GMM approach in asset pricing tests to estimate the premium of each risk factor

[Maxime Annet]

I am trying to apply a system-based method following the GMM approach but I have no idea how to implement it using stata. Any help would be greatly appreciated.

I am working on a new factor augmented version of the Fama-French 3-factor model. My aim is to estimate the premium and significance of each factor in my model simultaneously using GMM.


The system can be seen in the attached picture.

with i ranging from 1 to 12 for each portfolio I work on.

There are 5 sample moment conditions for each of 12 test equations as follows: (a) the mean regression error term is zero; and the regression error term is orthogonal to each regressor; namely, to rmt; SMBt; HMLt; and DEFt. In addition, there is one sample moment condition that defines each factor premium—relating to each of equations above, containing lambdas.

The parameters are the coefficients of the first equation above for all portfolios (i.e. b1,b2,....b12,s1,s2,....,s12,h1,h2,...,h12,d1,d2,. ..,d12) as well as the 4 lambdas premia ( λSMB, λHML, λDEF,λm)

Here is a small part of my data: DEF_factor is the new factor I introduce, Mkt_Rf is the excess return of the market portfolio, SMB and HML are the Fama-French factors. All other 3-letter variables correspond to the portfolios used in my analysis.

* Example generated by -dataex-. For more info, type help dataex
clear
input float(months Mkt_Rf SMB HML DEF_Factor SHh SHl SHm SLh SLl SLm BHh BHl BHm BLh BLl BLm)
507  -.08428764  -.00559345   .032425575  -.07813185   -.09492547  .018215122  -.05384039  -.03062054   -.03703001  -.05116905  -.06308237  .020132286   -.0884329   -.1184433  -.03412001   -.0828582
508  -.06315205 -.002620801   .021128025  -.10484485   -.16995627  -.06209308   -.1266427   -.2154937   -.04425187  -.16134925  -.14808667  -.07259968   -.1106821  -.27025634  -.09846856  -.12549312
509   -.1107862   .03947517    .06767985 -.027223393   -.06384285  -.07492935  -.07474729   -.1798431   -.10416538  -.11460046  -.06261309  -.06242198  -.15799314  -.21480024   -.1143972   -.2108242
510  -.11274572  -.01278664   .032860387  -.02036115   -.06284934   .02082293  -.06559436  -.07756396   -.01514564  -.12154429  -.04185171 -.020539403 -.021134976   -.1061659  -.06036251  -.06375475
511  -.05597799   .04694638    .03792885  -.12382965   -.18025766   -.0953157  -.19670726  -.28519288   -.12585828   -.2406542    -.284524  -.13522059  -.18730573   -.4247525  -.19349226   -.2272609
512   -.1727052  .010560396    .04608471   .13370809    .32251894   .01296526 .0031025305   .04141662     .0828689   .18028332   .13307826 -.008893575   .04647778     .260126   .09743346    .0242671
513   .05939813 -.007467624    .04858214   -.0124145 -.0012710652  .017550386  .017351441    -.200804   -.04413815   .15358874   .04304584   -.0526756 -.064032815  .007344689  -.00784542  -.01998902
514 -.017626012  .033545546   .005299048  -.05594885     -.094568  -.04037395   -.0822432  -.12420517    -.0692521    -.128752   -.1575013  .000733258   -.1101889  -.13551392  -.08955842  -.18057175
515    -.079896  .006123032   .028153917  -.02220239  -.030174743 -.035005312  -.06293606 -.072344825   -.05704848  -.04565262  -.09018745  -.04458411  -.04368405 -.006740072 -.022694156 -.072176725
516  -.05715353   .02923282  -.004789281  -.04751662    -.0908478 -.035295255  -.04533537  -.07675421  -.019248854  -.05787016  -.10553467 -.011164503  -.07648866   -.1324357 -.067759894  -.06460591
517 -.064474404  .005359494   .007752534  -.03234449   -.04627842   .05933344 -.015303615 -.002603471  -.012165518  -.10710526  -.05898568  -.05611139 -.017940866  -.12768039 .0004554915   .00678087
518 -.030088356   .04513578    .10802918   .09280714    .14439248   .09807388    .2041279   .16484226    .04946594   .19273894      .19169   .10080563   .10955758   .23759833   .12065089   .08189514
519   .11105676 .0012254386 -.0015531117  -.02212448    .04469951   .09605999 -.024199346  -.02428979    .03310702   .05453425 -.026247256   .02761525   .04167433 -.001939707 .0038582715  .034245998
520   .02003518   .02176811    .07190128   .06652844    .14458556   .04297268   .05236499   .05428386    .04319268   .12952381   .17888403    .0492203   .06973355   .08132066   .01059686   .04508882
521   .06124733  -.05688607    .06541783   .04658692   .021682767   .03458731  .030336613  .022342773  -.024634235   .04723833   .13537191  .033555243    .0912972   .10236192   .04219957   .04022808
522      .03758   .03195419   -.04240664  .026337083    .02918039 -.017502554   .03948096   .09825267   .015124202   .11902415    .0759192  .030651815  .016997904  .067860045  .022526823  .014354382
523   .03140718   .08937973    .03757022   .05101219    .13696127  .018665584  .068277486   .03013311       .02135  -.04987988   -.0849714 -.013344085  -.03521712   -.0780434  -.03077507  -.04288974
524    .0063791  -.01119391  .0013255086  .028412223    .02340335   .06100947   .06557961   .11172016   .024983367   .09106623   .10947536   .05305543   .08227125    .0399559    .0662006   .04545187
525   .05382015 .0028854716    .03150862  -.03586658   -.04720164 -.014998158   .04243352  .017400268  -.010204393   .02766948  .005756761  .016241942   .01491788   -.0856639  -.01580649   .02560175
526 -.005663334  -.02920251    .03026931  -.01460259   -.02874926  .007226138  -.04951923  -.04344221 .00017792462  -.05837573   .03937453 -.015704451  .001716505 -.034246746 -.016592767  -.02941052
527  -.01861251   .05008246   -.04154257 -.014751717    .08842237    .0881239   .04968941   .04095829     .1767033      .11914  .034308452   .05142882   .04691238   .09021956  .072423026   .06069831
528   .06583443  .032537013   -.02620695   .06219909    .07126294  -.00489649  .034134436   .14246863   -.02580186   .04952135  -.00907593  .004675042  .004417051 .0015182538   .02645699  .015102926
529    .0300992  .006737852 -.0027492754  -.02853841   -.08208693 -.013129297    -.044377  -.06251993  -.030761067 -.017743094  -.07295693   -.0493213  -.04384178  -.06075273  -.04923359  -.05783639
end