White Noise
White noise dibangkitkan dari suatu peubah acak, umumnya peubah acak normal.
set.seed(035)
wn <- rnorm(300)
ts.plot(wn)
Dapat terlihat bahwa white noise tidak memiliki autokorelasi
dari ACF. Perhatikan bahwa lag ke-0 adalah korelasi observasi ke-t
dengan dirinya sendiri. Nilai korelasi tersebut pasti 1. Sebagai
alternatif, lag pertama di plot ACF dapat ditetapkan sebagai 1
(alih-alih 0) dengan menambahkan argumen
xlim(1, lag akhir). Plot tersebut dapat disandingkan
bersamaan dengan membuat matriks \(1 \times
2\) dengan par(mfrow = c(1,2)).
par(mfrow = c(1, 2))
acf(wn)
acf(wn, xlim = c(1, 20))
Proses MA
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.
Pembangkitan Proses MA(2)
Akan dicoba membangkitkan proses MA, yaitu MA(2) dengan \(\theta1 = 0.4\) dan \(\theta2 = 0.6\) sebanyak 300 observasi dan \(c=0\). Karena diperlukan satu nilai awal untuk \(e_{t-1}\) dan \(e_{t-2}\), masukkan nilai pertama white noise sebagai nilai awal tersebut.
set.seed(035)
ma <- wn[c(1,2)]Nilai-nilai selanjutnya dapat dicari melalui loop. Bentuk loop dapat dilihat dari rumus MA(2) yang hendak dibangkitkan:
\[ y_t = e_t-0.4e_{t-1}-0.6e_{t-2} \]
for(i in 3:300){
ma[i] <- wn[i] + 0.4 * wn[i - 1]+ 0.6 * wn[i - 2]
}
ma## [1] 1.065125327 0.132881183 0.658183657 0.021134760 3.299420916
## [6] 0.915231810 2.256101722 -0.633709544 1.212692091 0.488247563
## [11] 1.365917354 0.859402939 -0.283312528 -0.541257282 -2.570059470
## [16] -2.842772162 -0.992370374 -1.067240744 2.218516529 2.200945279
## [21] 2.456432824 2.234504393 1.795219724 2.251167470 0.790990469
## [26] 1.215940621 0.526099293 1.094760371 1.080962605 0.908300966
## [31] 1.652803499 1.790913751 2.372726341 0.977329459 -0.011604417
## [36] 2.056545611 0.680957663 0.901810835 1.465150702 -0.401660786
## [41] 0.319160465 -0.522727520 -1.385340145 -1.252614832 0.060156387
## [46] 0.716966475 0.894508358 1.499279489 0.469600491 1.402106532
## [51] 0.485686038 1.584133220 -0.450447425 -1.134048977 -0.870574767
## [56] -1.499551006 1.219248831 -1.084224277 0.025365132 -1.345499887
## [61] 1.256230369 0.885180465 0.928511990 -0.615893983 -0.629149599
## [66] 2.333284252 1.260046662 0.928009646 0.300440456 -1.609279361
## [71] -2.243711296 -1.228093288 -0.384008230 2.222052781 1.366946810
## [76] 0.286202500 -0.437222316 0.251184461 0.177087208 0.875363006
## [81] 0.451368246 -0.956924380 -1.007689595 -2.619731719 -0.971496099
## [86] 0.016829578 0.261893053 0.721130841 0.072592286 0.729277777
## [91] 0.252451354 0.368767403 0.675968503 0.852165073 0.012702662
## [96] 1.669574565 1.868642026 0.915469178 0.359566884 -2.382471996
## [101] -1.268563372 -2.236629932 0.851566374 1.054256325 1.839758922
## [106] 1.089107790 2.282569970 0.501388394 2.310241027 1.924577740
## [111] 0.701628854 1.572163933 -0.811353906 1.491858309 0.024586469
## [116] 1.112575616 -0.571601276 0.377668248 -0.188704184 2.200551532
## [121] 0.137980542 0.950017590 -2.185559323 1.929037466 -0.806797583
## [126] -0.380090545 -1.144726418 0.324724378 -0.042361431 0.262863488
## [131] -2.286748789 -1.897223008 -0.380051401 0.816209921 2.132221753
## [136] 0.261903736 2.475658531 1.600373208 1.304524934 1.819312966
## [141] 1.362641279 -0.413939576 1.261613549 0.050621698 1.056754925
## [146] -0.986731556 -1.222474981 -0.376638636 1.363648342 2.452450632
## [151] 0.754805726 0.995028968 1.576494524 0.252185400 1.371836640
## [156] -0.192063167 1.124324950 1.373842963 0.161530570 -0.548495611
## [161] 0.122079347 -0.162928307 0.869627949 1.645302830 -0.133512418
## [166] 0.577205198 1.332200135 2.314074616 2.541704734 2.329764342
## [171] 0.760551363 1.260166357 -0.545679568 -0.526906104 0.141787476
## [176] -0.865162719 0.776325959 -0.800867390 -0.892046480 -0.198269829
## [181] 0.391609515 -0.323115699 1.411984335 1.539053907 1.726604278
## [186] 2.101460674 0.952816063 0.630389142 -0.434972955 0.284244921
## [191] -0.629639814 -0.326646935 0.071838968 0.089770764 2.391026520
## [196] 0.344158345 0.726821474 -0.426357782 0.673665279 0.476988095
## [201] 1.031459004 -0.824513344 -0.296015791 -1.270447231 0.741566980
## [206] -1.917525859 2.152745644 -0.934229115 0.986361102 -2.327510614
## [211] -0.836100979 -2.203163018 1.251274713 -0.483100429 0.405541344
## [216] -1.046801129 -0.118265714 -1.479388762 -0.972990734 -1.654317862
## [221] -0.369158859 1.188879940 0.758165612 0.004675097 -0.842823443
## [226] 0.789932300 -0.806046838 0.823141063 -1.638547001 -0.956585019
## [231] -3.511118437 0.160155860 -0.352221471 2.080835571 1.155445559
## [236] 0.007319216 -0.349601438 -1.389952187 -0.480352506 -0.374425649
## [241] 1.536508308 2.946040765 2.058586017 1.043034785 -0.514395852
## [246] 0.448597128 -0.107166162 -0.901895755 -0.643922756 -1.073635485
## [251] -1.848150055 -0.975969799 -1.018716345 -0.046728256 -0.457365043
## [256] 0.290596707 -0.817736130 -0.285379062 -1.197053507 -0.809829780
## [261] -0.284903059 1.427407937 2.072450303 2.578961105 0.990888560
## [266] 0.130744460 -1.055302300 -0.349721522 0.489784548 0.652814461
## [271] 0.864882450 -1.406581322 0.300966391 0.964790201 1.260471246
## [276] 1.745385800 0.068547273 3.790009529 0.959858473 2.336401436
## [281] -1.004590961 -1.634457108 -0.245794766 -1.413568572 -0.134170647
## [286] -0.816912736 0.107330937 -0.639551099 -0.525883926 -1.055692764
## [291] -0.158199423 -0.107872805 -0.227482501 0.463732156 -0.382036141
## [296] 0.326448357 0.727266022 1.184649221 2.168488530 1.460507679Selain menggunakan cara di atas, pembangkitan proses MA(2) dapat
dilakukan dengan fungsi arima.sim() sebagai berikut.
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] 0.658183657 0.021134760 3.299420916 0.915231810 2.256101722
## [6] -0.633709544 1.212692091 0.488247563 1.365917354 0.859402939
## [11] -0.283312528 -0.541257282 -2.570059470 -2.842772162 -0.992370374
## [16] -1.067240744 2.218516529 2.200945279 2.456432824 2.234504393
## [21] 1.795219724 2.251167470 0.790990469 1.215940621 0.526099293
## [26] 1.094760371 1.080962605 0.908300966 1.652803499 1.790913751
## [31] 2.372726341 0.977329459 -0.011604417 2.056545611 0.680957663
## [36] 0.901810835 1.465150702 -0.401660786 0.319160465 -0.522727520
## [41] -1.385340145 -1.252614832 0.060156387 0.716966475 0.894508358
## [46] 1.499279489 0.469600491 1.402106532 0.485686038 1.584133220
## [51] -0.450447425 -1.134048977 -0.870574767 -1.499551006 1.219248831
## [56] -1.084224277 0.025365132 -1.345499887 1.256230369 0.885180465
## [61] 0.928511990 -0.615893983 -0.629149599 2.333284252 1.260046662
## [66] 0.928009646 0.300440456 -1.609279361 -2.243711296 -1.228093288
## [71] -0.384008230 2.222052781 1.366946810 0.286202500 -0.437222316
## [76] 0.251184461 0.177087208 0.875363006 0.451368246 -0.956924380
## [81] -1.007689595 -2.619731719 -0.971496099 0.016829578 0.261893053
## [86] 0.721130841 0.072592286 0.729277777 0.252451354 0.368767403
## [91] 0.675968503 0.852165073 0.012702662 1.669574565 1.868642026
## [96] 0.915469178 0.359566884 -2.382471996 -1.268563372 -2.236629932
## [101] 0.851566374 1.054256325 1.839758922 1.089107790 2.282569970
## [106] 0.501388394 2.310241027 1.924577740 0.701628854 1.572163933
## [111] -0.811353906 1.491858309 0.024586469 1.112575616 -0.571601276
## [116] 0.377668248 -0.188704184 2.200551532 0.137980542 0.950017590
## [121] -2.185559323 1.929037466 -0.806797583 -0.380090545 -1.144726418
## [126] 0.324724378 -0.042361431 0.262863488 -2.286748789 -1.897223008
## [131] -0.380051401 0.816209921 2.132221753 0.261903736 2.475658531
## [136] 1.600373208 1.304524934 1.819312966 1.362641279 -0.413939576
## [141] 1.261613549 0.050621698 1.056754925 -0.986731556 -1.222474981
## [146] -0.376638636 1.363648342 2.452450632 0.754805726 0.995028968
## [151] 1.576494524 0.252185400 1.371836640 -0.192063167 1.124324950
## [156] 1.373842963 0.161530570 -0.548495611 0.122079347 -0.162928307
## [161] 0.869627949 1.645302830 -0.133512418 0.577205198 1.332200135
## [166] 2.314074616 2.541704734 2.329764342 0.760551363 1.260166357
## [171] -0.545679568 -0.526906104 0.141787476 -0.865162719 0.776325959
## [176] -0.800867390 -0.892046480 -0.198269829 0.391609515 -0.323115699
## [181] 1.411984335 1.539053907 1.726604278 2.101460674 0.952816063
## [186] 0.630389142 -0.434972955 0.284244921 -0.629639814 -0.326646935
## [191] 0.071838968 0.089770764 2.391026520 0.344158345 0.726821474
## [196] -0.426357782 0.673665279 0.476988095 1.031459004 -0.824513344
## [201] -0.296015791 -1.270447231 0.741566980 -1.917525859 2.152745644
## [206] -0.934229115 0.986361102 -2.327510614 -0.836100979 -2.203163018
## [211] 1.251274713 -0.483100429 0.405541344 -1.046801129 -0.118265714
## [216] -1.479388762 -0.972990734 -1.654317862 -0.369158859 1.188879940
## [221] 0.758165612 0.004675097 -0.842823443 0.789932300 -0.806046838
## [226] 0.823141063 -1.638547001 -0.956585019 -3.511118437 0.160155860
## [231] -0.352221471 2.080835571 1.155445559 0.007319216 -0.349601438
## [236] -1.389952187 -0.480352506 -0.374425649 1.536508308 2.946040765
## [241] 2.058586017 1.043034785 -0.514395852 0.448597128 -0.107166162
## [246] -0.901895755 -0.643922756 -1.073635485 -1.848150055 -0.975969799
## [251] -1.018716345 -0.046728256 -0.457365043 0.290596707 -0.817736130
## [256] -0.285379062 -1.197053507 -0.809829780 -0.284903059 1.427407937
## [261] 2.072450303 2.578961105 0.990888560 0.130744460 -1.055302300
## [266] -0.349721522 0.489784548 0.652814461 0.864882450 -1.406581322
## [271] 0.300966391 0.964790201 1.260471246 1.745385800 0.068547273
## [276] 3.790009529 0.959858473 2.336401436 -1.004590961 -1.634457108
## [281] -0.245794766 -1.413568572 -0.134170647 -0.816912736 0.107330937
## [286] -0.639551099 -0.525883926 -1.055692764 -0.158199423 -0.107872805
## [291] -0.227482501 0.463732156 -0.382036141 0.326448357 0.727266022
## [296] 1.184649221 2.168488530 1.460507679 0.906312503 0.015317578Karakteristik MA(2)
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 o o o
## 1 x x o o o o o o o o o o o o
## 2 x x o x o o o o o o o o o o
## 3 o x x x o o o o o o o o o o
## 4 x x o x o o o o o o o o o o
## 5 x x x x o x 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 x 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)
Scatterplot Antar Lag
Korelasi antara \(Y_t\) dengan \(Y_{t-1}\)
#Yt
yt_ma <- ma[-1]
yt_ma## [1] 0.132881183 0.658183657 0.021134760 3.299420916 0.915231810
## [6] 2.256101722 -0.633709544 1.212692091 0.488247563 1.365917354
## [11] 0.859402939 -0.283312528 -0.541257282 -2.570059470 -2.842772162
## [16] -0.992370374 -1.067240744 2.218516529 2.200945279 2.456432824
## [21] 2.234504393 1.795219724 2.251167470 0.790990469 1.215940621
## [26] 0.526099293 1.094760371 1.080962605 0.908300966 1.652803499
## [31] 1.790913751 2.372726341 0.977329459 -0.011604417 2.056545611
## [36] 0.680957663 0.901810835 1.465150702 -0.401660786 0.319160465
## [41] -0.522727520 -1.385340145 -1.252614832 0.060156387 0.716966475
## [46] 0.894508358 1.499279489 0.469600491 1.402106532 0.485686038
## [51] 1.584133220 -0.450447425 -1.134048977 -0.870574767 -1.499551006
## [56] 1.219248831 -1.084224277 0.025365132 -1.345499887 1.256230369
## [61] 0.885180465 0.928511990 -0.615893983 -0.629149599 2.333284252
## [66] 1.260046662 0.928009646 0.300440456 -1.609279361 -2.243711296
## [71] -1.228093288 -0.384008230 2.222052781 1.366946810 0.286202500
## [76] -0.437222316 0.251184461 0.177087208 0.875363006 0.451368246
## [81] -0.956924380 -1.007689595 -2.619731719 -0.971496099 0.016829578
## [86] 0.261893053 0.721130841 0.072592286 0.729277777 0.252451354
## [91] 0.368767403 0.675968503 0.852165073 0.012702662 1.669574565
## [96] 1.868642026 0.915469178 0.359566884 -2.382471996 -1.268563372
## [101] -2.236629932 0.851566374 1.054256325 1.839758922 1.089107790
## [106] 2.282569970 0.501388394 2.310241027 1.924577740 0.701628854
## [111] 1.572163933 -0.811353906 1.491858309 0.024586469 1.112575616
## [116] -0.571601276 0.377668248 -0.188704184 2.200551532 0.137980542
## [121] 0.950017590 -2.185559323 1.929037466 -0.806797583 -0.380090545
## [126] -1.144726418 0.324724378 -0.042361431 0.262863488 -2.286748789
## [131] -1.897223008 -0.380051401 0.816209921 2.132221753 0.261903736
## [136] 2.475658531 1.600373208 1.304524934 1.819312966 1.362641279
## [141] -0.413939576 1.261613549 0.050621698 1.056754925 -0.986731556
## [146] -1.222474981 -0.376638636 1.363648342 2.452450632 0.754805726
## [151] 0.995028968 1.576494524 0.252185400 1.371836640 -0.192063167
## [156] 1.124324950 1.373842963 0.161530570 -0.548495611 0.122079347
## [161] -0.162928307 0.869627949 1.645302830 -0.133512418 0.577205198
## [166] 1.332200135 2.314074616 2.541704734 2.329764342 0.760551363
## [171] 1.260166357 -0.545679568 -0.526906104 0.141787476 -0.865162719
## [176] 0.776325959 -0.800867390 -0.892046480 -0.198269829 0.391609515
## [181] -0.323115699 1.411984335 1.539053907 1.726604278 2.101460674
## [186] 0.952816063 0.630389142 -0.434972955 0.284244921 -0.629639814
## [191] -0.326646935 0.071838968 0.089770764 2.391026520 0.344158345
## [196] 0.726821474 -0.426357782 0.673665279 0.476988095 1.031459004
## [201] -0.824513344 -0.296015791 -1.270447231 0.741566980 -1.917525859
## [206] 2.152745644 -0.934229115 0.986361102 -2.327510614 -0.836100979
## [211] -2.203163018 1.251274713 -0.483100429 0.405541344 -1.046801129
## [216] -0.118265714 -1.479388762 -0.972990734 -1.654317862 -0.369158859
## [221] 1.188879940 0.758165612 0.004675097 -0.842823443 0.789932300
## [226] -0.806046838 0.823141063 -1.638547001 -0.956585019 -3.511118437
## [231] 0.160155860 -0.352221471 2.080835571 1.155445559 0.007319216
## [236] -0.349601438 -1.389952187 -0.480352506 -0.374425649 1.536508308
## [241] 2.946040765 2.058586017 1.043034785 -0.514395852 0.448597128
## [246] -0.107166162 -0.901895755 -0.643922756 -1.073635485 -1.848150055
## [251] -0.975969799 -1.018716345 -0.046728256 -0.457365043 0.290596707
## [256] -0.817736130 -0.285379062 -1.197053507 -0.809829780 -0.284903059
## [261] 1.427407937 2.072450303 2.578961105 0.990888560 0.130744460
## [266] -1.055302300 -0.349721522 0.489784548 0.652814461 0.864882450
## [271] -1.406581322 0.300966391 0.964790201 1.260471246 1.745385800
## [276] 0.068547273 3.790009529 0.959858473 2.336401436 -1.004590961
## [281] -1.634457108 -0.245794766 -1.413568572 -0.134170647 -0.816912736
## [286] 0.107330937 -0.639551099 -0.525883926 -1.055692764 -0.158199423
## [291] -0.107872805 -0.227482501 0.463732156 -0.382036141 0.326448357
## [296] 0.727266022 1.184649221 2.168488530 1.460507679#Yt-1
yt_1_ma <- ma[-300]
yt_1_ma## [1] 1.065125327 0.132881183 0.658183657 0.021134760 3.299420916
## [6] 0.915231810 2.256101722 -0.633709544 1.212692091 0.488247563
## [11] 1.365917354 0.859402939 -0.283312528 -0.541257282 -2.570059470
## [16] -2.842772162 -0.992370374 -1.067240744 2.218516529 2.200945279
## [21] 2.456432824 2.234504393 1.795219724 2.251167470 0.790990469
## [26] 1.215940621 0.526099293 1.094760371 1.080962605 0.908300966
## [31] 1.652803499 1.790913751 2.372726341 0.977329459 -0.011604417
## [36] 2.056545611 0.680957663 0.901810835 1.465150702 -0.401660786
## [41] 0.319160465 -0.522727520 -1.385340145 -1.252614832 0.060156387
## [46] 0.716966475 0.894508358 1.499279489 0.469600491 1.402106532
## [51] 0.485686038 1.584133220 -0.450447425 -1.134048977 -0.870574767
## [56] -1.499551006 1.219248831 -1.084224277 0.025365132 -1.345499887
## [61] 1.256230369 0.885180465 0.928511990 -0.615893983 -0.629149599
## [66] 2.333284252 1.260046662 0.928009646 0.300440456 -1.609279361
## [71] -2.243711296 -1.228093288 -0.384008230 2.222052781 1.366946810
## [76] 0.286202500 -0.437222316 0.251184461 0.177087208 0.875363006
## [81] 0.451368246 -0.956924380 -1.007689595 -2.619731719 -0.971496099
## [86] 0.016829578 0.261893053 0.721130841 0.072592286 0.729277777
## [91] 0.252451354 0.368767403 0.675968503 0.852165073 0.012702662
## [96] 1.669574565 1.868642026 0.915469178 0.359566884 -2.382471996
## [101] -1.268563372 -2.236629932 0.851566374 1.054256325 1.839758922
## [106] 1.089107790 2.282569970 0.501388394 2.310241027 1.924577740
## [111] 0.701628854 1.572163933 -0.811353906 1.491858309 0.024586469
## [116] 1.112575616 -0.571601276 0.377668248 -0.188704184 2.200551532
## [121] 0.137980542 0.950017590 -2.185559323 1.929037466 -0.806797583
## [126] -0.380090545 -1.144726418 0.324724378 -0.042361431 0.262863488
## [131] -2.286748789 -1.897223008 -0.380051401 0.816209921 2.132221753
## [136] 0.261903736 2.475658531 1.600373208 1.304524934 1.819312966
## [141] 1.362641279 -0.413939576 1.261613549 0.050621698 1.056754925
## [146] -0.986731556 -1.222474981 -0.376638636 1.363648342 2.452450632
## [151] 0.754805726 0.995028968 1.576494524 0.252185400 1.371836640
## [156] -0.192063167 1.124324950 1.373842963 0.161530570 -0.548495611
## [161] 0.122079347 -0.162928307 0.869627949 1.645302830 -0.133512418
## [166] 0.577205198 1.332200135 2.314074616 2.541704734 2.329764342
## [171] 0.760551363 1.260166357 -0.545679568 -0.526906104 0.141787476
## [176] -0.865162719 0.776325959 -0.800867390 -0.892046480 -0.198269829
## [181] 0.391609515 -0.323115699 1.411984335 1.539053907 1.726604278
## [186] 2.101460674 0.952816063 0.630389142 -0.434972955 0.284244921
## [191] -0.629639814 -0.326646935 0.071838968 0.089770764 2.391026520
## [196] 0.344158345 0.726821474 -0.426357782 0.673665279 0.476988095
## [201] 1.031459004 -0.824513344 -0.296015791 -1.270447231 0.741566980
## [206] -1.917525859 2.152745644 -0.934229115 0.986361102 -2.327510614
## [211] -0.836100979 -2.203163018 1.251274713 -0.483100429 0.405541344
## [216] -1.046801129 -0.118265714 -1.479388762 -0.972990734 -1.654317862
## [221] -0.369158859 1.188879940 0.758165612 0.004675097 -0.842823443
## [226] 0.789932300 -0.806046838 0.823141063 -1.638547001 -0.956585019
## [231] -3.511118437 0.160155860 -0.352221471 2.080835571 1.155445559
## [236] 0.007319216 -0.349601438 -1.389952187 -0.480352506 -0.374425649
## [241] 1.536508308 2.946040765 2.058586017 1.043034785 -0.514395852
## [246] 0.448597128 -0.107166162 -0.901895755 -0.643922756 -1.073635485
## [251] -1.848150055 -0.975969799 -1.018716345 -0.046728256 -0.457365043
## [256] 0.290596707 -0.817736130 -0.285379062 -1.197053507 -0.809829780
## [261] -0.284903059 1.427407937 2.072450303 2.578961105 0.990888560
## [266] 0.130744460 -1.055302300 -0.349721522 0.489784548 0.652814461
## [271] 0.864882450 -1.406581322 0.300966391 0.964790201 1.260471246
## [276] 1.745385800 0.068547273 3.790009529 0.959858473 2.336401436
## [281] -1.004590961 -1.634457108 -0.245794766 -1.413568572 -0.134170647
## [286] -0.816912736 0.107330937 -0.639551099 -0.525883926 -1.055692764
## [291] -0.158199423 -0.107872805 -0.227482501 0.463732156 -0.382036141
## [296] 0.326448357 0.727266022 1.184649221 2.168488530plot(y=yt_ma,x=yt_1_ma)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-1}\). Hal ini sesuai dengan teori yang ada
cor(yt_ma,yt_1_ma)## [1] 0.3786993Korelasi antara \(Y_t\) dengan \(Y_{t-1}\) dari hasil simulasi mendekati perhitungan teoritis yaitu
\[ \rho_1=\frac{\theta_1+\theta_1\theta_2}{1+(\theta_1)^2+(\theta_2)^2}=\frac{(0.4)+(0.4*0.6)}{1+(0.4)^2+(0.6)^2}=0.42 \]
Korelasi antara \(Y_t\) dengan \(Y_{t-2}\)
#Yt
yt_ma2 <- ma[-c(1,2)]
yt_ma2## [1] 0.658183657 0.021134760 3.299420916 0.915231810 2.256101722
## [6] -0.633709544 1.212692091 0.488247563 1.365917354 0.859402939
## [11] -0.283312528 -0.541257282 -2.570059470 -2.842772162 -0.992370374
## [16] -1.067240744 2.218516529 2.200945279 2.456432824 2.234504393
## [21] 1.795219724 2.251167470 0.790990469 1.215940621 0.526099293
## [26] 1.094760371 1.080962605 0.908300966 1.652803499 1.790913751
## [31] 2.372726341 0.977329459 -0.011604417 2.056545611 0.680957663
## [36] 0.901810835 1.465150702 -0.401660786 0.319160465 -0.522727520
## [41] -1.385340145 -1.252614832 0.060156387 0.716966475 0.894508358
## [46] 1.499279489 0.469600491 1.402106532 0.485686038 1.584133220
## [51] -0.450447425 -1.134048977 -0.870574767 -1.499551006 1.219248831
## [56] -1.084224277 0.025365132 -1.345499887 1.256230369 0.885180465
## [61] 0.928511990 -0.615893983 -0.629149599 2.333284252 1.260046662
## [66] 0.928009646 0.300440456 -1.609279361 -2.243711296 -1.228093288
## [71] -0.384008230 2.222052781 1.366946810 0.286202500 -0.437222316
## [76] 0.251184461 0.177087208 0.875363006 0.451368246 -0.956924380
## [81] -1.007689595 -2.619731719 -0.971496099 0.016829578 0.261893053
## [86] 0.721130841 0.072592286 0.729277777 0.252451354 0.368767403
## [91] 0.675968503 0.852165073 0.012702662 1.669574565 1.868642026
## [96] 0.915469178 0.359566884 -2.382471996 -1.268563372 -2.236629932
## [101] 0.851566374 1.054256325 1.839758922 1.089107790 2.282569970
## [106] 0.501388394 2.310241027 1.924577740 0.701628854 1.572163933
## [111] -0.811353906 1.491858309 0.024586469 1.112575616 -0.571601276
## [116] 0.377668248 -0.188704184 2.200551532 0.137980542 0.950017590
## [121] -2.185559323 1.929037466 -0.806797583 -0.380090545 -1.144726418
## [126] 0.324724378 -0.042361431 0.262863488 -2.286748789 -1.897223008
## [131] -0.380051401 0.816209921 2.132221753 0.261903736 2.475658531
## [136] 1.600373208 1.304524934 1.819312966 1.362641279 -0.413939576
## [141] 1.261613549 0.050621698 1.056754925 -0.986731556 -1.222474981
## [146] -0.376638636 1.363648342 2.452450632 0.754805726 0.995028968
## [151] 1.576494524 0.252185400 1.371836640 -0.192063167 1.124324950
## [156] 1.373842963 0.161530570 -0.548495611 0.122079347 -0.162928307
## [161] 0.869627949 1.645302830 -0.133512418 0.577205198 1.332200135
## [166] 2.314074616 2.541704734 2.329764342 0.760551363 1.260166357
## [171] -0.545679568 -0.526906104 0.141787476 -0.865162719 0.776325959
## [176] -0.800867390 -0.892046480 -0.198269829 0.391609515 -0.323115699
## [181] 1.411984335 1.539053907 1.726604278 2.101460674 0.952816063
## [186] 0.630389142 -0.434972955 0.284244921 -0.629639814 -0.326646935
## [191] 0.071838968 0.089770764 2.391026520 0.344158345 0.726821474
## [196] -0.426357782 0.673665279 0.476988095 1.031459004 -0.824513344
## [201] -0.296015791 -1.270447231 0.741566980 -1.917525859 2.152745644
## [206] -0.934229115 0.986361102 -2.327510614 -0.836100979 -2.203163018
## [211] 1.251274713 -0.483100429 0.405541344 -1.046801129 -0.118265714
## [216] -1.479388762 -0.972990734 -1.654317862 -0.369158859 1.188879940
## [221] 0.758165612 0.004675097 -0.842823443 0.789932300 -0.806046838
## [226] 0.823141063 -1.638547001 -0.956585019 -3.511118437 0.160155860
## [231] -0.352221471 2.080835571 1.155445559 0.007319216 -0.349601438
## [236] -1.389952187 -0.480352506 -0.374425649 1.536508308 2.946040765
## [241] 2.058586017 1.043034785 -0.514395852 0.448597128 -0.107166162
## [246] -0.901895755 -0.643922756 -1.073635485 -1.848150055 -0.975969799
## [251] -1.018716345 -0.046728256 -0.457365043 0.290596707 -0.817736130
## [256] -0.285379062 -1.197053507 -0.809829780 -0.284903059 1.427407937
## [261] 2.072450303 2.578961105 0.990888560 0.130744460 -1.055302300
## [266] -0.349721522 0.489784548 0.652814461 0.864882450 -1.406581322
## [271] 0.300966391 0.964790201 1.260471246 1.745385800 0.068547273
## [276] 3.790009529 0.959858473 2.336401436 -1.004590961 -1.634457108
## [281] -0.245794766 -1.413568572 -0.134170647 -0.816912736 0.107330937
## [286] -0.639551099 -0.525883926 -1.055692764 -0.158199423 -0.107872805
## [291] -0.227482501 0.463732156 -0.382036141 0.326448357 0.727266022
## [296] 1.184649221 2.168488530 1.460507679#Yt-2
yt_2_ma <- ma[-c(299,300)]
yt_2_ma## [1] 1.065125327 0.132881183 0.658183657 0.021134760 3.299420916
## [6] 0.915231810 2.256101722 -0.633709544 1.212692091 0.488247563
## [11] 1.365917354 0.859402939 -0.283312528 -0.541257282 -2.570059470
## [16] -2.842772162 -0.992370374 -1.067240744 2.218516529 2.200945279
## [21] 2.456432824 2.234504393 1.795219724 2.251167470 0.790990469
## [26] 1.215940621 0.526099293 1.094760371 1.080962605 0.908300966
## [31] 1.652803499 1.790913751 2.372726341 0.977329459 -0.011604417
## [36] 2.056545611 0.680957663 0.901810835 1.465150702 -0.401660786
## [41] 0.319160465 -0.522727520 -1.385340145 -1.252614832 0.060156387
## [46] 0.716966475 0.894508358 1.499279489 0.469600491 1.402106532
## [51] 0.485686038 1.584133220 -0.450447425 -1.134048977 -0.870574767
## [56] -1.499551006 1.219248831 -1.084224277 0.025365132 -1.345499887
## [61] 1.256230369 0.885180465 0.928511990 -0.615893983 -0.629149599
## [66] 2.333284252 1.260046662 0.928009646 0.300440456 -1.609279361
## [71] -2.243711296 -1.228093288 -0.384008230 2.222052781 1.366946810
## [76] 0.286202500 -0.437222316 0.251184461 0.177087208 0.875363006
## [81] 0.451368246 -0.956924380 -1.007689595 -2.619731719 -0.971496099
## [86] 0.016829578 0.261893053 0.721130841 0.072592286 0.729277777
## [91] 0.252451354 0.368767403 0.675968503 0.852165073 0.012702662
## [96] 1.669574565 1.868642026 0.915469178 0.359566884 -2.382471996
## [101] -1.268563372 -2.236629932 0.851566374 1.054256325 1.839758922
## [106] 1.089107790 2.282569970 0.501388394 2.310241027 1.924577740
## [111] 0.701628854 1.572163933 -0.811353906 1.491858309 0.024586469
## [116] 1.112575616 -0.571601276 0.377668248 -0.188704184 2.200551532
## [121] 0.137980542 0.950017590 -2.185559323 1.929037466 -0.806797583
## [126] -0.380090545 -1.144726418 0.324724378 -0.042361431 0.262863488
## [131] -2.286748789 -1.897223008 -0.380051401 0.816209921 2.132221753
## [136] 0.261903736 2.475658531 1.600373208 1.304524934 1.819312966
## [141] 1.362641279 -0.413939576 1.261613549 0.050621698 1.056754925
## [146] -0.986731556 -1.222474981 -0.376638636 1.363648342 2.452450632
## [151] 0.754805726 0.995028968 1.576494524 0.252185400 1.371836640
## [156] -0.192063167 1.124324950 1.373842963 0.161530570 -0.548495611
## [161] 0.122079347 -0.162928307 0.869627949 1.645302830 -0.133512418
## [166] 0.577205198 1.332200135 2.314074616 2.541704734 2.329764342
## [171] 0.760551363 1.260166357 -0.545679568 -0.526906104 0.141787476
## [176] -0.865162719 0.776325959 -0.800867390 -0.892046480 -0.198269829
## [181] 0.391609515 -0.323115699 1.411984335 1.539053907 1.726604278
## [186] 2.101460674 0.952816063 0.630389142 -0.434972955 0.284244921
## [191] -0.629639814 -0.326646935 0.071838968 0.089770764 2.391026520
## [196] 0.344158345 0.726821474 -0.426357782 0.673665279 0.476988095
## [201] 1.031459004 -0.824513344 -0.296015791 -1.270447231 0.741566980
## [206] -1.917525859 2.152745644 -0.934229115 0.986361102 -2.327510614
## [211] -0.836100979 -2.203163018 1.251274713 -0.483100429 0.405541344
## [216] -1.046801129 -0.118265714 -1.479388762 -0.972990734 -1.654317862
## [221] -0.369158859 1.188879940 0.758165612 0.004675097 -0.842823443
## [226] 0.789932300 -0.806046838 0.823141063 -1.638547001 -0.956585019
## [231] -3.511118437 0.160155860 -0.352221471 2.080835571 1.155445559
## [236] 0.007319216 -0.349601438 -1.389952187 -0.480352506 -0.374425649
## [241] 1.536508308 2.946040765 2.058586017 1.043034785 -0.514395852
## [246] 0.448597128 -0.107166162 -0.901895755 -0.643922756 -1.073635485
## [251] -1.848150055 -0.975969799 -1.018716345 -0.046728256 -0.457365043
## [256] 0.290596707 -0.817736130 -0.285379062 -1.197053507 -0.809829780
## [261] -0.284903059 1.427407937 2.072450303 2.578961105 0.990888560
## [266] 0.130744460 -1.055302300 -0.349721522 0.489784548 0.652814461
## [271] 0.864882450 -1.406581322 0.300966391 0.964790201 1.260471246
## [276] 1.745385800 0.068547273 3.790009529 0.959858473 2.336401436
## [281] -1.004590961 -1.634457108 -0.245794766 -1.413568572 -0.134170647
## [286] -0.816912736 0.107330937 -0.639551099 -0.525883926 -1.055692764
## [291] -0.158199423 -0.107872805 -0.227482501 0.463732156 -0.382036141
## [296] 0.326448357 0.727266022 1.184649221plot(y=yt_ma2,x=yt_2_ma)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-2}\). Hal ini sesuai dengan teori yang ada
cor(yt_ma2,yt_2_ma)## [1] 0.3897844Korelasi antara \(Y_t\) dengan \(Y_{t-3}\)
