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# Adv Topics in Econometrics ECON 310

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This 26 page Class Notes was uploaded by Aliya Schumm PhD on Wednesday October 28, 2015. The Class Notes belongs to ECON 310 at Vassar College taught by Staff in Fall. Since its upload, it has received 54 views. For similar materials see /class/230539/econ-310-vassar-college in Economcs at Vassar College.

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Date Created: 10/28/15

Economics 310 Handout VII The Normal Distribution and Properties of Es imators Thenrem LeiX NQJJZ Lety 4 bxwhereazmdb are cmmnts Theny Na ma a Thenrem Letxlgcz x be Indepmdentrandomvanablesx Im gf Lety 2 klx where kl are constants Thmy 412 1 k Za Z N z Thenmn Lemle x be randomvanablesx Nw ml my N Z Z Lay Ekyx Theny N21 sz a af xall Theurem deN mm mm x2 N 121 Theumm Letxlgcz in her 1ndepmdmt randomvanables each dxsmbuted 2 2 x 1 Then 2 X Zk Thanrem max 6quot ben 1ndepe1dmtrandomvanables ach dxsmbuted No1 n Then 3 x2 ngn 11 Thenrem Lem1 x2 xquot beh 1ndepmdmtrandtxnvanables ach dxstnbuted mo 039 t z x h quot 20 Tanzi1n 11 7 2 2 X1 quot1 Thenrem Letx z r1 x my n be Independmythen 1 I z 2 x2 m2 Thanrem deN 1201 NO1bemdependmt thm Miscellaneous Notes 2 N F NZ en 1 Letx 301 Thm Ex rz and a 2quot Let z N 1520 then for n gt 100 the vanable 22 7 12n 7 N N01 Multivariate Normal Let xbe an n rdlmenslonal muluvanate normal random vector thh mean yand variance rcovanance mamx n2 1 Th 0 1 70pm lbw an x 72 2mm 2 m1 2 Themm Letx N N1ELhen a Ax N Na ApAEA A n u Mum Thenrem Lem x N No1 and A be 1danpotmt ofrank r men x Ax N 1 r L h Themm Letx N No1 and A and Bbe 1danpotmt Thmx Ax and x Bx are 1ndepmdmt quadmue forms fo13 Thenrem Lem x N mo 03D and A and B be 1ndependent and manpotmt of ofmnkr mdrz respeeuvely Than N Frr2gt F2 Ordinary Least Squares Noa391 y x17 1 x xquotx39u rd oJX39X1quot and e39 Ne39iole39x39x e c u Mu whereM Is xx39x x39 SSE u Mu he smce xssymmetncxempotentoran he z k M d f k k 5 e737 c e39x39xquotxh c IrMu 17 aremdependentsmce Ir MM 0 k Testmg hypotheses rhvomhg qmdependent restrictions 0 Wme the restrictions as b can readdy be shown that SEEK SE 397 SSE MM 1 Propenies ofEsu minors Unbiased Le ibe an emumm cw nien is anunbiased esumaieiifm i 9 Now lhls de m on applies equally well wneuiei Qand 1 are scalars or vectors Wede ne mm 7 55 Emciency Let 9 be an Esllm lol eto nien M35193 E 79Z a Heisman irgisascaiai MSE E 7 9 7 9y VAR bzm bzm If 9 is aveaci De nirien Let i and 9 be two esllm lols of 9 Then i is ameie ef cient esummol mane If MsE 4visEe it9 is asealax MSE9 7MSEi is nennegabve de nite and im is a Vecvm Note If 9 and axe two unbiased esllm lols of 9 vhese de m ons m be sumd in ieinis ofvanances Vax ltVax9 irg is aseaian VARw wAR misncn itg is aveaei varc39 quot 2 vane3 for every e39 i r me mfolm non manix cm is defined as follows a 1n L09 I 5 7 7 E 7 555539 19 4 is called me Cianiei Rab Lewei Bound CRLB x any unbiased emumm em This means um fox any unbiased 8511mm VAR 7CRLB is nomeg nve de nite 111 Asympmdcrmpemes nfEs mamrs n nin n K n ccnsianmf Iijm Pdi gt e 0V gt o Weusethenommn minim c De nition An estimator 5 is said to be a consistent estimator of 9 if plim 9 Theorem Let plim JM c Let g x be a continuous mction Then p1img X gc Theorem A suf cient condition for an estimator to be consistent is its bias and variance each approach a limit of 0 as n approaches in nity De nition Let