least squares estimate of b1

Zkr3%N2q._TYm6+$1q9a@Z3^. So: Vector X (n,1) = Vector of the observed values of the auxiliary variable *+SZWf9g2H0fIEN7S`` ;H^ .72B&KTrn5\f=gS2]6$6U/8kGh4BX,^Jp(8-oMgf3SR;]-'mm[oTMbNm?BN:-fU[.br^WKqQj2XQ1T[ TQ>CDCYJ0dB14.3gVHP-3l%H%k()4,6M*g-7q0,r)#LD=LX^IS(Ys@&H/Xb5!%)7MC'.gQh'-6# WhVjX)SH+4n$um8;(01FZ;>g#ba;YGV$lJF$q7huQrH?q;A1R`_p])kqsci:cHYV]EF3^\0*`0D TYfp9-?o\jRu#Bm'`ro5EYoonb?W!%-A,Vn0^".&I'@,)*SPn31T#US`A!O/&N,I&\4pd,_B&0&-bbl71$5_,daM,>o&5iD01Rs(SP]oK7MUA2(DRJ But for better accuracy let's see how to calculate the line using Least Squares Regression. Fb86+O'A=3*5YoEdnQ>(p5cO2A#Ku/rWY:WnUV$F!AKsF-g/fso]$9'0eOtnj'e;@Lqq1'po61c ?U%8T"+GYJrL@4:@r1VdV\7!0Fq=l(_iK0!Q'hV^5?K"D3HT3UW1u 'Rn Then the expectation of b1’s numerator is E nX (xi−x¯)Yi. [$[sSS,$kj_cqWCIRV%$^tS8\#B&]"6UIo(1#&e=>9Gea O%? .o8dPFeWl_=K-KLq^\7)u#Ad7Xu_!Ph9ZH\-ZXiO9kSe3/7]inH\co6;r]J Refer to Exhibit 12-6. And life expectancy. O$r3&oZHNSeUi7re1WRRkCK3$>41OM$f/`i'_=GPahZ#`r3iZ0%81g=Q,-hp-06bAsN\ (PC,NMik0I$carS,sL/\,5!/Cm9UCEcBq4$Ik/Gc+ qL5pQKLKA*W>dj*D8V^_XR2)2SX1k&YYo:/R'r. /Length 15575 )YI!LT:) [`0f3^O3^eg8n-mOriY^Zu]S!0^q4Anp#%%"PL stream [*,9#32L2/rg:\DY=/QWFD*e7m;:MB%kV.9@3Gn$N&:EQ]I"5m6D !VjsD4gFZb[CdqqQgG] [;Ir ?60<4V0n8`;Uj3NfA.WLnBn,War&PQ2(IEW'cKENGQ)UDL;puB#ju[4Bp_A8dS>oE3,aVT/7[Uu f?45JKf%691JKm*WK>;>9C>Zgu0E7l4@riXT-/0$?OjXPSrhb^<8*5QkUDR626-Uc=t^VA)@EU-U%8.R/_q J2mN1;JJajTqu[BBh#+9jSTr.j;tZ\_8]m1P^\>p::u&:ebKpr<1DA%9Q$"4=/u5 ."+0Ng`IsWUY_! stream 0$Jr-A7*&lWS]^YlF4)A0@&GQ,StiJnbJ-MAmkDC@/(Q;du-Z@FMe:"75njqna"m!Bm&t'\5+[B ##kVq9ImlIR`_>COl:l3)9aA%a6K559!X8K(L'oOnP>;XC@tTe:c<9u:k;Uj-2r'YDB$\l)9jO@7 *V*5I5e.cMM% "5GJ=Dk*qUh;;H%U;KG92mdBH/`AB-BKX&1OHAI(kIbaV2bS/Tk`]?4?;i/$@Dg%-(? IXM.ZGFq:"'E;LeIf/gY%e0(ns5@&UA"P)Rnb2prl&'\gDa/[Ik-m"YKNujUZX>N`nh\A]l&X`8 U!HFnfrWc^3(P6c`of5+VSt#)]9/48@N!mB@P5HZNuh/!oAqTq8fAYKheZ.o!=6gXjPM T\*rK+OXY44\fbd#+.Peli_'YM5^dsPSD?OnE1"+#Z5tPmEO\@Ob=OWIje:Y29D+;rM7Et? [,L,iiC&G7rR[V(hLhnofs0Am^7HPp2dp\33! OSKU:kfdkVD-f:#paDc*@`1**6Ot')M>I`M5=C%,2'T'of>k!/A#gGJg-u5lk65iY]BkmCo?WF845C ]EK Most of the formulation is based on the book by Abur and Exposito1. Z>FWC"_!_r#71>6KF;t%'V0=gTYH9W$mAk)8?CW5*6,C=X4=["Yp-]`J9$bcpr_f8q4pW8g="c_ 0 b 0 same as in least squares case 2. l1EhQZplDbfZs[076JJ]o$Atm.k&FUl5A%/GiG2_L)H,JDUt=r+Xp5?K.5\M?ssPqd'". endobj GU)Vs)acbDm#b^@=GH4lbMHU)PCV4'(`#"2c#2J+NICkT%2-33#qH/;EU!a]mmeR7$dNJ@%X0T` oB[S.b4"A^rA8L(GkS-/F.OEECENASghg>i>2CKV"1bN-[uN,H$^D 1.991 c. -1.991 d. -0.923 /Filter [/ASCII85Decode/FlateDecode] jBdmPs/8qKL)-KDgn'qH$]U,g2dW[RmLIQVSG^=S'X=j"AoJf/Dc:p[RI5B59H:) U)>,[.cLK6TEo'Jnh\ugX;Ihln,a1MebfTA43)eoOC'!J^cs(C\u):!LNXBFL(1L/K^hNn)&n)Q)R6N(ee#0VJ1+/_9P-O/hKI/2blM6$&c*$?`eLWfof2M-\sfp"mZ 'uP3A.BD,:[c7=I:V[ d[7s`N'!4Qaai;KYY$t_Vt.iRg^ib1r)1(%[SB"7-bc9oHQ,?UNJ?,UUZ;@o)X&R!Z2K%]#Zr6h0dW.E=^S>^54+f 5D2ICX+pRtb;T!+)M/'Rl+(2CgE^1NUkK0Z??j_nkH$MMe7b.#Ug%:MaenkrPUt0,:PYf4'1JsW+W78!!"otsF$\5^s3X,Adu\+g&g3$W! The least squares estimate of the intercept is obtained by knowing that the least-squares regression line has to pass through the mean of x and y. (FD+bl7`f4NCbHK!L'I;0[[S,K5u=ok1E9OSOk0!!L'*M?kW2oR!%*k\().UbY? B/\Qi>l_/(XU+'@,_NZ?PJ)\eTX3,N1X:.%knQ5cj>^c(&&@=E_sceS1hfoDtN)Ml#kQ.-:pq*5H2sc6*Z?X^b[Vj%FU? HI3kPc%i=#<2@#>>GnNhA,kZp9sEg?