FOCUS AREAS

SAS/ETS Examples

Heteroscedastic Modeling of the Fed Funds Rate


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SAS Program

/*
Effective Federal Funds Rate
Averages of Daily Figures
Percent
Source: H.15 Release -- Federal Reserve Board of Governors
*/

data weeklyFedFunds;
   input Date yymmdd8. FFRate;
   format Date date7.;
   label Date   = 'Date'
         FFRate = 'Effective Federal Funds Rate, Averages of Daily Figures';
datalines;
19610104     2.50
19610111     1.61
19610118     1.96
19610125     0.66
19610201     1.36
19610208     2.25
19610215     2.79
19610222     2.54
19610301     2.50
19610308     2.61
19610315     2.00
19610322     1.82
19610329     1.61
19610405     1.79
19610412     2.11
19610419     1.64
19610426     1.14
19610503     0.54
19610510     1.64
19610517     2.29
19610524     2.50
19610531     2.14
19610607     2.50
19610614     1.00
19610621     2.29
19610628     1.18
19610705     1.04
19610712     1.36
19610719     1.18
19610726     0.86
19610802     1.64
19610809     2.25
19610816     2.75
19610823     1.64
19610830     1.50
19610906     1.29
19610913     2.36
19610920     1.86
19610927     1.96
19611004     1.64
19611011     2.43
19611018     2.64
19611025     2.36
19611101     2.14
19611108     2.71
19611115     2.75
19611122     2.79
19611129     2.29
19611206     2.25
19611213     1.61
19611220     2.36
19611227     2.86
19620103     2.61
19620110     2.00
19620117     2.79
19620124     1.46
19620131     2.14
19620207     2.04
19620214     2.61
19620221     2.36
19620228     2.46
19620307     2.75
19620314     2.89
19620321     2.82
19620328     2.96
19620404     2.71
19620411     2.79
19620418     2.93
19620425     2.86
19620502     2.61
19620509     2.71
19620516     2.46
19620523     2.43
19620530     1.71
19620606     2.46
19620613     2.46
19620620     2.86
19620627     2.89
19620704     2.82
19620711     2.71
19620718     2.86
19620725     2.50
19620801     2.75
19620808     2.89
19620815     3.00
19620822     2.89
19620829     2.93
19620905     2.96
19620912     3.00
19620919     2.89
19620926     2.79
19621003     2.89
19621010     2.82
19621017     3.00
19621024     2.89
19621031     2.82
19621107     2.95
19621114     3.00
19621121     2.96
19621128     2.89
19621205     2.79
19621212     2.88
19621219     2.98
19621226     3.00
19630102     2.93
19630109     2.77
19630116     3.00
19630123     2.95
19630130     2.98
19630206     3.00
19630213     3.00
19630220     3.00
19630227     3.00
19630306     3.00
19630313     3.00
19630320     2.95
19630327     2.98
19630403     2.75
19630410     2.98
19630417     2.89
19630424     3.00
19630501     2.95
19630508     3.00
19630515     3.00
19630522     3.00
19630529     3.00
19630605     3.00
19630612     3.00
19630619     3.00
19630626     2.96
19630703     3.00
19630710     2.98
19630717     3.04
19630724     2.64
19630731     3.41
19630807     3.50
19630814     3.46
19630821     3.50
19630828     3.48
19630904     3.50
19630911     3.50
19630918     3.43
19630925     3.48
19631002     3.50
19631009     3.48
19631016     3.50
19631023     3.50
19631030     3.50
19631106     3.50
19631113     3.43
19631120     3.50
19631127     3.50
19631204     3.32
19631211     3.16
19631218     3.50
19631225     3.50
19640101     3.43
19640108     3.50
19640115     3.50
19640122     3.46
19640129     3.50
19640205     3.50
19640212     3.50
19640219     3.43
19640226     3.50
19640304     3.29
19640311     3.50
19640318     3.43
19640325     3.46
19640401     3.50
19640408     3.45
19640415     3.50
19640422     3.46
19640429     3.45
19640506     3.50
19640513     3.50
19640520     3.50
19640527     3.48
19640603     3.50
19640610     3.50
19640617     3.50
19640624     3.50
19640701     3.50
19640708     3.50
19640715     3.50
19640722     3.14
19640729     3.50
19640805     3.50
19640812     3.50
19640819     3.50
19640826     3.50
19640902     3.50
19640909     3.43
19640916     3.46
19640923     3.50
19640930     3.39
19641007     3.36
19641014     3.07
19641021     3.48
19641028     3.46
19641104     3.50
19641111     3.50
19641118     3.59
19641125     3.34
19641202     3.71
19641209     3.39
19641216     4.00
