Lecture Notes Forecasting the process of estimating or predicting unknown situations

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Olivia Lyons
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1 Lecture Notes. Ecoomc Forecastg. Forecastg the process of estmatg or predctg ukow stuatos Eample usuall ecoomsts predct future ecoomc varables Forecastg apples to a varet of data () tme seres data predctg future data pots tme Eamples uemplomet, terest rates, echage rates etc. () cross-sectoal data predctg wth a class of a varable take at oe pot tme Eamples Usuall assocated wth households, dvduals, busesses, etc. How ma busesses bakrupted 9? How ma people are lvg povert 9? (3) pael data combes cross-sectoal ad tme seres data Eample eamg uemplomet over tme several regos Thus, forecastg s a eormous subject. Tme-seres aloe ca use advaced math ad s qute a comple topc Combes statstcs, calculus, ad ecoomcs Eamples ARIMA modelg ad Vector Auto Regressos. Tools for forecastg Forecastg etesvel uses ecoometrcs Foudato of ecoometrcs s least squares

2 regresso Also called ordar least squares, regresso, or multple () Method u,,, where ad are pared observatos or varables ad are ukow parameters s the total umber of observatos u s the error term assocated wth observato Error s also called stochastc, whte ose, or radom Has a epected value of zero Tme seres data the error s almost ever radom Istead the call t a ovato The error term cotas formato Whe we estmate the parameters, the are deoted b hats,,, The plot s below wth the data pots

3 We wat to mmze the errors Some errors are postve Other errors are egatve We caot add the errors because the ma cacel We square the errors terms to make them all postve () Dervato Startg wth the equato u,,, Solve for u t, whch elds u Square the errors to make them postve u Ths s ol for oe data pot. We wat to mmze the total errors of all the data pots. Sum over all the data pots Defe Sum of Squared Errors () u We wat to fd the mmum, thus we take the frst partal dervatves wth respect to the betas 3

4 4 The secod step s the Cha Rule from Calculus I ca put the frot of the summato because each term the summato has a Set the partal dervatve to zero, order to fd mmum value Now solve equato for, It s debatable whe ou should add hats to the estmators I added at ths step whe partal was set to zero Summato s a lear operator We ca appl the summato to all terms parethess s costat that s summed tmes s costat ad multpled b all s summato

5 5 Ca brg ths to the frot The last step works because we substtute the average for ad average for to the equato Repeatg these steps to get the estmator for Smlarl, set the partal to zero ad solve for, We substtute the estmator for to the equato,, whch elds

6 6 I dd ot break the last summato apart Ths s to solve for the estmator for What f we had the graph below

7 Whch le s the best ft? We could use algebra. Ths s ot stochastc, sce there s ol two pots. Desrable Propertes of Estmators No sgle estmator domates all other estmators terms of these propertes u,,, Estmators of ad are radom. Deped o the sample of s ad s. The have a probablt dstrbuto. Ubased A estmator s ubas whe the dstrbuto of the estmator has the true value of the parameter as ts mea value: (7) E ( ) where E s the epectato operator epected value s the mea 7

8 β s the true parameter value s the estmate of the parameter. E ( ) A estmator s bas, f the mea of the dstrbuto of the estmator does ot equal the true value. Mathematcall, ths s wrtte as: (8) ( ) E. The amout of bas s gve b: (9) bas () E( ) The desrablt of a ubased estmator s that the mea of the estmator s dstrbuto s cetered aroud the true parameter value. 8

9 . Effcec refers to varace Varace s how spread out the data s s a more effcet estmator tha ~, f the sample varace of s smaller tha the varace of ~. Ths s dffcult to determe the most effcet estmator. The based estmator has a smaller varace 3. Mmum Mea Squared Error accouts for both bas ad effcec MSE s defed as: () MSE bas( ) var( ) Bas s could be postve or egatve. That s wh t s squared. If the estmator s ubased, the the MSE = ) Eample choose the better estmator Estmator Estmator Bas Varace var(. Remember we are forecastg. Who cares f the estmator s based. 4. Cosstec large sample propertes As the sample sze approaches ft, the estmator collapse to the true parameter value Also called asmptotc

Econometric Methods. Review of Estimation

Econometric Methods. Review of Estimation Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators