Pertemuan 4
Kheni Hikmah Lestari
2023-08-28
Pembangkitan data model M(2) dengan theta1=0.4 dan theta2=0.6 sebanyak 300 data secara manual
data <- rnorm(300)
ts.plot(data)
Proses MA (2)
Proses MA dapat dituliskan sebagai berikut:
\[ y_{t} = c + e_t + \theta_{1}e_{t-1} + \theta_{2}e_{t-2} + \dots + \theta_{q}e_{t-q} = c+{e_t+\sum_{i=1}^p \theta_ie_{t-i}} \]
Terlihat bahwa \(e_t\), atau white noise, berperan penting dalam pembangkitan proses MA(2).
Pembangkitan Proses MA(2) manual
Akan dicoba membangkitkan proses MA paling sederhana, yaitu MA(1) dengan \(\theta1 = 0.4\) dan \(\theta2 = 0.6\) dan 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(1029)
ma <- data[c(1,2)]
ma## [1] -0.5690168 -0.9257456theta1= 0.4
theta2 =0.6Nilai-nilai selanjutnya dapat dicari melalui loop. Bentuk loop dapat dilihat dari rumus MA(1) yang hendak dibangkitkan:
\[ y_t = e_t+0.4e_{t-1}+0.6e_{t-2} \]
set.seed(1029)
for(i in 3:300){
ma[i] <- data[i] + 0.4 * data[i - 1] + 0.6 * data[i - 2]
}
ma## [1] -0.569016823 -0.925745616 -0.632801741 -0.113681936 -0.838825889
## [6] -0.228932635 -1.093658298 -0.757635672 -1.228501815 -1.007860736
## [11] -2.434925603 -1.358613058 -0.392994219 0.882701882 0.067594641
## [16] 1.140597370 -0.182002692 -1.076053700 -0.569618682 -2.413542676
## [21] -0.560580701 0.041987526 1.727771288 -0.317976737 -0.177969466
## [26] -1.183279546 1.531131241 0.806021380 1.195782368 -1.022105714
## [31] -0.385910314 -1.472913728 -0.928931224 -0.551238663 0.493167776
## [36] -1.060379610 2.670455911 0.029567154 0.876468817 0.196809621
## [41] 0.176167581 1.285340393 -2.136397607 0.383448008 -1.178690810
## [46] 2.172716418 0.370274410 -0.569939168 -0.842202031 -1.434030228
## [51] -1.985257132 -0.748416139 -0.691640462 1.154702476 1.930094322
## [56] 1.690258923 1.533171362 1.724733819 1.831992946 1.975479525
## [61] 3.178193231 0.179434806 2.295283500 0.434314093 2.702519607
## [66] -0.513299703 1.129950973 0.369158393 0.063723266 -1.157261888
## [71] -1.948506851 0.076944460 -1.323142023 -0.135325878 -1.559412210
## [76] -0.335927229 -0.525128456 -2.515051165 -1.813131544 0.195181320
## [81] 0.059363421 2.744453706 0.845875253 1.317199192 0.411904671
## [86] -1.173672539 0.495099578 0.474793962 1.465144736 1.883694074
## [91] 0.403094920 1.491998094 0.070824027 1.327404012 -1.226839108
## [96] 0.471932459 -0.515493626 -1.448579535 -1.548113160 0.352968304
## [101] 1.725373338 0.389559589 -0.302694277 -2.907258154 -1.606275356
## [106] -2.695390087 -2.056925062 -2.388603519 -0.034819633 0.668561477
## [111] 1.310707890 1.490689109 1.113712032 -0.590192825 -0.880951088
## [116] -0.413195350 0.271861395 0.664187901 0.875690081 0.027354715
## [121] 0.652416926 -1.012308414 -1.691867609 0.073725679 -0.555677678
## [126] 1.312655195 0.343343304 0.934066059 -0.445942939 -0.050310064
## [131] -0.105669642 -0.345247558 -0.066839236 -1.824100414 -0.254785654
## [136] -0.104251540 -0.016684151 1.061994505 -0.174267902 -1.836110795
## [141] -1.335313995 -0.607447909 -0.091499802 -0.017140403 -0.962088865
## [146] 3.004067288 0.382289451 2.421556583 0.865592706 0.442164618
## [151] 1.693698419 1.040701209 2.707898663 -0.261337264 1.680112883
## [156] 0.686831800 1.006510558 -0.287777543 -1.999572634 -2.142499929
## [161] -2.073710658 -0.784746002 -0.792554497 0.087088272 1.195277674
## [166] 0.551358585 -0.553081440 -1.100214936 -2.093871658 0.385861350
## [171] 0.025189245 1.759269388 0.575869974 -0.275148161 0.676640696
## [176] -0.136825898 0.058690258 -0.487721921 0.100659729 0.992323228
## [181] -0.177671193 -0.496756554 -1.963345417 -2.050508287 -1.733792720
## [186] -1.280197326 -0.090953136 1.028432333 0.078550592 0.959591725
## [191] -1.153256137 1.358588387 -0.568628121 1.798133651 0.009186167
## [196] -0.199465545 1.074402383 0.593974045 0.356009610 -0.597582471
## [201] -1.252520924 0.240999817 -0.853355366 0.818841600 -0.177271129
## [206] -0.302107605 -0.347924606 -1.704091081 -1.488014580 -3.172510098
## [211] -2.552614193 -0.873998740 -0.828563388 1.230771328 -0.845588720
## [216] 1.286157460 -0.187592788 0.291773847 -1.705146631 -0.829815915
## [221] 0.879833084 -1.754723702 -2.075356573 -2.907545106 -0.439134642
## [226] -0.167425962 0.162472011 -0.947636128 0.157347060 0.636491259
## [231] -1.025643117 -2.007399770 -2.541308773 -1.230362267 -1.507131933
## [236] 1.430386538 1.391466817 1.119453696 0.449577249 1.369710402
## [241] 1.520136970 2.398423880 1.180421535 0.622458417 0.728715002
## [246] 0.115039075 0.940423688 -0.010895815 0.627498239 0.382156515
## [251] 1.283816650 -0.187795082 -0.534204334 0.864200086 1.097481629
## [256] 1.193495187 0.726199396 -1.178612492 0.783518662 -0.083580750
## [261] 0.150312675 0.830724377 -1.042605201 0.008240644 0.123373382
## [266] -0.813487449 1.307059960 0.392111391 1.457140533 0.311981224
## [271] 0.749556513 -0.003925389 0.903851957 0.110285709 -0.608022447
## [276] 0.719879114 -0.458810236 0.351191065 0.010030728 -1.017365871
## [281] -1.326417513 0.907229152 -0.680304065 0.722030213 0.480038598
## [286] 0.433986923 1.382270034 0.320974947 -0.146237710 -0.294062200
## [291] -1.210628562 -0.295480574 -1.711979233 -1.155361183 -1.865836284
## [296] -1.223231019 -1.514064934 -0.541870440 -0.886283446 0.962672593Selain menggunakan cara di atas, pembangkitan proses MA(2) dapat
dilakukan dengan fungsi arima.sim() sebagai berikut.
Menggunakan fungsi Arima.sim()
ma2 <- arima.sim(list(order=c(0,0,2), ma=c(0.4, 0.6)), n=300)
ma2## Time Series:
## Start = 1
## End = 300
## Frequency = 1
## [1] -1.071082769 -0.850804055 0.499757168 -1.282264464 -0.547840914
## [6] -1.575171808 -2.048969551 -2.635133956 -1.571118561 -1.790433630
## [11] -1.717066338 -0.556133552 0.072669291 1.929914004 1.622290675
## [16] 1.146337528 0.859824636 1.116855057 -0.191069929 -0.512269221
## [21] -1.334764260 -0.621694643 -0.509740435 1.505990240 1.457148632
## [26] -0.220387312 1.708900173 -1.593188538 -0.457951490 -1.074571780
## [31] -1.424819544 0.444904719 1.162710619 0.801165189 1.235879033
## [36] 0.632147482 1.096573788 1.407632885 -1.045479149 -0.070897014
## [41] -1.294572191 -1.413649335 0.298450337 -1.586486923 -1.553027865
## [46] -1.772460310 0.126243449 -0.527127135 -0.067794069 -0.241172973
## [51] 1.021325829 -0.175471645 1.150838818 -0.752948351 -1.013927783
## [56] -0.753370742 0.148780792 0.501991662 1.015190456 -1.300195853
## [61] 1.166044768 -0.305120098 0.674591499 0.415204624 -1.047739572
## [66] -0.260909831 -0.986268839 0.038637905 -0.382428179 -0.406402097
## [71] 0.773056227 0.002205856 0.754852936 -0.786248286 -0.545091953
## [76] -0.616142986 0.563288411 0.524051538 0.624953430 -0.964802358
## [81] -0.066647152 -0.223942494 0.926525136 0.278816331 -0.071076048
## [86] -1.092350059 0.244387648 0.074371185 -0.077112085 -0.390230283
## [91] 0.068925502 -0.253560861 -0.039068952 1.312431066 0.970912635
## [96] 0.628067792 0.684822390 -0.333992958 -0.628030698 -0.662127240
## [101] 0.588982090 -0.141619294 0.763325657 0.655406947 0.985499521
## [106] 0.609121116 -0.542412732 0.961035406 0.725889562 1.426175801
## [111] 1.600950148 0.135300662 0.743149865 -1.570274359 -0.162972876
## [116] -1.202762256 0.947459128 1.144062871 0.869215089 -0.990393290
## [121] -0.366619070 -2.643551849 1.529864974 -2.126238994 0.938004833
## [126] 1.516173076 0.941782776 2.550289654 0.715490840 1.890331032
## [131] 1.227626992 0.968123780 0.080459000 0.758591268 -1.891570333
## [136] -1.216103591 -1.464820544 -0.673680094 0.925918181 0.698530817
## [141] 1.292040391 -0.518080662 -0.954875100 -0.462463739 -0.257139864
## [146] 1.507463953 -1.205522606 -1.132269730 -2.198648888 -2.700342733
## [151] -1.543950728 -1.003477892 -0.178897728 0.228618111 0.101802593
## [156] 0.018172527 0.814304386 -0.452914074 2.088948582 -0.365202676
## [161] -0.118731188 -1.315941822 -0.907063537 -0.319942335 2.131579204
## [166] 1.097535512 0.721015298 0.476986972 0.617178776 0.495925905
## [171] -0.549156353 -1.426458952 -1.561737708 0.203369328 1.571558550
## [176] 2.168363325 1.346315557 2.311187046 0.541474062 0.113853382
## [181] -0.429970046 -0.536558380 -0.528181183 2.184588916 0.322338368
## [186] 1.517541534 -0.631100826 0.926468572 1.173449965 2.665931678
## [191] -0.238634413 -0.168014854 -1.589562102 -0.114302136 -0.746017544
## [196] 0.276607256 0.889417328 0.397169584 0.237891005 0.459371406
## [201] 0.413055872 -1.068733926 1.161809897 -0.064600300 2.538444503
## [206] 0.920464388 1.635208158 0.594599195 0.665344758 -0.654894994
## [211] -1.141373249 -0.882992113 -2.258098967 -1.780558601 0.241481270
## [216] -0.937886801 1.416730487 -1.761404198 1.103185471 0.654019753
## [221] 1.556337164 1.622221485 0.408812516 1.230660997 0.050617969
## [226] 0.991719890 1.317263156 0.832131556 -1.039402294 -1.629890965
## [231] -1.699907568 -0.883984407 -1.378461611 -1.163270433 -0.870458086
## [236] 0.880456555 0.067434080 1.068496737 0.968127486 -0.452585120
## [241] 0.746971500 -1.012716859 0.641007849 -0.852465160 0.586467416
## [246] -1.785968236 0.270356272 1.332686762 0.916821442 -0.333308603
## [251] -0.259371538 -1.329931585 -0.099516973 -0.261575178 -0.395181243
## [256] 0.770076318 0.600685316 -0.493645849 -2.217544320 -0.130465339
## [261] -0.487397864 0.566375648 -0.454031504 -1.654570423 -1.138811290
## [266] -1.303406650 0.208605017 0.278875768 -0.588345571 0.243335371
## [271] -1.450925616 -1.112238460 -0.887506267 1.446570757 1.018451240
## [276] 1.232553523 0.239762425 -0.886935120 -0.732634174 0.906346302
## [281] 0.104700811 2.185706778 -0.828808713 0.959778859 0.035625865
## [286] 1.581985488 2.191549947 1.672841101 1.024249711 0.418745957
## [291] -1.748862238 0.568137229 -0.597046246 0.596620828 -0.195720732
## [296] -0.477767801 -0.998905336 0.427979455 1.250626072 -0.181410318Buatlah 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 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 x x x
## 1 x x x o o o o o o o o o o o
## 2 x x x 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 o x o o o o o o o o o o
## 6 x x x x o o o o o o o o o o
## 7 x x o x o 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(2)
Buatlah scatterplot antara \(Y_t\) dengan \(Y_{t-1}\), \(Y_t\) dengan \(Y_{t-2}\), dan \(Y_t\) dengan \(Y_{t-3}\)
Korelasi antara \(Y_t\) dengan \(Y_{t-1}\)
#Yt
yt1_ma2 <- ma2[-1]
yt1_ma2## [1] -0.850804055 0.499757168 -1.282264464 -0.547840914 -1.575171808
## [6] -2.048969551 -2.635133956 -1.571118561 -1.790433630 -1.717066338
## [11] -0.556133552 0.072669291 1.929914004 1.622290675 1.146337528
## [16] 0.859824636 1.116855057 -0.191069929 -0.512269221 -1.334764260
## [21] -0.621694643 -0.509740435 1.505990240 1.457148632 -0.220387312
## [26] 1.708900173 -1.593188538 -0.457951490 -1.074571780 -1.424819544
## [31] 0.444904719 1.162710619 0.801165189 1.235879033 0.632147482
## [36] 1.096573788 1.407632885 -1.045479149 -0.070897014 -1.294572191
## [41] -1.413649335 0.298450337 -1.586486923 -1.553027865 -1.772460310
## [46] 0.126243449 -0.527127135 -0.067794069 -0.241172973 1.021325829
## [51] -0.175471645 1.150838818 -0.752948351 -1.013927783 -0.753370742
## [56] 0.148780792 0.501991662 1.015190456 -1.300195853 1.166044768
## [61] -0.305120098 0.674591499 0.415204624 -1.047739572 -0.260909831
## [66] -0.986268839 0.038637905 -0.382428179 -0.406402097 0.773056227
## [71] 0.002205856 0.754852936 -0.786248286 -0.545091953 -0.616142986
## [76] 0.563288411 0.524051538 0.624953430 -0.964802358 -0.066647152
## [81] -0.223942494 0.926525136 0.278816331 -0.071076048 -1.092350059
## [86] 0.244387648 0.074371185 -0.077112085 -0.390230283 0.068925502
## [91] -0.253560861 -0.039068952 1.312431066 0.970912635 0.628067792
## [96] 0.684822390 -0.333992958 -0.628030698 -0.662127240 0.588982090
## [101] -0.141619294 0.763325657 0.655406947 0.985499521 0.609121116
## [106] -0.542412732 0.961035406 0.725889562 1.426175801 1.600950148
## [111] 0.135300662 0.743149865 -1.570274359 -0.162972876 -1.202762256
## [116] 0.947459128 1.144062871 0.869215089 -0.990393290 -0.366619070
## [121] -2.643551849 1.529864974 -2.126238994 0.938004833 1.516173076
## [126] 0.941782776 2.550289654 0.715490840 1.890331032 1.227626992
## [131] 0.968123780 0.080459000 0.758591268 -1.891570333 -1.216103591
## [136] -1.464820544 -0.673680094 0.925918181 0.698530817 1.292040391
## [141] -0.518080662 -0.954875100 -0.462463739 -0.257139864 1.507463953
## [146] -1.205522606 -1.132269730 -2.198648888 -2.700342733 -1.543950728
## [151] -1.003477892 -0.178897728 0.228618111 0.101802593 0.018172527
## [156] 0.814304386 -0.452914074 2.088948582 -0.365202676 -0.118731188
## [161] -1.315941822 -0.907063537 -0.319942335 2.131579204 1.097535512
## [166] 0.721015298 0.476986972 0.617178776 0.495925905 -0.549156353
## [171] -1.426458952 -1.561737708 0.203369328 1.571558550 2.168363325
## [176] 1.346315557 2.311187046 0.541474062 0.113853382 -0.429970046
## [181] -0.536558380 -0.528181183 2.184588916 0.322338368 1.517541534
## [186] -0.631100826 0.926468572 1.173449965 2.665931678 -0.238634413
## [191] -0.168014854 -1.589562102 -0.114302136 -0.746017544 0.276607256
## [196] 0.889417328 0.397169584 0.237891005 0.459371406 0.413055872
## [201] -1.068733926 1.161809897 -0.064600300 2.538444503 0.920464388
## [206] 1.635208158 0.594599195 0.665344758 -0.654894994 -1.141373249
## [211] -0.882992113 -2.258098967 -1.780558601 0.241481270 -0.937886801
## [216] 1.416730487 -1.761404198 1.103185471 0.654019753 1.556337164
## [221] 1.622221485 0.408812516 1.230660997 0.050617969 0.991719890
## [226] 1.317263156 0.832131556 -1.039402294 -1.629890965 -1.699907568
## [231] -0.883984407 -1.378461611 -1.163270433 -0.870458086 0.880456555
## [236] 0.067434080 1.068496737 0.968127486 -0.452585120 0.746971500
## [241] -1.012716859 0.641007849 -0.852465160 0.586467416 -1.785968236
## [246] 0.270356272 1.332686762 0.916821442 -0.333308603 -0.259371538
## [251] -1.329931585 -0.099516973 -0.261575178 -0.395181243 0.770076318
## [256] 0.600685316 -0.493645849 -2.217544320 -0.130465339 -0.487397864
## [261] 0.566375648 -0.454031504 -1.654570423 -1.138811290 -1.303406650
## [266] 0.208605017 0.278875768 -0.588345571 0.243335371 -1.450925616
## [271] -1.112238460 -0.887506267 1.446570757 1.018451240 1.232553523
## [276] 0.239762425 -0.886935120 -0.732634174 0.906346302 0.104700811
## [281] 2.185706778 -0.828808713 0.959778859 0.035625865 1.581985488
## [286] 2.191549947 1.672841101 1.024249711 0.418745957 -1.748862238
## [291] 0.568137229 -0.597046246 0.596620828 -0.195720732 -0.477767801
## [296] -0.998905336 0.427979455 1.250626072 -0.181410318#Yt-1
yt_1_ma2 <- ma2[-300]