19641223     4.00
19641230     4.00
19650106     4.00
19650113     4.00
19650120     3.57
19650127     4.00
19650203     4.00
19650210     4.00
19650217     4.00
19650224     3.93
19650303     4.00
19650310     4.05
19650317     4.04
19650324     4.09
19650331     4.02
19650407     4.11
19650414     4.13
19650421     4.09
19650428     4.02
19650505     4.13
19650512     4.07
19650519     4.07
19650526     4.11
19650602     4.11
19650609     4.11
19650616     4.13
19650623     3.80
19650630     4.13
19650707     4.13
19650714     4.09
19650721     4.04
19650728     4.09
19650804     4.11
19650811     4.13
19650818     4.13
19650825     4.13
19650901     4.05
19650908     4.11
19650915     4.13
19650922     4.13
19650929     3.71
19651006     4.09
19651013     4.13
19651020     4.07
19651027     4.00
19651103     4.13
19651110     4.04
19651117     4.13
19651124     4.11
19651201     4.13
19651208     3.68
19651215     4.50
19651222     4.43
19651229     4.63
19660105     4.61
19660112     4.59
19660119     3.86
19660126     4.55
19660202     4.54
19660209     4.63
19660216     4.59
19660223     4.63
19660302     4.63
19660309     4.68
19660316     4.57
19660323     4.68
19660330     4.68
19660406     4.75
19660413     4.70
19660420     4.77
19660427     4.48
19660504     4.73
19660511     4.79
19660518     4.98
19660525     4.84
19660601     4.95
19660608     4.98
19660615     5.18
19660622     5.18
19660629     5.43
19660706     5.36
19660713     5.45
19660720     4.70
19660727     5.46
19660803     5.63
19660810     5.75
19660817     5.36
19660824     5.71
19660831     5.34
19660907     5.89
19660914     5.80
19660921     5.29
19660928     4.50
19661005     5.86
19661012     5.50
19661019     5.64
19661026     5.04
19661102     5.86
19661109     5.82
19661116     5.96
19661123     5.50
19661130     5.64
19661207     5.36
19661214     5.43
19661221     5.25
19661228     5.57
19670104     5.14
19670111     5.39
19670118     4.82
19670125     5.18
19670201     4.21
19670208     4.93
19670215     5.18
19670222     5.18
19670301     4.68
19670308     4.71
19670315     4.54
19670322     4.71
19670329     4.25
19670405     4.55
19670412     3.93
19670419     3.93
19670426     4.00
19670503     4.00
19670510     3.84
19670517     4.02
19670524     3.93
19670531     3.95
19670607     3.93
19670614     4.00
19670621     3.98
19670628     4.07
19670705     3.73
19670712     3.98
19670719     3.54
19670726     3.93
19670802     3.75
19670809     4.02
19670816     4.05
19670823     3.98
19670830     3.59
19670906     4.02
19670913     3.98
19670920     4.00
19670927     4.00
19671004     3.96
19671011     4.04
19671018     4.04
19671025     3.50
19671101     3.89
19671108     3.96
19671115     4.09
19671122     4.02
19671129     4.38
19671206     4.36
19671213     4.52
19671220     4.48
19671227     4.63
19680103     4.54
19680110     4.63
19680117     4.61
19680124     4.70
19680131     4.55
19680207     4.73
19680214     4.66
19680221     4.70
19680228     4.75
19680306     4.82
19680313     4.64
19680320     5.13
19680327     5.39
19680403     5.52
19680410     5.66
19680417     5.71
19680424     5.63
19680501     6.14
19680508     6.04
19680515     6.34
19680522     6.07
19680529     6.02
19680605     5.93
19680612     6.18
19680619     6.18
19680626     6.14
19680703     5.73
19680710     5.95
19680717     6.13
19680724     6.11
19680731     6.05
19680807     6.11
19680814     6.13
19680821     6.05
19680828     5.93
19680904     5.80
19680911     5.84
19680918     5.68
19680925     5.70
19681002     5.93
19681009     5.89
19681016     5.96
19681023     5.84
19681030     5.91
19681106     6.07
19681113     6.00
19681120     5.45
19681127     5.73
19681204     5.71
19681211     5.84
19681218     6.00
19681225     6.25
19690101     5.95
19690108     6.43
19690115     6.36
19690122     6.46
19690129     6.27
19690205     6.32
19690212     6.75
19690219     6.75
19690226     6.48
19690305     6.75
19690312     6.75
19690319     6.82
19690326     6.88
19690402     6.66
19690409     7.04
19690416     7.63
19690423     7.48
19690430     7.79
19690507     8.23
19690514     8.30
19690521     8.91
19690528     8.93
19690604     9.20
