AR-MA manual
Diva Nisfu Mustika
White Noise
set.seed(1002)
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 paling sederhana, yaitu MA(2) dengan \(\theta_1 = 0.4\) dan \(\theta_2 = 0.6\)sebanyak 300 observasi dan \(c=0\). Karena diperlukan satu nilai awal untuk \(e_{t-2}\), masukkan nilai pertama white noise sebagai nilai awal tersebut.
set.seed(1002)
ma<- wn[c(1,2)]
ma## [1] -0.1382882 0.2846285Nilai-nilai selanjutnya dapat dicari melalui loop. Bentuk loop dapat dilihat dari rumus MA(2) yang hendak dibangkitkan:
\[ y_t = e_t+0.4e_{t-1}+0.6e_{t-2} \]
set.seed(1002)
for(i in 3:300){
ma[i] <- wn[i] + 0.4 * wn[i - 1] + 0.6 * wn[i - 2]
}
ma## [1] -0.1382881842 0.2846285351 -0.2913725396 1.9471847822 0.2039450696
## [6] 1.2931078276 0.0480891236 -1.3443863362 -0.2071210815 0.6596565425
## [11] 0.2030366571 1.2024391609 0.7593035708 0.7033449886 1.9403469147
## [16] 0.3925484551 0.6674976688 -0.3446283632 1.9084081647 0.3019882409
## [21] 0.0756044605 -1.2845018549 -0.7347703151 1.1684735578 -0.7022092018
## [26] 0.8164260215 -0.0005709686 1.9069194491 1.2030284748 0.0707856096
## [31] -1.3810693189 -1.6142778433 -0.6245852823 -0.8725018478 0.2341620404
## [36] -0.6939341378 -0.4827299922 1.3884701207 -0.9152117468 0.0387632465
## [41] -1.6400317856 1.0005821561 0.7046606538 0.1053001179 -0.7675444479
## [46] -0.3746294673 0.0867119079 0.0624423465 1.0284534633 0.5889407161
## [51] 2.0826487470 0.8222595799 0.9659892717 1.3617771939 0.2808865606
## [56] 0.2927791590 -1.2425251238 -0.6145897515 0.2305059778 -0.0008796766
## [61] -0.3538948232 -1.3370825139 -1.0768203043 -0.8889290122 0.7482329302
## [66] -0.4956215950 0.8665308492 0.6985899983 0.5059316503 3.4378466046
## [71] 1.1085522764 0.6293708571 -0.3546625549 -0.6117615042 1.0054429401
## [76] 2.0239920538 0.9722914878 1.7884856065 -0.9500941082 -1.2890124481
## [81] 0.1920572179 -0.3562315333 1.9317168346 0.0930827747 0.7722635840
## [86] -1.3645812477 1.1925189739 -0.1266541905 1.9254811223 0.3148387204
## [91] -0.4656714742 -0.5976814383 -0.1252779443 2.0231193708 2.4625345920
## [96] 2.1910596217 1.4214113024 0.8766225940 0.9267589084 -0.8232050274
## [101] -1.6264722750 -2.0437599459 0.0593472813 -0.4766026587 -0.4274172270
## [106] 0.5865156735 -2.5723728557 0.2018284826 -1.3864904483 -0.2671476548
## [111] -0.1622579378 -1.8886213965 0.0963531485 -0.5773190694 1.9487790570
## [116] 1.3912223790 1.6739861739 0.8671740946 0.9379216612 -1.2430639940
## [121] -1.2425258378 -3.7908666936 0.1054106417 -1.0097051137 -0.1444730816
## [126] -1.3360990732 -0.7395354999 -0.5234724890 1.1519528340 -0.7542851340
## [131] 0.5545346831 1.0773612788 2.1864016426 1.3575846676 -0.0603187191
## [136] -1.3182072883 -1.4831605057 -2.2369214890 -1.5564018418 -1.5200332576
## [141] 0.5518822885 -0.5403428041 -1.0113170122 -0.4044165025 0.3765462648
## [146] -0.5139685156 0.2227773739 -0.8298204850 0.1493382246 0.4209347510
## [151] 1.5354375656 1.5069008021 1.4404986178 -1.1516092916 -0.2291156817
## [156] -1.8238967982 -1.2169569496 -0.3471614692 0.8206307081 1.2674616742
## [161] 2.3185736810 -0.6890438106 0.2954115052 -0.0694363757 0.8270306299
## [166] 1.2112715185 -1.3932545390 -0.6249472403 -0.5380299092 -0.0956441836
## [171] 0.0714389431 -1.5618061241 0.0605293549 -0.5223492304 0.2232705480
## [176] 1.1938956098 0.3911999599 1.4792286808 0.5822047003 1.5292924790
## [181] 0.9925623705 2.2805912681 0.6918573713 1.2662349029 -0.6728839038
## [186] 0.5485919729 1.3758753904 1.2765154339 1.6191261090 -0.5721040622
## [191] 0.6914655381 -2.4633202551 -1.7141729685 -4.3145435835 -2.1094561668
## [196] -1.1148026417 0.5188166714 0.4280945822 0.6413359989 1.7023184786
## [201] 1.5466777022 2.4972648386 1.8464181654 0.4783098581 0.2668014391
## [206] 0.6609506233 -2.6271836829 -1.0271632093 0.7188278491 -1.4071705984
## [211] 2.5029596596 -2.1767000016 0.2261918704 -1.3393432359 0.7714065328
## [216] 0.4177407565 -1.8540075065 -1.2359973739 -1.3729928044 -0.6071069445
## [221] 0.2325083951 -0.8897303942 0.7797076467 -0.5835862514 0.3875298318
## [226] -1.5752881050 -1.6394575680 -0.6532492389 -0.3842339593 0.9968224736
## [231] -0.8086459427 0.8868701042 -0.3780727327 2.5758449791 1.7148315670
## [236] 2.7324272009 0.8332397531 1.0009507824 1.9685409632 2.3697016706
## [241] 1.3812051877 2.6212638095 1.9812571526 0.7688579152 0.2944286504
## [246] 0.2373890442 0.0905573537 0.1444155868 0.2068117180 -0.1686321278
## [251] 0.5768913360 0.6174192653 0.2471049651 2.1248481028 -0.8621236553
## [256] 1.8551046478 -0.7396138335 0.9953012828 1.1706687453 -0.2612307520
## [261] 1.0922108791 -1.6247741546 -0.0303707421 -2.2773534114 0.2039318009
## [266] -0.9852282256 -0.8710586780 -0.3454705217 -0.7734511977 -0.8588926361
## [271] 0.5258935590 -0.6277777871 0.1381115635 0.3688323198 0.5049496838
## [276] 0.1912585536 -0.4319266648 -1.3755588492 -0.3239822272 -0.4062433845
## [281] 0.7200298762 0.1349736491 -0.3088818566 -0.2619393812 -0.9094469352
## [286] -0.6455037033 0.4139233572 1.2062168717 2.4547921170 1.4016281232
## [291] 0.0525150278 -0.2901311561 0.0027978397 0.5333436989 0.5582367761
## [296] 0.1770709978 -0.0980639856 0.3733346775 -1.7322856400 0.7996808082Selain menggunakan cara di atas, pembangkitan proses MA(2) dapat
dilakukan dengan fungsi arima.sim() sebagai berikut.