Z 2 be a matrix of random variables each of which has a probability limit Then plim Z plim 2V Theorem Let A and B be two matrices such that plim A and plim B and the product AB exist Then plim AB plim Aplim B Theorem plim A 1 plim A 1 Convergence in distribution Let xn n 1 2 be a sequence of random variables Let FD n 12 be the sequence of cumulative distribution mctions CDF of the random variables xn n 12 This simply means that Fxnlt a Fna n 12 The sequence of random variables xn is said to converge in distribution to a random variable x with cumulative distribution mction F if linnanxFx at all points where F is continuous Alternatively we can say that xn converges in distribution to a random variable x if limnmPxnltaPxlta for every a at which F is continuous A familiar example of convergence in distribution is given by the central limit theorem which states that for any underlying population with nite mean and variance the distribution of z in u n IAZ converges to a standard normal distribution If xn converges in distribution to a random variable x with CDF Fx we say that Fx is the limiting distribution of xn Asymptotic distributions Most of the estimators which we are interested in this course have degenerate limiting distributions which is to say that in the limit the distribution collapses around a point This is not a very use ll property if we want to compare the asymptotic behaviors of two or more estimators For example we might have two estimators both of which are consistent Both distributions would collapse around the true value of the parameter which we are estimating But it is possible that the distribution of first estimator collapses more rapidly around the true value than the distribution of the second estimator so that for any large sample size n the variance of the first estimator is smaller than the variance of the second estimator assume that both estimators are unbiased Then we would say that in some sense the first estimator is more efficient than the second estimator at least for large sample sizes But the limiting distribution is not very use il for this purpose because in the limit the variances of both estimators are equal to zero In order to make the kinds of comparisons which we are considering we need a different concept the concept of an asymptotic distribution An asymptotic distribution is a distribution which is used to approximate the true distribution of an estimator for large but finite sample sizes The most common way of deriving an asymptotic distribution of an estimator is to find a transformation of that estimator which has a nondegenerate limiting distribution For example suppose that we want to find the asymptotic distribution of the sample mean The sample mean has a degenerate limiting distribution However we know that the mction zn has a standard normal distibution where inu IAZ Then using the relationship En p 1 2quot we can derive the following asymptotic n 2 distribution for the sample mean xquot Npo n Maximum Liklihood Estimators Suppose that the probability density function of a random variable x x Where 3 is a vector of parameters satisfies some regularity conditions The regularity conditions are conditions involving such things as the differentiability of the log of the density functions All the density functions which we will encounter in this course will satisfy these discussions For a full discussion of these conditions see Theil Principles ofEconometrics 1971 If the regularity conditions are met then the maximum likelihood estimator of 3 is 1 consistent II asymptotically normal 111 asymptotically ef cient IV achieves the Cramer Rao Lower Bound This last condition means that the asymptotic variance covariance matrix of the estimator is equal to the CRLB Maximum likelihood estimators have another desirable property the property of invariance This property simply says that if 6 is the maximum likelihood estimator of 0 and g0 is a continuous mction then MLEg 9 gMLE 9 FamnW Tv mums a Then the log of likelihood function 0mg sample 1 n n l a in 31mm Emory 2amp2 yquot 7 2 lnLA L M M j 1 Y 7 2 szmiw 321111 1111 jl h iwr liJYLT F E 1 13 903431 451 L IIWJ O Economics 310 Handout IX Time Series Models smonamy nr39 Au axe met E01 EON 5017 m2 Em w 0 EU MUM I E05 mm l V IY 15 called the aummvariance oflag s of39he tune senes y p 41 called the aummrrelan39on oflag s u De nition A cure senes e s called a whim noisepmcess 1r Ee 0123 5 EeeH 0V 0 Aumregressive and moving average models Amp 5 M 11 2139 pl xy 9 Amy an 2 Mmq Y1 u 2 WANm ake y u 3L2 ARMApq Amy Bays an ARDvJApdq My 15 an ARMApq Smn onarity An A necessary condmon fox smumanty 15 um I a 0 as A m An MAq model 15 necessanly stanonaxy 1f q 15 fume Y M t wode am can be expxessed as an AR Ah MAq 15 mvanble 1fthe mots ofBL 0 all he ontstde the sum male A mte AR 15 always mvemble Invemmhty and Stahehanty of m ARMApq Ah ARMApq ALy B Le an as u an BL u an he minds the ml eme em a t ways A Ly 3 Lye Assurmng fox stmphetty that the constant term 15 zem 311 y 2 pure MA form AL AL 311 2 pure AR fann N L th M oxdex1fthe pure MA form 15 of mte oxdex the pure AR 15 ofm mte oxdex Ifboth the AR and M s c r R t oxdex mtewemng hme penods An example should make ns clear quot yfwt e q 11 15 easyto venfy thaty 12 Conndexnow the palh l autoeonelahoh The pamal h pamal anteeenelaheh oflag 2 fax the senesy WM 122sz e as not zem because y Econ310 Handout 1 Some Useful Matrix Results Notation Capital letters denote matrices Small bold letters denote vectors Small unbold letters denote scalars Unprimed small bold letters denote column vectors primed small bold letters denote row vectors Things which you should know about matrices and vectors Matrix column vector row vector square matrix diagonal matrix scalar matrix identity matrix matrix equality matrix addition and subtraction scalar multiplication matrix multiplication A39 the transpose ofA 391 the inverse ofA AB39 B39A39 assuming that A and B are conformable for multiplication AB391 B39lA391 assuming both inverses exist A39391 A39139 a square matrix which has an inverse is said to be nonsingular A is symmetric if A A39 A is idempotent ifAA A The trace of a square matrix is the sum of the diagonal elements Two vectors x and y are said to be orthogonal iny 0 a set of vectors x1 x2 x is said to be linearly dependent if there exists a set of scalars a1 a2 an not all zero such the alx1 a22 auxquot 0 Otherwise the set of vectors is said to linearly independent The rank of a matrix is the maximum number of linearly independent rows or columns Note Given an n x m matrix A rankA 3 minimum of n and m For any matrix A rankA rankA39 rankA39A let A be a square matrix A non zero vector x is said to be an eigenvector of A if there exists a scalar such thatA x is called an eigenvalue of the matrix A x39Ax is called a quadratic form Note any quadratic form can be expressed as a quadratic form with a symmetric matrix A is