#Yt
yt_ma3 <- ma[-c(1,2,3)]
yt_ma3## [1] 0.021134760 3.299420916 0.915231810 2.256101722 -0.633709544
## [6] 1.212692091 0.488247563 1.365917354 0.859402939 -0.283312528
## [11] -0.541257282 -2.570059470 -2.842772162 -0.992370374 -1.067240744
## [16] 2.218516529 2.200945279 2.456432824 2.234504393 1.795219724
## [21] 2.251167470 0.790990469 1.215940621 0.526099293 1.094760371
## [26] 1.080962605 0.908300966 1.652803499 1.790913751 2.372726341
## [31] 0.977329459 -0.011604417 2.056545611 0.680957663 0.901810835
## [36] 1.465150702 -0.401660786 0.319160465 -0.522727520 -1.385340145
## [41] -1.252614832 0.060156387 0.716966475 0.894508358 1.499279489
## [46] 0.469600491 1.402106532 0.485686038 1.584133220 -0.450447425
## [51] -1.134048977 -0.870574767 -1.499551006 1.219248831 -1.084224277
## [56] 0.025365132 -1.345499887 1.256230369 0.885180465 0.928511990
## [61] -0.615893983 -0.629149599 2.333284252 1.260046662 0.928009646
## [66] 0.300440456 -1.609279361 -2.243711296 -1.228093288 -0.384008230
## [71] 2.222052781 1.366946810 0.286202500 -0.437222316 0.251184461
## [76] 0.177087208 0.875363006 0.451368246 -0.956924380 -1.007689595
## [81] -2.619731719 -0.971496099 0.016829578 0.261893053 0.721130841
## [86] 0.072592286 0.729277777 0.252451354 0.368767403 0.675968503
## [91] 0.852165073 0.012702662 1.669574565 1.868642026 0.915469178
## [96] 0.359566884 -2.382471996 -1.268563372 -2.236629932 0.851566374
## [101] 1.054256325 1.839758922 1.089107790 2.282569970 0.501388394
## [106] 2.310241027 1.924577740 0.701628854 1.572163933 -0.811353906
## [111] 1.491858309 0.024586469 1.112575616 -0.571601276 0.377668248
## [116] -0.188704184 2.200551532 0.137980542 0.950017590 -2.185559323
## [121] 1.929037466 -0.806797583 -0.380090545 -1.144726418 0.324724378
## [126] -0.042361431 0.262863488 -2.286748789 -1.897223008 -0.380051401
## [131] 0.816209921 2.132221753 0.261903736 2.475658531 1.600373208
## [136] 1.304524934 1.819312966 1.362641279 -0.413939576 1.261613549
## [141] 0.050621698 1.056754925 -0.986731556 -1.222474981 -0.376638636
## [146] 1.363648342 2.452450632 0.754805726 0.995028968 1.576494524
## [151] 0.252185400 1.371836640 -0.192063167 1.124324950 1.373842963
## [156] 0.161530570 -0.548495611 0.122079347 -0.162928307 0.869627949
## [161] 1.645302830 -0.133512418 0.577205198 1.332200135 2.314074616
## [166] 2.541704734 2.329764342 0.760551363 1.260166357 -0.545679568
## [171] -0.526906104 0.141787476 -0.865162719 0.776325959 -0.800867390
## [176] -0.892046480 -0.198269829 0.391609515 -0.323115699 1.411984335
## [181] 1.539053907 1.726604278 2.101460674 0.952816063 0.630389142
## [186] -0.434972955 0.284244921 -0.629639814 -0.326646935 0.071838968
## [191] 0.089770764 2.391026520 0.344158345 0.726821474 -0.426357782
## [196] 0.673665279 0.476988095 1.031459004 -0.824513344 -0.296015791
## [201] -1.270447231 0.741566980 -1.917525859 2.152745644 -0.934229115
## [206] 0.986361102 -2.327510614 -0.836100979 -2.203163018 1.251274713
## [211] -0.483100429 0.405541344 -1.046801129 -0.118265714 -1.479388762
## [216] -0.972990734 -1.654317862 -0.369158859 1.188879940 0.758165612
## [221] 0.004675097 -0.842823443 0.789932300 -0.806046838 0.823141063
## [226] -1.638547001 -0.956585019 -3.511118437 0.160155860 -0.352221471
## [231] 2.080835571 1.155445559 0.007319216 -0.349601438 -1.389952187
## [236] -0.480352506 -0.374425649 1.536508308 2.946040765 2.058586017
## [241] 1.043034785 -0.514395852 0.448597128 -0.107166162 -0.901895755
## [246] -0.643922756 -1.073635485 -1.848150055 -0.975969799 -1.018716345
## [251] -0.046728256 -0.457365043 0.290596707 -0.817736130 -0.285379062
## [256] -1.197053507 -0.809829780 -0.284903059 1.427407937 2.072450303
## [261] 2.578961105 0.990888560 0.130744460 -1.055302300 -0.349721522
## [266] 0.489784548 0.652814461 0.864882450 -1.406581322 0.300966391
## [271] 0.964790201 1.260471246 1.745385800 0.068547273 3.790009529
## [276] 0.959858473 2.336401436 -1.004590961 -1.634457108 -0.245794766
## [281] -1.413568572 -0.134170647 -0.816912736 0.107330937 -0.639551099
## [286] -0.525883926 -1.055692764 -0.158199423 -0.107872805 -0.227482501
## [291] 0.463732156 -0.382036141 0.326448357 0.727266022 1.184649221
## [296] 2.168488530 1.460507679#Yt-3
yt_3_ma <- ma[-c(298,299,300)]
yt_3_ma## [1] 1.065125327 0.132881183 0.658183657 0.021134760 3.299420916
## [6] 0.915231810 2.256101722 -0.633709544 1.212692091 0.488247563
## [11] 1.365917354 0.859402939 -0.283312528 -0.541257282 -2.570059470
## [16] -2.842772162 -0.992370374 -1.067240744 2.218516529 2.200945279
## [21] 2.456432824 2.234504393 1.795219724 2.251167470 0.790990469
## [26] 1.215940621 0.526099293 1.094760371 1.080962605 0.908300966
## [31] 1.652803499 1.790913751 2.372726341 0.977329459 -0.011604417
## [36] 2.056545611 0.680957663 0.901810835 1.465150702 -0.401660786
## [41] 0.319160465 -0.522727520 -1.385340145 -1.252614832 0.060156387
## [46] 0.716966475 0.894508358 1.499279489 0.469600491 1.402106532
## [51] 0.485686038 1.584133220 -0.450447425 -1.134048977 -0.870574767
## [56] -1.499551006 1.219248831 -1.084224277 0.025365132 -1.345499887
## [61] 1.256230369 0.885180465 0.928511990 -0.615893983 -0.629149599
## [66] 2.333284252 1.260046662 0.928009646 0.300440456 -1.609279361
## [71] -2.243711296 -1.228093288 -0.384008230 2.222052781 1.366946810
## [76] 0.286202500 -0.437222316 0.251184461 0.177087208 0.875363006
## [81] 0.451368246 -0.956924380 -1.007689595 -2.619731719 -0.971496099
## [86] 0.016829578 0.261893053 0.721130841 0.072592286 0.729277777
## [91] 0.252451354 0.368767403 0.675968503 0.852165073 0.012702662
## [96] 1.669574565 1.868642026 0.915469178 0.359566884 -2.382471996
## [101] -1.268563372 -2.236629932 0.851566374 1.054256325 1.839758922
## [106] 1.089107790 2.282569970 0.501388394 2.310241027 1.924577740
## [111] 0.701628854 1.572163933 -0.811353906 1.491858309 0.024586469
## [116] 1.112575616 -0.571601276 0.377668248 -0.188704184 2.200551532
## [121] 0.137980542 0.950017590 -2.185559323 1.929037466 -0.806797583
## [126] -0.380090545 -1.144726418 0.324724378 -0.042361431 0.262863488
## [131] -2.286748789 -1.897223008 -0.380051401 0.816209921 2.132221753
## [136] 0.261903736 2.475658531 1.600373208 1.304524934 1.819312966
## [141] 1.362641279 -0.413939576 1.261613549 0.050621698 1.056754925
## [146] -0.986731556 -1.222474981 -0.376638636 1.363648342 2.452450632
## [151] 0.754805726 0.995028968 1.576494524 0.252185400 1.371836640
## [156] -0.192063167 1.124324950 1.373842963 0.161530570 -0.548495611
## [161] 0.122079347 -0.162928307 0.869627949 1.645302830 -0.133512418
## [166] 0.577205198 1.332200135 2.314074616 2.541704734 2.329764342
## [171] 0.760551363 1.260166357 -0.545679568 -0.526906104 0.141787476
## [176] -0.865162719 0.776325959 -0.800867390 -0.892046480 -0.198269829
## [181] 0.391609515 -0.323115699 1.411984335 1.539053907 1.726604278
## [186] 2.101460674 0.952816063 0.630389142 -0.434972955 0.284244921
## [191] -0.629639814 -0.326646935 0.071838968 0.089770764 2.391026520
## [196] 0.344158345 0.726821474 -0.426357782 0.673665279 0.476988095
## [201] 1.031459004 -0.824513344 -0.296015791 -1.270447231 0.741566980
## [206] -1.917525859 2.152745644 -0.934229115 0.986361102 -2.327510614
## [211] -0.836100979 -2.203163018 1.251274713 -0.483100429 0.405541344
## [216] -1.046801129 -0.118265714 -1.479388762 -0.972990734 -1.654317862
## [221] -0.369158859 1.188879940 0.758165612 0.004675097 -0.842823443
## [226] 0.789932300 -0.806046838 0.823141063 -1.638547001 -0.956585019
## [231] -3.511118437 0.160155860 -0.352221471 2.080835571 1.155445559
## [236] 0.007319216 -0.349601438 -1.389952187 -0.480352506 -0.374425649
## [241] 1.536508308 2.946040765 2.058586017 1.043034785 -0.514395852
## [246] 0.448597128 -0.107166162 -0.901895755 -0.643922756 -1.073635485
## [251] -1.848150055 -0.975969799 -1.018716345 -0.046728256 -0.457365043
## [256] 0.290596707 -0.817736130 -0.285379062 -1.197053507 -0.809829780
## [261] -0.284903059 1.427407937 2.072450303 2.578961105 0.990888560
## [266] 0.130744460 -1.055302300 -0.349721522 0.489784548 0.652814461
## [271] 0.864882450 -1.406581322 0.300966391 0.964790201 1.260471246
## [276] 1.745385800 0.068547273 3.790009529 0.959858473 2.336401436
## [281] -1.004590961 -1.634457108 -0.245794766 -1.413568572 -0.134170647
## [286] -0.816912736 0.107330937 -0.639551099 -0.525883926 -1.055692764
## [291] -0.158199423 -0.107872805 -0.227482501 0.463732156 -0.382036141
## [296] 0.326448357 0.727266022plot(y=yt_ma3,x=yt_3_ma)
Berdasarkan scatterplot tersebut, terlihat bahwa cenderung tidak terdapat hubungan antara \(Y_t\) dengan \(Y_{t-3}\).
cor(yt_ma3,yt_3_ma)## [1] -0.07842954Korelasi antara \(Y_t\) dengan \(Y_{t-3}\) hasil simulasi mendekati teori yang ada yaitu 0.
Proses AR
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, yaitu AR(2) dengan \(\phi1 = 0.5\) dan \(\phi2 =0.2\) sebanyak 300 observasi dan \(c=0\).
set.seed(035)Nilai-nilai selanjutnya dapat dicari melalui loop. Bentuk loop dapat dilihat dari rumus AR(2) yang hendak dibangkitkan:
\[ Y_t = e_t+0.5Y_{t-1}+0.2Y_{t-2} \]
n<-length(wn)
n## [1] 300ar <- c(1:n) #bangkitkan data ar 1-300
for (i in 3:n) {ar[i]<-wn[i]+0.5*ar[i-1]+0.2*ar[i-2]} #looping dari 3-300
ar## [1] 1.000000000 2.000000000 1.165955987 0.938001648 4.040029883
## [6] 1.814697744 2.125920865 0.863714547 2.048268102 1.545945122
## [11] 1.694180277 1.601619567 0.371264692 0.004847193 -1.831910620
## [16] -2.693657731 -0.848960327 -1.308949603 1.013994421 1.918290789
## [21] 1.846186931 2.263708856 2.502983054 2.980469932 1.670192649
## [26] 2.009644523 1.825961756 1.867757240 1.866634014 1.656435730
## [31] 2.373983431 2.630476742 3.014418988 1.853562001 0.855327159
## [36] 3.232502351 1.899226305 0.992684756 2.515564069 0.760964948
## [41] 0.497271510 0.449843159 -0.848775762 -1.147182073 0.345806314
## [46] 0.712429910 0.358807807 1.386418838 0.848700937 1.431531414
## [51] 1.028995087 1.889603063 0.178539914 -0.931313695 -0.157885545
## [56] -1.034479287 0.814903596 -0.967592210 -0.646423203 -1.031097913
## [61] 1.012508610 0.830899026 0.339712179 -0.487081048 -0.308557477
## [66] 2.628643225 1.440297143 0.370609149 1.011299471 -0.719521142
## [71] -2.204256171 -0.875847780 -0.182803871 1.454982068 0.951672894
## [76] -0.084193829 -0.105019556 0.793754820 0.359673693 0.702568720
## [81] 0.738714864 -0.791640056 -0.924459617 -2.188829775 -1.217664591
## [86] -0.113462388 -0.448610736 -0.026409873 -0.029562012 0.547515984
## [91] 0.249246315 0.269786088 0.857606481 1.044384022 0.078048040
## [96] 1.826767358 2.535487860 0.958648831 0.651871556 -1.326317137
## [101] -0.863024298 -1.694919884 0.428903320 0.949023345 1.101248120
## [106] 0.969020809 2.571319697 1.097073289 2.406062099 3.039153274
## [111] 1.249777975 2.135263468 0.595822542 1.964001893 1.063195625
## [116] 1.308738982 0.164452921 0.772060309 0.480515551 2.313731813
## [121] 0.586367881 0.821151162 -1.283837828 2.136891063 0.046060676
## [126] -1.192182925 -0.615198781 0.775571069 -0.289266798 -0.298014894
## [131] -2.037803766 -2.058253261 -0.326273649 0.385105991 1.209314391
## [136] -0.065160402 2.334467051 2.352604180 1.193248716 2.347395232
## [141] 2.532856694 0.105614061 1.800803650 1.453752222 1.186045563
## [146] -0.461898370 -0.737351648 0.267153454 1.504638104 2.213862321
## [151] 0.688300452 1.224907443 2.583149433 0.792736505 1.482541286
## [156] 0.926229339 1.531657431 2.000256908 0.585096878 -0.196859960
## [161] 0.929270492 0.431738377 0.722352528 1.960689999 0.193659119
## [166] 0.530735676 2.178286382 2.734629384 2.604429048 2.934718476
## [171] 1.833690933 2.174430319 0.732749784 0.160451582 1.057609111
## [176] -0.252115568 0.688494155 -0.260449209 -1.024681055 -0.017274182
## [181] 0.578466907 -0.682446878 1.098520350 2.003139840 1.517300407
## [186] 2.188106855 1.761182249 1.185836932 0.344930942 1.013392706
## [191] 0.064673172 -0.249483969 0.460436878 0.331894176 2.245085434
## [196] 0.647315990 0.523908198 0.389529606 1.123226928 0.788167901
## [201] 1.096520643 -0.398925074 -0.120921393 -0.691496363 0.676592571
## [206] -1.805388468 1.559609098 -0.243064899 0.045276283 -1.898365165
## [211] -0.940204322 -1.929473218 0.530422647 -0.629244983 -0.609492025
## [216] -1.011877515 -0.273017437 -1.611431370 -1.537182882 -1.710921804
## [221] -0.877920476 0.665770994 0.165714575 -0.650828041 -0.793406642
## [226] 0.983621301 -0.776432286 0.169179685 -1.187775278 -1.286208196
## [231] -3.431067795 -0.356762324 -0.332976293 0.660739661 0.740816965
## [236] -0.220089489 -0.308649864 -1.016000070 -0.514958510 -0.366419173
## [241] 1.179740082 2.819694366 1.903573130 1.073784155 0.425320607
## [246] 1.338093471 0.578045818 -0.821233938 -0.282141322 -0.557374202
## [251] -2.090162609 -1.279272994 -0.974270278 -0.749447380 -1.074398976
## [256] -0.190689179 -1.023627691 -0.847809291 -1.278498663 -1.179979012
## [261] -0.592204081 1.016616724 1.690740927 2.178076820 1.185797922
## [266] 0.578184447 -0.204085336 0.226205484 0.915228004 0.690925870
## [271] 0.812397280 -1.088595469 0.402006579 1.614332024 1.025324528
## [276] 1.547182892 0.679887615 4.131904876 1.948039880 2.148843690
## [281] 0.472395798 -0.780902527 0.631961209 -0.756820300 -0.576300383
## [286] -0.576741758 -0.046839531 -0.838699282 -0.888703862 -1.063831326
## [291] -0.411168254 -0.374574747 -0.693606692 0.185382740 -0.416455123
## [296] -0.060793215 0.791690363 1.139969883 2.051071409 1.731167996Selain menggunakan cara di atas, pembangkitan proses AR dapat
dilakukan dengan fungsi arima.sim() sebagai berikut.
