Lecture Notes. Revew of Matrces he ablt to mapulate matrces s crtcal ecoomcs.. Matr a rectagular arra of umbers, parameters, or varables placed rows ad colums. Matrces are assocated wth lear equatos. lemets a j deotes the elemet row ad colum j. colum vector oe colum of elemets row vector oe row of elemets amples B C D a What s the dmeso of each matr? Wrte as G m,. b Whch matr s a square matr? Colum vector? Row vector? c What s the value for the followg elemets a,, b,, c,, ad d,?. Multplcato b a scalar multplg ever elemet of a matr b the scalar Scalar s a umber If w =, the w =?
. wo matrces ca be added or subtracted ol f the have the same dmesos. he commutatve law of addto hold for matrces, H + K = K + H Fd?. matr multplcato requres a coformablt codto. R ad H We wat to multpl S = H R Chec the dmesos for H, ad R,. he umber of colums H equals the umber of rows R herefore, the matrces are coformable for multplcato. he dmeso of S,. R H S Note he commutatve law of multplcato ever holds for matr multplcato, B B he dstrbutve law holds for matr multplcato, B+C=B+C or B+C=B+BC?. dett matr usuall deoted b I s a square matr wth oes ts prcple dagoal the dagoal rug orthwest to southeast ad zeros everwhere else.
Wrte dett matrces of dmesos:,, ad. matr multpled b the dett matr gets that same matr aga I H. raspose terchage the rows ad colums of a matr. Deoted b a prme smbol,, or a superscrpt cel =traspose = + B = + B B =B Fd. Iverse deoted b - ad has to be a square matr sgular matr oe or more rows are a learl combato of aother row. Or oe or more colums are a learl combato of aother colum Do ot eed them tetboo has deftos for determat, cofactor matr, ad adjot matr We ca use cel to calculate a verse, =mverse?,? However, ths returs a scalar, ou have to add a de commad, so =demverse?,?,, j he $ sg commad s had here
ample hus,.... I I..... se matrces to wrte a sstem of equatos. z z z z he soluto s:... z What about ths sstem of equatos?. uque verse does ot est for ths sstem, because the secod row s a lear combato of the frst row. Multple Lear Regresso regresso wth two varables s ot useful Ma ecoomc relatoshps have several varables We have N-pared observatos
s the depedet varable ad t s pared wth multple,, depedet varables ample quato s t t t t u t Lear quato OLS s used to estmate lear equatos. quatos must be lear the parameters. We ca wrte them as three equatos are: u u u where the β s are uow parameters u s are the error or resdual terms. ach equato has error term varables begs wth two, because of the tercept u u u u.
Note regresso, > If =, the we have equatos ad varables If equatos are depedet, the we ca solve for a uque soluto Note we use matrces to wrte these equatos quatos wrtte matr form u u. u u ad are colum vectors of dmeso β s colum vector of dmeso s a matr of dmeso. quato s re-wrtte usg. each row correspods to a dvdual observato, the colum of oes the matr represet the tercept term Ordar Least Squares OLS We solve for the resdual, û where the hat deotes a estmated value for the parameter. However, we use matrces
Resduals are a colum vector he objectve of the OLS estmator s to mmze the sum of the squared errors. u SSR SSR s a scalar, or a umber OLS s wrtte as.. m t w r. se matr algebra to reduce equato ppl the traspose to all terms wth parethess m Multpl matrces ' ' ' ' ' m m m he last step reles o the. Note - ths matr multplcato results a scalar.
lso ote the mddle term s lear ad the last term s quadratc Dervato of soluto. m Orgal problem to m. SSR.. ae partal dervatve Set partal to zero. Dvde both sdes b. Multpl b verse. ' ' Multpl b verse Secod order codto ae secod partal wth respect to b Ivolves a matr Do t worr, we are deed mmzg the squared resduals. lgebrac Propertes of the OLS stmator lgebrac Propert. he sum of the estmated resduals error terms s equal to zero: û.
hus, the mea equals zero lgebrac Propert. he pot, s alwas o the estmated regresso. lgebrac Propert. he sample covarace betwee each dvdual ad the OLS resdual û s equal to zero. lgebrac Propert. he mea of the varable,, wll equal the mea of the ŷ.. Fve ssumptos of the OLS stmator OLS estmator has fve assumptos ssumpto - Lear Parameters Lear parameters amples Lear parameters log amples Not lear parameters. log
Note equato ca be olear the s. Note there s olear least squares ssumpto B - Radom Sample of Observatos. he sample cossts of -pared observatos that are draw radoml from the populato. s a depedet varable, ad s are depedet varables, s or,, :. he umber of observatos s greater tha the umber of parameters to be estmated, usuall wrtte >. If =, the umber of observatos equatos wll equal the umber of uows.. he depedet varables s are ostochastc, whose values are fed. ssumpto C Zero Codtoal Mea here s o relatoshp betwee the error terms ad the depedet varables or. ssumpto D No Perfect Multcolleart Perfect Multolleart there s a eact lear relatoshp amog the depedet varables. Caot tae the verse of the matr, Correlatos could be used to determe presece of multcolleart If two varables are hghl correlated Multcolleart could stll be preset s more tha two varables are volved ssumpto - Homosedastct
he error terms all have the same varace ad are ot correlated wth each other. se the term depedet ad detcall dstrbuted d var u cov u u j ad for j where var represets the varace, cov the covarace, var u cov u u j ad for j We ca sa ~ do, u. Propertes of OLS hese propertes relate to the assumptos of OLS. OLS s a based stmator,. B addg the two assumptos B- ad C Proof ad Substtute to the estmator: sg propertes of matrces,
I. ag the epectato of both sdes of the equato, ad epected value of the error s zero, the ] [ ] [.. Gauss-Marov heorem OLS s oe of the strogest ad most used estmators for uow parameters. he Gauss-Marov heorem s Gve the assumptos, the OLS estmator s the Best Lear based stmator BL. ses all fve assumptos, Best meas the estmator gves the lowest varace for the parameter estmates Lear the estmator s lear based the estmator s ubsed If the error terms are dstrbuted ormall, ~ N, or ~ N, I, the the OLS estmator s the Best based stmator B. Note - Gauss-Marov heorem do ot mpl that OLS has the mmum varace amog all potetal estmators. Based estmators ma have smaller varace I forecastg, t s how well the estmator predcts. based stmator of - estmate the varaces of the betas
he smple formula for calculatg the varace of a radom umber s: var d d d where s the epectato operator. Statstcs reduces equato to: var d d d where - s the degrees ad freedom Lose oe pece of formato d s the mea of the observatos pplg ths cocept to the resduals var. We do ot ow the true varace, thus we estmate t: u SS OLS had parameters, thus we lose peces of formato to estmate the varaces Varace / Covarace Matr for cov w, z [ w [ w] z [ z]] [ w [ w] z [ z]] \ he varace / covarace matr s:
var cov cov cov cov var cov cov cov cov var cov cov cov cov var V where var deotes varace ad cov deotes covarace Square, smmetrc matr wth dmesos he dagoal s the varaces he varace tells how the estmated parameters var ad are gve b: V Dervato of the Varace / Covarace Matr. Start wth what we ow: ad Substtute oe equato to aother. se the matr traspose:. Substtutg these results to the covarace equato we obta:
' ' ] [ ] [ cov I I Note ssumg each error s homosedastct. hus, each error term has the same varace We estmate the varace b usg estmator,