19690611     9.13
19690618     8.54
19690625     8.34
19690702     9.00
19690709     9.07
19690716     9.23
19690723     8.50
19690730     8.05
19690806     9.57
19690813     9.18
19690820     8.79
19690827     8.82
19690903     9.57
19690910     8.57
19690917     9.07
19690924     9.61
19691001     9.11
19691008     9.43
19691015     9.68
19691022     8.68
19691029     8.39
19691105     9.07
19691112     9.32
19691119     8.79
19691126     8.32
19691203     8.91
19691210     8.75
19691217     9.14
19691224     9.18
19691231     8.71
19700107     8.45
19700114     8.96
19700121     9.30
19700128     9.04
19700204     9.21
19700211     9.18
19700218     9.39
19700225     8.41
19700304     8.32
19700311     7.71
19700318     7.82
19700325     7.45
19700401     7.93
19700408     7.68
19700415     8.02
19700422     8.21
19700429     8.43
19700506     8.46
19700513     7.96
19700520     7.84
19700527     7.64
19700603     7.84
19700610     7.98
19700617     7.80
19700624     7.21
19700701     7.23
19700708     7.34
19700715     7.59
19700722     7.16
19700729     6.89
19700805     6.93
19700812     6.82
19700819     6.64
19700826     6.34
19700902     6.46
19700909     6.64
19700916     6.46
19700923     5.84
19700930     6.05
19701007     6.36
19701014     6.21
19701021     6.18
19701028     6.11
19701104     6.07
19701111     5.80
19701118     5.70
19701125     5.16
19701202     5.50
19701209     4.91
19701216     5.07
19701223     4.84
19701230     4.82
19710106     3.82
19710113     4.27
19710120     4.13
19710127     4.23
19710203     4.09
19710210     3.59
19710217     4.14
19710224     3.46
19710303     3.41
19710310     3.29
19710317     3.93
19710324     3.70
19710331     4.02
19710407     3.98
19710414     4.20
19710421     4.27
19710428     4.14
19710505     4.41
19710512     4.59
19710519     4.55
19710526     4.68
19710602     4.82
19710609     4.77
19710616     4.89
19710623     4.96
19710630     5.07
19710707     5.18
19710714     5.13
19710721     5.46
19710728     5.38
19710804     5.57
19710811     5.59
19710818     5.59
19710825     5.48
19710901     5.59
19710908     5.73
19710915     5.59
19710922     5.46
19710929     5.43
19711006     5.32
19711013     5.29
19711020     5.14
19711027     5.11
19711103     5.16
19711110     4.93
19711117     4.88
19711124     4.86
19711201     4.68
19711208     4.59
19711215     4.20
19711222     3.89
19711229     4.05
19720105     3.57
19720112     3.71
19720119     3.54
19720126     3.43
19720202     3.23
19720209     3.25
19720216     3.43
19720223     3.34
19720301     3.18
19720308     3.43
19720315     3.88
19720322     3.91
19720329     4.09
19720405     4.16
19720412     4.18
19720419     4.04
19720426     4.20
19720503     4.25
19720510     4.20
19720517     4.32
19720524     4.24
19720531     4.38
19720607     4.48
19720614     4.46
19720621     4.39
19720628     4.49
19720705     4.61
19720712     4.62
19720719     4.46
19720726     4.54
19720802     4.56
19720809     4.69
19720816     4.87
19720823     4.75
19720830     4.90
19720906     4.89
19720913     4.69
19720920     4.93
19720927     4.99
19721004     5.15
19721011     5.09
19721018     4.91
19721025     5.01
19721101     5.06
19721108     5.25
19721115     4.89
19721122     4.97
19721129     5.03
19721206     5.17
19721213     5.29
19721220     5.38
19721227     5.34
19730103     5.61
19730110     5.66
19730117     5.86
19730124     6.03
19730131     6.35
19730207     6.21
19730214     6.58
19730221     6.79
19730228     6.75
19730307     7.02
19730314     7.13
19730321     6.96
19730328     7.11
19730404     7.18
19730411     6.84
19730418     7.23
19730425     7.14
19730502     7.43
19730509     7.60
19730516     7.81
19730523     8.06
19730530     7.95
19730606     8.43
19730613     8.17
19730620     8.55
19730627     8.59
19730704    10.21
19730711     9.52
19730718    10.22
19730725    10.58
19730801    10.57
19730808    10.39
19730815    10.39
19730822    10.52
19730829    10.79
19730905    10.79
19730912    10.74
19730919    10.80
19730926    10.84
19731003    10.72
19731010     9.87
19731017    10.07
19731024     9.98
19731031     9.90
19731107     9.71
19731114    10.03
19731121    10.23