positive de nite if x39Ax gt 0 for every X 0 A is positive semide nite if X Ax Z 0 for every x 0 If A is symmetric and idempotent then i rankA traceA ii A is positive semide nite Thm TzaceAB 7uace BA Cnrr naceABC 7 naceCAB 7 uaceaacA Thm IfA IS posmve de nue than A 15 nnnsinglllar Thm A 15 pnsiljve de niu at and only at all Its eigenvalues are posmve Thm TzaceA 7 sum ofthe elgenvalues ofA DEF A l l ll space DEF DEF ofthe vectm space DEF The dimmsinnof a vectm mace IS the numbezofvectozs In any basis Nate vecth J and b Tax by 7 aTx cm A hneaznansfmmanon can always be chzesented by a mam idempntmt Then only er the vectozy has us the space spanned bythe column vectms ofA Matrix calvllllls See GujannAppend1x B 6 Let x be an n vector Let a be an n vector Let A be an n X n matrix 9 9 9 ia39x a iAxA 7AXA39 if XX if t9 Eben ZAx assumlng that A IS symmemc 9 lt 3 39Ax A39 39Ax Ax 3 y y a y c Random Vector Suppose thatxx2 are n random variables Then the vector 1Ci Em r2 Em x isarandom vector Ex r Em The Variance covariance martix ofx which we can Val0 2 5i 032 01 0 02 0 W Varx Exr Exxr Ex39 2 2 2 Eu 02 53 where CE J and UV a A then EAx AEX and VarAx AVarxA39 A useful result is that ifA is a summetric matrix Ex39Ax Ex39AEx u39accAVarx Economics 310 Hxndom 11 Ordinxry Lens Squures The Model y zi liaxn ix UM A I 11 number of observanons K k m number of coefficients numb r of explanatory variables 1 0 1n mamx form y X u when y nxlX nxk kxlandu nxl Assumptions 1 Eux o 11 30mm 1 11112ka k IV 7 Fiequenuyio sunphfy vheexposmon we munLethal X15 nonslochasuc Whmh means um assumption 1 becomes Em 0 and assumpuon 11 becomes Eu The Ordinary Least Squares Soluriou y x i We ohoose mmnumze u39u lhesum ofsquaxed enois mi ya Xbrr Xx y39ya 2 x y3 X X t 72X39y 2X39X o 2 XXV X39y Nonnai equations X XquotX39y meexisoffnliiank 6 X39XquotX39y Xxquotx39x nj 17 X39xquotx39u and assummg nonstothau x E 17EX39xquotx39u 17x39xquotx395u 17 Note IEXas stochashcthls expression as Eg x 17EX39xquotx39ulx 17X X X Enlflt xi 7 xi Kail xw 39Elt imdis J EX XquotX39nX39XquotX n 1 a X39Xquot Theorem Gauserarkov Let c be an estimablefnnchon of 9 Let 3 be any solubon of lhe nonnalequauons Then e39 quot lsthe BLU39Ec39 Digression A mm on es mable runoh39ons The Gauserarkov 1heoaem as stamd an terms of estlmable funcuons ol Loosely speakang R s b naoae paecase than lhls Def A human c39 as es mahle lf c39 laes an due space spanned bythe mws ofohe manAx x om If X as ofmnk kthen my c39 laes an the mw space MK and eveay funcbon c39 as estlmable examples Wlll demonsham Example 1 Suppose lhae y 5 g p gm m and x x Clearly thlSls a Eveay mw of x as onhe form a b c c wheae a b and c can be chosen arbmanly Theaefoae lf c39 as an esumable functlon c must be ofthat foam also Suppose that we are lumested an ohe questlon whach ofohese functlons are eshmible k zjy u ag 7 Applyang ohe defanauon we can aeadaly see lhal 8 and2 are estlmable by choosang c to be 1000 and l n A m The sum of R A n 001 Example 11 Suppose y 9 52x gm m and mans case x3 x 1 Then are those wnh c39 ofthe form m1 bcd Example 111 Conmdex a CobbDouglas pmduchon funchon y AL K where K L 2 Wewould estimatedus as lny lnA alnL an ox lny l llnL zan c39 4 1n 13 See the now ntthe and owns handout End ofdlgzessmn Contmmng om above 3 7 x137 Xx39x a 7 y 7 y 7 y7xx39xquotX39y 7 I 7 XX39XquotX39y 7My where M 7 17 XX X quotX39 15 a symmemc xdempotemmatnx a My MIX nJ 7 NIX Mu 7 MnstnoeMx7 0 E01 7 MEu 7 0 quotxy Where XX39xquotx39 