Analysis of Movie Profitability STAT 469 IN CLASS ANALYSIS #2

Analysis of Movie Profitability STAT 469 IN CLASS ANALYSIS #2

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Problem Background and Research Questions ReleaseDate Movie ProductionBudget DomesticGross WWGross 1 2009-12-18 Avatar 425000000 760507625 2783918982 2 2011-05-20 Pirates: On Stranger Tides 410600000 241063875 1045663875 3 2015-05-01 Avengers: Age of Ultron 330600000 459005868 1408218722 4 2015-12-18 The Force Awakens 306000000 936662225 2058662225 5 2007-05-24 Pirates: At World s End 300000000 309420425 963420425 6 2017-11-17 Justice League 300000000 229024295 655953446...... 4617 2011-10-21 Martha Marcy May Marlene 1000000 2981038 5438911 4618 2005-08-03 Junebug 1000000 2678010 3553253 4619 2008-08-01 Frozen River 1000000 2511476 6030129 4620 2001-11-21 Sidewalks of New York 1000000 2402459 3575308 4621 1998-04-24 Two Girls and a Guy 1000000 2057193 2315026 4622 2000-09-29 The Broken Hearts Club 1000000 1744858 2022442 Primary Issue: We want to understand return on investment for movies Want to understand the risks associated with movie making. Specific Research Questions: 1. How much revenue should a movie studio expect to receive for each dollar of production budget invested in a film? 2. Are higher production budget films more variable in the domestic gross? 3. What films under vs. overperformed domestically relative to predicted gross?

Exploratory Data Analysis

Exploratory Data Analysis

Exploratory Data Analysis

Exploratory Data Analysis A multiple linear regression model seems reasonable based on our exploratory analysis (maybe some outliers), so lets plan out how we would use an MLR to answer the research questions.

Analysis with MLR

<latexit sha1_base64="rwexygccyefaenu1qbwu0b+ewtc=">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</latexit> <latexit sha1_base64="rwexygccyefaenu1qbwu0b+ewtc=">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</latexit> <latexit sha1_base64="rwexygccyefaenu1qbwu0b+ewtc=">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</latexit> <latexit sha1_base64="rwexygccyefaenu1qbwu0b+ewtc=">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</latexit> <latexit sha1_base64="rl6u/u7aug+zxmzgqzc6hlappaq=">aaab+nicbzdlssnafiyn9vbrldwlm2arxjverf0w3bisyc/qhjcznrrdjxdmttqs8yhuxcji1idx59s4bbpq1h8gpv5zdufm7yeck7ttb6o0srq2vlhergxt7+zumdx9toptyadfyhhlrk8vcb5bczkk6cysaogl6pjj62m9cw9s8ti6w0kcbkiheq84o6gtz6z2fudqzx2er8yav3numtw7bs9klyntqi0uanrmv38qszsecjmgsvuco0e3oxi5e5bx+qmchlixhujpy0rdug42oz23jruzsijy6hehnxn/t2q0vgos+rozpdhsi7wp+v+tl2jw6wy8slkeim0xbamwmlamovgdloghmgigthj9q8vgvfkgoq2kdsfz/piyte/rjubbs1rjviijta7jetkhdrkgdxjdmqrfghkgz+svvblpxovxbnzmw0tgmxna/sj4/ag+djrd</latexit> <latexit sha1_base64="rl6u/u7aug+zxmzgqzc6hlappaq=">aaab+nicbzdlssnafiyn9vbrldwlm2arxjverf0w3bisyc/qhjcznrrdjxdmttqs8yhuxcji1idx59s4bbpq1h8gpv5zdufm7yeck7ttb6o0srq2vlhergxt7+zumdx9toptyadfyhhlrk8vcb5bczkk6cysaogl6pjj62m9cw9s8ti6w0kcbkiheq84o6gtz6z2fudqzx2er8yav3numtw7bs9klyntqi0uanrmv38qszsecjmgsvuco0e3oxi5e5bx+qmchlixhujpy0rdug42oz23jruzsijy6hehnxn/t2q0vgos+rozpdhsi7wp+v+tl2jw6wy8slkeim0xbamwmlamovgdloghmgigthj9q8vgvfkgoq2kdsfz/piyte/rjubbs1rjviijta7jetkhdrkgdxjdmqrfghkgz+svvblpxovxbnzmw0tgmxna/sj4/ag+djrd</latexit> <latexit sha1_base64="rl6u/u7aug+zxmzgqzc6hlappaq=">aaab+nicbzdlssnafiyn9vbrldwlm2arxjverf0w3bisyc/qhjcznrrdjxdmttqs8yhuxcji1idx59s4bbpq1h8gpv5zdufm7yeck7ttb6o0srq2vlhergxt7+zumdx9toptyadfyhhlrk8vcb5bczkk6cysaogl6pjj62m9cw9s8ti6w0kcbkiheq84o6gtz6z2fudqzx2er8yav3numtw7bs9klyntqi0uanrmv38qszsecjmgsvuco0e3oxi5e5bx+qmchlixhujpy0rdug42oz23jruzsijy6hehnxn/t2q0vgos+rozpdhsi7wp+v+tl2jw6wy8slkeim0xbamwmlamovgdloghmgigthj9q8vgvfkgoq2kdsfz/piyte/rjubbs1rjviijta7jetkhdrkgdxjdmqrfghkgz+svvblpxovxbnzmw0tgmxna/sj4/ag+djrd</latexit> <latexit sha1_base64="rl6u/u7aug+zxmzgqzc6hlappaq=">aaab+nicbzdlssnafiyn9vbrldwlm2arxjverf0w3bisyc/qhjcznrrdjxdmttqs8yhuxcji1idx59s4bbpq1h8gpv5zdufm7yeck7ttb6o0srq2vlhergxt7+zumdx9toptyadfyhhlrk8vcb5bczkk6cysaogl6pjj62m9cw9s8ti6w0kcbkiheq84o6gtz6z2fudqzx2er8yav3numtw7bs9klyntqi0uanrmv38qszsecjmgsvuco0e3oxi5e5bx+qmchlixhujpy0rdug42oz23jruzsijy6hehnxn/t2q0vgos+rozpdhsi7wp+v+tl2jw6wy8slkeim0xbamwmlamovgdloghmgigthj9q8vgvfkgoq2kdsfz/piyte/rjubbs1rjviijta7jetkhdrkgdxjdmqrfghkgz+svvblpxovxbnzmw0tgmxna/sj4/ag+djrd</latexit> Model Specification: Analysis Plan w/ MLR y MVN (X, How would you use the MLR model to answer the research questions? 1. How much revenue should a movie studio expect to receive for each dollar of production budget invested in a film? Hypothesis testing and confidence intervals for PB 2. Are higher production budget films more variable in their domestic gross than low budget films? MLR assumes constant variance so can t get at this directly with an MLR (maybe could break up the data into pieces). 3. What films over vs. underperformed relative to their predicted domestic gross? Predict domestic gross and look at observed - predicted gross. 2 I)

<latexit sha1_base64="rwexygccyefaenu1qbwu0b+ewtc=">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</latexit> <latexit sha1_base64="rwexygccyefaenu1qbwu0b+ewtc=">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</latexit> <latexit sha1_base64="rwexygccyefaenu1qbwu0b+ewtc=">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</latexit> <latexit sha1_base64="rwexygccyefaenu1qbwu0b+ewtc=">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</latexit> Model Specification: y MVN (X, Analysis w/ MLR 2 I) Model Results: Q1: What is the effect of production budget on domestic gross? A1: 95% CI: (1.01, 1.08) Q2: More variable in domestic gross? A2: Can t answer this. Q3: Most over vs. under performing movies? A3: Over: Force Awakens Under: John Carter So are we done? These answers are only valid if our assumptions are OK.

<latexit sha1_base64="rwexygccyefaenu1qbwu0b+ewtc=">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</latexit> <latexit sha1_base64="rwexygccyefaenu1qbwu0b+ewtc=">aaacnxicbvbnsysxfm2oz4/6nlzduglwxqcpmrfrl4ibbzukthy6feumzdrgmhmsjfig+vnu/b+udofcebf+btntbb8oba7n3epupsqr3fjfv/dgrsd+ju9mtpwmf/+zms3pzdensjvlnaqe0g0chgkes5rlvrbgohliitgzudgr/lnlpg1x8antjawlorvzifowtmqxd0mj9pxews/hq6hhevcfcii7qh/na+92iw8luvdzsjtofbpedgmzsjz/c6muhp8bb3/b5yq/7vebv5ngscpoigq7fbt2fe0liy0vyewz8bpbykbbtgxls2fqwal0arqs6wgmkplw1r86xyto6ebiafdii/vqx0qg0piejg6ywnh89qrxj6+z2minlfe4ss2l6ecjkbxyklxuidtcm2pfzxggmrtdmt0hdds6okuuhodryd9jfwm9cpxks7k7naxjei2ijbsgarsndte+qqiaouga3afh9otdea/es/cygb3xhpkf9ane6xvn4qvy</latexit> <latexit sha1_base64="rwexygccyefaenu1qbwu0b+ewtc=">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</latexit> <latexit sha1_base64="rwexygccyefaenu1qbwu0b+ewtc=">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</latexit> MLR Assumptions Model Specification: y MVN (X, 2 I) Validation Model Assumptions: N and E assumptions don t hold! But should we care? Yes because if our variances are wrong then all our SEs are all wrong.

<latexit sha1_base64="rwexygccyefaenu1qbwu0b+ewtc=">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</latexit> <latexit sha1_base64="rwexygccyefaenu1qbwu0b+ewtc=">aaacnxicbvbnsysxfm2oz4/6nlzduglwxqcpmrfrl4ibbzukthy6feumzdrgmhmsjfig+vnu/b+udofcebf+btntbb8oba7n3epupsqr3fjfv/dgrsd+ju9mtpwmf/+zms3pzdensjvlnaqe0g0chgkes5rlvrbgohliitgzudgr/lnlpg1x8antjawlorvzifowtmqxd0mj9pxews/hq6hhevcfcii7qh/na+92iw8luvdzsjtofbpedgmzsjz/c6muhp8bb3/b5yq/7vebv5ngscpoigq7fbt2fe0liy0vyewz8bpbykbbtgxls2fqwal0arqs6wgmkplw1r86xyto6ebiafdii/vqx0qg0piejg6ywnh89qrxj6+z2minlfe4ss2l6ecjkbxyklxuidtcm2pfzxggmrtdmt0hdds6okuuhodryd9jfwm9cpxks7k7naxjei2ijbsgarsndte+qqiaouga3afh9otdea/es/cygb3xhpkf9ane6xvn4qvy</latexit> <latexit sha1_base64="rwexygccyefaenu1qbwu0b+ewtc=">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</latexit> <latexit sha1_base64="rwexygccyefaenu1qbwu0b+ewtc=">aaacnxicbvbnsysxfm2oz4/6nlzduglwxqcpmrfrl4ibbzukthy6feumzdrgmhmsjfig+vnu/b+udofcebf+btntbb8oba7n3epupsqr3fjfv/dgrsd+ju9mtpwmf/+zms3pzdensjvlnaqe0g0chgkes5rlvrbgohliitgzudgr/lnlpg1x8antjawlorvzifowtmqxd0mj9pxews/hq6hhevcfcii7qh/na+92iw8luvdzsjtofbpedgmzsjz/c6muhp8bb3/b5yq/7vebv5ngscpoigq7fbt2fe0liy0vyewz8bpbykbbtgxls2fqwal0arqs6wgmkplw1r86xyto6ebiafdii/vqx0qg0piejg6ywnh89qrxj6+z2minlfe4ss2l6ecjkbxyklxuidtcm2pfzxggmrtdmt0hdds6okuuhodryd9jfwm9cpxks7k7naxjei2ijbsgarsndte+qqiaouga3afh9otdea/es/cygb3xhpkf9ane6xvn4qvy</latexit> Model Specification: MLR Assumptions y MVN (X, 2 I) Validation Model Assumptions: N and E assumptions don t hold! What do we do? Try transforming the data.

An MLR (even after transformations) has issues Transformations

<latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> Criticisms of MLR Criticisms: 1. We can t find a reasonable transformation to make the assumptions work. For this problem it is a good idea to model log(dg)~log(pb) because of the interpretation of a percent return: log(dg 1 ) log(dg 0 )=( 0 + 1 log(pb (1 + 0.01))) ( 0 + 1 log(pb)) ) DG 1 DG 0 =(1.01) 1 1+ 1 /100! " Interpretation: % return on 1% investment If 1 > 1 then you get a positive return on your investment.

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Model Specification for a Heteroskedastic MLR I.E. WRITING A STATISTICAL MODEL IN TERMS OF POPULATION PARAMETERS

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sha1_base64="+nvawgb0bmr2bzoqzrnysedrlbq=">aaach3icbzbns8naeiy3ftb6ffxozbeonkoiur0irs8ek9gpaelybkft0s0m7g7eevppvphxvhhqrlz137htg2jrwmkz78wwm2+ycka044ytpewv1bx1wkzxc2t7z9fe22+oojuu6jtmswyfrafnauqaaq6trakjqg7nchazytcfqcowi3s9tmcpse+wlqnegymwkydx2auhx0swrerl9jia/dujhox5owa1a6oc6pzubhbjktvtwivg5lbcedqc+8vrxdsnqgjkivjt10m0nxgpgeuwknqpgotqaelb26agesg/m943wsdg6ebulm0tgk/v3x0zizqarqgpnpv11xxuiv6xa6e6e+lntcspbkfng7opxzrge7nwh0mgmg8necqz2rxtppgeamnp0zjgzp+8ci2zsmv47rxurer2fnahokknyeuxqipuuq3veuvp6aw9oxfr2xq1pqzpwemslfccod9hjb8bw6+iza==</latexit> <latexit sha1_base64="+nvawgb0bmr2bzoqzrnysedrlbq=">aaach3icbzbns8naeiy3ftb6ffxozbeonkoiur0irs8ek9gpaelybkft0s0m7g7eevppvphxvhhqrlz137htg2jrwmkz78wwm2+ycka044ytpewv1bx1wkzxc2t7z9fe22+oojuu6jtmswyfrafnauqaaq6trakjqg7nchazytcfqcowi3s9tmcpse+wlqnegymwkydx2auhx0swrerl9jia/dujhox5owa1a6oc6pzubhbjktvtwivg5lbcedqc+8vrxdsnqgjkivjt10m0nxgpgeuwknqpgotqaelb26agesg/m943wsdg6ebulm0tgk/v3x0zizqarqgpnpv11xxuiv6xa6e6e+lntcspbkfng7opxzrge7nwh0mgmg8necqz2rxtppgeamnp0zjgzp+8ci2zsmv47rxurer2fnahokknyeuxqipuuq3veuvp6aw9oxfr2xq1pqzpwemslfccod9hjb8bw6+iza==</latexit> Heteroskedastic MLR Model Notation: To keep it simple, I m only going to consider PB y i = log(dom. Gross of i th movie) x i = log(prod. Budget for the i th movie) Model Specification: y N X, 0 1 0 1 0 1 0 1 DG 1 20.45 1 x 1 1 19.87 DG 2 y = B C X <latexit sha1_base64="r3rme1rijjywwkvvi4xv9q9/fcm=">aaab9hicbvdlsgmxfl3js9zx1awbyffclrkrdsmu3lisyb/qdiwtztrqtdimmciw9dvcufderr/jzr8x085cww8edufcyz05qcyznq777aysrq1vbja2yts7u3v7lypdlpajirrjjjeqe2bnoro0azjhtbmriqoa03ywvsv99oqqzar4nglm/qgpbqszwczkfi/czhsewtpfz7f9stwtutogzeivpaofgv3kv28gsrjryqjhwnc9nzz+hpvhhnnpuzdogmmyxkpatvtgigo/m4weolordfaolx3cojn6eypdkdzpfnjjpkre9hlxp6+bmpdgz5iie0mfmr8ke46mrhkdamaujyanlmcimm2kyagrtiztqwxl8ba/vexafzxp8oflav2qqkmex3ac5+dbndthhhrqbajp8ayv8ozmnbfn3fmyj644xc4r/ihz+qm92zgy</latexit> @. A = 19.30 B C 1 x 2 = B C @. A @.. A = 1 17.53 B C @.. A DG n 14.37 1 x n 1 13.81 2 D What is this? <latexit sha1_base64="qiarjry+9pbosbkbjq2shatw1fs=">aaacbhicbzc7tsmwfiydrqxcaoxdlfoqu5ugbcxilvgyi0qvuhnvtuo0vp04sh1efxvg4vvygecilydg421w2gzq8kuwpv3nhpmchyecke0439bs8srq2nppo7y5tb2za+/tt5vijaetiriqxywu5symlc00p91euhrhtjt4dj3xo/dukibioz1oqb+hqcxcrpa2vt+uebewd5mhbq8ipiew5mgquw0cj6/6dtwpo1pbrxalqijczb795qwcpbgnnefiqz7rjnrpknsmcdope6micsijnka9gzgkqpkz6retegscaizcmhdrohv/t2qoumocydozl6rma7n5x62x6vdsz1icpjrgzpzrmhkobcwtgqgtlgg+nocizgzxsiziiqjnbmutgjt/8ik0t+uu4duzauo8ikmekuaqnaaxxiaguafn0aiepijn8arercfrxxq3pmats1yxcwd+ypr8axwal0e=</latexit> = <latexit sha1_base64="x7fjh+vam9b+g397ek89y8f59li=">aaab9hicbvdlsgmxfl3js9zx1awbyffclrkrdsmu3lisyb/qdiwt3mldm5kxyrtk0o9w40irt36mo//gtj2fth4ihm65l3tygkrwbvz321lzxvvf2cxsfbd3dvf2swehdr2nimgdxsjwrybqffxi3xajsjuopfegsbkm76z+c4rk81g+mngcfkt7koecuwmlvxnrmwjcrduhz7fdutmtudoqzellpaw5at3sv6cxszrcazigwrc9nzf+rpxhtock2ek1jpqnar/blkoaofazwegjobvkj4sxsk8amln/b2q00nocbxzyglivelpxp6+dmvdgz7hmuooszq+fqsamjtmgsi8rzeamlafmczuvsafvlbnbu9gw4c1+ezk0liqe5q+x5epvxkcbjueezsgda6jcpdsgdgye4ble4c0zos/ou/mxh11x8p0j+apn8wcluzgr</latexit> = 0 PB

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sha1_base64="uqkctaws575et39uhhnwl5oldtu=">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</latexit> <latexit sha1_base64="uqkctaws575et39uhhnwl5oldtu=">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</latexit> Heteroskedastic MLR Model Specification: y N X, 2 D D = 0 d 11 0 0 0 0 d 22 0 0 B @.. 0... 1 C A 0 0 0 d nn Key Model Properties: 1. Var(y i )= 2 d ii 2. y i is independent of y j because all the covariances are 0.