yt_1_ma2## [1] -1.071082769 -0.850804055 0.499757168 -1.282264464 -0.547840914
## [6] -1.575171808 -2.048969551 -2.635133956 -1.571118561 -1.790433630
## [11] -1.717066338 -0.556133552 0.072669291 1.929914004 1.622290675
## [16] 1.146337528 0.859824636 1.116855057 -0.191069929 -0.512269221
## [21] -1.334764260 -0.621694643 -0.509740435 1.505990240 1.457148632
## [26] -0.220387312 1.708900173 -1.593188538 -0.457951490 -1.074571780
## [31] -1.424819544 0.444904719 1.162710619 0.801165189 1.235879033
## [36] 0.632147482 1.096573788 1.407632885 -1.045479149 -0.070897014
## [41] -1.294572191 -1.413649335 0.298450337 -1.586486923 -1.553027865
## [46] -1.772460310 0.126243449 -0.527127135 -0.067794069 -0.241172973
## [51] 1.021325829 -0.175471645 1.150838818 -0.752948351 -1.013927783
## [56] -0.753370742 0.148780792 0.501991662 1.015190456 -1.300195853
## [61] 1.166044768 -0.305120098 0.674591499 0.415204624 -1.047739572
## [66] -0.260909831 -0.986268839 0.038637905 -0.382428179 -0.406402097
## [71] 0.773056227 0.002205856 0.754852936 -0.786248286 -0.545091953
## [76] -0.616142986 0.563288411 0.524051538 0.624953430 -0.964802358
## [81] -0.066647152 -0.223942494 0.926525136 0.278816331 -0.071076048
## [86] -1.092350059 0.244387648 0.074371185 -0.077112085 -0.390230283
## [91] 0.068925502 -0.253560861 -0.039068952 1.312431066 0.970912635
## [96] 0.628067792 0.684822390 -0.333992958 -0.628030698 -0.662127240
## [101] 0.588982090 -0.141619294 0.763325657 0.655406947 0.985499521
## [106] 0.609121116 -0.542412732 0.961035406 0.725889562 1.426175801
## [111] 1.600950148 0.135300662 0.743149865 -1.570274359 -0.162972876
## [116] -1.202762256 0.947459128 1.144062871 0.869215089 -0.990393290
## [121] -0.366619070 -2.643551849 1.529864974 -2.126238994 0.938004833
## [126] 1.516173076 0.941782776 2.550289654 0.715490840 1.890331032
## [131] 1.227626992 0.968123780 0.080459000 0.758591268 -1.891570333
## [136] -1.216103591 -1.464820544 -0.673680094 0.925918181 0.698530817
## [141] 1.292040391 -0.518080662 -0.954875100 -0.462463739 -0.257139864
## [146] 1.507463953 -1.205522606 -1.132269730 -2.198648888 -2.700342733
## [151] -1.543950728 -1.003477892 -0.178897728 0.228618111 0.101802593
## [156] 0.018172527 0.814304386 -0.452914074 2.088948582 -0.365202676
## [161] -0.118731188 -1.315941822 -0.907063537 -0.319942335 2.131579204
## [166] 1.097535512 0.721015298 0.476986972 0.617178776 0.495925905
## [171] -0.549156353 -1.426458952 -1.561737708 0.203369328 1.571558550
## [176] 2.168363325 1.346315557 2.311187046 0.541474062 0.113853382
## [181] -0.429970046 -0.536558380 -0.528181183 2.184588916 0.322338368
## [186] 1.517541534 -0.631100826 0.926468572 1.173449965 2.665931678
## [191] -0.238634413 -0.168014854 -1.589562102 -0.114302136 -0.746017544
## [196] 0.276607256 0.889417328 0.397169584 0.237891005 0.459371406
## [201] 0.413055872 -1.068733926 1.161809897 -0.064600300 2.538444503
## [206] 0.920464388 1.635208158 0.594599195 0.665344758 -0.654894994
## [211] -1.141373249 -0.882992113 -2.258098967 -1.780558601 0.241481270
## [216] -0.937886801 1.416730487 -1.761404198 1.103185471 0.654019753
## [221] 1.556337164 1.622221485 0.408812516 1.230660997 0.050617969
## [226] 0.991719890 1.317263156 0.832131556 -1.039402294 -1.629890965
## [231] -1.699907568 -0.883984407 -1.378461611 -1.163270433 -0.870458086
## [236] 0.880456555 0.067434080 1.068496737 0.968127486 -0.452585120
## [241] 0.746971500 -1.012716859 0.641007849 -0.852465160 0.586467416
## [246] -1.785968236 0.270356272 1.332686762 0.916821442 -0.333308603
## [251] -0.259371538 -1.329931585 -0.099516973 -0.261575178 -0.395181243
## [256] 0.770076318 0.600685316 -0.493645849 -2.217544320 -0.130465339
## [261] -0.487397864 0.566375648 -0.454031504 -1.654570423 -1.138811290
## [266] -1.303406650 0.208605017 0.278875768 -0.588345571 0.243335371
## [271] -1.450925616 -1.112238460 -0.887506267 1.446570757 1.018451240
## [276] 1.232553523 0.239762425 -0.886935120 -0.732634174 0.906346302
## [281] 0.104700811 2.185706778 -0.828808713 0.959778859 0.035625865
## [286] 1.581985488 2.191549947 1.672841101 1.024249711 0.418745957
## [291] -1.748862238 0.568137229 -0.597046246 0.596620828 -0.195720732
## [296] -0.477767801 -0.998905336 0.427979455 1.250626072Korelasi antara \(Y_t\) dengan \(Y_{t-2}\)
#Yt
yt2_ma2 <- ma2[-c(1,2)]
yt2_ma2## [1] 0.499757168 -1.282264464 -0.547840914 -1.575171808 -2.048969551
## [6] -2.635133956 -1.571118561 -1.790433630 -1.717066338 -0.556133552
## [11] 0.072669291 1.929914004 1.622290675 1.146337528 0.859824636
## [16] 1.116855057 -0.191069929 -0.512269221 -1.334764260 -0.621694643
## [21] -0.509740435 1.505990240 1.457148632 -0.220387312 1.708900173
## [26] -1.593188538 -0.457951490 -1.074571780 -1.424819544 0.444904719
## [31] 1.162710619 0.801165189 1.235879033 0.632147482 1.096573788
## [36] 1.407632885 -1.045479149 -0.070897014 -1.294572191 -1.413649335
## [41] 0.298450337 -1.586486923 -1.553027865 -1.772460310 0.126243449
## [46] -0.527127135 -0.067794069 -0.241172973 1.021325829 -0.175471645
## [51] 1.150838818 -0.752948351 -1.013927783 -0.753370742 0.148780792
## [56] 0.501991662 1.015190456 -1.300195853 1.166044768 -0.305120098
## [61] 0.674591499 0.415204624 -1.047739572 -0.260909831 -0.986268839
## [66] 0.038637905 -0.382428179 -0.406402097 0.773056227 0.002205856
## [71] 0.754852936 -0.786248286 -0.545091953 -0.616142986 0.563288411
## [76] 0.524051538 0.624953430 -0.964802358 -0.066647152 -0.223942494
## [81] 0.926525136 0.278816331 -0.071076048 -1.092350059 0.244387648
## [86] 0.074371185 -0.077112085 -0.390230283 0.068925502 -0.253560861
## [91] -0.039068952 1.312431066 0.970912635 0.628067792 0.684822390
## [96] -0.333992958 -0.628030698 -0.662127240 0.588982090 -0.141619294
## [101] 0.763325657 0.655406947 0.985499521 0.609121116 -0.542412732
## [106] 0.961035406 0.725889562 1.426175801 1.600950148 0.135300662
## [111] 0.743149865 -1.570274359 -0.162972876 -1.202762256 0.947459128
## [116] 1.144062871 0.869215089 -0.990393290 -0.366619070 -2.643551849
## [121] 1.529864974 -2.126238994 0.938004833 1.516173076 0.941782776
## [126] 2.550289654 0.715490840 1.890331032 1.227626992 0.968123780
## [131] 0.080459000 0.758591268 -1.891570333 -1.216103591 -1.464820544
## [136] -0.673680094 0.925918181 0.698530817 1.292040391 -0.518080662
## [141] -0.954875100 -0.462463739 -0.257139864 1.507463953 -1.205522606
## [146] -1.132269730 -2.198648888 -2.700342733 -1.543950728 -1.003477892
## [151] -0.178897728 0.228618111 0.101802593 0.018172527 0.814304386
## [156] -0.452914074 2.088948582 -0.365202676 -0.118731188 -1.315941822
## [161] -0.907063537 -0.319942335 2.131579204 1.097535512 0.721015298
## [166] 0.476986972 0.617178776 0.495925905 -0.549156353 -1.426458952
## [171] -1.561737708 0.203369328 1.571558550 2.168363325 1.346315557
## [176] 2.311187046 0.541474062 0.113853382 -0.429970046 -0.536558380
## [181] -0.528181183 2.184588916 0.322338368 1.517541534 -0.631100826
## [186] 0.926468572 1.173449965 2.665931678 -0.238634413 -0.168014854
## [191] -1.589562102 -0.114302136 -0.746017544 0.276607256 0.889417328
## [196] 0.397169584 0.237891005 0.459371406 0.413055872 -1.068733926
## [201] 1.161809897 -0.064600300 2.538444503 0.920464388 1.635208158
## [206] 0.594599195 0.665344758 -0.654894994 -1.141373249 -0.882992113
## [211] -2.258098967 -1.780558601 0.241481270 -0.937886801 1.416730487
## [216] -1.761404198 1.103185471 0.654019753 1.556337164 1.622221485
## [221] 0.408812516 1.230660997 0.050617969 0.991719890 1.317263156
## [226] 0.832131556 -1.039402294 -1.629890965 -1.699907568 -0.883984407
## [231] -1.378461611 -1.163270433 -0.870458086 0.880456555 0.067434080
## [236] 1.068496737 0.968127486 -0.452585120 0.746971500 -1.012716859
## [241] 0.641007849 -0.852465160 0.586467416 -1.785968236 0.270356272
## [246] 1.332686762 0.916821442 -0.333308603 -0.259371538 -1.329931585
## [251] -0.099516973 -0.261575178 -0.395181243 0.770076318 0.600685316
## [256] -0.493645849 -2.217544320 -0.130465339 -0.487397864 0.566375648
## [261] -0.454031504 -1.654570423 -1.138811290 -1.303406650 0.208605017
## [266] 0.278875768 -0.588345571 0.243335371 -1.450925616 -1.112238460
## [271] -0.887506267 1.446570757 1.018451240 1.232553523 0.239762425
## [276] -0.886935120 -0.732634174 0.906346302 0.104700811 2.185706778
## [281] -0.828808713 0.959778859 0.035625865 1.581985488 2.191549947
## [286] 1.672841101 1.024249711 0.418745957 -1.748862238 0.568137229
## [291] -0.597046246 0.596620828 -0.195720732 -0.477767801 -0.998905336
## [296] 0.427979455 1.250626072 -0.181410318#Yt-2
yt_2_ma2 <- ma2[-c(300,299)]
yt_2_ma2## [1] -1.071082769 -0.850804055 0.499757168 -1.282264464 -0.547840914
## [6] -1.575171808 -2.048969551 -2.635133956 -1.571118561 -1.790433630
## [11] -1.717066338 -0.556133552 0.072669291 1.929914004 1.622290675
## [16] 1.146337528 0.859824636 1.116855057 -0.191069929 -0.512269221
## [21] -1.334764260 -0.621694643 -0.509740435 1.505990240 1.457148632
## [26] -0.220387312 1.708900173 -1.593188538 -0.457951490 -1.074571780
## [31] -1.424819544 0.444904719 1.162710619 0.801165189 1.235879033
## [36] 0.632147482 1.096573788 1.407632885 -1.045479149 -0.070897014
## [41] -1.294572191 -1.413649335 0.298450337 -1.586486923 -1.553027865
## [46] -1.772460310 0.126243449 -0.527127135 -0.067794069 -0.241172973
## [51] 1.021325829 -0.175471645 1.150838818 -0.752948351 -1.013927783
## [56] -0.753370742 0.148780792 0.501991662 1.015190456 -1.300195853
## [61] 1.166044768 -0.305120098 0.674591499 0.415204624 -1.047739572
## [66] -0.260909831 -0.986268839 0.038637905 -0.382428179 -0.406402097
## [71] 0.773056227 0.002205856 0.754852936 -0.786248286 -0.545091953
## [76] -0.616142986 0.563288411 0.524051538 0.624953430 -0.964802358
## [81] -0.066647152 -0.223942494 0.926525136 0.278816331 -0.071076048
## [86] -1.092350059 0.244387648 0.074371185 -0.077112085 -0.390230283
## [91] 0.068925502 -0.253560861 -0.039068952 1.312431066 0.970912635
## [96] 0.628067792 0.684822390 -0.333992958 -0.628030698 -0.662127240
## [101] 0.588982090 -0.141619294 0.763325657 0.655406947 0.985499521
## [106] 0.609121116 -0.542412732 0.961035406 0.725889562 1.426175801
## [111] 1.600950148 0.135300662 0.743149865 -1.570274359 -0.162972876
## [116] -1.202762256 0.947459128 1.144062871 0.869215089 -0.990393290
## [121] -0.366619070 -2.643551849 1.529864974 -2.126238994 0.938004833
## [126] 1.516173076 0.941782776 2.550289654 0.715490840 1.890331032
## [131] 1.227626992 0.968123780 0.080459000 0.758591268 -1.891570333
## [136] -1.216103591 -1.464820544 -0.673680094 0.925918181 0.698530817
## [141] 1.292040391 -0.518080662 -0.954875100 -0.462463739 -0.257139864
## [146] 1.507463953 -1.205522606 -1.132269730 -2.198648888 -2.700342733
## [151] -1.543950728 -1.003477892 -0.178897728 0.228618111 0.101802593
## [156] 0.018172527 0.814304386 -0.452914074 2.088948582 -0.365202676
## [161] -0.118731188 -1.315941822 -0.907063537 -0.319942335 2.131579204
## [166] 1.097535512 0.721015298 0.476986972 0.617178776 0.495925905
## [171] -0.549156353 -1.426458952 -1.561737708 0.203369328 1.571558550
## [176] 2.168363325 1.346315557 2.311187046 0.541474062 0.113853382
## [181] -0.429970046 -0.536558380 -0.528181183 2.184588916 0.322338368
## [186] 1.517541534 -0.631100826 0.926468572 1.173449965 2.665931678
## [191] -0.238634413 -0.168014854 -1.589562102 -0.114302136 -0.746017544
## [196] 0.276607256 0.889417328 0.397169584 0.237891005 0.459371406
## [201] 0.413055872 -1.068733926 1.161809897 -0.064600300 2.538444503
## [206] 0.920464388 1.635208158 0.594599195 0.665344758 -0.654894994
## [211] -1.141373249 -0.882992113 -2.258098967 -1.780558601 0.241481270
## [216] -0.937886801 1.416730487 -1.761404198 1.103185471 0.654019753
## [221] 1.556337164 1.622221485 0.408812516 1.230660997 0.050617969
## [226] 0.991719890 1.317263156 0.832131556 -1.039402294 -1.629890965
## [231] -1.699907568 -0.883984407 -1.378461611 -1.163270433 -0.870458086
## [236] 0.880456555 0.067434080 1.068496737 0.968127486 -0.452585120
## [241] 0.746971500 -1.012716859 0.641007849 -0.852465160 0.586467416
## [246] -1.785968236 0.270356272 1.332686762 0.916821442 -0.333308603
## [251] -0.259371538 -1.329931585 -0.099516973 -0.261575178 -0.395181243
## [256] 0.770076318 0.600685316 -0.493645849 -2.217544320 -0.130465339
## [261] -0.487397864 0.566375648 -0.454031504 -1.654570423 -1.138811290
## [266] -1.303406650 0.208605017 0.278875768 -0.588345571 0.243335371
## [271] -1.450925616 -1.112238460 -0.887506267 1.446570757 1.018451240
## [276] 1.232553523 0.239762425 -0.886935120 -0.732634174 0.906346302
## [281] 0.104700811 2.185706778 -0.828808713 0.959778859 0.035625865
## [286] 1.581985488 2.191549947 1.672841101 1.024249711 0.418745957
## [291] -1.748862238 0.568137229 -0.597046246 0.596620828 -0.195720732
## [296] -0.477767801 -0.998905336 0.427979455Korelasi antara \(Y_t\) dengan \(Y_{t-3}\)
#Yt
yt3_ma2 <- ma2[-c(1,2,3)]
yt3_ma2## [1] -1.282264464 -0.547840914 -1.575171808 -2.048969551 -2.635133956
