Tugas 4 MPDW
Arfiah Kania Sektiaruni
2023-09-21
1. MA(2)
Membangiktkan model MA(2) dengan teta1 = 0.4 dan teta2 = 0.6 sebanyak 300 data (manual dan sengan arima.sim)
White Noise
wn <- rnorm(300)
ts.plot(wn)
Akan dicoba membangkitkan proses MA paling sederhana, yaitu MA(1) dengan \(\theta = 0.4\) dan \(\theta = 0.6\) sebanyak 300 observasi dan \(c=0\). Karena diperlukan satu nilai awal untuk \(e_{t-1}\) dan \(e_{t-2}\), masukkan nilai pertama white noise sebagai nilai awal tersebut.
set.seed(023)
ma <- wn[c(1,2)]
ma## [1] 0.3764867 2.8910833Nilai-nilai selanjutnya dapat dicari melalui loop. Bentuk loop dapat dilihat dari rumus MA(2) yang hendak dibangkitkan:
\[ y_t = e_t+0.4e_{t-1}+0.6e_{t-2} \]
for(i in 3:300){
ma[i] <- wn[i] + 0.4 * wn[i-1] + 0.6 * wn[i - 2]
}
ma## [1] 0.376486659 2.891083265 1.272939391 1.049329503 0.528455337
## [6] -0.660323932 -0.867766297 -1.167038680 -1.276285900 -0.756024562
## [11] 0.102562145 -1.673677061 -0.414898173 -2.635847349 0.643421450
## [16] -2.470738795 0.807116406 -2.435734193 0.789594500 -1.211021488
## [21] 0.309792804 -2.591360944 -1.551970879 -1.163496898 0.355554852
## [26] 0.035818895 0.190514914 -0.481533808 0.335096222 0.439786757
## [31] -0.586768006 -0.180757528 0.476906074 1.377054627 0.453350539
## [36] 1.089294028 -1.102237824 0.290631381 -1.579802794 -0.654088463
## [41] -1.806967638 -1.015694637 -2.271583889 -1.296964313 -1.282248111
## [46] -3.416281679 -1.394294840 -2.096391138 -0.385318133 0.433415090
## [51] -0.571718551 -0.781329468 -2.274489251 -2.495999349 -2.896649704
## [56] -1.507575269 0.167025748 -1.362883701 -1.163388675 -0.453860137
## [61] -0.082191647 2.081975145 -0.731771731 0.447599146 -0.949067464
## [66] -0.981865970 -1.253537117 -0.652734853 0.230514709 0.508389876
## [71] -0.335137033 -0.577333952 1.125884853 1.884175410 1.868599712
## [76] -0.373986078 0.540078189 0.326100544 0.922779061 -0.676952910
## [81] -1.021652887 -2.455669492 -0.525253954 -0.756449866 0.242026733
## [86] -0.793622237 -2.118250559 -1.171944294 -1.254756058 0.156409896
## [91] 0.881224244 1.963714779 1.319895662 -0.541030517 0.372948487
## [96] -1.269736070 1.185564889 -0.749225458 1.348942202 0.610003501
## [101] 1.471093899 1.131809788 -0.449774717 0.528073425 -1.069169086
## [106] 0.655515959 0.813411637 1.980547838 0.292419372 2.056655613
## [111] 1.452182364 1.443411029 1.538886178 -0.586702602 -0.802033556
## [116] -1.632108730 0.130561699 -0.916958534 1.760703550 1.559644810
## [121] 1.713929615 1.637894941 0.880972909 1.384500177 2.096981934
## [126] 0.884207740 -0.279617376 0.291208970 -0.768094492 -1.086892414
## [131] -0.560696861 0.318145290 0.006487837 -0.294477306 -2.566374703
## [136] -1.711312552 -1.945850023 1.209309432 0.787209035 1.817054541
## [141] 0.042340797 0.371645997 -1.666279378 -0.450065553 -1.926881708
## [146] 0.845610896 0.317984728 2.453425058 0.616148527 -1.080111772
## [151] -0.732158983 -2.282823453 -0.856121205 -0.284745928 -2.527788313
## [156] -1.583975853 -0.262485375 0.350113476 2.062510435 1.118959736
## [161] 0.810890964 0.361658001 -0.762405075 0.174734931 -1.333189977
## [166] -1.444625441 -2.294630100 -2.423037666 0.913607099 1.283219664
## [171] 2.669925684 2.144997162 1.016223907 2.781621020 2.229933096
## [176] 1.563078926 2.121133394 -0.124324765 0.666300793 -1.560166674
## [181] -1.736651137 -3.875143179 -3.522676167 -2.360833581 -1.000336996
## [186] -0.843211137 -0.135650765 -0.709671480 1.390898586 -0.096231517
## [191] 0.739315875 0.951040561 0.324689822 0.833814176 1.129219631
## [196] 1.207678300 -1.383559378 -0.683797515 0.092205088 0.132851910
## [201] -0.348714809 -0.020167567 -1.108436389 -2.394337109 -0.581490203
## [206] -2.833926427 -1.884126069 0.288866728 0.292297373 0.451081021
## [211] -0.030659568 0.245960795 1.332739654 0.941277885 1.933980662
## [216] 0.055740608 0.995855870 1.134146672 1.705224826 1.257882265
## [221] 0.688890766 0.979865547 0.082975177 0.745006291 0.426795522
## [226] 1.023379835 0.122487130 -0.648209322 -3.208880364 -0.473769727
## [231] -0.133838137 1.866038951 1.498457094 -0.849790736 -0.178248189
## [236] -1.921776375 -0.629875868 -1.580832048 0.598432589 -0.159793430
## [241] 0.880989211 0.987022925 0.514417044 0.074549124 0.662336368
## [246] -1.918440785 -0.045150758 -0.469879540 0.811847904 0.304228772
## [251] -0.298448803 -1.353749214 -1.554844881 -0.290228437 -1.934766491
## [256] -0.092434755 -3.369647389 -2.324057797 -1.958702604 -0.530963197
## [261] -0.908340921 0.324453692 -1.154031550 -1.415538487 -0.882397739
## [266] -0.111151421 -0.652326891 0.270556538 -0.805420362 0.272980086
## [271] -0.057273789 0.859319788 -1.786286310 -2.927631672 -2.497477692
## [276] -0.851240606 0.312379603 -0.187899331 -0.890509851 -1.178938323
## [281] 0.274957064 -1.013144834 -0.549619209 -0.103440636 -0.030645415
## [286] 1.420424337 -0.710422968 1.709037642 -1.476237380 1.602330567
## [291] -2.399768731 0.667900943 -1.152654377 0.723381614 -0.155364848
## [296] 1.416746072 0.894366166 1.313866117 -2.067594319 -1.228167575Selain menggunakan cara di atas, pembangkitan proses MA(1) dapat
dilakukan dengan fungsi arima.sim() sebagai berikut.
ma1 <- arima.sim(list(order=c(0,0,2), ma=c(0.4,0.6)), n=300)
ma1## Time Series:
## Start = 1
## End = 300
## Frequency = 1
## [1] 0.85532165 1.89788567 2.26192060 2.58216539 0.76287298 1.57246527
## [7] 0.28626760 2.20547776 0.87586260 -0.01375221 -0.57632971 -0.26184638
## [13] -1.19696950 0.11051650 -1.12674941 -0.46550299 -1.08834532 0.78946442
## [19] 0.99363469 0.54488810 1.06324806 -1.19017065 -0.52414013 -0.91294433
## [25] -0.83147786 1.03720732 -0.47097327 0.64811158 -0.88290527 -0.02312615
## [31] -1.43931644 -1.64968475 -1.39008210 -0.65330070 1.25738044 -0.71337628
## [37] 1.10849727 -0.22845302 1.65205692 1.88572609 2.10295151 0.19021663
## [43] -1.04254226 -2.70692797 -1.28888624 -0.96271449 0.56002069 -0.03743293
## [49] 1.47778120 0.12843854 -1.33627732 -0.24111047 0.30256183 0.05804925
## [55] -0.17148592 -0.45876310 0.74344799 1.50725901 0.56653033 0.41060169
## [61] -0.60597480 -0.10819194 1.62656718 -0.34906235 -0.31414295 -2.38467610
## [67] -0.36860151 -1.71455673 0.68886005 -0.86754890 0.13105888 0.72051062
## [73] 0.34002427 1.05509506 1.46491577 -1.64327112 -1.56753551 -2.47941910
## [79] -1.00433731 0.34550952 2.23416907 1.62923000 1.89627742 -1.56284182
## [85] -1.05529717 -2.02971830 -0.03406615 -0.27606109 1.80692276 1.70825979
## [91] 2.03699812 0.91170001 -0.06978773 -0.48516206 0.96619763 1.41314467
## [97] 1.97170910 1.50566833 1.04656140 1.42455384 -1.31470295 -0.15186704
## [103] -1.97176732 -0.67637421 -0.03167275 1.24923636 -1.37686269 -0.53243299
## [109] -1.57842549 1.08593208 2.17861716 2.13231699 1.64243148 0.22780396
## [115] 0.61092860 -0.88031633 0.14395071 -0.61974842 -1.17642515 -0.10902232
## [121] -0.58641830 0.05007131 -0.18167803 -1.29156689 1.68421564 0.22127221
## [127] 2.57116670 0.77449608 0.64684226 0.76842793 0.85890976 -0.25045341
## [133] 0.23407008 1.03166739 1.58558709 2.25104336 -1.56171520 -0.02213279
## [139] -0.91793337 1.32826007 0.88368762 1.43162530 -0.03439972 0.85605287
## [145] -0.29619502 -0.06515819 -1.12612274 0.95605918 1.35617739 -0.49863601
## [151] -0.81676837 -1.55758667 0.84907280 0.45287783 -0.12942991 -1.20114491
## [157] -3.29270192 -2.60482178 -2.60256467 -0.75556226 -1.43112671 -0.62130758
## [163] -0.06842639 -0.49351716 0.57141593 -0.89025605 -1.03204385 -1.20550630
## [169] -0.42569403 -1.27846186 0.14922865 -0.18388690 0.74158243 1.42967028
## [175] 0.70443578 -0.08261893 -0.66704427 -1.05893039 -1.41836248 -0.46432900
## [181] -0.29077092 0.42001813 -0.37550346 0.41111787 -0.10353333 1.71914871
## [187] 1.19777894 1.83449219 -0.74629489 -0.38710060 0.25720932 -0.34645704
## [193] -0.83306803 -0.42488436 -0.22880897 1.39802605 0.92000511 -0.76108532
## [199] 0.15447525 -2.89475132 -0.27701271 -0.68046076 -0.80148048 -0.16818554
## [205] -1.66924564 -0.79161896 -0.67295144 -0.37517414 -1.21735154 0.71308912
## [211] -0.62629048 0.16177483 -1.01314833 1.86629099 -0.38665491 -1.11289058
## [217] -1.76804712 -0.85877929 -0.24114300 -0.87086115 -2.49710754 0.04395285
## [223] -2.06543104 -0.24829907 -2.29193295 0.44615300 0.90533735 1.11020221
## [229] 0.93515351 0.04319453 -0.23340236 0.47901747 0.51488742 2.23274873
## [235] 1.28058420 2.57857296 0.38363226 0.36200755 0.12247535 0.38890149
## [241] -0.63584754 -0.06824732 0.15789941 -1.51064529 -0.56218556 -1.45000740
## [247] -1.28285092 -2.08369824 -0.48923616 -0.88959105 1.55212468 -0.47454819
## [253] 0.50064531 -0.57093744 1.39375016 0.22705038 -0.63103283 -1.50621237
## [259] -0.06023872 1.24129841 0.35066690 0.92613290 0.07356613 -0.58473557
## [265] 0.05487049 -2.39064625 0.37471455 1.50795275 0.79290943 1.47272342
## [271] -1.31279017 -0.60240804 -2.20975460 -0.50230713 -1.46652374 1.55308133
## [277] -0.67004161 -0.02049327 -0.09207832 -1.06134338 0.47711417 0.36473761
## [283] 0.45959761 0.47837574 0.50438837 0.25779129 0.20091368 -0.20691914
## [289] 0.54826422 -1.53089719 1.40360454 -0.27107495 -0.03246342 -1.41807822
## [295] -0.59876274 -1.33165255 0.34882268 2.01456340 0.58235213 2.48521968Buat plot time series, plot ACF, plot PACF, dan plot EACF. Identifikasi apakah data tersebut stasioner
Plot Time Series
ts.plot(ma)
Berdasarkan plot time series, terlihat bahwa data MA(2) yang dibangkitkan stasioner dalam rataan
Plot ACF
acf(ma,lag.max = 20)
Berdasarkan plot AFC tersebut, terlihat bahwa plot ACF cuts off di lag pertama dan kedua
Plot PACF
pacf(ma)
Berdasarkan plot PACF tersebut, terlihat bahwa plot PACF cenderung tails off dan membentuk gelombang sinus
Plot EACF
TSA::eacf(ma)## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x o o o o o o o o o o o o
## 1 x x x o o o o o o o o o o o
## 2 x x o o o o o o o o o o o o
## 3 x x o o o o o o o o o o o o
## 4 x x o o o x o o o o o o o o
## 5 x x x o o o o o o o o o o o
## 6 x x x o x o o o o o o o o o
## 7 x x o x x o o o o o o o o oBerdasarkan pola segitiga nol pada plot EACF, terlihat bahwa segitiga nol berada pada ordo AR(0) dan ordo MA(3)
Dapat disimpulkan bahwa data tersebut stasioner
Buat scatterplot dan korelasi antara \(Y_t\) dengan \(Y_{t-1}\), \(Y_t\) dengan \(Y_{t-2}\), serta \(Y_t\) dengan \(Y_{t-3}\)
Korelasi antara \(Y_t\) dengan \(Y_{t-1}\)
#Yt
yt_ma <- ma[-1]
yt_ma## [1] 2.891083265 1.272939391 1.049329503 0.528455337 -0.660323932
## [6] -0.867766297 -1.167038680 -1.276285900 -0.756024562 0.102562145
## [11] -1.673677061 -0.414898173 -2.635847349 0.643421450 -2.470738795
## [16] 0.807116406 -2.435734193 0.789594500 -1.211021488 0.309792804
## [21] -2.591360944 -1.551970879 -1.163496898 0.355554852 0.035818895
## [26] 0.190514914 -0.481533808 0.335096222 0.439786757 -0.586768006
## [31] -0.180757528 0.476906074 1.377054627 0.453350539 1.089294028
## [36] -1.102237824 0.290631381 -1.579802794 -0.654088463 -1.806967638
## [41] -1.015694637 -2.271583889 -1.296964313 -1.282248111 -3.416281679
## [46] -1.394294840 -2.096391138 -0.385318133 0.433415090 -0.571718551
## [51] -0.781329468 -2.274489251 -2.495999349 -2.896649704 -1.507575269
## [56] 0.167025748 -1.362883701 -1.163388675 -0.453860137 -0.082191647
## [61] 2.081975145 -0.731771731 0.447599146 -0.949067464 -0.981865970
## [66] -1.253537117 -0.652734853 0.230514709 0.508389876 -0.335137033
## [71] -0.577333952 1.125884853 1.884175410 1.868599712 -0.373986078
## [76] 0.540078189 0.326100544 0.922779061 -0.676952910 -1.021652887
## [81] -2.455669492 -0.525253954 -0.756449866 0.242026733 -0.793622237
## [86] -2.118250559 -1.171944294 -1.254756058 0.156409896 0.881224244
## [91] 1.963714779 1.319895662 -0.541030517 0.372948487 -1.269736070
## [96] 1.185564889 -0.749225458 1.348942202 0.610003501 1.471093899
## [101] 1.131809788 -0.449774717 0.528073425 -1.069169086 0.655515959
## [106] 0.813411637 1.980547838 0.292419372 2.056655613 1.452182364
## [111] 1.443411029 1.538886178 -0.586702602 -0.802033556 -1.632108730
## [116] 0.130561699 -0.916958534 1.760703550 1.559644810 1.713929615
## [121] 1.637894941 0.880972909 1.384500177 2.096981934 0.884207740
## [126] -0.279617376 0.291208970 -0.768094492 -1.086892414 -0.560696861
## [131] 0.318145290 0.006487837 -0.294477306 -2.566374703 -1.711312552
## [136] -1.945850023 1.209309432 0.787209035 1.817054541 0.042340797
## [141] 0.371645997 -1.666279378 -0.450065553 -1.926881708 0.845610896
## [146] 0.317984728 2.453425058 0.616148527 -1.080111772 -0.732158983
## [151] -2.282823453 -0.856121205 -0.284745928 -2.527788313 -1.583975853
## [156] -0.262485375 0.350113476 2.062510435 1.118959736 0.810890964
## [161] 0.361658001 -0.762405075 0.174734931 -1.333189977 -1.444625441
## [166] -2.294630100 -2.423037666 0.913607099 1.283219664 2.669925684
## [171] 2.144997162 1.016223907 2.781621020 2.229933096 1.563078926
## [176] 2.121133394 -0.124324765 0.666300793 -1.560166674 -1.736651137
## [181] -3.875143179 -3.522676167 -2.360833581 -1.000336996 -0.843211137
## [186] -0.135650765 -0.709671480 1.390898586 -0.096231517 0.739315875
## [191] 0.951040561 0.324689822 0.833814176 1.129219631 1.207678300
## [196] -1.383559378 -0.683797515 0.092205088 0.132851910 -0.348714809
## [201] -0.020167567 -1.108436389 -2.394337109 -0.581490203 -2.833926427
## [206] -1.884126069 0.288866728 0.292297373 0.451081021 -0.030659568
## [211] 0.245960795 1.332739654 0.941277885 1.933980662 0.055740608
## [216] 0.995855870 1.134146672 1.705224826 1.257882265 0.688890766
## [221] 0.979865547 0.082975177 0.745006291 0.426795522 1.023379835
## [226] 0.122487130 -0.648209322 -3.208880364 -0.473769727 -0.133838137
## [231] 1.866038951 1.498457094 -0.849790736 -0.178248189 -1.921776375
## [236] -0.629875868 -1.580832048 0.598432589 -0.159793430 0.880989211
## [241] 0.987022925 0.514417044 0.074549124 0.662336368 -1.918440785
## [246] -0.045150758 -0.469879540 0.811847904 0.304228772 -0.298448803
## [251] -1.353749214 -1.554844881 -0.290228437 -1.934766491 -0.092434755
## [256] -3.369647389 -2.324057797 -1.958702604 -0.530963197 -0.908340921
## [261] 0.324453692 -1.154031550 -1.415538487 -0.882397739 -0.111151421
## [266] -0.652326891 0.270556538 -0.805420362 0.272980086 -0.057273789
## [271] 0.859319788 -1.786286310 -2.927631672 -2.497477692 -0.851240606
## [276] 0.312379603 -0.187899331 -0.890509851 -1.178938323 0.274957064
## [281] -1.013144834 -0.549619209 -0.103440636 -0.030645415 1.420424337
## [286] -0.710422968 1.709037642 -1.476237380 1.602330567 -2.399768731
## [291] 0.667900943 -1.152654377 0.723381614 -0.155364848 1.416746072
## [296] 0.894366166 1.313866117 -2.067594319 -1.228167575#Yt-1
yt_1_ma <- ma[-300]
