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1 Rockefeller College Unverst at Alban PAD 705 Handout: Maxmum Lkelhood Estmaton Orgnal b Davd A. Wse John F. Kenned School of Government, Harvard Unverst Modfcatons b R. Karl Rethemeer Up to ths pont n the course, we have focused on usng the ordnar least squares model. OLS estmatons mnmze the squared dfferences between the actual ponts and a lne that passes through them. To be vald, the fve GaussMarkov condtons must be met. When one or more of the condtons are not met, we have generall found correctons that repar the problem. However, there are some tpes of GaussMarkov falures that cannot be corrected wthn the OLS framework. The most common falure that cannot be fxed n OLS s when the assumpton of normall dstrbuted error term cannot be mantaned. Regressons on qualtatve dependent varables are one promnent example of ths problem. Another s when there s truncaton or censorng of the error term. Truncaton and censorng occur when b nature or b polc the dependent varable cannot assume an value from negatve to postve nfnt (at least n theor. In the face of these falures, we must seek a dfferent form of estmaton: maxmum lkelhood estmaton (MLE. As we shall see, MLE s an extraordnarl flexble framework. It wll also turn out that OLS s a form of maxmum lkelhood estmaton. The Lnear Model and OLS Recall frst the standard lnear model: where ε s a dsturbance term and X β s shorthand for: X β + ε X β β + β X 0 + β X β k X k. It s mportant to keep n mnd that ths model smpl sas that the expected value of Y depends on X so that: and that the varance of, gven X, s: E( X X β Var( X Var( ε σ ε Ths s the varance of the dsturbance term. In other words, the varance of gven X s smpl the varance of the dsturbance term gven X. The emprcal counterpart s the resdual varance, whch s the varance around the estmated regresson lne. If we assume that the dsturbance term s normall dstrbuted wth the same varance for each, we can descrbe ths model b: ( X ~ ( X β, σ ε. Revsed: Aprl 0, 00

2 Maxmum Lkelhood Estmaton The sold lne n the fgure just below llustrates the expected value of gven X, that s E( X Xβ. The dsturbance term s assumed to be normall dstrbuted wth a varance equal to 4 and the sample values (the dots were randoml selected from ths dstrbuton. The varaton of the dots around the expected value of represents the varance of gven X. The estmated regresson lne s the dashed lne. The actual expected value of gven X s E( X 0 + *X. The estmated regresson lne s gven b *X, whch dffers somewhat from the true Apopulaton@ relatonshp between X and. The ordnar least squares estmate of β s found b mnmzng the sum of squared resduals and s gven b: Mn e ( X β ( The Maxmum Lkelhood Method The lnear model as set out above can be estmated b another method called maxmum lkelhood estmaton (MLE. But before settng out ths method for the general model above, we wll consder a smpler example. Recall that we are often concerned that estmators (e.g., the estmated mean, coeffcents n a regresson, etc. have partcular propertes, n partcular that the be unbased and have small varance. (Here the reference s to the varance of an estmate, not the resdual varance as mentoned above. In ths respect, maxmum lkelhood

3 Maxmum Lkelhood Estmaton estmates are ver attractve. Whle often based n small samples, the become unbased as the number of data ponts used n ther calculaton ncreases. Furthermore, these estmators have the smallest possble varance. More formall, the attan what s known as the CramerRao lower bound. o other possble estmator has a varance smaller than the CramerRao lower bound, and thus no estmator has a varance smaller than that of the maxmum lkelhood estmator. Whle maxmum lkelhood estmaton s a powerful technque, t does requre mportant a pror nformaton about the problem. Whle maxmum lkelhood alwas gves us the Abest@ estmator, we can onl use the technque f we know, or are wllng to assume, the dstrbuton followed b the observed data. Ths s a strong requrement, but one that we ma be comfortable wth n a wde varet of stuatons. And, n man cases nvolvng nonlnear models (such as qualtatve choce specfcatons, whch cannot be estmated usng ordnar least squares lnear models, we have no choce but to assume that the data follow a partcular dstrbuton. ow, we wll dscuss the basc deas behnd maxmum lkelhood estmaton and then turn to calculatng maxmum lkelhood estmators n practce. The basc prncpal behnd maxmum lkelhood estmaton s smple. Suppose that we observe a number of random draws from a known algebrac dstrbuton wth an unknown parameter. Ths parameter mght be the mean µ of a normal dstrbuton, or the parameter λ of the Posson dstrbuton, or a β from a multple regresson. We wsh to use the data, and the knowledge of the underlng algebrac dstrbuton, to estmate the unknown parameter. Maxmum lkelhood procedures estmate an unknown populaton parameter (.e., µ, λ, β, etc. wth the value of that parameter that s most lkel to have generated the observed sample. Let s elaborate on ths last, somewhat confusng statement. Consder a ver smple example: standardzed test scores. Let s assume the scores are drawn from a normal dstrbuton wth unknown mean and standard devaton 00. We pck a sngle test score at random from the group of examnees and wsh to estmate the mean of the underlng normal dstrbuton from whch we are drawng. Suppose the score chosen at random s 00. What s our maxmum lkelhood estmate of the populaton parameter µ? It s smpl 00 because, based on the rather lmted sample at our dsposal, the most lkel dstrbuton to have generated the observaton at 00 s a dstrbuton wth mean 00. As shown n the fgure below, a normal dstrbuton wth mean 00 s more lkel (n fact, eleven tmes more lkel to have generated the observaton at 00 than a normal dstrbuton wth mean 800 or mean 600. And workng wth a sample of one, we can construct no estmator whch s n an sense Abetter@ than ths maxmum lkelhood estmator.

4 Maxmum Lkelhood Estmaton ow consder the (slghtl more reasonable case of pckng more than a sngle observaton to use n calculatng a maxmum lkelhood estmator. To do ths, we wll need to return to the probablt dstrbuton functon (PDF for a normal dstrbuton. The PDF for a dstrbuton wth mean µ and varance 00 s: ( µ f( e σ π σ ( µ e 00 π * 00 Suppose we draw three examnees from the group wth scores, and. Snce these are ndependent draws, the jont PDF. for the three draws s smpl the product of the PDF's for each ndvdual draw: 4

5 Maxmum Lkelhood Estmaton f(,, f( * f( * f( ( µ e 00 π * 00 * ( µ e 00 π * 00 * ( µ e 00 π * 00 otce what we have wrtten s a functon whch gves the probablt of observng the three observatons we drew,, and, as a functon of the unknown populaton parameter µ. Such a jont probablt functon for a group of observed values s referred to as a "lkelhood functon" and s usuall denoted (n the equatons, I wll use L snce Mcrosoft Equaton.0 does not support. To fnd the maxmum lkelhood estmator, or MLE, we need onl to maxmze ths expresson over all possble values of µ. In other words, we fnd the µ whch maxmzes the probablt that we observe the partcular sample we have n fact drawn. Ths s qualtatvel no dfferent than what we showed graphcall whle workng wth onl a sngle observaton. To mplement ths maxmzaton, we take advantage of the fact that the logarthm of a functon s what s called a monotone transformaton. Ths means smpl that f A s greater than B, log(a s greater than log(b. In ths case, we wll use the fact to argue that maxmzng the logarthm of the lkelhood functon, or log lkelhood, gves the same answer as maxmzng the lkelhood functon tself. But the mathematcal problem of maxmzng a log lkelhood s much easer to solve. Ths results drectl from the propert that the logarthm of a product s the sum of two factors: So we wsh to maxmze: Log (A* B Log A+ Log B ln L ln(00 π ( µ (00 ( µ (00 ( µ (00 We take the dervatve of ths rather awful lookng expresson wth regard to the parameter µ: dln d _ µ µ µ µ Settng ths expresson equal to zero gves the frst order condton for a maxmum: µ µ µ 0 5

6 Maxmum Lkelhood Estmaton Solvng ths expresson for the maxmum lkelhood estmator, whch we denote here as µˆ : ˆ µ ˆ µ Y So the maxmum lkelhood estmator for ths problem s smpl the sample mean. After crawlng through all ths notaton, we stll end up wth our usual choce of estmator for the populaton mean, the sample mean. But we can now sa that ths s not onl the obvous, but also the best estmator, at least as the sample sze grows large. In the above example, we assumed that the varance of the normal dstrbuton was known. That assumpton s not necessar, although t helps smplf the algebra. We could have wrtten a lkelhood functon n terms of two unknown parameter, µ and σ, and maxmzed the lkelhood functon over both of these. An Exponental Example Consder another example of maxmum lkelhood estmaton, ths tme nvolvng the exponental dstrbuton. Ths probablt dstrbuton s often used to model the watng tme between consecutve events such as automoble accdents, presdental vetoes, and patent applcatons. The PDF of an exponental random varable s gven b: f( e Suppose we wsh to estmate the parameter the mean of the dstrbuton usng maxmum lkelhood and three data ponts:.0, 4.0 and.6. Obvousl, n a real applcaton we would want more than three ponts, but here an unrealstcall small number makes t possble to show what s gong on graphcall, and leaves the basc stor unchanged. The fgure below shows the P.D.F.s for several exponental dstrbutons wth varng values for the parameter. The maxmum lkelhood procedure searches for the value of whch s most lkel to generate the three observed data ponts. otce that the two ponts to the rght are more lkel to be drawn f s larger : 6

7 Maxmum Lkelhood Estmaton 0.4 Theta 0. Theta F(Y 0. Theta Y ow that we understand the basc problem graphcall, we can proceed analtcall. Frst, we wrte the lkelhood functon, or jont PDF for the three draws from the exponental wth unknown parameter : e L F(,, * e * e e ext, we maxmze ths lkelhood functon wth respect to, the parameter we wsh to estmate. But frst, takng the log of the lkelhood functon makes the algebra easer: ln L ln Takng the dervatve of the lkelhood functon wth respect to : dln L + + d ow set ths expresson equal to zero to fnd a frst order condton whch can be solved for the maxmum lkelhood estmator : 7

8 Maxmum Lkelhood Estmaton 0 + ˆ ˆ ˆ ˆ ˆ ˆ Y. Once agan, we fnd that the sample mean s the maxmum lkelhood estmator for. The General Lnear Model As t turns out, the least squares estmator s n fact a maxmum lkelhood estmator. Remember that we presented regresson as a wa of estmatng the expected value of a (random varable Y, when the expected value vares wth another varable X. We descrbed the relatonshp as: E( X X β Just above, we derved the MLE of the mean µ of a normall dstrbuted random varable Y. We wrote the PDF of each realzaton as: ( f( e σ σ π Maxmum lkelhood estmaton of the general lnear model can easl be understood f we replace µ wth X β and we assume that each s dstrbuted normall and condtonal on X wth mean X β and varance σ. As above, the model can be wrtten n standard statstcal notaton as ( X ~ (X β, σ. Then the PDF of condtonal upon X can be wrtten: µ ( X f( X e σ σ π Once we have the correct PDF, we can wrte the lkelhood functon for a sample of ndependent realzatons of X and Y: β 8

9 Maxmum Lkelhood Estmaton L f( X, X,..., X f( X * f( X *...* f( X σ π e ( X β σ ow, we take the logarthm: ln L ln( πσ σ ( X β The lkelhood functon s maxmzed wth respect to β and σ. otce that maxmzng ths expresson over possble values of the maxmum lkelhood estmator for βˆ nvolves pckng the βˆ that mnmzes the expresson ( X β summed over all observatons. Ths s, of course, just the famlar crtera weve alread used n defnng the least squares estmator above. So, least squares regresson estmates and maxmum lkelhood estmates for β are the same, assumng that the dstrbuton of Y gven X s normal (or, equvalentl, that the dsturbance term ε s normall dstrbuted. In practce, both β and σ would be estmated. The values would be found b takng the dervatve of ln wth respect to β and σ, settng the results equal to zero, and solvng for the estmated values of β and σ. To smplf, suppose that there s onl one X. Then d ln L [ X X β ] dβ σ Settng ths expresson equal to zero and solvng for β we fnd βˆ whch s n fact the least squares estmate. Smlarl, t can be shown that the MLE estmate of σ s gven b ˆ σ X X ( X β whch n large samples s the same as the least squares estmate. (Recall that the least squares estmate dvdes b k, where k s the number of parameters. 9

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