## [6] -1.571118561 -1.790433630 -1.717066338 -0.556133552 0.072669291
## [11] 1.929914004 1.622290675 1.146337528 0.859824636 1.116855057
## [16] -0.191069929 -0.512269221 -1.334764260 -0.621694643 -0.509740435
## [21] 1.505990240 1.457148632 -0.220387312 1.708900173 -1.593188538
## [26] -0.457951490 -1.074571780 -1.424819544 0.444904719 1.162710619
## [31] 0.801165189 1.235879033 0.632147482 1.096573788 1.407632885
## [36] -1.045479149 -0.070897014 -1.294572191 -1.413649335 0.298450337
## [41] -1.586486923 -1.553027865 -1.772460310 0.126243449 -0.527127135
## [46] -0.067794069 -0.241172973 1.021325829 -0.175471645 1.150838818
## [51] -0.752948351 -1.013927783 -0.753370742 0.148780792 0.501991662
## [56] 1.015190456 -1.300195853 1.166044768 -0.305120098 0.674591499
## [61] 0.415204624 -1.047739572 -0.260909831 -0.986268839 0.038637905
## [66] -0.382428179 -0.406402097 0.773056227 0.002205856 0.754852936
## [71] -0.786248286 -0.545091953 -0.616142986 0.563288411 0.524051538
## [76] 0.624953430 -0.964802358 -0.066647152 -0.223942494 0.926525136
## [81] 0.278816331 -0.071076048 -1.092350059 0.244387648 0.074371185
## [86] -0.077112085 -0.390230283 0.068925502 -0.253560861 -0.039068952
## [91] 1.312431066 0.970912635 0.628067792 0.684822390 -0.333992958
## [96] -0.628030698 -0.662127240 0.588982090 -0.141619294 0.763325657
## [101] 0.655406947 0.985499521 0.609121116 -0.542412732 0.961035406
## [106] 0.725889562 1.426175801 1.600950148 0.135300662 0.743149865
## [111] -1.570274359 -0.162972876 -1.202762256 0.947459128 1.144062871
## [116] 0.869215089 -0.990393290 -0.366619070 -2.643551849 1.529864974
## [121] -2.126238994 0.938004833 1.516173076 0.941782776 2.550289654
## [126] 0.715490840 1.890331032 1.227626992 0.968123780 0.080459000
## [131] 0.758591268 -1.891570333 -1.216103591 -1.464820544 -0.673680094
## [136] 0.925918181 0.698530817 1.292040391 -0.518080662 -0.954875100
## [141] -0.462463739 -0.257139864 1.507463953 -1.205522606 -1.132269730
## [146] -2.198648888 -2.700342733 -1.543950728 -1.003477892 -0.178897728
## [151] 0.228618111 0.101802593 0.018172527 0.814304386 -0.452914074
## [156] 2.088948582 -0.365202676 -0.118731188 -1.315941822 -0.907063537
## [161] -0.319942335 2.131579204 1.097535512 0.721015298 0.476986972
## [166] 0.617178776 0.495925905 -0.549156353 -1.426458952 -1.561737708
## [171] 0.203369328 1.571558550 2.168363325 1.346315557 2.311187046
## [176] 0.541474062 0.113853382 -0.429970046 -0.536558380 -0.528181183
## [181] 2.184588916 0.322338368 1.517541534 -0.631100826 0.926468572
## [186] 1.173449965 2.665931678 -0.238634413 -0.168014854 -1.589562102
## [191] -0.114302136 -0.746017544 0.276607256 0.889417328 0.397169584
## [196] 0.237891005 0.459371406 0.413055872 -1.068733926 1.161809897
## [201] -0.064600300 2.538444503 0.920464388 1.635208158 0.594599195
## [206] 0.665344758 -0.654894994 -1.141373249 -0.882992113 -2.258098967
## [211] -1.780558601 0.241481270 -0.937886801 1.416730487 -1.761404198
## [216] 1.103185471 0.654019753 1.556337164 1.622221485 0.408812516
## [221] 1.230660997 0.050617969 0.991719890 1.317263156 0.832131556
## [226] -1.039402294 -1.629890965 -1.699907568 -0.883984407 -1.378461611
## [231] -1.163270433 -0.870458086 0.880456555 0.067434080 1.068496737
## [236] 0.968127486 -0.452585120 0.746971500 -1.012716859 0.641007849
## [241] -0.852465160 0.586467416 -1.785968236 0.270356272 1.332686762
## [246] 0.916821442 -0.333308603 -0.259371538 -1.329931585 -0.099516973
## [251] -0.261575178 -0.395181243 0.770076318 0.600685316 -0.493645849
## [256] -2.217544320 -0.130465339 -0.487397864 0.566375648 -0.454031504
## [261] -1.654570423 -1.138811290 -1.303406650 0.208605017 0.278875768
## [266] -0.588345571 0.243335371 -1.450925616 -1.112238460 -0.887506267
## [271] 1.446570757 1.018451240 1.232553523 0.239762425 -0.886935120
## [276] -0.732634174 0.906346302 0.104700811 2.185706778 -0.828808713
## [281] 0.959778859 0.035625865 1.581985488 2.191549947 1.672841101
## [286] 1.024249711 0.418745957 -1.748862238 0.568137229 -0.597046246
## [291] 0.596620828 -0.195720732 -0.477767801 -0.998905336 0.427979455
## [296] 1.250626072 -0.181410318#Yt-3
yt_3_ma2 <- ma2[-c(300,299,298)]
yt_3_ma2## [1] -1.071082769 -0.850804055 0.499757168 -1.282264464 -0.547840914
## [6] -1.575171808 -2.048969551 -2.635133956 -1.571118561 -1.790433630
## [11] -1.717066338 -0.556133552 0.072669291 1.929914004 1.622290675
## [16] 1.146337528 0.859824636 1.116855057 -0.191069929 -0.512269221
## [21] -1.334764260 -0.621694643 -0.509740435 1.505990240 1.457148632
## [26] -0.220387312 1.708900173 -1.593188538 -0.457951490 -1.074571780
## [31] -1.424819544 0.444904719 1.162710619 0.801165189 1.235879033
## [36] 0.632147482 1.096573788 1.407632885 -1.045479149 -0.070897014
## [41] -1.294572191 -1.413649335 0.298450337 -1.586486923 -1.553027865
## [46] -1.772460310 0.126243449 -0.527127135 -0.067794069 -0.241172973
## [51] 1.021325829 -0.175471645 1.150838818 -0.752948351 -1.013927783
## [56] -0.753370742 0.148780792 0.501991662 1.015190456 -1.300195853
## [61] 1.166044768 -0.305120098 0.674591499 0.415204624 -1.047739572
## [66] -0.260909831 -0.986268839 0.038637905 -0.382428179 -0.406402097
## [71] 0.773056227 0.002205856 0.754852936 -0.786248286 -0.545091953
## [76] -0.616142986 0.563288411 0.524051538 0.624953430 -0.964802358
## [81] -0.066647152 -0.223942494 0.926525136 0.278816331 -0.071076048
## [86] -1.092350059 0.244387648 0.074371185 -0.077112085 -0.390230283
## [91] 0.068925502 -0.253560861 -0.039068952 1.312431066 0.970912635
## [96] 0.628067792 0.684822390 -0.333992958 -0.628030698 -0.662127240
## [101] 0.588982090 -0.141619294 0.763325657 0.655406947 0.985499521
## [106] 0.609121116 -0.542412732 0.961035406 0.725889562 1.426175801
## [111] 1.600950148 0.135300662 0.743149865 -1.570274359 -0.162972876
## [116] -1.202762256 0.947459128 1.144062871 0.869215089 -0.990393290
## [121] -0.366619070 -2.643551849 1.529864974 -2.126238994 0.938004833
## [126] 1.516173076 0.941782776 2.550289654 0.715490840 1.890331032
## [131] 1.227626992 0.968123780 0.080459000 0.758591268 -1.891570333
## [136] -1.216103591 -1.464820544 -0.673680094 0.925918181 0.698530817
## [141] 1.292040391 -0.518080662 -0.954875100 -0.462463739 -0.257139864
## [146] 1.507463953 -1.205522606 -1.132269730 -2.198648888 -2.700342733
## [151] -1.543950728 -1.003477892 -0.178897728 0.228618111 0.101802593
## [156] 0.018172527 0.814304386 -0.452914074 2.088948582 -0.365202676
## [161] -0.118731188 -1.315941822 -0.907063537 -0.319942335 2.131579204
## [166] 1.097535512 0.721015298 0.476986972 0.617178776 0.495925905
## [171] -0.549156353 -1.426458952 -1.561737708 0.203369328 1.571558550
## [176] 2.168363325 1.346315557 2.311187046 0.541474062 0.113853382
## [181] -0.429970046 -0.536558380 -0.528181183 2.184588916 0.322338368
## [186] 1.517541534 -0.631100826 0.926468572 1.173449965 2.665931678
## [191] -0.238634413 -0.168014854 -1.589562102 -0.114302136 -0.746017544
## [196] 0.276607256 0.889417328 0.397169584 0.237891005 0.459371406
## [201] 0.413055872 -1.068733926 1.161809897 -0.064600300 2.538444503
## [206] 0.920464388 1.635208158 0.594599195 0.665344758 -0.654894994
## [211] -1.141373249 -0.882992113 -2.258098967 -1.780558601 0.241481270
## [216] -0.937886801 1.416730487 -1.761404198 1.103185471 0.654019753
## [221] 1.556337164 1.622221485 0.408812516 1.230660997 0.050617969
## [226] 0.991719890 1.317263156 0.832131556 -1.039402294 -1.629890965
## [231] -1.699907568 -0.883984407 -1.378461611 -1.163270433 -0.870458086
## [236] 0.880456555 0.067434080 1.068496737 0.968127486 -0.452585120
## [241] 0.746971500 -1.012716859 0.641007849 -0.852465160 0.586467416
## [246] -1.785968236 0.270356272 1.332686762 0.916821442 -0.333308603
## [251] -0.259371538 -1.329931585 -0.099516973 -0.261575178 -0.395181243
## [256] 0.770076318 0.600685316 -0.493645849 -2.217544320 -0.130465339
## [261] -0.487397864 0.566375648 -0.454031504 -1.654570423 -1.138811290
## [266] -1.303406650 0.208605017 0.278875768 -0.588345571 0.243335371
## [271] -1.450925616 -1.112238460 -0.887506267 1.446570757 1.018451240
## [276] 1.232553523 0.239762425 -0.886935120 -0.732634174 0.906346302
## [281] 0.104700811 2.185706778 -0.828808713 0.959778859 0.035625865
## [286] 1.581985488 2.191549947 1.672841101 1.024249711 0.418745957
## [291] -1.748862238 0.568137229 -0.597046246 0.596620828 -0.195720732
## [296] -0.477767801 -0.998905336Plot
plot(y=yt1_ma2,x=yt_1_ma2)
plot(y=yt2_ma2,x=yt_2_ma2)
plot(y=yt3_ma2,x=yt_3_ma2)
Berdasarkan scatterplot pertama, terlihat bahwa terdapat hubungan positif yang rendah atau lemah antara \(Y_t\) dengan \(Y_{t-1}\).Berdasarkan scatterplot kedua, terlihat bahwa terdapat hubungan positif yang rendah atau lemah antara \(Y_t\) dengan \(Y_{t-2}\). Berdasarkan scatterplot ketiga, terlihat bahwa terdapat hubungan positif yang sangat lemah antara \(Y_t\) dengan \(Y_{t-3}\). Hal ini sesuai dengan teori yang ada.
Hitung autokorelasi masing-masing baik dari dta bangkitan maupun secara teoritis
cor(yt1_ma2,yt_1_ma2)## [1] 0.3369231cor(yt2_ma2,yt_2_ma2)## [1] 0.3545613cor(yt3_ma2,yt_3_ma2)## [1] -0.06551178Korelasi antara \(Y_t\) dengan \(Y_{t-1}\), \(Y_t\) dengan \(Y_{t-2}\), dan \(Y_t\) dengan \(Y_{t-3}\) dari hasil simulasi mendekati perhitungan teoritis. Korelasi antara \(Y_t\) dengan \(Y_{t-1}\) sebesar 0.3369231. Korelasi antara \(Y_t\) dengan \(Y_{t-2}\) sebesar 0.3545613 . Korelasi antara \(Y_t\) dengan \(Y_{t-3}\) sebesar -0.06551178 .
Proses AR (2)
Bangkitkan model AR(2) dengan theta1= 0.5 dan theta2 = 0.2 sebanyak 300 data(Manual dan dengan arima.sim)
Proses AR dapat dituliskan sebagai berikut:
\[ y_{t} = c + e_t + \phi_{1}Y_{t-1} + \phi_{2}Y_{t-2} + \dots + \phi_{q}Y_{t-q} = c+{e_t+\sum_{i=1}^p \phi_iY_{t-i}} \] Terlihat bahwa \(Y_t\) berperan penting dalam pembangkitan proses AR.
Pembangkitan Proses AR
Akan dicoba membangkitkan proses AR paling sederhana, yaitu AR(2) dengan \(\dots = 0.5\) dan \(\dots = 0.2\) sebanyak 300 observasi dan \(c=0\).
Nilai-nilai selanjutnya dapat dicari melalui loop. Bentuk loop dapat dilihat dari rumus AR(2) yang hendak dibangkitkan:
\[ Y_t = e_t+0.5Y_{t-1}+0.2Y_{t-2} \]
set.seed(1029)
n<-length(data)
n## [1] 300ar <- c(2:n)
for (i in 3:n) {ar[i]<-data[i]+0.5*ar[i-1]+0.2*ar[i-2]}
ar## [1] 2.000000000 3.000000000 1.978906600 1.999656094 0.345358400
## [6] 0.517656495 -0.113626221 -0.501334670 -1.017757562 -0.990428743
## [11] -2.534559286 -1.860891272 -0.570665319 0.115829627 -0.817975130
## [16] 0.595474158 0.016675170 -1.490412009 -0.593871753 -2.097053986
## [21] -1.215867292 -0.064709114 1.096328710 -0.909078093 -0.658636556
## [26] -0.658487767 1.383119545 0.716659625 0.661562683 -0.652714475
## [31] -0.145189295 -1.019496788 -1.170487022 -0.597885677 0.262645117
## [36] -1.481662647 2.102105623 0.564169109 -0.019049788 0.703071440
## [41] 0.716921619 1.276879641 -1.887212794 0.296149588 -0.200388434
## [46] 1.529532352 0.449376640 -0.871517629 -0.462053440 -0.951625041
## [51] -2.265260450 -1.064779599 -0.762130260 0.316451393 1.437718011
## [56] 1.353357394 1.409759397 2.179345291 2.454777567 2.483199682
## [61] 3.932880024 1.270419783 2.873938660 2.260102374 3.308447232
## [66] 0.610071610 1.732980229 1.948864327 0.540854532 -0.761208219
## [71] -1.184272508 0.550147295 -1.255629371 -0.912286813 -1.332571746
## [76] -0.697848527 -0.825743473 -3.073907488 -2.380464648 0.174392010
## [81] -0.714278511 1.364701988 0.905814360 0.884341437 0.752049277
## [86] -0.767362203 0.712700971 1.091436197 1.230396437 1.967189383
## [91] 0.854004443 1.782483652 0.973451042 1.628836800 -0.478821316
## [96] 0.682092402 0.384251986 -1.533064501 -1.576533453 0.729794702
## [101] 1.577205227 -0.381729281 -0.568170278 -2.200815976 -1.668515397
## [106] -2.683790651 -2.896091552 -3.039588312 -0.979622449 -0.244052773
## [111] -0.020347442 0.800457792 0.987532552 -0.688549489 -0.845580279
## [116] 0.111273825 0.308959934 0.268882684 0.781600782 0.182495890
## [121] 0.653596970 -0.654173676 -1.724863838 0.302340385 -0.350625232
## [126] 0.483175495 0.369691586 0.777443377 -0.300605258 -0.037395403
## [131] 0.290503342 -0.329654937 -0.208188282 -1.673084433 -0.470864916
## [136] 0.064608715 -0.576804975 0.611693464 -0.029690039 -2.172851337
## [141] -1.383434201 -0.249100061 -0.668969280 -0.820664476 -1.171031583
## [146] 2.766996691 0.501066622 1.374792168 1.813793551 0.871030459
## [151] 2.000591773 1.920770904 3.048504199 0.524107503 2.092786750
## [156] 2.180213079 1.370955871 0.271427294 -1.167005699 -1.530897157
## [161] -1.725742158 -1.062031677 -1.275390836 -0.667534156 0.772941330
## [166] 0.150067230 -1.099372863 -1.026552062 -1.826873785 0.008730764
## [171] -0.130577449 0.927067205 0.478785020 -0.461254839 0.871374452
## [176] 0.335748888 -0.199832380 -0.299083226 0.342867394 1.050780174
## [181] -0.278798166 -0.640392783 -1.531184509 -2.055397864 -1.909900642
## [186] -1.718805064 -0.845646843 0.315218674 -0.603132174 0.308632365
## [191] -0.983460991 1.007151830 -0.226316303 1.237392362 0.442891040
## [196] -0.367796889 1.392095628 1.123528665 0.103327458 -0.327094746
## [201] -0.711903522 0.409316288 -0.781939741 0.349009661 0.084129552
## [206] -0.611515728 -0.387111187 -1.546648665 -1.787537640 -3.262418268
## [211] -3.155528327 -1.401930062 -1.791874188 -0.258621035 -1.424472809
## [216] 0.346290587 -0.181370063 -0.367955468 -1.745014479 -0.958092022
## [221] 0.971439292 -2.173221526 -3.060443290 -2.524769242 -0.788774373
## [226] -1.165104584 -1.122949080 -1.429616690 -0.298417460 0.326039650
## [231] -1.611363891 -2.518696727 -2.382820696 -1.538098512 -2.334846069
## [236] 0.296806162 1.017700729 0.090049208 0.087646837 1.782808260
## [241] 1.837231215 2.269697053 1.727929757 1.253396836 1.591421361
## [246] 0.952478797 1.401029664 0.693859130 0.969590725 0.987032489
## [251] 1.620391783 0.228545169 -0.344004941 1.518283334 1.599405129
## [256] 0.946482576 1.036633702 -0.474277688 1.080356147 0.626809119
## [261] -0.058907119 1.053093854 -0.557684189 -0.205314157 0.607485615
## [266] -0.797224785 0.960898876 0.853853070 1.120298920 0.522715101
## [271] 1.017540271 0.521461576 1.085557669 0.563940275 -0.448479721
## [276] 1.037330390 0.079182972 0.048893823 0.339451497 -0.838630310
## [281] -1.450089235 1.064805118 -0.561751044 -0.198775462 0.803126497
## [286] 0.468365882 1.125536522 0.621190437 -0.034871756 0.062120321
## [291] -0.826325430 -0.329245596 -1.560217595 -1.552083048 -1.933273448
## [296] -1.738533911 -2.078281305 -1.322885664 -1.495551451 0.079323615Selain menggunakan cara di atas, pembangkitan proses AR dapat
dilakukan dengan fungsi arima.sim() sebagai berikut.