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sha1_base64="xmruoap9dupt2y7kz+aqki2mpew=">aaab7xicbzbnswmxeizn61etx1wpxojf8fr2rdrjwyvhcvyd2qvk07snzszlmiuupf/biwdfvpp/vplvtns9aoslgyd3zsjmgyvswpt9b6+wtr6xuvxclu3s7u0fla+pmlanhveg01kbdkqtl0lxbgquvj0ytuni8ly0vp3vw0/cwkhva04shsz0qmramiroanzxxjh2yhw/6s9fvihioqk56r3yv7evwrpzhuxsazubn2cyuyocst4tdvple8rgdmg7dhwnuq2z+bztcuacphlo455cmnd/t2q0tnysr64zpjiyy7wz+v+tk+lgjsyesllkii0+gqssocaz00lfgm5qthxqzotblbarnzshc6jkqgiwt16f5ku1chx/wald5xeu4qro4rwcuiya3eedgsdgez7hfd487b14797horxg5tph8efe5w+gxy8c</latexit> <latexit sha1_base64="xmruoap9dupt2y7kz+aqki2mpew=">aaab7xicbzbnswmxeizn61etx1wpxojf8fr2rdrjwyvhcvyd2qvk07snzszlmiuupf/biwdfvpp/vplvtns9aoslgyd3zsjmgyvswpt9b6+wtr6xuvxclu3s7u0fla+pmlanhveg01kbdkqtl0lxbgquvj0ytuni8ly0vp3vw0/cwkhva04shsz0qmramiroanzxxjh2yhw/6s9fvihioqk56r3yv7evwrpzhuxsazubn2cyuyocst4tdvple8rgdmg7dhwnuq2z+bztcuacphlo455cmnd/t2q0tnysr64zpjiyy7wz+v+tk+lgjsyesllkii0+gqssocaz00lfgm5qthxqzotblbarnzshc6jkqgiwt16f5ku1chx/wald5xeu4qro4rwcuiya3eedgsdgez7hfd487b14797horxg5tph8efe5w+gxy8c</latexit> Heteroskedastic MLR Variance Functions: f(x) :R! R + because variances are always positive. <latexit sha1_base64="ej4kpi+ci0xwbbeytqq4hulhkb4=">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</latexit> <latexit sha1_base64="ej4kpi+ci0xwbbeytqq4hulhkb4=">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</latexit> <latexit sha1_base64="ej4kpi+ci0xwbbeytqq4hulhkb4=">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</latexit> <latexit sha1_base64="ej4kpi+ci0xwbbeytqq4hulhkb4=">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</latexit> Fixed Weights: d ii = v i where v i is known (e.g. inverse sample size). Exponential: d ii =exp{2x i } where x i is a covariate. Group Specific (Ident): d ii = g, g =1,...,G and 1 = 1, 2,..., G > 0 For the exponential and the group specific variance functions, the parameters are estimated from the data. Exponential makes the most sense for the movie application so we are going to use: d ii =exp{2log(pb i ) } <latexit sha1_base64="pho1ftczu/ot+auiw1o8rqihkhm=">aaacfxicbzdlssnafiyn9v5vuzdubotsqsqrutec6mzlbvufjotj9lqdnfyyorflyeu48vxcufderedot3f6wwjrdwmf/zmhm+cpuyk0os63vzqanpmdm18oly4tr6zaa+snnwskq50nmlg3idmgrqx1fcjhnlxaoldctxh30a/f3ipsiomvszech7folnqcmzrwyo+1glyigu6cug8eui8/oj5molup4qhz2nkrif3qyreqeuvgv5x9zya6ce4ikmskwmb/ea2ezxheycxtuuk6kfo5uyi4hklszrpsxu9yb5ogyxab9vpbvqxdnk6lthnlxox04p6eyfmkds8ktwfeskvha33zv1ozw/ajn4s4zrbiplzuzitfhpyjoi2hgkpsgwbccfnxyrtmmy4mylijwr0/eriab/uu4avdytnrki55skm2sjw45jickutsi3xcysn5jq/kzxqyxqx362pywrjgmxvkj6zphyuiniy=</latexit>

