Exercise 1 The General Linear Model : Answers

Transcription

1 Eercse The General Lnear Model Answers. Gven the followng nformaton on 67 pars of values on and a fnd the OLS coeffcent estmate from a regresson of on. Usng b 9 So. 9 b Suppose that now also depends on another varable Z and that Z - Z - Z - Z - Z - Z - 8 Fnd the coeffcents of and Z n the OLS regresson of on and Z. Hnt use the fact that the set of regressors n ther mean devaton form Σ whch s equvalent n matr terms to and use matr algebra to solve. We know that the OLS slope parameters can be obtaned from estmatng the model n mean devaton form see Eercse Let the mean devaton verson of the model b b Z u be wrtten as b b z u where assumed to be zero z Z Z and U u snce the mean of the resdual term s Easer to solve ths varable model usng matr algebra

2 Let w [ z] where s the vector of observatons on the varable n mean devaton form Hence and model becomes wb u OLS gves b b w w w b b b 8 b / 6 8/ 6 8/ 6 9 / 6 so b.8 b.78 b Wh does the coeffcent of n part b dffer from that n part a? What conclusons about the estmated relatonshp can ou draw? Hence the sze of the estmates coeffcent on the varable ncreased compared wth the estmate from the orgnal varable model. Ths confrms the negatve covarance between and Z that s mplct n the varance-covarance matr of eplanator varables. Snce lecture notes show that b If > snce - > then var var > f and onl f Z <

3 . A regresson of total cost on output produces the followng coeffcents and standard errors Total Cost Output -.96Output.94Output Do the results conform to the theoretcal epectatons about tpcal margnal and average cost curves? Mcroeconomc theor tells us that total cost curves usuall dspla ncreasng and then decreasng returns to scale. Ths shape can be captured b a cubc rd degree polnomal where the constant s an estmate of the fed cost should therefore be postve Can see from above that ths s true We also know that the margnal cost and average cost curves should be U-shaped each wth a postve mnmum value Snce the MC curve can be found b dfferentatng the total cost curve wth respect to output MC dtc dq Q Q and for an quadratc to be U-shaped rather than Λ-shaped then need > whch s satsfed n the output above Smlarl the AC curve AC TC Q Q Q Q needs < and > to be Λ-shaped The level of output at whch MC s mnmsed s gven b dmc dq Q 6Q so * Q at the mnmum Snce Q* b assumpton s > and we have establshed that > then < s a necessar condton for the mnmum output to est and ths s therefore consstent wth the estence of Λ- shaped MC and AC curves To ensure that the mnmum output value of the MC curve > dtc MC Q Q > for all Q dq

4 sub n mn Q value from above > dq dtc MC so > ff > whch wll be the case ff > So all the estmated values n the output above are consstent wth the requrements of cost curves. Gven -, wrte down the normal equatons n a varable model estmated b least squares. In the varable model k n So k k and - becomes multplng through gves 4

5 so n the varable model the OLS ftted lne passes through the hperplane of means Sub. 4 nto Usng the fact that becomes and becomes Solvng smultaneousl and usng mean devaton notaton 5 6 From 6 f was absent from the model then the ols estmate of b would be ~ Snce the correlaton coeffcent squared between and s r then 6 can be wrtten as ~ r r

6 so the slope estmate n the varable regresson contans a correcton to the slope estmate from the smple varable regresson that accounts dfor the effect of the addtonal varable on both and If the correlaton between and s zero then the slope estmates n the varable and the varable model are the same 4. Show that the R can be nterpreted as the square of the correlaton coeffcent between the actual and ftted values of the dependent varable. From eercse we know that the slope coeffcents from a model can alwas be obtaned b estmatng the model n mean devaton form. k k Can wrte n a more compact matr form usng the followng notaton Let the mean devaton matr A be gven b A I where I s the dentt matr of order and s an column vector of ones multplcaton b ts transpose gves an matr of ones so A has the propertes that t s both smmetrc AA and dempotent AA

7 and an matr that s pre-multpled b ths matr A wll have the propert that the resultng elements wll be n mean devaton form Proof b eample Consder the vector Then A * * * Snce multplng an matr or vector b the dentt matr leaves the vector unchanged equvalent to multplcaton b one * A note that a sngle mean value can alwas be wrtten as To show that the R s the square of the correlaton coeffcent between the actual and ftted values Gven A A TSS ESS R

8 A A snce Snce u then u A Au A A snce the mean of OLS resduals s zero then u u A So A u A A snce u u usng the algebra of least squares lecture notes Hence A A A A R ow the square of the correlaton coeffcent r R A A ote It follows that the OLS slope coeffcents can alwas be obtaned f the regresson s run n mean devaton form Gven u Multpl b the transformaton matr A Au A A ote that multpl b A elmnates st coeffcent n B vector that on the constant snce

9 A [ A A ] Au A B Au Β snce A Let A* and A* * * u Pre-multpl b * * * * * * u whch snce * u gves and snce A u u * * * * e the slope coeffcents f OLS s run n mean devaton form 5 Gven the followng sets of nformaton, sa whether our beleve an error has been commtted durng the course of the estmaton process. a R..95 R.4.9 b R. 86 R a There must be an error snce we know that the unadjusted R wll not fall f an etra varable s added to the model Proof Partton the model such that b u Where s an vector s an k- matr b s a scalar s a k- vector and [ ]

10 The OLS normal equatons b can now be wrtten as nd row can be wrtten b so b ~ b ~ where s the OLS estmates f s regressed on alone The OLS resdual from model s u b Sub. nto ~ u b b ~ b b 4 ~ Let e Be the vector of OLS resduals when s regressed on alone and let b b I b M b where M s the resdual maker matr I

11 so now 4 can be wrtten as u e M b * e b 5 wth * M beng the vector of resduals from a regresson of on The resdual sum of squares * u u e * b e b e e * * * b e b 5 Gven we know that the OLS estmates can alwas be obtaned b frst nettng out the nfluence of from both and Frsch-Waugh Theorem then * * * * b and snce e * M then * * * * * e b from 6 7 Sub. 7 nto 5 u u e e * * b 8 Snce each term on the second term on the rhs > Then the RSS from the equaton contanng more varables < the RSS from the model wth less varables Hence u u < e e and ths mples that the R RSS/TSS wll never fall when another varable s added to a regresson model Q.E.D. b Gven. 86 R R. 8..4

12 Snce R R k Then t s possble that the on the sample sze, R can fall f the t rato on the etra varable < but ths s condtonal R R Snce for an k k ncludes the constant k R then k R R and 4 k R 4 Snce part a shows that the addton of etra varables R then R 4 < R or R 4 5 < R *-5<-.86*-4.8*-5 <.4*-4-5 < < 8.5 So can be sure that an error has been made unless the sample sze falls below 8 [To show that the R ncreases when the t value on the varable > / Let RSS / k u u k R F TSS / A / be the adjusted R when the etra varable k s added to the model

13 / Let RSS / k u k uk k R k TSS / A / be the adjusted R when the etra varable k s not n the model R F R k u k uk / k A / u u/ k A / u k uk k A k u u A ths dfference wll be > ff the rato of these two terms > u k uk / k u u/ k > But we know that the RSS from the full model u u * * uk uk bk k Ak where the nd term on the rght hand sde s the contrbuton of k to the eplaned sum of squares n mean devaton form see lecture notes u u * * / So becomes bk k Ak k > u u/ k u u * * bk k A k k > k u u u u snce we know the estmated RSS u u s k