set.seed(1029)
ar2 <- arima.sim(list(order=c(2,0,0), ar=(c(0.5 , 0.2))), n=300)
ar2## Time Series:
## Start = 1
## End = 300
## Frequency = 1
## [1] 1.215942714 1.448976036 -0.567634760 1.760737453 -0.607063821
## [6] -0.912629447 -0.443457900 -1.306035202 -0.016654070 1.144222717
## [11] 0.369409949 0.880922714 1.079163908 1.305988143 1.701478743
## [16] -0.620735893 0.152508811 -0.351893632 -1.511042651 0.201189839
## [21] -1.379577932 -2.347647069 -1.836182410 -0.087942212 -1.226340112
## [26] -1.152306602 -0.364895540 0.738735288 -0.613656330 0.664789030
## [31] -0.326806238 -1.324107384 -0.641439161 0.305046666 0.118406607
## [36] 0.563382016 -1.228593939 1.012109081 0.270103557 0.099920184
## [41] 0.608346110 -0.782796557 -0.390476612 -0.625595245 -0.170287456
## [46] -0.516654575 -0.708593450 0.665747135 -0.006263974 0.289813051
## [51] -0.588061489 -0.584351366 -0.447588542 0.446714031 0.365619752
## [56] 0.331966790 -0.888688922 -0.029368213 0.120880220 0.646621025
## [61] 0.201499798 -0.137839975 -0.886212400 0.337498849 0.257164336
## [66] -0.472196930 -0.466978768 0.254889320 -0.283251289 -0.352487508
## [71] 1.374653062 1.101826671 0.295384439 0.974065162 0.287989173
## [76] -0.549579526 -0.369092133 0.888312487 -0.153251138 0.364133492
## [81] 1.015738213 1.062608710 0.632214935 -0.262036441 1.334068670
## [86] 1.279458437 1.263599855 1.946920168 0.723555154 1.059824304
## [91] -0.717541313 0.061889511 -0.563505188 0.733243925 1.267501358
## [96] 0.642611209 -0.968620372 -0.022363562 -2.055771708 1.037798119
## [101] -1.736037439 -0.227035411 1.988347703 0.650866176 1.923108782
## [106] 1.505956361 2.142235783 1.949544576 1.537667552 0.839081200
## [111] 1.532861110 -1.087841443 -1.128088995 -0.676884189 -0.745187129
## [116] 0.427561515 0.497737723 0.891904648 -0.455386822 -0.938346995
## [121] -0.066568781 -0.142145777 1.095344855 -1.205456741 -1.533888024
## [126] -1.711772791 -2.891374586 -2.218265887 -1.481569787 -1.187536607
## [131] -0.783727580 -0.568251269 -0.510959369 0.436537830 -0.617051209
## [136] 1.677581182 0.030534935 -0.633288851 -0.821942990 -0.649687048
## [141] -0.457508013 1.827432814 1.026266159 0.206339861 0.931891234
## [146] 1.278373005 0.638946045 -0.362056898 -0.992845160 -1.192406587
## [151] 0.221789606 1.411490839 1.692898565 1.174498506 2.653038455
## [156] 1.384558562 0.371158807 0.479328556 0.281639582 -0.288695561
## [161] 2.326075351 0.777227483 1.054141649 0.168133008 1.306929029
## [166] 1.764357391 2.771369152 0.202440016 0.125242609 -0.352682679
## [171] 0.234872579 -0.580108757 0.052631560 1.057631894 0.300149844
## [176] 0.006766553 0.808234000 0.733498729 -1.118441582 1.211206899
## [181] 0.655926303 2.024807800 1.317805539 1.756623975 1.354838606
## [186] 1.193246864 0.019115747 -0.652477500 -0.336048571 -2.010754716
## [191] -2.160071919 0.221629264 -1.288122111 0.181474528 -1.660621833
## [196] 0.437937492 0.944323536 0.953922666 1.495890671 0.779012492
## [201] 1.485193549 0.726244001 1.242837208 1.954153125 1.233175373
## [206] -0.747487963 -1.059556184 -0.953260577 -0.903463609 -1.770485762
## [211] -1.649012049 -1.138969579 0.315161784 -0.512338696 0.323540624
## [216] 1.086034416 -0.565559093 0.534673502 -0.394619737 0.410022850
## [221] -0.597230088 0.358945158 -1.522173388 -0.131394790 1.688713510
## [226] 0.576619283 -0.845980204 0.166650344 -0.722309561 -0.457356161
## [231] -0.201039376 -0.578293541 0.491982350 0.634264557 -0.772492134
## [236] -2.304088376 0.093682664 -0.234647186 -0.444078242 -0.692401084
## [241] -1.712918894 -1.368530357 -1.414077708 -0.393093992 -0.203209892
## [246] -1.231620934 -0.158249786 -1.344817446 -1.707471389 -0.997219347
## [251] 1.158322496 0.523510138 0.469400119 0.502496636 -0.592632928
## [256] -0.651203532 1.207029270 0.190743529 1.644801983 -0.321950521
## [261] 0.815949463 0.829524893 1.576783124 2.454750015 2.016092872
## [266] 1.433629569 1.280909320 -0.846811926 1.013984553 0.332495685
## [271] 0.258994072 0.047376237 -0.276802485 -0.897743602 0.442647851
## [276] 1.374939782 -0.506804777 0.259224543 0.073019893 -0.123606980
## [281] -2.286072636 0.397494545 -0.908983209 0.888752871 -2.052915199
## [286] -1.661833090 2.089796346 -0.579996910 -0.043274191 -0.884093234
## [291] 0.192162173 1.744780008 2.041793006 2.406411753 2.471303241
## [296] 1.005071014 1.855499591 1.321010106 -0.044529774 -0.441597895Buatlah Plot time series, Plot ACF, plot PACF, dan Plot EACF. Identifikasi 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(ar2)## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x o o o o o o o o o o
## 1 x o o o o o o o o o o o o o
## 2 o 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 o x o o o o 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 x o x o o o o o o oBerdasarkan pola segitiga nol pada plot EACF, terlihat bahwa segitiga nol berada pada ordo AR(2) dan ordo MA(0)
Buatlah scatterplot antara \(Y_t\) dengan \(Y_{t-1}\), \(Y_t\) dengan \(Y_{t-2}\), dan \(Y_t\) dengan \(Y_{t-3}\)
Korelasi antara \(Y_t\) dengan \(Y_{t-1}\)
#Yt
yt_ar <- ar[-1]
yt_ar## [1] 3.000000000 1.978906600 1.999656094 0.345358400 0.517656495
## [6] -0.113626221 -0.501334670 -1.017757562 -0.990428743 -2.534559286
## [11] -1.860891272 -0.570665319 0.115829627 -0.817975130 0.595474158
## [16] 0.016675170 -1.490412009 -0.593871753 -2.097053986 -1.215867292
## [21] -0.064709114 1.096328710 -0.909078093 -0.658636556 -0.658487767
## [26] 1.383119545 0.716659625 0.661562683 -0.652714475 -0.145189295
## [31] -1.019496788 -1.170487022 -0.597885677 0.262645117 -1.481662647
## [36] 2.102105623 0.564169109 -0.019049788 0.703071440 0.716921619
## [41] 1.276879641 -1.887212794 0.296149588 -0.200388434 1.529532352
## [46] 0.449376640 -0.871517629 -0.462053440 -0.951625041 -2.265260450
## [51] -1.064779599 -0.762130260 0.316451393 1.437718011 1.353357394
## [56] 1.409759397 2.179345291 2.454777567 2.483199682 3.932880024
## [61] 1.270419783 2.873938660 2.260102374 3.308447232 0.610071610
## [66] 1.732980229 1.948864327 0.540854532 -0.761208219 -1.184272508
## [71] 0.550147295 -1.255629371 -0.912286813 -1.332571746 -0.697848527
## [76] -0.825743473 -3.073907488 -2.380464648 0.174392010 -0.714278511
## [81] 1.364701988 0.905814360 0.884341437 0.752049277 -0.767362203
## [86] 0.712700971 1.091436197 1.230396437 1.967189383 0.854004443
## [91] 1.782483652 0.973451042 1.628836800 -0.478821316 0.682092402
## [96] 0.384251986 -1.533064501 -1.576533453 0.729794702 1.577205227
## [101] -0.381729281 -0.568170278 -2.200815976 -1.668515397 -2.683790651
## [106] -2.896091552 -3.039588312 -0.979622449 -0.244052773 -0.020347442
## [111] 0.800457792 0.987532552 -0.688549489 -0.845580279 0.111273825
## [116] 0.308959934 0.268882684 0.781600782 0.182495890 0.653596970
## [121] -0.654173676 -1.724863838 0.302340385 -0.350625232 0.483175495
## [126] 0.369691586 0.777443377 -0.300605258 -0.037395403 0.290503342
## [131] -0.329654937 -0.208188282 -1.673084433 -0.470864916 0.064608715
## [136] -0.576804975 0.611693464 -0.029690039 -2.172851337 -1.383434201
## [141] -0.249100061 -0.668969280 -0.820664476 -1.171031583 2.766996691
## [146] 0.501066622 1.374792168 1.813793551 0.871030459 2.000591773
## [151] 1.920770904 3.048504199 0.524107503 2.092786750 2.180213079
## [156] 1.370955871 0.271427294 -1.167005699 -1.530897157 -1.725742158
## [161] -1.062031677 -1.275390836 -0.667534156 0.772941330 0.150067230
## [166] -1.099372863 -1.026552062 -1.826873785 0.008730764 -0.130577449
## [171] 0.927067205 0.478785020 -0.461254839 0.871374452 0.335748888
## [176] -0.199832380 -0.299083226 0.342867394 1.050780174 -0.278798166
## [181] -0.640392783 -1.531184509 -2.055397864 -1.909900642 -1.718805064
## [186] -0.845646843 0.315218674 -0.603132174 0.308632365 -0.983460991
## [191] 1.007151830 -0.226316303 1.237392362 0.442891040 -0.367796889
## [196] 1.392095628 1.123528665 0.103327458 -0.327094746 -0.711903522
## [201] 0.409316288 -0.781939741 0.349009661 0.084129552 -0.611515728
## [206] -0.387111187 -1.546648665 -1.787537640 -3.262418268 -3.155528327
## [211] -1.401930062 -1.791874188 -0.258621035 -1.424472809 0.346290587
## [216] -0.181370063 -0.367955468 -1.745014479 -0.958092022 0.971439292
## [221] -2.173221526 -3.060443290 -2.524769242 -0.788774373 -1.165104584
## [226] -1.122949080 -1.429616690 -0.298417460 0.326039650 -1.611363891
## [231] -2.518696727 -2.382820696 -1.538098512 -2.334846069 0.296806162
## [236] 1.017700729 0.090049208 0.087646837 1.782808260 1.837231215
## [241] 2.269697053 1.727929757 1.253396836 1.591421361 0.952478797
## [246] 1.401029664 0.693859130 0.969590725 0.987032489 1.620391783
## [251] 0.228545169 -0.344004941 1.518283334 1.599405129 0.946482576
## [256] 1.036633702 -0.474277688 1.080356147 0.626809119 -0.058907119
## [261] 1.053093854 -0.557684189 -0.205314157 0.607485615 -0.797224785
## [266] 0.960898876 0.853853070 1.120298920 0.522715101 1.017540271
## [271] 0.521461576 1.085557669 0.563940275 -0.448479721 1.037330390
## [276] 0.079182972 0.048893823 0.339451497 -0.838630310 -1.450089235
## [281] 1.064805118 -0.561751044 -0.198775462 0.803126497 0.468365882
## [286] 1.125536522 0.621190437 -0.034871756 0.062120321 -0.826325430
## [291] -0.329245596 -1.560217595 -1.552083048 -1.933273448 -1.738533911
## [296] -2.078281305 -1.322885664 -1.495551451 0.079323615#Yt-1
yt_1_ar <- ar[-300]
yt_1_ar## [1] 2.000000000 3.000000000 1.978906600 1.999656094 0.345358400
## [6] 0.517656495 -0.113626221 -0.501334670 -1.017757562 -0.990428743
## [11] -2.534559286 -1.860891272 -0.570665319 0.115829627 -0.817975130
## [16] 0.595474158 0.016675170 -1.490412009 -0.593871753 -2.097053986
## [21] -1.215867292 -0.064709114 1.096328710 -0.909078093 -0.658636556
## [26] -0.658487767 1.383119545 0.716659625 0.661562683 -0.652714475
## [31] -0.145189295 -1.019496788 -1.170487022 -0.597885677 0.262645117
## [36] -1.481662647 2.102105623 0.564169109 -0.019049788 0.703071440
## [41] 0.716921619 1.276879641 -1.887212794 0.296149588 -0.200388434
## [46] 1.529532352 0.449376640 -0.871517629 -0.462053440 -0.951625041
## [51] -2.265260450 -1.064779599 -0.762130260 0.316451393 1.437718011
## [56] 1.353357394 1.409759397 2.179345291 2.454777567 2.483199682
## [61] 3.932880024 1.270419783 2.873938660 2.260102374 3.308447232
## [66] 0.610071610 1.732980229 1.948864327 0.540854532 -0.761208219
## [71] -1.184272508 0.550147295 -1.255629371 -0.912286813 -1.332571746
## [76] -0.697848527 -0.825743473 -3.073907488 -2.380464648 0.174392010
## [81] -0.714278511 1.364701988 0.905814360 0.884341437 0.752049277
## [86] -0.767362203 0.712700971 1.091436197 1.230396437 1.967189383
## [91] 0.854004443 1.782483652 0.973451042 1.628836800 -0.478821316
## [96] 0.682092402 0.384251986 -1.533064501 -1.576533453 0.729794702
## [101] 1.577205227 -0.381729281 -0.568170278 -2.200815976 -1.668515397
## [106] -2.683790651 -2.896091552 -3.039588312 -0.979622449 -0.244052773
## [111] -0.020347442 0.800457792 0.987532552 -0.688549489 -0.845580279
## [116] 0.111273825 0.308959934 0.268882684 0.781600782 0.182495890
## [121] 0.653596970 -0.654173676 -1.724863838 0.302340385 -0.350625232
## [126] 0.483175495 0.369691586 0.777443377 -0.300605258 -0.037395403
## [131] 0.290503342 -0.329654937 -0.208188282 -1.673084433 -0.470864916
## [136] 0.064608715 -0.576804975 0.611693464 -0.029690039 -2.172851337
## [141] -1.383434201 -0.249100061 -0.668969280 -0.820664476 -1.171031583
## [146] 2.766996691 0.501066622 1.374792168 1.813793551 0.871030459
## [151] 2.000591773 1.920770904 3.048504199 0.524107503 2.092786750
## [156] 2.180213079 1.370955871 0.271427294 -1.167005699 -1.530897157
## [161] -1.725742158 -1.062031677 -1.275390836 -0.667534156 0.772941330
## [166] 0.150067230 -1.099372863 -1.026552062 -1.826873785 0.008730764
## [171] -0.130577449 0.927067205 0.478785020 -0.461254839 0.871374452
## [176] 0.335748888 -0.199832380 -0.299083226 0.342867394 1.050780174
## [181] -0.278798166 -0.640392783 -1.531184509 -2.055397864 -1.909900642
## [186] -1.718805064 -0.845646843 0.315218674 -0.603132174 0.308632365
## [191] -0.983460991 1.007151830 -0.226316303 1.237392362 0.442891040
## [196] -0.367796889 1.392095628 1.123528665 0.103327458 -0.327094746
## [201] -0.711903522 0.409316288 -0.781939741 0.349009661 0.084129552
## [206] -0.611515728 -0.387111187 -1.546648665 -1.787537640 -3.262418268
## [211] -3.155528327 -1.401930062 -1.791874188 -0.258621035 -1.424472809
## [216] 0.346290587 -0.181370063 -0.367955468 -1.745014479 -0.958092022
## [221] 0.971439292 -2.173221526 -3.060443290 -2.524769242 -0.788774373
## [226] -1.165104584 -1.122949080 -1.429616690 -0.298417460 0.326039650
## [231] -1.611363891 -2.518696727 -2.382820696 -1.538098512 -2.334846069
## [236] 0.296806162 1.017700729 0.090049208 0.087646837 1.782808260
## [241] 1.837231215 2.269697053 1.727929757 1.253396836 1.591421361
## [246] 0.952478797 1.401029664 0.693859130 0.969590725 0.987032489
## [251] 1.620391783 0.228545169 -0.344004941 1.518283334 1.599405129
## [256] 0.946482576 1.036633702 -0.474277688 1.080356147 0.626809119
## [261] -0.058907119 1.053093854 -0.557684189 -0.205314157 0.607485615
## [266] -0.797224785 0.960898876 0.853853070 1.120298920 0.522715101
## [271] 1.017540271 0.521461576 1.085557669 0.563940275 -0.448479721
## [276] 1.037330390 0.079182972 0.048893823 0.339451497 -0.838630310
## [281] -1.450089235 1.064805118 -0.561751044 -0.198775462 0.803126497
## [286] 0.468365882 1.125536522 0.621190437 -0.034871756 0.062120321
## [291] -0.826325430 -0.329245596 -1.560217595 -1.552083048 -1.933273448
## [296] -1.738533911 -2.078281305 -1.322885664 -1.495551451Korelasi antara \(Y_t\) dengan \(Y_{t-2}\)
#Yt