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] 2.00818916 1.82485226 1.86691142 1.86598921 1.65594416 2.37360869
## [7] 2.63019106 3.01420120 1.85339597 0.85520058 3.23240586 1.89915274
## [13] 0.99262868 2.51552132 0.76093236 0.49724666 0.44982422 -0.84879020
## [19] -1.14719308 0.34579792 0.71242351 0.35880293 1.38641512 0.84869810
## [25] 1.43152925 1.02899344 1.88960181 0.17853896 -0.93131443 -0.15788610
## [31] -1.03447971 0.81490327 -0.96759246 -0.64642339 -1.03109806 1.01250850
## [37] 0.83089894 0.33971212 -0.48708110 -0.30855751 2.62864320 1.44029712
## [43] 0.37060913 1.01129946 -0.71952115 -2.20425618 -0.87584779 -0.18280388
## [49] 1.45498206 0.95167289 -0.08419383 -0.10501956 0.79375482 0.35967369
## [55] 0.70256872 0.73871486 -0.79164006 -0.92445962 -2.18882978 -1.21766459
## [61] -0.11346239 -0.44861074 -0.02640987 -0.02956201 0.54751598 0.24924632
## [67] 0.26978609 0.85760648 1.04438402 0.07804804 1.82676736 2.53548786
## [73] 0.95864883 0.65187156 -1.32631714 -0.86302430 -1.69491988 0.42890332
## [79] 0.94902334 1.10124812 0.96902081 2.57131970 1.09707329 2.40606210
## [85] 3.03915327 1.24977797 2.13526347 0.59582254 1.96400189 1.06319563
## [91] 1.30873898 0.16445292 0.77206031 0.48051555 2.31373181 0.58636788
## [97] 0.82115116 -1.28383783 2.13689106 0.04606068 -1.19218293 -0.61519878
## [103] 0.77557107 -0.28926680 -0.29801489 -2.03780377 -2.05825326 -0.32627365
## [109] 0.38510599 1.20931439 -0.06516040 2.33446705 2.35260418 1.19324872
## [115] 2.34739523 2.53285669 0.10561406 1.80080365 1.45375222 1.18604556
## [121] -0.46189837 -0.73735165 0.26715345 1.50463810 2.21386232 0.68830045
## [127] 1.22490744 2.58314943 0.79273651 1.48254129 0.92622934 1.53165743
## [133] 2.00025691 0.58509688 -0.19685996 0.92927049 0.43173838 0.72235253
## [139] 1.96069000 0.19365912 0.53073568 2.17828638 2.73462938 2.60442905
## [145] 2.93471848 1.83369093 2.17443032 0.73274978 0.16045158 1.05760911
## [151] -0.25211557 0.68849415 -0.26044921 -1.02468105 -0.01727418 0.57846691
## [157] -0.68244688 1.09852035 2.00313984 1.51730041 2.18810685 1.76118225
## [163] 1.18583693 0.34493094 1.01339271 0.06467317 -0.24948397 0.46043688
## [169] 0.33189418 2.24508543 0.64731599 0.52390820 0.38952961 1.12322693
## [175] 0.78816790 1.09652064 -0.39892507 -0.12092139 -0.69149636 0.67659257
## [181] -1.80538847 1.55960910 -0.24306490 0.04527628 -1.89836517 -0.94020432
## [187] -1.92947322 0.53042265 -0.62924498 -0.60949203 -1.01187752 -0.27301744
## [193] -1.61143137 -1.53718288 -1.71092180 -0.87792048 0.66577099 0.16571458
## [199] -0.65082804 -0.79340664 0.98362130 -0.77643229 0.16917969 -1.18777528
## [205] -1.28620820 -3.43106780 -0.35676232 -0.33297629 0.66073966 0.74081696
## [211] -0.22008949 -0.30864986 -1.01600007 -0.51495851 -0.36641917 1.17974008
## [217] 2.81969437 1.90357313 1.07378416 0.42532061 1.33809347 0.57804582
## [223] -0.82123394 -0.28214132 -0.55737420 -2.09016261 -1.27927299 -0.97427028
## [229] -0.74944738 -1.07439898 -0.19068918 -1.02362769 -0.84780929 -1.27849866
## [235] -1.17997901 -0.59220408 1.01661672 1.69074093 2.17807682 1.18579792
## [241] 0.57818445 -0.20408534 0.22620548 0.91522800 0.69092587 0.81239728
## [247] -1.08859547 0.40200658 1.61433202 1.02532453 1.54718289 0.67988762
## [253] 4.13190488 1.94803988 2.14884369 0.47239580 -0.78090253 0.63196121
## [259] -0.75682030 -0.57630038 -0.57674176 -0.04683953 -0.83869928 -0.88870386
## [265] -1.06383133 -0.41116825 -0.37457475 -0.69360669 0.18538274 -0.41645512
## [271] -0.06079322 0.79169036 1.13996988 2.05107141 1.73116800 1.19740642
## [277] 0.70502814 -0.22680948 -1.79624497 0.32076174 0.11096382 -0.60337812
## [283] 0.62364119 0.62389796 -0.86150603 -1.47019425 0.93723550 0.11466766
## [289] -0.60234407 -0.36651407 1.54541404 0.03869346 2.11701028 2.27701311
## [295] 1.92245855 3.11631323 3.05576558 0.96483684 1.16321356 -0.05515815Karakteristik AR(2)
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 kedua, sejalan dengan teori yang ada
Plot EACF
TSA::eacf(ar)## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x x 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 o o o o o o o o o o o o o
## 3 x o o o o o o o o o o o o o
## 4 x x o o o o o o o o o o o o
## 5 x x o o o o o o o o o o o o
## 6 x x o x x o o o o o o o o o
## 7 x x o o x o 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(1)
Scatterplot Antar Lag
Korelasi antara \(Y_t\) dengan \(Y_{t-1}\)
#Yt
yt_ar <- ar[-1]
yt_ar## [1] 2.000000000 1.165955987 0.938001648 4.040029883 1.814697744
## [6] 2.125920865 0.863714547 2.048268102 1.545945122 1.694180277
## [11] 1.601619567 0.371264692 0.004847193 -1.831910620 -2.693657731
## [16] -0.848960327 -1.308949603 1.013994421 1.918290789 1.846186931
## [21] 2.263708856 2.502983054 2.980469932 1.670192649 2.009644523
## [26] 1.825961756 1.867757240 1.866634014 1.656435730 2.373983431
## [31] 2.630476742 3.014418988 1.853562001 0.855327159 3.232502351
## [36] 1.899226305 0.992684756 2.515564069 0.760964948 0.497271510
## [41] 0.449843159 -0.848775762 -1.147182073 0.345806314 0.712429910
## [46] 0.358807807 1.386418838 0.848700937 1.431531414 1.028995087
## [51] 1.889603063 0.178539914 -0.931313695 -0.157885545 -1.034479287
## [56] 0.814903596 -0.967592210 -0.646423203 -1.031097913 1.012508610
## [61] 0.830899026 0.339712179 -0.487081048 -0.308557477 2.628643225
## [66] 1.440297143 0.370609149 1.011299471 -0.719521142 -2.204256171
## [71] -0.875847780 -0.182803871 1.454982068 0.951672894 -0.084193829
## [76] -0.105019556 0.793754820 0.359673693 0.702568720 0.738714864
## [81] -0.791640056 -0.924459617 -2.188829775 -1.217664591 -0.113462388
## [86] -0.448610736 -0.026409873 -0.029562012 0.547515984 0.249246315
## [91] 0.269786088 0.857606481 1.044384022 0.078048040 1.826767358
## [96] 2.535487860 0.958648831 0.651871556 -1.326317137 -0.863024298
## [101] -1.694919884 0.428903320 0.949023345 1.101248120 0.969020809
## [106] 2.571319697 1.097073289 2.406062099 3.039153274 1.249777975
## [111] 2.135263468 0.595822542 1.964001893 1.063195625 1.308738982
## [116] 0.164452921 0.772060309 0.480515551 2.313731813 0.586367881
## [121] 0.821151162 -1.283837828 2.136891063 0.046060676 -1.192182925
## [126] -0.615198781 0.775571069 -0.289266798 -0.298014894 -2.037803766
## [131] -2.058253261 -0.326273649 0.385105991 1.209314391 -0.065160402
## [136] 2.334467051 2.352604180 1.193248716 2.347395232 2.532856694
## [141] 0.105614061 1.800803650 1.453752222 1.186045563 -0.461898370
## [146] -0.737351648 0.267153454 1.504638104 2.213862321 0.688300452
## [151] 1.224907443 2.583149433 0.792736505 1.482541286 0.926229339
## [156] 1.531657431 2.000256908 0.585096878 -0.196859960 0.929270492
## [161] 0.431738377 0.722352528 1.960689999 0.193659119 0.530735676
## [166] 2.178286382 2.734629384 2.604429048 2.934718476 1.833690933
## [171] 2.174430319 0.732749784 0.160451582 1.057609111 -0.252115568
## [176] 0.688494155 -0.260449209 -1.024681055 -0.017274182 0.578466907
## [181] -0.682446878 1.098520350 2.003139840 1.517300407 2.188106855
## [186] 1.761182249 1.185836932 0.344930942 1.013392706 0.064673172
## [191] -0.249483969 0.460436878 0.331894176 2.245085434 0.647315990
## [196] 0.523908198 0.389529606 1.123226928 0.788167901 1.096520643
## [201] -0.398925074 -0.120921393 -0.691496363 0.676592571 -1.805388468
## [206] 1.559609098 -0.243064899 0.045276283 -1.898365165 -0.940204322
## [211] -1.929473218 0.530422647 -0.629244983 -0.609492025 -1.011877515
## [216] -0.273017437 -1.611431370 -1.537182882 -1.710921804 -0.877920476
## [221] 0.665770994 0.165714575 -0.650828041 -0.793406642 0.983621301
## [226] -0.776432286 0.169179685 -1.187775278 -1.286208196 -3.431067795
## [231] -0.356762324 -0.332976293 0.660739661 0.740816965 -0.220089489
## [236] -0.308649864 -1.016000070 -0.514958510 -0.366419173 1.179740082
## [241] 2.819694366 1.903573130 1.073784155 0.425320607 1.338093471
## [246] 0.578045818 -0.821233938 -0.282141322 -0.557374202 -2.090162609
## [251] -1.279272994 -0.974270278 -0.749447380 -1.074398976 -0.190689179
## [256] -1.023627691 -0.847809291 -1.278498663 -1.179979012 -0.592204081
## [261] 1.016616724 1.690740927 2.178076820 1.185797922 0.578184447
## [266] -0.204085336 0.226205484 0.915228004 0.690925870 0.812397280
## [271] -1.088595469 0.402006579 1.614332024 1.025324528 1.547182892
## [276] 0.679887615 4.131904876 1.948039880 2.148843690 0.472395798
## [281] -0.780902527 0.631961209 -0.756820300 -0.576300383 -0.576741758
## [286] -0.046839531 -0.838699282 -0.888703862 -1.063831326 -0.411168254
## [291] -0.374574747 -0.693606692 0.185382740 -0.416455123 -0.060793215
## [296] 0.791690363 1.139969883 2.051071409 1.731167996#Yt-1
yt_1_ar <- ar[-300]
yt_1_ar## [1] 1.000000000 2.000000000 1.165955987 0.938001648 4.040029883
## [6] 1.814697744 2.125920865 0.863714547 2.048268102 1.545945122
## [11] 1.694180277 1.601619567 0.371264692 0.004847193 -1.831910620
## [16] -2.693657731 -0.848960327 -1.308949603 1.013994421 1.918290789
## [21] 1.846186931 2.263708856 2.502983054 2.980469932 1.670192649
## [26] 2.009644523 1.825961756 1.867757240 1.866634014 1.656435730
## [31] 2.373983431 2.630476742 3.014418988 1.853562001 0.855327159
## [36] 3.232502351 1.899226305 0.992684756 2.515564069 0.760964948
## [41] 0.497271510 0.449843159 -0.848775762 -1.147182073 0.345806314
## [46] 0.712429910 0.358807807 1.386418838 0.848700937 1.431531414
## [51] 1.028995087 1.889603063 0.178539914 -0.931313695 -0.157885545
## [56] -1.034479287 0.814903596 -0.967592210 -0.646423203 -1.031097913
## [61] 1.012508610 0.830899026 0.339712179 -0.487081048 -0.308557477
## [66] 2.628643225 1.440297143 0.370609149 1.011299471 -0.719521142
## [71] -2.204256171 -0.875847780 -0.182803871 1.454982068 0.951672894
## [76] -0.084193829 -0.105019556 0.793754820 0.359673693 0.702568720
## [81] 0.738714864 -0.791640056 -0.924459617 -2.188829775 -1.217664591
## [86] -0.113462388 -0.448610736 -0.026409873 -0.029562012 0.547515984
## [91] 0.249246315 0.269786088 0.857606481 1.044384022 0.078048040
## [96] 1.826767358 2.535487860 0.958648831 0.651871556 -1.326317137
## [101] -0.863024298 -1.694919884 0.428903320 0.949023345 1.101248120
## [106] 0.969020809 2.571319697 1.097073289 2.406062099 3.039153274
## [111] 1.249777975 2.135263468 0.595822542 1.964001893 1.063195625
## [116] 1.308738982 0.164452921 0.772060309 0.480515551 2.313731813
## [121] 0.586367881 0.821151162 -1.283837828 2.136891063 0.046060676
## [126] -1.192182925 -0.615198781 0.775571069 -0.289266798 -0.298014894
## [131] -2.037803766 -2.058253261 -0.326273649 0.385105991 1.209314391
## [136] -0.065160402 2.334467051 2.352604180 1.193248716 2.347395232
## [141] 2.532856694 0.105614061 1.800803650 1.453752222 1.186045563
## [146] -0.461898370 -0.737351648 0.267153454 1.504638104 2.213862321
## [151] 0.688300452 1.224907443 2.583149433 0.792736505 1.482541286
## [156] 0.926229339 1.531657431 2.000256908 0.585096878 -0.196859960
## [161] 0.929270492 0.431738377 0.722352528 1.960689999 0.193659119
## [166] 0.530735676 2.178286382 2.734629384 2.604429048 2.934718476
## [171] 1.833690933 2.174430319 0.732749784 0.160451582 1.057609111
## [176] -0.252115568 0.688494155 -0.260449209 -1.024681055 -0.017274182
## [181] 0.578466907 -0.682446878 1.098520350 2.003139840 1.517300407
## [186] 2.188106855 1.761182249 1.185836932 0.344930942 1.013392706
## [191] 0.064673172 -0.249483969 0.460436878 0.331894176 2.245085434
## [196] 0.647315990 0.523908198 0.389529606 1.123226928 0.788167901
## [201] 1.096520643 -0.398925074 -0.120921393 -0.691496363 0.676592571
## [206] -1.805388468 1.559609098 -0.243064899 0.045276283 -1.898365165
## [211] -0.940204322 -1.929473218 0.530422647 -0.629244983 -0.609492025
## [216] -1.011877515 -0.273017437 -1.611431370 -1.537182882 -1.710921804
## [221] -0.877920476 0.665770994 0.165714575 -0.650828041 -0.793406642
## [226] 0.983621301 -0.776432286 0.169179685 -1.187775278 -1.286208196
## [231] -3.431067795 -0.356762324 -0.332976293 0.660739661 0.740816965
## [236] -0.220089489 -0.308649864 -1.016000070 -0.514958510 -0.366419173
## [241] 1.179740082 2.819694366 1.903573130 1.073784155 0.425320607
## [246] 1.338093471 0.578045818 -0.821233938 -0.282141322 -0.557374202
## [251] -2.090162609 -1.279272994 -0.974270278 -0.749447380 -1.074398976
## [256] -0.190689179 -1.023627691 -0.847809291 -1.278498663 -1.179979012
## [261] -0.592204081 1.016616724 1.690740927 2.178076820 1.185797922
## [266] 0.578184447 -0.204085336 0.226205484 0.915228004 0.690925870
## [271] 0.812397280 -1.088595469 0.402006579 1.614332024 1.025324528
## [276] 1.547182892 0.679887615 4.131904876 1.948039880 2.148843690
## [281] 0.472395798 -0.780902527 0.631961209 -0.756820300 -0.576300383
## [286] -0.576741758 -0.046839531 -0.838699282 -0.888703862 -1.063831326
## [291] -0.411168254 -0.374574747 -0.693606692 0.185382740 -0.416455123
## [296] -0.060793215 0.791690363 1.139969883 2.051071409plot(y=yt_ar,x=yt_1_ar)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-1}\). Hal ini sesuai dengan teori yang ada
cor(yt_ar,yt_1_ar)## [1] 0.5926736Korelasi antara \(Y_t\) dengan \(Y_{t-1}\) dari hasil simulasi mendekati perhitungan teoritis yaitu
\[ \rho_1=\frac{\phi_1}{1-\phi_2}=\frac{0.5}{1-0.2}=0.626 \]
Korelasi antara \(Y_t\) dengan \(Y_{t-2}\)
#Yt
yt_ar2 <- ar[-c(1,2)]
yt_ar2## [1] 1.165955987 0.938001648 4.040029883 1.814697744 2.125920865
## [6] 0.863714547 2.048268102 1.545945122 1.694180277 1.601619567
## [11] 0.371264692 0.004847193 -1.831910620 -2.693657731 -0.848960327
## [16] -1.308949603 1.013994421 1.918290789 1.846186931 2.263708856
## [21] 2.502983054 2.980469932 1.670192649 2.009644523 1.825961756
## [26] 1.867757240 1.866634014 1.656435730 2.373983431 2.630476742
## [31] 3.014418988 1.853562001 0.855327159 3.232502351 1.899226305
## [36] 0.992684756 2.515564069 0.760964948 0.497271510 0.449843159
## [41] -0.848775762 -1.147182073 0.345806314 0.712429910 0.358807807
## [46] 1.386418838 0.848700937 1.431531414 1.028995087 1.889603063
## [51] 0.178539914 -0.931313695 -0.157885545 -1.034479287 0.814903596
## [56] -0.967592210 -0.646423203 -1.031097913 1.012508610 0.830899026
## [61] 0.339712179 -0.487081048 -0.308557477 2.628643225 1.440297143
## [66] 0.370609149 1.011299471 -0.719521142 -2.204256171 -0.875847780
## [71] -0.182803871 1.454982068 0.951672894 -0.084193829 -0.105019556
## [76] 0.793754820 0.359673693 0.702568720 0.738714864 -0.791640056
## [81] -0.924459617 -2.188829775 -1.217664591 -0.113462388 -0.448610736
## [86] -0.026409873 -0.029562012 0.547515984 0.249246315 0.269786088
## [91] 0.857606481 1.044384022 0.078048040 1.826767358 2.535487860
## [96] 0.958648831 0.651871556 -1.326317137 -0.863024298 -1.694919884
## [101] 0.428903320 0.949023345 1.101248120 0.969020809 2.571319697
## [106] 1.097073289 2.406062099 3.039153274 1.249777975 2.135263468
## [111] 0.595822542 1.964001893 1.063195625 1.308738982 0.164452921
## [116] 0.772060309 0.480515551 2.313731813 0.586367881 0.821151162
## [121] -1.283837828 2.136891063 0.046060676 -1.192182925 -0.615198781
## [126] 0.775571069 -0.289266798 -0.298014894 -2.037803766 -2.058253261
## [131] -0.326273649 0.385105991 1.209314391 -0.065160402 2.334467051
## [136] 2.352604180 1.193248716 2.347395232 2.532856694 0.105614061
## [141] 1.800803650 1.453752222 1.186045563 -0.461898370 -0.737351648
## [146] 0.267153454 1.504638104 2.213862321 0.688300452 1.224907443
## [151] 2.583149433 0.792736505 1.482541286 0.926229339 1.531657431
## [156] 2.000256908 0.585096878 -0.196859960 0.929270492 0.431738377
## [161] 0.722352528 1.960689999 0.193659119 0.530735676 2.178286382
## [166] 2.734629384 2.604429048 2.934718476 1.833690933 2.174430319
## [171] 0.732749784 0.160451582 1.057609111 -0.252115568 0.688494155
## [176] -0.260449209 -1.024681055 -0.017274182 0.578466907 -0.682446878
## [181] 1.098520350 2.003139840 1.517300407 2.188106855 1.761182249
## [186] 1.185836932 0.344930942 1.013392706 0.064673172 -0.249483969
## [191] 0.460436878 0.331894176 2.245085434 0.647315990 0.523908198
## [196] 0.389529606 1.123226928 0.788167901 1.096520643 -0.398925074
## [201] -0.120921393 -0.691496363 0.676592571 -1.805388468 1.559609098
## [206] -0.243064899 0.045276283 -1.898365165 -0.940204322 -1.929473218
## [211] 0.530422647 -0.629244983 -0.609492025 -1.011877515 -0.273017437
## [216] -1.611431370 -1.537182882 -1.710921804 -0.877920476 0.665770994
## [221] 0.165714575 -0.650828041 -0.793406642 0.983621301 -0.776432286
## [226] 0.169179685 -1.187775278 -1.286208196 -3.431067795 -0.356762324
## [231] -0.332976293 0.660739661 0.740816965 -0.220089489 -0.308649864
## [236] -1.016000070 -0.514958510 -0.366419173 1.179740082 2.819694366
## [241] 1.903573130 1.073784155 0.425320607 1.338093471 0.578045818
## [246] -0.821233938 -0.282141322 -0.557374202 -2.090162609 -1.279272994
## [251] -0.974270278 -0.749447380 -1.074398976 -0.190689179 -1.023627691
## [256] -0.847809291 -1.278498663 -1.179979012 -0.592204081 1.016616724
## [261] 1.690740927 2.178076820 1.185797922 0.578184447 -0.204085336
## [266] 0.226205484 0.915228004 0.690925870 0.812397280 -1.088595469
## [271] 0.402006579 1.614332024 1.025324528 1.547182892 0.679887615
## [276] 4.131904876 1.948039880 2.148843690 0.472395798 -0.780902527
## [281] 0.631961209 -0.756820300 -0.576300383 -0.576741758 -0.046839531
## [286] -0.838699282 -0.888703862 -1.063831326 -0.411168254 -0.374574747
## [291] -0.693606692 0.185382740 -0.416455123 -0.060793215 0.791690363
## [296] 1.139969883 2.051071409 1.731167996#Yt-2
yt_2_ar <- ar[-c(299,300)]
