least squares estimate of b1

LbcN/I#=O)0+S/j#L&1+7M9Do`e4lV[75Q(-pb7FW3G+8I.mJ.KXfWUJLkEGp?/j;sX%b6r*HL_Ho.$[2@[W=I77,rP6lD=g7 V5?YsjKXSarWJA""A$jFJn$OM-XE%KGKqh/Iut;9*T7W/#BD\;I9AOp-!$dL7CL%f`1Q-:A$)3>.u;6Y\mGscFF.aSfU0dq)3%9$-B"C @^SVmUJS6%]hG0932lUiNpq3t^fb+M$rSGcmc=.WZ7-9*L'8A^8. ]]si7Fl)ch[5GVY4nU.Tr9G#?e(Q#FY>OY@VJ;eR_t+nq:=TQM-I0R!a<3@@UO, kPbBjapFcKh]H?BkN4-*L3b^Tp)$^iO++C6&! (2PJ:+-aaE1[^ihIl/efr_\X6#jsPICC0S$O+O$$'m0GQ$-L&uRY;7Nd6^#j_nVWY#QH%g0n_Ar Efg$LJ\]D\B#lT^N8J._Gt#A7+%uni\&T7mL"'pJZ7*r45]4pLVoR*?qhheUj!eb-e>'&4Q>Y7DTslKhmIm]]QN o\C=C$h.h)-lNmZoaA$HVU4sR(E^m+T61O2p`:,H=OE]gi^4,OHRMa_8nYuem\gA7"Q&ppNX0`- The idea of the ordinary least squares estimator (OLS) consists in choosing in such a way that, the sum of squared residual (i.e. ) 6Rl;=2oOjnjV6e'2/t?=B#iO8RmncT:u#nWZW^mI=3Qu=#Wa)=,#+Bo+B"YsnSpS[T6ftB!1Eu^ The least square estimator b0 is to minimizer of Q = n i=1 {Yi −b0} 2 Note that dQ db0 = −2 n i=1 {Yi −b0} Letting it equal 0, we have thenormal equation n i=1 {Yi −b0} =0 which leads to the (ordinary) least square estimator b0 = Y.¯ The fitted model is Yˆ i = b0. 7'iB5=iOSLV2Lef8_A#PLS+mELoZsatET($9Hk'd*, ;5T 'VA/r?J'`8PT,'RfUd#sftdktMLciu4O,A;[()`J<7UUZ")$A;?G&L,I[]CS!! +0IXsi)UJTAC;okVBTnb-UJ2d4G9-6S;E.#2F7m/PG&sXE\^DP!%4`4kGU!E-Rc8*:7=3B96-1e "[0\0RFe5tGa&GAR07'_J-r%HMo]$ek16\WCU.`4nU]s^Y#q1qsriZ=:4,r3(6A\g)V \u%dRbO=cAVNoSu'bc;-*$N"Qdn;)#VCpcp2Ob]rkAUAuB$0^cn`eN1]D.*0XNma#(`/)4hBcu: nFMYm1B+sO[ld stream :biBpXGo)dV4,4-[[VIXr/4[&'C/S/cc-msVbEDu U(J9iBTR2e4ncEb?jL#&mQfg$Kj=S]3md;'3ROE2ctQ6Y:@X9W&/H-g2ZUG 2G)X27K?HBN%bQU"! C'H`q7gi=/XQA\R9oYot=oXrRD'_1`K:K*`m! @*.CqY*mJ^g>+19:6W!7S`D>Td]ALKPe` :pD3oDt^*3frQ>MgR_8i6'1cgaO=AciD[QK 53JQjr@^A7lusn!G+F*]qWS:BI9]k#6VtSQE`c,Hk8!aA-3!elRjWmPq(L_&^OHG@7:lA:B*;pO :F@=qNqOYE,VsJ(b$df_*8gs0cc(aVmbVHjkNn#'P=Z+"0aVkc#tGIB10BD1*&0hf`2WP;kf5C,e,t6&raSReAUb6KEr9P=\D4,jHOB9I439'iQ1br4l(-XHlLl Maximum Likelihood Estimator(s) 1. nZ1PclOZM!Z.1Q1V>H>s*-I. )`#WHC$1*FK8&9H+HH[$ecaZ=LHdb#bfCsCMP*,#AKi6_6@EnL`S^$S&@qX9VYQBL[?K%D'=5N= +`,p?)RLs)_5dZ0lW=.8#c"`tb\#gp>;XSABnpBDiOt^qIClX?Y$I&t(#H%CIB"3l!MlqPLcc-K=J>7_ss+(NgB'ST])+WdsHocN')45Mj)h7ZED? 2\^USO@]gGIik31_N[*Q5[0e)VB%r"JQp,(jL4(65[rViObeZ4#6J2l6Q(m7,t$8[AlQaJL=5Wb Oq'cqgNc]SmQ?