set.seed(1002)
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] -0.2913725396 1.9471847822 0.2039450696 1.2931078276 0.0480891236
## [6] -1.3443863362 -0.2071210815 0.6596565425 0.2030366571 1.2024391609
## [11] 0.7593035708 0.7033449886 1.9403469147 0.3925484551 0.6674976688
## [16] -0.3446283632 1.9084081647 0.3019882409 0.0756044605 -1.2845018549
## [21] -0.7347703151 1.1684735578 -0.7022092018 0.8164260215 -0.0005709686
## [26] 1.9069194491 1.2030284748 0.0707856096 -1.3810693189 -1.6142778433
## [31] -0.6245852823 -0.8725018478 0.2341620404 -0.6939341378 -0.4827299922
## [36] 1.3884701207 -0.9152117468 0.0387632465 -1.6400317856 1.0005821561
## [41] 0.7046606538 0.1053001179 -0.7675444479 -0.3746294673 0.0867119079
## [46] 0.0624423465 1.0284534633 0.5889407161 2.0826487470 0.8222595799
## [51] 0.9659892717 1.3617771939 0.2808865606 0.2927791590 -1.2425251238
## [56] -0.6145897515 0.2305059778 -0.0008796766 -0.3538948232 -1.3370825139
## [61] -1.0768203043 -0.8889290122 0.7482329302 -0.4956215950 0.8665308492
## [66] 0.6985899983 0.5059316503 3.4378466046 1.1085522764 0.6293708571
## [71] -0.3546625549 -0.6117615042 1.0054429401 2.0239920538 0.9722914878
## [76] 1.7884856065 -0.9500941082 -1.2890124481 0.1920572179 -0.3562315333
## [81] 1.9317168346 0.0930827747 0.7722635840 -1.3645812477 1.1925189739
## [86] -0.1266541905 1.9254811223 0.3148387204 -0.4656714742 -0.5976814383
## [91] -0.1252779443 2.0231193708 2.4625345920 2.1910596217 1.4214113024
## [96] 0.8766225940 0.9267589084 -0.8232050274 -1.6264722750 -2.0437599459
## [101] 0.0593472813 -0.4766026587 -0.4274172270 0.5865156735 -2.5723728557
## [106] 0.2018284826 -1.3864904483 -0.2671476548 -0.1622579378 -1.8886213965
## [111] 0.0963531485 -0.5773190694 1.9487790570 1.3912223790 1.6739861739
## [116] 0.8671740946 0.9379216612 -1.2430639940 -1.2425258378 -3.7908666936
## [121] 0.1054106417 -1.0097051137 -0.1444730816 -1.3360990732 -0.7395354999
## [126] -0.5234724890 1.1519528340 -0.7542851340 0.5545346831 1.0773612788
## [131] 2.1864016426 1.3575846676 -0.0603187191 -1.3182072883 -1.4831605057
## [136] -2.2369214890 -1.5564018418 -1.5200332576 0.5518822885 -0.5403428041
## [141] -1.0113170122 -0.4044165025 0.3765462648 -0.5139685156 0.2227773739
## [146] -0.8298204850 0.1493382246 0.4209347510 1.5354375656 1.5069008021
## [151] 1.4404986178 -1.1516092916 -0.2291156817 -1.8238967982 -1.2169569496
## [156] -0.3471614692 0.8206307081 1.2674616742 2.3185736810 -0.6890438106
## [161] 0.2954115052 -0.0694363757 0.8270306299 1.2112715185 -1.3932545390
## [166] -0.6249472403 -0.5380299092 -0.0956441836 0.0714389431 -1.5618061241
## [171] 0.0605293549 -0.5223492304 0.2232705480 1.1938956098 0.3911999599
## [176] 1.4792286808 0.5822047003 1.5292924790 0.9925623705 2.2805912681
## [181] 0.6918573713 1.2662349029 -0.6728839038 0.5485919729 1.3758753904
## [186] 1.2765154339 1.6191261090 -0.5721040622 0.6914655381 -2.4633202551
## [191] -1.7141729685 -4.3145435835 -2.1094561668 -1.1148026417 0.5188166714
## [196] 0.4280945822 0.6413359989 1.7023184786 1.5466777022 2.4972648386
## [201] 1.8464181654 0.4783098581 0.2668014391 0.6609506233 -2.6271836829
## [206] -1.0271632093 0.7188278491 -1.4071705984 2.5029596596 -2.1767000016
## [211] 0.2261918704 -1.3393432359 0.7714065328 0.4177407565 -1.8540075065
## [216] -1.2359973739 -1.3729928044 -0.6071069445 0.2325083951 -0.8897303942
## [221] 0.7797076467 -0.5835862514 0.3875298318 -1.5752881050 -1.6394575680
## [226] -0.6532492389 -0.3842339593 0.9968224736 -0.8086459427 0.8868701042
## [231] -0.3780727327 2.5758449791 1.7148315670 2.7324272009 0.8332397531
## [236] 1.0009507824 1.9685409632 2.3697016706 1.3812051877 2.6212638095
## [241] 1.9812571526 0.7688579152 0.2944286504 0.2373890442 0.0905573537
## [246] 0.1444155868 0.2068117180 -0.1686321278 0.5768913360 0.6174192653
## [251] 0.2471049651 2.1248481028 -0.8621236553 1.8551046478 -0.7396138335
## [256] 0.9953012828 1.1706687453 -0.2612307520 1.0922108791 -1.6247741546
## [261] -0.0303707421 -2.2773534114 0.2039318009 -0.9852282256 -0.8710586780
## [266] -0.3454705217 -0.7734511977 -0.8588926361 0.5258935590 -0.6277777871
## [271] 0.1381115635 0.3688323198 0.5049496838 0.1912585536 -0.4319266648
## [276] -1.3755588492 -0.3239822272 -0.4062433845 0.7200298762 0.1349736491
## [281] -0.3088818566 -0.2619393812 -0.9094469352 -0.6455037033 0.4139233572
## [286] 1.2062168717 2.4547921170 1.4016281232 0.0525150278 -0.2901311561
## [291] 0.0027978397 0.5333436989 0.5582367761 0.1770709978 -0.0980639856
## [296] 0.3733346775 -1.7322856400 0.7996808082 -1.0146825427 2.1128443720Karakteristik MA(2)
Plot Time Series
ts.plot(ma2)
Berdasarkan plot time series, terlihat bahwa data MA(2) yang dibangkitkan stasioner dalam rataan
Plot ACF
acf(ma2,lag.max = 20)
Berdasarkan plot AFC tersebut, terlihat bahwa plot ACF cuts off di lag kedua
Plot PACF
pacf(ma2)
Berdasarkan plot PACF tersebut, terlihat bahwa plot PACF cenderung tails off dan membentuk gelombang sinus
Plot EACF
TSA::eacf(ma2)## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x o o o o o o o o o o o o
## 1 x x x o o o o o o o o o o o
## 2 x x o o o o o o o o o o o o
## 3 x x o o o o o o o o o o o o
## 4 x x x o x x o o o o o o o o
## 5 x x x x x x o o o o o o o o
## 6 x x x x o o x x o o o o o o
## 7 x x x x x o o x 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_ma2 <- ma2[-1]
yt_ma2## [1] 1.9471847822 0.2039450696 1.2931078276 0.0480891236 -1.3443863362
## [6] -0.2071210815 0.6596565425 0.2030366571 1.2024391609 0.7593035708
## [11] 0.7033449886 1.9403469147 0.3925484551 0.6674976688 -0.3446283632
## [16] 1.9084081647 0.3019882409 0.0756044605 -1.2845018549 -0.7347703151
## [21] 1.1684735578 -0.7022092018 0.8164260215 -0.0005709686 1.9069194491
## [26] 1.2030284748 0.0707856096 -1.3810693189 -1.6142778433 -0.6245852823
## [31] -0.8725018478 0.2341620404 -0.6939341378 -0.4827299922 1.3884701207
## [36] -0.9152117468 0.0387632465 -1.6400317856 1.0005821561 0.7046606538
## [41] 0.1053001179 -0.7675444479 -0.3746294673 0.0867119079 0.0624423465
## [46] 1.0284534633 0.5889407161 2.0826487470 0.8222595799 0.9659892717
## [51] 1.3617771939 0.2808865606 0.2927791590 -1.2425251238 -0.6145897515
## [56] 0.2305059778 -0.0008796766 -0.3538948232 -1.3370825139 -1.0768203043
## [61] -0.8889290122 0.7482329302 -0.4956215950 0.8665308492 0.6985899983
## [66] 0.5059316503 3.4378466046 1.1085522764 0.6293708571 -0.3546625549
## [71] -0.6117615042 1.0054429401 2.0239920538 0.9722914878 1.7884856065
## [76] -0.9500941082 -1.2890124481 0.1920572179 -0.3562315333 1.9317168346
## [81] 0.0930827747 0.7722635840 -1.3645812477 1.1925189739 -0.1266541905
## [86] 1.9254811223 0.3148387204 -0.4656714742 -0.5976814383 -0.1252779443
## [91] 2.0231193708 2.4625345920 2.1910596217 1.4214113024 0.8766225940
## [96] 0.9267589084 -0.8232050274 -1.6264722750 -2.0437599459 0.0593472813
## [101] -0.4766026587 -0.4274172270 0.5865156735 -2.5723728557 0.2018284826
## [106] -1.3864904483 -0.2671476548 -0.1622579378 -1.8886213965 0.0963531485
## [111] -0.5773190694 1.9487790570 1.3912223790 1.6739861739 0.8671740946
## [116] 0.9379216612 -1.2430639940 -1.2425258378 -3.7908666936 0.1054106417
## [121] -1.0097051137 -0.1444730816 -1.3360990732 -0.7395354999 -0.5234724890
## [126] 1.1519528340 -0.7542851340 0.5545346831 1.0773612788 2.1864016426
## [131] 1.3575846676 -0.0603187191 -1.3182072883 -1.4831605057 -2.2369214890
## [136] -1.5564018418 -1.5200332576 0.5518822885 -0.5403428041 -1.0113170122
## [141] -0.4044165025 0.3765462648 -0.5139685156 0.2227773739 -0.8298204850
## [146] 0.1493382246 0.4209347510 1.5354375656 1.5069008021 1.4404986178
## [151] -1.1516092916 -0.2291156817 -1.8238967982 -1.2169569496 -0.3471614692
## [156] 0.8206307081 1.2674616742 2.3185736810 -0.6890438106 0.2954115052
## [161] -0.0694363757 0.8270306299 1.2112715185 -1.3932545390 -0.6249472403
## [166] -0.5380299092 -0.0956441836 0.0714389431 -1.5618061241 0.0605293549
## [171] -0.5223492304 0.2232705480 1.1938956098 0.3911999599 1.4792286808
## [176] 0.5822047003 1.5292924790 0.9925623705 2.2805912681 0.6918573713
## [181] 1.2662349029 -0.6728839038 0.5485919729 1.3758753904 1.2765154339
## [186] 1.6191261090 -0.5721040622 0.6914655381 -2.4633202551 -1.7141729685
## [191] -4.3145435835 -2.1094561668 -1.1148026417 0.5188166714 0.4280945822
## [196] 0.6413359989 1.7023184786 1.5466777022 2.4972648386 1.8464181654
## [201] 0.4783098581 0.2668014391 0.6609506233 -2.6271836829 -1.0271632093
## [206] 0.7188278491 -1.4071705984 2.5029596596 -2.1767000016 0.2261918704
## [211] -1.3393432359 0.7714065328 0.4177407565 -1.8540075065 -1.2359973739
## [216] -1.3729928044 -0.6071069445 0.2325083951 -0.8897303942 0.7797076467
## [221] -0.5835862514 0.3875298318 -1.5752881050 -1.6394575680 -0.6532492389
## [226] -0.3842339593 0.9968224736 -0.8086459427 0.8868701042 -0.3780727327
## [231] 2.5758449791 1.7148315670 2.7324272009 0.8332397531 1.0009507824
## [236] 1.9685409632 2.3697016706 1.3812051877 2.6212638095 1.9812571526
## [241] 0.7688579152 0.2944286504 0.2373890442 0.0905573537 0.1444155868
## [246] 0.2068117180 -0.1686321278 0.5768913360 0.6174192653 0.2471049651
## [251] 2.1248481028 -0.8621236553 1.8551046478 -0.7396138335 0.9953012828
## [256] 1.1706687453 -0.2612307520 1.0922108791 -1.6247741546 -0.0303707421
## [261] -2.2773534114 0.2039318009 -0.9852282256 -0.8710586780 -0.3454705217
## [266] -0.7734511977 -0.8588926361 0.5258935590 -0.6277777871 0.1381115635
## [271] 0.3688323198 0.5049496838 0.1912585536 -0.4319266648 -1.3755588492
## [276] -0.3239822272 -0.4062433845 0.7200298762 0.1349736491 -0.3088818566
## [281] -0.2619393812 -0.9094469352 -0.6455037033 0.4139233572 1.2062168717
## [286] 2.4547921170 1.4016281232 0.0525150278 -0.2901311561 0.0027978397
## [291] 0.5333436989 0.5582367761 0.1770709978 -0.0980639856 0.3733346775
## [296] -1.7322856400 0.7996808082 -1.0146825427 2.1128443720#Yt-1
yt_1_ma2 <- ma2[-300]
yt_1_ma2## [1] -0.2913725396 1.9471847822 0.2039450696 1.2931078276 0.0480891236
## [6] -1.3443863362 -0.2071210815 0.6596565425 0.2030366571 1.2024391609
## [11] 0.7593035708 0.7033449886 1.9403469147 0.3925484551 0.6674976688
## [16] -0.3446283632 1.9084081647 0.3019882409 0.0756044605 -1.2845018549
## [21] -0.7347703151 1.1684735578 -0.7022092018 0.8164260215 -0.0005709686
## [26] 1.9069194491 1.2030284748 0.0707856096 -1.3810693189 -1.6142778433
## [31] -0.6245852823 -0.8725018478 0.2341620404 -0.6939341378 -0.4827299922
## [36] 1.3884701207 -0.9152117468 0.0387632465 -1.6400317856 1.0005821561
## [41] 0.7046606538 0.1053001179 -0.7675444479 -0.3746294673 0.0867119079
## [46] 0.0624423465 1.0284534633 0.5889407161 2.0826487470 0.8222595799
## [51] 0.9659892717 1.3617771939 0.2808865606 0.2927791590 -1.2425251238
## [56] -0.6145897515 0.2305059778 -0.0008796766 -0.3538948232 -1.3370825139
## [61] -1.0768203043 -0.8889290122 0.7482329302 -0.4956215950 0.8665308492
## [66] 0.6985899983 0.5059316503 3.4378466046 1.1085522764 0.6293708571
## [71] -0.3546625549 -0.6117615042 1.0054429401 2.0239920538 0.9722914878
## [76] 1.7884856065 -0.9500941082 -1.2890124481 0.1920572179 -0.3562315333
## [81] 1.9317168346 0.0930827747 0.7722635840 -1.3645812477 1.1925189739
## [86] -0.1266541905 1.9254811223 0.3148387204 -0.4656714742 -0.5976814383
## [91] -0.1252779443 2.0231193708 2.4625345920 2.1910596217 1.4214113024
## [96] 0.8766225940 0.9267589084 -0.8232050274 -1.6264722750 -2.0437599459
## [101] 0.0593472813 -0.4766026587 -0.4274172270 0.5865156735 -2.5723728557
## [106] 0.2018284826 -1.3864904483 -0.2671476548 -0.1622579378 -1.8886213965
## [111] 0.0963531485 -0.5773190694 1.9487790570 1.3912223790 1.6739861739
## [116] 0.8671740946 0.9379216612 -1.2430639940 -1.2425258378 -3.7908666936
## [121] 0.1054106417 -1.0097051137 -0.1444730816 -1.3360990732 -0.7395354999
## [126] -0.5234724890 1.1519528340 -0.7542851340 0.5545346831 1.0773612788
## [131] 2.1864016426 1.3575846676 -0.0603187191 -1.3182072883 -1.4831605057
## [136] -2.2369214890 -1.5564018418 -1.5200332576 0.5518822885 -0.5403428041
## [141] -1.0113170122 -0.4044165025 0.3765462648 -0.5139685156 0.2227773739
## [146] -0.8298204850 0.1493382246 0.4209347510 1.5354375656 1.5069008021
## [151] 1.4404986178 -1.1516092916 -0.2291156817 -1.8238967982 -1.2169569496
## [156] -0.3471614692 0.8206307081 1.2674616742 2.3185736810 -0.6890438106
## [161] 0.2954115052 -0.0694363757 0.8270306299 1.2112715185 -1.3932545390
## [166] -0.6249472403 -0.5380299092 -0.0956441836 0.0714389431 -1.5618061241
## [171] 0.0605293549 -0.5223492304 0.2232705480 1.1938956098 0.3911999599
## [176] 1.4792286808 0.5822047003 1.5292924790 0.9925623705 2.2805912681
## [181] 0.6918573713 1.2662349029 -0.6728839038 0.5485919729 1.3758753904
## [186] 1.2765154339 1.6191261090 -0.5721040622 0.6914655381 -2.4633202551
## [191] -1.7141729685 -4.3145435835 -2.1094561668 -1.1148026417 0.5188166714
## [196] 0.4280945822 0.6413359989 1.7023184786 1.5466777022 2.4972648386
## [201] 1.8464181654 0.4783098581 0.2668014391 0.6609506233 -2.6271836829
## [206] -1.0271632093 0.7188278491 -1.4071705984 2.5029596596 -2.1767000016
## [211] 0.2261918704 -1.3393432359 0.7714065328 0.4177407565 -1.8540075065
## [216] -1.2359973739 -1.3729928044 -0.6071069445 0.2325083951 -0.8897303942
## [221] 0.7797076467 -0.5835862514 0.3875298318 -1.5752881050 -1.6394575680
## [226] -0.6532492389 -0.3842339593 0.9968224736 -0.8086459427 0.8868701042
## [231] -0.3780727327 2.5758449791 1.7148315670 2.7324272009 0.8332397531
## [236] 1.0009507824 1.9685409632 2.3697016706 1.3812051877 2.6212638095
## [241] 1.9812571526 0.7688579152 0.2944286504 0.2373890442 0.0905573537
## [246] 0.1444155868 0.2068117180 -0.1686321278 0.5768913360 0.6174192653
## [251] 0.2471049651 2.1248481028 -0.8621236553 1.8551046478 -0.7396138335
## [256] 0.9953012828 1.1706687453 -0.2612307520 1.0922108791 -1.6247741546
## [261] -0.0303707421 -2.2773534114 0.2039318009 -0.9852282256 -0.8710586780
## [266] -0.3454705217 -0.7734511977 -0.8588926361 0.5258935590 -0.6277777871
## [271] 0.1381115635 0.3688323198 0.5049496838 0.1912585536 -0.4319266648
## [276] -1.3755588492 -0.3239822272 -0.4062433845 0.7200298762 0.1349736491
## [281] -0.3088818566 -0.2619393812 -0.9094469352 -0.6455037033 0.4139233572
## [286] 1.2062168717 2.4547921170 1.4016281232 0.0525150278 -0.2901311561
## [291] 0.0027978397 0.5333436989 0.5582367761 0.1770709978 -0.0980639856
## [296] 0.3733346775 -1.7322856400 0.7996808082 -1.0146825427cor(yt_ma2,yt_1_ma2)## [1] 0.3298152Korelasi antara \(Y_t\) dengan \(Y_{t-1}\) hasil simulasi yaitu 0.3298152
plot(y=yt_ma2,x=yt_1_ma2)
Berdasarkan scatterplot tersebut, terlihat bahwa cenderung terdapat hubungan positif lemah antara \(Y_t\) dengan \(Y_{t-1}\).