19731128    10.09
19731205    10.17
19731212    10.04
19731219    10.18
19731226     9.52
19740102     9.87
19740109     9.76
19740116     9.77
19740123     9.60
19740130     9.47
19740206     9.13
19740213     8.93
19740220     9.07
19740227     8.81
19740306     8.98
19740313     9.03
19740320     9.33
19740327     9.61
19740403     9.93
19740410    10.02
19740417    10.36
19740424    10.78
19740501    11.17
19740508    11.29
19740515    11.46
19740522    10.95
19740529    11.54
19740605    11.45
19740612    11.60
19740619    11.85
19740626    11.97
19740703    13.55
19740710    13.34
19740717    13.04
19740724    12.60
19740731    12.29
19740807    12.09
19740814    12.02
19740821    12.23
19740828    11.84
19740904    11.64
19740911    11.48
19740918    11.41
19740925    11.12
19741002    11.04
19741009    10.43
19741016    10.11
19741023     9.81
19741030     9.72
19741106     9.63
19741113     9.37
19741120     9.34
19741127     9.46
19741204     9.02
19741211     8.86
19741218     8.72
19741225     8.45
19750101     7.35
19750108     7.70
19750115     7.22
19750122     7.17
19750129     6.99
19750205     6.46
19750212     6.28
19750219     6.29
19750226     6.15
19750305     5.88
19750312     5.44
19750319     5.38
19750326     5.53
19750402     5.59
19750409     5.28
19750416     5.44
19750423     5.54
19750430     5.71
19750507     5.42
19750514     5.20
19750521     5.13
19750528     5.14
19750604     5.24
19750611     5.15
19750618     5.31
19750625     5.72
19750702     6.31
19750709     6.06
19750716     5.93
19750723     6.14
19750730     6.25
19750806     6.09
19750813     6.08
19750820     6.15
19750827     6.23
19750903     6.06
19750910     6.15
19750917     6.28
19750924     6.29
19751001     6.36
19751008     6.06
19751015     5.82
19751022     5.73
19751029     5.65
19751105     5.17
19751112     5.24
19751119     5.24
19751126     5.28
19751203     5.25
19751210     5.26
19751217     5.17
19751224     5.18
19751231     5.18
19760107     5.12
19760114     4.76
19760121     4.81
19760128     4.80
19760204     4.82
19760211     4.73
19760218     4.70
19760225     4.80
19760303     4.95
19760310     4.86
19760317     4.77
19760324     4.79
19760331     4.84
19760407     4.73
19760414     4.77
19760421     4.78
19760428     4.93
19760505     5.03
19760512     5.02
19760519     5.28
19760526     5.50
19760602     5.54
19760609     5.44
19760616     5.47
19760623     5.48
19760630     5.58
19760707     5.37
19760714     5.27
19760721     5.30
19760728     5.28
19760804     5.36
19760811     5.25
19760818     5.29
19760825     5.28
19760901     5.28
19760908     5.25
19760915     5.22
19760922     5.21
19760929     5.32
19761006     5.17
19761013     5.02
19761020     4.97
19761027     4.99
19761103     5.06
19761110     4.98
19761117     5.02
19761124     4.90
19761201     4.78
19761208     4.67
19761215     4.68
19761222     4.63
19761229     4.66
19770105     4.47
19770112     4.55
19770119     4.65
19770126     4.72
19770202     4.60
19770209     4.66
19770216     4.70
19770223     4.74
19770302     4.68
19770309     4.63
19770316     4.62
19770323     4.77
19770330     4.74
19770406     4.60
19770413     4.65
19770420     4.71
19770427     4.82
19770504     5.15
19770511     5.31
19770518     5.34
19770525     5.45
19770601     5.36
19770608     5.31
19770615     5.37
19770622     5.43
19770629     5.43
19770706     5.35
19770713     5.33
19770720     5.35
19770727     5.45
19770803     5.80
19770810     5.70
19770817     5.94
19770824     5.99
19770831     6.02
19770907     5.97
19770914     6.05
19770921     6.10
19770928     6.35
19771005     6.41
19771012     6.41
19771019     6.50
19771026     6.49
19771102     6.50
19771109     6.58
19771116     6.42
19771123     6.51
19771130     6.55
19771207     6.51
19771214     6.49
19771221     6.54
19771228     6.65
19780104     6.69
19780111     6.58
19780118     6.78
19780125     6.72
19780201     6.80
19780208     6.75
19780215     6.76
19780222     6.78
19780301     6.80
19780308     6.76
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19950809     5.73
19950816     5.74
19950823     5.70
19950830     5.71
19950906     5.77
19950913     5.73