yt_2_ar## [1] 1.000000000 2.000000000 1.165955987 0.938001648 4.040029883
## [6] 1.814697744 2.125920865 0.863714547 2.048268102 1.545945122
## [11] 1.694180277 1.601619567 0.371264692 0.004847193 -1.831910620
## [16] -2.693657731 -0.848960327 -1.308949603 1.013994421 1.918290789
## [21] 1.846186931 2.263708856 2.502983054 2.980469932 1.670192649
## [26] 2.009644523 1.825961756 1.867757240 1.866634014 1.656435730
## [31] 2.373983431 2.630476742 3.014418988 1.853562001 0.855327159
## [36] 3.232502351 1.899226305 0.992684756 2.515564069 0.760964948
## [41] 0.497271510 0.449843159 -0.848775762 -1.147182073 0.345806314
## [46] 0.712429910 0.358807807 1.386418838 0.848700937 1.431531414
## [51] 1.028995087 1.889603063 0.178539914 -0.931313695 -0.157885545
## [56] -1.034479287 0.814903596 -0.967592210 -0.646423203 -1.031097913
## [61] 1.012508610 0.830899026 0.339712179 -0.487081048 -0.308557477
## [66] 2.628643225 1.440297143 0.370609149 1.011299471 -0.719521142
## [71] -2.204256171 -0.875847780 -0.182803871 1.454982068 0.951672894
## [76] -0.084193829 -0.105019556 0.793754820 0.359673693 0.702568720
## [81] 0.738714864 -0.791640056 -0.924459617 -2.188829775 -1.217664591
## [86] -0.113462388 -0.448610736 -0.026409873 -0.029562012 0.547515984
## [91] 0.249246315 0.269786088 0.857606481 1.044384022 0.078048040
## [96] 1.826767358 2.535487860 0.958648831 0.651871556 -1.326317137
## [101] -0.863024298 -1.694919884 0.428903320 0.949023345 1.101248120
## [106] 0.969020809 2.571319697 1.097073289 2.406062099 3.039153274
## [111] 1.249777975 2.135263468 0.595822542 1.964001893 1.063195625
## [116] 1.308738982 0.164452921 0.772060309 0.480515551 2.313731813
## [121] 0.586367881 0.821151162 -1.283837828 2.136891063 0.046060676
## [126] -1.192182925 -0.615198781 0.775571069 -0.289266798 -0.298014894
## [131] -2.037803766 -2.058253261 -0.326273649 0.385105991 1.209314391
## [136] -0.065160402 2.334467051 2.352604180 1.193248716 2.347395232
## [141] 2.532856694 0.105614061 1.800803650 1.453752222 1.186045563
## [146] -0.461898370 -0.737351648 0.267153454 1.504638104 2.213862321
## [151] 0.688300452 1.224907443 2.583149433 0.792736505 1.482541286
## [156] 0.926229339 1.531657431 2.000256908 0.585096878 -0.196859960
## [161] 0.929270492 0.431738377 0.722352528 1.960689999 0.193659119
## [166] 0.530735676 2.178286382 2.734629384 2.604429048 2.934718476
## [171] 1.833690933 2.174430319 0.732749784 0.160451582 1.057609111
## [176] -0.252115568 0.688494155 -0.260449209 -1.024681055 -0.017274182
## [181] 0.578466907 -0.682446878 1.098520350 2.003139840 1.517300407
## [186] 2.188106855 1.761182249 1.185836932 0.344930942 1.013392706
## [191] 0.064673172 -0.249483969 0.460436878 0.331894176 2.245085434
## [196] 0.647315990 0.523908198 0.389529606 1.123226928 0.788167901
## [201] 1.096520643 -0.398925074 -0.120921393 -0.691496363 0.676592571
## [206] -1.805388468 1.559609098 -0.243064899 0.045276283 -1.898365165
## [211] -0.940204322 -1.929473218 0.530422647 -0.629244983 -0.609492025
## [216] -1.011877515 -0.273017437 -1.611431370 -1.537182882 -1.710921804
## [221] -0.877920476 0.665770994 0.165714575 -0.650828041 -0.793406642
## [226] 0.983621301 -0.776432286 0.169179685 -1.187775278 -1.286208196
## [231] -3.431067795 -0.356762324 -0.332976293 0.660739661 0.740816965
## [236] -0.220089489 -0.308649864 -1.016000070 -0.514958510 -0.366419173
## [241] 1.179740082 2.819694366 1.903573130 1.073784155 0.425320607
## [246] 1.338093471 0.578045818 -0.821233938 -0.282141322 -0.557374202
## [251] -2.090162609 -1.279272994 -0.974270278 -0.749447380 -1.074398976
## [256] -0.190689179 -1.023627691 -0.847809291 -1.278498663 -1.179979012
## [261] -0.592204081 1.016616724 1.690740927 2.178076820 1.185797922
## [266] 0.578184447 -0.204085336 0.226205484 0.915228004 0.690925870
## [271] 0.812397280 -1.088595469 0.402006579 1.614332024 1.025324528
## [276] 1.547182892 0.679887615 4.131904876 1.948039880 2.148843690
## [281] 0.472395798 -0.780902527 0.631961209 -0.756820300 -0.576300383
## [286] -0.576741758 -0.046839531 -0.838699282 -0.888703862 -1.063831326
## [291] -0.411168254 -0.374574747 -0.693606692 0.185382740 -0.416455123
## [296] -0.060793215 0.791690363 1.139969883plot(y=yt_ar2,x=yt_2_ar)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-2}\). Hal ini sesuai dengan teori yang ada
cor(yt_ar2,yt_2_ar)## [1] 0.4820466Korelasi antara \(Y_t\) dengan \(Y_{t-2}\) dari hasil simulasi mendekati perhitungan teoritis yaitu
\[ \rho_1=\frac{\phi_1^2}{1-\phi_2}+\phi_2=\frac{0.5^2}{1-0.2}+0.2=0.5125 \]
Korelasi antara \(Y_t\) dengan \(Y_{t-3}\)
#Yt
yt_ar3 <- ar[-c(1,2,3)]
yt_ar3## [1] 0.938001648 4.040029883 1.814697744 2.125920865 0.863714547
## [6] 2.048268102 1.545945122 1.694180277 1.601619567 0.371264692
## [11] 0.004847193 -1.831910620 -2.693657731 -0.848960327 -1.308949603
## [16] 1.013994421 1.918290789 1.846186931 2.263708856 2.502983054
## [21] 2.980469932 1.670192649 2.009644523 1.825961756 1.867757240
## [26] 1.866634014 1.656435730 2.373983431 2.630476742 3.014418988
## [31] 1.853562001 0.855327159 3.232502351 1.899226305 0.992684756
## [36] 2.515564069 0.760964948 0.497271510 0.449843159 -0.848775762
## [41] -1.147182073 0.345806314 0.712429910 0.358807807 1.386418838
## [46] 0.848700937 1.431531414 1.028995087 1.889603063 0.178539914
## [51] -0.931313695 -0.157885545 -1.034479287 0.814903596 -0.967592210
## [56] -0.646423203 -1.031097913 1.012508610 0.830899026 0.339712179
## [61] -0.487081048 -0.308557477 2.628643225 1.440297143 0.370609149
## [66] 1.011299471 -0.719521142 -2.204256171 -0.875847780 -0.182803871
## [71] 1.454982068 0.951672894 -0.084193829 -0.105019556 0.793754820
## [76] 0.359673693 0.702568720 0.738714864 -0.791640056 -0.924459617
## [81] -2.188829775 -1.217664591 -0.113462388 -0.448610736 -0.026409873
## [86] -0.029562012 0.547515984 0.249246315 0.269786088 0.857606481
## [91] 1.044384022 0.078048040 1.826767358 2.535487860 0.958648831
## [96] 0.651871556 -1.326317137 -0.863024298 -1.694919884 0.428903320
## [101] 0.949023345 1.101248120 0.969020809 2.571319697 1.097073289
## [106] 2.406062099 3.039153274 1.249777975 2.135263468 0.595822542
## [111] 1.964001893 1.063195625 1.308738982 0.164452921 0.772060309
## [116] 0.480515551 2.313731813 0.586367881 0.821151162 -1.283837828
## [121] 2.136891063 0.046060676 -1.192182925 -0.615198781 0.775571069
## [126] -0.289266798 -0.298014894 -2.037803766 -2.058253261 -0.326273649
## [131] 0.385105991 1.209314391 -0.065160402 2.334467051 2.352604180
## [136] 1.193248716 2.347395232 2.532856694 0.105614061 1.800803650
## [141] 1.453752222 1.186045563 -0.461898370 -0.737351648 0.267153454
## [146] 1.504638104 2.213862321 0.688300452 1.224907443 2.583149433
## [151] 0.792736505 1.482541286 0.926229339 1.531657431 2.000256908
## [156] 0.585096878 -0.196859960 0.929270492 0.431738377 0.722352528
## [161] 1.960689999 0.193659119 0.530735676 2.178286382 2.734629384
## [166] 2.604429048 2.934718476 1.833690933 2.174430319 0.732749784
## [171] 0.160451582 1.057609111 -0.252115568 0.688494155 -0.260449209
## [176] -1.024681055 -0.017274182 0.578466907 -0.682446878 1.098520350
## [181] 2.003139840 1.517300407 2.188106855 1.761182249 1.185836932
## [186] 0.344930942 1.013392706 0.064673172 -0.249483969 0.460436878
## [191] 0.331894176 2.245085434 0.647315990 0.523908198 0.389529606
## [196] 1.123226928 0.788167901 1.096520643 -0.398925074 -0.120921393
## [201] -0.691496363 0.676592571 -1.805388468 1.559609098 -0.243064899
## [206] 0.045276283 -1.898365165 -0.940204322 -1.929473218 0.530422647
## [211] -0.629244983 -0.609492025 -1.011877515 -0.273017437 -1.611431370
## [216] -1.537182882 -1.710921804 -0.877920476 0.665770994 0.165714575
## [221] -0.650828041 -0.793406642 0.983621301 -0.776432286 0.169179685
## [226] -1.187775278 -1.286208196 -3.431067795 -0.356762324 -0.332976293
## [231] 0.660739661 0.740816965 -0.220089489 -0.308649864 -1.016000070
## [236] -0.514958510 -0.366419173 1.179740082 2.819694366 1.903573130
## [241] 1.073784155 0.425320607 1.338093471 0.578045818 -0.821233938
## [246] -0.282141322 -0.557374202 -2.090162609 -1.279272994 -0.974270278
## [251] -0.749447380 -1.074398976 -0.190689179 -1.023627691 -0.847809291
## [256] -1.278498663 -1.179979012 -0.592204081 1.016616724 1.690740927
## [261] 2.178076820 1.185797922 0.578184447 -0.204085336 0.226205484
## [266] 0.915228004 0.690925870 0.812397280 -1.088595469 0.402006579
## [271] 1.614332024 1.025324528 1.547182892 0.679887615 4.131904876
## [276] 1.948039880 2.148843690 0.472395798 -0.780902527 0.631961209
## [281] -0.756820300 -0.576300383 -0.576741758 -0.046839531 -0.838699282
## [286] -0.888703862 -1.063831326 -0.411168254 -0.374574747 -0.693606692
## [291] 0.185382740 -0.416455123 -0.060793215 0.791690363 1.139969883
## [296] 2.051071409 1.731167996#Yt-3
yt_3_ar <- ar[-c(298,299,300)]
yt_3_ar## [1] 1.000000000 2.000000000 1.165955987 0.938001648 4.040029883
## [6] 1.814697744 2.125920865 0.863714547 2.048268102 1.545945122
## [11] 1.694180277 1.601619567 0.371264692 0.004847193 -1.831910620
## [16] -2.693657731 -0.848960327 -1.308949603 1.013994421 1.918290789
## [21] 1.846186931 2.263708856 2.502983054 2.980469932 1.670192649
## [26] 2.009644523 1.825961756 1.867757240 1.866634014 1.656435730
## [31] 2.373983431 2.630476742 3.014418988 1.853562001 0.855327159
## [36] 3.232502351 1.899226305 0.992684756 2.515564069 0.760964948
## [41] 0.497271510 0.449843159 -0.848775762 -1.147182073 0.345806314
## [46] 0.712429910 0.358807807 1.386418838 0.848700937 1.431531414
## [51] 1.028995087 1.889603063 0.178539914 -0.931313695 -0.157885545
## [56] -1.034479287 0.814903596 -0.967592210 -0.646423203 -1.031097913
## [61] 1.012508610 0.830899026 0.339712179 -0.487081048 -0.308557477
## [66] 2.628643225 1.440297143 0.370609149 1.011299471 -0.719521142
## [71] -2.204256171 -0.875847780 -0.182803871 1.454982068 0.951672894
## [76] -0.084193829 -0.105019556 0.793754820 0.359673693 0.702568720
## [81] 0.738714864 -0.791640056 -0.924459617 -2.188829775 -1.217664591
## [86] -0.113462388 -0.448610736 -0.026409873 -0.029562012 0.547515984
## [91] 0.249246315 0.269786088 0.857606481 1.044384022 0.078048040
## [96] 1.826767358 2.535487860 0.958648831 0.651871556 -1.326317137
## [101] -0.863024298 -1.694919884 0.428903320 0.949023345 1.101248120
## [106] 0.969020809 2.571319697 1.097073289 2.406062099 3.039153274
## [111] 1.249777975 2.135263468 0.595822542 1.964001893 1.063195625
## [116] 1.308738982 0.164452921 0.772060309 0.480515551 2.313731813
## [121] 0.586367881 0.821151162 -1.283837828 2.136891063 0.046060676
## [126] -1.192182925 -0.615198781 0.775571069 -0.289266798 -0.298014894
## [131] -2.037803766 -2.058253261 -0.326273649 0.385105991 1.209314391
## [136] -0.065160402 2.334467051 2.352604180 1.193248716 2.347395232
## [141] 2.532856694 0.105614061 1.800803650 1.453752222 1.186045563
## [146] -0.461898370 -0.737351648 0.267153454 1.504638104 2.213862321
## [151] 0.688300452 1.224907443 2.583149433 0.792736505 1.482541286
## [156] 0.926229339 1.531657431 2.000256908 0.585096878 -0.196859960
## [161] 0.929270492 0.431738377 0.722352528 1.960689999 0.193659119
## [166] 0.530735676 2.178286382 2.734629384 2.604429048 2.934718476
## [171] 1.833690933 2.174430319 0.732749784 0.160451582 1.057609111
## [176] -0.252115568 0.688494155 -0.260449209 -1.024681055 -0.017274182
## [181] 0.578466907 -0.682446878 1.098520350 2.003139840 1.517300407
## [186] 2.188106855 1.761182249 1.185836932 0.344930942 1.013392706
## [191] 0.064673172 -0.249483969 0.460436878 0.331894176 2.245085434
## [196] 0.647315990 0.523908198 0.389529606 1.123226928 0.788167901
## [201] 1.096520643 -0.398925074 -0.120921393 -0.691496363 0.676592571
## [206] -1.805388468 1.559609098 -0.243064899 0.045276283 -1.898365165
## [211] -0.940204322 -1.929473218 0.530422647 -0.629244983 -0.609492025
## [216] -1.011877515 -0.273017437 -1.611431370 -1.537182882 -1.710921804
## [221] -0.877920476 0.665770994 0.165714575 -0.650828041 -0.793406642
## [226] 0.983621301 -0.776432286 0.169179685 -1.187775278 -1.286208196
## [231] -3.431067795 -0.356762324 -0.332976293 0.660739661 0.740816965
## [236] -0.220089489 -0.308649864 -1.016000070 -0.514958510 -0.366419173
## [241] 1.179740082 2.819694366 1.903573130 1.073784155 0.425320607
## [246] 1.338093471 0.578045818 -0.821233938 -0.282141322 -0.557374202
## [251] -2.090162609 -1.279272994 -0.974270278 -0.749447380 -1.074398976
## [256] -0.190689179 -1.023627691 -0.847809291 -1.278498663 -1.179979012
## [261] -0.592204081 1.016616724 1.690740927 2.178076820 1.185797922
## [266] 0.578184447 -0.204085336 0.226205484 0.915228004 0.690925870
## [271] 0.812397280 -1.088595469 0.402006579 1.614332024 1.025324528
## [276] 1.547182892 0.679887615 4.131904876 1.948039880 2.148843690
## [281] 0.472395798 -0.780902527 0.631961209 -0.756820300 -0.576300383
## [286] -0.576741758 -0.046839531 -0.838699282 -0.888703862 -1.063831326
## [291] -0.411168254 -0.374574747 -0.693606692 0.185382740 -0.416455123
## [296] -0.060793215 0.791690363plot(y=yt_ar3,x=yt_3_ar)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-3}\). Hal ini sesuai dengan teori yang ada
cor(yt_ar3,yt_3_ar)## [1] 0.3101654Korelasi antara \(Y_t\) dengan \(Y_{t-1}\) dari hasil simulasi mendekati perhitungan teoritis yaitu
\[ \rho_1=\frac{\phi_1^3+\phi_1\phi_2}{1-\phi_2}+\phi_1\phi_2=\frac{0.5^2+(0.5)(0.2)}{1-0.2}+(0.5)(0.2)=0.38125 \]
Fungsi pembangkitan ARMA
Bentuk umum data berpola ARMA.
\[ y_{t} = c + \sum_{i=1}^p \phi_{i}y_{t-i} + \sum_{j=1}^q e_{t-j}+ e_{t} \]
Pembangkitan Proses ARMA(2,2)
Pembangkitan proses ARMA(2,2) secara manual
set.seed(035)
n = length(wn)
phi1 = 0.5
phi2 = 0.2
theta1 = 0.4
theta2 = 0.6
y.arma=c(1:n)
for (i in 5:n){y.arma[i] = phi1*y.arma[i-1] +phi2*y.arma[i-2] + theta1*wn[i-1] + phi2*y.arma[i-2]+wn[i]}
y.arma## [1] 1.00000000 2.00000000 3.00000000 4.00000000 6.51984732 5.80214128
## [7] 5.76240857 4.80410177 5.67336680 5.58388293 5.71246954 5.73974999
## [13] 4.56461783 3.76974738 1.60169004 -0.23336275 0.67677656 0.24500555
## [19] 2.09317953 3.55298000 3.96723786 4.63546523 5.28930189 6.17583017
## [25] 5.39349157 5.61727649 5.68451752 5.83685720 5.98093763 5.90180419
## [31] 6.65554282 7.26966647 7.96631345 7.20103404 6.04080763 8.06520875
## [37] 7.53448767 6.43466235 7.62913112 6.34882703 5.56160057 5.21481600
## [43] 3.67850235 2.64315410 3.55702536 4.04039849 3.68402625 4.49607411
## [49] 4.23118909 4.67741335 4.46663037 5.25047544 3.87534893 2.25053612
## [55] 2.38806925 1.43380062 2.72813885 1.31492966 0.95585329 0.35928752
## [61] 2.01358014 2.34430626 1.91169185 0.95915586 0.78204851 3.60184102
## [67] 3.45356221 2.36732425 2.75291525 1.23926880 -0.84567207 -0.37564618
## [73] 0.31795315 2.00866016 2.08087540 1.09717213 0.78726833 1.59430349
## [79] 1.44110062 1.71577269 1.89541472 0.45870351 -0.20946941 -1.76007828
## [85] -1.52949347 -0.51098529 -0.64238231 -0.36433595 -0.27752064 0.31243029
## [91] 0.25363920 0.28001174 0.92858962 1.40706874 0.68395438 2.23740511
## [97] 3.63032549 2.67827405 2.18693736 0.18697545 -0.09957045 -1.10523537
## [103] 0.45726454 1.43968103 1.87312438 1.95740900 3.68595075 2.99017212
## [109] 4.15977159 5.42996681 4.27456399 4.91140811 3.80478239 4.81728645
## [115] 4.38820153 4.49016820 3.45154533 3.66890372 3.44789923 5.13521100
## [121] 4.04778885 3.87655923 1.77179651 4.32642690 3.15214666 1.35780570
## [127] 1.25440547 2.48060428 1.71669457 1.32048826 -0.60741929 -1.48763992
## [133] -0.26827330 0.67486634 1.69609736 0.80396287 2.90686924 3.82349625
## [139] 3.10391013 4.18162489 4.96498585 2.97305328 4.06182893 4.24893347
## [145] 4.06109806 2.42405431 1.55458617 2.17767913 3.52048932 4.64684157
## [151] 3.57530322 3.79654966 5.33662564 4.17632723 4.49482914 4.17217411
## [157] 4.66661777 5.36017466 4.24504431 3.08858701 3.79720841 3.50478657
## [163] 3.59449594 4.84058036 3.53219859 3.37163700 4.98959190 6.13246450
## [169] 6.47926171 7.09877756 6.42077058 6.65871583 5.44471905 4.45655506
## [175] 4.98067476 3.79228231 4.16623700 3.28697030 2.05611540 2.47713989
## [181] 3.07191879 1.87540594 3.10323070 4.42176693 4.38434978 5.10812091
## [187] 5.08300058 4.59784058 3.67894632 4.04227965 3.22321293 2.53961548
## [193] 2.93753750 2.86508517 4.65523757 3.72686768 3.26011976 3.01941250
## [199] 3.63667956 3.50422549 3.74403528 2.36000539 2.09492623 1.38390942
## [205] 2.35594998 0.14476329 2.53959226 1.59670362 1.40435892 -0.58957465
## [211] -0.48207688 -1.55659712 0.28019160 -0.31782467 -0.65121442 -1.19440121
## [217] -0.73537965 -1.97606957 -2.46806921 -2.91525201 -2.40789432 -0.80914157
## [223] -0.78054910 -1.57740544 -1.94879371 -0.29532301 -1.43254451 -0.91755453
## [229] -2.00460511 -2.54231233 -4.91386944 -2.87800990 -2.22652901 -1.15324082
## [235] -0.63075751 -1.30850403 -1.54238190 -2.25095721 -2.01472276 -1.79157557
## [241] -0.19803148 2.07383874 2.13672797 1.55906937 0.96516327 1.81997128
## [247] 1.48425644 0.02181518 0.06632643 -0.20502087 -1.93184981 -1.87266900
## [253] -1.67476242 -1.55954696 -1.95708274 -1.30788886 -1.95162091 -2.08218493
## [259] -2.59075236 -2.75936536 -2.31096560 -0.60912031 0.69141281 1.75179065
## [265] 1.36282498 0.83524340 0.05228260 0.28071506 1.08925740 1.18216243
## [271] 1.38590123 -0.35360820 0.50818954 2.05722926 2.02206678 2.60068220
## [277] 2.09506076 5.45082025 4.70255415 4.77849192 3.41801070 1.77687935
## [283] 2.60482960 1.46771947 1.08486094 0.86257776 1.16713368 0.37138335
## [289] -0.08741364 -0.53088754 -0.18261686 -0.14049253 -0.54986842 0.10633555
## [295] -0.29436433 -0.14246021 0.76054527 1.44172310 2.65034139 2.90859761set.seed(1234)
n = length(wn)
phi1 = 0.7 #ARMA 1 1
theta1 = 0.5
y.arma=c(1:n)
for (i in 3:n){y.arma[i] = phi1*y.arma[i-1] + theta1*wn[i-1]+wn[i]}
y.arma## [1] 1.000000000 2.000000000 1.432396579 0.940679254 3.973825166
## [6] 4.057679020 3.054482567 1.781235370 2.156998697 2.454580578
## [11] 2.404294581 2.384123659 1.123175616 -0.099076708 -2.228495384
## [16] -4.292912206 -3.030123812 -2.034699230 0.241102881 2.760986160
## [21] 3.453474713 3.716510858 4.081927457 4.634531474 3.561651420
## [26] 2.911290742 2.814231591 2.766360053 2.780438809 2.579655962
## [31] 3.152981574 3.905504364 4.514335926 3.592484272 1.750529931
## [36] 3.322328504 3.654602836 2.010747967 2.745185574 2.045964079
## [41] 0.698174058 0.344574366 -0.907442380 -2.034548311 -0.741412727
## [46] 0.794550511 0.874098530 1.643113718 1.766174036 2.008083981
## [51] 1.914096569 2.500411431 1.322626984 -0.958695693 -1.098275798
## [56] -1.402035174 -0.002341160 -0.487926789 -1.251230645 -1.553033229
## [61] 0.313035012 1.578659916 1.092255036 -0.197657945 -0.682878421
## [66] 2.335843584 3.262946621 1.502638080 1.152148090 -0.223821311
## [71] -2.853076765 -2.650346948 -0.974179344 1.387613645 2.092848966
## [76] 0.744338903 -0.157733333 0.626061420 0.853594871 0.953397417
## [81] 1.164864411 -0.328358262 -1.556988961 -2.996355508 -2.819942614
## [86] -1.010002857 -0.388780692 -0.125731824 0.095646781 0.671214247
## [91] 0.735040181 0.540888223 1.069315798 1.646576709 0.817750210
## [96] 1.843459048 3.686349173 2.709243137 1.224695479 -1.153971062
## [101] -2.060011124 -2.605272114 -0.873794512 1.186383563 1.908202109
## [106] 1.834811448 3.265223722 2.836545718 3.137648157 4.484692034
## [111] 3.196627125 2.764677273 1.664781217 2.043502387 2.012000805
## [116] 1.773756439 0.731244208 0.588679150 0.687713244 2.431258652
## [121] 1.994810909 1.128287927 -0.989275007 1.016243747 1.253043206
## [126] -1.148269877 -1.653404105 0.150064444 0.211836057 -0.437216757
## [131] -2.291242544 -3.499089660 -1.828823227 0.234924076 1.726409712
## [136] 1.002656066 2.453624222 3.978531878 2.934226898 3.109235475
## [141] 3.937099186 1.685930765 1.606430150 2.277441330 1.959331446
## [146] 0.075364777 -1.363692030 -0.598181256 1.463909304 3.192114944
## [151] 2.218978420 1.631490482 3.194071335 2.408548045 1.883616929
## [156] 1.629714818 1.926040574 2.783427979 1.751627685 -0.024001933
## [161] 0.449149853 0.776220569 0.867221201 2.280535521 1.421801508
## [166] 0.571450979 2.295086464 4.082992943 4.429221668 4.586761783
## [171] 3.598968171 3.112641820 1.792966290 0.253665248 0.687994350
## [176] 0.084002287 0.255326497 -0.074029566 -1.361112564 -0.921700249
## [181] 0.420428128 -0.277905701 0.645403670 2.704176814 2.984134707
## [186] 3.265736186 3.164098412 2.264327695 0.918617631 0.946680203
## [191] 0.453546698 -0.422521093 0.034229915 0.461655585 2.385996146
## [196] 2.122117225 0.965912393 0.549867541 1.207643919 1.405839502
## [201] 1.536203216 0.209418929 -0.546579223 -1.004237600 0.067933386
## [206] -1.434569595 0.320093360 0.725766007 0.032027241 -1.922527802
## [211] -2.282041621 -2.677165656 -0.730664851 -0.178426998 -0.780133793
## [216] -1.327853193 -0.865318762 -1.700860420 -2.503739578 -2.711093648
## [221] -1.922810596 0.243436676 0.902276643 -0.231039291 -1.096282804
## [226] 0.492524500 -0.009549343 -0.200793777 -1.077298523 -2.038804792
## [231] -4.340650243 -2.697646278 -0.548727360 0.780280334 1.472528763
## [236] 0.546645426 -0.325439666 -1.218849262 -1.207251606 -0.723430279
## [241] 1.006670224 3.740748001 4.027855611 2.506446610 1.041255889
## [246] 1.393412410 1.254661817 -0.587644789 -1.087422639 -1.006819359
## [251] -2.585849172 -2.810334892 -1.945194108 -1.326394147 -1.436525955
## [256] -0.761578999 -0.998308712 -1.353375362 -1.746160038 -1.918414088
## [261] -1.274988615 0.782965116 2.623306246 3.616134074 2.854596685
## [266] 1.427193465 0.043533342 -0.122083970 0.863789181 1.214194363
## [271] 1.227860184 -0.631532782 -0.474737731 1.690643845 2.136731806
## [276] 2.276245088 1.650429936 4.488441077 4.629280918 3.461994377
## [281] 1.605983429 -0.818497775 -0.368449712 -0.710568540 -1.279990653
## [286] -1.195362201 -0.548575870 -0.905538562 -1.443828891 -1.692412920
## [291] -1.112070631 -0.585429706 -0.811998639 -0.173340846 -0.188213229
## [296] -0.206604040 0.815934067 1.780126189 2.946978399 3.201897391Pembangkitan 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] -2.75189059 -1.74814288 -2.98064692 -1.12111691 -1.75293106 -1.33898765
## [7] -2.09046996 -3.56829323 -4.32225209 -6.49932813 -7.02769509 -6.95242810
## [13] -6.26996407 -3.43890672 -3.74182891 -2.97181976 -3.49823015 -3.96328709
## [19] -4.21591372 -4.99194225 -5.61517562 -5.49498965 -5.32810268 -6.08211738
## [25] -5.70927763 -6.49640104 -6.19781946 -5.63181804 -4.16631811 -1.43386858
## [31] -1.32659085 1.33518960 -0.57718012 1.13545868 2.56923465 2.89049818
## [37] 2.80495292 1.68426168 2.77538382 1.29220743 3.17981524 3.04188012
## [43] 3.84589813 3.47054592 2.25362466 1.27634579 1.31729358 3.02358708
## [49] 2.83893413 1.81428668 0.10302896 -0.45121970 -0.85290753 -0.66635425
## [55] -0.93510416 -2.14779642 -2.08562341 -1.51503473 -0.24121274 0.91556687
## [61] 0.64537308 0.48311177 -1.14217407 -1.12039092 -1.27141453 0.91636167
## [67] 2.04091439 2.13232979 2.23257264 0.25301429 1.21071226 1.29939237
## [73] 3.92904462 4.07112121 3.78513073 2.83160983 1.41191959 0.28488816
## [79] -0.04006883 -1.28812485 -0.74220308 -0.74424287 -0.32976863 -0.31752860
## [85] -0.98342470 -2.04217397 -1.20184647 -1.31314616 0.45270778 -0.93468943
## [91] -0.20680752 -0.29613748 0.15497575 0.22025744 -0.73310547 1.30806278
## [97] 1.61203256 4.42693491 3.79591780 3.27848542 0.68061022 -0.62425847
## [103] -1.11211846 0.04164014 0.59240357 -0.15810424 0.39101368 -1.92189820
## [109] -1.50741039 -2.59060462 -3.89098912 -2.51252738 -3.45825773 -2.26278511
## [115] -0.93523383 0.07413764 0.83334886 1.28389670 1.34833602 3.06177019
## [121] 2.98115137 3.71727357 3.17046413 2.39423349 1.98646516 2.92352912
## [127] 1.69753466 2.18824295 2.31910758 1.98048956 1.12411345 -0.47375475
## [133] -1.38212484 -2.31108442 -2.26562282 -2.62211380 -2.26938378 -1.57393069
## [139] -0.56321701 1.57589801 1.65267412 1.80012273 2.53204975 3.72627716
## [145] 3.97712776 3.44105725 0.58660225 1.46428641 -0.51695507 -0.57765858
## [151] 1.69949100 1.51248377 2.98314953 -0.81043866 -1.02122299 -1.37591544
## [157] -0.14779506 1.31440815 3.22462748 4.38519006 3.98666624 4.07252161
## [163] 2.61160764 1.92537895 -1.70502573 -2.72222986 -3.24361938 -0.27688538
## [169] 0.87342985 2.50264542 1.00860632 1.15325179 -1.81437043 -0.92501023
## [175] -1.18146222 -0.02037563 0.94652009 1.17246770 2.08111275 3.60870710
## [181] 4.52620181 4.56792960 2.75523948 2.28308227 0.79090755 2.43630393
## [187] 0.12650863 0.38802680 -0.48922578 -0.39939179 -0.71118176 -1.07369267
## [193] -0.53512569 0.48185240 1.52990415 3.93599432 4.75299655 5.28489601
## [199] 3.75245965 5.76106696 5.08023491 5.03046078 3.85049062 2.27094879
## [205] 1.61300623 0.02267186 -0.08744134 -0.23338993 -1.01395405 -4.02632785
## [211] -4.41061552 -5.02289664 -2.94041531 -2.16023161 -0.87458164 -1.76626866
## [217] -0.53794493 -0.27753067 0.71955739 2.19354067 2.33905070 2.84874980
## [223] 2.50865574 2.52684453 3.16087589 3.22420516 3.72442687 1.90727460
## [229] 1.27139490 1.66409882 0.23091397 1.10296539 0.37155340 1.69901525
## [235] 1.20826458 1.54849208 -0.89323198 -0.40844829 -1.97195019 -1.36977976
## [241] -2.76244287 -1.88853615 -1.38886217 -0.87915921 1.38576118 1.73484421
## [247] 3.45643730 4.90637915 3.88230417 4.70024354 4.04594276 3.83665310
## [253] 3.63604751 1.69745094 1.83280517 0.13589632 1.24990445 -0.25440835
## [259] 0.60581161 -0.95359799 -0.74837042 -2.72398269 -2.15183291 -1.51561139
## [265] -0.53368698 -1.67162994 -2.15962923 -1.06604857 1.13495225 0.89677762
## [271] 0.92454452 0.14354987 0.49726828 0.03722489 -0.87570167 -1.33912012
## [277] -0.47863083 -0.18724669 -0.22304216 -0.89877996 0.52922646 -0.01897051
## [283] 1.64938777 2.32971329 2.98499263 2.59721428 1.95311423 0.73468623
## [289] 1.76477012 1.07548761 3.11749005 1.74249123 1.55297139 0.67230168
## [295] 0.33991628 -0.10447830 -1.75292012 -1.80251894 -2.90523171 -0.99909951Karakteristik ARMA(2,2)
Plot Time Series
par(mfrow = c(1, 2))
ts.plot(y.arma)
ts.plot(arma22)
