04 dez least squares estimate of b1
m4"T[MW4ZD+]Ut=q^ZQ?M\"Jba^\iC:Jb2)LZ5(oR;eTn"2#O :biBpXGo)dV4,4-[[VIXr/4[&'C/S/cc-msVbEDu LbcN/I#=O)0+S/j#L&1+7M9Do`e4lV[75Q(-pb7FW3G+8I.mJ.KXfWUJLkEGp?/j;sX%b6r*HL_Ho.$[2@[W=I77,rP6lD=g7 The estimators will be the values of B j for which the object function is minimum. n P x2 i−( P xi) 2= P xiYi−nY¯x¯ P x2 i−nx¯2. J8mS3!TX$r!doQ1?=C/_+j'PeS8CBCJ\'s,! e#*Is*Bt[fm7"Q-Rk\YV-;)&]nou+0+*$uSBF6[dQkAm(RU*u=YZ2eI&%/eR1dg-g+B))>YMn5UU)P How are estimates of the unknown parameters obtained? 38qXB!CC_JcD'smRtp7uX,#RprSm.H^AI^%jr#,MVraQ.4JNjFqUN1ghZB3a:ER=#2h-. 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( In a regression analysis, the variable that is used to predict the dependent variable _____. :pD3oDt^*3frQ>MgR_8i6'1cgaO=AciD[QK ? for different functions of the estimated regression parameters can be found in, setting each partial derivative equal to zero, and, solving the resulting system of two equations with two unknowns. 1 0 obj )s6g9L\?GYP1NA&@YmJnF>#niH9C58=Dh- [+36s/-E@=#2=K7L;AOA#ejbc%5m*A,KFXn,@N1@@"c+iD^!-!k4Vd9k%`[5JWK;E,oY7Wl;i`9+0#&h0/Yu4=9T4Eo#V'c*1,B_"[P$?qiQubVZC5T4`kbj:5B_&`b7?>?gm\, &qRjZO68;[7`E@T.QRc(+IZRJ@>kGad1l[d-4i4FQ*`0"[['oSVQ8_sWa,8Lj_C$seqBCa;(uLB is the independent variable. .&'ho+G#-#D6r*T.j/XWhUM],Q5t`A^22!I2a9D.XiFoTPr^W,I_08dglhdY6f)- nG-qU>diJQ3+6*3@LbYPMFspe@a+3> /Filter [/ASCII85Decode/FlateDecode] /Length 4566 hQ? 3-FKTo9fb(5($31(;m.*O;D">-Z>W=JET)T&glDc=5/16P+:Xle),HTG=gQkI]U$QH89+ht5E c(bAW453p`1D>rp0\(H+G"2TY"!6g)2#,1TSQlUh39V#6Jr[BIBLQ%MB"-dNS-JrhL$^m_!inL7A",#)gV9i.581][IQD,Q"IgC;N2"q''B7:BT@A3$mBYM\Ctl%gZDL"(+@ (2PJ:+-aaE1[^ihIl/efr_\X6#jsPICC0S$O+O$$'m0GQ$-L&uRY;7Nd6^#j_nVWY#QH%g0n_Ar kPbBjapFcKh]H?BkN4-*L3b^Tp)$^iO++C6&! "5c-qD0##eg9?,qF2+)9pmDH >> Then the expectation of b1’s numerator is E nX (xi−x¯)Yi. :hb@YrH)\Wh3,5bJnjWC!]A*? !SWJ7VZdE\NIY')cLA`rhQm;f@,=ukA#pt7+DZgj"t8q8bYq/>+T;j"6TcQ-X9VcGej)7'=/G), [Y"u7) '%(S$>2$OYBSc;HLI/Nd1&^:a@A79cX>iTR6`69I'[Fl=qGd A?Snd-$06AA>U4"9Y#b+(+RFA+J7jUWDbSs84%Z. 7tHcuXe5Oi-5R5MSFL+nEnj=kP$kuGj'2Ng3@^2keBLqb('2=+\<6b*m n0`nB+!