15a symmetrmldempotentmamx Let SSE the rssldual sum ofsquaxes n MMII 7 Win since Mls symmemc and 1dsmpotent Then ESSE 7 Eu 1Vu 7 Tra2M511 E mmmu 7 JszceM 7 5101 7 k 1 ssE Is an unbtased estnnaton of 53 39I 1 7 k These nesults followed om two theonems whmh are easy to pmve Theorem Let A be a symmetn39c idempotent matnx then Ex Ax TraceA2 Ex AEx where Z E x7 Exx7 1300 Thzmzm LetAbe an nxn matnx ofrankk lt n then rankI7 A 7 n7 k The Geometry omLs Consldex OLS whece Ihe dacals expxessed m xaw dam om y an nrveclox m n space x k lmeaxly dependem Vecloxs m akrdlmznsloml space 9 Xi XX Xquot x y 1s ayeccox 111 he krdlmensmml subspace spanned by vhe columns ofmamx x Smce x 39 39 ls symmemc and ldmpolml we know Lhax 5r 1s vhe msull of pmjecnng vhe Vecvmy onto vhe space spanned by vhe columns of x u I Xx39X x39 y ls ayecvox m the 11k dlmznslonal subspace onhogonal co the space spamed by Ihe columns ofvhe mamx N my hypovheses ls vhe dlmenslon of vhe vecvcx space comammg vhe enox vecvcx 551 Ihevmalvananonln y 2 y if y39y S 55R 2 f f I i I i x ys i y Ihexegesslonsumof squaxes vhe yananon explamed by vhe xegesslon SSE u39u y y 7 x y vhe unexplamed vanaucmny 551 SSR SSE 7 ssn 39 E If he daxa 1s expxessed as demanons fmm means yecvox each ofwhose elemenns equal co 1 x k 71 hneaxly dependencyecvoxs m akrl dlmenslonal subspace onhogonal vo vhe space spanned by vhe yecvox each of whose elmml equal co 1 y X XX X I x y 1s ayeccox 111 he krl dimensional subspace spanned by vhe columns ofmamx X Smce X XX ls symmemc and ldmpolml we know Lhax 5 1s vhe msull of pmjecnng vhe Vecvmy onto vhe space spanned by vhe columns of X 1 e Xx39Xquotx39y ls ayecvox mvhe neuron nrk dlmznslondl subspace onhogoml co vhe space spanned by vhe columxs ofvhe mamx X Xquot X39U39 XX39XquotX39y X39XG39XY X39IX W quot X b 0 55139 he mlalvananonln y Zy y39y vhesquaxedlengvh oftheyveaox 55R 2 5393 x39y lhexegesslonsumofsquams lhevananonexplmned bythexegesslo SSE in y y YX39y vheunaxplaxnedvananomny 551 SSRSSE Dismnces 1n vhax Vanable x x Z x 7 1 So from vhe above we can see vhax when vhe vanables SSR vhe Veclox The equauon ss39r ssR SSE as just vhe fan111131 Pavhagoxean Theoxem Also vhe R1 SSRSST as just vhe cosme squaxed ofvhe angle whlch whey veaox makes wth whey Veclox Thls gwes from vhe xegessxon Obvlously a good in means vhe coneanon 1s neax 1 ox V1 and vhe R1 as close to l A nm on Es mable Funminus and Mumoohneamy Suppose vhax we have vhe followmg daxa y x2 x3 30 12 24 40 13 26 50 18 36 so 20 40 70 27 54 75 32 4 Cleaxly vhexe 1s mulucohneamy because x3 2x2 The normal eqummns would be X39X X 6 122 244 325 112 1790 55 80 X39y 7270 244 55 80 11160 14540 It as easylo Venfy vhaxvhe normal equanons do not have a umtu soluuon Two posslble soluuons fox 1m 93 axe 1067 214 o and 1067 0 107 Any malon ofvhe foxm c39 whexe c as ofvhe foxma b 2b So vhe conslammmls esumable bunhe coef aems ofx2 and x3 axe not But l9 1 2 Q as esumable The esumaxes we get from vhe two soluuons axe bovh equal 214 Economics 310 Handout VIII Time Series and Distributed Lag Models A Lag Operators A lag upstate generates thelagged Value of a Vanable gt2 IV 11202 LUXJ 11024 1706 rm Pmpemes ofLag Opexatoxs L5 c for a constant c L My 13w W yn yr L L y my m yea 1 M y 7 ya 17 LN Antoxegesswe Processes Def My 15 a functlon oflts own lagged Values and an error term then y 15 called an oflags In geneml an aulexegaesswe process ofoxdex p ARp can be wnuen as y an alyael azyaeztumayw t 7 or y e an my angyapL y u aL a2L2apL y u 1 aL aJ2 apL l39 n a A y an u where1U We have been Imng antoxegxesslve processes fox some 11116 F 7 an axm Li a L r 0 example where a pa V Lag polynomials can lie manipulated algebmcally For example it am v Ir 9L V wwkwpz Note 17 pLy 117 pL which can befoundby polynomial long division Distributed Lag Models and Auturegres ve Models Model 1 Distributed Lags y a ux 599 2x72 lx4 147 Estimation ynoinial Dlslnbuled Lags Pol Geometne Dlslnbuled Lags Model 2 Geometric Distributed Lags larger Specificallylet s assurnethat A aim32 1AA 11 Substituting into the distnliuted lag model we get Ax lagmm 147 y 11 43th it where BtL 1 111 Zsz 23113 note that BtL 1 7 ALquot 17 Amy alr 1L ux 17 1L y e Ay alr 1 ux v where v 17 1L y 110 1 13oz Mei V i 132 ayki V This transformation is know as a Knyck transfurma nn Model 3 Adap ve Expee mtions Suppose am y depends upon the expected level ofa Vanable x y a ux u whae is formedby x 7 47 17 1W 7 x11 17 MM 17 1m xquot ofx at same me xx Ihe fmme Ihe expectatl ms bemg formed mday Altematlvely x mlght be i might bexewmn x 7x 1 e mxm 7 4 Substmmng for the expemd level ofx m the above We get mu 1700 2c 17 an yxa5n 17 AL 17 Amy an A y 1117 1 nue 1x lyH v Wherev 1e g 317 A 521w sh V Mode smok AdjusnnentModel Suppose um yfxs Ihe ophrnal levelofy sumlet Suppose am y a ux Agents are 4 optlrnal level Thls can be modeled l39 7 7 17 My 7 ykm u u ALM 1 1y ll 1 1a 7030 u y an l 13007 m Ay 1 Now lhat models 2 3 and 4 lead 00 a smnlar equahon to be extlrnated Fox example 17 1L l171 1 1 L w Estimation Eaeh ofmodels 2 3 and 4 leadlo the equahon y A 3 133 V Lumodelsz and 3 the enoneuu ls ofthe fenn V 1 7 AL whlleln model 4 V u Ifwe Imewlhe F dl onhuh u V For example senal eonelauoh Howevex usually we have no pnm knowledge ofthe dlslnbnllon ofthe u appmpn le esnmauou pmcedule Then V and V conslslem eshrnaloxs There w l be small sample blas Case 11 If V 15 senally eonelaled then there 15 contempomneous eonelauoh lselweeh V and V and OLS Wlll produce blased aud moonslslem eshm les In lhls case a allemahve lest fox senal eonelanoh should be used Economics 310 Handout VIa Truncated and Censored Regression Model I The Truncated Regression Model A Truncated distribu nns Suppose that x u a random vanable wuh pdf x fgtlt x x gt a 7 f l P06 gt a EOC X gt a gt Ex Vaer gt a lt Varx NP702 ar am gt a m we where a U Vormx gt o 039217 6a 1a M 6a AMI101 11 17 ltDa 0 lt 6a lt 1 1a is called the inverse Mills ratio Truncated Regression Model The truncated egremon model occurs when pail oflhe data u mlsslng For example 15 Some appmpnale But f we are lnlelesled m thls Ielahonshlp for all workers employed 0 not then OLS would gve mlsleadlng esulls y Xg u u N N00 2 ylx whet33 up gt a dig3 o M whoea w Lang 5 and andltD Elm gt a X UM D Estimation The above model can be estimated by OLS or by maximum likelihood Which is more appropriate depends upon What use you intend to use the results for If you are interested in the y gt a subpopulation use the OLS results If you are interested in the Whole population use the MLE estimates E Example g X 100uniform g y 1002X50invnormuniform reg y X Source 1 SS df MS Number of obs 753 r F 1 751 118282 Model 1 266299579 1 266299579 Prob gt F 00000 Residual l 169080410 751 2251 40359 R squared 06116 Adj R squared 06111 Total l 435379989 752 578962752 Root MSE 47449 y l Coef Std Err t Pgtltl 95 Conf Interval X 1 2056637 0597997 3439 0000 1939242 2174031 cons l 9890442 3498202 2827 0 000 92037 1057718 The above gives the results for the whole population Now suppose the data is truncated ie missing for y lt 130 Then the OLS estimation for the subsample would be OLS Estimation reg y X if y gt130 Source 1 SS df MS Number of obs 621 r F 1 619 60787 Model 1 112535339 1 112535339 Prob gt F 00000 Residual l 114596529 619 185131711 R squared 04955 Adj R squared 04946 Total 1 227131869 620 366341724 Root MSE 43027 y l Coef Std Err t Pgtltl 95 Conf Interval X 1 1617359 0655998 2465 0 000 1488534 1746185 cons 1 1335722 4167651 3205 0 000 1253878 1417567 Notice that the coefficient of X is estimated to be 1617 significantly below the true value of 20 MLE estimation truncreg y x 11130 note 132 obs truncated Fitting full model Iteration 0 log likelihood 31748276 Iteration 4 log likelihood 3153 3351 Truncated gss1en 11 133 133 13332 e 325 621 uppe 1 3313 3312317 337 35 Lug 112113333 3153 3351 232 gt 3312 3 3333 y Cuef 23 2 z sgt7zx 351 Cunf 1nv31 q1 x 7 2 133655 1356376 23 23 3 333 1 326633 2 333731 sens 7 32 71333 7 652736 12 12 3 333 77 71337 137 7125 51933 sens 33 32576 1 756333 27 51 3 333 33 33333 51 76313 The truncated xegsessmn can be truncated both mm above andbelow y 1 a y lt b The staid mmand 31 suchacasewonldbeezunuzeg y x 13 1313 The Censored Regression Tobit Model The 39 F31 exmnple salesy r m 1 a 0Tm 0 dependent vanable 1n 3 segsessmn equamn 1s slrml y 11nnted The TnhitMndel y A 1Xz l 13 0 ify lt 0 y 15 unobservable buty 1f 20 Exmnpbs g 15 P1 3332 a 925 53 25q33 3 5633 15 ee2 23 2 rgtttw 351 Can 1v1 P1 3 313233 1 357751 7 363 3 333 13 33333 6 131332 39 7 1123 233 31 77353 35 253 3 333 1356 323 1133 332 g 15 P1 12 15 lt 1333 3332 a 925 16 25q33 3 5376 15 ee2 23 2 rgtttw 351 Can 1v1 P1 23 5 373373 3 366 3 331 35 52625 12 37375 39 7 1723 231 1536 7 331 3 333 1223 212 2215 733 tobit tickets price ul Tobit estimates Number of obs 50 LR chi21 7227 Prob gt chi2 00000 Log likelihood 10056717 Pseudo R2 02643 tickets 1 Coef Std Err t Pgt1t1 95 Conf Interval price 1 3497672 4717039 7415 0000 4445597 2549748 cons 1 2217171 1976837 11216 0000 1819911 2614432 se 1 1039596 1872889 Ancillary parameter Obs summary 16 uncensored observations 34 right censored observations at ticketsgt 1000 reg insure income Source 1 SS df MS Number of obs 200 1 F 1 198 52713 Model 1 166757822 1 166757822 Prob gt F 00000 Residual 1 626368196 198 316347574 R squared 07269 1 Adj R squared 07256 Total 1 229394642 199 115273689 Root MSE 17786 insure 1 Coef Std Err t Pgt1t1 95 Conf Interval income 1 9704973 0422701 22959 0000 8871398 1053855 cons 1 2342143 2490869 9403 0000 2833346 1850939 reg insure income if insure gt 0 Source 1 SS df MS Number of obs 107 1 F 1 105 90259 Model 1 100691287 1 100691287 Prob gt F 00000 Residual 1 117136334 105 111558413 R squared 08958 1 Adj R squared 08948 Total 1 112404921 106 106042378 Root MSE 10562 insure 1 Coef Std Err t Pgt1t1 95 Conf Interval income 1 1940817 0646011 30043 0000 1812725 2068909 cons 1 9509566 4887191 19458 0000 1047861 8540526 tobit insure income ll Tobit estimates Number of obs 200 LR chi21 46822 Prob gt chi2 00000 Log likelihood 41640595 Pseudo R2 03599 insure 1 Coef Std Err t Pgt1t1 95 Conf Interval income 1 20489 0597651 34283 0000 1931046 2166754 cons 1 1042937 441718 23611 0000 1130042 9558318 se 1 1073973 7259328 Ancillary parameter Obs summary 93 left censored observations at insurelt 0 107 uncensored observations

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