yt_ar2 <- ar[-c(1,2)]
yt_ar2## [1] 1.978906600 1.999656094 0.345358400 0.517656495 -0.113626221
## [6] -0.501334670 -1.017757562 -0.990428743 -2.534559286 -1.860891272
## [11] -0.570665319 0.115829627 -0.817975130 0.595474158 0.016675170
## [16] -1.490412009 -0.593871753 -2.097053986 -1.215867292 -0.064709114
## [21] 1.096328710 -0.909078093 -0.658636556 -0.658487767 1.383119545
## [26] 0.716659625 0.661562683 -0.652714475 -0.145189295 -1.019496788
## [31] -1.170487022 -0.597885677 0.262645117 -1.481662647 2.102105623
## [36] 0.564169109 -0.019049788 0.703071440 0.716921619 1.276879641
## [41] -1.887212794 0.296149588 -0.200388434 1.529532352 0.449376640
## [46] -0.871517629 -0.462053440 -0.951625041 -2.265260450 -1.064779599
## [51] -0.762130260 0.316451393 1.437718011 1.353357394 1.409759397
## [56] 2.179345291 2.454777567 2.483199682 3.932880024 1.270419783
## [61] 2.873938660 2.260102374 3.308447232 0.610071610 1.732980229
## [66] 1.948864327 0.540854532 -0.761208219 -1.184272508 0.550147295
## [71] -1.255629371 -0.912286813 -1.332571746 -0.697848527 -0.825743473
## [76] -3.073907488 -2.380464648 0.174392010 -0.714278511 1.364701988
## [81] 0.905814360 0.884341437 0.752049277 -0.767362203 0.712700971
## [86] 1.091436197 1.230396437 1.967189383 0.854004443 1.782483652
## [91] 0.973451042 1.628836800 -0.478821316 0.682092402 0.384251986
## [96] -1.533064501 -1.576533453 0.729794702 1.577205227 -0.381729281
## [101] -0.568170278 -2.200815976 -1.668515397 -2.683790651 -2.896091552
## [106] -3.039588312 -0.979622449 -0.244052773 -0.020347442 0.800457792
## [111] 0.987532552 -0.688549489 -0.845580279 0.111273825 0.308959934
## [116] 0.268882684 0.781600782 0.182495890 0.653596970 -0.654173676
## [121] -1.724863838 0.302340385 -0.350625232 0.483175495 0.369691586
## [126] 0.777443377 -0.300605258 -0.037395403 0.290503342 -0.329654937
## [131] -0.208188282 -1.673084433 -0.470864916 0.064608715 -0.576804975
## [136] 0.611693464 -0.029690039 -2.172851337 -1.383434201 -0.249100061
## [141] -0.668969280 -0.820664476 -1.171031583 2.766996691 0.501066622
## [146] 1.374792168 1.813793551 0.871030459 2.000591773 1.920770904
## [151] 3.048504199 0.524107503 2.092786750 2.180213079 1.370955871
## [156] 0.271427294 -1.167005699 -1.530897157 -1.725742158 -1.062031677
## [161] -1.275390836 -0.667534156 0.772941330 0.150067230 -1.099372863
## [166] -1.026552062 -1.826873785 0.008730764 -0.130577449 0.927067205
## [171] 0.478785020 -0.461254839 0.871374452 0.335748888 -0.199832380
## [176] -0.299083226 0.342867394 1.050780174 -0.278798166 -0.640392783
## [181] -1.531184509 -2.055397864 -1.909900642 -1.718805064 -0.845646843
## [186] 0.315218674 -0.603132174 0.308632365 -0.983460991 1.007151830
## [191] -0.226316303 1.237392362 0.442891040 -0.367796889 1.392095628
## [196] 1.123528665 0.103327458 -0.327094746 -0.711903522 0.409316288
## [201] -0.781939741 0.349009661 0.084129552 -0.611515728 -0.387111187
## [206] -1.546648665 -1.787537640 -3.262418268 -3.155528327 -1.401930062
## [211] -1.791874188 -0.258621035 -1.424472809 0.346290587 -0.181370063
## [216] -0.367955468 -1.745014479 -0.958092022 0.971439292 -2.173221526
## [221] -3.060443290 -2.524769242 -0.788774373 -1.165104584 -1.122949080
## [226] -1.429616690 -0.298417460 0.326039650 -1.611363891 -2.518696727
## [231] -2.382820696 -1.538098512 -2.334846069 0.296806162 1.017700729
## [236] 0.090049208 0.087646837 1.782808260 1.837231215 2.269697053
## [241] 1.727929757 1.253396836 1.591421361 0.952478797 1.401029664
## [246] 0.693859130 0.969590725 0.987032489 1.620391783 0.228545169
## [251] -0.344004941 1.518283334 1.599405129 0.946482576 1.036633702
## [256] -0.474277688 1.080356147 0.626809119 -0.058907119 1.053093854
## [261] -0.557684189 -0.205314157 0.607485615 -0.797224785 0.960898876
## [266] 0.853853070 1.120298920 0.522715101 1.017540271 0.521461576
## [271] 1.085557669 0.563940275 -0.448479721 1.037330390 0.079182972
## [276] 0.048893823 0.339451497 -0.838630310 -1.450089235 1.064805118
## [281] -0.561751044 -0.198775462 0.803126497 0.468365882 1.125536522
## [286] 0.621190437 -0.034871756 0.062120321 -0.826325430 -0.329245596
## [291] -1.560217595 -1.552083048 -1.933273448 -1.738533911 -2.078281305
## [296] -1.322885664 -1.495551451 0.079323615#Yt-2
yt_2_ar <- ar[-c(299,300)]
yt_2_ar## [1] 2.000000000 3.000000000 1.978906600 1.999656094 0.345358400
## [6] 0.517656495 -0.113626221 -0.501334670 -1.017757562 -0.990428743
## [11] -2.534559286 -1.860891272 -0.570665319 0.115829627 -0.817975130
## [16] 0.595474158 0.016675170 -1.490412009 -0.593871753 -2.097053986
## [21] -1.215867292 -0.064709114 1.096328710 -0.909078093 -0.658636556
## [26] -0.658487767 1.383119545 0.716659625 0.661562683 -0.652714475
## [31] -0.145189295 -1.019496788 -1.170487022 -0.597885677 0.262645117
## [36] -1.481662647 2.102105623 0.564169109 -0.019049788 0.703071440
## [41] 0.716921619 1.276879641 -1.887212794 0.296149588 -0.200388434
## [46] 1.529532352 0.449376640 -0.871517629 -0.462053440 -0.951625041
## [51] -2.265260450 -1.064779599 -0.762130260 0.316451393 1.437718011
## [56] 1.353357394 1.409759397 2.179345291 2.454777567 2.483199682
## [61] 3.932880024 1.270419783 2.873938660 2.260102374 3.308447232
## [66] 0.610071610 1.732980229 1.948864327 0.540854532 -0.761208219
## [71] -1.184272508 0.550147295 -1.255629371 -0.912286813 -1.332571746
## [76] -0.697848527 -0.825743473 -3.073907488 -2.380464648 0.174392010
## [81] -0.714278511 1.364701988 0.905814360 0.884341437 0.752049277
## [86] -0.767362203 0.712700971 1.091436197 1.230396437 1.967189383
## [91] 0.854004443 1.782483652 0.973451042 1.628836800 -0.478821316
## [96] 0.682092402 0.384251986 -1.533064501 -1.576533453 0.729794702
## [101] 1.577205227 -0.381729281 -0.568170278 -2.200815976 -1.668515397
## [106] -2.683790651 -2.896091552 -3.039588312 -0.979622449 -0.244052773
## [111] -0.020347442 0.800457792 0.987532552 -0.688549489 -0.845580279
## [116] 0.111273825 0.308959934 0.268882684 0.781600782 0.182495890
## [121] 0.653596970 -0.654173676 -1.724863838 0.302340385 -0.350625232
## [126] 0.483175495 0.369691586 0.777443377 -0.300605258 -0.037395403
## [131] 0.290503342 -0.329654937 -0.208188282 -1.673084433 -0.470864916
## [136] 0.064608715 -0.576804975 0.611693464 -0.029690039 -2.172851337
## [141] -1.383434201 -0.249100061 -0.668969280 -0.820664476 -1.171031583
## [146] 2.766996691 0.501066622 1.374792168 1.813793551 0.871030459
## [151] 2.000591773 1.920770904 3.048504199 0.524107503 2.092786750
## [156] 2.180213079 1.370955871 0.271427294 -1.167005699 -1.530897157
## [161] -1.725742158 -1.062031677 -1.275390836 -0.667534156 0.772941330
## [166] 0.150067230 -1.099372863 -1.026552062 -1.826873785 0.008730764
## [171] -0.130577449 0.927067205 0.478785020 -0.461254839 0.871374452
## [176] 0.335748888 -0.199832380 -0.299083226 0.342867394 1.050780174
## [181] -0.278798166 -0.640392783 -1.531184509 -2.055397864 -1.909900642
## [186] -1.718805064 -0.845646843 0.315218674 -0.603132174 0.308632365
## [191] -0.983460991 1.007151830 -0.226316303 1.237392362 0.442891040
## [196] -0.367796889 1.392095628 1.123528665 0.103327458 -0.327094746
## [201] -0.711903522 0.409316288 -0.781939741 0.349009661 0.084129552
## [206] -0.611515728 -0.387111187 -1.546648665 -1.787537640 -3.262418268
## [211] -3.155528327 -1.401930062 -1.791874188 -0.258621035 -1.424472809
## [216] 0.346290587 -0.181370063 -0.367955468 -1.745014479 -0.958092022
## [221] 0.971439292 -2.173221526 -3.060443290 -2.524769242 -0.788774373
## [226] -1.165104584 -1.122949080 -1.429616690 -0.298417460 0.326039650
## [231] -1.611363891 -2.518696727 -2.382820696 -1.538098512 -2.334846069
## [236] 0.296806162 1.017700729 0.090049208 0.087646837 1.782808260
## [241] 1.837231215 2.269697053 1.727929757 1.253396836 1.591421361
## [246] 0.952478797 1.401029664 0.693859130 0.969590725 0.987032489
## [251] 1.620391783 0.228545169 -0.344004941 1.518283334 1.599405129
## [256] 0.946482576 1.036633702 -0.474277688 1.080356147 0.626809119
## [261] -0.058907119 1.053093854 -0.557684189 -0.205314157 0.607485615
## [266] -0.797224785 0.960898876 0.853853070 1.120298920 0.522715101
## [271] 1.017540271 0.521461576 1.085557669 0.563940275 -0.448479721
## [276] 1.037330390 0.079182972 0.048893823 0.339451497 -0.838630310
## [281] -1.450089235 1.064805118 -0.561751044 -0.198775462 0.803126497
## [286] 0.468365882 1.125536522 0.621190437 -0.034871756 0.062120321
## [291] -0.826325430 -0.329245596 -1.560217595 -1.552083048 -1.933273448
## [296] -1.738533911 -2.078281305 -1.322885664Korelasi antara \(Y_t\) dengan \(Y_{t-3}\)
#Yt
yt_ar3 <- ar[-c(1,2,3)]
yt_ar3## [1] 1.999656094 0.345358400 0.517656495 -0.113626221 -0.501334670