19950920     5.78
19950927     5.80
19951004     6.00
19951011     5.72
19951018     5.71
19951025     5.76
19951101     5.76
19951108     5.71
19951115     5.74
19951122     5.81
19951129     5.91
19951206     5.75
19951213     5.73
19951220     5.90
19951227     5.48
19960103     5.35
19960110     5.53
19960117     5.61
19960124     5.44
19960131     5.53
19960207     5.21
19960214     5.09
19960221     5.17
19960228     5.31
19960306     5.57
19960313     5.24
19960320     5.36
19960327     5.22
19960403     5.30
19960410     5.08
19960417     5.24
19960424     5.24
19960501     5.30
19960508     5.22
19960515     5.26
19960522     5.22
19960529     5.19
19960605     5.33
19960612     5.24
19960619     5.45
19960626     5.21
19960703     5.53
19960710     5.26
19960717     5.23
19960724     5.25
19960731     5.53
19960807     5.38
19960814     5.10
19960821     5.18
19960828     5.21
19960904     5.39
19960911     5.16
19960918     5.22
19960925     5.34
19961002     5.40
19961009     5.14
19961016     5.22
19961023     5.22
19961030     5.27
19961106     5.32
19961113     5.21
19961120     5.41
19961127     5.30
19961204     5.52
19961211     5.22
19961218     5.38
19961225     5.18
19970101     5.37
19970108     5.28
19970115     5.19
19970122     5.19
19970129     5.18
19970205     5.30
19970212     5.05
19970219     5.22
19970226     5.16
19970305     5.36
19970312     5.19
19970319     5.26
19970326     5.40
19970402     5.86
19970409     5.37
19970416     5.48
19970423     5.48
19970430     5.61
19970507     5.55
19970514     5.49
19970521     5.52
19970528     5.43
19970604     5.54
19970611     5.48
19970618     5.62
19970625     5.42
19970702     5.82
19970709     5.48
19970716     5.44
19970723     5.43
19970730     5.57
19970806     5.62
19970813     5.45
19970820     5.59
19970827     5.56
19970903     5.64
19970910     5.48
19970917     5.58
19970924     5.45
19971001     5.58
19971008     5.46
19971015     5.45
19971022     5.54
19971029     5.50
19971105     5.60
19971112     5.50
19971119     5.51
19971126     5.49
19971203     5.58
19971210     5.40
19971217     5.66
19971224     5.44
19971231     5.45
19980107     5.74
19980114     5.45
19980121     5.53
19980128     5.53
19980204     5.52
19980211     5.43
19980218     5.54
19980225     5.51
19980304     5.60
19980311     5.45
19980318     5.47
19980325     5.43
19980401     5.60
19980408     5.48
19980415     5.47
19980422     5.37
19980429     5.40
19980506     5.35
19980513     5.49
19980520     5.60
19980527     5.45
19980603     5.63
19980610     5.43
19980617     5.58
19980624     5.42
19980701     5.88
19980708     5.47
19980715     5.49
19980722     5.50
19980729     5.54
19980805     5.61
19980812     5.50
19980819     5.59
19980826     5.48
19980902     5.61
19980909     5.47
19980916     5.54
19980923     5.42
19980930     5.58
19981007     5.22
19981014     5.14
19981021     4.87
19981028     4.95
19981104     5.22
19981111     4.80
19981118     4.89
19981125     4.54
19981202     4.86
19981209     4.68
19981216     4.97
19981223     4.69
19981230     4.48
19990106     4.30
19990113     4.75
19990120     4.64
19990127     4.66
19990203     4.75
19990210     4.77
19990217     4.75
19990224     4.75
19990303     4.85
19990310     4.80
19990317     4.79
19990324     4.79
19990331     4.84
19990407     4.80
19990414     4.68
19990421     4.61
19990428     4.79
19990505     4.90
19990512     4.70
19990519     4.76
19990526     4.73
19990602     4.65
19990609     4.71
19990616     4.73
19990623     4.71
19990630     4.95
19990707     5.00
19990714     4.97
19990721     4.96
19990728     5.01
19990804     5.06
19990811     4.96
19990818     5.03
19990825     5.02
19990901     5.34
19990908     5.16
19990915     5.24
19990922     5.16
19990929     5.27
19991006     5.27
19991013     5.17
19991020     5.18
19991027     5.18
19991103     5.27
19991110     5.20
19991117     5.44
19991124     5.52
19991201     5.63
19991208     5.45
19991215     5.44
19991222     5.46
19991229     5.01
20000105     4.72
20000112     5.62
20000119     5.59
20000126     5.43
20000202     5.66
20000209     5.71
20000216     5.75
20000223     5.72
20000301     5.77
;
run;

proc gplot data=weeklyFedFunds;
   plot ffrate*date / haxis=axis1 vaxis=axis2 cframe=ligr ;
   symbol1 c=green i=join;
   axis1 order=('01jan60'd, '01jan65'd, '01jan70'd, '01jan75'd,
                '01jan80'd, '01jan85'd, '01jan90'd, '01jan95'd,
                '01jan00'd, '01jan05'd);
   axis2 label=(angle=90 'Fed Fund Rate');
   title1 "Effective Weekly Federal Funds Rate";
run;
quit;


/* Heteroscedastic Modeling of the Fed Funds Rate */


proc model data=weeklyFedFunds;
      id date;
      ffrate = lag(ffrate) + kappa  * (theta - lag(ffrate));
      lag_ffrate = lag( ffrate );
      label kappa = "Speed of Mean Reversion";
      label theta = "Long term Mean";
   fit ffrate / fiml breusch=( lag_ffrate ) out=resid outresid;
run;


proc gplot data=resid;
   plot ffrate*date / haxis=axis1 vaxis=axis2 cframe=ligr ;
   symbol1 c=blue i=join;
   axis1 order=('01jan60'd, '01jan65'd, '01jan70'd, '01jan75'd,
                '01jan80'd, '01jan85'd, '01jan90'd, '01jan95'd,
                '01jan00'd, '01jan05'd);
   axis2 label=(angle=90 'Fed Fund Residuals');
   title1 "Effective Weekly Federal Funds Rate";
   title2 "Vasicek Model";
run;
quit;


proc model data=weeklyFedFunds;
      id date;
      ffrate = lag(ffrate) + kappa * (theta - lag(ffrate));
      h.ffrate = sigma**2 * lag(ffrate);
      label kappa = "Speed of Mean Reversion";
      label theta = "Long term Mean";
      label sigma ="Constant part of variance";
   fit ffrate / fiml out=resid outresid;
run;


proc gplot data=resid;
   plot ffrate*date / haxis=axis1 vaxis=axis2 cframe=ligr ;
   symbol1 c=blue i=join;
   axis1 order=('01jan60'd, '01jan65'd, '01jan70'd, '01jan75'd,
                '01jan80'd, '01jan85'd, '01jan90'd, '01jan95'd,
                '01jan00'd, '01jan05'd);
   axis2 label=(angle=90 'Fed Fund Residuals');
   title1 "Effective Weekly Federal Funds Rate";
   title2 "Cox Ingersoll Ross Model";
run;
quit;



proc model data=weeklyFedFunds;
      id date;
      ffrate = lag(ffrate) + kappa * (theta - lag(ffrate));
      if ( _OBS_ = 1 ) then
         h.ffrate = arch0 + arch1 * mse.ffrate + garch1 * mse.ffrate;
      else
         h.ffrate = arch0 + arch1 * zlag(resid.ffrate**2)
                    + garch1 * zlag(h.ffrate);
   fit ffrate / fiml out=resid outresid;
run;


proc gplot data=resid;
   plot ffrate*date / haxis=axis1 vaxis=axis2 cframe=ligr ;
   symbol1 c=blue i=join;
   axis1 order=('01jan60'd, '01jan65'd, '01jan70'd, '01jan75'd,
                '01jan80'd, '01jan85'd, '01jan90'd, '01jan95'd,
                '01jan00'd, '01jan05'd);
   axis2 label=(angle=90 'Fed Fund Residuals');
   title1 "Effective Weekly Federal Funds Rate";
   title2 "Garch(1,1) Model";
run;
quit;