Providing software solutions since 1976
SAS/ETS Examples
Heteroscedastic Modeling of the Fed Funds Rate
Contents | Back to Example
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
19780315 6.77
19780322 6.77
19780329 6.82
19780405 6.86
19780412 6.74
19780419 6.78
19780426 7.00
19780503 7.27
19780510 7.32
19780517 7.34
19780524 7.43
19780531 7.36
19780607 7.47
19780614 7.49
19780621 7.53
19780628 7.78
19780705 7.72
19780712 7.72
19780719 7.94
19780726 7.88
19780802 7.89
19780809 7.83
19780816 7.87
19780823 8.14
19780830 8.28
19780906 8.30
19780913 8.33
19780920 8.36
19780927 8.62
19781004 8.85
19781011 8.71
19781018 8.78
19781025 9.24
19781101 9.29
19781108 9.77
19781115 9.68
19781122 9.68
19781129 9.85
19781206 9.87
19781213 9.79
19781220 9.75
19781227 10.25
19790103 10.59
19790110 9.97
19790117 10.05
19790124 10.05
19790131 10.12
19790207 10.06
19790214 10.15
19790221 9.97
19790228 10.06
19790307 10.07
19790314 10.21
19790321 10.09
19790328 10.00
19790404 9.95
19790411 9.93
19790418 9.96
19790425 10.09
19790502 10.22
19790509 10.25
19790516 10.25
19790523 10.17
19790530 10.28
19790606 10.23
19790613 10.23
19790620 10.28
19790627 10.32
19790704 10.42
19790711 10.28
19790718 10.35
19790725 10.63
19790801 10.75
19790808 10.67
19790815 10.80
19790822 11.04
19790829 11.16
19790905 11.02
19790912 11.30
19790919 11.37
19790926 11.61
19791003 11.91
19791010 12.00
19791017 13.22
19791024 15.14
19791031 15.61
19791107 13.77
19791114 13.30
19791121 13.10
19791128 12.46
19791205 13.77
19791212 13.79
19791219 13.90
19791226 13.49
19800102 14.04
19800109 13.94
19800116 13.91
19800123 13.77
19800130 13.54
19800206 12.80
19800213 13.64
19800220 14.87
19800227 14.62
19800305 16.17
19800312 16.45
19800319 16.24
19800326 17.78
19800402 19.39
19800409 19.04
19800416 18.35
19800423 17.56
19800430 15.12
19800507 12.96
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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;