## [6] -1.017757562 -0.990428743 -2.534559286 -1.860891272 -0.570665319
## [11] 0.115829627 -0.817975130 0.595474158 0.016675170 -1.490412009
## [16] -0.593871753 -2.097053986 -1.215867292 -0.064709114 1.096328710
## [21] -0.909078093 -0.658636556 -0.658487767 1.383119545 0.716659625
## [26] 0.661562683 -0.652714475 -0.145189295 -1.019496788 -1.170487022
## [31] -0.597885677 0.262645117 -1.481662647 2.102105623 0.564169109
## [36] -0.019049788 0.703071440 0.716921619 1.276879641 -1.887212794
## [41] 0.296149588 -0.200388434 1.529532352 0.449376640 -0.871517629
## [46] -0.462053440 -0.951625041 -2.265260450 -1.064779599 -0.762130260
## [51] 0.316451393 1.437718011 1.353357394 1.409759397 2.179345291
## [56] 2.454777567 2.483199682 3.932880024 1.270419783 2.873938660
## [61] 2.260102374 3.308447232 0.610071610 1.732980229 1.948864327
## [66] 0.540854532 -0.761208219 -1.184272508 0.550147295 -1.255629371
## [71] -0.912286813 -1.332571746 -0.697848527 -0.825743473 -3.073907488
## [76] -2.380464648 0.174392010 -0.714278511 1.364701988 0.905814360
## [81] 0.884341437 0.752049277 -0.767362203 0.712700971 1.091436197
## [86] 1.230396437 1.967189383 0.854004443 1.782483652 0.973451042
## [91] 1.628836800 -0.478821316 0.682092402 0.384251986 -1.533064501
## [96] -1.576533453 0.729794702 1.577205227 -0.381729281 -0.568170278
## [101] -2.200815976 -1.668515397 -2.683790651 -2.896091552 -3.039588312
## [106] -0.979622449 -0.244052773 -0.020347442 0.800457792 0.987532552
## [111] -0.688549489 -0.845580279 0.111273825 0.308959934 0.268882684
## [116] 0.781600782 0.182495890 0.653596970 -0.654173676 -1.724863838
## [121] 0.302340385 -0.350625232 0.483175495 0.369691586 0.777443377
## [126] -0.300605258 -0.037395403 0.290503342 -0.329654937 -0.208188282
## [131] -1.673084433 -0.470864916 0.064608715 -0.576804975 0.611693464
## [136] -0.029690039 -2.172851337 -1.383434201 -0.249100061 -0.668969280
## [141] -0.820664476 -1.171031583 2.766996691 0.501066622 1.374792168
## [146] 1.813793551 0.871030459 2.000591773 1.920770904 3.048504199
## [151] 0.524107503 2.092786750 2.180213079 1.370955871 0.271427294
## [156] -1.167005699 -1.530897157 -1.725742158 -1.062031677 -1.275390836
## [161] -0.667534156 0.772941330 0.150067230 -1.099372863 -1.026552062
## [166] -1.826873785 0.008730764 -0.130577449 0.927067205 0.478785020
## [171] -0.461254839 0.871374452 0.335748888 -0.199832380 -0.299083226
## [176] 0.342867394 1.050780174 -0.278798166 -0.640392783 -1.531184509
## [181] -2.055397864 -1.909900642 -1.718805064 -0.845646843 0.315218674
## [186] -0.603132174 0.308632365 -0.983460991 1.007151830 -0.226316303
## [191] 1.237392362 0.442891040 -0.367796889 1.392095628 1.123528665
## [196] 0.103327458 -0.327094746 -0.711903522 0.409316288 -0.781939741
## [201] 0.349009661 0.084129552 -0.611515728 -0.387111187 -1.546648665
## [206] -1.787537640 -3.262418268 -3.155528327 -1.401930062 -1.791874188
## [211] -0.258621035 -1.424472809 0.346290587 -0.181370063 -0.367955468
## [216] -1.745014479 -0.958092022 0.971439292 -2.173221526 -3.060443290
## [221] -2.524769242 -0.788774373 -1.165104584 -1.122949080 -1.429616690
## [226] -0.298417460 0.326039650 -1.611363891 -2.518696727 -2.382820696
## [231] -1.538098512 -2.334846069 0.296806162 1.017700729 0.090049208
## [236] 0.087646837 1.782808260 1.837231215 2.269697053 1.727929757
## [241] 1.253396836 1.591421361 0.952478797 1.401029664 0.693859130
## [246] 0.969590725 0.987032489 1.620391783 0.228545169 -0.344004941
## [251] 1.518283334 1.599405129 0.946482576 1.036633702 -0.474277688
## [256] 1.080356147 0.626809119 -0.058907119 1.053093854 -0.557684189
## [261] -0.205314157 0.607485615 -0.797224785 0.960898876 0.853853070
## [266] 1.120298920 0.522715101 1.017540271 0.521461576 1.085557669
## [271] 0.563940275 -0.448479721 1.037330390 0.079182972 0.048893823
## [276] 0.339451497 -0.838630310 -1.450089235 1.064805118 -0.561751044
## [281] -0.198775462 0.803126497 0.468365882 1.125536522 0.621190437
## [286] -0.034871756 0.062120321 -0.826325430 -0.329245596 -1.560217595
## [291] -1.552083048 -1.933273448 -1.738533911 -2.078281305 -1.322885664
## [296] -1.495551451 0.079323615#Yt-3
yt_3_ar <- ar[-c(298,299,300)]
yt_3_ar## [1] 2.000000000 3.000000000 1.978906600 1.999656094 0.345358400
## [6] 0.517656495 -0.113626221 -0.501334670 -1.017757562 -0.990428743
## [11] -2.534559286 -1.860891272 -0.570665319 0.115829627 -0.817975130
## [16] 0.595474158 0.016675170 -1.490412009 -0.593871753 -2.097053986
## [21] -1.215867292 -0.064709114 1.096328710 -0.909078093 -0.658636556
## [26] -0.658487767 1.383119545 0.716659625 0.661562683 -0.652714475
## [31] -0.145189295 -1.019496788 -1.170487022 -0.597885677 0.262645117
## [36] -1.481662647 2.102105623 0.564169109 -0.019049788 0.703071440
## [41] 0.716921619 1.276879641 -1.887212794 0.296149588 -0.200388434
## [46] 1.529532352 0.449376640 -0.871517629 -0.462053440 -0.951625041
## [51] -2.265260450 -1.064779599 -0.762130260 0.316451393 1.437718011
## [56] 1.353357394 1.409759397 2.179345291 2.454777567 2.483199682
## [61] 3.932880024 1.270419783 2.873938660 2.260102374 3.308447232
## [66] 0.610071610 1.732980229 1.948864327 0.540854532 -0.761208219
## [71] -1.184272508 0.550147295 -1.255629371 -0.912286813 -1.332571746
## [76] -0.697848527 -0.825743473 -3.073907488 -2.380464648 0.174392010
## [81] -0.714278511 1.364701988 0.905814360 0.884341437 0.752049277
## [86] -0.767362203 0.712700971 1.091436197 1.230396437 1.967189383
## [91] 0.854004443 1.782483652 0.973451042 1.628836800 -0.478821316
## [96] 0.682092402 0.384251986 -1.533064501 -1.576533453 0.729794702
## [101] 1.577205227 -0.381729281 -0.568170278 -2.200815976 -1.668515397
## [106] -2.683790651 -2.896091552 -3.039588312 -0.979622449 -0.244052773
## [111] -0.020347442 0.800457792 0.987532552 -0.688549489 -0.845580279
## [116] 0.111273825 0.308959934 0.268882684 0.781600782 0.182495890
## [121] 0.653596970 -0.654173676 -1.724863838 0.302340385 -0.350625232
## [126] 0.483175495 0.369691586 0.777443377 -0.300605258 -0.037395403
## [131] 0.290503342 -0.329654937 -0.208188282 -1.673084433 -0.470864916
## [136] 0.064608715 -0.576804975 0.611693464 -0.029690039 -2.172851337
## [141] -1.383434201 -0.249100061 -0.668969280 -0.820664476 -1.171031583
## [146] 2.766996691 0.501066622 1.374792168 1.813793551 0.871030459
## [151] 2.000591773 1.920770904 3.048504199 0.524107503 2.092786750
## [156] 2.180213079 1.370955871 0.271427294 -1.167005699 -1.530897157
## [161] -1.725742158 -1.062031677 -1.275390836 -0.667534156 0.772941330
## [166] 0.150067230 -1.099372863 -1.026552062 -1.826873785 0.008730764
## [171] -0.130577449 0.927067205 0.478785020 -0.461254839 0.871374452
## [176] 0.335748888 -0.199832380 -0.299083226 0.342867394 1.050780174
## [181] -0.278798166 -0.640392783 -1.531184509 -2.055397864 -1.909900642
## [186] -1.718805064 -0.845646843 0.315218674 -0.603132174 0.308632365
## [191] -0.983460991 1.007151830 -0.226316303 1.237392362 0.442891040
## [196] -0.367796889 1.392095628 1.123528665 0.103327458 -0.327094746
## [201] -0.711903522 0.409316288 -0.781939741 0.349009661 0.084129552
## [206] -0.611515728 -0.387111187 -1.546648665 -1.787537640 -3.262418268
## [211] -3.155528327 -1.401930062 -1.791874188 -0.258621035 -1.424472809
## [216] 0.346290587 -0.181370063 -0.367955468 -1.745014479 -0.958092022
## [221] 0.971439292 -2.173221526 -3.060443290 -2.524769242 -0.788774373
## [226] -1.165104584 -1.122949080 -1.429616690 -0.298417460 0.326039650
## [231] -1.611363891 -2.518696727 -2.382820696 -1.538098512 -2.334846069
## [236] 0.296806162 1.017700729 0.090049208 0.087646837 1.782808260
## [241] 1.837231215 2.269697053 1.727929757 1.253396836 1.591421361
## [246] 0.952478797 1.401029664 0.693859130 0.969590725 0.987032489
## [251] 1.620391783 0.228545169 -0.344004941 1.518283334 1.599405129
## [256] 0.946482576 1.036633702 -0.474277688 1.080356147 0.626809119
## [261] -0.058907119 1.053093854 -0.557684189 -0.205314157 0.607485615
## [266] -0.797224785 0.960898876 0.853853070 1.120298920 0.522715101
## [271] 1.017540271 0.521461576 1.085557669 0.563940275 -0.448479721
## [276] 1.037330390 0.079182972 0.048893823 0.339451497 -0.838630310
## [281] -1.450089235 1.064805118 -0.561751044 -0.198775462 0.803126497
## [286] 0.468365882 1.125536522 0.621190437 -0.034871756 0.062120321
## [291] -0.826325430 -0.329245596 -1.560217595 -1.552083048 -1.933273448
## [296] -1.738533911 -2.078281305Plot
plot(y=yt_ar,x=yt_1_ar)
plot(y=yt_ar2,x=yt_2_ar)
plot(y=yt_ar3,x=yt_3_ar)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif yang kuat antara \(Y_t\) dengan \(Y_{t-1}\).Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif yang sedang antara \(Y_t\) dengan \(Y_{t-2}\). Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif yang lemah antara \(Y_t\) dengan \(Y_{t-3}\).Hal ini sesuai dengan teori yang ada.
Hitung autokorelasi masing-masing baik dari data bangkitan maupun secara teoritispp
cor(yt_ar,yt_1_ar)## [1] 0.6222727cor(yt_ar2,yt_2_ar)## [1] 0.5130163cor(yt_ar3,yt_3_ar)## [1] 0.3897613Korelasi antara \(Y_t\) dengan \(Y_{t-1}\), \(Y_t\) dengan \(Y_{t-2}\), dan \(Y_t\) dengan \(Y_{t-3}\) dari hasil simulasi mendekati perhitungan teoritis. Korelasi antara \(Y_t\) dengan \(Y_{t-1}\) sebesar 0.5257299. Korelasi antara \(Y_t\) dengan \(Y_{t-2}\) sebesar 0.4170124 . Korelasi antara \(Y_t\) dengan \(Y_{t-3}\) sebesar 0.2605118 .
Fungsi pembangkitan ARMA
Setelah mengetahui cara membangkitkan data berpola AR, MA, dan ARMA sederhana, bagaimana cara melakukan pembangkitan data berpola tersebut yang lebih kompleks? Apakah dapat dibuat suatu fungsi yang fleksibel yang memungkinan pembangkitan dengan berapapun jumlah koefisien?
Pertama, lihat kembali bentuk umum data berpola ARMA.
\[ y_{t} = c + \sum_{i=1}^p \phi_{i}y_{t-i} + \sum_{j=1}^q e_{t-j}+ e_{t} \]
Komponen \(c\) dan \(e_{t}\) cukup mudah untuk dibuat dan dicari. Bagaimana untuk komponen AR dan MA? Bayangkan ada koefisien dan data sebagai berikut:
\[ \begin{aligned} \begin{bmatrix} \phi_1 \ \phi_2 \ \end{bmatrix}&= \begin{bmatrix} 0.5 \ 0.2 \end{bmatrix} \\ \begin{bmatrix} y_{t-1} \ y_{t-2} \end{bmatrix}&= \begin{bmatrix} 1 \ 2 \end{bmatrix} \end{aligned} \]
Maka dari itu,
\[ \begin{aligned} \begin{bmatrix} \phi_1 \ \phi_2 \ \end{bmatrix} \begin{bmatrix} y_{t-1} \\ y_{t-2} \end{bmatrix} &= \phi_1 \ y_{t-1}+\phi_2 \ y_{t-2} \begin{bmatrix} 0.5 \ 0.2 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \end{bmatrix} & = 0.5 \cdot1+0.2 \cdot 2= 0.9 \end{aligned} \]
Jika koefisien dan white noise/nilai deret waktu sebelumnya dapat diekstrak dalam bentuk vektor, dapat dilakukan perkalian matriks untuk mencari nilai bagian AR dan MA:
Bangkitkan model ARIMA(2,2) dengan theta1= 0.4 dan theta2 = 0.6, dots1=0.5 dan dots2=0.2 sebanyak 300 data(Manual dan dengan arima.sim)
set.seed(1029)
coefs <- c(0.5, 0.2)
e <- c(1, 2)
coefs %*% e## [,1]
## [1,] 0.9Atau, dapat dilakukan perkalian elementwise yang dijumlahkan:
coefs * e## [1] 0.5 0.4sum(coefs * e)## [1] 0.9Dari prinsip ini, dapat dibuat fungsi umum untuk membangkitkan data ARMA. Input dari fungsi adalah jumlah data yang hendak dibangkitkan, koefisien MA, dan koefisien AR
arma.sim <- function(n, macoef, arcoef){
manum <- length(macoef)
arnum <- length(arcoef)
stopifnot(manum < n & arnum < n)
data <- rnorm(n, sd = 0.5)
init <- max(manum, arnum)
arma <- data[1:init]
for(i in {init+1}:n){
mastart <- i - manum
maend <- i-1
arstart <- i - arnum
arend <- i-1
arma[i] <- sum(arcoef * arma[arstart:arend]) + sum(macoef * data[mastart:maend]) + data[i]
}
return(arma)
}Terlihat bahwa komponen \(\sum_{i=1}^q
y_{t-1}\) disimulasikan melalui
sum(arcoef * arma[arstart:arend]). Jadi, koefisien
dikalikan dengan data \(y\) dari \(t-q\) di mana q adalah jumlah koefisien AR,
sampai data \(t-1\). Lalu komponen
\(\sum_{j=1}^q e_{t-j}\) disimulasikan
melalui sum(macoef * data[mastart:maend]). Koefisien
dikalikan dengan white noise \(e\) dari \(t-p\), p jumlah koefisien MA, sampai \(t-1\).
# beberapa contoh pembangkitan melalui fungsi
ma3 <- arma.sim(300, c(0.4, 0.6), 0)
ar2 <- arma.sim(300, 0, c(0.5, 0.2))
par(mfrow = c(2, 2))
acf(ma3)
pacf(ma3)
acf(ar2)
pacf(ar2)
#contoh untuk ARMA
arma22 <- arma.sim(300, c(0.5, 0.2), c(0.4,0.6))
arma22 |> arima(c(2,0,2))## Warning in arima(arma22, c(2, 0, 2)): possible convergence problem: optim gave
## code = 1##
## Call:
## arima(x = arma22, order = c(2, 0, 2))
##
## Coefficients:
## ar1 ar2 ma1 ma2 intercept
## 0.5878 0.4076 0.2362 0.5171 11.6274
## s.e. 0.0947 0.0947 0.0901 0.0469 6.0713
##
## sigma^2 estimated as 0.2283: log likelihood = -207.26, aic = 426.53set.seed(1029)
n = length(data)
phi1 = 0.5
theta1 = 0.4
phi2 = 0.2
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*data[i-1]+data[i]+ theta2*data[i-2]+data[i]}Pembangkitan 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] 0.74304034 2.40422722 -0.24246686 -0.09833948 -1.17223489 -2.03060488