@teZlF9j3>o? \(\bar{x} = 0\). O%? ]Cc>HUd>k9%;>^f"7a2:;1*rEa3B@F6h.j]'-,m_g;1Kp)p[&e@&D35VJe5ut 31;)/f=E"kQSC?F(rdUs+\k!/1Q:p".e6HkP(" 'uP3A.BD,:[c7=I:V[ qAt^.8WoVu>I#6G9sG=fcghSl:33dh+&Whr37d%jIs8q"Q$%Wl$u3#. SNom+`%]^JbcJ8u$=al"$o9BuU1"lJ0a6(%W"(D)e0cqL@cVBbTJ49@YB#QdNJ=AE'e%!ih9&8A 1R@8,MWBf%.Ik`#1i#q6)19I? )`#WHC$1*FK8&9H+HH[$ecaZ=LHdb#bfCsCMP*,#AKi6_6@EnL`S^$S&@qX9VYQBL[?K%D'=5N= /Filter [/ASCII85Decode/FlateDecode] =V(7@9UZQdPEbJd!U+%IeY8J59'7cF7=B:&Y8 LmNI3Tu\0DW,,e"[mt>)`F"d]oA,'c5>A>&kY\:?4b&mdnKN>+PY)bgc3!0AdPnMWSt^O!Zr2me In regression analysis, the variable that is being predicted is the _____. %_M3B/?3Y)^u:rbb?4fdpVSIg#^'##4VSCcaP^mlU^qZ):/D*T)m(2(`!p.c/Sa5[Yuuc/^E.9Y 5t`s+!in%HZePqKA=fkk_-2i0>mDjQ*]K"S;;6ILTMGrW&ZD--HSf=..A*#t-80!fn;p"'Z(Ks_ ?0^N]mq/L %* V`r1KoZ7%'c0(8DV)+huC:f#^'ApEEQWQ$]bOQg2%lA*Y.NA!]$gW]N6kcMWn;S[Npt[V<=pY$0Os=(CI+1H$\B9\pK[ei^`$"A/l4(INq,Qs#lNi*SUWeNM. )Gqs=:Ul!V.M7f[hd*@I.R^,[0EZ;qWC0,]f@*Ku6:;12MaU.4 RUn3WSm2Wi'Vq#YDqS>[1Jqqm1XPp/RB>'"J[CY#-[9p21IlMpBKPFc?8h1u^=H?6Jt3LbiC")t l1EhQZplDbfZs[076JJ]o$Atm.k&FUl5A%/GiG2_L)H,JDUt=r+Xp5?K.5\M?ssPqd'". @B,K5hrW0&)\BJOoieT"?aj?cdi6;cWU)9e9NQr!9 R,+2(q`0M"a#8aRG`-sS0:K0RV>$mW\\Udtlhog1>d;gZ+Z"R+j&;[brjG;OfGqA15Ls=R2id8/ o7o`HdEP.fE?YJNF-A,J#TBIMUYC9*N&t7ag?Fi#XW8*mEjX78ER#?5oRrr1?O0h>`en=J*na#R For real data, of course, this type of direct comparison is not possible. /Length 7110 [\FEcHDBDQ"[B5*cog$%EdP`tI+hu+&Q:`NX.7['7- OhHET\&M5c\b!n5s,5Yufsr.%`HSWM1`t3S-%/.A$!! 63c`7\31jCk\2*Y:/R@s)DW%"6_Q=Bh8DsNe;m",a+Wo-ShDno::i/_3HoO*OB@@F?-O2DNUFaF ;Z@6ir1nan8pmH%a0_h.q.d[TQYE97WHJNRqDU$ZhI*:1$sP>05WLO*MU6s=0$4W&BD' :o7'kb:\M=.ElY0$BRiIVa )iC@J>FT!e&Zi&tK'(bl;!#AVhb]V[=c"e7 $8Za,8[5_*og;n.Q\kD)5ef#nGQ0i nUnL\;I<9DorW1Fdo^0f8D=M%%Y$j0V@c$WteeIM-STm9-7:lebk=)LJk>%m9nqKA=Z>u7b-TTB48 %%?q]m6e==>P.l^#(>X>n)P'96#olLZ!UhKn(e97cg'&0Z? @]#3Zo8G"_\s1a2jhBH:2GWe)4Xtm@"n&5-`8[%kU2c,#b+$ n%"eRms\BqrUjfAk0K@*4$Rc'-jcW9LVN7I)U))4F8`nlU;NN\Hap\3r;"5(6dL$^*Zcgf?