Model Fitting I.E. ESTIMATING THE PARAMETERS OF YOUR MODEL USING YOUR OBSERVED DATA

<latexit sha1_base64="l4zkdqnllbzwibcvjullic1+ix8=">aaacjnicbzdlsgmxfiyz9vbrrerstbavxeizkajuhiiblxxsbtq1znk0du0mq3jgleofxo2v4szfrcsdj2km7ujbdwq+/vmfcs4friibcn0vj7oyura+kd3mbw3v7o7l9w/qrswashpvqulmqawtpgq14cbym9kmyecwrjc8sfunr6ynv+e9jclwlqqf8h6nbkzuyv/7mlbpir8o0zuebrjobwxicxygfcp7kjyuls15yda1dfift+rocy+dn4ccmle1k5/4xuvjyukgghjt8twi2gnrwklg45wfgxyroir91riyeslmo5meocynvunintl2hycn6u+jhehjrjkwznrns9hlxf96rrh6v+2eh1emlkszj3qxwkbwmhnucs0oijefqjw3u2i6ijpqsmnmbaje4snluc+xpmt354xkxtyoldpcx+gueegsvdatqqiaougzvaijendendfnw/mcwtpofoyq/snn+wejyauy</latexit> <latexit sha1_base64="l4zkdqnllbzwibcvjullic1+ix8=">aaacjnicbzdlsgmxfiyz9vbrrerstbavxeizkajuhiiblxxsbtq1znk0du0mq3jgleofxo2v4szfrcsdj2km7ujbdwq+/vmfcs4friibcn0vj7oyura+kd3mbw3v7o7l9w/qrswashpvqulmqawtpgq14cbym9kmyecwrjc8sfunr6ynv+e9jclwlqqf8h6nbkzuyv/7mlbpir8o0zuebrjobwxicxygfcp7kjyuls15yda1dfift+rocy+dn4ccmle1k5/4xuvjyukgghjt8twi2gnrwklg45wfgxyroir91riyeslmo5meocynvunintl2hycn6u+jhehjrjkwznrns9hlxf96rrh6v+2eh1emlkszj3qxwkbwmhnucs0oijefqjw3u2i6ijpqsmnmbaje4snluc+xpmt354xkxtyoldpcx+gueegsvdatqqiaougzvaijendendfnw/mcwtpofoyq/snn+wejyauy</latexit> <latexit sha1_base64="l4zkdqnllbzwibcvjullic1+ix8=">aaacjnicbzdlsgmxfiyz9vbrrerstbavxeizkajuhiiblxxsbtq1znk0du0mq3jgleofxo2v4szfrcsdj2km7ujbdwq+/vmfcs4friibcn0vj7oyura+kd3mbw3v7o7l9w/qrswashpvqulmqawtpgq14cbym9kmyecwrjc8sfunr6ynv+e9jclwlqqf8h6nbkzuyv/7mlbpir8o0zuebrjobwxicxygfcp7kjyuls15yda1dfift+rocy+dn4ccmle1k5/4xuvjyukgghjt8twi2gnrwklg45wfgxyroir91riyeslmo5meocynvunintl2hycn6u+jhehjrjkwznrns9hlxf96rrh6v+2eh1emlkszj3qxwkbwmhnucs0oijefqjw3u2i6ijpqsmnmbaje4snluc+xpmt354xkxtyoldpcx+gueegsvdatqqiaougzvaijendendfnw/mcwtpofoyq/snn+wejyauy</latexit> <latexit sha1_base64="l4zkdqnllbzwibcvjullic1+ix8=">aaacjnicbzdlsgmxfiyz9vbrrerstbavxeizkajuhiiblxxsbtq1znk0du0mq3jgleofxo2v4szfrcsdj2km7ujbdwq+/vmfcs4friibcn0vj7oyura+kd3mbw3v7o7l9w/qrswashpvqulmqawtpgq14cbym9kmyecwrjc8sfunr6ynv+e9jclwlqqf8h6nbkzuyv/7mlbpir8o0zuebrjobwxicxygfcp7kjyuls15yda1dfift+rocy+dn4ccmle1k5/4xuvjyukgghjt8twi2gnrwklg45wfgxyroir91riyeslmo5meocynvunintl2hycn6u+jhehjrjkwznrns9hlxf96rrh6v+2eh1emlkszj3qxwkbwmhnucs0oijefqjw3u2i6ijpqsmnmbaje4snluc+xpmt354xkxtyoldpcx+gueegsvdatqqiaougzvaijendendfnw/mcwtpofoyq/snn+wejyauy</latexit> Estimation for the Heteroskedastic MLR <latexit sha1_base64="pho1ftczu/ot+auiw1o8rqihkhm=">aaacfxicbzdlssnafiyn9v5vuzdubotsqsqrutec6mzlbvufjotj9lqdnfyyorflyeu48vxcufderedot3f6wwjrdwmf/zmhm+cpuyk0os63vzqanpmdm18oly4tr6zaa+snnwskq50nmlg3idmgrqx1fcjhnlxaoldctxh30a/f3ipsiomvszech7folnqcmzrwyo+1glyigu6cug8eui8/oj5molup4qhz2nkrif3qyreqeuvgv5x9zya6ce4ikmskwmb/ea2ezxheycxtuuk6kfo5uyi4hklszrpsxu9yb5ogyxab9vpbvqxdnk6lthnlxox04p6eyfmkds8ktwfeskvha33zv1ozw/ajn4s4zrbiplzuzitfhpyjoi2hgkpsgwbccfnxyrtmmy4mylijwr0/eriab/uu4avdytnrki55skm2sjw45jickutsi3xcysn5jq/kzxqyxqx362pywrjgmxvkj6zphyuiniy=</latexit> Model Specification: y N X, 2 D( ) d ii =exp{2log(pb i ) } What are the parameters that we need to estimate?, 2, How do we estimate them? Maximum likelihood estimation Bayesian estimation (not here but take Stat 451)

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sha1_base64="pelq9/ezxygtlz98tfxeyvkp3is=">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</latexit> Estimation for the Heteroskedastic MLR Model Specification: y N X, What is our log-likelihood? L(, 2, ) = log(prob(y, 2, )) = log(pdf of y) 1 = log 2 1/2 exp 2 (y µ)0 1 (y µ) 1 = log 2 2 D( ) 1/2 exp 2 (y X )0 ( 2 D( )) 1 (y X ) = n 2 log(2 2 ) = n 2 log(2 2 ) 2 D( ) d ii =exp{2log(pb i ) } <latexit sha1_base64="pho1ftczu/ot+auiw1o8rqihkhm=">aaacfxicbzdlssnafiyn9v5vuzdubotsqsqrutec6mzlbvufjotj9lqdnfyyorflyeu48vxcufderedot3f6wwjrdwmf/zmhm+cpuyk0os63vzqanpmdm18oly4tr6zaa+snnwskq50nmlg3idmgrqx1fcjhnlxaoldctxh30a/f3ipsiomvszech7folnqcmzrwyo+1glyigu6cug8eui8/oj5molup4qhz2nkrif3qyreqeuvgv5x9zya6ce4ikmskwmb/ea2ezxheycxtuuk6kfo5uyi4hklszrpsxu9yb5ogyxab9vpbvqxdnk6lthnlxox04p6eyfmkds8ktwfeskvha33zv1ozw/ajn4s4zrbiplzuzitfhpyjoi2hgkpsgwbccfnxyrtmmy4mylijwr0/eriab/uu4avdytnrki55skm2sjw45jickutsi3xcysn5jq/kzxqyxqx362pywrjgmxvkj6zphyuiniy=</latexit> 1 2 log( D( ) ) 1 2 2 (y X )0 [D( )] 1 (y X ) 1 2 log( D( ) ) 1 nx ( 2 d ii ) 1 (y i x 0 i ) 2 2 i=1 Weighted least squares

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sha1_base64="0ujnzl7pnulkragwv+dcsna/qja=">aaab7xicdvdjsgnbeo1xjxglevtsgarpq88ystwcxjxgmaskq+jpdji2pt1dd40qhvydfw+kepv/vpk3dhzbrr8upn6roqpemehhgjapz219y3nro7et393bpzgshb23tjxqxpsslrhuhnrwkrrvggdjo4nmnaolb4etq7nfvufaifjdwjthqurhsgwfo2clvg/ghgi/ucruuxrzrpywcx2/wvoijz5x8ws+9lyyqbgt0ogx3nudmkurv8aknabrkqscjgoqtpjzvpcanla2ospetvtrijsgw1w7w+dwgebhrg0pwav1+0rgi2omuwg7iwpj89ubi3953rsg1satkkmbk7zcnewlhhjpx8cdotkdobwemi3srzinqaymbeb5g8lxp/h/0vjtlc65krxr5vucoxskztaf8laf1de1aqamyugopaan9ozezqpz4rwuw9ec1cwj+ghn7rmnqi9n</latexit> <latexit sha1_base64="0ujnzl7pnulkragwv+dcsna/qja=">aaab7xicdvdjsgnbeo1xjxglevtsgarpq88ystwcxjxgmaskq+jpdji2pt1dd40qhvydfw+kepv/vpk3dhzbrr8upn6roqpemehhgjapz219y3nro7et393bpzgshb23tjxqxpsslrhuhnrwkrrvggdjo4nmnaolb4etq7nfvufaifjdwjthqurhsgwfo2clvg/ghgi/ucruuxrzrpywcx2/wvoijz5x8ws+9lyyqbgt0ogx3nudmkurv8aknabrkqscjgoqtpjzvpcanla2ospetvtrijsgw1w7w+dwgebhrg0pwav1+0rgi2omuwg7iwpj89ubi3953rsg1satkkmbk7zcnewlhhjpx8cdotkdobwemi3srzinqaymbeb5g8lxp/h/0vjtlc65krxr5vucoxskztaf8laf1de1aqamyugopaan9ozezqpz4rwuw9ec1cwj+ghn7rmnqi9n</latexit> Estimation for the Heteroskedastic MLR Model Specification: y N X, How do we maximize a function? Take derivatives, set equal to zero and solve. The maximum likelihood estimates are: b =(X 0 [D( )] 1 X) 1 X 0 [D( )] The unbiased MLE for the variance is: 2 D( ) d ii =exp{2log(pb i ) } <latexit sha1_base64="pho1ftczu/ot+auiw1o8rqihkhm=">aaacfxicbzdlssnafiyn9v5vuzdubotsqsqrutec6mzlbvufjotj9lqdnfyyorflyeu48vxcufderedot3f6wwjrdwmf/zmhm+cpuyk0os63vzqanpmdm18oly4tr6zaa+snnwskq50nmlg3idmgrqx1fcjhnlxaoldctxh30a/f3ipsiomvszech7folnqcmzrwyo+1glyigu6cug8eui8/oj5molup4qhz2nkrif3qyreqeuvgv5x9zya6ce4ikmskwmb/ea2ezxheycxtuuk6kfo5uyi4hklszrpsxu9yb5ogyxab9vpbvqxdnk6lthnlxox04p6eyfmkds8ktwfeskvha33zv1ozw/ajn4s4zrbiplzuzitfhpyjoi2hgkpsgwbccfnxyrtmmy4mylijwr0/eriab/uu4avdytnrki55skm2sjw45jickutsi3xcysn5jq/kzxqyxqx362pywrjgmxvkj6zphyuiniy=</latexit> s 2 = (y Xb ) 0 [D( )] 1 (y X b ) n (P + 1) No closed form MLE for. There are non-paper and pencil ways to maximize functions (not going to get into those here but we will later). 1 y Weighting each observation in calculating coefficients

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sha1_base64="un5vnxxps/nsvbwlbecuwve0kra=">aaab+xicbzdlsgnbeev74ivg16hln41bcwoymrldcae3lioybyrd6olukiy9d7pramhin7hxoyhb/8sdf2mnmyumxmg43kqiqq8fs6hrcb6t3nr6xuzwfruws7u3f2afhjv0lcgodr7jslv8pkgkeoooueirvsacx0lth93p6s0xkc2i8aknmxgbg4silzhdy3vtu4ndqebp7+iluyqxk1276jscuegqubkusaza1/7q9ckebbail0zrtuve6kvmoeaspovooifmfmqg0dyysgc0l84vn9iz4/rop1lmhujn7u+jlavatwlfdaymh3q5njp/q7ut7n96qqjjbchki0x9rfkm6cwg2hmkomqjacavmldspmskctrhfuwi7vkxv6fxvxinp14xq5usjjw5iafkgrjkhltja6mroufktj7jk3mzuuvferc+fq05k5s5jn9kff4ayd2rew==</latexit> <latexit sha1_base64="un5vnxxps/nsvbwlbecuwve0kra=">aaab+xicbzdlsgnbeev74ivg16hln41bcwoymrldcae3lioybyrd6olukiy9d7pramhin7hxoyhb/8sdf2mnmyumxmg43kqiqq8fs6hrcb6t3nr6xuzwfruws7u3f2afhjv0lcgodr7jslv8pkgkeoooueirvsacx0lth93p6s0xkc2i8aknmxgbg4silzhdy3vtu4ndqebp7+iluyqxk1276jscuegqubkusaza1/7q9ckebbail0zrtuve6kvmoeaspovooifmfmqg0dyysgc0l84vn9iz4/rop1lmhujn7u+jlavatwlfdaymh3q5njp/q7ut7n96qqjjbchki0x9rfkm6cwg2hmkomqjacavmldspmskctrhfuwi7vkxv6fxvxinp14xq5usjjw5iafkgrjkhltja6mroufktj7jk3mzuuvferc+fq05k5s5jn9kff4ayd2rew==</latexit> <latexit sha1_base64="un5vnxxps/nsvbwlbecuwve0kra=">aaab+xicbzdlsgnbeev74ivg16hln41bcwoymrldcae3lioybyrd6olukiy9d7pramhin7hxoyhb/8sdf2mnmyumxmg43kqiqq8fs6hrcb6t3nr6xuzwfruws7u3f2afhjv0lcgodr7jslv8pkgkeoooueirvsacx0lth93p6s0xkc2i8aknmxgbg4silzhdy3vtu4ndqebp7+iluyqxk1276jscuegqubkusaza1/7q9ckebbail0zrtuve6kvmoeaspovooifmfmqg0dyysgc0l84vn9iz4/rop1lmhujn7u+jlavatwlfdaymh3q5njp/q7ut7n96qqjjbchki0x9rfkm6cwg2hmkomqjacavmldspmskctrhfuwi7vkxv6fxvxinp14xq5usjjw5iafkgrjkhltja6mroufktj7jk3mzuuvferc+fq05k5s5jn9kff4ayd2rew==</latexit> <latexit sha1_base64="un5vnxxps/nsvbwlbecuwve0kra=">aaab+xicbzdlsgnbeev74ivg16hln41bcwoymrldcae3lioybyrd6olukiy9d7pramhin7hxoyhb/8sdf2mnmyumxmg43kqiqq8fs6hrcb6t3nr6xuzwfruws7u3f2afhjv0lcgodr7jslv8pkgkeoooueirvsacx0lth93p6s0xkc2i8aknmxgbg4silzhdy3vtu4ndqebp7+iluyqxk1276jscuegqubkusaza1/7q9ckebbail0zrtuve6kvmoeaspovooifmfmqg0dyysgc0l84vn9iz4/rop1lmhujn7u+jlavatwlfdaymh3q5njp/q7ut7n96qqjjbchki0x9rfkm6cwg2hmkomqjacavmldspmskctrhfuwi7vkxv6fxvxinp14xq5usjjw5iafkgrjkhltja6mroufktj7jk3mzuuvferc+fq05k5s5jn9kff4ayd2rew==</latexit> Estimation for the Heteroskedastic MLR Model Specification: y N X, Parameter Estimates: b = 0.19 1.01 2 D( ) s 2 = 333 2 <latexit sha1_base64="bgbzjhxjdf6va1zngusfjqkqtu4=">aaab8nicbvbnswmxej2tx7v+vt16crbbu9ltrb0ibs8ek9ha2g5lns22odlksbjcwfozvhhqxku/xpv/xrtdg7y+ghi8n8pmvddhtbvx/xyka+sbm1vf7dlo7t7+qfnwqk1lqghtecml6oryu84ebrlmoo0kiui45pqxhn/o/mcnqjst4sfmehreechyxag2vvj1r4zuul1e79x65ypbdedaq8tlsqvynpvlr+5akjsmwhcotfy9nzfbhpvhhnnpqztqmmayxkpqwypwthwqzu+eojordfakls1h0fz9pzhhwotjhnrogjurxvzm4n+en5roosiysfjdbvksilkojesz/9gakuomn1iciwl2vkrgwgfibeolg4k3/piqadeqxr3q3l9ugpd5heu4gvm4bw+uoaf30iqwejdwdk/w5hjnxxl3phatbsefoyy/cd5/alwlj4c=</latexit> d ii =exp{2log(pb i ) } <latexit sha1_base64="pho1ftczu/ot+auiw1o8rqihkhm=">aaacfxicbzdlssnafiyn9v5vuzdubotsqsqrutec6mzlbvufjotj9lqdnfyyorflyeu48vxcufderedot3f6wwjrdwmf/zmhm+cpuyk0os63vzqanpmdm18oly4tr6zaa+snnwskq50nmlg3idmgrqx1fcjhnlxaoldctxh30a/f3ipsiomvszech7folnqcmzrwyo+1glyigu6cug8eui8/oj5molup4qhz2nkrif3qyreqeuvgv5x9zya6ce4ikmskwmb/ea2ezxheycxtuuk6kfo5uyi4hklszrpsxu9yb5ogyxab9vpbvqxdnk6lthnlxox04p6eyfmkds8ktwfeskvha33zv1ozw/ajn4s4zrbiplzuzitfhpyjoi2hgkpsgwbccfnxyrtmmy4mylijwr0/eriab/uu4avdytnrki55skm2sjw45jickutsi3xcysn5jq/kzxqyxqx362pywrjgmxvkj6zphyuiniy=</latexit> = 0.336 How do we interpret: b PB =1.01 : A 1% increase in investment is met by a 1.01% increase in revenue on average. Warning! These estimates are only valid if our assumptions are valid.