## [7] -0.80484470 0.35416729 0.81727989 1.71535244 1.65327968 2.26628412
## [13] 2.87143088 0.84349311 0.92513572 -0.66330571 -1.56027505 -0.61434833
## [19] -2.20571610 -2.77875558 -3.60298132 -2.23099833 -2.36322256 -1.69560501
## [25] -1.56161999 -0.09860517 -0.53709823 0.86256867 -0.42908366 -1.05595588
## [31] -1.36716541 -0.74599309 -0.14443797 0.79377285 -0.93219702 0.85870083
## [37] -0.06220909 0.81522712 0.81037637 -0.47950596 -0.33858754 -1.25146380
## [43] -0.65481150 -0.96012669 -1.01742774 0.07231702 -0.16512118 0.68675575
## [49] -0.47589465 -0.64568813 -1.03416598 -0.08293220 0.27575224 0.74624311
## [55] -0.53653035 -0.18566371 -0.42408042 0.67735218 0.53267634 0.33073256
## [61] -0.82044851 -0.09969010 -0.13956357 -0.16683189 -0.50155894 -0.21522035
## [67] -0.46148282 -0.31285443 1.06370729 1.44019539 1.56090694 1.75331494
## [73] 0.85484590 0.15005524 -0.41613044 0.41092792 -0.01938142 0.83582053
## [79] 1.06944093 1.68738409 1.66670135 0.62841476 1.60858305 1.65586404
## [85] 2.57582443 3.22003517 2.26048313 2.51739847 0.14052150 0.41076757
## [91] -0.96927417 0.54497556 1.22269581 1.58955811 0.04892493 -0.02424499
## [97] -2.64588936 0.20207130 -2.55438122 -0.29877151 0.85591107 1.30998401
## [103] 3.37646387 2.66571958 3.89848360 3.71001271 3.60282685 2.62387497
## [109] 2.79109412 0.02875172 -0.64350891 -1.78082465 -1.69279420 -0.27664385
## [115] 0.22165005 1.34753665 0.20001767 -0.58535893 -0.71513967 -0.73178149
## [121] 0.99854527 -0.85260627 -1.35886381 -3.04860204 -4.49641652 -4.40187940
## [127] -4.10370089 -3.11112405 -2.14768409 -1.59426426 -1.20849642 -0.10879668
## [133] -0.74901170 1.69268340 0.33133668 0.38547383 -1.05693757 -1.35843756
## [139] -1.21054863 1.25461738 1.48273448 1.71330601 1.63018687 1.77493342
## [145] 1.70942999 0.66054532 -0.75430029 -1.80677879 -0.85088012 0.78476273
## [151] 2.39056866 2.69855244 4.13857700 3.15047305 2.51680531 1.45852722
## [157] 0.69606629 0.11155740 2.37958088 1.53444029 2.76067785 1.05612616
## [163] 2.00666722 2.38800881 4.26126953 2.36960211 1.86904011 -0.18112163
## [169] 0.16894507 -0.69776933 -0.03848840 0.73061926 0.75478154 0.76140563
## [175] 0.99103053 1.06085226 -0.34010169 1.20392950 0.46934411 3.01390246
## [181] 2.52128444 3.49863087 2.84817152 2.78915669 1.30931766 0.07111692
## [187] -0.58557012 -2.53666065 -3.16600295 -1.84885233 -2.49551356 -0.20079676
## [193] -2.36090529 -0.11742652 0.12312543 1.59441458 2.44405386 1.94972236
## [199] 2.69433295 1.78772892 2.42445094 2.88703441 2.76053895 0.91827406
## [205] -0.61864614 -1.82557583 -1.92050155 -2.70382755 -2.89928452 -2.86086586
## [211] -1.12983328 -1.06965573 0.30770222 0.90804745 0.06297905 0.96007051
## [217] -0.52008579 0.57297906 -0.66999279 0.36606683 -1.73693338 -0.52489705
## [223] 0.72285156 1.17326781 0.39789562 0.17422983 -1.16323755 -0.64628978
## [229] -0.81736758 -0.93312299 0.14004131 0.48408137 -0.22359690 -2.23252650
## [235] -1.29144797 -1.57962715 -0.48172752 -1.01082069 -2.25632627 -2.46913856
## [241] -2.98924119 -1.77984329 -1.20889411 -1.54876129 -0.77282409 -2.14708992
## [247] -2.34034824 -2.48709837 -0.26504808 0.38850753 1.37379767 1.00436277
## [253] -0.10999420 -0.58675872 0.59096810 0.28283312 2.44531696 0.45041639
## [259] 1.67405044 0.96273437 2.39816276 3.58317820 3.94406275 3.71291673
## [265] 3.06401687 0.52572954 1.44380538 0.23000235 1.00038308 0.35047128
## [271] -0.10245555 -0.98003885 -0.08253108 1.01335276 0.30875985 0.88146650
## [277] -0.12737316 0.06113570 -2.29170349 -0.59109870 -2.12162897 0.76365631
## [283] -2.24280398 -1.94974745 0.19331399 -0.74117823 0.97860485 -1.24940106
## [289] -0.18743963 1.29118894 2.85500231 4.26999696 4.65894375 3.43743936
## [295] 3.74030994 2.66625255 1.59717402 0.33319626 -1.06567575 -0.89432153Buatlah Plot time series, Plot ACF, plot PACF, dan Plot EACF. Identifikasi apakah data tersebut stasioner
Plot Time Series
par(mfrow = c(1, 2))
ts.plot(y.arma)
ts.plot(arma22)
par(mfrow = c(2, 2))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(2, 2))Berdasarkan plot ACF tersebut, terlihat bahwa model ARMA(2,2) hasil simulasi memiliki plot ACF yang tails off, sesuai dengan teori yang ada
Plot PACF
par(mfrow = c(2, 2))
pacf(y.arma)
pacf(arma22)
par(mfrow = c(2, 2))
Berdasarkan plot PACF tersebut, terlihat bahwa model ARMA(2,2) hasil simulasi memiliki plot PACF yang cut 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 x x x
## 1 x x o o o o o o o o o o o o
## 2 x x x o o o x o o o o o o o
## 3 x x o o o o o o o o o o o o
## 4 x o o o o o o o x o o o o o
## 5 x x o o o o o o x o o o o o
## 6 x x o o x o o o x o o o o o
## 7 x x o x x x o o x 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 o o o o o o o o o
## 1 x x o o o o o o o o o o o o
## 2 x x x o o o o o o o o o o o
## 3 o x x o o o o o o o o o o o
## 4 x x x o x o o o o o o o o o
## 5 x x x x x o o o o o o o o o
## 6 x x x o x x x o o o o o o o
## 7 x x x x x x o o o o o o o oBerdasarkan pola segitiga nol pada plot EACF, terlihat bahwa segitiga nol berada pada ordo AR(2) dan ordo MA(2)
Buatlah scatterplot antara \(Y_t\) dengan \(Y_{t-1}\), \(Y_t\) dengan \(Y_{t-2}\), dan \(Y_t\) dengan \(Y_{t-3}\)
Korelasi antara \(Y_t\) dengan \(Y_{t-1}\)
#Yt
yt_arma <- arma22[-1]
yt_arma## [1] 2.40422722 -0.24246686 -0.09833948 -1.17223489 -2.03060488 -0.80484470
## [7] 0.35416729 0.81727989 1.71535244 1.65327968 2.26628412 2.87143088
## [13] 0.84349311 0.92513572 -0.66330571 -1.56027505 -0.61434833 -2.20571610
## [19] -2.77875558 -3.60298132 -2.23099833 -2.36322256 -1.69560501 -1.56161999
## [25] -0.09860517 -0.53709823 0.86256867 -0.42908366 -1.05595588 -1.36716541
## [31] -0.74599309 -0.14443797 0.79377285 -0.93219702 0.85870083 -0.06220909
## [37] 0.81522712 0.81037637 -0.47950596 -0.33858754 -1.25146380 -0.65481150
## [43] -0.96012669 -1.01742774 0.07231702 -0.16512118 0.68675575 -0.47589465
## [49] -0.64568813 -1.03416598 -0.08293220 0.27575224 0.74624311 -0.53653035
## [55] -0.18566371 -0.42408042 0.67735218 0.53267634 0.33073256 -0.82044851
## [61] -0.09969010 -0.13956357 -0.16683189 -0.50155894 -0.21522035 -0.46148282
## [67] -0.31285443 1.06370729 1.44019539 1.56090694 1.75331494 0.85484590
## [73] 0.15005524 -0.41613044 0.41092792 -0.01938142 0.83582053 1.06944093
## [79] 1.68738409 1.66670135 0.62841476 1.60858305 1.65586404 2.57582443
## [85] 3.22003517 2.26048313 2.51739847 0.14052150 0.41076757 -0.96927417
## [91] 0.54497556 1.22269581 1.58955811 0.04892493 -0.02424499 -2.64588936
## [97] 0.20207130 -2.55438122 -0.29877151 0.85591107 1.30998401 3.37646387
## [103] 2.66571958 3.89848360 3.71001271 3.60282685 2.62387497 2.79109412
## [109] 0.02875172 -0.64350891 -1.78082465 -1.69279420 -0.27664385 0.22165005
## [115] 1.34753665 0.20001767 -0.58535893 -0.71513967 -0.73178149 0.99854527
## [121] -0.85260627 -1.35886381 -3.04860204 -4.49641652 -4.40187940 -4.10370089
## [127] -3.11112405 -2.14768409 -1.59426426 -1.20849642 -0.10879668 -0.74901170
## [133] 1.69268340 0.33133668 0.38547383 -1.05693757 -1.35843756 -1.21054863
## [139] 1.25461738 1.48273448 1.71330601 1.63018687 1.77493342 1.70942999
## [145] 0.66054532 -0.75430029 -1.80677879 -0.85088012 0.78476273 2.39056866
## [151] 2.69855244 4.13857700 3.15047305 2.51680531 1.45852722 0.69606629
## [157] 0.11155740 2.37958088 1.53444029 2.76067785 1.05612616 2.00666722
## [163] 2.38800881 4.26126953 2.36960211 1.86904011 -0.18112163 0.16894507
## [169] -0.69776933 -0.03848840 0.73061926 0.75478154 0.76140563 0.99103053
## [175] 1.06085226 -0.34010169 1.20392950 0.46934411 3.01390246 2.52128444
## [181] 3.49863087 2.84817152 2.78915669 1.30931766 0.07111692 -0.58557012
## [187] -2.53666065 -3.16600295 -1.84885233 -2.49551356 -0.20079676 -2.36090529
## [193] -0.11742652 0.12312543 1.59441458 2.44405386 1.94972236 2.69433295
## [199] 1.78772892 2.42445094 2.88703441 2.76053895 0.91827406 -0.61864614
## [205] -1.82557583 -1.92050155 -2.70382755 -2.89928452 -2.86086586 -1.12983328
## [211] -1.06965573 0.30770222 0.90804745 0.06297905 0.96007051 -0.52008579
## [217] 0.57297906 -0.66999279 0.36606683 -1.73693338 -0.52489705 0.72285156
## [223] 1.17326781 0.39789562 0.17422983 -1.16323755 -0.64628978 -0.81736758
## [229] -0.93312299 0.14004131 0.48408137 -0.22359690 -2.23252650 -1.29144797
## [235] -1.57962715 -0.48172752 -1.01082069 -2.25632627 -2.46913856 -2.98924119
## [241] -1.77984329 -1.20889411 -1.54876129 -0.77282409 -2.14708992 -2.34034824
## [247] -2.48709837 -0.26504808 0.38850753 1.37379767 1.00436277 -0.10999420
## [253] -0.58675872 0.59096810 0.28283312 2.44531696 0.45041639 1.67405044
## [259] 0.96273437 2.39816276 3.58317820 3.94406275 3.71291673 3.06401687
## [265] 0.52572954 1.44380538 0.23000235 1.00038308 0.35047128 -0.10245555
## [271] -0.98003885 -0.08253108 1.01335276 0.30875985 0.88146650 -0.12737316
## [277] 0.06113570 -2.29170349 -0.59109870 -2.12162897 0.76365631 -2.24280398
## [283] -1.94974745 0.19331399 -0.74117823 0.97860485 -1.24940106 -0.18743963
## [289] 1.29118894 2.85500231 4.26999696 4.65894375 3.43743936 3.74030994
## [295] 2.66625255 1.59717402 0.33319626 -1.06567575 -0.89432153#Yt-1
yt_1_arma <- arma22[-200]
yt_1_arma## [1] 0.74304034 2.40422722 -0.24246686 -0.09833948 -1.17223489 -2.03060488
## [7] -0.80484470 0.35416729 0.81727989 1.71535244 1.65327968 2.26628412
## [13] 2.87143088 0.84349311 0.92513572 -0.66330571 -1.56027505 -0.61434833
## [19] -2.20571610 -2.77875558 -3.60298132 -2.23099833 -2.36322256 -1.69560501
## [25] -1.56161999 -0.09860517 -0.53709823 0.86256867 -0.42908366 -1.05595588
## [31] -1.36716541 -0.74599309 -0.14443797 0.79377285 -0.93219702 0.85870083
## [37] -0.06220909 0.81522712 0.81037637 -0.47950596 -0.33858754 -1.25146380
## [43] -0.65481150 -0.96012669 -1.01742774 0.07231702 -0.16512118 0.68675575
## [49] -0.47589465 -0.64568813 -1.03416598 -0.08293220 0.27575224 0.74624311
## [55] -0.53653035 -0.18566371 -0.42408042 0.67735218 0.53267634 0.33073256
## [61] -0.82044851 -0.09969010 -0.13956357 -0.16683189 -0.50155894 -0.21522035
## [67] -0.46148282 -0.31285443 1.06370729 1.44019539 1.56090694 1.75331494
## [73] 0.85484590 0.15005524 -0.41613044 0.41092792 -0.01938142 0.83582053
## [79] 1.06944093 1.68738409 1.66670135 0.62841476 1.60858305 1.65586404
## [85] 2.57582443 3.22003517 2.26048313 2.51739847 0.14052150 0.41076757
## [91] -0.96927417 0.54497556 1.22269581 1.58955811 0.04892493 -0.02424499
## [97] -2.64588936 0.20207130 -2.55438122 -0.29877151 0.85591107 1.30998401
## [103] 3.37646387 2.66571958 3.89848360 3.71001271 3.60282685 2.62387497
## [109] 2.79109412 0.02875172 -0.64350891 -1.78082465 -1.69279420 -0.27664385
## [115] 0.22165005 1.34753665 0.20001767 -0.58535893 -0.71513967 -0.73178149
## [121] 0.99854527 -0.85260627 -1.35886381 -3.04860204 -4.49641652 -4.40187940
## [127] -4.10370089 -3.11112405 -2.14768409 -1.59426426 -1.20849642 -0.10879668
## [133] -0.74901170 1.69268340 0.33133668 0.38547383 -1.05693757 -1.35843756
## [139] -1.21054863 1.25461738 1.48273448 1.71330601 1.63018687 1.77493342
## [145] 1.70942999 0.66054532 -0.75430029 -1.80677879 -0.85088012 0.78476273
## [151] 2.39056866 2.69855244 4.13857700 3.15047305 2.51680531 1.45852722
## [157] 0.69606629 0.11155740 2.37958088 1.53444029 2.76067785 1.05612616
## [163] 2.00666722 2.38800881 4.26126953 2.36960211 1.86904011 -0.18112163
## [169] 0.16894507 -0.69776933 -0.03848840 0.73061926 0.75478154 0.76140563
## [175] 0.99103053 1.06085226 -0.34010169 1.20392950 0.46934411 3.01390246
## [181] 2.52128444 3.49863087 2.84817152 2.78915669 1.30931766 0.07111692
## [187] -0.58557012 -2.53666065 -3.16600295 -1.84885233 -2.49551356 -0.20079676
## [193] -2.36090529 -0.11742652 0.12312543 1.59441458 2.44405386 1.94972236
## [199] 2.69433295 2.42445094 2.88703441 2.76053895 0.91827406 -0.61864614
## [205] -1.82557583 -1.92050155 -2.70382755 -2.89928452 -2.86086586 -1.12983328
## [211] -1.06965573 0.30770222 0.90804745 0.06297905 0.96007051 -0.52008579
## [217] 0.57297906 -0.66999279 0.36606683 -1.73693338 -0.52489705 0.72285156
## [223] 1.17326781 0.39789562 0.17422983 -1.16323755 -0.64628978 -0.81736758
## [229] -0.93312299 0.14004131 0.48408137 -0.22359690 -2.23252650 -1.29144797
## [235] -1.57962715 -0.48172752 -1.01082069 -2.25632627 -2.46913856 -2.98924119
## [241] -1.77984329 -1.20889411 -1.54876129 -0.77282409 -2.14708992 -2.34034824
## [247] -2.48709837 -0.26504808 0.38850753 1.37379767 1.00436277 -0.10999420
## [253] -0.58675872 0.59096810 0.28283312 2.44531696 0.45041639 1.67405044
## [259] 0.96273437 2.39816276 3.58317820 3.94406275 3.71291673 3.06401687
## [265] 0.52572954 1.44380538 0.23000235 1.00038308 0.35047128 -0.10245555
## [271] -0.98003885 -0.08253108 1.01335276 0.30875985 0.88146650 -0.12737316
## [277] 0.06113570 -2.29170349 -0.59109870 -2.12162897 0.76365631 -2.24280398
## [283] -1.94974745 0.19331399 -0.74117823 0.97860485 -1.24940106 -0.18743963
## [289] 1.29118894 2.85500231 4.26999696 4.65894375 3.43743936 3.74030994
## [295] 2.66625255 1.59717402 0.33319626 -1.06567575 -0.89432153plot(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.