par(mfrow = c(1, 1))Berdasarkan plot time series tersebut, terlihat bahwa model ARMA(2,2) cenderung stasioner dalam rataan
Plot ACF
par(mfrow = c(1, 2))
acf(y.arma)
acf(arma22)
par(mfrow = c(1, 1))Berdasarkan plot ACF tersebut, terlihat bahwa model ARMA(2,2) hasil simulasi memiliki plot ACF yang tails off, sesuai dengan teori yang ada
Plot PACF
par(mfrow = c(1, 2))
pacf(y.arma)
pacf(arma22)
par(mfrow = c(1, 1))Berdasarkan plot PACF tersebut, terlihat bahwa model ARMA(2,2) hasil simulasi memiliki plot PACF yang tails off, sesuai dengan teori
Plot EACF
TSA::eacf(y.arma)## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x o o o o o o o o o x
## 1 x o o o o o o o o o o o o o
## 2 x x o o o o o o o o o o o o
## 3 x x x o o o o o o o o o o o
## 4 x x x o o o o o o o o o o o
## 5 x o o o x o o o o o o o o o
## 6 x o x x o x o o o o o o o o
## 7 x x o x o x o o o o o o o oTSA::eacf(arma22)## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x x x x o o o o o x o
## 1 o x o o o o x o o o o o o o
## 2 x x o o o o o o o o o o o o
## 3 x x x x o o o o o o o o o o
## 4 x x x x o o o o o o o o o o
## 5 x x x x x x x o o o o o o o
## 6 o o x o x x o o o o o o o o
## 7 x o x x x x o x 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)
Scatterplot Antar Lag
Korelasi antara \(Y_t\) dengan \(Y_{t-1}\)
#Yt
yt_arma <- arma22[-1]
yt_arma## [1] -1.74814288 -2.98064692 -1.12111691 -1.75293106 -1.33898765 -2.09046996
## [7] -3.56829323 -4.32225209 -6.49932813 -7.02769509 -6.95242810 -6.26996407
## [13] -3.43890672 -3.74182891 -2.97181976 -3.49823015 -3.96328709 -4.21591372
## [19] -4.99194225 -5.61517562 -5.49498965 -5.32810268 -6.08211738 -5.70927763
## [25] -6.49640104 -6.19781946 -5.63181804 -4.16631811 -1.43386858 -1.32659085
## [31] 1.33518960 -0.57718012 1.13545868 2.56923465 2.89049818 2.80495292
## [37] 1.68426168 2.77538382 1.29220743 3.17981524 3.04188012 3.84589813
## [43] 3.47054592 2.25362466 1.27634579 1.31729358 3.02358708 2.83893413
## [49] 1.81428668 0.10302896 -0.45121970 -0.85290753 -0.66635425 -0.93510416
## [55] -2.14779642 -2.08562341 -1.51503473 -0.24121274 0.91556687 0.64537308
## [61] 0.48311177 -1.14217407 -1.12039092 -1.27141453 0.91636167 2.04091439
## [67] 2.13232979 2.23257264 0.25301429 1.21071226 1.29939237 3.92904462
## [73] 4.07112121 3.78513073 2.83160983 1.41191959 0.28488816 -0.04006883
## [79] -1.28812485 -0.74220308 -0.74424287 -0.32976863 -0.31752860 -0.98342470
## [85] -2.04217397 -1.20184647 -1.31314616 0.45270778 -0.93468943 -0.20680752
## [91] -0.29613748 0.15497575 0.22025744 -0.73310547 1.30806278 1.61203256
## [97] 4.42693491 3.79591780 3.27848542 0.68061022 -0.62425847 -1.11211846
## [103] 0.04164014 0.59240357 -0.15810424 0.39101368 -1.92189820 -1.50741039
## [109] -2.59060462 -3.89098912 -2.51252738 -3.45825773 -2.26278511 -0.93523383
## [115] 0.07413764 0.83334886 1.28389670 1.34833602 3.06177019 2.98115137
## [121] 3.71727357 3.17046413 2.39423349 1.98646516 2.92352912 1.69753466
## [127] 2.18824295 2.31910758 1.98048956 1.12411345 -0.47375475 -1.38212484
## [133] -2.31108442 -2.26562282 -2.62211380 -2.26938378 -1.57393069 -0.56321701
## [139] 1.57589801 1.65267412 1.80012273 2.53204975 3.72627716 3.97712776
## [145] 3.44105725 0.58660225 1.46428641 -0.51695507 -0.57765858 1.69949100
## [151] 1.51248377 2.98314953 -0.81043866 -1.02122299 -1.37591544 -0.14779506
## [157] 1.31440815 3.22462748 4.38519006 3.98666624 4.07252161 2.61160764
## [163] 1.92537895 -1.70502573 -2.72222986 -3.24361938 -0.27688538 0.87342985
## [169] 2.50264542 1.00860632 1.15325179 -1.81437043 -0.92501023 -1.18146222
## [175] -0.02037563 0.94652009 1.17246770 2.08111275 3.60870710 4.52620181
## [181] 4.56792960 2.75523948 2.28308227 0.79090755 2.43630393 0.12650863
## [187] 0.38802680 -0.48922578 -0.39939179 -0.71118176 -1.07369267 -0.53512569
## [193] 0.48185240 1.52990415 3.93599432 4.75299655 5.28489601 3.75245965
## [199] 5.76106696 5.08023491 5.03046078 3.85049062 2.27094879 1.61300623
## [205] 0.02267186 -0.08744134 -0.23338993 -1.01395405 -4.02632785 -4.41061552
## [211] -5.02289664 -2.94041531 -2.16023161 -0.87458164 -1.76626866 -0.53794493
## [217] -0.27753067 0.71955739 2.19354067 2.33905070 2.84874980 2.50865574
## [223] 2.52684453 3.16087589 3.22420516 3.72442687 1.90727460 1.27139490
## [229] 1.66409882 0.23091397 1.10296539 0.37155340 1.69901525 1.20826458
## [235] 1.54849208 -0.89323198 -0.40844829 -1.97195019 -1.36977976 -2.76244287
## [241] -1.88853615 -1.38886217 -0.87915921 1.38576118 1.73484421 3.45643730
## [247] 4.90637915 3.88230417 4.70024354 4.04594276 3.83665310 3.63604751
## [253] 1.69745094 1.83280517 0.13589632 1.24990445 -0.25440835 0.60581161
## [259] -0.95359799 -0.74837042 -2.72398269 -2.15183291 -1.51561139 -0.53368698
## [265] -1.67162994 -2.15962923 -1.06604857 1.13495225 0.89677762 0.92454452
## [271] 0.14354987 0.49726828 0.03722489 -0.87570167 -1.33912012 -0.47863083
## [277] -0.18724669 -0.22304216 -0.89877996 0.52922646 -0.01897051 1.64938777
## [283] 2.32971329 2.98499263 2.59721428 1.95311423 0.73468623 1.76477012
## [289] 1.07548761 3.11749005 1.74249123 1.55297139 0.67230168 0.33991628
## [295] -0.10447830 -1.75292012 -1.80251894 -2.90523171 -0.99909951#Yt-1
yt_1_arma <- arma22[-300]
yt_1_arma## [1] -2.75189059 -1.74814288 -2.98064692 -1.12111691 -1.75293106 -1.33898765
## [7] -2.09046996 -3.56829323 -4.32225209 -6.49932813 -7.02769509 -6.95242810
## [13] -6.26996407 -3.43890672 -3.74182891 -2.97181976 -3.49823015 -3.96328709
## [19] -4.21591372 -4.99194225 -5.61517562 -5.49498965 -5.32810268 -6.08211738
## [25] -5.70927763 -6.49640104 -6.19781946 -5.63181804 -4.16631811 -1.43386858
## [31] -1.32659085 1.33518960 -0.57718012 1.13545868 2.56923465 2.89049818
## [37] 2.80495292 1.68426168 2.77538382 1.29220743 3.17981524 3.04188012
## [43] 3.84589813 3.47054592 2.25362466 1.27634579 1.31729358 3.02358708
## [49] 2.83893413 1.81428668 0.10302896 -0.45121970 -0.85290753 -0.66635425
## [55] -0.93510416 -2.14779642 -2.08562341 -1.51503473 -0.24121274 0.91556687
## [61] 0.64537308 0.48311177 -1.14217407 -1.12039092 -1.27141453 0.91636167
## [67] 2.04091439 2.13232979 2.23257264 0.25301429 1.21071226 1.29939237
## [73] 3.92904462 4.07112121 3.78513073 2.83160983 1.41191959 0.28488816
## [79] -0.04006883 -1.28812485 -0.74220308 -0.74424287 -0.32976863 -0.31752860
## [85] -0.98342470 -2.04217397 -1.20184647 -1.31314616 0.45270778 -0.93468943
## [91] -0.20680752 -0.29613748 0.15497575 0.22025744 -0.73310547 1.30806278
## [97] 1.61203256 4.42693491 3.79591780 3.27848542 0.68061022 -0.62425847
## [103] -1.11211846 0.04164014 0.59240357 -0.15810424 0.39101368 -1.92189820
## [109] -1.50741039 -2.59060462 -3.89098912 -2.51252738 -3.45825773 -2.26278511
## [115] -0.93523383 0.07413764 0.83334886 1.28389670 1.34833602 3.06177019
## [121] 2.98115137 3.71727357 3.17046413 2.39423349 1.98646516 2.92352912
## [127] 1.69753466 2.18824295 2.31910758 1.98048956 1.12411345 -0.47375475
## [133] -1.38212484 -2.31108442 -2.26562282 -2.62211380 -2.26938378 -1.57393069
## [139] -0.56321701 1.57589801 1.65267412 1.80012273 2.53204975 3.72627716
## [145] 3.97712776 3.44105725 0.58660225 1.46428641 -0.51695507 -0.57765858
## [151] 1.69949100 1.51248377 2.98314953 -0.81043866 -1.02122299 -1.37591544
## [157] -0.14779506 1.31440815 3.22462748 4.38519006 3.98666624 4.07252161
## [163] 2.61160764 1.92537895 -1.70502573 -2.72222986 -3.24361938 -0.27688538
## [169] 0.87342985 2.50264542 1.00860632 1.15325179 -1.81437043 -0.92501023
## [175] -1.18146222 -0.02037563 0.94652009 1.17246770 2.08111275 3.60870710
## [181] 4.52620181 4.56792960 2.75523948 2.28308227 0.79090755 2.43630393
## [187] 0.12650863 0.38802680 -0.48922578 -0.39939179 -0.71118176 -1.07369267
## [193] -0.53512569 0.48185240 1.52990415 3.93599432 4.75299655 5.28489601
## [199] 3.75245965 5.76106696 5.08023491 5.03046078 3.85049062 2.27094879
## [205] 1.61300623 0.02267186 -0.08744134 -0.23338993 -1.01395405 -4.02632785
## [211] -4.41061552 -5.02289664 -2.94041531 -2.16023161 -0.87458164 -1.76626866
## [217] -0.53794493 -0.27753067 0.71955739 2.19354067 2.33905070 2.84874980
## [223] 2.50865574 2.52684453 3.16087589 3.22420516 3.72442687 1.90727460
## [229] 1.27139490 1.66409882 0.23091397 1.10296539 0.37155340 1.69901525
## [235] 1.20826458 1.54849208 -0.89323198 -0.40844829 -1.97195019 -1.36977976
## [241] -2.76244287 -1.88853615 -1.38886217 -0.87915921 1.38576118 1.73484421
## [247] 3.45643730 4.90637915 3.88230417 4.70024354 4.04594276 3.83665310
## [253] 3.63604751 1.69745094 1.83280517 0.13589632 1.24990445 -0.25440835
## [259] 0.60581161 -0.95359799 -0.74837042 -2.72398269 -2.15183291 -1.51561139
## [265] -0.53368698 -1.67162994 -2.15962923 -1.06604857 1.13495225 0.89677762
## [271] 0.92454452 0.14354987 0.49726828 0.03722489 -0.87570167 -1.33912012
## [277] -0.47863083 -0.18724669 -0.22304216 -0.89877996 0.52922646 -0.01897051
## [283] 1.64938777 2.32971329 2.98499263 2.59721428 1.95311423 0.73468623
## [289] 1.76477012 1.07548761 3.11749005 1.74249123 1.55297139 0.67230168
## [295] 0.33991628 -0.10447830 -1.75292012 -1.80251894 -2.90523171plot(y=yt_arma,x=yt_1_arma)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-1}\). Hal ini sesuai dengan teori yang ada
cor(yt_arma,yt_1_arma)## [1] 0.8918177Korelasi antara \(Y_t\) dengan \(Y_{t-2}\)
#Yt
yt_arma2 <- arma22[-c(1,2)]
yt_arma2## [1] -2.98064692 -1.12111691 -1.75293106 -1.33898765 -2.09046996 -3.56829323
## [7] -4.32225209 -6.49932813 -7.02769509 -6.95242810 -6.26996407 -3.43890672
## [13] -3.74182891 -2.97181976 -3.49823015 -3.96328709 -4.21591372 -4.99194225
## [19] -5.61517562 -5.49498965 -5.32810268 -6.08211738 -5.70927763 -6.49640104
## [25] -6.19781946 -5.63181804 -4.16631811 -1.43386858 -1.32659085 1.33518960
## [31] -0.57718012 1.13545868 2.56923465 2.89049818 2.80495292 1.68426168
## [37] 2.77538382 1.29220743 3.17981524 3.04188012 3.84589813 3.47054592
## [43] 2.25362466 1.27634579 1.31729358 3.02358708 2.83893413 1.81428668
## [49] 0.10302896 -0.45121970 -0.85290753 -0.66635425 -0.93510416 -2.14779642
## [55] -2.08562341 -1.51503473 -0.24121274 0.91556687 0.64537308 0.48311177
## [61] -1.14217407 -1.12039092 -1.27141453 0.91636167 2.04091439 2.13232979
## [67] 2.23257264 0.25301429 1.21071226 1.29939237 3.92904462 4.07112121
## [73] 3.78513073 2.83160983 1.41191959 0.28488816 -0.04006883 -1.28812485
## [79] -0.74220308 -0.74424287 -0.32976863 -0.31752860 -0.98342470 -2.04217397
## [85] -1.20184647 -1.31314616 0.45270778 -0.93468943 -0.20680752 -0.29613748
## [91] 0.15497575 0.22025744 -0.73310547 1.30806278 1.61203256 4.42693491
## [97] 3.79591780 3.27848542 0.68061022 -0.62425847 -1.11211846 0.04164014
## [103] 0.59240357 -0.15810424 0.39101368 -1.92189820 -1.50741039 -2.59060462
## [109] -3.89098912 -2.51252738 -3.45825773 -2.26278511 -0.93523383 0.07413764
## [115] 0.83334886 1.28389670 1.34833602 3.06177019 2.98115137 3.71727357
## [121] 3.17046413 2.39423349 1.98646516 2.92352912 1.69753466 2.18824295
## [127] 2.31910758 1.98048956 1.12411345 -0.47375475 -1.38212484 -2.31108442
## [133] -2.26562282 -2.62211380 -2.26938378 -1.57393069 -0.56321701 1.57589801
## [139] 1.65267412 1.80012273 2.53204975 3.72627716 3.97712776 3.44105725
## [145] 0.58660225 1.46428641 -0.51695507 -0.57765858 1.69949100 1.51248377
## [151] 2.98314953 -0.81043866 -1.02122299 -1.37591544 -0.14779506 1.31440815
## [157] 3.22462748 4.38519006 3.98666624 4.07252161 2.61160764 1.92537895
## [163] -1.70502573 -2.72222986 -3.24361938 -0.27688538 0.87342985 2.50264542
## [169] 1.00860632 1.15325179 -1.81437043 -0.92501023 -1.18146222 -0.02037563
## [175] 0.94652009 1.17246770 2.08111275 3.60870710 4.52620181 4.56792960
## [181] 2.75523948 2.28308227 0.79090755 2.43630393 0.12650863 0.38802680
## [187] -0.48922578 -0.39939179 -0.71118176 -1.07369267 -0.53512569 0.48185240
## [193] 1.52990415 3.93599432 4.75299655 5.28489601 3.75245965 5.76106696
## [199] 5.08023491 5.03046078 3.85049062 2.27094879 1.61300623 0.02267186
## [205] -0.08744134 -0.23338993 -1.01395405 -4.02632785 -4.41061552 -5.02289664
## [211] -2.94041531 -2.16023161 -0.87458164 -1.76626866 -0.53794493 -0.27753067
## [217] 0.71955739 2.19354067 2.33905070 2.84874980 2.50865574 2.52684453
## [223] 3.16087589 3.22420516 3.72442687 1.90727460 1.27139490 1.66409882
## [229] 0.23091397 1.10296539 0.37155340 1.69901525 1.20826458 1.54849208
## [235] -0.89323198 -0.40844829 -1.97195019 -1.36977976 -2.76244287 -1.88853615
## [241] -1.38886217 -0.87915921 1.38576118 1.73484421 3.45643730 4.90637915
## [247] 3.88230417 4.70024354 4.04594276 3.83665310 3.63604751 1.69745094
## [253] 1.83280517 0.13589632 1.24990445 -0.25440835 0.60581161 -0.95359799
## [259] -0.74837042 -2.72398269 -2.15183291 -1.51561139 -0.53368698 -1.67162994
## [265] -2.15962923 -1.06604857 1.13495225 0.89677762 0.92454452 0.14354987
## [271] 0.49726828 0.03722489 -0.87570167 -1.33912012 -0.47863083 -0.18724669
## [277] -0.22304216 -0.89877996 0.52922646 -0.01897051 1.64938777 2.32971329
## [283] 2.98499263 2.59721428 1.95311423 0.73468623 1.76477012 1.07548761
## [289] 3.11749005 1.74249123 1.55297139 0.67230168 0.33991628 -0.10447830
## [295] -1.75292012 -1.80251894 -2.90523171 -0.99909951#Yt-2
yt_2_arma <- arma22[-c(299,300)]