Model Validation I.E. CHECKING OUR MODEL ASSUMPTIONS, MODEL FIT AND PREDICTIONS

Validating the Assumptions of an MLR <latexit sha1_base64="pho1ftczu/ot+auiw1o8rqihkhm=">aaacfxicbzdlssnafiyn9v5vuzdubotsqsqrutec6mzlbvufjotj9lqdnfyyorflyeu48vxcufderedot3f6wwjrdwmf/zmhm+cpuyk0os63vzqanpmdm18oly4tr6zaa+snnwskq50nmlg3idmgrqx1fcjhnlxaoldctxh30a/f3ipsiomvszech7folnqcmzrwyo+1glyigu6cug8eui8/oj5molup4qhz2nkrif3qyreqeuvgv5x9zya6ce4ikmskwmb/ea2ezxheycxtuuk6kfo5uyi4hklszrpsxu9yb5ogyxab9vpbvqxdnk6lthnlxox04p6eyfmkds8ktwfeskvha33zv1ozw/ajn4s4zrbiplzuzitfhpyjoi2hgkpsgwbccfnxyrtmmy4mylijwr0/eriab/uu4avdytnrki55skm2sjw45jickutsi3xcysn5jq/kzxqyxqx362pywrjgmxvkj6zphyuiniy=</latexit> Model Specification: y N X, 2 D( ) d ii =exp{2log(pb i ) } Model Assumptions: 1. Linearity 2. Independence 3. Normality 4. Equal Variance (but only after normalizing)

Validating the Assumptions of an MLR Model Specification: y N X, Checking the Linearity Assumption: 2 D( ) d ii =exp{2log(pb i ) } <latexit sha1_base64="pho1ftczu/ot+auiw1o8rqihkhm=">aaacfxicbzdlssnafiyn9v5vuzdubotsqsqrutec6mzlbvufjotj9lqdnfyyorflyeu48vxcufderedot3f6wwjrdwmf/zmhm+cpuyk0os63vzqanpmdm18oly4tr6zaa+snnwskq50nmlg3idmgrqx1fcjhnlxaoldctxh30a/f3ipsiomvszech7folnqcmzrwyo+1glyigu6cug8eui8/oj5molup4qhz2nkrif3qyreqeuvgv5x9zya6ce4ikmskwmb/ea2ezxheycxtuuk6kfo5uyi4hklszrpsxu9yb5ogyxab9vpbvqxdnk6lthnlxox04p6eyfmkds8ktwfeskvha33zv1ozw/ajn4s4zrbiplzuzitfhpyjoi2hgkpsgwbccfnxyrtmmy4mylijwr0/eriab/uu4avdytnrki55skm2sjw45jickutsi3xcysn5jq/kzxqyxqx362pywrjgmxvkj6zphyuiniy=</latexit>

