The Mathematical Derivation of Least Squares


 Benedict George
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1 Pscholog 885 Prof. Federco The Mathematcal Dervaton of Least Squares Back when the powers that e forced ou to learn matr algera and calculus, I et ou all asked ourself the ageold queston: When the hell wll I use ths stuff? Well, at long last, that when s now! Gven the centralt of the lnear regresson model to research n the socal and ehavoral scences, our decson to ecome a pschologst more or less ensures that ou wll regularl use a tool that s crtcall dependent on matr algera and dfferental calculus n order to do some quanttatve heav lftng. As ou know, oth varate and multple OLS regresson requres us to estmate values for a crtcal set of parameters: a regresson constant and one regresson coeffcent for each ndependent varale n our model. The regresson constant tells us the predcted value of the dependent varale (DV, hereafter) when all of the ndependent varales (IVs, hereafter) equal. The unstandardzed regresson coeffcent for each IV tells us how much the predcted value of the DV would change wth a oneunt ncrease n the IV, when all other IVs are at. OLS estmates these parameters fndng the values for the constant and coeffcents that mnmze the sum of the squared errors of predcton,.e., the dfferences etween a case s actual score on the DV and the score we predct for them usng actual scores on the IVs. For oth the varate and multple regresson cases, ths handout wll show how ths s done hopefull sheddng lght on the conceptual underpnnngs of regresson tself. The Bvarate Case For the case n whch there s onl one IV, the classcal OLS regresson model can e epressed as follows: + + e () where s case s score on the DV, s case s score on the IV, s the regresson constant, s the regresson coeffcent for the effect of, and e s the error we make n predctng from. Page
2 Pscholog 885 Prof. Federco ow, n runnng the regresson model, what are trng to do s to mnmze the sum of the squared errors of predcton.e., of the e values across all cases. Mathematcall, ths quantt can e epressed as: SSE e () Specfcall, what we want to do s fnd the values of and that mnmze the quantt n Equaton aove. So, how do we do ths? The ke s to thnk ack to dfferental calculus and rememer how one goes aout fndng the mnmum value of a mathematcal functon. Ths nvolves takng the dervatve of that functon. As ou ma recall, f s some mathematcal functon of varale, the dervatve of wth respect to s the amount of change n that occurs wth a tn change n. Roughl, t s the nstantaneous rate of change n wth respect to changes n. So, what does ths have to do wth the mnmum of a mathematcal functon? Well, the dervatve of functon wth respect to the etent to whch changes wth a tn change n equals zero when s at ts mnmum value. If we fnd the value of for whch the dervatve of equals zero, then we have found the value of for whch s nether ncreasng nor decreasng wth respect to. Thus, f we want to fnd the values of and that mnmze SSE, we need to epress SSE n terms of and, take the dervatves of SSE wth respect to and, set these dervatves to zero, and solve for and. Formall, for the mathematcall nclned, the dervatve of wth respect to d/d s defned as: d d Δ lm Δ Δ In plan Englsh, t s the value that the change n Δ relatve to the change n Δ converges on as the sze of Δ approaches zero. It s an nstantaneous rate of change n. ote that the value of for whch the dervatve of equals zero can also ndcate a mamum. However, we can e sure that we have found a mnmum f the second dervatve of wth respect to.e., the dervatve of the dervatve of wth respect to has a postve value at the value of for whch the dervatve of equals zero. As we wll see elow, ths s the case wth regard to the dervatves of SSE wth respect to the regresson constant and coeffcent. Page
3 Pscholog 885 Prof. Federco However, snce SSE s a functon of two crtcal varales and we wll need to take the partal dervatves of SSE wth respect to and. In practce, ths means we wll need to take the dervatve of SSE wth regard to each of these crtcal varales one at a tme, whle treatng the other crtcal varale as a constant (keepng n mnd that the dervatve of a constant alwas equals zero). In effect, what ths does s take the dervatve of SSE wth respect to one varale whle holdng the other constant. We egn rearrangng the asc OLS equaton for the varate case so that we can epress e n terms of,,, and. Ths gves us: e (3) Susttutng ths epresson ack nto Equaton (), we get SSE ( ) (4) where the sample sze for the data. It s ths epresson that we actuall need to dfferentate wth respect to and. Let s start takng the partal dervatve of SSE wth respect to the regresson constant,,.e., ( ) In dong ths, we can move the summaton operator (Σ) out front, snce the dervatve of a sum s equal to the sum of the dervatves: ( ) We then focus on dfferentatng the squared quantt n parentheses. Snce ths quantt s a composte we do the math n parentheses and then square the result we need to use the chan rule n order to otan the partal dervatve of SSE wth respect to the regresson constant. 3 In order to do ths, we treat,, and as constants. Ths gves us: 3 Use of the chan rule n ths contet s a twostep procedure. In the frst step, we take the partal dervatve of the quantt n parentheses wth respect to. Here, we treat,, and as Page 3
4 Pscholog 885 Prof. Federco [ ( )] Further rearrangement gves us a fnal result of: ( ) (5) For the tme eng, let s put ths result asde and take the partal dervatve of SSE wth respect to the regresson coeffcent,,.e., ( ) Agan, we can move the summaton operator (Σ) out front: ( ) We then dfferentate the squared quantt n parentheses, agan usng the chan rule. Ths tme, however, we treat,, and as constants. Wth some susequent rearrangement, ths gves us: ( ) (6) constants, meanng that the dervatves of frst and last terms n ths quantt equal zero. In order to take the dervatve of the mddle term ( ), we sutract one from the value of the eponent on  (.e., ) and multpl ths result the eponent on  (.e., ) from the orgnal epresson. Snce rasng to the power of zero gves us, the dervatve for the quantt n parentheses s . In the second step, we take the dervatve of ( ) wth respect to ( ). We do ths sutractng one from the value of the eponent on the quantt n parentheses (.e., ) and multpl ths result the eponent on the quantt n parentheses (.e., ) from the orgnal epresson. Ths gves us ( ). Multplng ths the result from the frst step, we get a fnal result of ( ). Page 4
5 Pscholog 885 Prof. Federco Wth that, we have our two partal dervatves of SSE n Equatons (5) and (6). 4 The net step s to set each one of them to zero: ( ) (7) ( ) (8) Equatons (7) and (8) form a sstem of equatons wth two unknowns our OLS estmates, and. The net step s to solve for these two unknowns. We start solvng Equaton (7) for. Frst, we get rd of the  multplng each sde of the equaton /: ( ) et, we dstrute the summaton operator though all of the terms n the epresson n parentheses: Then, we add the mddle summaton term on the rght to oth sdes of the equaton, gvng us: Snce and the same for all cases n the orgnal OLS equaton, ths further smplfes to: 4 The second partal dervatves of SSE wth respect to and are and, respectvel. Snce oth of these values are necessarl postve (.e., ecause,, and the square of wll alwas e postve), we can e sure that the values of and that satsf the equatons generated settng each partal dervatve to zero refer to mnmum rather than mamum values of SSE. Page 5
6 Pscholog 885 Prof. Federco Page 6 To solate on the left sde of the equaton, we then dvde oth sdes : (9) Equaton (9) wll come n hand later on, so keep t n mnd. Rght now, though, t s mportant to note that the frst term on the rght of Equaton (9) s smpl the mean of, whle everthng followng n the second term on the rght s the mean of. Ths smplfes the equaton for to the form from lecture: () ow, we need to solve Equaton (8) for. Agan, we get rd of the  multplng each sde of the equaton /: ( ) et, we dstrute through all of the terms n parentheses: ( ) We then dstrute the summaton operator through all of the terms n the epresson n parentheses: et, we rng all of the constants n these terms (.e., and ) out n front of the summaton operators, as follows:
7 Pscholog 885 Prof. Federco Page 7 We then add the last term on the rght sde of the equaton to oth sdes: et, we go ack to the value for from Equaton (9) and susttute t nto the result we just otaned. Ths gves us: Multplng out the last term on the rght, we get: If we then add the last term on the rght to oth sdes of the equaton, we get: + On the left sde of the equaton, we can then factor out : +
8 Pscholog 885 Prof. Federco Fnall, f we dvde oth sdes of the equaton the quantt n the large rackets on the left sde, we can solate and otan the leastsquare estmator for the regresson coeffcent n the varate case. Ths s the form from lecture: () The epresson on the rght, as ou wll recall, s the rato of the sum of the crossproducts of and over the sum of squares for. The Multple Regresson Case: Dervng OLS wth Matrces The foregong math s all well and good f ou have onl one ndependent varale n our analss. However, n the socal scences, ths wll rarel e the case: rather, we wll usuall e trng to predct a dependent varale usng scores from several ndependent varales. Dervng a more general form of the leastsquares estmator for stuatons lke ths requres the use of matr operatons. As ou wll recall from lecture, the asc OLS regresson equaton can e represented n the followng matr form: Y XB + e () where Y s an column matr of cases scores on the DV, X s an (k+) matr of cases scores on the IVs (where the frst column s a placeholder column of ones for the constant and the remanng columns correspond to each IV), B s a (k+) column matr contanng the regresson constant and coeffcents, and e s an column matr of cases errors of predcton. As efore, what we want to do s fnd the values for the elements of B that mnmze the sum of the squared errors. The quantt that we are trng to mnmze can e epressed as follows: SSE e e (3) Page 8
9 Pscholog 885 Prof. Federco If ou work out the matr operatons for the epresson on the rght, ou ll notce that the result s a scalar a sngle numer consstng of the sum of the squared errors of predcton (.e., multplng a matr a matr produces a matr,.e., a scalar). In order to take the dervatve of the quantt wth regard to the B matr, we frst of all need to epress e n terms of Y, X, and B: e Y XB Susttutng the epresson on the rght sde nto Equaton (3), we get: SSE ( Y XB) ( Y XB) et, the transposton operator on the frst quantt n ths product (Y  XB) can dstruted: 5 ( )( ) SSE Y B X Y XB When ths product s computed, we get the followng: SSE Y Y Y XB B X Y + B X XB ow, f multpled out, the two mddle terms Y XB and B X Y  are dentcal: the produce the same scalar value. As such, the equaton can e further smplfed to: SSE Y Y Y XB + B X XB (4) We now have an equaton whch epresses SSE n terms of Y, X, and B. The net step as n the varate case s to take the dervatve of SSE wth respect to the matr B. Snce we re reall dealng wth a set of varales n ths dfferentaton prolem the constant and one regresson coeffcent for each IV we agan use the partal dervatve operator: B B ( Y Y Y XB + B X XB) 5 Rememer that for an two matrces A and B that can e multpled together, (AB) B A. Page 9
10 Pscholog 885 Prof. Federco Ths looks lke a comple prolem, ut t s actuall qute smlar to takng the dervatve of a polnomal n the scalar contet. Frst, snce we are treatng all matrces esdes B as the equvalent of constants, the frst term n parentheses ased completel on the Y matr has a dervatve of zero. Second, the mddle term known as a lnear form n B s the equvalent of a scalar term n whch the varale we are dfferentatng wth respect to s rased to the frst power (.e. a lnear term), whch means we otan the dervatve droppng the B and takng the transpose of all the matrces n the epresson whch reman, gvng us X Y. Fnall, the thrd term known as a quadratc form n B s the equvalent of a scalar term n whch the varale we are dfferentatng wth respect to s rased to the second power (.e., a quadratc term). Ths means we otan the dervatve droppng the B from the term and multplng two, gvng us X XB. Thus, the full partal dervatve s B X Y + X XB (5) The net step s to set ths partal dervatve to zero and solve for the matr B. Ths wll gve us an epresson for the matr of estmates that mnmze the sum of the squared errors of predcton. We start wth the followng: X Y + X XB We then sutract X XB from each sde of the equaton: X XB X Y et, we elmnate the  on each term multplng each sde of the equaton /: X XB X Y Fnall, we need to solve for B premultplng each sde of the equaton the nverse of (X X),.e., (X X) . Rememer that ths s the matr equvalent of dvdng each sde of the equaton (X X): Page
11 Pscholog 885 Prof. Federco ( X X) X Y B (6) Equaton (6) s, of course, the famlar OLS estmator we dscussed n lecture. To te ths ack to the varate case, note closel what the epresson on the rght does. Whle X Y gves the sum of the crossproducts of X and Y, X X gves us the sum of squares for X. Snce premultplng X Y (X X)  s the matr equvalent of dvdng X Y X X, ths epresson s ascall dong the same thng as the scalar epresson for n Equaton (): dvdng the sum of the cross products of the IV (or IVs) and the DV the sum of squares for the IV (or IVs). Page
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