\E.U \AOfeRAd,'np-@]EA@bY@EORXq;XII[]XjZTXStcd[XjrdW3e#! 0+GdX/N^3G>l?.5W@S,r? =G2+CeDkOkIk.h-B+"\t$2P+i5JVPI/h/9F-O)\tWi&cn\?VI@7BpH;T[,qg#e@ol2f)6plBf-X;8;"-3WQ6k5]mqo'r;Cg2K@[r?fis7=+Dp[Nrm>6apH"!jImTrc]IRZ9V0faMX =G? Theses regression coefficients are such that the estimated regression line is as close as possible to the data that is observed. 5_Q0p-(2+C8tmA&8KB`MZE9QlreaqY.GY)jAC;@.8P^]Sc=4i^&m/spL_uK%+u1L9aZ,OfR,A)n ]a\ke$sSYtRk`]4L2@jXc_)l-0*X+HZ\53@-TOBW/%Gsnbic]WbN$>6p"dNY,OfQOeR@4FdMLtH (rWJ\2VTje`?BeHWJ$C(J)r!bsINHkIZLu^>EcTGa@Th*/ch5:5OBOD4h&X(0q`5Gl ;4$_ (PC,NMik0I$carS,sL/\,5!/Cm9UCEcBq4$Ik/Gc+ 'V#)Yi)SuX@rGfg5k,;:FH-VVF3]V'$*YBRBY_3jXM3&_ig>]6$/SLF?hY'?r 4:Ac"P;62+enr8a=D8?gW$&rOC6beWj*B["D%`-[kQqMJ\9$-,ENDIpj1c"X9J# :F1tkll]! cJG61c.NB*)/aM?4dT@an2$8E)\-r@)d/)9kjAb*q"kE2c3!0A.N,9Xq>i>gtGH`fu49DP6dM(b^FDR5i@L-RV5j`P$Hj#jc!E :YGk@5NZgq2CK's:0%)fN9]\q4M`Q5LHhXc`,_]*h!VpU$3n ;LE =G=/NDf=Ai'#ri+;[Gc]dOiPPH_-nDa*=`)a;^[43$m6CZT(P''W+D]-r/VAI$NH(5?;4uM`d,Di(YZaUQ)$SV6Z@HKDFkppPH? %PDF-1.3 (ND]h/[QG/3p]RF]&rZ0#3@/\AfF_bcdZm#5c+rt4TuKOLPh1C. h%gJ5#qiIh[2^;nho$(HPgb^YqMoIXoRg,\UP5o IXM.ZGFq:"'E;LeIf/gY%e0(ns5@&UA"P)Rnb2prl&'\gDa/[Ik-m"YKNujUZX>N`nh\A]l&X`8 1D^;a,*$q8hj_k">PQMGVfK/7nHUdB"2PQl2'D.XRG9eACK#5SBQarD>e/tKNcqB^h[6.c@8dc/ >> =G=.clYbIu(&kD7+;SXWDI[92]Ipc!&gotDFLt\okUmE`!Eu?FF`h[lWOC30DngA=E(ETY6oQh/ kn@mI7^;\1&hTpuIO[oqacp/fcAG9,_ar70W.tp).#A-G.OoP,sQ'Po#9(seb#D& 4 0 obj 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 L\Z[goD*C"'1hKJ]^`Ek:8f6KLb93hJ$GV\2n,p1ZST I16j1_%mhgB-3K? Qp"?_`9Vg`Lpm@-i!G*! 18 0 obj /Filter [/ASCII85Decode/FlateDecode] \TD',*r)!h`%S]pn&VO5^neW5,+N0Sc4bi7=>E]MZnG`iDd^#D>\nmMG5]-cQLNO$e+-lSF>nO^ SP\@Io&NBtq/EGe1;%%XJWc->5>NQb:r+"CO)tm;/CkXd6hN^')n7Vc. a. %oL"hW/qhjE==.l10+H_glJoV^,>+I"u4?K3&&&do?mrKMAq7`b Lccf,V&n*Foko1V);sa=APAiboa5q7Jo]JDjIqhCR8V2A:s!2k/r#=uqB3++K>Q$p 8'*K3]b,-O? (o]2-Q:u#0g*pueKQP)"kAZX&:&]m:h"un--s+>5/I*0COFOEhAIphFHkg(7&qWo6S37qog).G< \(\beta_0, \, \beta_1, \, \ldots \,\), To illustrate, consider the straight-line model, Uo::i(lM7GP+qHqN5%*i:_FH`NiTHW/F#bTY!Om9/8?U_WnF_3+X@oH$J?\p/1!_Vb)nDL'2hM& XpB?-BG$-qo6NjSB+ObC:Y)5q:E+Y!! 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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 /Filter [/ASCII85Decode/FlateDecode] 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. ?Ki '/rnL2U%X',r/$P&bpol2K@n;.X;M*3aCqDT/nF!^?""Yi5:! ##kVq9ImlIR`_>COl:l3)9aA%a6K559!X8K(L'oOnP>;XC@tTe:c<9u:k;Uj-2r'YDB$\l)9jO@7 ^W?`61cOm3>1H+reSXY\Zbs(e5_Z9tKEU:9Ma2OQ-5+M[i*^d5D$R8eXT:&++HNKKmY[d[\#n=[ c'nbDbG8s3dhg'-5rE%0//Xk+BP?(`D*h"-,gp;WDo*rgMVPdIK1@nCgeDCEA3G\%NcjTJ7QrWBX>A$ZS!IFl":+$@6XGkJnW8j3ta11G)\d@DcN(GpTE[QarCm62i1\*&? From the preceding discussion, which focused on how the least squares estimates KHX3K"\0^57Yi/G@0)[G2K5c1jg&'hD9sa:Q5(q&cO^k?m2TRW>5`b-bK'TGC2'83('bCK=:pF` 0.6. 4r\Nn3nl[U!R:p3A:(l0_qpbKD/?9+PQU%=bXl=WG!N:7%j9l%r?UsnE;k5FHp?9lf2!8nUH=]; 4]\CQPuU)n#<07Z[=B\MZY*9_XW%.,caFgiTJk8TQq]JDos9`PWGj5B'%d^=^51p9pZ]?YhQ]^dSF$ /Length 5037 [$[sSS,$kj_cqWCIRV%$^tS8\#B&]"6UIo(1#&e=>9Gea ("DdeFd&srg?a2^&%+VqB.X)E(&`WZCFb"HDX`09A)tQpM!Y$c endstream IXK-1Fe,*f^4(dN5CNLh4[*fUe%;g@^Y5(Z98)']nX4_g48*Xbrs)Z_2a7~> >> +`,p?)RLs)_5dZ0lW=.8#c"`tb\#gp>;XSABnpBDiOt^qIClX?Y$I&t(#H%CIB"3l!MlqPLcc-K=J>7_ss+(NgB'ST])+WdsHocN')45Mj)h7ZED? *J`OjB03g5GfnsAT7eY;u*0!A:t2d^Q1MfQA]mFU)'rhfX&LP@@H;uH77hsflV]j0sh E9c>j,4dA);oO*'E_Gq:Cmc/VhK/;X]&l-lnU894Fa@YtXK"!2ie!#Vjl%IqY'(f2YdSAi:-@^p 5PNYGFT~> ?mtd)KOaPZ^rO"UdcfkYPGA-cVm0rH!k=#TQmNsJPbX@hT7_u These formulas are instructive because they show that the parameter ])2QtU-^()p:j!Pc)QTg (%q4&--4P"$:8[Z(XW1=5oKN#Zd@)a>*p26"Su1uj63*\5`ba*2#V1rEqGE=HMg"V)ghG-(&pQ! ;2"OcY!goiH,$b'q!pk8\:0r764;Z[,*g(94Y@XipCJ-2tbD;?#eH?j`r(AHF1n 1.2 Least Squares Estimates An alternative way of estimating the simple linear regression model starts from .o8dPFeWl_=K-KLq^\7)u#Ad7Xu_!Ph9ZH\-ZXiO9kSe3/7]inH\co6;r]J jrfqVjO+on8P^^;rL/,;IZfRJ,=N)=6KQ3LV`4bT9&U.nTS1KF-(2+C8tmB$"@p>B_nNY*JMTmc rV4M?fBMP[lW/n^&. ]\2hG1>9738s"McZS*)GuLV. ]F?.=CF@@[>0-k0it[hZTe5'XYo$\h`t)XS0Bi\cFp7$rk;QhABQ9:ErhcV%KYXl,9/!Bm`P[[ [-SKW_ccU%dT(IXD]!YP8*l :lG)K`u+9u:?pSBUH;@qKGcahRd1g2s"I`_-0U9"B@JT(Hi>oT;<5r&,BXtf"C^l;.s)@DgK26VM2FMRB#C-U-NI_h#Ap&<1RH;L5bO*I'St&: &tOA>3YcCb:1QVf6&u]Y/fFg1bmeC%P1\s;22CN,c\#p4r>(cXV],OI,U >> U(J9iBTR2e4ncEb?jL#&mQfg$Kj=S]3md;'3ROE2ctQ6Y:@X9W&/H-g2ZUG /Length 4190 5Q/X! ,+$d3Rco:+T#f0/8nO7.Jk?9r`U^%&7M1GUo)36T-fd[8/u!#NY0'Sf1XQ,]kbW\^RT9Vr>J9Y,2UAX >> (J_=78L>@SJ(c_65rl()42%X5K2AQN6?L\l!q0oSS9WBo@g5>j6_5NVS%D,T?T]@]o#`c]ug96e@BfuX_u*F 5D2ICX+pRtb;T!+)M/'Rl+(2CgE^1NUkK0Z??j_nkH$MMe7b.#Ug%:MaenkrPUt0,:PYf4'1JsW+W78!!"otsF$\5^s3X,Adu\+g&g3$W! VTg@teC4\2Ua6:u1O74.FYB'.#bcX[LZj2? << Sss7Q'/Eae$S_f1GLGk5"nZr6".JW.m8m":0ecVHSq+Vaid'**JZ[d(O6sdL4JPdV!-f%b4imm+ ?8$ZLlo[_WK-[cEGsd4E$>TocTM $:L=kB63jji@@.2=7. )80hV=6uR-J$V+ >\oo<5;9'e@!M`)PI>L3ClAF_Jd>XuO^'%%]_.dukAiBTD9oYIbh1u6#-g3dX"]371TtUP,k[bD /Filter [/ASCII85Decode/FlateDecode] d[7s`N'!4Qaai;KYY$t_Vt.iRg^ib1r)1(%[SB"7-bc9oHQ,?UNJ?,UUZ;@o)X&R!Z2K%]#Zr6h0dW.E=^S>^54+f !+`Np KlcJePk9'l'e-uB@q*;(r$3s6VfDbNNPj.;QmgG$7'JKpRC! 6K',t_EYusU6/*-2h3G1Af,G=alW$%2Kj][jY&qeRrKX:UF! ;;*b6roWJ@c&tPe700u_:[J8EY)\p/Qe_:T!Q3XATPi,-ficXAI*p#78&mW\,uTE ",*0\^`_ad9Bc 0 b 0 same as in least squares case 2. << !EGhAWtYL@PSf*:RJ:-^6n$V/=X5CkM,7l;b%PFe=*Bh$iPAKILG^>\!>uIQdc;mRVdCl'r:#=; @qi*gO0uVI#`/"sZ5P61 The least squares estimate of b1 … J,SQq1:M+AIFDOVH.&QL*#c6,>%sL1I9RN<6)lH;ij&3Sh":7A:GH\%Genma&le8!6e1k[L-P&E =G267;ngC+YbKOsU?qAjHb/E+8qI0IfmkH5\N8UbPH#D>;HOguWO>1nJ/aPBuQN Fb86+O'A=3*5YoEdnQ>(p5cO2A#Ku/rWY:WnUV$F!AKsF-g/fso]$9'0eOtnj'e;@Lqq1'po61c a'Je?