Validating the Assumptions of an MLR Model Specification: y N X, 2 D( ) d ii =exp{2log(pb i ) } <latexit sha1_base64="pho1ftczu/ot+auiw1o8rqihkhm=">aaacfxicbzdlssnafiyn9v5vuzdubotsqsqrutec6mzlbvufjotj9lqdnfyyorflyeu48vxcufderedot3f6wwjrdwmf/zmhm+cpuyk0os63vzqanpmdm18oly4tr6zaa+snnwskq50nmlg3idmgrqx1fcjhnlxaoldctxh30a/f3ipsiomvszech7folnqcmzrwyo+1glyigu6cug8eui8/oj5molup4qhz2nkrif3qyreqeuvgv5x9zya6ce4ikmskwmb/ea2ezxheycxtuuk6kfo5uyi4hklszrpsxu9yb5ogyxab9vpbvqxdnk6lthnlxox04p6eyfmkds8ktwfeskvha33zv1ozw/ajn4s4zrbiplzuzitfhpyjoi2hgkpsgwbccfnxyrtmmy4mylijwr0/eriab/uu4avdytnrki55skm2sjw45jickutsi3xcysn5jq/kzxqyxqx362pywrjgmxvkj6zphyuiniy=</latexit> Checking the Independent Assumption: Movies released on the same week may be negatively correlated but we don t see this in the data (it will have a minimum influence on our analysis).

Validating the Assumptions of an MLR Model Specification: y N X, 2 D( ) Checking the Normality Assumption: d ii =exp{2log(pb i ) } <latexit sha1_base64="pho1ftczu/ot+auiw1o8rqihkhm=">aaacfxicbzdlssnafiyn9v5vuzdubotsqsqrutec6mzlbvufjotj9lqdnfyyorflyeu48vxcufderedot3f6wwjrdwmf/zmhm+cpuyk0os63vzqanpmdm18oly4tr6zaa+snnwskq50nmlg3idmgrqx1fcjhnlxaoldctxh30a/f3ipsiomvszech7folnqcmzrwyo+1glyigu6cug8eui8/oj5molup4qhz2nkrif3qyreqeuvgv5x9zya6ce4ikmskwmb/ea2ezxheycxtuuk6kfo5uyi4hklszrpsxu9yb5ogyxab9vpbvqxdnk6lthnlxox04p6eyfmkds8ktwfeskvha33zv1ozw/ajn4s4zrbiplzuzitfhpyjoi2hgkpsgwbccfnxyrtmmy4mylijwr0/eriab/uu4avdytnrki55skm2sjw45jickutsi3xcysn5jq/kzxqyxqx362pywrjgmxvkj6zphyuiniy=</latexit>

Validating the Assumptions of an MLR Model Specification: y N X, 2 D( ) Checking the Equal Variance Assumption: Raw Residuals d ii =exp{2log(pb i ) } <latexit sha1_base64="pho1ftczu/ot+auiw1o8rqihkhm=">aaacfxicbzdlssnafiyn9v5vuzdubotsqsqrutec6mzlbvufjotj9lqdnfyyorflyeu48vxcufderedot3f6wwjrdwmf/zmhm+cpuyk0os63vzqanpmdm18oly4tr6zaa+snnwskq50nmlg3idmgrqx1fcjhnlxaoldctxh30a/f3ipsiomvszech7folnqcmzrwyo+1glyigu6cug8eui8/oj5molup4qhz2nkrif3qyreqeuvgv5x9zya6ce4ikmskwmb/ea2ezxheycxtuuk6kfo5uyi4hklszrpsxu9yb5ogyxab9vpbvqxdnk6lthnlxox04p6eyfmkds8ktwfeskvha33zv1ozw/ajn4s4zrbiplzuzitfhpyjoi2hgkpsgwbccfnxyrtmmy4mylijwr0/eriab/uu4avdytnrki55skm2sjw45jickutsi3xcysn5jq/kzxqyxqx362pywrjgmxvkj6zphyuiniy=</latexit> Standardized Residuals Q: How do you standardize with heteroskedasticity?

<latexit sha1_base64="sf6odkdrg3bh3fc8jqrazhpzyig=">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</latexit> <latexit sha1_base64="sf6odkdrg3bh3fc8jqrazhpzyig=">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</latexit> <latexit sha1_base64="sf6odkdrg3bh3fc8jqrazhpzyig=">aaacnxicfvfrixmxfm2mr7w+qn7tdwal0svlmsmy68eiccqyvldbhayomfroj2zmqxjht4t8k3+j3/w3ztpb9ifecjycey7n5istltqyrb+d8nr1gzdv7dzu3bl77/6d/snhx6zqticpqfslt1juqmkspihrwumtgrepgll6+q7tz76dnriqv+k6hkxbv6xmpodoqat/k+uc7dolligcos3hh3p0jtoyokzgmauu7jfjcjicbrqsbequiegrev7ezu/eosdqfmaou9vq+je9ylc7r823ewvwvjm7+p9wa4f568ec03kv427sh0t70abovrb3yec6mit9x2xziaaaeoxixszjqmaf5rqluob6rdfqc3hkvzd3sor+/4xdpovoc88savzpf0qkg/b8howfmesi9co2mho515l/6s0bzn4srczrbqeuw6osurqr2n4vxuonatxaay609ltskxpnbfop7fkq4stpvgqor/uxx19ed8yhxrw75cl5toykjodktd6qczksetwjxshh4fp4lhwffg6pttiw6gyekwsvzv4aijfrhg==</latexit> <latexit sha1_base64="sf6odkdrg3bh3fc8jqrazhpzyig=">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</latexit> Validating the Assumptions of an MLR Our L-I-N-E assumptions check out OK, but how good are our predictions? Run a cross-validation. Is there anything different about a cross validation with heteroskedasticity? Yes! We have to plug in x s into the mean equation (the regression line) but also into the variance function so we get the predictive variances (e.g. width of the prediction intervals) correct. ŷ new N ˆ0 + ˆPB log(pb new ),s 2 exp{2 log(pb new )ˆ }

Validating the Assumptions of an MLR Our L-I-N-E assumptions check out OK, but how good are our predictions? Run a cross-validation. Prediction Diagnostics: RPMSE = 53,648,941 Coverage = 0.94 Width/2 = 116,518,415 Are these results good? Min. 1st Qu. Median Mean 3rd Qu. Max. 1632 8,000,000 26,655,881 49,808,759 62,462,374 936,662,225 They aren t ideal - just OK but keep in mind we are ONLY use production budget to predict domestic gross (lots more variables impact movie success).

Validating the Assumptions of an MLR Our L-I-N-E assumptions check out OK, but how good are our predictions? Run a cross-validation. Prediction Bounds: