/* This program looks at various transfer function models that examine the relationship between the quality spread and the growth rate in U.S. industrial production. Both monthly and quarterly models are entertained. ip = U.S. Industrial production index, baa = Moody's Baa Corporate Bond, t10 = ten year U.S. government bond rate (secondary market). */ data qspread; input date:yymmn6. ip baa t10; format date yymmn6.; spread = baa - t10; lip = log(ip); datalines; 195401 22.5 3.71 2.48 195402 22.6 3.61 2.47 195403 22.5 3.51 2.37 195404 22.3 3.47 2.29 195405 22.5 3.47 2.37 195406 22.5 3.49 2.38 195407 22.6 3.5 2.3 195408 22.5 3.49 2.36 195409 22.6 3.47 2.38 195410 22.8 3.46 2.43 195411 23.2 3.45 2.48 195412 23.5 3.45 2.51 195501 24 3.45 2.61 195502 24.3 3.47 2.65 195503 24.9 3.48 2.68 195504 25.2 3.49 2.75 195505 25.6 3.5 2.76 195506 25.7 3.51 2.78 195507 25.9 3.52 2.9 195508 25.8 3.56 2.97 195509 26 3.59 2.97 195510 26.4 3.59 2.88 195511 26.5 3.58 2.89 195512 26.6 3.62 2.96 195601 26.8 3.6 2.9 195602 26.5 3.58 2.84 195603 26.5 3.6 2.96 195604 26.7 3.68 3.18 195605 26.5 3.73 3.07 195606 26.3 3.76 3 195607 25.5 3.8 3.11 195608 26.5 3.93 3.33 195609 27.1 4.07 3.38 195610 27.3 4.17 3.34 195611 27.1 4.24 3.49 195612 27.5 4.37 3.59 195701 27.4 4.49 3.46 195702 27.7 4.47 3.34 195703 27.6 4.43 3.41 195704 27.3 4.44 3.48 195705 27.2 4.52 3.6 195706 27.2 4.63 3.8 195707 27.4 4.73 3.93 195708 27.4 4.82 3.93 195709 27.2 4.93 3.92 195710 26.8 4.99 3.97 195711 26.1 5.09 3.72 195712 25.6 5.03 3.21 195801 25.1 4.83 3.09 195802 24.6 4.66 3.05 195803 24.3 4.68 2.98 195804 23.9 4.67 2.88 195805 24.1 4.62 2.92 195806 24.8 4.55 2.97 195807 25.1 4.53 3.2 195808 25.7 4.67 3.54 195809 25.9 4.87 3.76 195810 26.2 4.92 3.8 195811 27 4.87 3.74 195812 27 4.85 3.86 195901 27.4 4.87 4.02 195902 27.9 4.89 3.96 195903 28.3 4.85 3.99 195904 28.9 4.86 4.12 195905 29.4 4.96 4.31 195906 29.4 5.04 4.34 195907 28.7 5.08 4.4 195908 27.7 5.09 4.43 195909 27.7 5.18 4.68 195910 27.5 5.28 4.53 195911 27.7 5.26 4.53 195912 29.4 5.28 4.69 196001 30.1 5.34 4.72 196002 29.9 5.34 4.49 196003 29.6 5.25 4.25 196004 29.4 5.2 4.28 196005 29.3 5.28 4.35 196006 29 5.26 4.15 196007 28.9 5.22 3.9 196008 28.8 5.08 3.8 196009 28.5 5.01 3.8 196010 28.5 5.11 3.89 196011 28.1 5.08 3.93 196012 27.6 5.1 3.84 196101 27.6 5.1 3.84 196102 27.6 5.07 3.78 196103 27.7 5.02 3.74 196104 28.3 5.01 3.78 196105 28.7 5.01 3.71 196106 29.1 5.03 3.88 196107 29.5 5.09 3.92 196108 29.7 5.11 4.04 196109 29.7 5.12 3.98 196110 30.3 5.13 3.92 196111 30.8 5.11 3.94 196112 31 5.1 4.06 196201 30.8 5.08 4.08 196202 31.3 5.07 4.04 196203 31.4 5.04 3.93 196204 31.5 5.02 3.84 196205 31.5 5 3.87 196206 31.4 5.02 3.91 196207 31.7 5.05 4.01 196208 31.7 5.06 3.98 196209 31.9 5.03 3.98 196210 32 4.99 3.93 196211 32.1 4.96 3.92 196212 32.1 4.92 3.86 196301 32.3 4.91 3.83 196302 32.7 4.89 3.92 196303 32.9 4.88 3.93 196304 33.2 4.87 3.97 196305 33.6 4.85 3.93 196306 33.7 4.84 3.99 196307 33.6 4.84 4.02 196308 33.6 4.83 4 196309 34 4.84 4.08 196310 34.2 4.83 4.11 196311 34.4 4.84 4.12 196312 34.3 4.85 4.13 196401 34.6 4.83 4.17 196402 34.8 4.83 4.15 196403 34.8 4.83 4.22 196404 35.4 4.85 4.23 196405 35.6 4.85 4.2 196406 35.7 4.85 4.17 196407 35.9 4.83 4.19 196408 36.2 4.82 4.19 196409 36.3 4.82 4.2 196410 35.8 4.81 4.19 196411 36.9 4.81 4.15 196412 37.4 4.81 4.18 196501 37.8 4.8 4.19 196502 38 4.78 4.21 196503 38.5 4.78 4.21 196504 38.7 4.8 4.2 196505 39 4.81 4.21 196506 39.3 4.85 4.21 196507 39.6 4.88 4.2 196508 39.8 4.88 4.25 196509 39.9 4.91 4.29 196510 40.3 4.93 4.35 196511 40.5 4.95 4.45 196512 41 5.02 4.62 196601 41.4 5.06 4.61 196602 41.6 5.12 4.83 196603 42.2 5.32 4.87 196604 42.3 5.41 4.75 196605 42.7 5.48 4.78 196606 42.9 5.58 4.81 196607 43.1 5.68 5.02 196608 43.2 5.83 5.22 196609 43.6 6.09 5.18 196610 43.9 6.1 5.01 196611 43.6 6.13 5.16 196612 43.7 6.18 4.84 196701 43.9 5.97 4.58 196702 43.4 5.82 4.63 196703 43.1 5.85 4.54 196704 43.5 5.83 4.59 196705 43.2 5.96 4.85 196706 43.1 6.15 5.02 196707 43 6.26 5.16 196708 43.9 6.33 5.28 196709 43.8 6.4 5.3 196710 44.2 6.52 5.48 196711 44.8 6.72 5.75 196712 45.3 6.93 5.7 196801 45.2 6.84 5.53 196802 45.4 6.8 5.56 196803 45.5 6.85 5.74 196804 45.6 6.97 5.64 196805 46.1 7.03 5.87 196806 46.3 7.07 5.72 196807 46.2 6.98 5.5 196808 46.3 6.82 5.42 196809 46.5 6.79 5.46 196810 46.6 6.84 5.58 196811 47.2 7.01 5.7 196812 47.4 7.23 6.03 196901 47.6 7.32 6.04 196902 47.9 7.3 6.19 196903 48.3 7.51 6.3 196904 48.1 7.54 6.17 196905 48 7.52 6.32 196906 48.4 7.7 6.57 196907 48.7 7.84 6.72 196908 48.8 7.86 6.69 196909 48.8 8.05 7.16 196910 48.8 8.22 7.1 196911 48.3 8.25 7.14 196912 48.2 8.65 7.65 197001 47.3 8.86 7.79 197002 47.3 8.78 7.24 197003 47.2 8.63 7.07 197004 47.1 8.7 7.39 197005 47 8.98 7.91 197006 46.9 9.25 7.84 197007 47 9.4 7.46 197008 46.9 9.44 7.53 197009 46.6 9.39 7.39 197010 45.7 9.33 7.33 197011 45.4 9.38 6.84 197012 46.4 9.12 6.39 197101 46.8 8.74 6.24 197102 46.7 8.39 6.11 197103 46.7 8.46 5.7 197104 46.9 8.45 5.83 197105 47.2 8.62 6.39 197106 47.4 8.75 6.52 197107 47.2 8.76 6.73 197108 46.9 8.76 6.58 197109 47.7 8.59 6.14 197110 48.1 8.48 5.93 197111 48.3 8.38 5.81 197112 48.8 8.38 5.93 197201 50 8.23 5.95 197202 50.4 8.23 6.08 197203 50.8 8.24 6.07 197204 51.3 8.24 6.19 197205 51.3 8.23 6.13 197206 51.5 8.2 6.11 197207 51.5 8.23 6.11 197208 52.2 8.19 6.21 197209 52.6 8.09 6.55 197210 53.3 8.06 6.48 197211 53.9 7.99 6.28 197212 54.5 7.93 6.36 197301 54.9 7.9 6.46 197302 55.7 7.97 6.64 197303 55.7 8.03 6.71 197304 55.5 8.09 6.67 197305 55.9 8.06 6.85 197306 56 8.13 6.9 197307 56.2 8.24 7.13 197308 56.2 8.53 7.4 197309 56.6 8.63 7.09 197310 57 8.41 6.79 197311 57.3 8.42 6.73 197312 57.1 8.48 6.74 197401 56.7 8.48 6.99 197402 56.5 8.53 6.96 197403 56.5 8.62 7.21 197404 56.5 8.87 7.51 197405 56.8 9.05 7.58 197406 56.7 9.27 7.54 197407 56.8 9.48 7.8 197408 56.2 9.77 8.04 197409 56.3 10.18 8.04 197410 56.1 10.48 7.9 197411 54.2 10.6 7.68 197412 52.3 10.63 7.43 197501 51.6 10.81 7.5 197502 50.4 10.65 7.39 197503 49.8 10.48 7.73 197504 49.8 10.58 8.23 197505 49.8 10.69 8.06 197506 50.1 10.62 7.86 197507 50.6 10.55 8.06 197508 51.1 10.59 8.4 197509 51.8 10.61 8.43 197510 52 10.62 8.14 197511 52.1 10.56 8.05 197512 52.8 10.56 8 197601 53.5 10.41 7.74 197602 54 10.24 7.79 197603 54 10.12 7.73 197604 54.4 9.94 7.56 197605 54.6 9.86 7.89 197606 54.6 9.89 7.86 197607 54.9 9.82 7.83 197608 55.3 9.64 7.77 197609 55.4 9.4 7.59 197610 55.5 9.29 7.41 197611 56.3 9.23 7.29 197612 56.9 9.12 6.87 197701 56.6 9.08 7.21 197702 57.4 9.12 7.39 197703 58.1 9.12 7.46 197704 58.7 9.07 7.37 197705 59.1 9.01 7.46 197706 59.5 8.91 7.28 197707 59.7 8.87 7.33 197708 59.8 8.82 7.4 197709 60.1 8.8 7.34 197710 60.3 8.89 7.52 197711 60.3 8.95 7.58 197712 60.4 8.99 7.69 197801 59.5 9.17 7.96 197802 59.8 9.2 8.03 197803 60.9 9.22 8.04 197804 62.1 9.32 8.15 197805 62.4 9.49 8.35 197806 62.8 9.6 8.46 197807 62.8 9.6 8.64 197808 63 9.48 8.41 197809 63.2 9.42 8.42 197810 63.7 9.59 8.64 197811 64.2 9.83 8.81 197812 64.6 9.94 9.01 197901 64.1 10.13 9.1 197902 64.5 10.08 9.1 197903 64.7 10.26 9.12 197904 64 10.33 9.18 197905 64.5 10.47 9.25 197906 64.5 10.38 8.91 197907 64.4 10.29 8.95 197908 63.9 10.35 9.03 197909 64 10.54 9.33 197910 64.3 11.4 10.3 197911 64.3 11.99 10.65 197912 64.4 12.06 10.39 198001 64.6 12.42 10.8 198002 64.7 13.57 12.41 198003 64.4 14.45 12.75 198004 63.2 14.19 11.47 198005 61.6 13.17 10.18 198006 60.8 12.71 9.78 198007 60.4 12.65 10.25 198008 60.6 13.15 11.1 198009 61.6 13.7 11.51 198010 62.3 14.23 11.75 198011 63.4 14.64 12.68 198012 63.7 15.14 12.84 198101 63.4 15.03 12.57 198102 63.1 15.37 13.19 198103 63.4 15.34 13.12 198104 63.1 15.56 13.68 198105 63.6 15.95 14.1 198106 63.9 15.8 13.47 198107 64.3 16.17 14.28 198108 64.3 16.34 14.94 198109 63.9 16.92 15.32 198110 63.4 17.11 15.15 198111 62.7 16.39 13.39 198112 62.1 16.55 13.72 198201 60.9 17.1 14.59 198202 62.1 17.18 14.43 198203 61.6 16.82 13.86 198204 61.1 16.78 13.87 198205 60.7 16.64 13.62 198206 60.5 16.92 14.3 198207 60.2 16.8 13.95 198208 59.7 16.32 13.05 198209 59.5 15.63 12.34 198210 59 14.73 10.91 198211 58.7 14.3 10.55 198212 58.2 14.14 10.54 198301 59.4 13.94 10.46 198302 59 13.95 10.72 198303 59.5 13.61 10.51 198304 60.2 13.29 10.4 198305 60.7 13.09 10.38 198306 61 13.37 10.85 198307 62 13.39 11.38 198308 62.7 13.64 11.85 198309 63.6 13.55 11.65 198310 64.2 13.46 11.54 198311 64.3 13.61 11.69 198312 64.7 13.75 11.83 198401 66 13.65 11.67 198402 66.3 13.59 11.84 198403 66.6 13.99 12.32 198404 67 14.31 12.63 198405 67.3 14.74 13.41 198406 67.6 15.05 13.56 198407 67.8 15.15 13.36 198408 67.9 14.63 12.72 198409 67.7 14.35 12.52 198410 67.7 13.94 12.16 198411 68 13.48 11.57 198412 68 13.4 11.5 198501 67.8 13.26 11.38 198502 68.1 13.23 11.51 198503 68.2 13.69 11.86 198504 68.1 13.51 11.43 198505 68.2 13.15 10.85 198506 68.2 12.4 10.16 198507 67.7 12.43 10.31 198508 68.1 12.5 10.33 198509 68.4 12.48 10.37 198510 68.1 12.36 10.24 198511 68.3 11.99 9.78 198512 69 11.58 9.26 198601 69.4 11.44 9.19 198602 68.9 11.11 8.7 198603 68.4 10.5 7.78 198604 68.4 10.19 7.3 198605 68.6 10.29 7.71 198606 68.3 10.34 7.8 198607 68.8 10.16 7.3 198608 68.6 10.18 7.17 198609 68.8 10.2 7.45 198610 69.1 10.24 7.43 198611 69.4 10.07 7.25 198612 70 9.97 7.11 198701 69.8 9.72 7.08 198702 70.7 9.65 7.25 198703 70.8 9.61 7.25 198704 71.2 10.04 8.02 198705 71.7 10.51 8.61 198706 72 10.52 8.4 198707 72.5 10.61 8.45 198708 73 10.8 8.76 198709 73.1 11.31 9.42 198710 74.2 11.62 9.52 198711 74.6 11.23 8.86 198712 74.9 11.29 8.99 198801 75 11.07 8.67 198802 75.2 10.62 8.21 198803 75.4 10.57 8.37 198804 75.8 10.9 8.72 198805 75.7 11.04 9.09 198806 75.9 11 8.92 198807 76 11.11 9.06 198808 76.4 11.21 9.26 198809 76.2 10.9 8.98 198810 76.6 10.41 8.8 198811 76.8 10.48 8.96 198812 77.1 10.65 9.11 198901 77.3 10.65 9.1 198902 76.9 10.61 9.17 198903 77.1 10.67 9.36 198904 77.1 10.61 9.18 198905 76.5 10.46 8.86 198906 76.6 10.03 8.28 198907 75.9 9.87 8.02 198908 76.6 9.88 8.11 198909 76.3 9.91 8.19 198910 76.3 9.81 8.01 198911 76.5 9.81 7.87 198912 77 9.82 7.84 199001 76.6 9.94 8.21 199002 77.2 10.14 8.47 199003 77.6 10.21 8.59 199004 77.6 10.3 8.79 199005 77.6 10.41 8.76 199006 77.9 10.22 8.48 199007 77.7 10.2 8.47 199008 77.9 10.41 8.75 199009 78.1 10.64 8.89 199010 77.5 10.74 8.72 199011 76.6 10.62 8.39 199012 76.1 10.43 8.08 199101 75.7 10.45 8.09 199102 75.2 10.07 7.85 199103 74.9 10.09 8.11 199104 75 9.94 8.04 199105 75.8 9.86 8.07 199106 76.5 9.96 8.28 199107 76.5 9.89 8.27 199108 76.6 9.65 7.9 199109 77.3 9.51 7.65 199110 77.1 9.49 7.53 199111 77 9.45 7.42 199112 76.8 9.26 7.09 199201 76.3 9.13 7.03 199202 76.9 9.23 7.34 199203 77.5 9.25 7.54 199204 78 9.21 7.48 199205 78.4 9.13 7.39 199206 78.3 9.05 7.26 199207 78.9 8.84 6.84 199208 78.6 8.65 6.59 199209 78.7 8.62 6.42 199210 79.3 8.84 