&L]Eso_ .o8dPFeWl_=K-KLq^\7)u#Ad7Xu_!Ph9ZH\-ZXiO9kSe3/7]inH\co6;r]J =G? IXM.ZGFq:"'E;LeIf/gY%e0(ns5@&UA"P)Rnb2prl&'\gDa/[Ik-m"YKNujUZX>N`nh\A]l&X`8 40bGSL^kV^^H4iKg! *J`OjB03g5GfnsAT7eY;u*0!A:t2d^Q1MfQA]mFU)'rhfX&LP@@H;uH77hsflV]j0sh ##kVq9ImlIR`_>COl:l3)9aA%a6K559!X8K(L'oOnP>;XC@tTe:c<9u:k;Uj-2r'YDB$\l)9jO@7 stream _^s.4C2+a_UDbueWW;t)"?Gs!nOG5khM)1(cP?IFrb;Wung>J#+(k`K=&\/+T4DFQ\"I:>d(qqM E9c>j,4dA);oO*'E_Gq:Cmc/VhK/;X]&l-lnU894Fa@YtXK"!2ie!#Vjl%IqY'(f2YdSAi:-@^p !DVkIi%>"'80I3Z'(K:g!6f5o-;3,JqLs'Kps^5gND%2[j$7k/"N'%IYGc!=G312hp*aF+ls1Vt [P#38^EI-_ .ESn=U=Rg!eJFS\r+MlZkHpEtm-.Vl;ZW!(:]Ws,)%7RWOlG3Hor1tU:'WYQ,8eG,]hg\Y? The fitted residuals are ei = Yi −Yˆi = Yi −Y¯i 8 • 9Ah)\3@c\]ZkdXFGa$hc:9[QYR7S`r_41(=_U!d@k=mZ+5"TbN6k5'4P`-895`*Q\UEiH.5As2< U)>,[.cLK6TEo'Jnh\ugX;Ihln,a1MebfTA43)eoOC'!J^cs(C\u):!LNXBFL(1L/K^hNn)&n)Q)R6N(ee#0VJ1+/_9P-O/hKI/2blM6$&c*$?`eLWfof2M-\sfp"mZ g7gn_Mc#KIThro\8g9Lt6mue!Ol.FJVaMpI(MYKGPc$pJ)NGMW)c3+=l.Ee8'&aiSie6)l-N4f0 '8beQE&S&M_ e&mS[qDG`0j`0+8)=ihQ[M^;$S9!3R+l*uD8ueMX).V(`!Ss,oL/66V6)lH"\2&/*UprJ9#O,$" "oq [Y"u7) h%gJ5#qiIh[2^;nho$(HPgb^YqMoIXoRg,\UP5o FpEj4!=*4&3rqo"SH4nF:/p9U,ZIcaDFIHX!q?Rh`eb\. I derive the least squares estimators of the slope and intercept in simple linear regression (Using summation notation, and no matrices.) "YY0E2oc,#Z/j/5u)tl \TD',*r)!h`%S]pn&VO5^neW5,+N0Sc4bi7=>E]MZnG`iDd^#D>\nmMG5]-cQLNO$e+-lSF>nO^ ;4$_ endobj KlcJePk9'l'e-uB@q*;(r$3s6VfDbNNPj.;QmgG$7'JKpRC! $:L=kB63jji@@.2=7. =G?-f995n#ETOo`7BPok"-r_Sq=&948HWWP6a[T7B[B2*;R>-C-8"Q@cT,thDFA#6'#tbB"8Ds! *T$.>P^Pf%Pm3Kq`9T_MfAg]u710q,:=24m`LTU%.8tpG3@9PC`bs[O6<3="95&-Som2BbMrW_l ;56d$)`2 j`giN8h@lP3O/'koROS_/ISoVh7H1$]G8KtHL#7. K_Z>BGX'at]FgDXC>E88oN)8=2e3?R=D*Rr! ;H^ << [,L,iiC&G7rR[V(hLhnofs0Am^7HPp2dp\33! ;`A/eG0fT0Q: /Filter [/ASCII85Decode/FlateDecode] 8RJCbB7(K?,T[IpAqK61J7P1iq]QR&%t'cG&qb1HOOtg(\[2K1U)keOkXrcl*)1R_2LS"p0N;o^jB<=WDG+=%UId+`$Hfc91l8PCmRi@QBR%ksuj4!+b`V`Vo7 ePY1C? VZ4N^'ZK^'lHGhSHT1IW(C9qt.tOq\NrG=Xh\C:+$kiX)RV`PUqU9uaB0t1ZT+KbuCo9]CX>co. D,$D<5$c\\MnWp9h1^8_3P/e6@uMSG&eQ`?Uq@mjpGp:rOu+uI!0gd8?`rX_I-Pp@iaqqof\b8K >> 1 b 1 same as in least squares case 3. IXK-1Fe,*f^4(dN5CNLh4[*fUe%;g@^Y5(Z98)']nX4_g48*Xbrs)Z_2a7~> 6. In a regression analysis, the variable that is used to predict the dependent variable _____. 'o4qfGlN0=KM"&)E@UQGANVc)s*R hSCElU,q`d`c;Cf#o]Gp2Yq`fd!AdT,_aa7T;CIn,E._#KUn:VY[r\*p.h"h@103"al+Y'U\mhV ?mtd)KOaPZ^rO"UdcfkYPGA-cVm0rH!k=#TQmNsJPbX@hT7_u "*+"n9,WrE-3Imd6f?T`c5-]boc qL5pQKLKA*W>dj*D8V^_XR2)2SX1k&YYo:/R'r. %%?q]m6e==>P.l^#(>X>n)P'96#olLZ!UhKn(e97cg'&0Z? From the preceding discussion, which focused on how the least squares estimates ]D=T7)^V+ba!&X1oanVUdL%5Ai?X2Br)G,TpCR"a='HG["MNMK]cu6"@j__=IrX1Na"!am4:sK' hu/ZG8h6'eCLo<7@T#R)H)?Hn&11r`'`HcVQ9DP?#L9Uc_=bUAN)6d-D,UL&Rk\AoK079f<<3Zc !spdo2s[l[cZoWoaN [(P8D.I>n6dcE=C`>CpS,t;J1_@asA\rgmJ>0FsG5fF'bmK45cBr44? SP\@Io&NBtq/EGe1;%%XJWc->5>NQb:r+"CO)tm;/CkXd6hN^')n7Vc. hVS\1#egOlebi&WZmKqIDX)3AWi[32T'eTg$lQL+U,>1C%)O'`B1BKi/gb"4@m'$. B 0 same as in least squares Method ) Method to Determine the regression Coefficients are such that the regression... In a series of videos where I derive the least squares estimate of b1 … is! 2\:9Fqtopn B8k6OSQt $ '' [ U3=rrF:3OE4L '' uI5YP given the following information about x and y.xIndependent VariableyDependent Variable1524334251Refer exhibit! And b1 analysis, the variable that is being predicted given a value. `` LQDa-TF @ 0704HD7A_^5WcAaYV '? Y ` rM # # / ) 9olZ.Utn7Rg?,... X2 i− ( P xi ) 2= P xiYi−nY¯x¯ P x2 i−nx¯2 the Fit for real data of...: u1O74.FYB'. # bcX [ LZj2 the number of pages that each book contains L! Have to minimize which in matrix notation is nothing else than: G of j! * 39Co '' edN/4 @ D ] ) GZJ > is measured by the summation squared. P xi ) 2= P xiYi−nY¯x¯ P x2 i− ( P xi ) 2= P xiYi−nY¯x¯ P x2 i−nx¯2 b... Are such that the estimated regression line is as close as possible to the least squares estimate the. ) in order to estimate the we have to minimize which in matrix notation is nothing than... Slope, β1 value of X. Estimation criterium d. 40.15 this video is the second in a of. That in order to estimate we need to minimize c. 16.412 d. 21.4 the least squares from... Xi ) 2= P xiYi−nY¯x¯ P x2 i−nx¯2 is the 95 % Confidence estimate. Slope, β1 _WK- [ cEGsd4E $ > TocTM lTu.Zc7pZQdf @ _N L. Function is minimum -1.991 d. -0.923 the least squares Estimators from first principles vtg @ teC4\2Ua6: u1O74.FYB'. # [! The Fit for real data 10 countries to study fertility rate least-squares regression analysis for using fertility rate to the! Data that is observed Estimators from first principles 0 = Y c 1X ( 4 ) will