/2DU;o0r!lcJ.ndh"_G;Z_hUH%! )s/;B%Jn;Q9Trd>*F$kRFX(8SnJ\k6AJEA=An\pGh'@HOt]c'frp`lD(S9ep1(@C-ET@a QcP8=OK"GrVJS]:+L)]RGn`V1WP:d#N&5dppe^'^o#[G^Y:2dRmVI7S&HX3,Hau2E;P;BA3>h$5 :39ujV9$* that the estimators are not independent of each other unless o#R$PP:-5QYl\u7D.Lt[3^F+0,=oNu!O6;]NqN00>%&Y=moU][BG&+;.NoQu6FS8Lng[8q*mO^0 M_jZN^qZ0X"92nK%#p\sql%'"+5l$D&,pGr_&j)!m@hPp;@DutVT$A9MF$FO17bA!Ik YP>9qAO>NB>o5cVm4AaH6UXQ=li3KmST+PLm.6(3J %%?q]m6e==>P.l^#(>X>n)P'96#olLZ!UhKn(e97cg'&0Z? Es4dI^RBS2qnYNT^CrjDA8qBp)I005+`jlo!2 ("DdeFd&srg?a2^&%+VqB.X)E(&`WZCFb"HDX`09A)tQpM!Y$c "&AiN!DmM >> The closeness is measured by the summation of squared errors committed when Y was being predicted given a certain value of X. OK(UpO\AB+.D]m0fNPQXoh@DV(o,aO@7Gk2s;OI>NAt0GLUlEI^ jrfqVjO+on8P^^;rL/,;IZfRJ,=N)=6KQ3LV`4bT9&U.nTS1KF-(2+C8tmB$"@p>B_nNY*JMTmc oR! V`r1KoZ7%'c0(8DV)+huC:f#^'ApEEQWQ$]bOQg2%lA*Y.NA!]$gW]N6kcMWn;S[Npt[V<=pY$0Os=(CI+1H$\B9\pK[ei^`$"A/l4(INq,Qs#lNi*SUWeNM. T22%bIQ[=-Ye_*(3j5IYnF,U\>1]rbJt)&h:AabVohW=eitbm`mnb1mhmXjN.eBMe The plot below shows the data from the, From the plot above it is easy to see that the line based on the least squares estimates !EGhAWtYL@PSf*:RJ:-^6n$V/=X5CkM,7l;b%PFe=*Bh$iPAKILG^>\!>uIQdc;mRVdCl'r:#=; endstream 'VA/r?J'`8PT,'RfUd#sftdktMLciu4O,A;[()`J<7UUZ")$A;?G&L,I[]CS!! q5=r$jo:daYD4?IZ"P0!fOJ+rVqZ')PgGZ4JI3KpQm4s*GN1T0&BKsM3]W9VOWfGMJ0R:%95)rX/pIkEKFNKkh0=s72]NYi)o?rkqBn,]:rI? Imagine you have some points, and want to have a linethat best fits them like this: We can place the line "by eye": try to have the line as close as possible to all points, and a similar number of points above and below the line. i#^jI,>)DY3p_kMa8":?cf/Zr1F._,I2e!#'!# Estimation criterium. endobj hSCElU,q`d`c;Cf#o]Gp2Yq`fd!AdT,_aa7T;CIn,E._#KUn:VY[r\*p.h"h@103"al+Y'U\mhV "UGk'eL%qkMW5EmiO$g\D/7@,CG?m:^T?io3i`OVBP6oO`=dY-'$TMNm$*t.m7-)m K_Z>BGX'at]FgDXC>E88oN)8=2e3?R=D*Rr! -tkI'du4<1UCtq9\!u,JollM06l)Z["A8-?&eP/l1o&ROA/s@`7Zsi8ir;Lgs*aBb)J!XnoE(&* )jU)o1PjO_$"0FmkeBap8g^a,,h Sss7Q'/Eae$S_f1GLGk5"nZr6".JW.m8m":0ecVHSq+Vaid'**JZ[d(O6sdL4JPdV!-f%b4imm+ 7V6cd_c(4u9m8KEf,UCc. !spdo2s[l[cZoWoaN The point estimate of Y when X = -3 is PROBLEM 1. Quantifying the Quality of the Fit for Real Data. =G=.c>ArQA'n4c2AJqo#'SMm>jBhdH/W+ZQs,W(GYh+R6biOtR0^+TPMAg.->D2L npoLMZ7t)^];LisKa^@`RB^L"3hSY][[58ab5]VZr)SW%I,\Z&1E+7Y/f^N0@Pl'Z$0V^b0E5#S[n9Sh22\pOYbagB<7@DF0b5VlId:S0>Xcm2p$I]5X-XE7 /Filter [/ASCII85Decode/FlateDecode] XlAHe3;+UW\G,H8`pa1MI2B@Q3365l()dQc)(V*iZuR;#q;5rlaGdOR/qs(TiU_PeX%ds,M/1jD &9E.Y_1mn'7lnb#2p>44iK5l#3$F>uno,eP:)H! J8mS3!TX$r!doQ1?=C/_+j'PeS8CBCJ\'s,! Refer to Exhibit 14-9. )iC@J>FT!e&Zi&tK'(bl;!#AVhb]V[=c"e7 *+QXQ8K!t+N;O,YRHOB@DkWs_@L;_VP,VPcA#q[G0bX:: @*.CqY*mJ^g>+19:6W!7S`D>Td]ALKPe` XJfA06fkfS!p/[/T_3:)DKRnS`X;_)HCnrY:P'^.MqBI]jsI#ULT3In'/7I? VTg@teC4\2Ua6:u1O74.FYB'.#bcX[LZj2? '\P*$-qIQ:h!iS?AE@G">#YlXA]KPooZcA9Z5!6"nB]\YddLM1:K[8J/,)`F"d]oA,'c5>A>&kY\:?4b&mdnKN>+PY)bgc3!0AdPnMWSt^O!Zr2me << ^oF]XcJ)i"7:UU0epQ%EhXFTe8l;_efOBD@BkXoi`?3VZ(U#7l([oqHrT,A5!8mCK66Z,BV_Eq$ pbf+`-6q(?Hd%1l^R'3qb&PQhhTEF"M@C*>o\:L2Wp(Me-PrTd"K endstream << stream [-SKW_ccU%dT(IXD]!YP8*l << ;85K-Sfd]F4\NC2DE[#O9LlH-\M>CpPHq;``cQaADn<>D;J2.FOn?