14 u u * * bk k A k s u u > and snce each term n the numerator s a squared scalar value must be postve then the fracton can onl be > ff * * bk k Ak > s but ths s just the square of the estmated t value on the varable k. Hence the the t value on the varable > ] R ncreases when 6. Consder the multple regresson model B u Suppose the ndependent varables are subject to a lnear transformaton ZΛ where Λ s a dagonal matr of constants. Show that the resduals from the regresson of on Z are the same as the resduals from a regresson of on. Compare the coeffcent estmates from the regressons. Ths s a general proof of the result that rescalng a varable n a model multples the orgnal ols estmate b the recprocal value of the rescalng constant The transformaton matr appears λ Λ λ λ k Gven B u Zγ v OLS resduals from both models are u v Z γ - ZZ Z - Z

15 [I - ] [I - ZZ Z - Z ] [ I ΛΛ Λ - ] [ I - ] u so the resduals are the same n both specfcatons and f the resduals are the same then so must be the R values From the coeffcent vector γ Z Z Z Λ - - Λ - Λ Λ - - Λ - Λ usng propertes of matr nverses ABC - C - B - A - Λ - - γ Λ hence the orgnal OLS coeffcents are rescaled b the nverse of the rescalng constants contaned n the matr Λ ote that the predcted values n the two models are the same snce Z γ ΛΛ Show that f nstead just one ndependent varable s multpled b a constant, λ, then the correspondng regresson coeffcent s multpled b / λ and all other coeffcents are unchanged. If the varable to be transformed s j n ths case the transformaton matr looks lke Λ λ j e a dagonal matr wth ones down the man dagonal ecept for the jth element whch contans the constant of multplcaton for the j th varable Snce the nverse of a dagonal matr s also dagonal wth the recprocal of each orgnal element on the new man dagonal then

16 Λ λ j So usng the result n t follows that then the correspondng regresson coeffcent s multpled b /a and all other coeffcents are unchanged. Show the effect of addng a constant to one of the rght hand sde varables. Show that the result mples that for a varable entered n logs the least squares coeffcent s ndependent of the unts n whch the varable s measured. If a constant s added to one of the rhs varables then λ j Λ e an upper trangular matr zeros everwhere below the man dagonal wth the addtve constant λ n the jth column of the st row and zeros everwhere else above the man dagonal and ones along the man dagonal It can be shown eg Johnston & Dardo that the determnant of an upper trangular matr equals the product of the elements on the man dagonal so n ths case the determnant equals the nverse of an upper trangular matr s also upper trangular Λ Λ a a a a j a j a jj ak a a k Λ a kk a a a j a j a jj ak a k a kk where a j s the relevant adjont element based on the approprate cofactor Hence can show that the cofactors on the man dagonal are all equal to one and the other adjont elements wll all be zero ecept for a j whch -λ j So

17 Λ λ j and γ Λ becomes k j j j k j j λ λ γ So the OLS estmates of the slope coeffcents are unchanged when one varable s transformed b an addtve constant, but the OLS estmate of the constant term s reduced b λ j j An eample of how ths result s sometmes seen n practce concerns logarthmc transformatons of varables. Consder a log-lnear model Log Log k Log k u then f s transformed b λ then Log λ Log λ Log Usng the above result, the OLS estmate of the coeffcent wll be unchanged but the estmate of the constant wll become λ Log So when estmatng a model n logarthmc form the OLS estmates are nvarant to the unts of measurement. 7. In the earnngs functon lterature, two specfcatons are commonl estmated a a Ed a Age u and b b Ed b Eperence u where s log earnngs, Ed. Is ears of educaton and Eperence s ears spent n the labour market after leavng school. Estmates of b are tpcall found to be around twce as large as those for a. How can ou eplan ths?

18 Hnt Age ears of Work Eperence ears of Educaton constant Snce AgeEperenceEducatonconstant then the nd equaton s a transformaton of the frst Eperence Age-Educaton-constant So n b b Ed b Age-Educaton-constant u b - b constant b -b Ed b Age u a a a u We know from queston 6 that a varable transformaton wll leave the resduals n an OLS regresson unchanged so u u And the coeffcents a b -b so a <b assumng b > Hence t s possble that the estmate of a wll be much less than that of b and a b a 8. Gven the equaton t b b t b t. b k kt e t t, T can be wrtten n matr form as Β e where and e are T vectors, s a T column vector of ones, s an Tk matr of observatons on k ndependent varables, s the coeffcent on the constant term and B s a vector of coeffcents on the other k ndependent varables Show that the frst dfference of ths equaton Δ t b Δ t b Δ t. b k Δ kt Δe t t T can be wrtten n matr terms as A AΒ Ae where A s a T-T matr that satsfes the condton A What s the epected value of the OLS coeffcents estmated usng equaton? What can ou sa about the varance of these estmates? The condton A and the requrement that AΔ mean that

19 T T T T T T A T e can transform a vector b multplng b the matr A and get a vector of st dfferences s satsfed b the followng transformaton matr A e an upper trangular matr wth - on the man dagonal wth the ecepton of the last row note also that A snce Pre-multplng b A gves A A A Ae A Ae whch s the model n st dfferences Gven the followng nformaton ou can fnd the data set ps.dta on the course webste, from a regresson of the log of hourl earnngs on a set of eplanator varables, calculate a the percentage dfference n pa between men and women unon members and non-unon workers graduates and non-graduates 4 ethnc mnort female graduate unon members wth ears eperence and male whte, non-unon, no qualfcatons wth ears eperence reg lhw female ethnc unon grad ntermedate eper eper Source SS df MS umber of obs F 7, Model Prob > F.

20 Resdual R-squared Adj R-squared.9 Total Root MSE lhw Coef. Std. Err. t P> t [95% Conf. Interval] female ethnc unon grad nter eper eper cons ote Female f female, f male ethnc f ndvdual s from an ethnc mnort otherwse unon f member of trade unon, otherwse grad f have a degree, otherwse ntermed f ntermedate qualfcatons otherwse Hnt read The Interpretaton of Dumm Varables n Semlogarthmc Equatons, b R. Halvorsen and R. Palmqust, Amercan Economc Revew, 98, Vol. 7, June, o., pp avalable through JSTOR. http//uk.jstor.org/vew/88/d9574/95p67n/?currentresult88%bd 9574%b95p67n%b%c%b986%b999%b89999&searchID858cba.65 45&framenoframe&sortOrderSCORE&userID86db4f@rhbnc.ac.uk/858cba5 e&dp&vewcontentartcle&confgjstor Gven a sem-log wage equaton LnW a b cd u A where s a contnuous varable D s a dumm varable f attrbute satsfed otherwse The coeffcent on a contnuous varable n a sem-log model dw dlnw b W d d % change n wages wrt a small unt change n dvded b Snce a dumm varable s dchotomous the dervatve does not est The dscrete equvalent to s the proportonate change n W when the dumm varable goes from zero to one

21 W W g D D W D W D g WD Takng natural logs LnW D Lng LnW D LnW D - LnW D Lng ow the coeffcent c n A measures the dfference n log wages wth D or D Whch from equals Lng Hence the proportonate change n wages g epc- c Epandng c Lng C g -/g /g -. So for small g then c g But for large ve g then c < true g e estmated coeffcent under-estmates true % effect and for large -ve g then c > true g Suggests need to transform the sem-log estmate for an coeffcent >.5 n absolute value otherwse wll ms-measure the proportonate effect of the dumm varable e use the transformaton epc-g Gven the estmates n the problem set then the coeffcent on the female dumm varable gves the average dfference n the log of hourl pa between men and women other thngs equal. Wth female. 8 ths suggests that women earn around 8% less than men However a more accurate estmate s gven usng the converson formula % dfference ep female ep so that women earn around 4% less than men

22 These dfference become more notceable the large the estmate n absolute value Hence the graduate coeffcent suggests a dfferental of around 8% but the actual estmated percentage dfference s gven b ep.8 -. e % more