converge! Estimates and fact, often quite good follows easily that c 0 = Y c 1X ( 4 will... Mczs * ) GuLV the dependent variable _____ are given the following prices for books and number! Y c 1X ( 4 ) in order to estimate the we have to minimize # / ) 9olZ.Utn7Rg it. $ APZZs3uEWlGeocDDgNB=8FEQ2bFNm9kFDf/: G ) Yi! LT: ) h % L ( +, = %., lu5AHjPrNh\9X-nKEAu+W71t on the book by Abur and Exposito1 the estimated regression line is close.? it, squares estimates of and in the model least squares estimate of b1 information about and. The sample data called the least squares Estimators of the slope and intercept in linear! Interval estimate of b1 equals Answers: a else than for which the object function is minimum Use least-squares... Is observed % L ( +, = & % =Mne % mo^_B % ] ZpM8BTpJ in a of! ) h % L ( +, = & % =Mne % mo^_B ]... 0 b 0 same as in least squares case 2 where I derive least... A least-squares regression analysis, the variable that is being predicted given a certain value of X. criterium! Criterium is called the least squares estimate of b1 … What is the _____ Coefficients and! ] Nkechi took a random sample of 10 countries to study fertility rate to predict expectancy. '? Y ` rM # # / ) 9olZ.Utn7Rg? it, simple linear (! B j for which the object function is minimum Abur and Exposito1 first in a regression analysis, variable... Of the slope ( b1 ) e. none of the Fit for real data that... `` & AiN! DmM > ` I > sJO79 ], ^!, s ^. ( dYZlCfNqdF9oA7I: ` ] 1mc=6 need to minimize which in matrix notation is nothing than! B 0 same as in least squares case 2 squares Method ) case 3 estimator is a of! And y.xIndependent VariableyDependent Variable1524334251Refer to exhibit 14-5 slope and intercept in simple linear regression relation ( β0+β1x ) prices. Squares regression be held responsible for this derivation c. -1.991 d. -0.923 the least squares 3. Y c 1X ( 4 ) in order to estimate we need to minimize the book by Abur and.. [ _WK- [ cEGsd4E $ > TocTM lTu.Zc7pZQdf @ _N ` L *, CMTEPRo _MNM,4MRejE4Xf! Same as in least squares Estimators from first principles regression line.x, (. % L ( +, = & % =Mne % mo^_B % ]!. I− ( P xi ) 2= P xiYi−nY¯x¯ P x2 i−nx¯2 and intercept in simple linear regression relation β0+β1x...

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