yt_1_ma## [1] 0.376486659 2.891083265 1.272939391 1.049329503 0.528455337
## [6] -0.660323932 -0.867766297 -1.167038680 -1.276285900 -0.756024562
## [11] 0.102562145 -1.673677061 -0.414898173 -2.635847349 0.643421450
## [16] -2.470738795 0.807116406 -2.435734193 0.789594500 -1.211021488
## [21] 0.309792804 -2.591360944 -1.551970879 -1.163496898 0.355554852
## [26] 0.035818895 0.190514914 -0.481533808 0.335096222 0.439786757
## [31] -0.586768006 -0.180757528 0.476906074 1.377054627 0.453350539
## [36] 1.089294028 -1.102237824 0.290631381 -1.579802794 -0.654088463
## [41] -1.806967638 -1.015694637 -2.271583889 -1.296964313 -1.282248111
## [46] -3.416281679 -1.394294840 -2.096391138 -0.385318133 0.433415090
## [51] -0.571718551 -0.781329468 -2.274489251 -2.495999349 -2.896649704
## [56] -1.507575269 0.167025748 -1.362883701 -1.163388675 -0.453860137
## [61] -0.082191647 2.081975145 -0.731771731 0.447599146 -0.949067464
## [66] -0.981865970 -1.253537117 -0.652734853 0.230514709 0.508389876
## [71] -0.335137033 -0.577333952 1.125884853 1.884175410 1.868599712
## [76] -0.373986078 0.540078189 0.326100544 0.922779061 -0.676952910
## [81] -1.021652887 -2.455669492 -0.525253954 -0.756449866 0.242026733
## [86] -0.793622237 -2.118250559 -1.171944294 -1.254756058 0.156409896
## [91] 0.881224244 1.963714779 1.319895662 -0.541030517 0.372948487
## [96] -1.269736070 1.185564889 -0.749225458 1.348942202 0.610003501
## [101] 1.471093899 1.131809788 -0.449774717 0.528073425 -1.069169086
## [106] 0.655515959 0.813411637 1.980547838 0.292419372 2.056655613
## [111] 1.452182364 1.443411029 1.538886178 -0.586702602 -0.802033556
## [116] -1.632108730 0.130561699 -0.916958534 1.760703550 1.559644810
## [121] 1.713929615 1.637894941 0.880972909 1.384500177 2.096981934
## [126] 0.884207740 -0.279617376 0.291208970 -0.768094492 -1.086892414
## [131] -0.560696861 0.318145290 0.006487837 -0.294477306 -2.566374703
## [136] -1.711312552 -1.945850023 1.209309432 0.787209035 1.817054541
## [141] 0.042340797 0.371645997 -1.666279378 -0.450065553 -1.926881708
## [146] 0.845610896 0.317984728 2.453425058 0.616148527 -1.080111772
## [151] -0.732158983 -2.282823453 -0.856121205 -0.284745928 -2.527788313
## [156] -1.583975853 -0.262485375 0.350113476 2.062510435 1.118959736
## [161] 0.810890964 0.361658001 -0.762405075 0.174734931 -1.333189977
## [166] -1.444625441 -2.294630100 -2.423037666 0.913607099 1.283219664
## [171] 2.669925684 2.144997162 1.016223907 2.781621020 2.229933096
## [176] 1.563078926 2.121133394 -0.124324765 0.666300793 -1.560166674
## [181] -1.736651137 -3.875143179 -3.522676167 -2.360833581 -1.000336996
## [186] -0.843211137 -0.135650765 -0.709671480 1.390898586 -0.096231517
## [191] 0.739315875 0.951040561 0.324689822 0.833814176 1.129219631
## [196] 1.207678300 -1.383559378 -0.683797515 0.092205088 0.132851910
## [201] -0.348714809 -0.020167567 -1.108436389 -2.394337109 -0.581490203
## [206] -2.833926427 -1.884126069 0.288866728 0.292297373 0.451081021
## [211] -0.030659568 0.245960795 1.332739654 0.941277885 1.933980662
## [216] 0.055740608 0.995855870 1.134146672 1.705224826 1.257882265
## [221] 0.688890766 0.979865547 0.082975177 0.745006291 0.426795522
## [226] 1.023379835 0.122487130 -0.648209322 -3.208880364 -0.473769727
## [231] -0.133838137 1.866038951 1.498457094 -0.849790736 -0.178248189
## [236] -1.921776375 -0.629875868 -1.580832048 0.598432589 -0.159793430
## [241] 0.880989211 0.987022925 0.514417044 0.074549124 0.662336368
## [246] -1.918440785 -0.045150758 -0.469879540 0.811847904 0.304228772
## [251] -0.298448803 -1.353749214 -1.554844881 -0.290228437 -1.934766491
## [256] -0.092434755 -3.369647389 -2.324057797 -1.958702604 -0.530963197
## [261] -0.908340921 0.324453692 -1.154031550 -1.415538487 -0.882397739
## [266] -0.111151421 -0.652326891 0.270556538 -0.805420362 0.272980086
## [271] -0.057273789 0.859319788 -1.786286310 -2.927631672 -2.497477692
## [276] -0.851240606 0.312379603 -0.187899331 -0.890509851 -1.178938323
## [281] 0.274957064 -1.013144834 -0.549619209 -0.103440636 -0.030645415
## [286] 1.420424337 -0.710422968 1.709037642 -1.476237380 1.602330567
## [291] -2.399768731 0.667900943 -1.152654377 0.723381614 -0.155364848
## [296] 1.416746072 0.894366166 1.313866117 -2.067594319plot(y=yt_ma,x=yt_1_ma)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif lemah antara \(Y_t\) dengan \(Y_{t-1}\). Hal ini sesuai dengan teori yang ada
cor(yt_ma,yt_1_ma)## [1] 0.4385996Korelasi antara \(Y_t\) dengan \(Y_{t-1}\) hasil simulasi mendekati teori yang ada yaitu
\[ \rho_1=\frac{-\theta_{1}+\theta_{1}\theta_{2}}{1+\theta_{1}^2+\theta_{2}^2}=\frac{(-0.4)+(0.4)(0.6)}{1+(0.4)^2+(0.6)^2}=0.105 \]
Korelasi antara \(Y_t\) dengan \(Y_{t-2}\)
#Yt
yt_ma2 <- ma[-c(1,2)]
yt_ma2## [1] 1.272939391 1.049329503 0.528455337 -0.660323932 -0.867766297
## [6] -1.167038680 -1.276285900 -0.756024562 0.102562145 -1.673677061
## [11] -0.414898173 -2.635847349 0.643421450 -2.470738795 0.807116406
## [16] -2.435734193 0.789594500 -1.211021488 0.309792804 -2.591360944
## [21] -1.551970879 -1.163496898 0.355554852 0.035818895 0.190514914
## [26] -0.481533808 0.335096222 0.439786757 -0.586768006 -0.180757528
## [31] 0.476906074 1.377054627 0.453350539 1.089294028 -1.102237824
## [36] 0.290631381 -1.579802794 -0.654088463 -1.806967638 -1.015694637
## [41] -2.271583889 -1.296964313 -1.282248111 -3.416281679 -1.394294840
## [46] -2.096391138 -0.385318133 0.433415090 -0.571718551 -0.781329468
## [51] -2.274489251 -2.495999349 -2.896649704 -1.507575269 0.167025748
## [56] -1.362883701 -1.163388675 -0.453860137 -0.082191647 2.081975145
## [61] -0.731771731 0.447599146 -0.949067464 -0.981865970 -1.253537117
## [66] -0.652734853 0.230514709 0.508389876 -0.335137033 -0.577333952
## [71] 1.125884853 1.884175410 1.868599712 -0.373986078 0.540078189
## [76] 0.326100544 0.922779061 -0.676952910 -1.021652887 -2.455669492
## [81] -0.525253954 -0.756449866 0.242026733 -0.793622237 -2.118250559
## [86] -1.171944294 -1.254756058 0.156409896 0.881224244 1.963714779
## [91] 1.319895662 -0.541030517 0.372948487 -1.269736070 1.185564889
## [96] -0.749225458 1.348942202 0.610003501 1.471093899 1.131809788
## [101] -0.449774717 0.528073425 -1.069169086 0.655515959 0.813411637
## [106] 1.980547838 0.292419372 2.056655613 1.452182364 1.443411029
## [111] 1.538886178 -0.586702602 -0.802033556 -1.632108730 0.130561699
## [116] -0.916958534 1.760703550 1.559644810 1.713929615 1.637894941
## [121] 0.880972909 1.384500177 2.096981934 0.884207740 -0.279617376
## [126] 0.291208970 -0.768094492 -1.086892414 -0.560696861 0.318145290
## [131] 0.006487837 -0.294477306 -2.566374703 -1.711312552 -1.945850023
## [136] 1.209309432 0.787209035 1.817054541 0.042340797 0.371645997
## [141] -1.666279378 -0.450065553 -1.926881708 0.845610896 0.317984728
## [146] 2.453425058 0.616148527 -1.080111772 -0.732158983 -2.282823453
## [151] -0.856121205 -0.284745928 -2.527788313 -1.583975853 -0.262485375
## [156] 0.350113476 2.062510435 1.118959736 0.810890964 0.361658001
## [161] -0.762405075 0.174734931 -1.333189977 -1.444625441 -2.294630100
## [166] -2.423037666 0.913607099 1.283219664 2.669925684 2.144997162
## [171] 1.016223907 2.781621020 2.229933096 1.563078926 2.121133394
## [176] -0.124324765 0.666300793 -1.560166674 -1.736651137 -3.875143179
## [181] -3.522676167 -2.360833581 -1.000336996 -0.843211137 -0.135650765
## [186] -0.709671480 1.390898586 -0.096231517 0.739315875 0.951040561
## [191] 0.324689822 0.833814176 1.129219631 1.207678300 -1.383559378
## [196] -0.683797515 0.092205088 0.132851910 -0.348714809 -0.020167567
## [201] -1.108436389 -2.394337109 -0.581490203 -2.833926427 -1.884126069
## [206] 0.288866728 0.292297373 0.451081021 -0.030659568 0.245960795
## [211] 1.332739654 0.941277885 1.933980662 0.055740608 0.995855870
## [216] 1.134146672 1.705224826 1.257882265 0.688890766 0.979865547
## [221] 0.082975177 0.745006291 0.426795522 1.023379835 0.122487130
## [226] -0.648209322 -3.208880364 -0.473769727 -0.133838137 1.866038951
## [231] 1.498457094 -0.849790736 -0.178248189 -1.921776375 -0.629875868
## [236] -1.580832048 0.598432589 -0.159793430 0.880989211 0.987022925
## [241] 0.514417044 0.074549124 0.662336368 -1.918440785 -0.045150758
## [246] -0.469879540 0.811847904 0.304228772 -0.298448803 -1.353749214
## [251] -1.554844881 -0.290228437 -1.934766491 -0.092434755 -3.369647389
## [256] -2.324057797 -1.958702604 -0.530963197 -0.908340921 0.324453692
## [261] -1.154031550 -1.415538487 -0.882397739 -0.111151421 -0.652326891
## [266] 0.270556538 -0.805420362 0.272980086 -0.057273789 0.859319788
## [271] -1.786286310 -2.927631672 -2.497477692 -0.851240606 0.312379603
## [276] -0.187899331 -0.890509851 -1.178938323 0.274957064 -1.013144834
## [281] -0.549619209 -0.103440636 -0.030645415 1.420424337 -0.710422968
## [286] 1.709037642 -1.476237380 1.602330567 -2.399768731 0.667900943
## [291] -1.152654377 0.723381614 -0.155364848 1.416746072 0.894366166
## [296] 1.313866117 -2.067594319 -1.228167575#Yt-2
yt_2_ma <- ma[-c(299,300)]
yt_2_ma## [1] 0.376486659 2.891083265 1.272939391 1.049329503 0.528455337
## [6] -0.660323932 -0.867766297 -1.167038680 -1.276285900 -0.756024562
## [11] 0.102562145 -1.673677061 -0.414898173 -2.635847349 0.643421450
## [16] -2.470738795 0.807116406 -2.435734193 0.789594500 -1.211021488
## [21] 0.309792804 -2.591360944 -1.551970879 -1.163496898 0.355554852
## [26] 0.035818895 0.190514914 -0.481533808 0.335096222 0.439786757
## [31] -0.586768006 -0.180757528 0.476906074 1.377054627 0.453350539
## [36] 1.089294028 -1.102237824 0.290631381 -1.579802794 -0.654088463
## [41] -1.806967638 -1.015694637 -2.271583889 -1.296964313 -1.282248111
## [46] -3.416281679 -1.394294840 -2.096391138 -0.385318133 0.433415090
## [51] -0.571718551 -0.781329468 -2.274489251 -2.495999349 -2.896649704
## [56] -1.507575269 0.167025748 -1.362883701 -1.163388675 -0.453860137
## [61] -0.082191647 2.081975145 -0.731771731 0.447599146 -0.949067464
## [66] -0.981865970 -1.253537117 -0.652734853 0.230514709 0.508389876
## [71] -0.335137033 -0.577333952 1.125884853 1.884175410 1.868599712
## [76] -0.373986078 0.540078189 0.326100544 0.922779061 -0.676952910
## [81] -1.021652887 -2.455669492 -0.525253954 -0.756449866 0.242026733
## [86] -0.793622237 -2.118250559 -1.171944294 -1.254756058 0.156409896
## [91] 0.881224244 1.963714779 1.319895662 -0.541030517 0.372948487
## [96] -1.269736070 1.185564889 -0.749225458 1.348942202 0.610003501
## [101] 1.471093899 1.131809788 -0.449774717 0.528073425 -1.069169086
## [106] 0.655515959 0.813411637 1.980547838 0.292419372 2.056655613
## [111] 1.452182364 1.443411029 1.538886178 -0.586702602 -0.802033556
## [116] -1.632108730 0.130561699 -0.916958534 1.760703550 1.559644810
## [121] 1.713929615 1.637894941 0.880972909 1.384500177 2.096981934
## [126] 0.884207740 -0.279617376 0.291208970 -0.768094492 -1.086892414
## [131] -0.560696861 0.318145290 0.006487837 -0.294477306 -2.566374703
## [136] -1.711312552 -1.945850023 1.209309432 0.787209035 1.817054541
## [141] 0.042340797 0.371645997 -1.666279378 -0.450065553 -1.926881708
## [146] 0.845610896 0.317984728 2.453425058 0.616148527 -1.080111772
## [151] -0.732158983 -2.282823453 -0.856121205 -0.284745928 -2.527788313
## [156] -1.583975853 -0.262485375 0.350113476 2.062510435 1.118959736
## [161] 0.810890964 0.361658001 -0.762405075 0.174734931 -1.333189977
## [166] -1.444625441 -2.294630100 -2.423037666 0.913607099 1.283219664
## [171] 2.669925684 2.144997162 1.016223907 2.781621020 2.229933096
## [176] 1.563078926 2.121133394 -0.124324765 0.666300793 -1.560166674
## [181] -1.736651137 -3.875143179 -3.522676167 -2.360833581 -1.000336996
## [186] -0.843211137 -0.135650765 -0.709671480 1.390898586 -0.096231517
## [191] 0.739315875 0.951040561 0.324689822 0.833814176 1.129219631
## [196] 1.207678300 -1.383559378 -0.683797515 0.092205088 0.132851910
## [201] -0.348714809 -0.020167567 -1.108436389 -2.394337109 -0.581490203
## [206] -2.833926427 -1.884126069 0.288866728 0.292297373 0.451081021
## [211] -0.030659568 0.245960795 1.332739654 0.941277885 1.933980662
## [216] 0.055740608 0.995855870 1.134146672 1.705224826 1.257882265
## [221] 0.688890766 0.979865547 0.082975177 0.745006291 0.426795522
## [226] 1.023379835 0.122487130 -0.648209322 -3.208880364 -0.473769727
## [231] -0.133838137 1.866038951 1.498457094 -0.849790736 -0.178248189
## [236] -1.921776375 -0.629875868 -1.580832048 0.598432589 -0.159793430
## [241] 0.880989211 0.987022925 0.514417044 0.074549124 0.662336368
## [246] -1.918440785 -0.045150758 -0.469879540 0.811847904 0.304228772
## [251] -0.298448803 -1.353749214 -1.554844881 -0.290228437 -1.934766491
## [256] -0.092434755 -3.369647389 -2.324057797 -1.958702604 -0.530963197
## [261] -0.908340921 0.324453692 -1.154031550 -1.415538487 -0.882397739
## [266] -0.111151421 -0.652326891 0.270556538 -0.805420362 0.272980086
## [271] -0.057273789 0.859319788 -1.786286310 -2.927631672 -2.497477692
## [276] -0.851240606 0.312379603 -0.187899331 -0.890509851 -1.178938323
## [281] 0.274957064 -1.013144834 -0.549619209 -0.103440636 -0.030645415
## [286] 1.420424337 -0.710422968 1.709037642 -1.476237380 1.602330567
## [291] -2.399768731 0.667900943 -1.152654377 0.723381614 -0.155364848
## [296] 1.416746072 0.894366166 1.313866117plot(y=yt_ma2,x=yt_2_ma)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan yang positif antara \(Y_t\) dengan \(Y_{t-2}\).
cor(yt_ma2,yt_2_ma)## [1] 0.4618698Korelasi antara \(Y_t\) dengan \(Y_{t-2}\) hasil simulasi mendekati teori yang ada yaitu
\[ \rho_2=\frac{-\theta_{2}}{1+\theta_{1}^2+\theta_{2}^2}=\frac{-(-0.4)}{1+(0.4)^2+(0.6)^2}=0.2631 \]
Korelasi antara \(Y_t\) dengan \(Y_{t-3}\)
#Yt
yt_ma3 <- ma[-c(1,2,3)]
yt_ma3## [1] 1.049329503 0.528455337 -0.660323932 -0.867766297 -1.167038680
## [6] -1.276285900 -0.756024562 0.102562145 -1.673677061 -0.414898173
## [11] -2.635847349 0.643421450 -2.470738795 0.807116406 -2.435734193
## [16] 0.789594500 -1.211021488 0.309792804 -2.591360944 -1.551970879
## [21] -1.163496898 0.355554852 0.035818895 0.190514914 -0.481533808
## [26] 0.335096222 0.439786757 -0.586768006 -0.180757528 0.476906074
## [31] 1.377054627 0.453350539 1.089294028 -1.102237824 0.290631381
## [36] -1.579802794 -0.654088463 -1.806967638 -1.015694637 -2.271583889
## [41] -1.296964313 -1.282248111 -3.416281679 -1.394294840 -2.096391138
## [46] -0.385318133 0.433415090 -0.571718551 -0.781329468 -2.274489251
## [51] -2.495999349 -2.896649704 -1.507575269 0.167025748 -1.362883701
## [56] -1.163388675 -0.453860137 -0.082191647 2.081975145 -0.731771731
## [61] 0.447599146 -0.949067464 -0.981865970 -1.253537117 -0.652734853
## [66] 0.230514709 0.508389876 -0.335137033 -0.577333952 1.125884853
## [71] 1.884175410 1.868599712 -0.373986078 0.540078189 0.326100544
## [76] 0.922779061 -0.676952910 -1.021652887 -2.455669492 -0.525253954
## [81] -0.756449866 0.242026733 -0.793622237 -2.118250559 -1.171944294
## [86] -1.254756058 0.156409896 0.881224244 1.963714779 1.319895662
## [91] -0.541030517 0.372948487 -1.269736070 1.185564889 -0.749225458
## [96] 1.348942202 0.610003501 1.471093899 1.131809788 -0.449774717
## [101] 0.528073425 -1.069169086 0.655515959 0.813411637 1.980547838
## [106] 0.292419372 2.056655613 1.452182364 1.443411029 1.538886178
## [111] -0.586702602 -0.802033556 -1.632108730 0.130561699 -0.916958534
## [116] 1.760703550 1.559644810 1.713929615 1.637894941 0.880972909
## [121] 1.384500177 2.096981934 0.884207740 -0.279617376 0.291208970
## [126] -0.768094492 -1.086892414 -0.560696861 0.318145290 0.006487837
## [131] -0.294477306 -2.566374703 -1.711312552 -1.945850023 1.209309432
## [136] 0.787209035 1.817054541 0.042340797 0.371645997 -1.666279378
## [141] -0.450065553 -1.926881708 0.845610896 0.317984728 2.453425058
## [146] 0.616148527 -1.080111772 -0.732158983 -2.282823453 -0.856121205
## [151] -0.284745928 -2.527788313 -1.583975853 -0.262485375 0.350113476
## [156] 2.062510435 1.118959736 0.810890964 0.361658001 -0.762405075
## [161] 0.174734931 -1.333189977 -1.444625441 -2.294630100 -2.423037666
## [166] 0.913607099 1.283219664 2.669925684 2.144997162 1.016223907
## [171] 2.781621020 2.229933096 1.563078926 2.121133394 -0.124324765
## [176] 0.666300793 -1.560166674 -1.736651137 -3.875143179 -3.522676167
## [181] -2.360833581 -1.000336996 -0.843211137 -0.135650765 -0.709671480
## [186] 1.390898586 -0.096231517 0.739315875 0.951040561 0.324689822
## [191] 0.833814176 1.129219631 1.207678300 -1.383559378 -0.683797515
## [196] 0.092205088 0.132851910 -0.348714809 -0.020167567 -1.108436389
## [201] -2.394337109 -0.581490203 -2.833926427 -1.884126069 0.288866728
## [206] 0.292297373 0.451081021 -0.030659568 0.245960795 1.332739654
## [211] 0.941277885 1.933980662 0.055740608 0.995855870 1.134146672
## [216] 1.705224826 1.257882265 0.688890766 0.979865547 0.082975177
## [221] 0.745006291 0.426795522 1.023379835 0.122487130 -0.648209322
## [226] -3.208880364 -0.473769727 -0.133838137 1.866038951 1.498457094
## [231] -0.849790736 -0.178248189 -1.921776375 -0.629875868 -1.580832048
## [236] 0.598432589 -0.159793430 0.880989211 0.987022925 0.514417044
## [241] 0.074549124 0.662336368 -1.918440785 -0.045150758 -0.469879540
## [246] 0.811847904 0.304228772 -0.298448803 -1.353749214 -1.554844881
## [251] -0.290228437 -1.934766491 -0.092434755 -3.369647389 -2.324057797
## [256] -1.958702604 -0.530963197 -0.908340921 0.324453692 -1.154031550
## [261] -1.415538487 -0.882397739 -0.111151421 -0.652326891 0.270556538
## [266] -0.805420362 0.272980086 -0.057273789 0.859319788 -1.786286310
## [271] -2.927631672 -2.497477692 -0.851240606 0.312379603 -0.187899331
## [276] -0.890509851 -1.178938323 0.274957064 -1.013144834 -0.549619209
## [281] -0.103440636 -0.030645415 1.420424337 -0.710422968 1.709037642
## [286] -1.476237380 1.602330567 -2.399768731 0.667900943 -1.152654377
## [291] 0.723381614 -0.155364848 1.416746072 0.894366166 1.313866117
## [296] -2.067594319 -1.228167575#Yt-3
yt_3_ma <- ma[-c(298,299,300)]
yt_3_ma## [1] 0.376486659 2.891083265 1.272939391 1.049329503 0.528455337
## [6] -0.660323932 -0.867766297 -1.167038680 -1.276285900 -0.756024562
## [11] 0.102562145 -1.673677061 -0.414898173 -2.635847349 0.643421450
## [16] -2.470738795 0.807116406 -2.435734193 0.789594500 -1.211021488
## [21] 0.309792804 -2.591360944 -1.551970879 -1.163496898 0.355554852
## [26] 0.035818895 0.190514914 -0.481533808 0.335096222 0.439786757
## [31] -0.586768006 -0.180757528 0.476906074 1.377054627 0.453350539
## [36] 1.089294028 -1.102237824 0.290631381 -1.579802794 -0.654088463
## [41] -1.806967638 -1.015694637 -2.271583889 -1.296964313 -1.282248111
## [46] -3.416281679 -1.394294840 -2.096391138 -0.385318133 0.433415090
## [51] -0.571718551 -0.781329468 -2.274489251 -2.495999349 -2.896649704
## [56] -1.507575269 0.167025748 -1.362883701 -1.163388675 -0.453860137
## [61] -0.082191647 2.081975145 -0.731771731 0.447599146 -0.949067464
## [66] -0.981865970 -1.253537117 -0.652734853 0.230514709 0.508389876
## [71] -0.335137033 -0.577333952 1.125884853 1.884175410 1.868599712
## [76] -0.373986078 0.540078189 0.326100544 0.922779061 -0.676952910
## [81] -1.021652887 -2.455669492 -0.525253954 -0.756449866 0.242026733
## [86] -0.793622237 -2.118250559 -1.171944294 -1.254756058 0.156409896
## [91] 0.881224244 1.963714779 1.319895662 -0.541030517 0.372948487
## [96] -1.269736070 1.185564889 -0.749225458 1.348942202 0.610003501
## [101] 1.471093899 1.131809788 -0.449774717 0.528073425 -1.069169086
## [106] 0.655515959 0.813411637 1.980547838 0.292419372 2.056655613
## [111] 1.452182364 1.443411029 1.538886178 -0.586702602 -0.802033556
## [116] -1.632108730 0.130561699 -0.916958534 1.760703550 1.559644810
## [121] 1.713929615 1.637894941 0.880972909 1.384500177 2.096981934
## [126] 0.884207740 -0.279617376 0.291208970 -0.768094492 -1.086892414
## [131] -0.560696861 0.318145290 0.006487837 -0.294477306 -2.566374703
## [136] -1.711312552 -1.945850023 1.209309432 0.787209035 1.817054541
## [141] 0.042340797 0.371645997 -1.666279378 -0.450065553 -1.926881708
## [146] 0.845610896 0.317984728 2.453425058 0.616148527 -1.080111772
## [151] -0.732158983 -2.282823453 -0.856121205 -0.284745928 -2.527788313
## [156] -1.583975853 -0.262485375 0.350113476 2.062510435 1.118959736
## [161] 0.810890964 0.361658001 -0.762405075 0.174734931 -1.333189977
## [166] -1.444625441 -2.294630100 -2.423037666 0.913607099 1.283219664
## [171] 2.669925684 2.144997162 1.016223907 2.781621020 2.229933096
## [176] 1.563078926 2.121133394 -0.124324765 0.666300793 -1.560166674
## [181] -1.736651137 -3.875143179 -3.522676167 -2.360833581 -1.000336996
## [186] -0.843211137 -0.135650765 -0.709671480 1.390898586 -0.096231517
## [191] 0.739315875 0.951040561 0.324689822 0.833814176 1.129219631
## [196] 1.207678300 -1.383559378 -0.683797515 0.092205088 0.132851910
## [201] -0.348714809 -0.020167567 -1.108436389 -2.394337109 -0.581490203
## [206] -2.833926427 -1.884126069 0.288866728 0.292297373 0.451081021
## [211] -0.030659568 0.245960795 1.332739654 0.941277885 1.933980662
## [216] 0.055740608 0.995855870 1.134146672 1.705224826 1.257882265
## [221] 0.688890766 0.979865547 0.082975177 0.745006291 0.426795522
## [226] 1.023379835 0.122487130 -0.648209322 -3.208880364 -0.473769727
## [231] -0.133838137 1.866038951 1.498457094 -0.849790736 -0.178248189
## [236] -1.921776375 -0.629875868 -1.580832048 0.598432589 -0.159793430
## [241] 0.880989211 0.987022925 0.514417044 0.074549124 0.662336368
## [246] -1.918440785 -0.045150758 -0.469879540 0.811847904 0.304228772
## [251] -0.298448803 -1.353749214 -1.554844881 -0.290228437 -1.934766491
## [256] -0.092434755 -3.369647389 -2.324057797 -1.958702604 -0.530963197
## [261] -0.908340921 0.324453692 -1.154031550 -1.415538487 -0.882397739
## [266] -0.111151421 -0.652326891 0.270556538 -0.805420362 0.272980086
## [271] -0.057273789 0.859319788 -1.786286310 -2.927631672 -2.497477692
## [276] -0.851240606 0.312379603 -0.187899331 -0.890509851 -1.178938323
## [281] 0.274957064 -1.013144834 -0.549619209 -0.103440636 -0.030645415
## [286] 1.420424337 -0.710422968 1.709037642 -1.476237380 1.602330567
## [291] -2.399768731 0.667900943 -1.152654377 0.723381614 -0.155364848
## [296] 1.416746072 0.894366166plot(y=yt_ma3,x=yt_3_ma)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan negatif yang lemah antara \(Y_t\) dengan \(Y_{t-3}\).
cor(yt_ma3,yt_3_ma)## [1] 0.02760745Korelasi antara \(Y_t\) dengan \(Y_{t-3}\) hasil simulasi mendekati teori yang ada yaitu 0.
2. AR(2)
Bangkitkan model AR(2) dengan 𝜙1 = 0.5 dan 𝜙2 = 0.2 sebanyak300 data (Manual dan dengan arima.sim)
Pembangkitan Proses AR
Akan dicoba membangkitkan proses AR paling sederhana, yaitu AR(2) dengan \(\phi_1 = 0.5\) dan \(\phi_2 = 0.2\) sebanyak 300 observasi dan \(c=0\).
set.seed(023)Nilai-nilai selanjutnya dapat dicari melalui loop. Bentuk loop dapat dilihat dari rumus AR(2) yang hendak dibangkitkan:
\[ Y_t = e_t+0.5Y_{t-1}+0.2Y_{t-2} \]
n<-length(wn)
n## [1] 300ar <- c(1:n)
for (i in 3:n) {ar[i]<-wn[i]+0.5*ar[i-1]+0.2*ar[i-2]}
ar## [1] 1.000000000 2.000000000 1.090614090 0.303740953 1.220706614
## [6] 0.055431893 -0.860069177 -0.763814548 -1.013104606 -1.024746573
## [11] -0.190749421 -1.964440185 -1.084168557 -2.546830537 -0.163805092
## [16] -2.625470071 -0.520565413 -2.330562320 -0.356683887 -1.293448679
## [21] -0.696296818 -2.817508246 -2.228774904 -1.242679372 -0.477166856
## [26] -0.948397369 -0.548562602 -0.677155524 -0.040557833 0.248900606
## [31] -0.876912990 -0.414899587 0.700514003 1.226727082 0.173029045
## [36] 1.077658886 -0.478859773 0.240171913 -1.029739786 -0.857744472
## [41] -1.652991974 -1.371925429 -2.527689711 -2.006414485 -1.697042317
## [46] -4.309856156 -2.551632262 -2.375228952 -1.953904993 -0.773732617
## [51] -1.467093687 -1.800803685 -2.689637876 -3.055145914 -3.524593363
## [56] -2.487164257 -0.860483997 -2.657462237 -2.625111721 -0.810322492
## [61] -0.751295282 0.852472193 -1.119201409 -0.217546339 -0.513197817
## [66] -1.312676356 -1.499136967 -0.761234010 -0.106182052 -0.077174738
## [71] -0.790783457 -0.772674611 1.164705837 1.845422234 1.431690401
## [76] -0.250034220 1.069761837 1.248576592 0.910474198 -0.459115417
## [81] -0.646828743 -1.932719897 -0.654367150 -0.736231374 -0.512775984
## [86] -1.178239465 -2.491811288 -1.468680169 -1.412525932 -0.779383689
## [91] 0.228675243 1.429457566 0.951437457 -0.738407735 0.679455127
## [96] -0.521027869 0.831148632 -0.392321001 1.027024400 1.044483278
## [101] 1.320793202 1.398183204 -0.053968464 0.870279347 -0.281542656
## [106] 0.600579792 1.254008083 1.983280000 0.434380958 2.252030355
## [111] 2.494636578 1.645526980 2.112408810 0.513638425 -0.248494834
## [116] -0.759507313 0.552981593 -0.742550338 1.257420039 1.952889009
## [121] 1.441923171 1.780223030 1.663598806 1.977092938 2.811486998
## [126] 1.615732162 0.270585613 1.300737311 0.259218777 -1.024407605
## [131] -0.188230925 0.758795331 -0.238155081 -0.664508544 -2.319440026
## [136] -1.809798337 -1.944033708 0.415737494 0.251533355 0.803151768
## [141] -0.002954494 0.356191367 -1.294687668 -0.555517088 -1.588497703
## [146] 0.348626193 0.304087875 1.743817428 0.671570924 -1.204329630
## [151] -0.287764963 -1.606278981 -1.336252858 -0.350985659 -2.940554116
## [156] -2.508363697 -0.218935815 -0.329636209 0.767289279 0.877416652
## [161] 0.593640716 0.497552323 -0.405884944 0.365509442 -0.955155275
## [166] -1.707772857 -2.184188936 -2.618999860 0.324972582 0.804592356
## [171] 1.428067784 1.936095067 1.268961662 3.160515013 3.199276524
## [176] 1.965440416 3.031093235 1.380683356 1.328931023 -0.315741963
## [181] -1.145620675 -3.255881063 -3.579650362 -2.540847296 -1.913200938
## [186] -2.277334637 -1.375824057 -1.423704990 0.428721624 -0.564711654
## [191] -0.109006398 1.045151871 0.287862455 0.544382218 1.510162304
## [196] 1.484624045 -1.295720697 -0.471106852 1.049615156 0.017815817
## [201] -0.891363346 0.229453872 -0.774494504 -2.854257508 -0.731788183
## [206] -2.603027190 -3.175628345 -0.128681079 -0.162409980 -1.058525795
## [211] -0.534004363 0.327107996 1.050522326 0.650963765 1.849193088
## [216] 0.548918607 1.054338023 1.910605778 2.115910041 1.553870795
## [221] 1.273646596 1.829774557 0.855603421 1.135061923 1.217332953
## [226] 1.462802992 0.559296780 -0.286046290 -2.647399257 -0.293231260
## [231] 0.324736612 0.916821592 1.096076142 -0.835335279 -0.093636347
## [236] -1.237545429 -0.920801410 -1.561225148 0.144966732 -0.331461714
## [241] 0.115087968 0.932564405 0.476096447 -0.060396091 0.929262499
## [246] -1.520610939 -0.348883293 0.145226538 0.429826559 0.003117157
## [251] -0.370782310 -1.210689600 -1.548993672 -0.343513104 -2.163874992
## [256] -0.974019981 -3.350688609 -3.327816731 -2.251139147 -1.480657835
## [261] -2.272831866 -0.861468811 -1.618398733 -2.446439648 -1.403459929
## [266] -0.480577260 -1.543547546 -0.614570295 -0.909202432 -0.339243436
## [271] -0.328120063 0.475111109 -1.911167985 -3.379165529 -2.311992834
## [276] -1.075837285 -0.846228618 -1.341411876 -1.541662624 -1.515489559
## [281] -0.179543314 -1.474802475 -1.422076341 -0.200772981 -0.348277583
## [286] 0.708385868 -0.816871759 1.329235053 -0.952546203 1.015823972
## [291] -1.700592384 0.092217559 -0.531606579 0.127451215 -0.205326825
## [296] 1.179778754 1.038048079 1.118985127 -1.739628715 -1.089895751Selain menggunakan cara di atas, pembangkitan proses AR dapat
dilakukan dengan fungsi arima.sim() sebagai berikut.
ar1 <- arima.sim(list(order=c(2,0,0), ar=c(0.5,0.2)), n=300)
ar1## Time Series:
## Start = 1
## End = 300
## Frequency = 1
## [1] -0.473844877 -0.884413028 0.705892023 -0.484519445 0.065542897
## [6] -0.617337681 -0.197378846 -1.368822819 -1.973814435 -1.462783464
## [11] -0.948654294 0.540763481 -1.262284523 0.338093474 0.149464836
## [16] 1.184608206 1.951295324 2.158527257 0.483897189 -0.542212787
## [21] -1.803532892 -0.917892765 -0.841770384 -0.090985832 -0.443400631
## [26] 1.021618421 0.183693445 -1.701642201 -0.113006453 0.823987347
## [31] -0.461531550 -0.629576062 -0.129859974 0.779873697 1.316595687
## [36] 0.417319239 0.469783285 -0.048569116 0.109567287 1.875833692
## [41] -0.145474710 -0.668049887 -1.696421266 -0.234814007 -1.670069798
## [46] -0.155990192 -0.841933837 -0.584740670 0.570738668 0.175392594
## [51] 0.635253070 1.640074368 -1.471176476 -1.780224758 -1.663747649
## [56] -1.176905256 -0.292458059 1.594453241 1.200293217 1.445053398
## [61] -1.087597856 -0.805621823 -1.199605434 -0.232783104 -0.496066348
## [66] 1.251348060 1.700197634 1.740309082 1.161675689 0.494554890
## [71] 0.197299460 1.537290451 1.854745765 1.984045967 1.940970080
## [76] 1.733127458 2.186179529 -0.467056341 0.255696259 -0.814062911
## [81] -0.724059617 0.099851689 1.125373095 -1.657125926 -0.972163714
## [86] -0.904592700 0.695242477 1.860782626 1.718941889 1.597810157
## [91] 0.834323092 1.251290709 -0.108610987 0.391611459 0.016744165
## [96] -1.144190320 -0.091011149 -0.313325191 -0.395842000 -0.330484264
## [101] -1.375431830 1.424750323 0.465748546 1.770469287 1.234746603
## [106] 0.764177550 1.326564516 1.520390724 0.074829824 0.533270741
## [111] 1.806964715 1.870511636 2.288324042 -0.956346047 0.352197981
## [116] 0.402580172 1.209265875 0.943171881 1.479333534 0.432729911
## [121] 1.106976375 0.403284008 0.095732322 -0.724627701 1.150535030
## [126] 1.700928796 -0.822520783 -0.888958220 -0.697561184 0.848661882
## [131] 0.240749634 -0.646835562 -1.075194216 -3.077529941 -2.914443488
## [136] -2.764697104 -1.747627955 -2.529742200 -1.925074322 -0.750847662
## [141] -1.354604448 -0.448972288 -1.180563418 -1.665157144 -1.469071336
## [146] -0.742060506 -1.833280965 -0.643753313 -0.339877588 0.050651310
## [151] 1.038090957 0.591710734 -0.252603861 -0.410094078 -0.699998032
## [156] -1.431327643 -0.653610368 -0.385076970 -0.115671763 -0.730186502
## [161] 0.136471076 -0.034013433 1.397101667 1.308528422 1.689376123
## [166] -0.312245943 -0.091307422 1.109514109 -0.133172469 -1.140454428
## [171] -0.101636836 0.071654384 0.976165477 0.827814255 -0.858504747
## [176] 0.282603334 -2.263080759 -0.786735542 -0.302149464 -1.500406606
## [181] -0.828325508 -1.661221934 -1.398488496 -0.975368384 -0.923676679
## [186] -1.845417791 0.174824075 -0.707765854 -0.756066528 -1.102218800
## [191] 1.659310283 -0.372517301 -1.991575490 -1.394429555 -0.542346487
## [196] -0.817992500 -1.613063389 -2.868238591 -0.296177064 -2.232524214
## [201] -1.695814346 -2.471745458 -0.345759130 0.452798172 0.081872856
## [206] 0.424783241 0.199870491 -0.212924497 0.588993300 0.743296422
## [211] 2.132352859 1.543426715 2.659576895 1.240394264 0.796516091
## [216] 1.149898529 1.135087175 -0.300795099 0.207199109 0.808099294
## [221] -1.449366864 -0.826101853 -0.910802589 -1.662498298 -2.555630281
## [226] -0.857536739 -1.205264329 0.432466289 -0.822788291 -0.229031629
## [231] -0.409577471 1.137835143 0.236982771 -1.018024812 -1.272183345
## [236] 0.242741579 1.161597736 -0.187314292 0.614661724 0.643031287
## [241] -0.575152488 0.079842720 -1.949520180 0.022399799 1.861418699
## [246] 0.243334794 0.899342666 -0.561495785 -0.522588685 -1.778763924
## [251] -0.681113062 -1.444845461 0.806178461 -0.772729306 -0.889776737
## [256] 0.106456231 -1.069638128 -0.081983568 0.504147039 0.132721291
## [261] 0.231308129 0.682712768 0.390732679 0.207268037 0.022848960
## [266] 0.739499456 -1.335867265 1.155672687 0.395417437 -0.642945835
## [271] -1.282604742 -0.309494703 -1.302950397 -0.284116719 1.975221251
## [276] 0.304437087 1.856300742 0.856060938 -0.808413609 -1.540636262
## [281] -0.977149533 -0.007811559 -0.688677928 0.524259023 1.669615398
## [286] 0.619897457 1.181548319 1.751590183 -1.920436642 -0.042981489
## [291] -0.460509699 0.634765141 0.487619074 0.388611835 -2.403988536
## [296] -0.375178707 -0.723742110 -1.975609414 -1.510747832 -0.733001614Buat plot time series, plot ACF, plot PACF dan plot EACF. Idenifikasi apakah data tersebut stasioner
Plot Time Series
ts.plot(ar)
Berdasarkan plot time series tersebut terlihat bahwa data cenderung stasioner pada rataan
Plot ACF
acf(ar)
Berdasarkan plot ACF tersebut terlihat bahwa plot ACF cenderung tails off dan cenderung membentuk pola grafik sinus
Plot PACF
pacf(ar)
Berdasarkan plot PACF tersebut, terlihat bahwa plot PACF cuts off pada lag pertama, sejalan dengan teori yang ada
Plot EACF
TSA::eacf(ar)## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x x x o o o o o o o o
## 1 x x o x x x o o o o o o o o
## 2 x o o o o x o o o o o o o o
## 3 x o x o o x o o o o o o o o
## 4 x x o o o x o o o o o o o o
## 5 x x o o o o o o o o o o o o
## 6 x x x x x o o o o o o o o o
## 7 x x x o o o o o o o o o o oBerdasarkan pola segitiga nol pada plot EACF, terlihat bahwa segitiga nol berada pada ordo AR(1) dan ordo MA(1)
Dapat disimpulkan bahwa data tersebut stasioner
Buat scatterplot dan korelasi antara \(Y_t\) dengan \(Y_{t-1}\), \(Y_t\) dengan \(Y_{t-2}\), serta \(Y_t\) dengan \(Y_{t-3}\)
Scatterplot Antar Lag
Korelasi antara \(Y_t\) dengan \(Y_{t-1}\)
#Yt
yt_ar <- ar[-1]
yt_ar## [1] 2.000000000 1.090614090 0.303740953 1.220706614 0.055431893
## [6] -0.860069177 -0.763814548 -1.013104606 -1.024746573 -0.190749421
## [11] -1.964440185 -1.084168557 -2.546830537 -0.163805092 -2.625470071
## [16] -0.520565413 -2.330562320 -0.356683887 -1.293448679 -0.696296818
## [21] -2.817508246 -2.228774904 -1.242679372 -0.477166856 -0.948397369
## [26] -0.548562602 -0.677155524 -0.040557833 0.248900606 -0.876912990
## [31] -0.414899587 0.700514003 1.226727082 0.173029045 1.077658886
## [36] -0.478859773 0.240171913 -1.029739786 -0.857744472 -1.652991974
## [41] -1.371925429 -2.527689711 -2.006414485 -1.697042317 -4.309856156
## [46] -2.551632262 -2.375228952 -1.953904993 -0.773732617 -1.467093687
## [51] -1.800803685 -2.689637876 -3.055145914 -3.524593363 -2.487164257
## [56] -0.860483997 -2.657462237 -2.625111721 -0.810322492 -0.751295282
## [61] 0.852472193 -1.119201409 -0.217546339 -0.513197817 -1.312676356
## [66] -1.499136967 -0.761234010 -0.106182052 -0.077174738 -0.790783457
## [71] -0.772674611 1.164705837 1.845422234 1.431690401 -0.250034220
## [76] 1.069761837 1.248576592 0.910474198 -0.459115417 -0.646828743
## [81] -1.932719897 -0.654367150 -0.736231374 -0.512775984 -1.178239465
## [86] -2.491811288 -1.468680169 -1.412525932 -0.779383689 0.228675243
## [91] 1.429457566 0.951437457 -0.738407735 0.679455127 -0.521027869
## [96] 0.831148632 -0.392321001 1.027024400 1.044483278 1.320793202
## [101] 1.398183204 -0.053968464 0.870279347 -0.281542656 0.600579792
## [106] 1.254008083 1.983280000 0.434380958 2.252030355 2.494636578
## [111] 1.645526980 2.112408810 0.513638425 -0.248494834 -0.759507313
## [116] 0.552981593 -0.742550338 1.257420039 1.952889009 1.441923171
## [121] 1.780223030 1.663598806 1.977092938 2.811486998 1.615732162
## [126] 0.270585613 1.300737311 0.259218777 -1.024407605 -0.188230925
## [131] 0.758795331 -0.238155081 -0.664508544 -2.319440026 -1.809798337
## [136] -1.944033708 0.415737494 0.251533355 0.803151768 -0.002954494
## [141] 0.356191367 -1.294687668 -0.555517088 -1.588497703 0.348626193
## [146] 0.304087875 1.743817428 0.671570924 -1.204329630 -0.287764963
## [151] -1.606278981 -1.336252858 -0.350985659 -2.940554116 -2.508363697
## [156] -0.218935815 -0.329636209 0.767289279 0.877416652 0.593640716
## [161] 0.497552323 -0.405884944 0.365509442 -0.955155275 -1.707772857
## [166] -2.184188936 -2.618999860 0.324972582 0.804592356 1.428067784
## [171] 1.936095067 1.268961662 3.160515013 3.199276524 1.965440416
## [176] 3.031093235 1.380683356 1.328931023 -0.315741963 -1.145620675
## [181] -3.255881063 -3.579650362 -2.540847296 -1.913200938 -2.277334637
## [186] -1.375824057 -1.423704990 0.428721624 -0.564711654 -0.109006398
## [191] 1.045151871 0.287862455 0.544382218 1.510162304 1.484624045
## [196] -1.295720697 -0.471106852 1.049615156 0.017815817 -0.891363346
## [201] 0.229453872 -0.774494504 -2.854257508 -0.731788183 -2.603027190
## [206] -3.175628345 -0.128681079 -0.162409980 -1.058525795 -0.534004363
## [211] 0.327107996 1.050522326 0.650963765 1.849193088 0.548918607
## [216] 1.054338023 1.910605778 2.115910041 1.553870795 1.273646596
## [221] 1.829774557 0.855603421 1.135061923 1.217332953 1.462802992
## [226] 0.559296780 -0.286046290 -2.647399257 -0.293231260 0.324736612
## [231] 0.916821592 1.096076142 -0.835335279 -0.093636347 -1.237545429
## [236] -0.920801410 -1.561225148 0.144966732 -0.331461714 0.115087968
## [241] 0.932564405 0.476096447 -0.060396091 0.929262499 -1.520610939
## [246] -0.348883293 0.145226538 0.429826559 0.003117157 -0.370782310
## [251] -1.210689600 -1.548993672 -0.343513104 -2.163874992 -0.974019981
## [256] -3.350688609 -3.327816731 -2.251139147 -1.480657835 -2.272831866
## [261] -0.861468811 -1.618398733 -2.446439648 -1.403459929 -0.480577260
## [266] -1.543547546 -0.614570295 -0.909202432 -0.339243436 -0.328120063
## [271] 0.475111109 -1.911167985 -3.379165529 -2.311992834 -1.075837285
## [276] -0.846228618 -1.341411876 -1.541662624 -1.515489559 -0.179543314
## [281] -1.474802475 -1.422076341 -0.200772981 -0.348277583 0.708385868
## [286] -0.816871759 1.329235053 -0.952546203 1.015823972 -1.700592384
## [291] 0.092217559 -0.531606579 0.127451215 -0.205326825 1.179778754
## [296] 1.038048079 1.118985127 -1.739628715 -1.089895751#Yt-1
yt_1_ar <- ar[-300]
yt_1_ar## [1] 1.000000000 2.000000000 1.090614090 0.303740953 1.220706614
## [6] 0.055431893 -0.860069177 -0.763814548 -1.013104606 -1.024746573
## [11] -0.190749421 -1.964440185 -1.084168557 -2.546830537 -0.163805092
## [16] -2.625470071 -0.520565413 -2.330562320 -0.356683887 -1.293448679
## [21] -0.696296818 -2.817508246 -2.228774904 -1.242679372 -0.477166856
## [26] -0.948397369 -0.548562602 -0.677155524 -0.040557833 0.248900606
## [31] -0.876912990 -0.414899587 0.700514003 1.226727082 0.173029045
## [36] 1.077658886 -0.478859773 0.240171913 -1.029739786 -0.857744472
## [41] -1.652991974 -1.371925429 -2.527689711 -2.006414485 -1.697042317
## [46] -4.309856156 -2.551632262 -2.375228952 -1.953904993 -0.773732617
## [51] -1.467093687 -1.800803685 -2.689637876 -3.055145914 -3.524593363
## [56] -2.487164257 -0.860483997 -2.657462237 -2.625111721 -0.810322492
## [61] -0.751295282 0.852472193 -1.119201409 -0.217546339 -0.513197817
## [66] -1.312676356 -1.499136967 -0.761234010 -0.106182052 -0.077174738
## [71] -0.790783457 -0.772674611 1.164705837 1.845422234 1.431690401
## [76] -0.250034220 1.069761837 1.248576592 0.910474198 -0.459115417
## [81] -0.646828743 -1.932719897 -0.654367150 -0.736231374 -0.512775984
## [86] -1.178239465 -2.491811288 -1.468680169 -1.412525932 -0.779383689
## [91] 0.228675243 1.429457566 0.951437457 -0.738407735 0.679455127
## [96] -0.521027869 0.831148632 -0.392321001 1.027024400 1.044483278
## [101] 1.320793202 1.398183204 -0.053968464 0.870279347 -0.281542656
## [106] 0.600579792 1.254008083 1.983280000 0.434380958 2.252030355
## [111] 2.494636578 1.645526980 2.112408810 0.513638425 -0.248494834
## [116] -0.759507313 0.552981593 -0.742550338 1.257420039 1.952889009
## [121] 1.441923171 1.780223030 1.663598806 1.977092938 2.811486998
## [126] 1.615732162 0.270585613 1.300737311 0.259218777 -1.024407605
## [131] -0.188230925 0.758795331 -0.238155081 -0.664508544 -2.319440026
## [136] -1.809798337 -1.944033708 0.415737494 0.251533355 0.803151768
## [141] -0.002954494 0.356191367 -1.294687668 -0.555517088 -1.588497703
## [146] 0.348626193 0.304087875 1.743817428 0.671570924 -1.204329630
## [151] -0.287764963 -1.606278981 -1.336252858 -0.350985659 -2.940554116
## [156] -2.508363697 -0.218935815 -0.329636209 0.767289279 0.877416652
## [161] 0.593640716 0.497552323 -0.405884944 0.365509442 -0.955155275
## [166] -1.707772857 -2.184188936 -2.618999860 0.324972582 0.804592356
## [171] 1.428067784 1.936095067 1.268961662 3.160515013 3.199276524
## [176] 1.965440416 3.031093235 1.380683356 1.328931023 -0.315741963
## [181] -1.145620675 -3.255881063 -3.579650362 -2.540847296 -1.913200938
## [186] -2.277334637 -1.375824057 -1.423704990 0.428721624 -0.564711654
## [191] -0.109006398 1.045151871 0.287862455 0.544382218 1.510162304
## [196] 1.484624045 -1.295720697 -0.471106852 1.049615156 0.017815817
## [201] -0.891363346 0.229453872 -0.774494504 -2.854257508 -0.731788183
## [206] -2.603027190 -3.175628345 -0.128681079 -0.162409980 -1.058525795
## [211] -0.534004363 0.327107996 1.050522326 0.650963765 1.849193088
## [216] 0.548918607 1.054338023 1.910605778 2.115910041 1.553870795
## [221] 1.273646596 1.829774557 0.855603421 1.135061923 1.217332953
## [226] 1.462802992 0.559296780 -0.286046290 -2.647399257 -0.293231260
## [231] 0.324736612 0.916821592 1.096076142 -0.835335279 -0.093636347
## [236] -1.237545429 -0.920801410 -1.561225148 0.144966732 -0.331461714
## [241] 0.115087968 0.932564405 0.476096447 -0.060396091 0.929262499
## [246] -1.520610939 -0.348883293 0.145226538 0.429826559 0.003117157
## [251] -0.370782310 -1.210689600 -1.548993672 -0.343513104 -2.163874992
## [256] -0.974019981 -3.350688609 -3.327816731 -2.251139147 -1.480657835
## [261] -2.272831866 -0.861468811 -1.618398733 -2.446439648 -1.403459929
## [266] -0.480577260 -1.543547546 -0.614570295 -0.909202432 -0.339243436
## [271] -0.328120063 0.475111109 -1.911167985 -3.379165529 -2.311992834
## [276] -1.075837285 -0.846228618 -1.341411876 -1.541662624 -1.515489559
## [281] -0.179543314 -1.474802475 -1.422076341 -0.200772981 -0.348277583
## [286] 0.708385868 -0.816871759 1.329235053 -0.952546203 1.015823972
## [291] -1.700592384 0.092217559 -0.531606579 0.127451215 -0.205326825
## [296] 1.179778754 1.038048079 1.118985127 -1.739628715plot(y=yt_ar,x=yt_1_ar)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-1}\). Hal ini sesuai dengan teori yang ada
cor(yt_ar,yt_1_ar)## [1] 0.6621678Korelasi antara \(Y_t\) dengan \(Y_{t-1}\) dari hasil simulasi mendekati perhitungan teoritis yaitu
\[ \rho_1=\frac{\phi_{1}}{1-\phi_{2}}=\frac{0.5}{1-0.2}=0.625 \]
Korelasi antara \(Y_t\) dengan \(Y_{t-2}\)
#Yt
yt_ar2 <- ar[-c(1,2)]
yt_ar2## [1] 1.090614090 0.303740953 1.220706614 0.055431893 -0.860069177
## [6] -0.763814548 -1.013104606 -1.024746573 -0.190749421 -1.964440185
## [11] -1.084168557 -2.546830537 -0.163805092 -2.625470071 -0.520565413
## [16] -2.330562320 -0.356683887 -1.293448679 -0.696296818 -2.817508246
## [21] -2.228774904 -1.242679372 -0.477166856 -0.948397369 -0.548562602
## [26] -0.677155524 -0.040557833 0.248900606 -0.876912990 -0.414899587
## [31] 0.700514003 1.226727082 0.173029045 1.077658886 -0.478859773
## [36] 0.240171913 -1.029739786 -0.857744472 -1.652991974 -1.371925429
## [41] -2.527689711 -2.006414485 -1.697042317 -4.309856156 -2.551632262
## [46] -2.375228952 -1.953904993 -0.773732617 -1.467093687 -1.800803685
## [51] -2.689637876 -3.055145914 -3.524593363 -2.487164257 -0.860483997
## [56] -2.657462237 -2.625111721 -0.810322492 -0.751295282 0.852472193
## [61] -1.119201409 -0.217546339 -0.513197817 -1.312676356 -1.499136967
## [66] -0.761234010 -0.106182052 -0.077174738 -0.790783457 -0.772674611
## [71] 1.164705837 1.845422234 1.431690401 -0.250034220 1.069761837
## [76] 1.248576592 0.910474198 -0.459115417 -0.646828743 -1.932719897
## [81] -0.654367150 -0.736231374 -0.512775984 -1.178239465 -2.491811288
## [86] -1.468680169 -1.412525932 -0.779383689 0.228675243 1.429457566
## [91] 0.951437457 -0.738407735 0.679455127 -0.521027869 0.831148632
## [96] -0.392321001 1.027024400 1.044483278 1.320793202 1.398183204
## [101] -0.053968464 0.870279347 -0.281542656 0.600579792 1.254008083
## [106] 1.983280000 0.434380958 2.252030355 2.494636578 1.645526980
## [111] 2.112408810 0.513638425 -0.248494834 -0.759507313 0.552981593
## [116] -0.742550338 1.257420039 1.952889009 1.441923171 1.780223030
## [121] 1.663598806 1.977092938 2.811486998 1.615732162 0.270585613
## [126] 1.300737311 0.259218777 -1.024407605 -0.188230925 0.758795331
## [131] -0.238155081 -0.664508544 -2.319440026 -1.809798337 -1.944033708
## [136] 0.415737494 0.251533355 0.803151768 -0.002954494 0.356191367
## [141] -1.294687668 -0.555517088 -1.588497703 0.348626193 0.304087875
## [146] 1.743817428 0.671570924 -1.204329630 -0.287764963 -1.606278981
## [151] -1.336252858 -0.350985659 -2.940554116 -2.508363697 -0.218935815
## [156] -0.329636209 0.767289279 0.877416652 0.593640716 0.497552323
## [161] -0.405884944 0.365509442 -0.955155275 -1.707772857 -2.184188936
## [166] -2.618999860 0.324972582 0.804592356 1.428067784 1.936095067
## [171] 1.268961662 3.160515013 3.199276524 1.965440416 3.031093235
## [176] 1.380683356 1.328931023 -0.315741963 -1.145620675 -3.255881063
## [181] -3.579650362 -2.540847296 -1.913200938 -2.277334637 -1.375824057
## [186] -1.423704990 0.428721624 -0.564711654 -0.109006398 1.045151871
## [191] 0.287862455 0.544382218 1.510162304 1.484624045 -1.295720697
## [196] -0.471106852 1.049615156 0.017815817 -0.891363346 0.229453872
## [201] -0.774494504 -2.854257508 -0.731788183 -2.603027190 -3.175628345
## [206] -0.128681079 -0.162409980 -1.058525795 -0.534004363 0.327107996
## [211] 1.050522326 0.650963765 1.849193088 0.548918607 1.054338023
## [216] 1.910605778 2.115910041 1.553870795 1.273646596 1.829774557
## [221] 0.855603421 1.135061923 1.217332953 1.462802992 0.559296780
## [226] -0.286046290 -2.647399257 -0.293231260 0.324736612 0.916821592
## [231] 1.096076142 -0.835335279 -0.093636347 -1.237545429 -0.920801410
## [236] -1.561225148 0.144966732 -0.331461714 0.115087968 0.932564405
## [241] 0.476096447 -0.060396091 0.929262499 -1.520610939 -0.348883293
## [246] 0.145226538 0.429826559 0.003117157 -0.370782310 -1.210689600
## [251] -1.548993672 -0.343513104 -2.163874992 -0.974019981 -3.350688609
## [256] -3.327816731 -2.251139147 -1.480657835 -2.272831866 -0.861468811
## [261] -1.618398733 -2.446439648 -1.403459929 -0.480577260 -1.543547546
## [266] -0.614570295 -0.909202432 -0.339243436 -0.328120063 0.475111109
## [271] -1.911167985 -3.379165529 -2.311992834 -1.075837285 -0.846228618
## [276] -1.341411876 -1.541662624 -1.515489559 -0.179543314 -1.474802475
## [281] -1.422076341 -0.200772981 -0.348277583 0.708385868 -0.816871759
## [286] 1.329235053 -0.952546203 1.015823972 -1.700592384 0.092217559
## [291] -0.531606579 0.127451215 -0.205326825 1.179778754 1.038048079
## [296] 1.118985127 -1.739628715 -1.089895751#Yt-2
yt_2_ar <- ar[-c(299,300)]
yt_2_ar## [1] 1.000000000 2.000000000 1.090614090 0.303740953 1.220706614
## [6] 0.055431893 -0.860069177 -0.763814548 -1.013104606 -1.024746573
## [11] -0.190749421 -1.964440185 -1.084168557 -2.546830537 -0.163805092
## [16] -2.625470071 -0.520565413 -2.330562320 -0.356683887 -1.293448679
## [21] -0.696296818 -2.817508246 -2.228774904 -1.242679372 -0.477166856
## [26] -0.948397369 -0.548562602 -0.677155524 -0.040557833 0.248900606
## [31] -0.876912990 -0.414899587 0.700514003 1.226727082 0.173029045
## [36] 1.077658886 -0.478859773 0.240171913 -1.029739786 -0.857744472
## [41] -1.652991974 -1.371925429 -2.527689711 -2.006414485 -1.697042317
## [46] -4.309856156 -2.551632262 -2.375228952 -1.953904993 -0.773732617
## [51] -1.467093687 -1.800803685 -2.689637876 -3.055145914 -3.524593363
## [56] -2.487164257 -0.860483997 -2.657462237 -2.625111721 -0.810322492
## [61] -0.751295282 0.852472193 -1.119201409 -0.217546339 -0.513197817
## [66] -1.312676356 -1.499136967 -0.761234010 -0.106182052 -0.077174738
## [71] -0.790783457 -0.772674611 1.164705837 1.845422234 1.431690401
## [76] -0.250034220 1.069761837 1.248576592 0.910474198 -0.459115417
## [81] -0.646828743 -1.932719897 -0.654367150 -0.736231374 -0.512775984
## [86] -1.178239465 -2.491811288 -1.468680169 -1.412525932 -0.779383689
## [91] 0.228675243 1.429457566 0.951437457 -0.738407735 0.679455127
## [96] -0.521027869 0.831148632 -0.392321001 1.027024400 1.044483278
## [101] 1.320793202 1.398183204 -0.053968464 0.870279347 -0.281542656
## [106] 0.600579792 1.254008083 1.983280000 0.434380958 2.252030355
## [111] 2.494636578 1.645526980 2.112408810 0.513638425 -0.248494834
## [116] -0.759507313 0.552981593 -0.742550338 1.257420039 1.952889009
## [121] 1.441923171 1.780223030 1.663598806 1.977092938 2.811486998
## [126] 1.615732162 0.270585613 1.300737311 0.259218777 -1.024407605
## [131] -0.188230925 0.758795331 -0.238155081 -0.664508544 -2.319440026
## [136] -1.809798337 -1.944033708 0.415737494 0.251533355 0.803151768
## [141] -0.002954494 0.356191367 -1.294687668 -0.555517088 -1.588497703
## [146] 0.348626193 0.304087875 1.743817428 0.671570924 -1.204329630
## [151] -0.287764963 -1.606278981 -1.336252858 -0.350985659 -2.940554116
## [156] -2.508363697 -0.218935815 -0.329636209 0.767289279 0.877416652
## [161] 0.593640716 0.497552323 -0.405884944 0.365509442 -0.955155275
## [166] -1.707772857 -2.184188936 -2.618999860 0.324972582 0.804592356
## [171] 1.428067784 1.936095067 1.268961662 3.160515013 3.199276524
## [176] 1.965440416 3.031093235 1.380683356 1.328931023 -0.315741963
## [181] -1.145620675 -3.255881063 -3.579650362 -2.540847296 -1.913200938
## [186] -2.277334637 -1.375824057 -1.423704990 0.428721624 -0.564711654
## [191] -0.109006398 1.045151871 0.287862455 0.544382218 1.510162304
## [196] 1.484624045 -1.295720697 -0.471106852 1.049615156 0.017815817
## [201] -0.891363346 0.229453872 -0.774494504 -2.854257508 -0.731788183
## [206] -2.603027190 -3.175628345 -0.128681079 -0.162409980 -1.058525795
## [211] -0.534004363 0.327107996 1.050522326 0.650963765 1.849193088
## [216] 0.548918607 1.054338023 1.910605778 2.115910041 1.553870795
## [221] 1.273646596 1.829774557 0.855603421 1.135061923 1.217332953
## [226] 1.462802992 0.559296780 -0.286046290 -2.647399257 -0.293231260
## [231] 0.324736612 0.916821592 1.096076142 -0.835335279 -0.093636347
## [236] -1.237545429 -0.920801410 -1.561225148 0.144966732 -0.331461714
## [241] 0.115087968 0.932564405 0.476096447 -0.060396091 0.929262499
## [246] -1.520610939 -0.348883293 0.145226538 0.429826559 0.003117157
## [251] -0.370782310 -1.210689600 -1.548993672 -0.343513104 -2.163874992
## [256] -0.974019981 -3.350688609 -3.327816731 -2.251139147 -1.480657835
## [261] -2.272831866 -0.861468811 -1.618398733 -2.446439648 -1.403459929
## [266] -0.480577260 -1.543547546 -0.614570295 -0.909202432 -0.339243436
## [271] -0.328120063 0.475111109 -1.911167985 -3.379165529 -2.311992834
## [276] -1.075837285 -0.846228618 -1.341411876 -1.541662624 -1.515489559
## [281] -0.179543314 -1.474802475 -1.422076341 -0.200772981 -0.348277583
## [286] 0.708385868 -0.816871759 1.329235053 -0.952546203 1.015823972
## [291] -1.700592384 0.092217559 -0.531606579 0.127451215 -0.205326825
## [296] 1.179778754 1.038048079 1.118985127plot(y=yt_ar2,x=yt_2_ar)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif yang lemah antara \(Y_t\) dengan \(Y_{t-2}\). Hal ini sesuai dengan teori yang ada
cor(yt_ar2,yt_2_ar)## [1] 0.5738651Korelasi antara \(Y_t\) dengan \(Y_{t-2}\) dari hasil simulasi mendekati perhitungan teoritis yaitu
\[ \rho_2=\frac{\phi_{2}(1-\phi_{2})+\phi_{1}^2}{1-\phi_{2}}=\frac{0.2(1-0.2)+0.5^2}{1-0.2}=0.512 \]
Korelasi antara \(Y_t\) dengan \(Y_{t-3}\)
#Yt
yt_ar3 <- ar[-c(1,2,3)]
yt_ar3## [1] 0.303740953 1.220706614 0.055431893 -0.860069177 -0.763814548
## [6] -1.013104606 -1.024746573 -0.190749421 -1.964440185 -1.084168557
## [11] -2.546830537 -0.163805092 -2.625470071 -0.520565413 -2.330562320
## [16] -0.356683887 -1.293448679 -0.696296818 -2.817508246 -2.228774904
## [21] -1.242679372 -0.477166856 -0.948397369 -0.548562602 -0.677155524
## [26] -0.040557833 0.248900606 -0.876912990 -0.414899587 0.700514003
## [31] 1.226727082 0.173029045 1.077658886 -0.478859773 0.240171913
## [36] -1.029739786 -0.857744472 -1.652991974 -1.371925429 -2.527689711
## [41] -2.006414485 -1.697042317 -4.309856156 -2.551632262 -2.375228952
## [46] -1.953904993 -0.773732617 -1.467093687 -1.800803685 -2.689637876
## [51] -3.055145914 -3.524593363 -2.487164257 -0.860483997 -2.657462237
## [56] -2.625111721 -0.810322492 -0.751295282 0.852472193 -1.119201409
## [61] -0.217546339 -0.513197817 -1.312676356 -1.499136967 -0.761234010
## [66] -0.106182052 -0.077174738 -0.790783457 -0.772674611 1.164705837
## [71] 1.845422234 1.431690401 -0.250034220 1.069761837 1.248576592
## [76] 0.910474198 -0.459115417 -0.646828743 -1.932719897 -0.654367150
## [81] -0.736231374 -0.512775984 -1.178239465 -2.491811288 -1.468680169
## [86] -1.412525932 -0.779383689 0.228675243 1.429457566 0.951437457
## [91] -0.738407735 0.679455127 -0.521027869 0.831148632 -0.392321001
## [96] 1.027024400 1.044483278 1.320793202 1.398183204 -0.053968464
## [101] 0.870279347 -0.281542656 0.600579792 1.254008083 1.983280000
## [106] 0.434380958 2.252030355 2.494636578 1.645526980 2.112408810
## [111] 0.513638425 -0.248494834 -0.759507313 0.552981593 -0.742550338
## [116] 1.257420039 1.952889009 1.441923171 1.780223030 1.663598806
## [121] 1.977092938 2.811486998 1.615732162 0.270585613 1.300737311
## [126] 0.259218777 -1.024407605 -0.188230925 0.758795331 -0.238155081
## [131] -0.664508544 -2.319440026 -1.809798337 -1.944033708 0.415737494
## [136] 0.251533355 0.803151768 -0.002954494 0.356191367 -1.294687668
## [141] -0.555517088 -1.588497703 0.348626193 0.304087875 1.743817428
## [146] 0.671570924 -1.204329630 -0.287764963 -1.606278981 -1.336252858
## [151] -0.350985659 -2.940554116 -2.508363697 -0.218935815 -0.329636209
## [156] 0.767289279 0.877416652 0.593640716 0.497552323 -0.405884944
## [161] 0.365509442 -0.955155275 -1.707772857 -2.184188936 -2.618999860
## [166] 0.324972582 0.804592356 1.428067784 1.936095067 1.268961662
## [171] 3.160515013 3.199276524 1.965440416 3.031093235 1.380683356
## [176] 1.328931023 -0.315741963 -1.145620675 -3.255881063 -3.579650362
## [181] -2.540847296 -1.913200938 -2.277334637 -1.375824057 -1.423704990
## [186] 0.428721624 -0.564711654 -0.109006398 1.045151871 0.287862455
## [191] 0.544382218 1.510162304 1.484624045 -1.295720697 -0.471106852
## [196] 1.049615156 0.017815817 -0.891363346 0.229453872 -0.774494504
## [201] -2.854257508 -0.731788183 -2.603027190 -3.175628345 -0.128681079
## [206] -0.162409980 -1.058525795 -0.534004363 0.327107996 1.050522326
## [211] 0.650963765 1.849193088 0.548918607 1.054338023 1.910605778
## [216] 2.115910041 1.553870795 1.273646596 1.829774557 0.855603421
## [221] 1.135061923 1.217332953 1.462802992 0.559296780 -0.286046290
## [226] -2.647399257 -0.293231260 0.324736612 0.916821592 1.096076142
## [231] -0.835335279 -0.093636347 -1.237545429 -0.920801410 -1.561225148
## [236] 0.144966732 -0.331461714 0.115087968 0.932564405 0.476096447
## [241] -0.060396091 0.929262499 -1.520610939 -0.348883293 0.145226538
## [246] 0.429826559 0.003117157 -0.370782310 -1.210689600 -1.548993672
## [251] -0.343513104 -2.163874992 -0.974019981 -3.350688609 -3.327816731
## [256] -2.251139147 -1.480657835 -2.272831866 -0.861468811 -1.618398733
## [261] -2.446439648 -1.403459929 -0.480577260 -1.543547546 -0.614570295
## [266] -0.909202432 -0.339243436 -0.328120063 0.475111109 -1.911167985
## [271] -3.379165529 -2.311992834 -1.075837285 -0.846228618 -1.341411876
## [276] -1.541662624 -1.515489559 -0.179543314 -1.474802475 -1.422076341
## [281] -0.200772981 -0.348277583 0.708385868 -0.816871759 1.329235053
## [286] -0.952546203 1.015823972 -1.700592384 0.092217559 -0.531606579
## [291] 0.127451215 -0.205326825 1.179778754 1.038048079 1.118985127
## [296] -1.739628715 -1.089895751#Yt-3
yt_3_ar <- ar[-c(298,299,300)]
yt_3_ar## [1] 1.000000000 2.000000000 1.090614090 0.303740953 1.220706614
## [6] 0.055431893 -0.860069177 -0.763814548 -1.013104606 -1.024746573
## [11] -0.190749421 -1.964440185 -1.084168557 -2.546830537 -0.163805092
## [16] -2.625470071 -0.520565413 -2.330562320 -0.356683887 -1.293448679
## [21] -0.696296818 -2.817508246 -2.228774904 -1.242679372 -0.477166856
## [26] -0.948397369 -0.548562602 -0.677155524 -0.040557833 0.248900606
## [31] -0.876912990 -0.414899587 0.700514003 1.226727082 0.173029045
## [36] 1.077658886 -0.478859773 0.240171913 -1.029739786 -0.857744472
## [41] -1.652991974 -1.371925429 -2.527689711 -2.006414485 -1.697042317
## [46] -4.309856156 -2.551632262 -2.375228952 -1.953904993 -0.773732617
## [51] -1.467093687 -1.800803685 -2.689637876 -3.055145914 -3.524593363
## [56] -2.487164257 -0.860483997 -2.657462237 -2.625111721 -0.810322492
## [61] -0.751295282 0.852472193 -1.119201409 -0.217546339 -0.513197817
## [66] -1.312676356 -1.499136967 -0.761234010 -0.106182052 -0.077174738
## [71] -0.790783457 -0.772674611 1.164705837 1.845422234 1.431690401
## [76] -0.250034220 1.069761837 1.248576592 0.910474198 -0.459115417
## [81] -0.646828743 -1.932719897 -0.654367150 -0.736231374 -0.512775984
## [86] -1.178239465 -2.491811288 -1.468680169 -1.412525932 -0.779383689
## [91] 0.228675243 1.429457566 0.951437457 -0.738407735 0.679455127
## [96] -0.521027869 0.831148632 -0.392321001 1.027024400 1.044483278
## [101] 1.320793202 1.398183204 -0.053968464 0.870279347 -0.281542656
## [106] 0.600579792 1.254008083 1.983280000 0.434380958 2.252030355
## [111] 2.494636578 1.645526980 2.112408810 0.513638425 -0.248494834
## [116] -0.759507313 0.552981593 -0.742550338 1.257420039 1.952889009
## [121] 1.441923171 1.780223030 1.663598806 1.977092938 2.811486998
## [126] 1.615732162 0.270585613 1.300737311 0.259218777 -1.024407605
## [131] -0.188230925 0.758795331 -0.238155081 -0.664508544 -2.319440026
## [136] -1.809798337 -1.944033708 0.415737494 0.251533355 0.803151768
## [141] -0.002954494 0.356191367 -1.294687668 -0.555517088 -1.588497703
## [146] 0.348626193 0.304087875 1.743817428 0.671570924 -1.204329630
## [151] -0.287764963 -1.606278981 -1.336252858 -0.350985659 -2.940554116
## [156] -2.508363697 -0.218935815 -0.329636209 0.767289279 0.877416652
## [161] 0.593640716 0.497552323 -0.405884944 0.365509442 -0.955155275
## [166] -1.707772857 -2.184188936 -2.618999860 0.324972582 0.804592356
## [171] 1.428067784 1.936095067 1.268961662 3.160515013 3.199276524
## [176] 1.965440416 3.031093235 1.380683356 1.328931023 -0.315741963
## [181] -1.145620675 -3.255881063 -3.579650362 -2.540847296 -1.913200938
## [186] -2.277334637 -1.375824057 -1.423704990 0.428721624 -0.564711654
## [191] -0.109006398 1.045151871 0.287862455 0.544382218 1.510162304
## [196] 1.484624045 -1.295720697 -0.471106852 1.049615156 0.017815817
## [201] -0.891363346 0.229453872 -0.774494504 -2.854257508 -0.731788183
## [206] -2.603027190 -3.175628345 -0.128681079 -0.162409980 -1.058525795
## [211] -0.534004363 0.327107996 1.050522326 0.650963765 1.849193088
## [216] 0.548918607 1.054338023 1.910605778 2.115910041 1.553870795
## [221] 1.273646596 1.829774557 0.855603421 1.135061923 1.217332953
## [226] 1.462802992 0.559296780 -0.286046290 -2.647399257 -0.293231260
## [231] 0.324736612 0.916821592 1.096076142 -0.835335279 -0.093636347
## [236] -1.237545429 -0.920801410 -1.561225148 0.144966732 -0.331461714
## [241] 0.115087968 0.932564405 0.476096447 -0.060396091 0.929262499
## [246] -1.520610939 -0.348883293 0.145226538 0.429826559 0.003117157
## [251] -0.370782310 -1.210689600 -1.548993672 -0.343513104 -2.163874992
## [256] -0.974019981 -3.350688609 -3.327816731 -2.251139147 -1.480657835
## [261] -2.272831866 -0.861468811 -1.618398733 -2.446439648 -1.403459929
## [266] -0.480577260 -1.543547546 -0.614570295 -0.909202432 -0.339243436
## [271] -0.328120063 0.475111109 -1.911167985 -3.379165529 -2.311992834
## [276] -1.075837285 -0.846228618 -1.341411876 -1.541662624 -1.515489559
## [281] -0.179543314 -1.474802475 -1.422076341 -0.200772981 -0.348277583
## [286] 0.708385868 -0.816871759 1.329235053 -0.952546203 1.015823972
## [291] -1.700592384 0.092217559 -0.531606579 0.127451215 -0.205326825
## [296] 1.179778754 1.038048079plot(y=yt_ar3,x=yt_3_ar)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif yang lemah antara \(Y_t\) dengan \(Y_{t-3}\). Hal ini sesuai dengan teori yang ada
cor(yt_ar3,yt_3_ar)## [1] 0.41631872. ARMA(2,2)
Bangkitkan model AR(2) dengan 𝜃1 = 0.5, 𝜃2 = 0.2 ,𝜙1 = 0.3 dan𝜙2 = 0.7 sebanyak 300 data (Manual dan arima.sim)
Pembangkitan proses ARMA
set.seed(023)
n = length(wn)
phi1 = 0.5
phi2 = 0.2
theta1 = 0.4
theta2 = 0.6
y.arma=c(1:n)
for (i in 3:n){y.arma[i] = phi1*y.arma[i-1] + phi2*y.arma[i-2] + theta1*wn[i-1] + theta2*wn[i-2] + wn[i]}
head(y.arma)## [1] 1.000000 2.000000 2.472939 2.685799 2.365943 1.059807Pembangkitan ARMA(p,q) juga dapat dilakukan dengan fungsi
arima.sim sebagai berikut.
arma22 <- arima.sim(list(order=c(2,0,2), ar = c(0.5,0.2), ma = c(0.4,0.6)), n=300)
arma22## Time Series:
## Start = 1
## End = 300
## Frequency = 1
## [1] 0.06795477 -0.73270762 0.29534872 -0.88177243 -0.40494262 -1.81814223
## [7] -2.63974439 -3.07358274 -2.71804095 -0.71635658 -1.61516276 0.15764457
## [13] -0.47266329 1.44725419 2.51482053 3.64981261 2.51808704 0.94646378
## [19] -1.73007867 -1.96463282 -2.29104663 -0.97842919 -0.98485685 0.78966693
## [25] 0.32630064 -1.01519362 -0.68344715 -0.24220047 -0.19974042 -0.31979622
## [31] -0.65860929 0.35018410 1.55062920 1.41188175 1.42666841 0.38973575
## [37] 0.37200962 1.89051914 0.67059914 0.39926045 -2.05092604 -1.31421244
## [43] -2.78184816 -0.96490651 -1.90637179 -1.01510832 -0.16831790 0.05284366
## [49] 1.04785331 1.99941115 -0.43399489 -1.38465073 -3.25854344 -2.91053917
## [55] -1.76146875 0.77132686 1.66259968 2.88184263 0.21059943 -0.37362893
## [61] -2.17441288 -1.19599837 -1.30894285 0.91325166 1.90309705 3.17119697
## [67] 2.87791790 2.00341061 1.09212683 1.91294317 2.58804162 3.64831854
## [73] 3.84743593 3.69994307 4.04401256 1.44729195 1.38058144 -0.99201821
## [79] -0.89626703 -0.67820990 0.73087800 -1.14706567 -0.95979023 -2.28773374
## [85] -0.24989283 1.59612400 2.88040043 3.40185649 2.50481229 2.54370604
## [91] 0.89249915 1.09894149 0.10822216 -0.90252578 -0.53864078 -1.03624384
## [97] -0.57577877 -0.67681618 -1.74513073 0.67628703 0.21038958 2.81161890
## [103] 2.22238345 2.32035776 2.37308350 2.50952306 1.47892482 1.47543711
## [109] 2.06517091 2.91325997 4.12070753 1.08129055 1.34265399 -0.03034826
## [115] 1.58161673 1.66842633 2.58216181 1.59036645 2.16766846 1.10571250
## [121] 0.92123175 -0.44436437 0.91812334 1.72636619 0.54817175 -0.19740926
## [127] -1.54665694 0.03626248 0.16167768 -0.04133858 -1.18947866 -3.89570896
## [133] -4.79057199 -5.77699246 -4.60217289 -4.88761164 -3.98554798 -3.03872271
## [139] -2.80998811 -1.44132266 -2.17291500 -2.40676588 -2.84347224 -2.32878333
## [145] -3.01154797 -1.82230200 -1.69734749 -0.47155171 0.85442493 1.03733790
## [151] 0.60693501 -0.15610918 -1.01559798 -1.95738330 -1.64614024 -1.50531770
## [157] -0.66186877 -1.00750139 -0.22500658 -0.41753690 1.46537894 1.84696103
## [163] 3.05104849 1.14862156 0.79741988 0.88564357 0.25584872 -0.52801495
## [169] -0.63772209 -0.65327301 0.94384513 1.26127308 0.05832024 0.43588999
## [175] -2.66514227 -1.52240584 -1.97469214 -2.09330772 -1.60977783 -2.89279610
## [181] -2.55997257 -2.53149694 -2.15291713 -2.80010949 -1.11754905 -1.74508690
## [187] -0.93427842 -1.82930492 0.76478285 -0.37012447 -1.14499624 -2.41457013
## [193] -2.29506360 -1.87158883 -2.26566828 -4.00425945 -2.41131053 -4.07193819
## [199] -2.76653027 -4.48958572 -2.35194592 -1.16855276 0.05553665 0.72921129
## [205] 0.41890750 0.12189364 0.62374580 0.85113904 2.78306741 2.84234571
## [211] 4.55635930 3.23028105 2.88841993 2.21274152 2.07295624 0.84317889
## [217] 0.76793337 0.71050188 -1.00180768 -0.92098902 -2.11086345 -2.52248045
## [223] -3.76711115 -2.87728783 -3.08165719 -0.56416149 -1.37296037 -0.29866717
## [229] -0.99486310 0.83658518 0.44637035 -0.24053062 -1.53720361 -0.87694665
## [235] 0.49538436 0.42296975 1.23669465 0.77650740 0.05085706 0.23560050
## [241] -2.26267459 -0.70950264 0.70066651 1.00134215 2.11352780 -0.05575784
## [247] -0.20758140 -2.32469687 -1.70617184 -2.78454904 -0.18042756 -1.31716520
## [253] -0.71516138 -0.71309205 -1.56092168 -0.44596508 -0.17042927 0.28518997
## [259] 0.58688487 0.85486879 0.80260266 0.77318877 0.34019578 0.87299986
## [265] -1.02635811 1.06502545 0.05616615 0.20862475 -1.30253261 -1.20830410
## [271] -2.19631112 -0.99099370 1.07980433 0.92405556 3.16320833 1.78124349
## [277] 0.64779121 -1.35036514 -2.07845220 -1.32305313 -1.27809227 0.24410092
## [283] 1.46611225 1.60229903 2.43127654 2.59614799 -0.51087158 0.23979796
## [289] -1.62996428 0.42477237 0.46521931 0.96451855 -1.95597236 -1.10360702
## [295] -2.31620671 -2.49021348 -2.73523686 -2.52266639 -0.96657601 -0.45523786##Buat plot time series, plot ACF, plot PACF dan plot EACF. Idenifikasi apakah data tersebut stasioner
Plot Time Series
par(mfrow = c(1, 2))
ts.plot(y.arma)
ts.plot(arma22)
par(mfrow = c(1, 1))Berdasarkan plot time series tersebut, terlihat bahwa model ARMA(2,2) cenderung stasioner dalam rataan
Plot ACF
par(mfrow = c(1, 2))
acf(y.arma)
acf(arma22)
par(mfrow = c(1, 1))Berdasarkan plot ACF tersebut, terlihat bahwa model ARMA(1,1) hasil simulasi memiliki plot ACF yang tails off, sesuai dengan teori yang ada
Plot PACF
par(mfrow = c(1, 2))
pacf(y.arma)
pacf(arma22)
par(mfrow = c(1, 1))Berdasarkan plot PACF tersebut, terlihat bahwa model ARMA(1,1) hasil simulasi memiliki plot PACF yang tails off, sesuai dengan teori
Plot EACF
TSA::eacf(y.arma)## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x x x x o o o o o o o
## 1 o x x x o x o o o o o o o o
## 2 x x x x o x o o o o o o o o
## 3 x x x x o x o o o o o o o o
## 4 x x x x x x o o o o o o o o
## 5 x x x o o o o o o o o o o o
## 6 x x x o o o o o o o o o o o
## 7 x x x o o o o o o o o o o oTSA::eacf(arma22)## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x x x x x o o o o o o
## 1 o x x o x o o o o o o o o o
## 2 o x o o x o o o o o o o o o
## 3 x x o x x o o o o o o o o o
## 4 x x o x x o o o o o o o o o
## 5 x x x o x o o o o o o o o o
## 6 x x o o x x o o o o o o o o
## 7 x o o x o x o x o o o o o oBerdasarkan pola segitiga nol pada plot EACF, terlihat bahwa segitiga nol berada pada ordo AR(1) dan ordo MA(5)
Dapat disimpulkan bahwa data tersebut stasioner
Buat scatterplot dan korelasi antara \(Y_t\) dengan \(Y_{t-1}\), \(Y_t\) dengan \(Y_{t-2}\), serta \(Y_t\) dengan \(Y_{t-3}\)
Korelasi antara \(Y_t\) dengan \(Y_{t-1}\)
#Yt
yt_arma <- arma22[-1]
yt_arma## [1] -0.73270762 0.29534872 -0.88177243 -0.40494262 -1.81814223 -2.63974439
## [7] -3.07358274 -2.71804095 -0.71635658 -1.61516276 0.15764457 -0.47266329
## [13] 1.44725419 2.51482053 3.64981261 2.51808704 0.94646378 -1.73007867
## [19] -1.96463282 -2.29104663 -0.97842919 -0.98485685 0.78966693 0.32630064
## [25] -1.01519362 -0.68344715 -0.24220047 -0.19974042 -0.31979622 -0.65860929
## [31] 0.35018410 1.55062920 1.41188175 1.42666841 0.38973575 0.37200962
## [37] 1.89051914 0.67059914 0.39926045 -2.05092604 -1.31421244 -2.78184816
## [43] -0.96490651 -1.90637179 -1.01510832 -0.16831790 0.05284366 1.04785331
## [49] 1.99941115 -0.43399489 -1.38465073 -3.25854344 -2.91053917 -1.76146875
## [55] 0.77132686 1.66259968 2.88184263 0.21059943 -0.37362893 -2.17441288
## [61] -1.19599837 -1.30894285 0.91325166 1.90309705 3.17119697 2.87791790
## [67] 2.00341061 1.09212683 1.91294317 2.58804162 3.64831854 3.84743593
## [73] 3.69994307 4.04401256 1.44729195 1.38058144 -0.99201821 -0.89626703
## [79] -0.67820990 0.73087800 -1.14706567 -0.95979023 -2.28773374 -0.24989283
## [85] 1.59612400 2.88040043 3.40185649 2.50481229 2.54370604 0.89249915
## [91] 1.09894149 0.10822216 -0.90252578 -0.53864078 -1.03624384 -0.57577877
## [97] -0.67681618 -1.74513073 0.67628703 0.21038958 2.81161890 2.22238345
## [103] 2.32035776 2.37308350 2.50952306 1.47892482 1.47543711 2.06517091
## [109] 2.91325997 4.12070753 1.08129055 1.34265399 -0.03034826 1.58161673
## [115] 1.66842633 2.58216181 1.59036645 2.16766846 1.10571250 0.92123175
## [121] -0.44436437 0.91812334 1.72636619 0.54817175 -0.19740926 -1.54665694
## [127] 0.03626248 0.16167768 -0.04133858 -1.18947866 -3.89570896 -4.79057199
## [133] -5.77699246 -4.60217289 -4.88761164 -3.98554798 -3.03872271 -2.80998811
## [139] -1.44132266 -2.17291500 -2.40676588 -2.84347224 -2.32878333 -3.01154797
## [145] -1.82230200 -1.69734749 -0.47155171 0.85442493 1.03733790 0.60693501
## [151] -0.15610918 -1.01559798 -1.95738330 -1.64614024 -1.50531770 -0.66186877
## [157] -1.00750139 -0.22500658 -0.41753690 1.46537894 1.84696103 3.05104849
## [163] 1.14862156 0.79741988 0.88564357 0.25584872 -0.52801495 -0.63772209
## [169] -0.65327301 0.94384513 1.26127308 0.05832024 0.43588999 -2.66514227
## [175] -1.52240584 -1.97469214 -2.09330772 -1.60977783 -2.89279610 -2.55997257
## [181] -2.53149694 -2.15291713 -2.80010949 -1.11754905 -1.74508690 -0.93427842
## [187] -1.82930492 0.76478285 -0.37012447 -1.14499624 -2.41457013 -2.29506360
## [193] -1.87158883 -2.26566828 -4.00425945 -2.41131053 -4.07193819 -2.76653027
## [199] -4.48958572 -2.35194592 -1.16855276 0.05553665 0.72921129 0.41890750
## [205] 0.12189364 0.62374580 0.85113904 2.78306741 2.84234571 4.55635930
## [211] 3.23028105 2.88841993 2.21274152 2.07295624 0.84317889 0.76793337
## [217] 0.71050188 -1.00180768 -0.92098902 -2.11086345 -2.52248045 -3.76711115
## [223] -2.87728783 -3.08165719 -0.56416149 -1.37296037 -0.29866717 -0.99486310
## [229] 0.83658518 0.44637035 -0.24053062 -1.53720361 -0.87694665 0.49538436
## [235] 0.42296975 1.23669465 0.77650740 0.05085706 0.23560050 -2.26267459
## [241] -0.70950264 0.70066651 1.00134215 2.11352780 -0.05575784 -0.20758140
## [247] -2.32469687 -1.70617184 -2.78454904 -0.18042756 -1.31716520 -0.71516138
## [253] -0.71309205 -1.56092168 -0.44596508 -0.17042927 0.28518997 0.58688487
## [259] 0.85486879 0.80260266 0.77318877 0.34019578 0.87299986 -1.02635811
## [265] 1.06502545 0.05616615 0.20862475 -1.30253261 -1.20830410 -2.19631112
## [271] -0.99099370 1.07980433 0.92405556 3.16320833 1.78124349 0.64779121
## [277] -1.35036514 -2.07845220 -1.32305313 -1.27809227 0.24410092 1.46611225
## [283] 1.60229903 2.43127654 2.59614799 -0.51087158 0.23979796 -1.62996428
## [289] 0.42477237 0.46521931 0.96451855 -1.95597236 -1.10360702 -2.31620671
## [295] -2.49021348 -2.73523686 -2.52266639 -0.96657601 -0.45523786#Yt-1
yt_1_arma <- arma22[-300]
yt_1_arma## [1] 0.06795477 -0.73270762 0.29534872 -0.88177243 -0.40494262 -1.81814223
## [7] -2.63974439 -3.07358274 -2.71804095 -0.71635658 -1.61516276 0.15764457
## [13] -0.47266329 1.44725419 2.51482053 3.64981261 2.51808704 0.94646378
## [19] -1.73007867 -1.96463282 -2.29104663 -0.97842919 -0.98485685 0.78966693
## [25] 0.32630064 -1.01519362 -0.68344715 -0.24220047 -0.19974042 -0.31979622
## [31] -0.65860929 0.35018410 1.55062920 1.41188175 1.42666841 0.38973575
## [37] 0.37200962 1.89051914 0.67059914 0.39926045 -2.05092604 -1.31421244
## [43] -2.78184816 -0.96490651 -1.90637179 -1.01510832 -0.16831790 0.05284366
## [49] 1.04785331 1.99941115 -0.43399489 -1.38465073 -3.25854344 -2.91053917
## [55] -1.76146875 0.77132686 1.66259968 2.88184263 0.21059943 -0.37362893
## [61] -2.17441288 -1.19599837 -1.30894285 0.91325166 1.90309705 3.17119697
## [67] 2.87791790 2.00341061 1.09212683 1.91294317 2.58804162 3.64831854
## [73] 3.84743593 3.69994307 4.04401256 1.44729195 1.38058144 -0.99201821
## [79] -0.89626703 -0.67820990 0.73087800 -1.14706567 -0.95979023 -2.28773374
## [85] -0.24989283 1.59612400 2.88040043 3.40185649 2.50481229 2.54370604
## [91] 0.89249915 1.09894149 0.10822216 -0.90252578 -0.53864078 -1.03624384
## [97] -0.57577877 -0.67681618 -1.74513073 0.67628703 0.21038958 2.81161890
## [103] 2.22238345 2.32035776 2.37308350 2.50952306 1.47892482 1.47543711
## [109] 2.06517091 2.91325997 4.12070753 1.08129055 1.34265399 -0.03034826
## [115] 1.58161673 1.66842633 2.58216181 1.59036645 2.16766846 1.10571250
## [121] 0.92123175 -0.44436437 0.91812334 1.72636619 0.54817175 -0.19740926
## [127] -1.54665694 0.03626248 0.16167768 -0.04133858 -1.18947866 -3.89570896
## [133] -4.79057199 -5.77699246 -4.60217289 -4.88761164 -3.98554798 -3.03872271
## [139] -2.80998811 -1.44132266 -2.17291500 -2.40676588 -2.84347224 -2.32878333
## [145] -3.01154797 -1.82230200 -1.69734749 -0.47155171 0.85442493 1.03733790
## [151] 0.60693501 -0.15610918 -1.01559798 -1.95738330 -1.64614024 -1.50531770
## [157] -0.66186877 -1.00750139 -0.22500658 -0.41753690 1.46537894 1.84696103
## [163] 3.05104849 1.14862156 0.79741988 0.88564357 0.25584872 -0.52801495
## [169] -0.63772209 -0.65327301 0.94384513 1.26127308 0.05832024 0.43588999
## [175] -2.66514227 -1.52240584 -1.97469214 -2.09330772 -1.60977783 -2.89279610
## [181] -2.55997257 -2.53149694 -2.15291713 -2.80010949 -1.11754905 -1.74508690
## [187] -0.93427842 -1.82930492 0.76478285 -0.37012447 -1.14499624 -2.41457013
## [193] -2.29506360 -1.87158883 -2.26566828 -4.00425945 -2.41131053 -4.07193819
## [199] -2.76653027 -4.48958572 -2.35194592 -1.16855276 0.05553665 0.72921129
## [205] 0.41890750 0.12189364 0.62374580 0.85113904 2.78306741 2.84234571
## [211] 4.55635930 3.23028105 2.88841993 2.21274152 2.07295624 0.84317889
## [217] 0.76793337 0.71050188 -1.00180768 -0.92098902 -2.11086345 -2.52248045
## [223] -3.76711115 -2.87728783 -3.08165719 -0.56416149 -1.37296037 -0.29866717
## [229] -0.99486310 0.83658518 0.44637035 -0.24053062 -1.53720361 -0.87694665
## [235] 0.49538436 0.42296975 1.23669465 0.77650740 0.05085706 0.23560050
## [241] -2.26267459 -0.70950264 0.70066651 1.00134215 2.11352780 -0.05575784
## [247] -0.20758140 -2.32469687 -1.70617184 -2.78454904 -0.18042756 -1.31716520
## [253] -0.71516138 -0.71309205 -1.56092168 -0.44596508 -0.17042927 0.28518997
## [259] 0.58688487 0.85486879 0.80260266 0.77318877 0.34019578 0.87299986
## [265] -1.02635811 1.06502545 0.05616615 0.20862475 -1.30253261 -1.20830410
## [271] -2.19631112 -0.99099370 1.07980433 0.92405556 3.16320833 1.78124349
## [277] 0.64779121 -1.35036514 -2.07845220 -1.32305313 -1.27809227 0.24410092
## [283] 1.46611225 1.60229903 2.43127654 2.59614799 -0.51087158 0.23979796
## [289] -1.62996428 0.42477237 0.46521931 0.96451855 -1.95597236 -1.10360702
## [295] -2.31620671 -2.49021348 -2.73523686 -2.52266639 -0.96657601plot(y=yt_arma,x=yt_1_arma)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-1}\). Hal ini sesuai dengan teori yang ada
cor(yt_arma,yt_1_arma)## [1] 0.7981987Korelasi antara \(Y_t\) dengan \(Y_{t-2}\)
#Yt
yt_arma2 <- arma22[-c(1,2)]
yt_arma2## [1] 0.29534872 -0.88177243 -0.40494262 -1.81814223 -2.63974439 -3.07358274
## [7] -2.71804095 -0.71635658 -1.61516276 0.15764457 -0.47266329 1.44725419
## [13] 2.51482053 3.64981261 2.51808704 0.94646378 -1.73007867 -1.96463282
## [19] -2.29104663 -0.97842919 -0.98485685 0.78966693 0.32630064 -1.01519362
## [25] -0.68344715 -0.24220047 -0.19974042 -0.31979622 -0.65860929 0.35018410
## [31] 1.55062920 1.41188175 1.42666841 0.38973575 0.37200962 1.89051914
## [37] 0.67059914 0.39926045 -2.05092604 -1.31421244 -2.78184816 -0.96490651
## [43] -1.90637179 -1.01510832 -0.16831790 0.05284366 1.04785331 1.99941115
## [49] -0.43399489 -1.38465073 -3.25854344 -2.91053917 -1.76146875 0.77132686
## [55] 1.66259968 2.88184263 0.21059943 -0.37362893 -2.17441288 -1.19599837
## [61] -1.30894285 0.91325166 1.90309705 3.17119697 2.87791790 2.00341061
## [67] 1.09212683 1.91294317 2.58804162 3.64831854 3.84743593 3.69994307
## [73] 4.04401256 1.44729195 1.38058144 -0.99201821 -0.89626703 -0.67820990
## [79] 0.73087800 -1.14706567 -0.95979023 -2.28773374 -0.24989283 1.59612400
## [85] 2.88040043 3.40185649 2.50481229 2.54370604 0.89249915 1.09894149
## [91] 0.10822216 -0.90252578 -0.53864078 -1.03624384 -0.57577877 -0.67681618
## [97] -1.74513073 0.67628703 0.21038958 2.81161890 2.22238345 2.32035776
## [103] 2.37308350 2.50952306 1.47892482 1.47543711 2.06517091 2.91325997
## [109] 4.12070753 1.08129055 1.34265399 -0.03034826 1.58161673 1.66842633
## [115] 2.58216181 1.59036645 2.16766846 1.10571250 0.92123175 -0.44436437
## [121] 0.91812334 1.72636619 0.54817175 -0.19740926 -1.54665694 0.03626248
## [127] 0.16167768 -0.04133858 -1.18947866 -3.89570896 -4.79057199 -5.77699246
## [133] -4.60217289 -4.88761164 -3.98554798 -3.03872271 -2.80998811 -1.44132266
## [139] -2.17291500 -2.40676588 -2.84347224 -2.32878333 -3.01154797 -1.82230200
## [145] -1.69734749 -0.47155171 0.85442493 1.03733790 0.60693501 -0.15610918
## [151] -1.01559798 -1.95738330 -1.64614024 -1.50531770 -0.66186877 -1.00750139
## [157] -0.22500658 -0.41753690 1.46537894 1.84696103 3.05104849 1.14862156
## [163] 0.79741988 0.88564357 0.25584872 -0.52801495 -0.63772209 -0.65327301
## [169] 0.94384513 1.26127308 0.05832024 0.43588999 -2.66514227 -1.52240584
## [175] -1.97469214 -2.09330772 -1.60977783 -2.89279610 -2.55997257 -2.53149694
## [181] -2.15291713 -2.80010949 -1.11754905 -1.74508690 -0.93427842 -1.82930492
## [187] 0.76478285 -0.37012447 -1.14499624 -2.41457013 -2.29506360 -1.87158883
## [193] -2.26566828 -4.00425945 -2.41131053 -4.07193819 -2.76653027 -4.48958572
## [199] -2.35194592 -1.16855276 0.05553665 0.72921129 0.41890750 0.12189364
## [205] 0.62374580 0.85113904 2.78306741 2.84234571 4.55635930 3.23028105
## [211] 2.88841993 2.21274152 2.07295624 0.84317889 0.76793337 0.71050188
## [217] -1.00180768 -0.92098902 -2.11086345 -2.52248045 -3.76711115 -2.87728783
## [223] -3.08165719 -0.56416149 -1.37296037 -0.29866717 -0.99486310 0.83658518
## [229] 0.44637035 -0.24053062 -1.53720361 -0.87694665 0.49538436 0.42296975
## [235] 1.23669465 0.77650740 0.05085706 0.23560050 -2.26267459 -0.70950264
## [241] 0.70066651 1.00134215 2.11352780 -0.05575784 -0.20758140 -2.32469687
## [247] -1.70617184 -2.78454904 -0.18042756 -1.31716520 -0.71516138 -0.71309205
## [253] -1.56092168 -0.44596508 -0.17042927 0.28518997 0.58688487 0.85486879
## [259] 0.80260266 0.77318877 0.34019578 0.87299986 -1.02635811 1.06502545
## [265] 0.05616615 0.20862475 -1.30253261 -1.20830410 -2.19631112 -0.99099370
## [271] 1.07980433 0.92405556 3.16320833 1.78124349 0.64779121 -1.35036514
## [277] -2.07845220 -1.32305313 -1.27809227 0.24410092 1.46611225 1.60229903
## [283] 2.43127654 2.59614799 -0.51087158 0.23979796 -1.62996428 0.42477237
## [289] 0.46521931 0.96451855 -1.95597236 -1.10360702 -2.31620671 -2.49021348
## [295] -2.73523686 -2.52266639 -0.96657601 -0.45523786#Yt-2
yt_2_arma <- arma22[-c(299,300)]
yt_2_arma## [1] 0.06795477 -0.73270762 0.29534872 -0.88177243 -0.40494262 -1.81814223
## [7] -2.63974439 -3.07358274 -2.71804095 -0.71635658 -1.61516276 0.15764457
## [13] -0.47266329 1.44725419 2.51482053 3.64981261 2.51808704 0.94646378
## [19] -1.73007867 -1.96463282 -2.29104663 -0.97842919 -0.98485685 0.78966693
## [25] 0.32630064 -1.01519362 -0.68344715 -0.24220047 -0.19974042 -0.31979622
## [31] -0.65860929 0.35018410 1.55062920 1.41188175 1.42666841 0.38973575
## [37] 0.37200962 1.89051914 0.67059914 0.39926045 -2.05092604 -1.31421244
## [43] -2.78184816 -0.96490651 -1.90637179 -1.01510832 -0.16831790 0.05284366
## [49] 1.04785331 1.99941115 -0.43399489 -1.38465073 -3.25854344 -2.91053917
## [55] -1.76146875 0.77132686 1.66259968 2.88184263 0.21059943 -0.37362893
## [61] -2.17441288 -1.19599837 -1.30894285 0.91325166 1.90309705 3.17119697
## [67] 2.87791790 2.00341061 1.09212683 1.91294317 2.58804162 3.64831854
## [73] 3.84743593 3.69994307 4.04401256 1.44729195 1.38058144 -0.99201821
## [79] -0.89626703 -0.67820990 0.73087800 -1.14706567 -0.95979023 -2.28773374
## [85] -0.24989283 1.59612400 2.88040043 3.40185649 2.50481229 2.54370604
## [91] 0.89249915 1.09894149 0.10822216 -0.90252578 -0.53864078 -1.03624384
## [97] -0.57577877 -0.67681618 -1.74513073 0.67628703 0.21038958 2.81161890
## [103] 2.22238345 2.32035776 2.37308350 2.50952306 1.47892482 1.47543711
## [109] 2.06517091 2.91325997 4.12070753 1.08129055 1.34265399 -0.03034826
## [115] 1.58161673 1.66842633 2.58216181 1.59036645 2.16766846 1.10571250
## [121] 0.92123175 -0.44436437 0.91812334 1.72636619 0.54817175 -0.19740926
## [127] -1.54665694 0.03626248 0.16167768 -0.04133858 -1.18947866 -3.89570896
## [133] -4.79057199 -5.77699246 -4.60217289 -4.88761164 -3.98554798 -3.03872271
## [139] -2.80998811 -1.44132266 -2.17291500 -2.40676588 -2.84347224 -2.32878333
## [145] -3.01154797 -1.82230200 -1.69734749 -0.47155171 0.85442493 1.03733790
## [151] 0.60693501 -0.15610918 -1.01559798 -1.95738330 -1.64614024 -1.50531770
## [157] -0.66186877 -1.00750139 -0.22500658 -0.41753690 1.46537894 1.84696103
## [163] 3.05104849 1.14862156 0.79741988 0.88564357 0.25584872 -0.52801495
## [169] -0.63772209 -0.65327301 0.94384513 1.26127308 0.05832024 0.43588999
## [175] -2.66514227 -1.52240584 -1.97469214 -2.09330772 -1.60977783 -2.89279610
## [181] -2.55997257 -2.53149694 -2.15291713 -2.80010949 -1.11754905 -1.74508690
## [187] -0.93427842 -1.82930492 0.76478285 -0.37012447 -1.14499624 -2.41457013
## [193] -2.29506360 -1.87158883 -2.26566828 -4.00425945 -2.41131053 -4.07193819
## [199] -2.76653027 -4.48958572 -2.35194592 -1.16855276 0.05553665 0.72921129
## [205] 0.41890750 0.12189364 0.62374580 0.85113904 2.78306741 2.84234571
## [211] 4.55635930 3.23028105 2.88841993 2.21274152 2.07295624 0.84317889
## [217] 0.76793337 0.71050188 -1.00180768 -0.92098902 -2.11086345 -2.52248045
## [223] -3.76711115 -2.87728783 -3.08165719 -0.56416149 -1.37296037 -0.29866717
## [229] -0.99486310 0.83658518 0.44637035 -0.24053062 -1.53720361 -0.87694665
## [235] 0.49538436 0.42296975 1.23669465 0.77650740 0.05085706 0.23560050
## [241] -2.26267459 -0.70950264 0.70066651 1.00134215 2.11352780 -0.05575784
## [247] -0.20758140 -2.32469687 -1.70617184 -2.78454904 -0.18042756 -1.31716520
## [253] -0.71516138 -0.71309205 -1.56092168 -0.44596508 -0.17042927 0.28518997
## [259] 0.58688487 0.85486879 0.80260266 0.77318877 0.34019578 0.87299986
## [265] -1.02635811 1.06502545 0.05616615 0.20862475 -1.30253261 -1.20830410
## [271] -2.19631112 -0.99099370 1.07980433 0.92405556 3.16320833 1.78124349
## [277] 0.64779121 -1.35036514 -2.07845220 -1.32305313 -1.27809227 0.24410092
## [283] 1.46611225 1.60229903 2.43127654 2.59614799 -0.51087158 0.23979796
## [289] -1.62996428 0.42477237 0.46521931 0.96451855 -1.95597236 -1.10360702
## [295] -2.31620671 -2.49021348 -2.73523686 -2.52266639plot(y=yt_arma2,x=yt_2_arma)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-1}\). Hal ini sesuai dengan teori yang ada
cor(yt_arma2,yt_2_arma)## [1] 0.6546106Korelasi antara \(Y_t\) dengan \(Y_{t-3}\)
#Yt
yt_arma3 <- arma22[-c(1,2,3)]
yt_arma3## [1] -0.88177243 -0.40494262 -1.81814223 -2.63974439 -3.07358274 -2.71804095
## [7] -0.71635658 -1.61516276 0.15764457 -0.47266329 1.44725419 2.51482053
## [13] 3.64981261 2.51808704 0.94646378 -1.73007867 -1.96463282 -2.29104663
## [19] -0.97842919 -0.98485685 0.78966693 0.32630064 -1.01519362 -0.68344715
## [25] -0.24220047 -0.19974042 -0.31979622 -0.65860929 0.35018410 1.55062920
## [31] 1.41188175 1.42666841 0.38973575 0.37200962 1.89051914 0.67059914
## [37] 0.39926045 -2.05092604 -1.31421244 -2.78184816 -0.96490651 -1.90637179
## [43] -1.01510832 -0.16831790 0.05284366 1.04785331 1.99941115 -0.43399489
## [49] -1.38465073 -3.25854344 -2.91053917 -1.76146875 0.77132686 1.66259968
## [55] 2.88184263 0.21059943 -0.37362893 -2.17441288 -1.19599837 -1.30894285
## [61] 0.91325166 1.90309705 3.17119697 2.87791790 2.00341061 1.09212683
## [67] 1.91294317 2.58804162 3.64831854 3.84743593 3.69994307 4.04401256
## [73] 1.44729195 1.38058144 -0.99201821 -0.89626703 -0.67820990 0.73087800
## [79] -1.14706567 -0.95979023 -2.28773374 -0.24989283 1.59612400 2.88040043
## [85] 3.40185649 2.50481229 2.54370604 0.89249915 1.09894149 0.10822216
## [91] -0.90252578 -0.53864078 -1.03624384 -0.57577877 -0.67681618 -1.74513073
## [97] 0.67628703 0.21038958 2.81161890 2.22238345 2.32035776 2.37308350
## [103] 2.50952306 1.47892482 1.47543711 2.06517091 2.91325997 4.12070753
## [109] 1.08129055 1.34265399 -0.03034826 1.58161673 1.66842633 2.58216181
## [115] 1.59036645 2.16766846 1.10571250 0.92123175 -0.44436437 0.91812334
## [121] 1.72636619 0.54817175 -0.19740926 -1.54665694 0.03626248 0.16167768
## [127] -0.04133858 -1.18947866 -3.89570896 -4.79057199 -5.77699246 -4.60217289
## [133] -4.88761164 -3.98554798 -3.03872271 -2.80998811 -1.44132266 -2.17291500
## [139] -2.40676588 -2.84347224 -2.32878333 -3.01154797 -1.82230200 -1.69734749
## [145] -0.47155171 0.85442493 1.03733790 0.60693501 -0.15610918 -1.01559798
## [151] -1.95738330 -1.64614024 -1.50531770 -0.66186877 -1.00750139 -0.22500658
## [157] -0.41753690 1.46537894 1.84696103 3.05104849 1.14862156 0.79741988
## [163] 0.88564357 0.25584872 -0.52801495 -0.63772209 -0.65327301 0.94384513
## [169] 1.26127308 0.05832024 0.43588999 -2.66514227 -1.52240584 -1.97469214
## [175] -2.09330772 -1.60977783 -2.89279610 -2.55997257 -2.53149694 -2.15291713
## [181] -2.80010949 -1.11754905 -1.74508690 -0.93427842 -1.82930492 0.76478285
## [187] -0.37012447 -1.14499624 -2.41457013 -2.29506360 -1.87158883 -2.26566828
## [193] -4.00425945 -2.41131053 -4.07193819 -2.76653027 -4.48958572 -2.35194592
## [199] -1.16855276 0.05553665 0.72921129 0.41890750 0.12189364 0.62374580
## [205] 0.85113904 2.78306741 2.84234571 4.55635930 3.23028105 2.88841993
## [211] 2.21274152 2.07295624 0.84317889 0.76793337 0.71050188 -1.00180768
## [217] -0.92098902 -2.11086345 -2.52248045 -3.76711115 -2.87728783 -3.08165719
## [223] -0.56416149 -1.37296037 -0.29866717 -0.99486310 0.83658518 0.44637035
## [229] -0.24053062 -1.53720361 -0.87694665 0.49538436 0.42296975 1.23669465
## [235] 0.77650740 0.05085706 0.23560050 -2.26267459 -0.70950264 0.70066651
## [241] 1.00134215 2.11352780 -0.05575784 -0.20758140 -2.32469687 -1.70617184
## [247] -2.78454904 -0.18042756 -1.31716520 -0.71516138 -0.71309205 -1.56092168
## [253] -0.44596508 -0.17042927 0.28518997 0.58688487 0.85486879 0.80260266
## [259] 0.77318877 0.34019578 0.87299986 -1.02635811 1.06502545 0.05616615
## [265] 0.20862475 -1.30253261 -1.20830410 -2.19631112 -0.99099370 1.07980433
## [271] 0.92405556 3.16320833 1.78124349 0.64779121 -1.35036514 -2.07845220
## [277] -1.32305313 -1.27809227 0.24410092 1.46611225 1.60229903 2.43127654
## [283] 2.59614799 -0.51087158 0.23979796 -1.62996428 0.42477237 0.46521931
## [289] 0.96451855 -1.95597236 -1.10360702 -2.31620671 -2.49021348 -2.73523686
## [295] -2.52266639 -0.96657601 -0.45523786#Yt-3
yt_3_arma <- arma22[-c(288,289,300)]
yt_3_arma## [1] 0.06795477 -0.73270762 0.29534872 -0.88177243 -0.40494262 -1.81814223
## [7] -2.63974439 -3.07358274 -2.71804095 -0.71635658 -1.61516276 0.15764457
## [13] -0.47266329 1.44725419 2.51482053 3.64981261 2.51808704 0.94646378
## [19] -1.73007867 -1.96463282 -2.29104663 -0.97842919 -0.98485685 0.78966693
## [25] 0.32630064 -1.01519362 -0.68344715 -0.24220047 -0.19974042 -0.31979622
## [31] -0.65860929 0.35018410 1.55062920 1.41188175 1.42666841 0.38973575
## [37] 0.37200962 1.89051914 0.67059914 0.39926045 -2.05092604 -1.31421244
## [43] -2.78184816 -0.96490651 -1.90637179 -1.01510832 -0.16831790 0.05284366
## [49] 1.04785331 1.99941115 -0.43399489 -1.38465073 -3.25854344 -2.91053917
## [55] -1.76146875 0.77132686 1.66259968 2.88184263 0.21059943 -0.37362893
## [61] -2.17441288 -1.19599837 -1.30894285 0.91325166 1.90309705 3.17119697
## [67] 2.87791790 2.00341061 1.09212683 1.91294317 2.58804162 3.64831854
## [73] 3.84743593 3.69994307 4.04401256 1.44729195 1.38058144 -0.99201821
## [79] -0.89626703 -0.67820990 0.73087800 -1.14706567 -0.95979023 -2.28773374
## [85] -0.24989283 1.59612400 2.88040043 3.40185649 2.50481229 2.54370604
## [91] 0.89249915 1.09894149 0.10822216 -0.90252578 -0.53864078 -1.03624384
## [97] -0.57577877 -0.67681618 -1.74513073 0.67628703 0.21038958 2.81161890
## [103] 2.22238345 2.32035776 2.37308350 2.50952306 1.47892482 1.47543711
## [109] 2.06517091 2.91325997 4.12070753 1.08129055 1.34265399 -0.03034826
## [115] 1.58161673 1.66842633 2.58216181 1.59036645 2.16766846 1.10571250
## [121] 0.92123175 -0.44436437 0.91812334 1.72636619 0.54817175 -0.19740926
## [127] -1.54665694 0.03626248 0.16167768 -0.04133858 -1.18947866 -3.89570896
## [133] -4.79057199 -5.77699246 -4.60217289 -4.88761164 -3.98554798 -3.03872271
## [139] -2.80998811 -1.44132266 -2.17291500 -2.40676588 -2.84347224 -2.32878333
## [145] -3.01154797 -1.82230200 -1.69734749 -0.47155171 0.85442493 1.03733790
## [151] 0.60693501 -0.15610918 -1.01559798 -1.95738330 -1.64614024 -1.50531770
## [157] -0.66186877 -1.00750139 -0.22500658 -0.41753690 1.46537894 1.84696103
## [163] 3.05104849 1.14862156 0.79741988 0.88564357 0.25584872 -0.52801495
## [169] -0.63772209 -0.65327301 0.94384513 1.26127308 0.05832024 0.43588999
## [175] -2.66514227 -1.52240584 -1.97469214 -2.09330772 -1.60977783 -2.89279610
## [181] -2.55997257 -2.53149694 -2.15291713 -2.80010949 -1.11754905 -1.74508690
## [187] -0.93427842 -1.82930492 0.76478285 -0.37012447 -1.14499624 -2.41457013
## [193] -2.29506360 -1.87158883 -2.26566828 -4.00425945 -2.41131053 -4.07193819
## [199] -2.76653027 -4.48958572 -2.35194592 -1.16855276 0.05553665 0.72921129
## [205] 0.41890750 0.12189364 0.62374580 0.85113904 2.78306741 2.84234571
## [211] 4.55635930 3.23028105 2.88841993 2.21274152 2.07295624 0.84317889
## [217] 0.76793337 0.71050188 -1.00180768 -0.92098902 -2.11086345 -2.52248045
## [223] -3.76711115 -2.87728783 -3.08165719 -0.56416149 -1.37296037 -0.29866717
## [229] -0.99486310 0.83658518 0.44637035 -0.24053062 -1.53720361 -0.87694665
## [235] 0.49538436 0.42296975 1.23669465 0.77650740 0.05085706 0.23560050
## [241] -2.26267459 -0.70950264 0.70066651 1.00134215 2.11352780 -0.05575784
## [247] -0.20758140 -2.32469687 -1.70617184 -2.78454904 -0.18042756 -1.31716520
## [253] -0.71516138 -0.71309205 -1.56092168 -0.44596508 -0.17042927 0.28518997
## [259] 0.58688487 0.85486879 0.80260266 0.77318877 0.34019578 0.87299986
## [265] -1.02635811 1.06502545 0.05616615 0.20862475 -1.30253261 -1.20830410
## [271] -2.19631112 -0.99099370 1.07980433 0.92405556 3.16320833 1.78124349
## [277] 0.64779121 -1.35036514 -2.07845220 -1.32305313 -1.27809227 0.24410092
## [283] 1.46611225 1.60229903 2.43127654 2.59614799 -0.51087158 0.42477237
## [289] 0.46521931 0.96451855 -1.95597236 -1.10360702 -2.31620671 -2.49021348
## [295] -2.73523686 -2.52266639 -0.96657601plot(y=yt_arma3,x=yt_3_arma)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif yang lemah antara \(Y_t\) dengan \(Y_{t-1}\). Hal ini sesuai dengan teori yang ada
cor(yt_arma3,yt_3_arma)## [1] 0.4003797