yt_2_arma## [1] -2.75189059 -1.74814288 -2.98064692 -1.12111691 -1.75293106 -1.33898765
## [7] -2.09046996 -3.56829323 -4.32225209 -6.49932813 -7.02769509 -6.95242810
## [13] -6.26996407 -3.43890672 -3.74182891 -2.97181976 -3.49823015 -3.96328709
## [19] -4.21591372 -4.99194225 -5.61517562 -5.49498965 -5.32810268 -6.08211738
## [25] -5.70927763 -6.49640104 -6.19781946 -5.63181804 -4.16631811 -1.43386858
## [31] -1.32659085 1.33518960 -0.57718012 1.13545868 2.56923465 2.89049818
## [37] 2.80495292 1.68426168 2.77538382 1.29220743 3.17981524 3.04188012
## [43] 3.84589813 3.47054592 2.25362466 1.27634579 1.31729358 3.02358708
## [49] 2.83893413 1.81428668 0.10302896 -0.45121970 -0.85290753 -0.66635425
## [55] -0.93510416 -2.14779642 -2.08562341 -1.51503473 -0.24121274 0.91556687
## [61] 0.64537308 0.48311177 -1.14217407 -1.12039092 -1.27141453 0.91636167
## [67] 2.04091439 2.13232979 2.23257264 0.25301429 1.21071226 1.29939237
## [73] 3.92904462 4.07112121 3.78513073 2.83160983 1.41191959 0.28488816
## [79] -0.04006883 -1.28812485 -0.74220308 -0.74424287 -0.32976863 -0.31752860
## [85] -0.98342470 -2.04217397 -1.20184647 -1.31314616 0.45270778 -0.93468943
## [91] -0.20680752 -0.29613748 0.15497575 0.22025744 -0.73310547 1.30806278
## [97] 1.61203256 4.42693491 3.79591780 3.27848542 0.68061022 -0.62425847
## [103] -1.11211846 0.04164014 0.59240357 -0.15810424 0.39101368 -1.92189820
## [109] -1.50741039 -2.59060462 -3.89098912 -2.51252738 -3.45825773 -2.26278511
## [115] -0.93523383 0.07413764 0.83334886 1.28389670 1.34833602 3.06177019
## [121] 2.98115137 3.71727357 3.17046413 2.39423349 1.98646516 2.92352912
## [127] 1.69753466 2.18824295 2.31910758 1.98048956 1.12411345 -0.47375475
## [133] -1.38212484 -2.31108442 -2.26562282 -2.62211380 -2.26938378 -1.57393069
## [139] -0.56321701 1.57589801 1.65267412 1.80012273 2.53204975 3.72627716
## [145] 3.97712776 3.44105725 0.58660225 1.46428641 -0.51695507 -0.57765858
## [151] 1.69949100 1.51248377 2.98314953 -0.81043866 -1.02122299 -1.37591544
## [157] -0.14779506 1.31440815 3.22462748 4.38519006 3.98666624 4.07252161
## [163] 2.61160764 1.92537895 -1.70502573 -2.72222986 -3.24361938 -0.27688538
## [169] 0.87342985 2.50264542 1.00860632 1.15325179 -1.81437043 -0.92501023
## [175] -1.18146222 -0.02037563 0.94652009 1.17246770 2.08111275 3.60870710
## [181] 4.52620181 4.56792960 2.75523948 2.28308227 0.79090755 2.43630393
## [187] 0.12650863 0.38802680 -0.48922578 -0.39939179 -0.71118176 -1.07369267
## [193] -0.53512569 0.48185240 1.52990415 3.93599432 4.75299655 5.28489601
## [199] 3.75245965 5.76106696 5.08023491 5.03046078 3.85049062 2.27094879
## [205] 1.61300623 0.02267186 -0.08744134 -0.23338993 -1.01395405 -4.02632785
## [211] -4.41061552 -5.02289664 -2.94041531 -2.16023161 -0.87458164 -1.76626866
## [217] -0.53794493 -0.27753067 0.71955739 2.19354067 2.33905070 2.84874980
## [223] 2.50865574 2.52684453 3.16087589 3.22420516 3.72442687 1.90727460
## [229] 1.27139490 1.66409882 0.23091397 1.10296539 0.37155340 1.69901525
## [235] 1.20826458 1.54849208 -0.89323198 -0.40844829 -1.97195019 -1.36977976
## [241] -2.76244287 -1.88853615 -1.38886217 -0.87915921 1.38576118 1.73484421
## [247] 3.45643730 4.90637915 3.88230417 4.70024354 4.04594276 3.83665310
## [253] 3.63604751 1.69745094 1.83280517 0.13589632 1.24990445 -0.25440835
## [259] 0.60581161 -0.95359799 -0.74837042 -2.72398269 -2.15183291 -1.51561139
## [265] -0.53368698 -1.67162994 -2.15962923 -1.06604857 1.13495225 0.89677762
## [271] 0.92454452 0.14354987 0.49726828 0.03722489 -0.87570167 -1.33912012
## [277] -0.47863083 -0.18724669 -0.22304216 -0.89877996 0.52922646 -0.01897051
## [283] 1.64938777 2.32971329 2.98499263 2.59721428 1.95311423 0.73468623
## [289] 1.76477012 1.07548761 3.11749005 1.74249123 1.55297139 0.67230168
## [295] 0.33991628 -0.10447830 -1.75292012 -1.80251894plot(y=yt_arma2,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
cor(yt_arma2,yt_2_arma)## [1] 0.7836583Korelasi antara \(Y_t\) dengan \(Y_{t-3}\)
#Yt
yt_arma3 <- arma22[-c(1,2,3)]
yt_arma3## [1] -1.12111691 -1.75293106 -1.33898765 -2.09046996 -3.56829323 -4.32225209
## [7] -6.49932813 -7.02769509 -6.95242810 -6.26996407 -3.43890672 -3.74182891
## [13] -2.97181976 -3.49823015 -3.96328709 -4.21591372 -4.99194225 -5.61517562
## [19] -5.49498965 -5.32810268 -6.08211738 -5.70927763 -6.49640104 -6.19781946
## [25] -5.63181804 -4.16631811 -1.43386858 -1.32659085 1.33518960 -0.57718012
## [31] 1.13545868 2.56923465 2.89049818 2.80495292 1.68426168 2.77538382
## [37] 1.29220743 3.17981524 3.04188012 3.84589813 3.47054592 2.25362466
## [43] 1.27634579 1.31729358 3.02358708 2.83893413 1.81428668 0.10302896
## [49] -0.45121970 -0.85290753 -0.66635425 -0.93510416 -2.14779642 -2.08562341
## [55] -1.51503473 -0.24121274 0.91556687 0.64537308 0.48311177 -1.14217407
## [61] -1.12039092 -1.27141453 0.91636167 2.04091439 2.13232979 2.23257264
## [67] 0.25301429 1.21071226 1.29939237 3.92904462 4.07112121 3.78513073
## [73] 2.83160983 1.41191959 0.28488816 -0.04006883 -1.28812485 -0.74220308
## [79] -0.74424287 -0.32976863 -0.31752860 -0.98342470 -2.04217397 -1.20184647
## [85] -1.31314616 0.45270778 -0.93468943 -0.20680752 -0.29613748 0.15497575
## [91] 0.22025744 -0.73310547 1.30806278 1.61203256 4.42693491 3.79591780
## [97] 3.27848542 0.68061022 -0.62425847 -1.11211846 0.04164014 0.59240357
## [103] -0.15810424 0.39101368 -1.92189820 -1.50741039 -2.59060462 -3.89098912
## [109] -2.51252738 -3.45825773 -2.26278511 -0.93523383 0.07413764 0.83334886
## [115] 1.28389670 1.34833602 3.06177019 2.98115137 3.71727357 3.17046413
## [121] 2.39423349 1.98646516 2.92352912 1.69753466 2.18824295 2.31910758
## [127] 1.98048956 1.12411345 -0.47375475 -1.38212484 -2.31108442 -2.26562282
## [133] -2.62211380 -2.26938378 -1.57393069 -0.56321701 1.57589801 1.65267412
## [139] 1.80012273 2.53204975 3.72627716 3.97712776 3.44105725 0.58660225
## [145] 1.46428641 -0.51695507 -0.57765858 1.69949100 1.51248377 2.98314953
## [151] -0.81043866 -1.02122299 -1.37591544 -0.14779506 1.31440815 3.22462748
## [157] 4.38519006 3.98666624 4.07252161 2.61160764 1.92537895 -1.70502573
## [163] -2.72222986 -3.24361938 -0.27688538 0.87342985 2.50264542 1.00860632
## [169] 1.15325179 -1.81437043 -0.92501023 -1.18146222 -0.02037563 0.94652009
## [175] 1.17246770 2.08111275 3.60870710 4.52620181 4.56792960 2.75523948
## [181] 2.28308227 0.79090755 2.43630393 0.12650863 0.38802680 -0.48922578
## [187] -0.39939179 -0.71118176 -1.07369267 -0.53512569 0.48185240 1.52990415
## [193] 3.93599432 4.75299655 5.28489601 3.75245965 5.76106696 5.08023491
## [199] 5.03046078 3.85049062 2.27094879 1.61300623 0.02267186 -0.08744134
## [205] -0.23338993 -1.01395405 -4.02632785 -4.41061552 -5.02289664 -2.94041531
## [211] -2.16023161 -0.87458164 -1.76626866 -0.53794493 -0.27753067 0.71955739
## [217] 2.19354067 2.33905070 2.84874980 2.50865574 2.52684453 3.16087589
## [223] 3.22420516 3.72442687 1.90727460 1.27139490 1.66409882 0.23091397
## [229] 1.10296539 0.37155340 1.69901525 1.20826458 1.54849208 -0.89323198
## [235] -0.40844829 -1.97195019 -1.36977976 -2.76244287 -1.88853615 -1.38886217
## [241] -0.87915921 1.38576118 1.73484421 3.45643730 4.90637915 3.88230417
## [247] 4.70024354 4.04594276 3.83665310 3.63604751 1.69745094 1.83280517
## [253] 0.13589632 1.24990445 -0.25440835 0.60581161 -0.95359799 -0.74837042
## [259] -2.72398269 -2.15183291 -1.51561139 -0.53368698 -1.67162994 -2.15962923
## [265] -1.06604857 1.13495225 0.89677762 0.92454452 0.14354987 0.49726828
## [271] 0.03722489 -0.87570167 -1.33912012 -0.47863083 -0.18724669 -0.22304216
## [277] -0.89877996 0.52922646 -0.01897051 1.64938777 2.32971329 2.98499263
## [283] 2.59721428 1.95311423 0.73468623 1.76477012 1.07548761 3.11749005
## [289] 1.74249123 1.55297139 0.67230168 0.33991628 -0.10447830 -1.75292012
## [295] -1.80251894 -2.90523171 -0.99909951#Yt-3
yt_3_arma <- arma22[-c(298,299,300)]
yt_3_arma## [1] -2.75189059 -1.74814288 -2.98064692 -1.12111691 -1.75293106 -1.33898765
## [7] -2.09046996 -3.56829323 -4.32225209 -6.49932813 -7.02769509 -6.95242810
## [13] -6.26996407 -3.43890672 -3.74182891 -2.97181976 -3.49823015 -3.96328709
## [19] -4.21591372 -4.99194225 -5.61517562 -5.49498965 -5.32810268 -6.08211738
## [25] -5.70927763 -6.49640104 -6.19781946 -5.63181804 -4.16631811 -1.43386858
## [31] -1.32659085 1.33518960 -0.57718012 1.13545868 2.56923465 2.89049818
## [37] 2.80495292 1.68426168 2.77538382 1.29220743 3.17981524 3.04188012
## [43] 3.84589813 3.47054592 2.25362466 1.27634579 1.31729358 3.02358708
## [49] 2.83893413 1.81428668 0.10302896 -0.45121970 -0.85290753 -0.66635425
## [55] -0.93510416 -2.14779642 -2.08562341 -1.51503473 -0.24121274 0.91556687
## [61] 0.64537308 0.48311177 -1.14217407 -1.12039092 -1.27141453 0.91636167
## [67] 2.04091439 2.13232979 2.23257264 0.25301429 1.21071226 1.29939237
## [73] 3.92904462 4.07112121 3.78513073 2.83160983 1.41191959 0.28488816
## [79] -0.04006883 -1.28812485 -0.74220308 -0.74424287 -0.32976863 -0.31752860
## [85] -0.98342470 -2.04217397 -1.20184647 -1.31314616 0.45270778 -0.93468943
## [91] -0.20680752 -0.29613748 0.15497575 0.22025744 -0.73310547 1.30806278
## [97] 1.61203256 4.42693491 3.79591780 3.27848542 0.68061022 -0.62425847
## [103] -1.11211846 0.04164014 0.59240357 -0.15810424 0.39101368 -1.92189820
## [109] -1.50741039 -2.59060462 -3.89098912 -2.51252738 -3.45825773 -2.26278511
## [115] -0.93523383 0.07413764 0.83334886 1.28389670 1.34833602 3.06177019
## [121] 2.98115137 3.71727357 3.17046413 2.39423349 1.98646516 2.92352912
## [127] 1.69753466 2.18824295 2.31910758 1.98048956 1.12411345 -0.47375475
## [133] -1.38212484 -2.31108442 -2.26562282 -2.62211380 -2.26938378 -1.57393069
## [139] -0.56321701 1.57589801 1.65267412 1.80012273 2.53204975 3.72627716
## [145] 3.97712776 3.44105725 0.58660225 1.46428641 -0.51695507 -0.57765858
## [151] 1.69949100 1.51248377 2.98314953 -0.81043866 -1.02122299 -1.37591544
## [157] -0.14779506 1.31440815 3.22462748 4.38519006 3.98666624 4.07252161
## [163] 2.61160764 1.92537895 -1.70502573 -2.72222986 -3.24361938 -0.27688538
## [169] 0.87342985 2.50264542 1.00860632 1.15325179 -1.81437043 -0.92501023
## [175] -1.18146222 -0.02037563 0.94652009 1.17246770 2.08111275 3.60870710
## [181] 4.52620181 4.56792960 2.75523948 2.28308227 0.79090755 2.43630393
## [187] 0.12650863 0.38802680 -0.48922578 -0.39939179 -0.71118176 -1.07369267
## [193] -0.53512569 0.48185240 1.52990415 3.93599432 4.75299655 5.28489601
## [199] 3.75245965 5.76106696 5.08023491 5.03046078 3.85049062 2.27094879
## [205] 1.61300623 0.02267186 -0.08744134 -0.23338993 -1.01395405 -4.02632785
## [211] -4.41061552 -5.02289664 -2.94041531 -2.16023161 -0.87458164 -1.76626866
## [217] -0.53794493 -0.27753067 0.71955739 2.19354067 2.33905070 2.84874980
## [223] 2.50865574 2.52684453 3.16087589 3.22420516 3.72442687 1.90727460
## [229] 1.27139490 1.66409882 0.23091397 1.10296539 0.37155340 1.69901525
## [235] 1.20826458 1.54849208 -0.89323198 -0.40844829 -1.97195019 -1.36977976
## [241] -2.76244287 -1.88853615 -1.38886217 -0.87915921 1.38576118 1.73484421
## [247] 3.45643730 4.90637915 3.88230417 4.70024354 4.04594276 3.83665310
## [253] 3.63604751 1.69745094 1.83280517 0.13589632 1.24990445 -0.25440835
## [259] 0.60581161 -0.95359799 -0.74837042 -2.72398269 -2.15183291 -1.51561139
## [265] -0.53368698 -1.67162994 -2.15962923 -1.06604857 1.13495225 0.89677762
## [271] 0.92454452 0.14354987 0.49726828 0.03722489 -0.87570167 -1.33912012
## [277] -0.47863083 -0.18724669 -0.22304216 -0.89877996 0.52922646 -0.01897051
## [283] 1.64938777 2.32971329 2.98499263 2.59721428 1.95311423 0.73468623
## [289] 1.76477012 1.07548761 3.11749005 1.74249123 1.55297139 0.67230168
## [295] 0.33991628 -0.10447830 -1.75292012plot(y=yt_arma3,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
cor(yt_arma3,yt_3_arma)## [1] 0.6044325