Solutions: Wednesday, November 14
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1 Amhers College Deparmen of Economics Economics 360 Fall 2012 Soluions: Wednesday, November 14 Judicial Daa: Cross secion daa of judicial and economic saisics for he fify saes in JudExp CrimesAll GdpPC Pop UnemRae Sae Year Sae and local expendiures for he judicial sysem per 100,000 persons in sae Crimes per 100,000 persons in sae Real per capia GDP in sae (chained 2000 dollars) Populaion in sae (persons) Unemploymen rae in sae (percen) Name of sae Year 1. We wish o explain sae and local judicial expendiures. To do so consider he following linear model: JudExp CrimesAll + β GDP GdpPC + e a. Develop a heory regarding how each explanaory variable influences he dependen variable. Wha does your heory imply abou he sign of each coefficien? A sae wih a higher crime rae should have more cour cases and hence would spend more on is judicial sysem: β Crimes > 0. A sae wih a higher per capia GDP has a larger ax base and hence, can afford o spend more on is judicial sysem: β GDP > 0. b. Using he ordinary leas squares esimaion procedure, esimae he value of each coefficien using he Judicial Daa. Inerpre he coefficien esimaes. Wha are he criical resuls? [Link o MIT-JudExpenses-2000.wf1 goes here.] Dependen Variable: JudExp Explanaory Variables: CrimesAll and GdpPC Dependen Variable: JUDEXP Mehod: Leas Squares Sample: 1 50 Included observaions: 50 Coefficien Sd. Error -Saisic Prob. CRIMESALL GDPPC C Esimaed Equaion: JudExp 2, CrimesAll +.298GdpPC
2 2 Inerpreing he Coefficien Esimaes b CrimesAll.321 We esimae ha a 1 per 100,000 persons rise in he crime rae increases judicial expendiures by $.321 per 100,000 persons. b GDP.298 We esimae ha a $1 rise in per capia sae income increases judicial expendiures by.298 per 100,000 persons. Criical Resuls CrimeAll coefficien esimae equals.321. This evidence, he posiive sign of he coefficien esimae, suggess ha higher crime raes increase judicial expendiures hereby supporing he heory. GdpPC coefficien esimae equals.298. This evidence, he posiive sign of he coefficien esimae, suggess ha higher per capia GDP increases judicial expendiures hereby supporing he heory. c. Formulae he null and alernaive hypoheses. Crime Rae GDP H 0 : β Crimes 0 H 0 : β GDP 0 H 1 : β Crimes > 0 H 1 : β GDP > 0 d. Calculae Prob[Resuls IF H 0 True] s and assess your heories. Crime Rae < GDP <.0001 A he 1 percen significance level we rejec he GDP null hypohesis. Even a a 10 percen significance level we canno rejec he crime rae null hypohesis. 2. Consider he possibiliy of heeroskedasiciy in he judicial expendiure model. a. Inuiively, is here reason o suspec ha heeroskedasiciy migh exis? More specifically, is here reason o suspec ha he variance of he error erm s probabiliy disribuion may be correlaed wih per capia real GDP? Yes. A sae wih high per capia GDP has more discreion in deermining is judicial expendiures. b. Consider he ordinary leas squares (OLS) esimaes of he parameers ha you compued in he previous quesion. Plo he residuals versus per capia real GDP. Does your graph appear o confirm your suspicions concerning he presence of heeroskedasiciy? [Link o MIT-JudExpenses-2000.wf1 goes here.]
3 3 10,000 8,000 6,000 RESID 4,000 2, ,000-4,000 20,000 30,000 40,000 50,000 60,000 GDPPC c. Based on your suspicions, formulae a linear model of heeroskedasiciy. Var[e ] V GdpPC where V is a consan Heeroskedasiciy Model: ResSqr α Cons + α GDP GdpPC where ResSqr Square of he residual for sae d. Use he Breusch-Pagan-Godfrey approach o es for he presence of heeroskedasiciy. Heeroskedasiciy Tes: Breusch-Pagan-Godfrey Tes Equaion: Dependen Variable: RESID^2 Mehod: Leas Squares Dae: 04/19/10 Time: 11:52 Sample: 1 50 Included observaions: 50 Coefficien Sd. Error -Saisic Prob. C GDPPC H 0 : α GDP 0 Per capia GDP does no affec he squared deviaion of he residual H 1 : α GDP > 0 Higher per capia GDP increases he squared deviaion of he residual Prob[Resuls IF H 0 True] A he 5 percen significance level we rejec he null hypohesis.
4 4 3. Apply he generalized leas squares (GLS) esimaion procedure o he judicial expendiure model. To simplify he mahemaics use he following equaion o model he variance of he error erm s probabiliy disribuion: Var[e ] V GdpPC where V equals a consan. a. Apply his heeroskedasiciy model o manipulae algebraically he original model o derive a new, weaked model in which he error erms do no suffer from heeroskedasiciy. Original Model: JudExp CrimesAll + β GDP GdpPC + e Divide boh sides of he equaion by GdpPC. JudExp GdpPC β Cons GdpPC β Cons GdpPC CrimesAll GdpPC e GdpPC + β GDP GdpPC + GdpPC CrimesAll GdpPC GdpPC + β GDP GdpPC + ε Var[ε ] Var[ e GdpPC ] 1 GdpPC Var[e ] where ε Arihmeic of variances: Var[cx] c 2 Var[x] e GdpPC Var[e ] V GdpPC where V equals a consan 1 GdpPC V GdpPC V
5 5 b. Use he ordinary leas squares (OLS) esimaion procedure o esimae he parameers of he weaked model. [Link o MIT-JudExpenses-2000.wf1 goes here.] Generae adjused variables: 1 AdjCons GdpPC AdjCrimesAll CrimesAll GdpPC AdjGdpPC Dependen Variable: ADJJUDEXP Mehod: Leas Squares Sample: 1 50 Included observaions: 50 Coefficien Sd. Error -Saisic Prob. ADJCRIMESALL ADJGDPPC ADJCONST GdpPC GdpPC 4. How, if a all, does accouning for heeroskedasiciy affec he assessmen of your heories? The impac on he assessmen is minimal.
6 6 5. Consider he following equaions: y + β x x + e e ρe 1 Esy b Cons + b x x Res y Esy Sar wih he las equaion, he equaion for Res. Using algebra and he oher equaions, show ha Res (β Cons b Cons ) + (β x b x )x + ρe 1 Res y Esy Subsiuing for y y + β x x + e + β x x + e Esy Subsiuing for e e ρe 1 + β x x + ρe 1 Esy Subsiuing for Esy 6. Consider he following equaions: y + β x x + e Esy b Cons + b x x + β x x + ρe 1 (b Cons + b x x ) Rearranging erms (β Cons b Cons ) + (β x b x )x + ρe 1 y 1 + β x x 1 + e 1 e ρe 1 Muliply he y 1 equaion by ρ. Then, subrac i from he y equaion. Using algebra and he e equaion show ha (y ρy 1 ) (β Cons ρβ Cons ) + β x (x ρx 1 ) y 1 + β x x 1 + e 1 y 1 equaion ρy 1 ρβ Cons + ρβ x x 1 + ρe 1 Muliplying by ρ y + β x x + e y equaion ρy -1 ρβ Cons + ρβ x x -1 + ρe -1 ρy 1 equaion Subracing y ρy 1 ρβ Cons + β x x ρβ x x 1 + e ρe 1 Facoring ou β x y ρy 1 ρβ Cons + β x (x ρx 1 ) + e ρe 1 Subsiuing for e y ρy 1 ρβ Cons + β x (x ρx 1 ) + ρe 1 ρe 1 Simplifying (y ρy 1 ) (β Cons ρβ Cons ) + β x (x ρx 1 )
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