes the form lnˆ Y ln(prîce ) γˆ + γˆ X γˆ + γˆ weght (,..., N) (3c) where ˆγ and $γ are the OLS coeffcent estmates for the LOG-LIN model and l$ny ln( $ prce) denotes the predcted values of lny. ECON 35* -- Wnter 8: Stata Tutoral 6 Page 6 of 3 pages

7 ECONOMICS 35* -- Stata Tutoral 6 To make a scatterplot of the sample data (lny, X ) and a graph of the OLS sample regresson functon (3c), use the followng graph twoway command: graph twoway scatter lnprce weght lne lnyhatsl weght, yttle("ln(car prce)," "observed and estmated") xttle("car weght (pounds)") ttle("log-lin Model of Car Prce on Car Weght") subttle("ols Regresson and Scatterplot of Sample Data") legend(label( "Sample data ponts") label( "Sample regresson lne")) Unts of Measurement and Re-scalng of Varables The coeffcent estmates n lnear (LIN-LIN) regresson models depend on the unts of measurement for the dependent varable Y and the ndependent varable X. Ths secton presents some analytcal results on how changng unts of measurement for Y and/or X affects the OLS estmates of the slope coeffcent β and the ntercept coeffcent β n a smple lnear regresson equaton. It then llustrates these results wth a smple lnear regresson model. Analyss: (There are no Stata commands n ths secton.) The term "re-scalng a varable" means multplyng that varable by a constant; ths s what happens when we change the unts n whch a varable s measured. Wrte the orgnal regresson equaton, expressed n terms of the orgnal varables Y and X, as equaton (4): Y β + β X + u. (4) Re-scale the orgnal varables Y and X by multplyng each by some arbtrarlyselected constant. Create the re-scaled varable X& by multplyng X by the constant c: X & cx (,, N), where c s a specfed constant. ECON 35* -- Wnter 8: Stata Tutoral 6 Page 7 of 3 pages

8 ECONOMICS 35* -- Stata Tutoral 6 Smlarly, create the re-scaled varable Y& by multplyng Y by the constant d: Y & dy (,, N), where d s a specfed constant. The new regresson equaton wrtten n terms of the re-scaled varables can be wrtten as: X& and Y& Y & & + & β + β X u. (5) Questons: How s the OLS estmate of the slope coeffcent OLS estmate of β n equaton (4)? How s the OLS estmate of the ntercept coeffcent the OLS estmate of β n equaton (4)? Answers: β n equaton (5) related to the β n equaton (5) related to The formula for the OLS estmator of β n the orgnal equaton (4) s: β ˆ x y x where x X X and y Y Y (,, N). The formula for the OLS estmator of β n the orgnal equaton (4) s: β ˆ Y ˆ X where Y Y N and X X N. β The formula for the OLS estmator of β n the re-scaled equaton (5) s: β ˆ x& y& x& where x& X& X& and y& Y& Y& (,, N). (6) To see how the new slope coeffcent estmator βˆ s related to the orgnal slope coeffcent estmator ˆβ, we need to determne how the re-scaled devatons-from- ECON 35* -- Wnter 8: Stata Tutoral 6 Page 8 of 3 pages

9 ECONOMICS 35* -- Stata Tutoral 6 means varables x& X& X& and y& Y& Y& are related to the orgnal devatons-from-means varables x X X and y Y Y. Here s the algebra: X & cx X & cx x & X& X& cx cx c(x X) cx ; Y & dy Y & dy y & Y& Y& dy dy d(y Y) dy. Thus, we see that x & cx and mply the followng results: y & dy (,, N). These two equaltes n turn x & y& cx dy cdx y x & y& cd x y ; x & (cx ) c x x& c x. Now substtute these results nto expresson (6) for βˆ : βˆ x& y& x& cd c x y x d c x y x d βˆ. c The formula for the OLS estmator of β n the re-scaled equaton (5) s: β ˆ Y& ˆ X&. (7) β To see how the new ntercept coeffcent estmator βˆ s related to the orgnal ntercept coeffcent estmator ˆβ, substtute nto expresson (7) for βˆ the d prevous results showng that β ˆ ˆ β, Y & dy and X & cx : c β ˆ Y& βˆ X& dy d βˆ c cx dy dˆ β X d(y βˆ X) dˆ β. ECON 35* -- Wnter 8: Stata Tutoral 6 Page 9 of 3 pages

10 ECONOMICS 35* -- Stata Tutoral 6 Results: d β ˆ ˆ β βˆ s affected by the re-scalng of both Y and X. (8) c β ˆ dβˆ ˆ s affected only by the re-scalng of Y. (9) β Some Examples To llustrate the effects of varable re-scalng.e., of changng the unts of measurement for Y and/or X we nvestgate how changng the unts of measurement for the varables n regresson equaton () affect the OLS coeffcent estmates. For convenence, the orgnal equaton () s rewrtten here as: prce β + β weght + u () where prce car prce measured n US dollars and weght car weght measured n pounds.. Re-scale only the dependent varable. Re-scale the dependent varable prce so that t s measured n hundreds of US dollars nstead of US dollars. Generate the re-scaled prce varable newp car prce measured n hundreds of US dollars, where newp prce /. Enter the command: generate newp prce/ Compare the sample values of the orgnal prce varable wth those of the rescaled prce varable newp. Enter the commands: summarze prce newp regress newp prce ECON 35* -- Wnter 8: Stata Tutoral 6 Page of 3 pages

11 ECONOMICS 35* -- Stata Tutoral 6 Estmate by OLS the regresson equaton wth newp as dependent varable and weght as the ndependent varable. Enter the command: regress newp weght Carefully compare the results of ths command wth those from OLS estmaton of the orgnal regresson equaton (). Whch results have changed as a result of re-scalng only the dependent varable?. Re-scale only the ndependent varable. Re-scale the ndependent varable weght so that t s measured n klograms nstead of pounds, where klogram. pounds. Generate the re-scaled weght varable neww car weght measured n klograms, where neww weght /.. Enter the command: generate neww weght/. Compare the sample values of the orgnal weght varable wth those of the rescaled weght varable neww. Enter the commands: summarze weght neww regress neww weght regress weght neww Estmate by OLS the regresson equaton wth prce as dependent varable and neww as the ndependent varable. Enter the command: regress prce neww Carefully compare the results of ths command wth those from OLS estmaton of the orgnal regresson equaton (). Whch results have changed as a result of re-scalng only the ndependent varable? ECON 35* -- Wnter 8: Stata Tutoral 6 Page of 3 pages

12 ECONOMICS 35* -- Stata Tutoral 6 3. Re-scale both the dependent varable and the ndependent varable. Re-scale both the dependent varable prce and the ndependent varable weght as above. The re-scaled dependent varable s newp car prce measured n hundreds of US dollars, where newp prce /. The re-scaled ndependent varable s neww car weght measured n klograms, where neww weght /.. Estmate by OLS the regresson equaton wth newp as dependent varable and neww as the ndependent varable. Enter the command: regress newp neww Carefully compare the results of ths command wth those from OLS estmaton of the orgnal regresson equaton (). Whch results have changed as a result of re-scalng both the dependent and ndependent varables? ECON 35* -- Wnter 8: Stata Tutoral 6 Page of 3 pages

13 ECONOMICS 35* -- Stata Tutoral 6 Preparng to End Your Stata Sesson Before you end your Stata sesson, you should do two 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 auto6.dta: save auto6 Second, 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 35, and D:\courses on the computers n MC B. There s one fle you wll want to be sure you have: the Stata log fle 35tutoral6.log. If you saved the Stata-format data set auto6.dta, 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 (e.g., 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 35* -- Wnter 8: Stata Tutoral 6 Page 3 of 3 pages