Korelasi antara \(Y_t\) dengan \(Y_{t-2}\)
#Yt
yt_ma2 <- ma2[-c(1,2)]
yt_ma2## [1] 0.2039450696 1.2931078276 0.0480891236 -1.3443863362 -0.2071210815
## [6] 0.6596565425 0.2030366571 1.2024391609 0.7593035708 0.7033449886
## [11] 1.9403469147 0.3925484551 0.6674976688 -0.3446283632 1.9084081647
## [16] 0.3019882409 0.0756044605 -1.2845018549 -0.7347703151 1.1684735578
## [21] -0.7022092018 0.8164260215 -0.0005709686 1.9069194491 1.2030284748
## [26] 0.0707856096 -1.3810693189 -1.6142778433 -0.6245852823 -0.8725018478
## [31] 0.2341620404 -0.6939341378 -0.4827299922 1.3884701207 -0.9152117468
## [36] 0.0387632465 -1.6400317856 1.0005821561 0.7046606538 0.1053001179
## [41] -0.7675444479 -0.3746294673 0.0867119079 0.0624423465 1.0284534633
## [46] 0.5889407161 2.0826487470 0.8222595799 0.9659892717 1.3617771939
## [51] 0.2808865606 0.2927791590 -1.2425251238 -0.6145897515 0.2305059778
## [56] -0.0008796766 -0.3538948232 -1.3370825139 -1.0768203043 -0.8889290122
## [61] 0.7482329302 -0.4956215950 0.8665308492 0.6985899983 0.5059316503
## [66] 3.4378466046 1.1085522764 0.6293708571 -0.3546625549 -0.6117615042
## [71] 1.0054429401 2.0239920538 0.9722914878 1.7884856065 -0.9500941082
## [76] -1.2890124481 0.1920572179 -0.3562315333 1.9317168346 0.0930827747
## [81] 0.7722635840 -1.3645812477 1.1925189739 -0.1266541905 1.9254811223
## [86] 0.3148387204 -0.4656714742 -0.5976814383 -0.1252779443 2.0231193708
## [91] 2.4625345920 2.1910596217 1.4214113024 0.8766225940 0.9267589084
## [96] -0.8232050274 -1.6264722750 -2.0437599459 0.0593472813 -0.4766026587
## [101] -0.4274172270 0.5865156735 -2.5723728557 0.2018284826 -1.3864904483
## [106] -0.2671476548 -0.1622579378 -1.8886213965 0.0963531485 -0.5773190694
## [111] 1.9487790570 1.3912223790 1.6739861739 0.8671740946 0.9379216612
## [116] -1.2430639940 -1.2425258378 -3.7908666936 0.1054106417 -1.0097051137
## [121] -0.1444730816 -1.3360990732 -0.7395354999 -0.5234724890 1.1519528340
## [126] -0.7542851340 0.5545346831 1.0773612788 2.1864016426 1.3575846676
## [131] -0.0603187191 -1.3182072883 -1.4831605057 -2.2369214890 -1.5564018418
## [136] -1.5200332576 0.5518822885 -0.5403428041 -1.0113170122 -0.4044165025
## [141] 0.3765462648 -0.5139685156 0.2227773739 -0.8298204850 0.1493382246
## [146] 0.4209347510 1.5354375656 1.5069008021 1.4404986178 -1.1516092916
## [151] -0.2291156817 -1.8238967982 -1.2169569496 -0.3471614692 0.8206307081
## [156] 1.2674616742 2.3185736810 -0.6890438106 0.2954115052 -0.0694363757
## [161] 0.8270306299 1.2112715185 -1.3932545390 -0.6249472403 -0.5380299092
## [166] -0.0956441836 0.0714389431 -1.5618061241 0.0605293549 -0.5223492304
## [171] 0.2232705480 1.1938956098 0.3911999599 1.4792286808 0.5822047003
## [176] 1.5292924790 0.9925623705 2.2805912681 0.6918573713 1.2662349029
## [181] -0.6728839038 0.5485919729 1.3758753904 1.2765154339 1.6191261090
## [186] -0.5721040622 0.6914655381 -2.4633202551 -1.7141729685 -4.3145435835
## [191] -2.1094561668 -1.1148026417 0.5188166714 0.4280945822 0.6413359989
## [196] 1.7023184786 1.5466777022 2.4972648386 1.8464181654 0.4783098581
## [201] 0.2668014391 0.6609506233 -2.6271836829 -1.0271632093 0.7188278491
## [206] -1.4071705984 2.5029596596 -2.1767000016 0.2261918704 -1.3393432359
## [211] 0.7714065328 0.4177407565 -1.8540075065 -1.2359973739 -1.3729928044
## [216] -0.6071069445 0.2325083951 -0.8897303942 0.7797076467 -0.5835862514
## [221] 0.3875298318 -1.5752881050 -1.6394575680 -0.6532492389 -0.3842339593
## [226] 0.9968224736 -0.8086459427 0.8868701042 -0.3780727327 2.5758449791
## [231] 1.7148315670 2.7324272009 0.8332397531 1.0009507824 1.9685409632
## [236] 2.3697016706 1.3812051877 2.6212638095 1.9812571526 0.7688579152
## [241] 0.2944286504 0.2373890442 0.0905573537 0.1444155868 0.2068117180
## [246] -0.1686321278 0.5768913360 0.6174192653 0.2471049651 2.1248481028
## [251] -0.8621236553 1.8551046478 -0.7396138335 0.9953012828 1.1706687453
## [256] -0.2612307520 1.0922108791 -1.6247741546 -0.0303707421 -2.2773534114
## [261] 0.2039318009 -0.9852282256 -0.8710586780 -0.3454705217 -0.7734511977
## [266] -0.8588926361 0.5258935590 -0.6277777871 0.1381115635 0.3688323198
## [271] 0.5049496838 0.1912585536 -0.4319266648 -1.3755588492 -0.3239822272
## [276] -0.4062433845 0.7200298762 0.1349736491 -0.3088818566 -0.2619393812
## [281] -0.9094469352 -0.6455037033 0.4139233572 1.2062168717 2.4547921170
## [286] 1.4016281232 0.0525150278 -0.2901311561 0.0027978397 0.5333436989
## [291] 0.5582367761 0.1770709978 -0.0980639856 0.3733346775 -1.7322856400
## [296] 0.7996808082 -1.0146825427 2.1128443720#Yt-2
yt_2_ma2 <- ma2[-c(299,300)]
yt_2_ma2## [1] -0.2913725396 1.9471847822 0.2039450696 1.2931078276 0.0480891236
## [6] -1.3443863362 -0.2071210815 0.6596565425 0.2030366571 1.2024391609
## [11] 0.7593035708 0.7033449886 1.9403469147 0.3925484551 0.6674976688
## [16] -0.3446283632 1.9084081647 0.3019882409 0.0756044605 -1.2845018549
## [21] -0.7347703151 1.1684735578 -0.7022092018 0.8164260215 -0.0005709686
## [26] 1.9069194491 1.2030284748 0.0707856096 -1.3810693189 -1.6142778433
## [31] -0.6245852823 -0.8725018478 0.2341620404 -0.6939341378 -0.4827299922
## [36] 1.3884701207 -0.9152117468 0.0387632465 -1.6400317856 1.0005821561
## [41] 0.7046606538 0.1053001179 -0.7675444479 -0.3746294673 0.0867119079
## [46] 0.0624423465 1.0284534633 0.5889407161 2.0826487470 0.8222595799
## [51] 0.9659892717 1.3617771939 0.2808865606 0.2927791590 -1.2425251238
## [56] -0.6145897515 0.2305059778 -0.0008796766 -0.3538948232 -1.3370825139
## [61] -1.0768203043 -0.8889290122 0.7482329302 -0.4956215950 0.8665308492
## [66] 0.6985899983 0.5059316503 3.4378466046 1.1085522764 0.6293708571
## [71] -0.3546625549 -0.6117615042 1.0054429401 2.0239920538 0.9722914878
## [76] 1.7884856065 -0.9500941082 -1.2890124481 0.1920572179 -0.3562315333
## [81] 1.9317168346 0.0930827747 0.7722635840 -1.3645812477 1.1925189739
## [86] -0.1266541905 1.9254811223 0.3148387204 -0.4656714742 -0.5976814383
## [91] -0.1252779443 2.0231193708 2.4625345920 2.1910596217 1.4214113024
## [96] 0.8766225940 0.9267589084 -0.8232050274 -1.6264722750 -2.0437599459
## [101] 0.0593472813 -0.4766026587 -0.4274172270 0.5865156735 -2.5723728557
## [106] 0.2018284826 -1.3864904483 -0.2671476548 -0.1622579378 -1.8886213965
## [111] 0.0963531485 -0.5773190694 1.9487790570 1.3912223790 1.6739861739
## [116] 0.8671740946 0.9379216612 -1.2430639940 -1.2425258378 -3.7908666936
## [121] 0.1054106417 -1.0097051137 -0.1444730816 -1.3360990732 -0.7395354999
## [126] -0.5234724890 1.1519528340 -0.7542851340 0.5545346831 1.0773612788
## [131] 2.1864016426 1.3575846676 -0.0603187191 -1.3182072883 -1.4831605057
## [136] -2.2369214890 -1.5564018418 -1.5200332576 0.5518822885 -0.5403428041
## [141] -1.0113170122 -0.4044165025 0.3765462648 -0.5139685156 0.2227773739
## [146] -0.8298204850 0.1493382246 0.4209347510 1.5354375656 1.5069008021
## [151] 1.4404986178 -1.1516092916 -0.2291156817 -1.8238967982 -1.2169569496
## [156] -0.3471614692 0.8206307081 1.2674616742 2.3185736810 -0.6890438106
## [161] 0.2954115052 -0.0694363757 0.8270306299 1.2112715185 -1.3932545390
## [166] -0.6249472403 -0.5380299092 -0.0956441836 0.0714389431 -1.5618061241
## [171] 0.0605293549 -0.5223492304 0.2232705480 1.1938956098 0.3911999599
## [176] 1.4792286808 0.5822047003 1.5292924790 0.9925623705 2.2805912681
## [181] 0.6918573713 1.2662349029 -0.6728839038 0.5485919729 1.3758753904
## [186] 1.2765154339 1.6191261090 -0.5721040622 0.6914655381 -2.4633202551
## [191] -1.7141729685 -4.3145435835 -2.1094561668 -1.1148026417 0.5188166714
## [196] 0.4280945822 0.6413359989 1.7023184786 1.5466777022 2.4972648386
## [201] 1.8464181654 0.4783098581 0.2668014391 0.6609506233 -2.6271836829
## [206] -1.0271632093 0.7188278491 -1.4071705984 2.5029596596 -2.1767000016
## [211] 0.2261918704 -1.3393432359 0.7714065328 0.4177407565 -1.8540075065
## [216] -1.2359973739 -1.3729928044 -0.6071069445 0.2325083951 -0.8897303942
## [221] 0.7797076467 -0.5835862514 0.3875298318 -1.5752881050 -1.6394575680
## [226] -0.6532492389 -0.3842339593 0.9968224736 -0.8086459427 0.8868701042
## [231] -0.3780727327 2.5758449791 1.7148315670 2.7324272009 0.8332397531
## [236] 1.0009507824 1.9685409632 2.3697016706 1.3812051877 2.6212638095
## [241] 1.9812571526 0.7688579152 0.2944286504 0.2373890442 0.0905573537
## [246] 0.1444155868 0.2068117180 -0.1686321278 0.5768913360 0.6174192653
## [251] 0.2471049651 2.1248481028 -0.8621236553 1.8551046478 -0.7396138335
## [256] 0.9953012828 1.1706687453 -0.2612307520 1.0922108791 -1.6247741546
## [261] -0.0303707421 -2.2773534114 0.2039318009 -0.9852282256 -0.8710586780
## [266] -0.3454705217 -0.7734511977 -0.8588926361 0.5258935590 -0.6277777871
## [271] 0.1381115635 0.3688323198 0.5049496838 0.1912585536 -0.4319266648
## [276] -1.3755588492 -0.3239822272 -0.4062433845 0.7200298762 0.1349736491
## [281] -0.3088818566 -0.2619393812 -0.9094469352 -0.6455037033 0.4139233572
## [286] 1.2062168717 2.4547921170 1.4016281232 0.0525150278 -0.2901311561
## [291] 0.0027978397 0.5333436989 0.5582367761 0.1770709978 -0.0980639856
## [296] 0.3733346775 -1.7322856400 0.7996808082cor(yt_ma2,yt_2_ma2)## [1] 0.3985644Korelasi antara \(Y_t\) dengan \(Y_{t-2}\) hasil simulasi 0.3985644
plot(y=yt_ma2,x=yt_2_ma2)
Berdasarkan scatterplot tersebut, terlihat bahwa cenderung terdapat hubungan psitif lemah antara \(Y_t\) dengan \(Y_{t-2}\).
Korelasi antara \(Y_t\) dengan \(Y_{t-3}\)
#Yt
yt_ma2 <- ma2[-c(1,2,3)]
yt_ma2## [1] 1.2931078276 0.0480891236 -1.3443863362 -0.2071210815 0.6596565425
## [6] 0.2030366571 1.2024391609 0.7593035708 0.7033449886 1.9403469147
## [11] 0.3925484551 0.6674976688 -0.3446283632 1.9084081647 0.3019882409
## [16] 0.0756044605 -1.2845018549 -0.7347703151 1.1684735578 -0.7022092018
## [21] 0.8164260215 -0.0005709686 1.9069194491 1.2030284748 0.0707856096
## [26] -1.3810693189 -1.6142778433 -0.6245852823 -0.8725018478 0.2341620404
## [31] -0.6939341378 -0.4827299922 1.3884701207 -0.9152117468 0.0387632465
## [36] -1.6400317856 1.0005821561 0.7046606538 0.1053001179 -0.7675444479
## [41] -0.3746294673 0.0867119079 0.0624423465 1.0284534633 0.5889407161
## [46] 2.0826487470 0.8222595799 0.9659892717 1.3617771939 0.2808865606
## [51] 0.2927791590 -1.2425251238 -0.6145897515 0.2305059778 -0.0008796766
## [56] -0.3538948232 -1.3370825139 -1.0768203043 -0.8889290122 0.7482329302
## [61] -0.4956215950 0.8665308492 0.6985899983 0.5059316503 3.4378466046
## [66] 1.1085522764 0.6293708571 -0.3546625549 -0.6117615042 1.0054429401
## [71] 2.0239920538 0.9722914878 1.7884856065 -0.9500941082 -1.2890124481
## [76] 0.1920572179 -0.3562315333 1.9317168346 0.0930827747 0.7722635840
## [81] -1.3645812477 1.1925189739 -0.1266541905 1.9254811223 0.3148387204
## [86] -0.4656714742 -0.5976814383 -0.1252779443 2.0231193708 2.4625345920
## [91] 2.1910596217 1.4214113024 0.8766225940 0.9267589084 -0.8232050274
## [96] -1.6264722750 -2.0437599459 0.0593472813 -0.4766026587 -0.4274172270
## [101] 0.5865156735 -2.5723728557 0.2018284826 -1.3864904483 -0.2671476548
## [106] -0.1622579378 -1.8886213965 0.0963531485 -0.5773190694 1.9487790570
## [111] 1.3912223790 1.6739861739 0.8671740946 0.9379216612 -1.2430639940
## [116] -1.2425258378 -3.7908666936 0.1054106417 -1.0097051137 -0.1444730816
## [121] -1.3360990732 -0.7395354999 -0.5234724890 1.1519528340 -0.7542851340
## [126] 0.5545346831 1.0773612788 2.1864016426 1.3575846676 -0.0603187191
## [131] -1.3182072883 -1.4831605057 -2.2369214890 -1.5564018418 -1.5200332576
## [136] 0.5518822885 -0.5403428041 -1.0113170122 -0.4044165025 0.3765462648
## [141] -0.5139685156 0.2227773739 -0.8298204850 0.1493382246 0.4209347510
## [146] 1.5354375656 1.5069008021 1.4404986178 -1.1516092916 -0.2291156817
## [151] -1.8238967982 -1.2169569496 -0.3471614692 0.8206307081 1.2674616742
## [156] 2.3185736810 -0.6890438106 0.2954115052 -0.0694363757 0.8270306299
## [161] 1.2112715185 -1.3932545390 -0.6249472403 -0.5380299092 -0.0956441836
## [166] 0.0714389431 -1.5618061241 0.0605293549 -0.5223492304 0.2232705480
## [171] 1.1938956098 0.3911999599 1.4792286808 0.5822047003 1.5292924790
## [176] 0.9925623705 2.2805912681 0.6918573713 1.2662349029 -0.6728839038
## [181] 0.5485919729 1.3758753904 1.2765154339 1.6191261090 -0.5721040622
## [186] 0.6914655381 -2.4633202551 -1.7141729685 -4.3145435835 -2.1094561668
## [191] -1.1148026417 0.5188166714 0.4280945822 0.6413359989 1.7023184786
## [196] 1.5466777022 2.4972648386 1.8464181654 0.4783098581 0.2668014391
## [201] 0.6609506233 -2.6271836829 -1.0271632093 0.7188278491 -1.4071705984
## [206] 2.5029596596 -2.1767000016 0.2261918704 -1.3393432359 0.7714065328
## [211] 0.4177407565 -1.8540075065 -1.2359973739 -1.3729928044 -0.6071069445
## [216] 0.2325083951 -0.8897303942 0.7797076467 -0.5835862514 0.3875298318
## [221] -1.5752881050 -1.6394575680 -0.6532492389 -0.3842339593 0.9968224736
## [226] -0.8086459427 0.8868701042 -0.3780727327 2.5758449791 1.7148315670
## [231] 2.7324272009 0.8332397531 1.0009507824 1.9685409632 2.3697016706
## [236] 1.3812051877 2.6212638095 1.9812571526 0.7688579152 0.2944286504
## [241] 0.2373890442 0.0905573537 0.1444155868 0.2068117180 -0.1686321278
## [246] 0.5768913360 0.6174192653 0.2471049651 2.1248481028 -0.8621236553
## [251] 1.8551046478 -0.7396138335 0.9953012828 1.1706687453 -0.2612307520
## [256] 1.0922108791 -1.6247741546 -0.0303707421 -2.2773534114 0.2039318009
## [261] -0.9852282256 -0.8710586780 -0.3454705217 -0.7734511977 -0.8588926361
## [266] 0.5258935590 -0.6277777871 0.1381115635 0.3688323198 0.5049496838
## [271] 0.1912585536 -0.4319266648 -1.3755588492 -0.3239822272 -0.4062433845
## [276] 0.7200298762 0.1349736491 -0.3088818566 -0.2619393812 -0.9094469352
## [281] -0.6455037033 0.4139233572 1.2062168717 2.4547921170 1.4016281232
## [286] 0.0525150278 -0.2901311561 0.0027978397 0.5333436989 0.5582367761
## [291] 0.1770709978 -0.0980639856 0.3733346775 -1.7322856400 0.7996808082
## [296] -1.0146825427 2.1128443720#Yt-2
yt_3_ma2 <- ma2[-c(298,299,300)]
yt_3_ma2## [1] -0.2913725396 1.9471847822 0.2039450696 1.2931078276 0.0480891236
## [6] -1.3443863362 -0.2071210815 0.6596565425 0.2030366571 1.2024391609
## [11] 0.7593035708 0.7033449886 1.9403469147 0.3925484551 0.6674976688
## [16] -0.3446283632 1.9084081647 0.3019882409 0.0756044605 -1.2845018549
## [21] -0.7347703151 1.1684735578 -0.7022092018 0.8164260215 -0.0005709686
## [26] 1.9069194491 1.2030284748 0.0707856096 -1.3810693189 -1.6142778433
## [31] -0.6245852823 -0.8725018478 0.2341620404 -0.6939341378 -0.4827299922
## [36] 1.3884701207 -0.9152117468 0.0387632465 -1.6400317856 1.0005821561
## [41] 0.7046606538 0.1053001179 -0.7675444479 -0.3746294673 0.0867119079
## [46] 0.0624423465 1.0284534633 0.5889407161 2.0826487470 0.8222595799
## [51] 0.9659892717 1.3617771939 0.2808865606 0.2927791590 -1.2425251238
## [56] -0.6145897515 0.2305059778 -0.0008796766 -0.3538948232 -1.3370825139
## [61] -1.0768203043 -0.8889290122 0.7482329302 -0.4956215950 0.8665308492
## [66] 0.6985899983 0.5059316503 3.4378466046 1.1085522764 0.6293708571
## [71] -0.3546625549 -0.6117615042 1.0054429401 2.0239920538 0.9722914878
## [76] 1.7884856065 -0.9500941082 -1.2890124481 0.1920572179 -0.3562315333
## [81] 1.9317168346 0.0930827747 0.7722635840 -1.3645812477 1.1925189739
## [86] -0.1266541905 1.9254811223 0.3148387204 -0.4656714742 -0.5976814383
## [91] -0.1252779443 2.0231193708 2.4625345920 2.1910596217 1.4214113024
## [96] 0.8766225940 0.9267589084 -0.8232050274 -1.6264722750 -2.0437599459
## [101] 0.0593472813 -0.4766026587 -0.4274172270 0.5865156735 -2.5723728557
## [106] 0.2018284826 -1.3864904483 -0.2671476548 -0.1622579378 -1.8886213965
## [111] 0.0963531485 -0.5773190694 1.9487790570 1.3912223790 1.6739861739
## [116] 0.8671740946 0.9379216612 -1.2430639940 -1.2425258378 -3.7908666936
## [121] 0.1054106417 -1.0097051137 -0.1444730816 -1.3360990732 -0.7395354999
## [126] -0.5234724890 1.1519528340 -0.7542851340 0.5545346831 1.0773612788
## [131] 2.1864016426 1.3575846676 -0.0603187191 -1.3182072883 -1.4831605057
## [136] -2.2369214890 -1.5564018418 -1.5200332576 0.5518822885 -0.5403428041
## [141] -1.0113170122 -0.4044165025 0.3765462648 -0.5139685156 0.2227773739
## [146] -0.8298204850 0.1493382246 0.4209347510 1.5354375656 1.5069008021
## [151] 1.4404986178 -1.1516092916 -0.2291156817 -1.8238967982 -1.2169569496
## [156] -0.3471614692 0.8206307081 1.2674616742 2.3185736810 -0.6890438106
## [161] 0.2954115052 -0.0694363757 0.8270306299 1.2112715185 -1.3932545390
## [166] -0.6249472403 -0.5380299092 -0.0956441836 0.0714389431 -1.5618061241
## [171] 0.0605293549 -0.5223492304 0.2232705480 1.1938956098 0.3911999599
## [176] 1.4792286808 0.5822047003 1.5292924790 0.9925623705 2.2805912681
## [181] 0.6918573713 1.2662349029 -0.6728839038 0.5485919729 1.3758753904
## [186] 1.2765154339 1.6191261090 -0.5721040622 0.6914655381 -2.4633202551
## [191] -1.7141729685 -4.3145435835 -2.1094561668 -1.1148026417 0.5188166714
## [196] 0.4280945822 0.6413359989 1.7023184786 1.5466777022 2.4972648386
## [201] 1.8464181654 0.4783098581 0.2668014391 0.6609506233 -2.6271836829
## [206] -1.0271632093 0.7188278491 -1.4071705984 2.5029596596 -2.1767000016
## [211] 0.2261918704 -1.3393432359 0.7714065328 0.4177407565 -1.8540075065
## [216] -1.2359973739 -1.3729928044 -0.6071069445 0.2325083951 -0.8897303942
## [221] 0.7797076467 -0.5835862514 0.3875298318 -1.5752881050 -1.6394575680
## [226] -0.6532492389 -0.3842339593 0.9968224736 -0.8086459427 0.8868701042
## [231] -0.3780727327 2.5758449791 1.7148315670 2.7324272009 0.8332397531
## [236] 1.0009507824 1.9685409632 2.3697016706 1.3812051877 2.6212638095
## [241] 1.9812571526 0.7688579152 0.2944286504 0.2373890442 0.0905573537
## [246] 0.1444155868 0.2068117180 -0.1686321278 0.5768913360 0.6174192653
## [251] 0.2471049651 2.1248481028 -0.8621236553 1.8551046478 -0.7396138335
## [256] 0.9953012828 1.1706687453 -0.2612307520 1.0922108791 -1.6247741546
## [261] -0.0303707421 -2.2773534114 0.2039318009 -0.9852282256 -0.8710586780
## [266] -0.3454705217 -0.7734511977 -0.8588926361 0.5258935590 -0.6277777871
## [271] 0.1381115635 0.3688323198 0.5049496838 0.1912585536 -0.4319266648
## [276] -1.3755588492 -0.3239822272 -0.4062433845 0.7200298762 0.1349736491
## [281] -0.3088818566 -0.2619393812 -0.9094469352 -0.6455037033 0.4139233572
## [286] 1.2062168717 2.4547921170 1.4016281232 0.0525150278 -0.2901311561
## [291] 0.0027978397 0.5333436989 0.5582367761 0.1770709978 -0.0980639856
## [296] 0.3733346775 -1.7322856400cor(yt_ma2,yt_3_ma2)## [1] -0.01106738Korelasi antara \(Y_t\) dengan \(Y_{t-3}\) hasil simulasi mendekati teori yang ada yaitu -0.01106738
plot(y=yt_ma2,x=yt_3_ma2)
Berdasarkan scatterplot tersebut, terlihat bahwa cenderung tidak terdapat hubungan antara \(Y_t\) dengan \(Y_{t-3}\).
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 \(\phi_1 = 0.5\) dan \(\phi_2 = 0.2\) sebanyak 300 observasi dan \(c=0\).
Nilai-nilai selanjutnya dapat dicari melalui loop. Bentuk loop dapat dilihat dari rumus AR(2) yang hendak dibangkitkan:
set.seed(1002)
n<-length(wn)
n## [1] 300ar <- c(2:n)
for (i in 3:n) {ar[i]<-wn[i]+0.5*ar[i-1]+0.2*ar[i-2]}
ar## [1] 2.0000000000 3.0000000000 1.5777489568 3.2941825568 1.5978135339
## [6] 1.7535972732 1.3450053906 -0.5581342021 0.3261690081 1.5254340049
## [11] 0.2396559060 0.9782860694 1.4280061994 0.9246097195 2.1477140618
## [16] 1.0824346015 0.8689107540 0.4528603017 2.4489628800 0.9163907073
## [21] -0.0461928397 -0.4874501497 -0.1321727478 1.3451599187 -0.7320328580
## [26] 0.3654710899 0.6776827513 1.7848402105 1.2970110168 0.1448946455
## [31] -0.8664208476 -1.0228538455 -0.3428852101 -1.0140721643 -0.2913230970
## [36] -0.7732908559 -0.9282881060 1.2179080447 -0.9365726180 -0.7440200072
## [41] -1.1757087785 0.8220666502 0.6268927577 -0.5324756731 -0.7748702931
## [46] -0.0087541161 0.1136969301 -0.2827849295 0.8811245850 0.7757645000
## [51] 1.8901856384 1.1570183945 1.1041794437 2.0521537420 0.9317553126
## [56] 0.5339543378 -0.4631607973 -0.1673712310 0.6211143001 -0.0172205828
## [61] -0.5990201049 -1.1776015215 -1.0067865409 -0.9837814718 0.3318411004
## [66] -0.7895722587 0.2265540591 0.8872056975 0.2891248820 3.2806598247
## [71] 1.7431179347 0.3638828245 0.6144478598 0.4329707110 1.2733000647
## [76] 2.3418840437 1.1900824926 1.9749275569 0.0520940933 -0.9455311177
## [81] 0.9803676582 0.1876985183 1.4013626213 0.4547553092 0.7264361466
## [86] -0.8278470766 1.3054175689 0.5000733545 1.4869953924 0.7602406157
## [91] -0.3403685081 -0.1586993606 0.3942567023 2.0561737540 2.4881506489
## [96] 2.1594135594 1.9683793594 1.8338159151 1.8356198584 0.0007075591
## [101] -1.0604699200 -1.2323549937 0.3687490214 -0.5961504203 -1.1563333613
## [106] 0.5823544504 -2.4651580155 -0.6721029368 -0.8781344511 -1.0874180244
## [111] -0.6465909279 -2.1501403282 -0.5079376822 -0.5742794875 1.0982944086
## [116] 1.1648745588 1.2916360209 1.1118011217 1.3652222306 -0.6982953603
## [121] -1.0079190711 -3.0997989464 -0.1045110079 -0.8670011155 -1.5091448797
## [126] -1.7253033946 -0.9522386083 -0.9511508590 0.4105727942 -1.0918847612
## [131] -0.1124743113 1.3263703587 1.9758860667 1.1161327929 0.1466411658
## [136] -0.6167685819 -0.9129288077 -2.0152010753 -1.7921113090 -1.7171293502
## [141] -0.1367362503 -1.1334169855 -1.9648739708 -0.6322357996 0.2591906744
## [146] -1.2442644354 -0.4295216849 -0.6012956061 -0.2666046035 0.2020034393
## [151] 1.3289238978 1.4259272819 1.3620753210 -0.7713557220 0.1226569167
## [156] -0.9686600663 -1.4680204754 -0.3461845617 0.7262443519 0.7352360331
## [161] 1.9391358235 -0.4077467286 0.2333486824 0.8605477752 0.9441682128
## [166] 1.1733216766 -1.1097460174 -0.5085365235 0.1922280646 -0.2556189579
## [171] -0.3189819666 -1.5305582475 -0.1027979128 -0.3784047944 -0.4139003189
## [176] 1.0054563402 0.4183960863 1.1172863248 0.9426594865 1.6797464363
## [181] 1.4467814506 2.5816050898 1.4120843743 1.6424688459 0.3635698141
## [186] 1.1028398582 2.2070336998 1.6119036231 1.8016172551 0.2566905276
## [191] 1.2341785606 -1.5131975128 -1.7985919608 -3.6919725607 -2.5458486583
## [196] -1.4960416420 -0.7403983695 -0.7571961238 -0.1603016025 1.3768511657
## [201] 1.3398408079 2.2041005425 2.3028273087 0.9421341274 0.8987884932
## [206] 1.7019690895 -2.0023942272 -1.1132007272 1.7625894950 -1.5649450191
## [211] 1.3306436751 -1.1944467094 -0.5425499034 -0.8368576901 0.5020182359
## [216] 0.2858115497 -2.3089419094 -1.4337098680 -0.8857248322 -1.2520379937
## [221] -0.5374332169 -1.2016867375 0.1849641988 -0.6792216656 -0.2385181208
## [226] -1.5372115969 -1.9813836327 -0.7160887685 -0.6723286910 0.1354161862
## [231] -1.1705183538 0.4013194768 -0.1330584065 2.0537282110 1.9588552681
## [236] 2.5151632283 1.4574347337 1.6344739973 3.0313260751 3.2015842648
## [241] 1.8910906862 3.5180968134 3.5352098463 1.5215641097 1.3033507636
## [246] 1.8289626578 1.0152030100 0.5580043396 0.9109794516 0.4161176122
## [251] 0.7701732172 1.0243457770 0.4629471262 2.3088735960 -0.2421535396
## [256] 1.6679559758 0.4085380936 0.8876103835 1.7824870110 0.0948950650
## [261] 1.1315195004 -0.7467429074 -0.0813903669 -1.6947790487 -0.0972479954
## [266] -0.7765347037 -1.5830454318 -0.5887958482 -0.8224746430 -1.5181223592
## [271] 0.1248683775 -0.6948623436 -0.6319318000 0.3098869089 0.4132612494
## [276] -0.1529102684 -0.4879447321 -1.1995455649 -0.3548623894 -0.4055887408
## [281] 0.2360627975 -0.0390954987 -0.5567111742 -0.2687581847 -0.8115092727
## [286] -0.8891446997 0.3183772907 1.0752590349 2.0633864637 1.4072009691
## [291] 0.2273615401 0.3642827445 0.7760965501 0.7933583066 0.6480641244
## [296] 0.4218363672 0.2091145687 0.6513462250 -1.4709084168 0.6524046217Selain menggunakan cara di atas, pembangkitan proses AR dapat
dilakukan dengan fungsi arima.sim() sebagai berikut.