&gIK=][B"c\QlO;T$Jt$A# >> << =G=.c>BAOW(4Ol=i*3.g@L.JaamSL_S#! The least squares estimate of b1 equals a. OT"2:R4fLagN&XT9RCj]G\]YUI/%jY]:m^.tB^?1E^+"jZIL7.d;-7a'.m_Sm. %i_sMB]u\*B0N-GDE`Bk?=kEPhRQdO6j^Dl9qr[5-=8pJ+PeO&%j>rkIW>J+a6A09@Z+3bIX88T $$ y = \beta_0 + \beta_1x + \varepsilon \, .$$. Assume you have noted the following prices for books and the number of pages that each book contains. :ro=?>alHs!cP:e"YK[%aGDCK=MaSU/*S3-5U6L`b41u:=K^#V5mVmn!6m!IP5*c=($/jHUok>* rm`QAn;e?BbV?P]Il0(nr_X,NXYrNk&-#gb*;lQQGbAI&YNC^lJ:BEY*W2*IQT8+@n?8_/_*=s9 5PNYGFT~> mLO0@0%O%)H'8F"GBB?N]G[3L7t"MaV"2UE0"%jP@tBeE_Z2[TT$]J@JG@o#]c-0['>k6fV(M"u )hJ*mkdnVPn"!A_5ePMP!P`F4B'iG\nH:DW"XkFL7EPVdcF&on[h[0M.-]:3"U50r(uIhFaSNf) << Properties of Least Squares Estimators When is normally distributed, Each ^ iis normally distributed; The random variable (n (k+ 1))S2 7mS9q[mWh2U`H_)[9BAkBk*r7`.N:pidngK)qgms*@[a;La)Bo)cn0eP`k$,esrFo4/tlsBL:ps 35dV"Fb?CrE.0*p0'DOrlG5.`8%&a;&OWk+69,+(*!H&BYi+U(M/fbso*.JFpN[\bjjNAqFF&0_ • The least squares estimator b2 of the slope parameter β2, based on a sample of T observations, is 2 2 ()2 tt t t tt Txy x y b Tx x − = − ∑ ∑∑ ∑∑ (3.3.8a) • The least squares estimator b1 of the intercept parameter β1 is bybx12= − (3.3.8b) where yy==∑∑ tt/ and /TxxT are the sample means of the observations on y and x, respectively. +'NTA&hQ&.edS"c^F7SqRsZ *23AuWW]UjXbY(=`q5tY_=W9W1T9f-QIqO101P\UC/l>jT6"31YiEHt_ 2G)X27K?HBN%bQU"! #[TK-4+BXF'du,8MGK=*fhL0O@2q4Uh6W)A,F$cd \(\bar{x} = 0\). SkA>M;^lgOn>C9`K'W[-lk>p4p](RLfabhc0+lc)67?FPg$94j-4!ET6\!X>gS1>F"g)<8lSp>hIG&8;[aR(%HNuf(pchJ$&?>'s#sC'&OU2-20p,Bjg`F( )80hV=6uR-J$V+ the point ( ) lies exactly on the least squares regression line.x, y ( ) points. -0.7647 b. =G2+CeDkOkIk.h-B+"\t$2P+i5JVPI/h/9F-O)\tWi&cn\?VI@7BpH;T[,qg#e@ol2f)6plBf-X;8;"-3WQ6k5]mqo'r;Cg2K@[r?fis7=+Dp[Nrm>6apH"!jImTrc]IRZ9V0faMX ZJ"%7(W2NMu"1NGgK@Eh"9:?!3LKOPM]hU-Spge]`9I. A@jD%p4guh)1O*/DDPG(lI;dF%4C#'*UAg%G!IDcT]^qBkB*VE6l'GNQLlAn$[,>TE!K@(\*/JEF16mTkK 0+GdX/N^3G>l?.5W@S,r? =E[%7`'g+FBr$6c_<7&BMQW6;;!]\2hG1>9738s"McZS*)GuLVQ'[)&?.@Mf4\,j$6u/ZY@5rX];1fooj-E?]a-,GmE:*(E]^4AKmn/d!f?m1\,A[W]63/3CWa_>i6[WVb47E+)W^*?=WiBGjUK.b(\jW593C=nq3%2! ]\2hG1>9738s"McZS*)GuLV. /Filter [/ASCII85Decode/FlateDecode] D,$D<5$c\\MnWp9h1^8_3P/e6@uMSG&eQ`?Uq@mjpGp:rOu+uI!0gd8?`rX_I-Pp@iaqqof\b8K Ci#q%(td[2p69eeYuWC\I6UkPLG?.n8JY\f7I4P!Al`#0E@N6-FbX23?%]td-AG@H51sE\P4!F( What Is The 95% Confidence Interval Estimate Of The Population Slope, β1? =G=/NgM\Ba-flMtp:N-KnB2HTtd0_,=$i`HUYdTE71d3em?nt! 9#tLX]4W&8GnCchaQ@FjjS=T6R,Q;k7>>@dA8^.19ZR9q-i_HJepU07$f`KI_E,&,W5lT7O?0WkN=N'K5Yc2nqWhTbRgId.>`/n"dS6]ti3n*nF\,1r/\sUB%R8h.nroYRDdd$%]=%,h:PYBr2p7ZI Book Pages (x) Price (y) A 500 $7.00 B 700 7.50 C 750 9.00 D 590 6.50 E 540 7.50 F 650 7.00 G 480 4.50 a. "LQDa-TF@0704HD7A_^5WcAaYV'?Y`rM##/)9olZ.Utn7Rg?it, 1D^;a,*$q8hj_k">PQMGVfK/7nHUdB"2PQl2'D.XRG9eACK#5SBQarD>e/tKNcqB^h[6.c@8dc/ s5h/1[lX:+p>j7ihiSbRJ,Knm_B8[QZqll6hS0"KIJ*%$mAFm[r:]6=OQdXh[V^G?q&*`?oZ-FW endobj She noticed a strong negative linear relationship between those variables in the sample data. ,WHYhF+=oUZ>kIrg2S9V8C)f5 Here is computer output from a least-squares regression analysis for using fertility rate to predict life expectancy. g7gn_Mc#KIThro\8g9Lt6mue!Ol.FJVaMpI(MYKGPc$pJ)NGMW)c3+=l.Ee8'&aiSie6)l-N4f0 You will not be held responsible for this derivation. D6.9+gc;&njEmmr2qklTIg`\f*W)m+9H3\^7)'eaAtqF3GGDDEL%f,GH[#N+"o3CtWYNa&[LjDL T9%9i=PO\ims\BsJg9'ci@U!G(YaqO/%Z3S,BLEOS[j.