Korelasi antara \(Y_t\) dengan \(Y_{t-2}\)
#Yt
yt2_arma <- arma22[-c(1,2)]
yt2_arma## [1] -0.24246686 -0.09833948 -1.17223489 -2.03060488 -0.80484470 0.35416729
## [7] 0.81727989 1.71535244 1.65327968 2.26628412 2.87143088 0.84349311
## [13] 0.92513572 -0.66330571 -1.56027505 -0.61434833 -2.20571610 -2.77875558
## [19] -3.60298132 -2.23099833 -2.36322256 -1.69560501 -1.56161999 -0.09860517
## [25] -0.53709823 0.86256867 -0.42908366 -1.05595588 -1.36716541 -0.74599309
## [31] -0.14443797 0.79377285 -0.93219702 0.85870083 -0.06220909 0.81522712
## [37] 0.81037637 -0.47950596 -0.33858754 -1.25146380 -0.65481150 -0.96012669
## [43] -1.01742774 0.07231702 -0.16512118 0.68675575 -0.47589465 -0.64568813
## [49] -1.03416598 -0.08293220 0.27575224 0.74624311 -0.53653035 -0.18566371
## [55] -0.42408042 0.67735218 0.53267634 0.33073256 -0.82044851 -0.09969010
## [61] -0.13956357 -0.16683189 -0.50155894 -0.21522035 -0.46148282 -0.31285443
## [67] 1.06370729 1.44019539 1.56090694 1.75331494 0.85484590 0.15005524
## [73] -0.41613044 0.41092792 -0.01938142 0.83582053 1.06944093 1.68738409
## [79] 1.66670135 0.62841476 1.60858305 1.65586404 2.57582443 3.22003517
## [85] 2.26048313 2.51739847 0.14052150 0.41076757 -0.96927417 0.54497556
## [91] 1.22269581 1.58955811 0.04892493 -0.02424499 -2.64588936 0.20207130
## [97] -2.55438122 -0.29877151 0.85591107 1.30998401 3.37646387 2.66571958
## [103] 3.89848360 3.71001271 3.60282685 2.62387497 2.79109412 0.02875172
## [109] -0.64350891 -1.78082465 -1.69279420 -0.27664385 0.22165005 1.34753665
## [115] 0.20001767 -0.58535893 -0.71513967 -0.73178149 0.99854527 -0.85260627
## [121] -1.35886381 -3.04860204 -4.49641652 -4.40187940 -4.10370089 -3.11112405
## [127] -2.14768409 -1.59426426 -1.20849642 -0.10879668 -0.74901170 1.69268340
## [133] 0.33133668 0.38547383 -1.05693757 -1.35843756 -1.21054863 1.25461738
## [139] 1.48273448 1.71330601 1.63018687 1.77493342 1.70942999 0.66054532
## [145] -0.75430029 -1.80677879 -0.85088012 0.78476273 2.39056866 2.69855244
## [151] 4.13857700 3.15047305 2.51680531 1.45852722 0.69606629 0.11155740
## [157] 2.37958088 1.53444029 2.76067785 1.05612616 2.00666722 2.38800881
## [163] 4.26126953 2.36960211 1.86904011 -0.18112163 0.16894507 -0.69776933
## [169] -0.03848840 0.73061926 0.75478154 0.76140563 0.99103053 1.06085226
## [175] -0.34010169 1.20392950 0.46934411 3.01390246 2.52128444 3.49863087
## [181] 2.84817152 2.78915669 1.30931766 0.07111692 -0.58557012 -2.53666065
## [187] -3.16600295 -1.84885233 -2.49551356 -0.20079676 -2.36090529 -0.11742652
## [193] 0.12312543 1.59441458 2.44405386 1.94972236 2.69433295 1.78772892
## [199] 2.42445094 2.88703441 2.76053895 0.91827406 -0.61864614 -1.82557583
## [205] -1.92050155 -2.70382755 -2.89928452 -2.86086586 -1.12983328 -1.06965573
## [211] 0.30770222 0.90804745 0.06297905 0.96007051 -0.52008579 0.57297906
## [217] -0.66999279 0.36606683 -1.73693338 -0.52489705 0.72285156 1.17326781
## [223] 0.39789562 0.17422983 -1.16323755 -0.64628978 -0.81736758 -0.93312299
## [229] 0.14004131 0.48408137 -0.22359690 -2.23252650 -1.29144797 -1.57962715
## [235] -0.48172752 -1.01082069 -2.25632627 -2.46913856 -2.98924119 -1.77984329
## [241] -1.20889411 -1.54876129 -0.77282409 -2.14708992 -2.34034824 -2.48709837
## [247] -0.26504808 0.38850753 1.37379767 1.00436277 -0.10999420 -0.58675872
## [253] 0.59096810 0.28283312 2.44531696 0.45041639 1.67405044 0.96273437
## [259] 2.39816276 3.58317820 3.94406275 3.71291673 3.06401687 0.52572954
## [265] 1.44380538 0.23000235 1.00038308 0.35047128 -0.10245555 -0.98003885
## [271] -0.08253108 1.01335276 0.30875985 0.88146650 -0.12737316 0.06113570
## [277] -2.29170349 -0.59109870 -2.12162897 0.76365631 -2.24280398 -1.94974745
## [283] 0.19331399 -0.74117823 0.97860485 -1.24940106 -0.18743963 1.29118894
## [289] 2.85500231 4.26999696 4.65894375 3.43743936 3.74030994 2.66625255
## [295] 1.59717402 0.33319626 -1.06567575 -0.89432153#Yt-2
yt_2_arma <- arma22[-c(200,199)]
yt_2_arma## [1] 0.74304034 2.40422722 -0.24246686 -0.09833948 -1.17223489 -2.03060488
## [7] -0.80484470 0.35416729 0.81727989 1.71535244 1.65327968 2.26628412
## [13] 2.87143088 0.84349311 0.92513572 -0.66330571 -1.56027505 -0.61434833
## [19] -2.20571610 -2.77875558 -3.60298132 -2.23099833 -2.36322256 -1.69560501
## [25] -1.56161999 -0.09860517 -0.53709823 0.86256867 -0.42908366 -1.05595588
## [31] -1.36716541 -0.74599309 -0.14443797 0.79377285 -0.93219702 0.85870083
## [37] -0.06220909 0.81522712 0.81037637 -0.47950596 -0.33858754 -1.25146380
## [43] -0.65481150 -0.96012669 -1.01742774 0.07231702 -0.16512118 0.68675575
## [49] -0.47589465 -0.64568813 -1.03416598 -0.08293220 0.27575224 0.74624311
## [55] -0.53653035 -0.18566371 -0.42408042 0.67735218 0.53267634 0.33073256
## [61] -0.82044851 -0.09969010 -0.13956357 -0.16683189 -0.50155894 -0.21522035
## [67] -0.46148282 -0.31285443 1.06370729 1.44019539 1.56090694 1.75331494
## [73] 0.85484590 0.15005524 -0.41613044 0.41092792 -0.01938142 0.83582053
## [79] 1.06944093 1.68738409 1.66670135 0.62841476 1.60858305 1.65586404
## [85] 2.57582443 3.22003517 2.26048313 2.51739847 0.14052150 0.41076757
## [91] -0.96927417 0.54497556 1.22269581 1.58955811 0.04892493 -0.02424499
## [97] -2.64588936 0.20207130 -2.55438122 -0.29877151 0.85591107 1.30998401
## [103] 3.37646387 2.66571958 3.89848360 3.71001271 3.60282685 2.62387497
## [109] 2.79109412 0.02875172 -0.64350891 -1.78082465 -1.69279420 -0.27664385
## [115] 0.22165005 1.34753665 0.20001767 -0.58535893 -0.71513967 -0.73178149
## [121] 0.99854527 -0.85260627 -1.35886381 -3.04860204 -4.49641652 -4.40187940
## [127] -4.10370089 -3.11112405 -2.14768409 -1.59426426 -1.20849642 -0.10879668
## [133] -0.74901170 1.69268340 0.33133668 0.38547383 -1.05693757 -1.35843756
## [139] -1.21054863 1.25461738 1.48273448 1.71330601 1.63018687 1.77493342
## [145] 1.70942999 0.66054532 -0.75430029 -1.80677879 -0.85088012 0.78476273
## [151] 2.39056866 2.69855244 4.13857700 3.15047305 2.51680531 1.45852722
## [157] 0.69606629 0.11155740 2.37958088 1.53444029 2.76067785 1.05612616
## [163] 2.00666722 2.38800881 4.26126953 2.36960211 1.86904011 -0.18112163
## [169] 0.16894507 -0.69776933 -0.03848840 0.73061926 0.75478154 0.76140563
## [175] 0.99103053 1.06085226 -0.34010169 1.20392950 0.46934411 3.01390246
## [181] 2.52128444 3.49863087 2.84817152 2.78915669 1.30931766 0.07111692
## [187] -0.58557012 -2.53666065 -3.16600295 -1.84885233 -2.49551356 -0.20079676
## [193] -2.36090529 -0.11742652 0.12312543 1.59441458 2.44405386 1.94972236
## [199] 2.42445094 2.88703441 2.76053895 0.91827406 -0.61864614 -1.82557583
## [205] -1.92050155 -2.70382755 -2.89928452 -2.86086586 -1.12983328 -1.06965573
## [211] 0.30770222 0.90804745 0.06297905 0.96007051 -0.52008579 0.57297906
## [217] -0.66999279 0.36606683 -1.73693338 -0.52489705 0.72285156 1.17326781
## [223] 0.39789562 0.17422983 -1.16323755 -0.64628978 -0.81736758 -0.93312299
## [229] 0.14004131 0.48408137 -0.22359690 -2.23252650 -1.29144797 -1.57962715
## [235] -0.48172752 -1.01082069 -2.25632627 -2.46913856 -2.98924119 -1.77984329
## [241] -1.20889411 -1.54876129 -0.77282409 -2.14708992 -2.34034824 -2.48709837
## [247] -0.26504808 0.38850753 1.37379767 1.00436277 -0.10999420 -0.58675872
## [253] 0.59096810 0.28283312 2.44531696 0.45041639 1.67405044 0.96273437
## [259] 2.39816276 3.58317820 3.94406275 3.71291673 3.06401687 0.52572954
## [265] 1.44380538 0.23000235 1.00038308 0.35047128 -0.10245555 -0.98003885
## [271] -0.08253108 1.01335276 0.30875985 0.88146650 -0.12737316 0.06113570
## [277] -2.29170349 -0.59109870 -2.12162897 0.76365631 -2.24280398 -1.94974745
## [283] 0.19331399 -0.74117823 0.97860485 -1.24940106 -0.18743963 1.29118894
## [289] 2.85500231 4.26999696 4.65894375 3.43743936 3.74030994 2.66625255
## [295] 1.59717402 0.33319626 -1.06567575 -0.89432153plot(y=yt2_arma,x=yt_2_arma)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan
positif antara \(Y_t\) dengan \(Y_{t-2}\). Hal ini sesuai dengan teori yang
ada.
Korelasi antara \(Y_t\) dengan \(Y_{t-3}\)
#Yt
yt3_arma <- arma22[-c(1,2,3)]
yt3_arma## [1] -0.09833948 -1.17223489 -2.03060488 -0.80484470 0.35416729 0.81727989
## [7] 1.71535244 1.65327968 2.26628412 2.87143088 0.84349311 0.92513572
## [13] -0.66330571 -1.56027505 -0.61434833 -2.20571610 -2.77875558 -3.60298132
## [19] -2.23099833 -2.36322256 -1.69560501 -1.56161999 -0.09860517 -0.53709823
## [25] 0.86256867 -0.42908366 -1.05595588 -1.36716541 -0.74599309 -0.14443797
## [31] 0.79377285 -0.93219702 0.85870083 -0.06220909 0.81522712 0.81037637
## [37] -0.47950596 -0.33858754 -1.25146380 -0.65481150 -0.96012669 -1.01742774
## [43] 0.07231702 -0.16512118 0.68675575 -0.47589465 -0.64568813 -1.03416598
## [49] -0.08293220 0.27575224 0.74624311 -0.53653035 -0.18566371 -0.42408042
## [55] 0.67735218 0.53267634 0.33073256 -0.82044851 -0.09969010 -0.13956357
## [61] -0.16683189 -0.50155894 -0.21522035 -0.46148282 -0.31285443 1.06370729
## [67] 1.44019539 1.56090694 1.75331494 0.85484590 0.15005524 -0.41613044
## [73] 0.41092792 -0.01938142 0.83582053 1.06944093 1.68738409 1.66670135
## [79] 0.62841476 1.60858305 1.65586404 2.57582443 3.22003517 2.26048313
## [85] 2.51739847 0.14052150 0.41076757 -0.96927417 0.54497556 1.22269581
## [91] 1.58955811 0.04892493 -0.02424499 -2.64588936 0.20207130 -2.55438122
## [97] -0.29877151 0.85591107 1.30998401 3.37646387 2.66571958 3.89848360
## [103] 3.71001271 3.60282685 2.62387497 2.79109412 0.02875172 -0.64350891
## [109] -1.78082465 -1.69279420 -0.27664385 0.22165005 1.34753665 0.20001767
## [115] -0.58535893 -0.71513967 -0.73178149 0.99854527 -0.85260627 -1.35886381
## [121] -3.04860204 -4.49641652 -4.40187940 -4.10370089 -3.11112405 -2.14768409
## [127] -1.59426426 -1.20849642 -0.10879668 -0.74901170 1.69268340 0.33133668
## [133] 0.38547383 -1.05693757 -1.35843756 -1.21054863 1.25461738 1.48273448
## [139] 1.71330601 1.63018687 1.77493342 1.70942999 0.66054532 -0.75430029
## [145] -1.80677879 -0.85088012 0.78476273 2.39056866 2.69855244 4.13857700
## [151] 3.15047305 2.51680531 1.45852722 0.69606629 0.11155740 2.37958088
## [157] 1.53444029 2.76067785 1.05612616 2.00666722 2.38800881 4.26126953
## [163] 2.36960211 1.86904011 -0.18112163 0.16894507 -0.69776933 -0.03848840
## [169] 0.73061926 0.75478154 0.76140563 0.99103053 1.06085226 -0.34010169
## [175] 1.20392950 0.46934411 3.01390246 2.52128444 3.49863087 2.84817152
## [181] 2.78915669 1.30931766 0.07111692 -0.58557012 -2.53666065 -3.16600295
## [187] -1.84885233 -2.49551356 -0.20079676 -2.36090529 -0.11742652 0.12312543
## [193] 1.59441458 2.44405386 1.94972236 2.69433295 1.78772892 2.42445094
## [199] 2.88703441 2.76053895 0.91827406 -0.61864614 -1.82557583 -1.92050155
## [205] -2.70382755 -2.89928452 -2.86086586 -1.12983328 -1.06965573 0.30770222
## [211] 0.90804745 0.06297905 0.96007051 -0.52008579 0.57297906 -0.66999279
## [217] 0.36606683 -1.73693338 -0.52489705 0.72285156 1.17326781 0.39789562
## [223] 0.17422983 -1.16323755 -0.64628978 -0.81736758 -0.93312299 0.14004131
## [229] 0.48408137 -0.22359690 -2.23252650 -1.29144797 -1.57962715 -0.48172752
## [235] -1.01082069 -2.25632627 -2.46913856 -2.98924119 -1.77984329 -1.20889411
## [241] -1.54876129 -0.77282409 -2.14708992 -2.34034824 -2.48709837 -0.26504808
## [247] 0.38850753 1.37379767 1.00436277 -0.10999420 -0.58675872 0.59096810
## [253] 0.28283312 2.44531696 0.45041639 1.67405044 0.96273437 2.39816276
## [259] 3.58317820 3.94406275 3.71291673 3.06401687 0.52572954 1.44380538
## [265] 0.23000235 1.00038308 0.35047128 -0.10245555 -0.98003885 -0.08253108
## [271] 1.01335276 0.30875985 0.88146650 -0.12737316 0.06113570 -2.29170349
## [277] -0.59109870 -2.12162897 0.76365631 -2.24280398 -1.94974745 0.19331399
## [283] -0.74117823 0.97860485 -1.24940106 -0.18743963 1.29118894 2.85500231
## [289] 4.26999696 4.65894375 3.43743936 3.74030994 2.66625255 1.59717402
## [295] 0.33319626 -1.06567575 -0.89432153#Yt-3
yt_3_arma <- arma22[-c(200,199,198)]
yt_3_arma## [1] 0.74304034 2.40422722 -0.24246686 -0.09833948 -1.17223489 -2.03060488
## [7] -0.80484470 0.35416729 0.81727989 1.71535244 1.65327968 2.26628412
## [13] 2.87143088 0.84349311 0.92513572 -0.66330571 -1.56027505 -0.61434833
## [19] -2.20571610 -2.77875558 -3.60298132 -2.23099833 -2.36322256 -1.69560501
## [25] -1.56161999 -0.09860517 -0.53709823 0.86256867 -0.42908366 -1.05595588
## [31] -1.36716541 -0.74599309 -0.14443797 0.79377285 -0.93219702 0.85870083
## [37] -0.06220909 0.81522712 0.81037637 -0.47950596 -0.33858754 -1.25146380
## [43] -0.65481150 -0.96012669 -1.01742774 0.07231702 -0.16512118 0.68675575
## [49] -0.47589465 -0.64568813 -1.03416598 -0.08293220 0.27575224 0.74624311
## [55] -0.53653035 -0.18566371 -0.42408042 0.67735218 0.53267634 0.33073256
## [61] -0.82044851 -0.09969010 -0.13956357 -0.16683189 -0.50155894 -0.21522035
## [67] -0.46148282 -0.31285443 1.06370729 1.44019539 1.56090694 1.75331494
## [73] 0.85484590 0.15005524 -0.41613044 0.41092792 -0.01938142 0.83582053
## [79] 1.06944093 1.68738409 1.66670135 0.62841476 1.60858305 1.65586404
## [85] 2.57582443 3.22003517 2.26048313 2.51739847 0.14052150 0.41076757
## [91] -0.96927417 0.54497556 1.22269581 1.58955811 0.04892493 -0.02424499
## [97] -2.64588936 0.20207130 -2.55438122 -0.29877151 0.85591107 1.30998401
## [103] 3.37646387 2.66571958 3.89848360 3.71001271 3.60282685 2.62387497
## [109] 2.79109412 0.02875172 -0.64350891 -1.78082465 -1.69279420 -0.27664385
## [115] 0.22165005 1.34753665 0.20001767 -0.58535893 -0.71513967 -0.73178149
## [121] 0.99854527 -0.85260627 -1.35886381 -3.04860204 -4.49641652 -4.40187940
## [127] -4.10370089 -3.11112405 -2.14768409 -1.59426426 -1.20849642 -0.10879668
## [133] -0.74901170 1.69268340 0.33133668 0.38547383 -1.05693757 -1.35843756
## [139] -1.21054863 1.25461738 1.48273448 1.71330601 1.63018687 1.77493342
## [145] 1.70942999 0.66054532 -0.75430029 -1.80677879 -0.85088012 0.78476273
## [151] 2.39056866 2.69855244 4.13857700 3.15047305 2.51680531 1.45852722
## [157] 0.69606629 0.11155740 2.37958088 1.53444029 2.76067785 1.05612616
## [163] 2.00666722 2.38800881 4.26126953 2.36960211 1.86904011 -0.18112163
## [169] 0.16894507 -0.69776933 -0.03848840 0.73061926 0.75478154 0.76140563
## [175] 0.99103053 1.06085226 -0.34010169 1.20392950 0.46934411 3.01390246
## [181] 2.52128444 3.49863087 2.84817152 2.78915669 1.30931766 0.07111692
## [187] -0.58557012 -2.53666065 -3.16600295 -1.84885233 -2.49551356 -0.20079676
## [193] -2.36090529 -0.11742652 0.12312543 1.59441458 2.44405386 2.42445094
## [199] 2.88703441 2.76053895 0.91827406 -0.61864614 -1.82557583 -1.92050155
## [205] -2.70382755 -2.89928452 -2.86086586 -1.12983328 -1.06965573 0.30770222
## [211] 0.90804745 0.06297905 0.96007051 -0.52008579 0.57297906 -0.66999279
## [217] 0.36606683 -1.73693338 -0.52489705 0.72285156 1.17326781 0.39789562
## [223] 0.17422983 -1.16323755 -0.64628978 -0.81736758 -0.93312299 0.14004131
## [229] 0.48408137 -0.22359690 -2.23252650 -1.29144797 -1.57962715 -0.48172752
## [235] -1.01082069 -2.25632627 -2.46913856 -2.98924119 -1.77984329 -1.20889411
## [241] -1.54876129 -0.77282409 -2.14708992 -2.34034824 -2.48709837 -0.26504808
## [247] 0.38850753 1.37379767 1.00436277 -0.10999420 -0.58675872 0.59096810
## [253] 0.28283312 2.44531696 0.45041639 1.67405044 0.96273437 2.39816276
## [259] 3.58317820 3.94406275 3.71291673 3.06401687 0.52572954 1.44380538
## [265] 0.23000235 1.00038308 0.35047128 -0.10245555 -0.98003885 -0.08253108
## [271] 1.01335276 0.30875985 0.88146650 -0.12737316 0.06113570 -2.29170349
## [277] -0.59109870 -2.12162897 0.76365631 -2.24280398 -1.94974745 0.19331399
## [283] -0.74117823 0.97860485 -1.24940106 -0.18743963 1.29118894 2.85500231
## [289] 4.26999696 4.65894375 3.43743936 3.74030994 2.66625255 1.59717402
## [295] 0.33319626 -1.06567575 -0.89432153plot(y=yt3_arma,x=yt_3_arma)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-3}\). Hal ini sesuai dengan teori yang ada
Hitung autokorelasi masing-masing baik dari dta bangkitan maupun secara teoritispp
cor(yt_arma,yt_1_arma)## [1] 0.8570346cor(yt2_arma,yt_2_arma)## [1] 0.7772623cor(yt3_arma,yt_3_arma)## [1] 0.5938008Korelasi antara \(Y_t\) dengan \(Y_{t-1}\), \(Y_t\) dengan \(Y_{t-2}\), dan \(Y_t\) dengan \(Y_{t-3}\) dari hasil simulasi mendekati perhitungan teoritis. Korelasi antara \(Y_t\) dengan \(Y_{t-1}\) sebesar 0.8570346. Korelasi antara \(Y_t\) dengan \(Y_{t-2}\) sebesar 0.7772623 . Korelasi antara \(Y_t\) dengan \(Y_{t-3}\) sebesar 0.5938008 .