6.59 199211 79.6 8.96 6.87 199212 79.6 8.81 6.77 199301 80 8.67 6.6 199302 80.4 8.39 6.26 199303 80.4 8.15 5.98 199304 80.6 8.14 5.97 199305 80.3 8.21 6.04 199306 80.5 8.07 5.96 199307 80.8 7.93 5.81 199308 80.8 7.6 5.68 199309 81.2 7.34 5.36 199310 81.7 7.31 5.33 199311 82.1 7.66 5.72 199312 82.6 7.69 5.77 199401 82.9 7.65 5.75 199402 83 7.76 5.97 199403 83.7 8.13 6.48 199404 84.2 8.52 6.97 199405 84.7 8.62 7.18 199406 85.3 8.65 7.1 199407 85.5 8.8 7.3 199408 85.9 8.74 7.24 199409 86.1 8.98 7.46 199410 86.8 9.2 7.74 199411 87.4 9.32 7.96 199412 88.3 9.1 7.81 199501 88.6 9.08 7.78 199502 88.6 8.85 7.47 199503 88.7 8.7 7.2 199504 88.6 8.6 7.06 199505 88.8 8.2 6.63 199506 89.1 7.9 6.17 199507 88.7 8.04 6.28 199508 89.9 8.19 6.49 199509 90.3 7.93 6.2 199510 90.2 7.75 6.04 199511 90.4 7.68 5.93 199512 90.9 7.49 5.71 199601 90 7.47 5.65 199602 91.4 7.63 5.81 199603 91.1 8.03 6.27 199604 92 8.19 6.51 199605 92.6 8.3 6.74 199606 93.4 8.4 6.91 199607 93.3 8.35 6.87 199608 93.9 8.18 6.64 199609 94.5 8.35 6.83 199610 94.6 8.07 6.53 199611 95.5 7.79 6.2 199612 96.2 7.89 6.3 199701 96.4 8.09 6.58 199702 97.6 7.94 6.42 199703 98.1 8.18 6.69 199704 98.3 8.34 6.89 199705 98.7 8.2 6.71 199706 99.2 8.02 6.49 199707 99.8 7.75 6.22 199708 100.8 7.82 6.3 199709 101.7 7.7 6.21 199710 102.5 7.57 6.03 199711 103.2 7.42 5.88 199712 103.6 7.32 5.81 199801 104.1 7.19 5.54 199802 104.2 7.25 5.57 199803 104.5 7.32 5.65 199804 105.2 7.33 5.64 199805 105.7 7.3 5.65 199806 105.3 7.13 5.5 199807 105.1 7.15 5.46 199808 107 7.14 5.34 199809 106.7 7.09 4.81 199810 107.5 7.18 4.53 199811 107.3 7.34 4.83 199812 107.5 7.23 4.65 199901 108.2 7.29 4.72 199902 108.6 7.39 5 199903 109 7.53 5.23 199904 109.3 7.48 5.18 199905 110 7.72 5.54 199906 110 8.02 5.9 199907 110.7 7.95 5.79 199908 111.4 8.15 5.94 199909 111 8.2 5.92 199910 112.3 8.38 6.11 199911 112.8 8.15 6.03 199912 113.9 8.19 6.28 200001 113.8 8.33 6.66 200002 114.3 8.29 6.52 200003 114.9 8.37 6.26 200004 115.7 8.4 5.99 200005 116.4 8.9 6.44 200006 116.4 8.48 6.1 200007 115.9 8.35 6.05 200008 115.5 8.26 5.83 200009 115.9 8.35 5.8 200010 115.4 8.34 5.74 200011 115.2 8.28 5.72 200012 115 8.02 5.24 200101 113.8 7.93 5.16 200102 113.1 7.87 5.1 200103 112.7 7.84 4.89 200104 112.5 8.07 5.14 200105 111.9 8.07 5.39 200106 111.3 7.97 5.28 200107 110.8 7.97 5.24 200108 110.7 7.85 4.97 200109 110.1 8.03 4.73 200110 109.7 7.91 4.57 200111 109.2 7.81 4.65 200112 109.2 8.05 5.09 200201 109.9 7.87 5.04 200202 109.7 7.89 4.91 200203 110.3 8.11 5.28 200204 110.7 8.03 5.21 200205 111 8.09 5.16 200206 111.8 7.95 4.93 200207 111.7 7.9 4.65 200208 111.6 7.58 4.26 200209 111.6 7.4 3.87 200210 111.1 7.73 3.94 200211 111.2 7.62 4.05 200212 110.7 7.45 4.03 200301 110.9 7.35 4.05 200302 111 7.06 3.9 200303 110.6 6.95 3.81 200304 109.5 6.85 3.96 200305 109.6 6.38 3.57 200306 109.9 6.19 3.33 200307 110.6 6.62 3.98 200308 110.5 7.01 4.45 200309 111.3 6.79 4.27 200310 111.6 6.73 4.29 200311 112.7 6.66 4.3 200312 112.9 6.6 4.27 200401 113.2 6.44 4.15 200402 114.4 6.27 4.08 200403 114.1 6.11 3.83 200404 114.7 6.46 4.35 200405 115.5 6.75 4.72 200406 115.1 6.78 4.73 200407 115.9 6.62 4.5 200408 116 6.46 4.28 200409 115.7 6.27 4.13 200410 116.6 6.21 4.1 200411 116.9 6.2 4.19 200412 117.9 6.15 4.23 200501 117.8 6.02 4.22 200502 118.3 5.82 4.17 200503 118.6 6.06 4.5 200504 118.1 6.05 4.34 200505 118.4 6.01 4.14 200506 119.4 5.86 4 200507 119.4 5.95 4.18 ; proc expand data=qspread out=qtr from = month to = quarter; convert ip spread / method = aggregate observed = total; id date; run; data qtr; set qtr; ip = ip/3; spread = spread/3; lip = log(ip); run; /* Fitting a Transfer Function Model Using the Quarterly Data. */ proc arima data=qtr; identify var=lip stationarity=(adf=(4)); identify var=lip(1) stationarity=(adf=(4)); run; proc arima data=qtr; identify var=lip(1) noprint; estimate p = 1; estimate p= 2; estimate q = 1; estimate q = 2; estimate p = 1 q = 1; run; proc arima data = qtr; identify var=spread stationarity=(adf=(4)); identify var=spread(1) stationarity=(adf=(4)); run; proc arima data = qtr; identify var=spread(1) noprint; estimate p = 1; estimate p= 2; estimate q = 1; estimate q = 2; estimate p = 1 q = 1; run; proc arima data=qtr; identify var=spread(1) noprint; estimate q = 1 method=ml noprint; identify var=lip(1)crosscor=(spread(1)); /* Produces CCF */ estimate input = ((1)spread) method=ml; /* Plausible model 1 */ estimate input = ((1)/(1)spread) method=ml; /* Plausible model 2 */ estimate input = (/(1,2)spread) method=ml; /* Plausible model 3 */ /* The preceding models can still be improved by adding an autocorrelated error term. ARMA(1,1) does better than AR(1) in producting white noise residuals. */ estimate p=1 q=1 input = ((1)spread) method=ml; /* Plausible model 1 + ARMA(1,1) error */ estimate p=1 q=1 input = ((1)/(1)spread) method=ml; /* Plausible model 2 + ARMA(1,1) error */ estimate p=1 q=1 input = (/(1,2)spread) method=ml; /* Plausible model 3 + ARMA(1,1) error */ /* It appears that the best model is the first one if one adopts the SBC criterion. However, in Overfit 4 below, the overfitting coefficient (Den(1,1)) is statistically significant. In that overfitting model the AIC is a little smaller while the SBC is larger so we can see that the SBC criterion picks the more parsimonious model. Thus, it is a close call as to whether the CCF cuts off after lag 1 or if, instead, it damps out. I am going to choose the more parsimonious model "estimate p = 1 q = 1 input = ((1)spread)" and assume that, in the population, the CCF cuts off after lag one. */ estimate p = 2 q = 1 input = ((1)spread) method=ml; /* Overfit 1 */ estimate p = 1 q = 2 input = (/(1)spread) method=ml; /* Overfit 2 */ estimate p = 1 q = 1 input = ((1,2)/spread) method=ml; /* Overfit 3 */ estimate p = 1 q = 1 input = ((1)/(1)spread) method=ml; /* Overfit 4 */ run; /* For the monthly data the best model appears to be "estimate p=1 input = (/(1)spread)" */ /* Fitting a Transfer Function model using the monthly data. */ proc arima data = qspread; identify var=spread(1) stationarity=(adf=(4)); estimate p = 1; estimate p= 2; estimate q = 1; estimate q = 2; estimate p = 1 q = 1; estimate p = 1 q = 2; estimate q = 3; run; /* From the above B-J model estimations it appears that the MA(2) is to be preferred for spread(1). */ proc arima data=qspread; identify var=spread(1) noprint; estimate q = 2 method=ml noprint; identify var= lip(1) crosscor=(spread(1)); /* Produces CCF */ estimate input = ((1,2)/(1)spread) method=ml; /* Plausible model 1 */ estimate input = (/(1,2)spread) method=ml; /* Plausible model 2 */ estimate input = ((1,2,3,4,5)spread) method=ml; /* Plausible model 3 */ run; /* All three models suffer from having residuals that are not white noise. However, we note that the SBC of the second tentative model is the lowest. Let's try the plausible models again while at the same time soaking up the autocorrelation in the residuals by adding a p=1 error process. */ proc arima data=qspread; identify var=spread(1) noprint; estimate q = 2 method=ml noprint; identify var=lip(1) crosscor=(spread(1))noprint; estimate p=1 input = ((1,2)/(1)spread) method=ml; /* Plausible model 1 with AR(1) error */ estimate p=1 input = (/(1,2)spread) method=ml; /* Plausible model 2 with AR(1) error */ estimate p=1 input = ((1,2,3,4,5)spread) method=ml; /* Plausible model 3 with AR(1) error */ run; /* The "estimate p=1 input = (/(1,2)spread)" appears to be the best although some autocorrelation still exists at the long lags that we cannot account for. */ proc arima data=qspread; identify var=spread(1) noprint; estimate q = 2 method=ml noprint; identify var=lip(1) crosscor=(spread(1))noprint; estimate p=2 input = (/(1,2)spread) method=ml; /* Overfit 1 */ estimate p=1 q=1 input = (/(1,2)spread) method=ml; /* Overfit 2 */ estimate p=1 input = ((1)/(1,2)spread) method=ml; /* Overfit 3 */ estimate p=1 input = (/(1,2,3)spread) method=ml; /* Overfit 4 */ run; /* Thus, from the overfits, we conclude that the model "estimate p=1 input = (/1,2)spread" is the best one to choose. */