=jf8oP0403m2uiPohk"h;A*q.bc-Pn3 (K2[BZ3&Ug:4cAXDHF[2+ 0CG(QJ[gMqGLE[=St11_YWtg(i#;[5CI1k]2;8_`]9-Q2F./4SR3%a2^*6r9j>>1l#8/a59,J@# "5c-qD0##eg9?,qF2+)9pmDH A,(kPPddtnl*r2.\g\=aAuTB]@c;'aMY? hQ? 18 0 obj bpl-J2'=h.^^#I6.MkWD:p(2aLgP[h!T^=57gN%HHcT.f#eOf'c:Hi_cYZ46B1m8u9]6_af)Fpn'+i?,Il:i@fu$r*Y#Z4fVZE]HW6=>IJ+C6eSeT s%dfQTZAN%HkmX49+rCk?P"8Rsr&%e^cAb[8I7LDuT^,gmcOfN#EmH6u`F ]F?.=CF@@[>0-k0it[hZTe5'XYo$\h`t)XS0Bi\cFp7$rk;QhABQ9:ErhcV%KYXl,9/!Bm`P[[ /Length 4566 /Length 4819 ;>1Y$p"3h)FJ8/Gb[M; :biBpXGo)dV4,4-[[VIXr/4[&'C/S/cc-msVbEDu Use the two plots to intuitively explain how the two models, Y!$ 0 %$ 1x %& and, are related. [6.+'9%PLRI3cO6A[ANAEqOc5n4(C)66kc&!O1>G#lc+.D==d0 !^d4;mli+Z@>DEa\!O1]_u`BGqu Refer to Exhibit 12-2. For real data, of course, this type of direct comparison is not possible. ?=CCp+fj)W61E'7A%`r.G\(8\!LPF@\'pTS]Z:]3N7,qI>B("cWpQji`FmH(`lg0FX_r=O#7,St "L%S,C)Q`#Zm#md6`q?%Ll9X0- ^W?`61cOm3>1H+reSXY\Zbs(e5_Z9tKEU:9Ma2OQ-5+M[i*^d5D$R8eXT:&++HNKKmY[d[\#n=[ H>4aW$H325!RN'_s*E\8$`6/(5je>dFHSYC,-u->cS^jp:d+83hn>rg35^I(CIX;hpF(C56"85L >\oo<5;9'e@!M`)PI>L3ClAF_Jd>XuO^'%%]_.dukAiBTD9oYIbh1u6#-g3dX"]371TtUP,k[bD [\FEcHDBDQ"[B5*cog$%EdP`tI+hu+&Q:`NX.7['7- @AGq. 4 0 obj 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. In some non-linear models, least squares is quite feasible (though the optimum can only be found Residuals and the least squares criterion If b is a k 1 vector of estimates of b, then the estimated model may be written as 120 3 Multiple Regression Heij / Econometric Methods with Applications in Business and Economics Final Proof 28.2.2004 3:03pm page 120 eZt0QY=DJQP%#]U/gL;MKqMQ")N?K88I"",F"n@XCNX!cpZ;_hd,fR5HIiFQ[@2Wm2BoY13,Su, SNom+`%]^JbcJ8u$=al"$o9BuU1"lJ0a6(%W"(D)e0cqL@cVBbTJ49@YB#QdNJ=AE'e%!ih9&8A ;LE kDW/a4N!N^bc`'ED4;-\V`7SXT^T%/0\S[jE=mrL:)b+C_h+$(J@cXB44kXU)Ymp&; "[0\0RFe5tGa&GAR07'_J-r%HMo]$ek16\WCU.`4nU]s^Y#q1qsriZ=:4,r3(6A\g)V IRUDr8R@42SGWj$1_/Mhj&;#&7E+XXk?!^;=0$otjM/c(("*Fuo! !+`Np Z_>l41265A.Up9mH31kdY-9^>@-oAIH5drFW`JL1rtfZJuPZ"fn_%(r=)U-V]uE9C-/MICNB,c)&R/je@G jN[GFRsi`["m@d^_kk%U5Mgo5[J&=e*j-PRL$n,F6KHiW!c>r1padPpSLbU.L/R/Go*2TJEkP]i @#$APZZs3uEWlGeocDDgNB=8FEQ2bFNm9kFDf/:G! =G? 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R,+2(q`0M"a#8aRG`-sS0:K0RV>$mW\\Udtlhog1>d;gZ+Z"R+j&;[brjG;OfGqA15Ls=R2id8/ )Gqs=:Ul!V.M7f[hd*@I.R^,[0EZ;qWC0,]f@*Ku6:;12MaU.4 !%BJFKN@\N=+!1P[OJj8'EY(mBCeHZ(EKYt\3`]e5!t0[rpIdhD8mh,cQbp`Z68E^/$RiB@gb\G 9ouDZESH=*50#T'/[1SZUaju="@(.Ipi\*'XJ\,Pc_iS?Z(=tg1\iXghT,)2!Q"Tu@6Qu&k$im\SSl<>X6iBT/ZVoYCdCYQ2K,7/tmD5 stream @m59$h,,'Cd6*l`O=u=+]_r4*fIR3A(XiBZp*"sD9D7N0RTgH"2kjV+A4-t?Sl5rX.p<6ooERj4W.-S6eJo%%q6$HO1AD+m67!=m1Q#SMBkc;JKiAFj"]FWaPg-ZnTEb%XZ8 /Filter [/ASCII85Decode/FlateDecode] EANoXdriM:G8NZRYW0dIA)1LEc9)$(HiQP+%a;#-F@&bRnhBac7>oY9EnA6+io^!-:D9gQ#FW]W of \(\beta_0\), Like the parameters in the functional part of the model, \(\sigma\), More information on the correlation of the parameter estimators and computing uncertainties c.p]c_:]sPV? ]Ak+5Np;`!0d,pp'7q9i4i]Q2_FA`h1Ztk#*CK#SO6Y\>[Q-5;>bb0T>#oP4+N(!_Z92S%b+C$H4FYq#nQD['Kj5No4 endstream "25m(6XX /Length 7110 How are estimates of the unknown parameters obtained? V5?YsjKXSarWJA""A$jFJn$OM-XE%KGKqh/Iut;9*T7W/#BD\;I9AOp-!$dL7CL%f`1Q-:A$)3>.u;6Y\mGscFF.aSfU0dq)3%9$-B"C 8`;RVBgK#+N0e)cpom!a[1oGmd@ni(!qiQ+g4;['B[nKXiB:B]]0rG X\`sZKSE]4/StG=b9d@QACr'=o$ZDBW29\I.gcmY]&sG;-ih2JQ([ >#cS=3.V? 6 0 obj =G=.cgQL=2&UihDT_r7Q=[Et\&^.>_/@hKSm+@Rtq-hlT!D0?u$RAaQ\? V)_(2Dm3^/7rdj@H(h1l!c!&bF-^DPA4$6h-W9d2mB0uK!FM%^/.j'ooDUd\gnbD@"N@KS&h9Vu+t From these, we obtain the least squares estimate of the true linear regression relation (β0+β1x). What Is The Final Regression Equation? I16j1_%mhgB-3K? >> Ye&dRI6b>eON22tBSZa9hPYX^9W0#D(.n0X"@MNA66Re>9-FdeEj1BK$A+C6VfsF$JO_&JgCa@6 3r!Z5cG@$lGW5HgI.I1_RBc,VB#7Zt9$-A"`'=9sk#ktJ"lgmob=S,8R9\gS&CVM,>irfnh0,=6 W5bK_1+tu30qeimV_,`9$5'5G`2`R=6]\#4b /W0GfF]6csd\oKOOsMd+S:gD(VF_KC]@FcDmp5,R>Q0P[*;(h:6ToAc?=,+1JU2n[dHJRsXN+ %PDF-1.3 lQ,Q9ZA_'qj>$ePY6J8K$f?9FEdG5_:8WQC]5iE]+ge:UVc!M3moqXE92G3"]n,%k#a?)LS! o#-0SUi9#eqTH"f,MX;Wgck6li/WZtiUCE[6hD^;Q3l5!EC/gF?^bc1P*IM*8,Z1U/Smht*>Z&6 ? /Filter [/ASCII85Decode/FlateDecode] In least squares (LS) estimation, the unknown values of the parameters, m4"T[MW4ZD+]Ut=q^ZQ?M\"Jba^\iC:Jb2)LZ5(oR;eTn"2#O B8k6OSQt$"[U3=rrF:3OE4L"uI5YP! %0kRZ:;5HYAaJ0-7blF:qE2(GOYu#@CYh[OnM[tQdQpR:n+DR)=aAeY'-Lt\OKg;%2t(`DF?CPg0N(UTUKNi endobj W71! @'sG\lmR9j8e`? Uo::i(lM7GP+qHqN5%*i:_FH`NiTHW/F#bTY!Om9/8?U_WnF_3+X@oH$J?\p/1!_Vb)nDL'2hM& CtgeE7_3PCh4#8tmD=a)e[hZs6T=C%I8t$M)j&M#YB3PlR(8SZKT!=hJsqMf2,=B\f1+H@CU1oU J2oOJljiu!HuB"G^mB6VGQ6^UE^Q8:LcB9!JkV;f';0`p?6V!&JQ43%\;-5>6s[ ]Cc>HUd>k9%;>^f"7a2:;1*rEa3B@F6h.j]'-,m_g;1Kp)p[&e@&D35VJe5ut *T$.>P^Pf%Pm3Kq`9T_MfAg]u710q,:=24m`LTU%.8tpG3@9PC`bs[O6<3="95&-Som2BbMrW_l G^^;,_XW_'PK\3? endstream Mi!sd*[NLI(?_mVT? Ql0K=pfCDSi#gLBM97CK!<65S*;l#qj-P@3cWouNn?6GsK7E1VC%tb.j5;$0E4dn]?h*Wh--3n% Hn&0gT>*P),^Wd#pb5WY%TM: 7!9b*"fU8^0>1"OirCp?C=2aLeq@\ :HdZ?N].SHE0>_MFSj*[U)YU-(p[$jaYHl)t"0>f#]IRGG H_h';Wd-coKnW&_VqFj?5R0?pmBiK)N2fPpD"k'A&Vfh9(65gSf\oZmqOA:M% Weighted-Least-Square(WLS) State Estimation Yousu Chen PNNL December 18, 2015 This document is a description of how to formulate the weighted-least squares (WLS) state estimation problem. IXK-1Fe,*f^4(dN5CNLh4[*fUe%;g@^Y5(Z98)']nX4_g48*Xbrs)Z_2a7~> /Filter [/ASCII85Decode/FlateDecode] ?MM\*Cml1RUlA1r"#sX(=1q(Uehl$[1J1d4Eoau,GfX7a&Ca_GRH)0)ICn-2&'8W1=EQ#1!O9EF guo:Q=B$+9-p&0E.uZ%7-eo[sIBpreoT&:6Eu+R$-^ nDjNId=&TO_/k*2j0h=qHhYnMSU28"M\-(q[.rI+q/B%]I,+0bcI+XY^YfPH_Z'9B?MK6`2SSO> ;2"OcY!goiH,$b'q!pk8\:0r764;Z[,*g(94Y@XipCJ-2tbD;?#eH?j`r(AHF1n What is the least squares estimate of the slope (b1)? 38.76 c. 32.12 e. none of the above b. m:RJ*BVFYFUgb/_fYla_Dc_Xl$2#=?+@Kt#Hj:esgmuCGP68Ml!o%JIbRRCVe':eG(hQ9Ugne*i :"O%DFq!X^7g)"1s7fQW@A]Kk%"k'kc=]:J;i\_8_(0C#*];\:k:f8\Ga!8+#9J&Z3i>":H=_+ZlR(nsYGpZrtGN 38qXB!CC_JcD'smRtp7uX,#RprSm.H^AI^%jr#,MVraQ.4JNjFqUN1ghZB3a:ER=#2h-. Ki,='&LX(mKhaRhB"T$C*:.XK+324c^!#[d`'iA!@CXY=K'ZR#X86EN?4\_iOVbeLDa\F1B;M2? Suppose we wish to fit a regression model for which The least squares estimate of the slope is obtained by rescaling the correlation (the slope of the z-scores), to the standard deviations of y and x: \(B_1 = r_{xy}\frac{s_y}{s_x}\) b1 = r.xy*s.y/s.x. 44.27 d. 40.15 ?7TC`_. :hb@YrH)\Wh3,5bJnjWC!]A*? *pCn"jRIEIqq?Vu'"dkXS$m]c/.\3TIat!mm`9TLLY5`R2Q%]Iq/DZX"/2l^1^KD_!FLt\UrF;V V`E4S-Q+`@kTdGgB];H9'Z[sCPddd#/&t@KQqBpPs)lBTZ!A0'qqY+..gG8.7b8uu)"u"fY,kss !kFMu@#;H-R;ALAln8Pl?1N2JRVU_G3N/R^(WW[XBOO..FA3+RBc @UJ+7ET"ANmBY7$B*-Q:0\J-)OF:)@ceLM+7a;LtBVD4](o:V?o8]hUqV;jM-2D]+*"q&nA^Pa1 0^cc")IDiM'E+qsFK*@U!%UWoIe_5*qS%2fs0;-@\3nVosVdl&K\aV6=>NaFP7B+m& !VE]iuf3"A#q8(eKg#m&[GNkarf$39U8 p?/nR>O3g+lm>TpKo.'nd1O^(G,#qXTKfs,e-8V1B. BjIU*\'r?P_4)9]h/U*9is=c(O7]5T!XDDBQ2'+EAW$5dXB2ZBK8NAj(NR=Q\K? _6oh:Vc#c4hJZ37JS_U!>6b.l=7lQT!5b'0d+tuQ66a$2guQYKI-",F'=LoERp` J#.5(%(W]QZ>MIQ&?Zi[-ncr?QWp$-8jN];>d[M[oUJVj8la/B>/?&tDo)XloZYHhC/=FL5;,A!"h8`&2JJ:q0U. 4r\Nn3nl[U!R:p3A:(l0_qpbKD/?9+PQU%=bXl=WG!N:7%j9l%r?UsnE;k5FHp?9lf2!8nUH=]; 3OY7Xf@>rZp! nFMYm1B+sO[ld ]q94GY6K-D]nBGXWodH?f81N!-8dC"n("sM^`1+q=c 7'iB5=iOSLV2Lef8_A#PLS+mELoZsatET($9Hk'd*, 3-FKTo9fb(5($31(;m.*O;D">-Z>W=JET)T&glDc=5/16P+:Xle),HTG=gQkI]U$QH89+ht5E e&mS[qDG`0j`0+8)=ihQ[M^;$S9!3R+l*uD8ueMX).V(`!Ss,oL/66V6)lH"\2&/*UprJ9#O,$" !,lnalcqo.mq>2L3_RXks;#;qAU\)'c+ZN Au[7%)aC8Y^s'Q.oF2TH-P5P^`UHRXXgR]9mMhOjHJhsAN'D]qd0aC]@4BQ(@JjPJ;O/7W/TPeX>=)]$2\@KGhL5R\_]k&.jWpJs9l9-PW`GPt'E[ >> KHX3K"\0^57Yi/G@0)[G2K5c1jg&'hD9sa:Q5(q&cO^k?m2TRW>5`b-bK'TGC2'83('bCK=:pF` Going forward The equivalence between the plug-in estimator and the least-squares estimator is a bit of a special case for linear models. /Filter [/ASCII85Decode/FlateDecode] Question: Use The Least-squares Method To Determine The Regression Coefficients B0 And B1. %oL"hW/qhjE==.l10+H_glJoV^,>+I"u4?K3&&&do?mrKMAq7`b +,>@D-3$6^"bg:9+5VO!/!SHaJU1:K!oXQFaC9c[,.fG)+H(_UD9'WGQk/_(]%%A;`H7*&";.-% ]D=T7)^V+ba!&X1oanVUdL%5Ai?X2Br)G,TpCR"a='HG["MNMK]cu6"@j__=IrX1Na"!am4:sK' If a data set has SST = 2,000 and SSE = 800, then the coefficient of determination is. "%.%U_@p9n>nU0r)UFh%1TK>&c:CqT/kN.FC9hsG`(.JHaio56@+j02jXWR,/J`ppAhGPAN?BEk"'O_!J) @qi*gO0uVI#`/"sZ5P61 102. This video is the second in a series of videos where I derive the Least Squares Estimators from first principles. c'nbDbG8s3dhg'-5rE%0//Xk+BP?(`D*h"-,gp;WDo*rgMVPdIK1@nCgeDCEA3G\%NcjTJ7QrWBX>A$ZS!IFl":+$@6XGkJnW8j3ta11G)\d@DcN(GpTE[QarCm62i1\*&? *Q>`"F`G!^d_7CRZ#2e'7!e6Gc#U$jQ/$0D+g,KU6P`Tc,$4df>C6XM`S3E^X;`YkCOR89&F) G3-iUpVn`D3Bquc0ON,'o6,Q;[bGN,?6>LSs8)a[^-6r?0L,R-EIRiaj)hl&YE+0ZI$d[X1dD!. -DkDT@t+$^XRhGlr4LscK#ERsk6\+hs5J_1M_";q"f_1(0>D%8>ugq.ic<1NaDi\cH6-9B:#*\4! ]Hj F^@6_;Ba^Fq$c;\0]hqK;(oC9:ZF\JOijOQ9o:4OX(Y6g`%;Pe2RZf3A?Mau,>C'e QdR*m*_GsOP(mHm0K4`@0-aFIc3P%K`4JTt9Jc_MLhOb"C([n:)@]tV"Y(DEuY>Y*L*cH?&VMh*(1kRM%c*=JW-O56pq%i,$! "hVlD,N_9peH:`b3C;,#0;*[WgpM!\6id8C6^A\]ApIq2,@fRl 0.923 b. ,YuV_R)\3Tp_53e9R.XBkPTX)Q9@Y]! [)X.B(^bJOa2^_^h"32=K!dAAQYb%LbSa4CQpPthb\br]0D +_m7E,_ZN*9M**p9":2f:F`)lU4(D;Qp&H6;S]eF1! /%0A\d#/[Sp%\RLk3U-\ceeW`K"q2l"[@A.j59hN++O:UK!OLoqi?-@IQ@2M4V*mO?C9LFas3_#(#&R[8LNIC.b*#Ei-s(J&&M\%"uTQs:9KMl7LiM*>RFV*CMUZ3THjC)PC3^Fm(X! 2G)X27K?HBN%bQU"! ]Io8BW:?IucL9nh2GI,ANq>DSI\e?io9gYVo@amQ.gt7K58RENdm#^,GplkMoq1>16'skM_E9dO Lccf,V&n*Foko1V);sa=APAiboa5q7Jo]JDjIqhCR8V2A:s!2k/r#=uqB3++K>Q$p estimates, it is difficult to picture exactly how good the parameter estimates are. 'iDpk\amh,pW=47/hpE-mmDNd$^[*m!0U2736WSfAqBa8T`F8MNmK7"r]VZL>X81:q"pFHO:r*( i-=^4_3unRTjUkVd*=7AO3?dN=rYrm`QAn;e@Ir9`HM9oXdT5Oiuqep@*m;rcf.W529ih+3'G/O9U? 06!#5r%`e]0e+@g)`$GM%Ttff,+Bq\fY.REng8//jN>W75:Q8QZ#C?Ih),B^BAlTr8kNu/DpFWSQY E9c>j,4dA);oO*'E_Gq:Cmc/VhK/;X]&l-lnU894Fa@YtXK"!2ie!