PBAF 528 Week Theory Is the variable s place in the equation certain and theoretically sound? Most important! 2. T-test

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1 PBAF 528 Week 6 How do we choose our model? How do you decde whch ndependent varables? If you want to read more about ths, try Studenmund, A.H. Usng Econometrcs Chapter 7. (ether 3 rd or 4 th Edtons) 1. Theory Is the varable s place n the equaton certan and theoretcally sound? Most mportant! 2. T-test 3. Is the varable s estmated coeffcent sgnfcant n the expected drecton (one-sded test)? 2 R Does the overall ft of the equaton mprove when then varable s added to the equaton? 4. Bas Do other varables coeffcents change sgnfcantly when the varable s added to the equaton? If all of these are true, then the varable belongs n the equaton. Dummy Varables 1. Intercept Dummy Varables A dummy varable that changes the constant or ntercept term Y = β 0 + β 1 X + β 2 D + ε 2. Seasonal Dummes (Mult-alternatve Dummes) Dummy varables used to represent qualtatve varables that take on more than two alternatves Use one less dummy varable than there are alternatves. Each dummy wll represent one condton.

2 3. Slope Dummes A dummy varable that changes the slope of the relatonshp between x and y 4. Dummy Dependent Varables Dummy varable s used as the dependent varable Example #1: More than 2 categores (more than one dummy varable) Educaton can be thought of as: (1) not havng earned a hgh school dploma, (2) havng earned only a hgh school dploma, and (3) havng more educaton than a hgh school dploma. Then we use two dummes. We ll call them D 1 and D 2. D 1 =1 f you have only a hgh school dploma 0 otherwse D 2 =1 f you have more educaton than a hgh school dploma 0 otherwse. What are all the possbltes? you have more than a hgh school degree you have a hgh school dploma and nothng beyond that you have not earned any dploma D 1 = and D 2 = D 1 = and D 2 = D 1 = and D 2 = CAUTION: Don t nclude too many dummes or you ll have to explan each data pont! CAUTION: Don t nclude a dummy that only takes a value of 1 for one data pont and zero for all other observatons. Ths one-tme dummy acts to elmnate that observaton from the data set, mprovng the ft artfcally. 2

3 Some deas for usng dummy varables: Could use dummy for seasonal changes f you have data where each case s at a dfferent tme pont. Illustraton #1: If the data has been recorded quarterly, you wll need 3 dummy varables. D 1 = { 1 n Quarter 1 0 otherwse D 2 = { 1 n Quarter 2 0 otherwse D 3 = { 1 n Quarter 3 0 otherwse Quarter 1 Quarter 2 Quarter 3 Quarter 4 D 1 D 2 D 3 Illustraton #2: Dummy for Tme Seres Data If you were nterested to study the mpact of a partcular event on a gven varable, a dummy varable could be used for ths. For example, the mpact of Sept 11 on arlne travel could be modeled wth a dummy varable. D = 0 for D = 1 for 2002 to present The sgn of the coeffcent of D wll gve the drecton of any shft n Interacton Terms Interacton terms are products of two or more ndependent varables. Allow for dfferences n effect of an explanatory factor across categores or levels of another factor Used when the change n Y wth respect to one ndependent varable depends on the level of another ndependent varable. Can nteract wth dummes or contnuous varables. 3

4 For a contnuous(ndependent) and a dummy: Allows the slope between the dependent and ndependent varable to be dfferent dependng on whether the condton specfed by the dummy s met. Used whenever the mpact of an ndependent varable on the dependent varable s hypotheszed to change f some qualtatve condton s met. 1. What does the regresson equaton look lke wth an nteracton n t? Ths s called a slope dummy varable Y = β 0 + β 1 X + β 2 D +β 3 X D + ε I Ths one has an nteracton between an ndependent varable, X, and a dummy varable, D. What effect does the nteracton have on the slope (that s, the change n Y brought about by a change n X)? When D=0, Y/ X=β 1 Ths s the slope for the reference group When D=1, Y/ X=β 1 +β 3 Ths s the slope for the ndcated group The slope (or the coeffcent of X) changes when the condton specfed by D s met. 4

5 from AH Studenmund Usng Econometrcs: A Practcal Gude p You need both the slope dummy and the ntercept dummy n the equaton. The above regresson lne has both a slope dummy and an ntercept dummy (a dummy that does not get multpled by anythng). Ths s necessary n most cases snce just ncludng a slope dummy would bas the slope by forcng t to explan more than t should, for example, changes n the mean between two groups. An ntercept dummy best explans ths sort of change. So, the model should nclude an ntercept dummy (plan dummy term) where there s a slope dummy (a dummy multpled by a predctor). Thnk carefully about your hypotheses about the drecton of the relatonshp between the dummes and the outcomes snce these terms make the model very flexble. 3. How do you test for sgnfcance of nteracton terms? To test for dfferences n slopes between the categores, use the t-test on the nteracton term. To test overall dfferences n the regresson relatonshp wth and wthout the ncluson of an nteracton use an F-test (#2). 5

6 Example #2: Dummy Varable Interactng wth a Contnuous (Independent) Varable Does extensve meda coverage of a mltary crss nfluence publc opnon on how to respond to the crss? Poltcal scentsts at UCLA came up wth a model concernng the 1990 Persan Gulf War, precptated by Iraq leader Saddam Hussen s nvason of Kuwat. They developed a model to analyze the level of support Amercans had for mltary (rather than dplomatc) response to the crss. The dependent varable ranges from 0 (preference for a dplomatc response) to 4 (preference for mltary response. Here s the model they developed based on data from 1,763 U.S. Ctzens. E(y)=β 0 + β 1 x 1 + β 2 x 2 +β 3 x 3 +β 4 x 4 +β 5 x 5 +β 6 x 6 +β 7 x 7 +β 8 x 2 x 3 +β 9 x 2 x 4 where: x 1 = Level of TV news exposure n a selected week (number of days) x 2 = Knowledge of seven poltcal fgures (1 pont for each correct answer) x 3 = Dummy varable for Gender (1 f male, 0 f female) x 4 = Dummy varable for Race (1 f nonwhte, 0 f whte) x 5 = Partsanshp (0-6 scale, where 0 = strong Democrat and 6 = strong Republcan) x 6 = Defense spendng atttude (1-7 scale, where 1 = greatly decrease spendng and 7 = greatly ncreased spendng) x 7 = Educaton The regresson results: Varable β estmate Standard Error Two-Taled p-value TV news exposure (x 1 ) Poltcal knowledge (x 2 ) Gender (x 3 ) <.001 Race (x 4 ) <.001 Partsanshp (x 5 ) <.001 Defense spendng (x 6 ) <.001 Educaton (x 7 ) <.001 Knowledge X Gender (x 2 x 3 ) Knowledge X Race (x 2 x 4 ) R 2 =.194, F=46.88 (p<.001) Source: Iyengar, S. and Smon, A News coverage of the Gulf Crss and publc opnon. Communcaton Research 20,: 380 (Table 2) 6

7 1) Interpret the β estmates for TV news exposure. 2) Is there enough support to say that TV news exposure s assocated wth support for a mltary resoluton of the crss? 3) Is there suffcent evdence to say that the relatonshp between support for a mltary resoluton and gender depends on poltcal knowledge? 4) What s the effect of knowledge on support for mltary resoluton for men? E(y)=β 0 + β 1 x 1 + β 2 x 2 +β 3 (1) +β 4 x 4 +β 5 x 5 +β 6 x 6 +β 7 x 7 +β 8 x 2 (1) +β 9 x 2 x 4 5) What s the effect of knowledge on support for mltary resoluton for women? E(y)=β 0 + β 1 x 1 + β 2 x 2 +β 3 (0) +β 4 x 4 +β 5 x 5 +β 6 x 6 +β 7 x 7 +β 8 x 2 (0) +β 9 x 2 x 4 6) What test would you use to answer the followng queston: Overall, does gender affect support for a mltary resoluton? 7

8 Example #3: Interactng 2 contnuous varables Fowles and Loeb hypotheszed that drunk drvng fataltes are more lkely at hgh alttude because hgher elevatons dmnsh the oxygen ntake of the bran, whch ncreases the mpact of a gven amount of alcohol. F = Traffc fataltes per vehcle mle (by state) B = per capta consumpton of beer S = average hghway drvng speed D = dummy (1=state has a vehcle nspecton program, 0=no nspecton program) A = average alttude of metro areas (1000s of feet) Fˆ = B S 0.24D 0.35A B A (t-statstcs) (-0.8) (1.53) (-0.96) (-1.07) (1.97) Hypotheszed relatonshp ? n=48 adjusted R 2 =.499 The nteracton n ths model s between two contnuous varables, consumpton rate of beer and alttude. The effect on the outcome of each the two varables nvolved n the nteracton depends on the nteracton coeffcent and the coeffcent on the orgnal varable. 1) Does the average alttude of metropoltan areas n the state affect the relatonshp between per capta beer consumpton and the rate of traffc fataltes? 2) How much hgher a fatalty rate do we expect for an average alttude of metro area ncrease by 1000 feet? 3) Does the alttude affect the overall regresson relatonshp explanng fatalty rate? (How would you approach ths?) 8

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