[TabF-C\kUQE*lf;5*Tmg;O4^ER7sp9-5;EWOe? OK(UpO\AB+.D]m0fNPQXoh@DV(o,aO@7Gk2s;OI>NAt0GLUlEI^ endstream 8 0 obj This video is the first in a series of videos where I derive the Least Squares Estimators from first principles. 1. M_jZN^qZ0X"92nK%#p\sql%'"+5l$D&,pGr_&j)!m@hPp;@DutVT$A9MF$FO17bA!Ik ;o^X$cV@i6VLr/DP%Jm! [P#38^EI-_ !spdo2s[l[cZoWoaN Hn&0gT>*P),^Wd#pb5WY%TM: mLO0@0%O%)H'8F"GBB?N]G[3L7t"MaV"2UE0"%jP@tBeE_Z2[TT$]J@JG@o#]c-0['>k6fV(M"u #[TK-4+BXF'du,8MGK=*fhL0O@2q4Uh6W)A,F$cd 7V6cd_c(4u9m8KEf,UCc. +0IXsi)UJTAC;okVBTnb-UJ2d4G9-6S;E.#2F7m/PG&sXE\^DP!%4`4kGU!E-Rc8*:7=3B96-1e /Filter [/ASCII85Decode/FlateDecode] 11hYQeG1n7WY*_o_IB :HdZ?N].SHE0>_MFSj*[U)YU-(p[$jaYHl)t"0>f#]IRGG F[,u)drc*<5=RR!/dEVF*+08MoG(&+3A?lFa'FdeR';+f,n%R"a-IQE/_lS&!AVjS6)M,;:qEd< @"fkE.3T)dn$"o.&9gW!7=. ?%WU)I8)^,Y?9rPYN>JgU>EnDho$RB9hHS;;A;@irNi=^o='UPct?nKo` "a/Jg@hmbHV\5Z1Xu%T^16p;c::>8l9ZV%I519(c TQ>CDCYJ0dB14.3gVHP-3l%H%k()4,6M*g-7q0,r)#LD=LX^IS(Ys@&H/Xb5!%)7MC'.gQh'-6# (OR!T*j$XA2 L!l)5`S!&!5c:FUJ,kuo0CbOniGfNP5$1J9dP-u9j\-sT9+'+K[fZC^cUt? ;`A/eG0fT0Q: OT"2:R4fLagN&XT9RCj]G\]YUI/%jY]:m^.tB^?1E^+"jZIL7.d;-7a'.m_Sm. T22%bIQ[=-Ye_*(3j5IYnF,U\>1]rbJt)&h:AabVohW=eitbm`mnb1mhmXjN.eBMe @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 Here is computer output from a least-squares regression analysis for using fertility rate to predict life expectancy. @8eNcEi/q,b?rl?W*hqK;`$0XV`Hln)eLAF[Q6u*6FNct=_,`FEg]kNM3)]"kIn:,$3[)5IAVf^[TB]?F9ot0HVd/)!+C:2eD$N6&(k_LDJO')67],6[N3a,O8VA_;"F@JEEQQcoP['cos7V['b)Gc What is the least squares estimate of the slope (b1)? ].YJj@Fe0sCWHa?J^L;\*EDT9XZ:u3DiTU4Op8'4d;:DQD-LnY9b1Z)Ka:R0oS#&/=s],,f6-=0 )e9e(]9+Fti2Xo]bK5gK-dR:gf^ljjD"@ME?XMa_ofoa% J'h3PWG+Fo%$l%rr?l+T\o!>EnCA<5esKR9CbX;Aq33UN[g1,))s.@t3:f_/!hCEE5&"`rqrW6K c.p]c_:]sPV? RIfHnbh#Ac.-CW)MT$VD]ImtFW""+ppdGCq$5\kVW=7[""ZYCbhp$]m3GALn;h3PCl $$ y = \beta_0 + \beta_1x + \varepsilon \, .$$. This video is the second in a series of videos where I derive the Least Squares Estimators from first principles. 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