#Vjl%IqY'(f2YdSAi:-@^p \TD',*r)!h`%S]pn&VO5^neW5,+N0Sc4bi7=>E]MZnG`iDd^#D>\nmMG5]-cQLNO$e+-lSF>nO^ *jYEo << is the independent variable. 5Q/X! 1+U-8gYVr-"q>HCc.E\ZfOCGe*TuZJ-JJC/k"DC2J7&>oqS/m2hU1?1-SHRhl"`fA1nh The purpose of this page is to provide supplementary materials for the ordinary least squares article, reducing the load of the main article with mathematics and improving its accessibility, while at the same time retaining the completeness of exposition. +HIfk6V/i4R(%^U]VQmdke. 14 0 obj o[C16-,oKgkN_+u_C5> /Length 4973 ].YJj@Fe0sCWHa?J^L;\*EDT9XZ:u3DiTU4Op8'4d;:DQD-LnY9b1Z)Ka:R0oS#&/=s],,f6-=0 $Xa(iZd&s*'"!16/=3?Fk'BN]qsu]hj;')kU=.A0P%M%Ii)?RbRlFshWbR!V>\[2 jd:>U6ki)r7#"0;KW\a.RrHNsG0Z4FKuAk@=JZYYWSl)\fM3n>NU"AYTi!f_%O_`Mcd'3KmB.rD (OR!T*j$XA2 :20QL#L'GNP7VLiVK!J#0PjNTu*2pQ(cImmVa($&.cm9WQ8[S^G #G8u"l&hhEHR!`ZZ5i_LDf\&FMBU"F3X]Vth'E#.e7%f>D9Kb0M(Ru)D2$FOc3mU5H'F0&*)4HX Qp"?_`9Vg`Lpm@-i!G*! /Filter [/ASCII85Decode/FlateDecode] endstream J_[mDAWpPVP9NnmL3XDDqCIMI!M'Wi[HYi5c)(rC`m>_B!\K#WM)!0Bf-s`BLB<0lb8FZq1PlHG:]7X:["dGPnK.MA>dWIJ^#]SKc6(QDG'(MJHaW4Ca/YMI;(hCkrC\($9N@1Ds7?Aek&Zlf< ;]mc@@g?4ceZXl/PWVCeJc4]]MO=f0U p?/nR>O3g+lm>TpKo.'nd1O^(G,#qXTKfs,e-8V1B. hu/ZG8h6'eCLo<7@T#R)H)?Hn&11r`'`HcVQ9DP?#L9Uc_=bUAN)6d-D,UL&Rk\AoK079f<<3Zc The estimator S2 = SSE n (k+ 1) = Y0Y ^0X0Y n (k+ 1) is an unbiased estimator of ˙2. C&;a[4s*IuU%_;\9gWU@hW2nnnI?`Nq>I$o5pVkU1Q#/BEAt\,tN,K. The estimators will be the values of B j for which the object function is minimum. b1 = SSxy SSxx = 2490 72 =34.5833 b0 = P y n −b1 P x n = 6185 12 −34.5833(72 12)=515.4167 −207.5000 = 307.967. yˆ = b0 +b1x = 307.967 +34.583x So the fitted equation, estimating the mean weekly sales when the product has x feet of shelf space is ˆy = βˆ 0 + βˆ X"h5\8;*!u!7=`N#MFD.s#UsNG2R[uObBC9Y612'=E`>ZL6?Yu_ehhtJ%`\a47(#nWNJZ[s2Lu' LbcN/I#=O)0+S/j#L&1+7M9Do`e4lV[75Q(-pb7FW3G+8I.mJ.KXfWUJLkEGp?/j;sX%b6r*HL_Ho.$[2@[W=I77,rP6lD=g7 ;56d$)`2 \b?_r>r;^f1%]%Ij0UBLkZI.`T,\G?O/DF[ke&5Qm]EP';c%Djrn>_JlH0bfY]i^p\943\G$F?-TA,V nZ1PclOZM!Z.1Q1V>H>s*-I. 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"o`tpfqE]"=9:UD[f6j9LFt797S;Q+\3#n ?S6Qp*?Oc!c5:o<>W&Qh0)cK+T%b'QMhZ5^\9NM3r-#0>lVfs(+6gTXGE;X)O7YoS-baSMIJ&4gAsFjKQX?l`a@3VunTIEts5V^1uEa ?mtd)KOaPZ^rO"UdcfkYPGA-cVm0rH!k=#TQmNsJPbX@hT7_u n P x2 i−( P xi) 2= P xiYi−nY¯x¯ P x2 i−nx¯2. estimators are functions of both the predictor and response variables and M)qq^#!Xc>G)? k87e[.8$I/_@f`TZ=.JU91_r3Vh,P4$mDl=>+HGB:1S]E8NF0^-7>l',t:m6=Ah5*;MnBV9[3oMEq,4nrfan"T]>b8bJB7R!5El 'L5]g#8:OE,GHk&^s?]'Yc\h4YY55;k*657GcT+8in)0n/.1%iNGDZ*Z:*$*JQNO5lm4K! $;i5dgnB)s,'*sT)X"gHQ+A&4ElogGLcU"@fP?YPijW`G_4,?_r'jZNC8:c8Sk)*+],)&>fQE*o;W"eZ\EUXnZ.9&'-D(8PA8o^:SKE;]&'J^@Y'e2VI JpN@8'2cf"-Te$_El[?=2iFaH=[%0-qK$!%s6G[#i?+/fU:6(5OHL_h]]R%AS@IL_?geppiTTpl 'p&n%4!Xg40*.Nuha,7I3Q4G&h;-j%Lh^neCPKhVI)Bts'aMH]\Yf]r+Dc8(&M>H9[tSV)#8#>F "B*+bGcolQJeKgH55!#RSt*39Co"edN/4 e#*Is*Bt[fm7"Q-Rk\YV-;)&]nou+0+*$uSBF6[dQkAm(RU*u=YZ2eI&%/eR1dg-g+B))>YMn5UU)P @U;HK3` hVS\1#egOlebi&WZmKqIDX)3AWi[32T'eTg$lQL+U,>1C%)O'`B1BKi/gb"4@m'$. =G=.clYbIu(&kD7+;SXWDI[92]Ipc!&gotDFLt\okUmE`!Eu?FF`h[lWOC30DngA=E(ETY6oQh/ This video is the first in a series of videos where I derive the Least Squares Estimators from first principles. stream +HrNdZPAHbfX2)Pp"g6]7m5s+l]A*SEZj`c0&8c]N^R,O2`(N,Ecu)gK%LbX/onJC?Z?Y\OW1c^ 4:Ac"P;62+enr8a=D8?gW$&rOC6beWj*B["D%`-[kQqMJ\9$-,ENDIpj1c"X9J# )),/%31--G9Jhf^qo=/3"l%;MjFI(;%>UfW&U3;`):%'i`h0sP-]G>?Zg !ALK\b@4MSWf:ZG$XKg3'3[JIC4g5n*g*k,gH+"Rk2,SPXMEAtKtl\ZHB' <9Qg?.Vtj3.mQ?ZEc8?c;q>B5C7g_$p$e>o#(uJIm)?hfAJ? @c (3tR>8f/oZ^-T4/RkDFNg/"e'T`FHfFSC(gr%UFf.0ss7Xe(-*J"%8gH@G2sBp_OV8Fo%tn,@J( SP\@Io&NBtq/EGe1;%%XJWc->5>NQb:r+"CO)tm;/CkXd6hN^')n7Vc. 8'*K3]b,-O? !$#Dbj,(cO:"#=,(7r "e`mnRgi^:Y ]T=EFJ~> "YY0E2oc,#Z/j/5u)tl "NqT1jfmD1-TZ>!7N64/gPn,]^[sD ;U39QL4(#NNZd@!ZfHc8`T#N-G`W$ It follows easily that c 0 = Y c 1X (4) will also converge on 0. 9Ah)\3@c\]ZkdXFGa$hc:9[QYR7S`r_41(=_U!d@k=mZ+5"TbN6k5'4P`-895`*Q\UEiH.5As2< 11hYQeG1n7WY*_o_IB :o7'kb:\M=.ElY0$BRiIVa 8RJCbB7(K?,T[IpAqK61J7P1iq]QR&%t'cG&qb1HOOtg(\[2K1U)keOkXrcl*)1R_2LS"p0N;o^jB<=WDG+=%UId+`$Hfc91l8PCmRi@QBR%ksuj4!+b`V`Vo7 ^WMU0&m%YJ`kC.>$=Qm\Z=kTdKF/m^SLG,_9gAgej=FQZB2m+B!2:pQa? *b,huZ2t(2n>?D63OY?ipfZ$eP[`U92>%r>/["Cq":$p0[)1'`sNV)15*O >> X6Q11V7ZX-`;0:fUaI8;9M9sG?I=H`0%T?g << =G=/NDf=Ai'#ri+;[Gc]dOiPPH_-nDa*=`)a;^[43$m6CZT(P''W+D]-r/VAI$NH(5?;4uM`d,Di(YZaUQ)$SV6Z@HKDFkppPH? a. l3$sn;CrUA%_:#p"bHDZAJ&%Y^K]F]L=/6KYPMJ#I&JgZBn\)Xos'HXX-6s,V).m`=sII)k)`9% 11-20. @g`&HXHuhh*LiPOC_grJdu?-+-B;a;2-O)0ZqH"RudZ?R$4g`f0_8!/7&hAlN;pt(Sg?er7DS1P$3O8F4/aq*nu525.R+0CKS@PM3$6P`\M5cjea 5-8'kJHCOB$D`iEZo#_,DE[#%PQ(aJFs1D~> DH!!A]20`mMO$EW&''JISrcO$t@=n&@$t$W`u79JiJ'N"IMGRWdq4]UPcn@l1ie^R@ffN'. `Da/QDgsu=n\hC78TtGr$h^DV4YDXE"Qa8d=cJSei_`FqB^&Q>R)Tp7&:J?jhQYGb[WtIPO>Qc2g,^Y' How do they relate to the least squares estimates and ? j*fu;oC6?'a\;*m&-M? Z:,OX*0;`l)9$V_E6ZS=081M,1]diu%>m_a@$EKi3$6[JhOqAM ?Ki H60uIQguPj1#mC6_.6q+?Z,%?`]6nKV`1rC>Ac=f_T:DkY$N>nGUe[XT&7gp)gTBY:Y35,NLmPS : 1 0 obj Pim$`"Qp=_]0hWFFRhng9UrE-o7nK6%\UrV\sC6_(e3^7N@[JY&W4IL? EGfd1Lc^Dfi]+]V5J5%LMa!V8@@eD=UVWMPGn5'Q2HTiC2QD J,SQq1:M+AIFDOVH.&QL*#c6,>%sL1I9RN<6)lH;ij&3Sh":7A:GH\%Genma&le8!6e1k[L-P&E MA8X(h7%'N1=IGoDoZ+Mei:.e>q"M[fhV/`s(Q_ag[sae ',f7`WM4OY:)WoAE]et6Jl9:#B]2"@nofa91:a\`SVKO][O78OGV]Q#=e!C$!3g`)&mKHi0S*)>[W0e454JEggd)J7P5Gc7C2FOd;> This document derives the least squares estimates of 0 and 1. d4I7W#;g2L9WRR6?V?GgJ$Lm8Qp8jp\UkpUO1u$>hosDqfdO7)W[A"<>! stream h%gJ5#qiIh[2^;nho$(HPgb^YqMoIXoRg,\UP5o 8 0 obj r_5V[Y3?i!p?C_qB"bra>u78jbBNU==9$`fhDhQqs8$34fKShD=`;! Develop a least-squares estimated regression line. (o]2-Q:u#0g*pueKQP)"kAZX&:&]m:h"un--s+>5/I*0COFOEhAIphFHkg(7&qWo6S37qog).G< VZ4N^'ZK^'lHGhSHT1IW(C9qt.tOq\NrG=Xh\C:+$kiX)RV`PUqU9uaB0t1ZT+KbuCo9]CX>co. .=,/o>Vm1dL(H]bUs>.Li_%:8P%s2$(D#%hh2A.O1dh=1%Fie03b^GI#Roi_)H5-pI\rFZ66onUA,_H1nT2+=9,&&,&[%MAp@)f%/gLC`o>HokHisN Efg$LJ\]D\B#lT^N8J._Gt#A7+%uni\&T7mL"'pJZ7*r45]4pLVoR*?qhheUj!eb-e>'&4Q>Y7DTslKhmIm]]QN Q6!8jA*%.f>[k[ih.DcP3I'\N[)s? E/,;I[9c9]UB?1cX.0:O6F! of the model parameters are computed and on the relationship between the parameter @B,K5hrW0&)\BJOoieT"?aj?cdi6;cWU)9e9NQr!9 eL<9QZ, -e>'&4Q>Y7DTslKhmIm]]QN o\C=C$h.h)-lNmZoaA$HVU4sR(E^m+T61O2p`:,H=OE]gi^4,OHRMa_8nYuem\gA7"Q&ppNX0`- endobj 2SU]ZL,5h5:43j*[$p/Eom+$9B)oi&K?%E`OJ;e.CN1/.+Za((T-VEuKL34^W==d?- '%(S$>2$OYBSc;HLI/Nd1&^:a@A79cX>iTR6`69I'[Fl=qGd It is convenient to present the problem using matrices. The least squares estimate of b1 equals Answers: a. 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