set.seed(1002)
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] 0.3617754456 0.6748653860 1.7826923989 1.2953736379 0.1436463938
## [6] -0.8673724492 -1.0235792966 -0.3434382560 -1.0144937775 -0.2916445128
## [11] -0.7735358864 -0.9284749044 1.2177656394 -0.9366811803 -0.7441027694
## [16] -1.1757718721 0.8220185510 0.6268560893 -0.5325036271 -0.7748916038
## [21] -0.0087703622 0.1136845449 -0.2827943713 0.8811173871 0.7757590126
## [26] 1.8901814551 1.1570152054 1.1041770125 2.0521518885 0.9317538996
## [31] 0.5339532607 -0.4631616185 -0.1673718570 0.6211138228 -0.0172209466
## [36] -0.5990203823 -1.1776017330 -1.0067867021 -0.9837815947 0.3318410068
## [41] -0.7895723302 0.2265540046 0.8872056560 0.2891248503 3.2806598006
## [46] 1.7431179163 0.3638828105 0.6144478491 0.4329707028 1.2733000585
## [51] 2.3418840390 1.1900824890 1.9749275542 0.0520940912 -0.9455311193
## [56] 0.9803676570 0.1876985174 1.4013626206 0.4547553087 0.7264361462
## [61] -0.8278470769 1.3054175687 0.5000733543 1.4869953923 0.7602406156
## [66] -0.3403685082 -0.1586993606 0.3942567023 2.0561737539 2.4881506489
## [71] 2.1594135594 1.9683793593 1.8338159151 1.8356198584 0.0007075591
## [76] -1.0604699200 -1.2323549937 0.3687490214 -0.5961504203 -1.1563333613
## [81] 0.5823544503 -2.4651580155 -0.6721029368 -0.8781344511 -1.0874180244
## [86] -0.6465909279 -2.1501403282 -0.5079376822 -0.5742794875 1.0982944086
## [91] 1.1648745588 1.2916360209 1.1118011217 1.3652222306 -0.6982953603
## [96] -1.0079190711 -3.0997989464 -0.1045110079 -0.8670011155 -1.5091448797
## [101] -1.7253033946 -0.9522386083 -0.9511508590 0.4105727942 -1.0918847612
## [106] -0.1124743113 1.3263703587 1.9758860667 1.1161327929 0.1466411658
## [111] -0.6167685819 -0.9129288077 -2.0152010753 -1.7921113090 -1.7171293502
## [116] -0.1367362503 -1.1334169855 -1.9648739708 -0.6322357996 0.2591906744
## [121] -1.2442644354 -0.4295216849 -0.6012956061 -0.2666046035 0.2020034393
## [126] 1.3289238978 1.4259272819 1.3620753210 -0.7713557220 0.1226569167
## [131] -0.9686600663 -1.4680204754 -0.3461845617 0.7262443519 0.7352360331
## [136] 1.9391358235 -0.4077467286 0.2333486824 0.8605477752 0.9441682128
## [141] 1.1733216766 -1.1097460174 -0.5085365235 0.1922280646 -0.2556189579
## [146] -0.3189819666 -1.5305582475 -0.1027979128 -0.3784047944 -0.4139003189
## [151] 1.0054563402 0.4183960863 1.1172863248 0.9426594865 1.6797464363
## [156] 1.4467814506 2.5816050898 1.4120843743 1.6424688459 0.3635698141
## [161] 1.1028398582 2.2070336998 1.6119036231 1.8016172551 0.2566905276
## [166] 1.2341785606 -1.5131975128 -1.7985919608 -3.6919725607 -2.5458486583
## [171] -1.4960416420 -0.7403983695 -0.7571961238 -0.1603016025 1.3768511657
## [176] 1.3398408079 2.2041005425 2.3028273087 0.9421341274 0.8987884932
## [181] 1.7019690895 -2.0023942272 -1.1132007272 1.7625894950 -1.5649450191
## [186] 1.3306436751 -1.1944467094 -0.5425499034 -0.8368576901 0.5020182359
## [191] 0.2858115497 -2.3089419094 -1.4337098680 -0.8857248322 -1.2520379937
## [196] -0.5374332169 -1.2016867375 0.1849641988 -0.6792216656 -0.2385181208
## [201] -1.5372115969 -1.9813836327 -0.7160887685 -0.6723286910 0.1354161862
## [206] -1.1705183538 0.4013194768 -0.1330584065 2.0537282110 1.9588552681
## [211] 2.5151632283 1.4574347337 1.6344739973 3.0313260751 3.2015842648
## [216] 1.8910906862 3.5180968134 3.5352098463 1.5215641097 1.3033507636
## [221] 1.8289626578 1.0152030100 0.5580043396 0.9109794516 0.4161176122
## [226] 0.7701732172 1.0243457770 0.4629471262 2.3088735960 -0.2421535396
## [231] 1.6679559758 0.4085380936 0.8876103835 1.7824870110 0.0948950650
## [236] 1.1315195004 -0.7467429074 -0.0813903669 -1.6947790487 -0.0972479954
## [241] -0.7765347037 -1.5830454318 -0.5887958482 -0.8224746430 -1.5181223592
## [246] 0.1248683775 -0.6948623436 -0.6319318000 0.3098869089 0.4132612494
## [251] -0.1529102684 -0.4879447321 -1.1995455649 -0.3548623894 -0.4055887408
## [256] 0.2360627975 -0.0390954987 -0.5567111742 -0.2687581847 -0.8115092727
## [261] -0.8891446997 0.3183772907 1.0752590349 2.0633864637 1.4072009691
## [266] 0.2273615401 0.3642827445 0.7760965501 0.7933583066 0.6480641244
## [271] 0.4218363672 0.2091145687 0.6513462250 -1.4709084168 0.6524046217
## [276] -0.3826550834 1.4633142879 0.7169196898 -0.3157065182 -0.8610409829
## [281] 0.2494666233 -1.2014939550 -0.1312093416 -0.0839659963 1.4356598933
## [286] 0.5068877271 0.7486908492 1.0053755569 1.6392870638 2.3148615884
## [291] 2.5494925896 1.5025221432 -0.0820571327 0.8241497586 -0.6585846350
## [296] 1.2699916168 0.2566299169 0.5587826461 -0.3150394184 1.7862124219Karakteristik 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 pertama, sejalan dengan teori yang ada
Plot EACF
TSA::eacf(ar)## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x x o o o o o o o o o
## 1 x x o o x o o o o o o o o o
## 2 o x o o x o o o o o o o o o
## 3 x x o o x o o o o o o o o o
## 4 x x o x x o o o o o o o o o
## 5 x x x x x o o o o o o o o o
## 6 x x x x x o o o o o o o o o
## 7 x x x x x o o o o o o o o oBerdasarkan pola segitiga nol pada plot EACF, tidak terlihat bahwa segitiga nol berada pada ordo AR(2) dan ordo MA(0)
Scatterplot Antar Lag
Korelasi antara \(Y_t\) dengan \(Y_{t-1}\)
#Yt
yt_ar <- ar[-1]
yt_ar## [1] 3.0000000000 1.5777489568 3.2941825568 1.5978135339 1.7535972732
## [6] 1.3450053906 -0.5581342021 0.3261690081 1.5254340049 0.2396559060
## [11] 0.9782860694 1.4280061994 0.9246097195 2.1477140618 1.0824346015
## [16] 0.8689107540 0.4528603017 2.4489628800 0.9163907073 -0.0461928397
## [21] -0.4874501497 -0.1321727478 1.3451599187 -0.7320328580 0.3654710899
## [26] 0.6776827513 1.7848402105 1.2970110168 0.1448946455 -0.8664208476
## [31] -1.0228538455 -0.3428852101 -1.0140721643 -0.2913230970 -0.7732908559
## [36] -0.9282881060 1.2179080447 -0.9365726180 -0.7440200072 -1.1757087785
## [41] 0.8220666502 0.6268927577 -0.5324756731 -0.7748702931 -0.0087541161
## [46] 0.1136969301 -0.2827849295 0.8811245850 0.7757645000 1.8901856384
## [51] 1.1570183945 1.1041794437 2.0521537420 0.9317553126 0.5339543378
## [56] -0.4631607973 -0.1673712310 0.6211143001 -0.0172205828 -0.5990201049
## [61] -1.1776015215 -1.0067865409 -0.9837814718 0.3318411004 -0.7895722587
## [66] 0.2265540591 0.8872056975 0.2891248820 3.2806598247 1.7431179347
## [71] 0.3638828245 0.6144478598 0.4329707110 1.2733000647 2.3418840437
## [76] 1.1900824926 1.9749275569 0.0520940933 -0.9455311177 0.9803676582
## [81] 0.1876985183 1.4013626213 0.4547553092 0.7264361466 -0.8278470766
## [86] 1.3054175689 0.5000733545 1.4869953924 0.7602406157 -0.3403685081
## [91] -0.1586993606 0.3942567023 2.0561737540 2.4881506489 2.1594135594
## [96] 1.9683793594 1.8338159151 1.8356198584 0.0007075591 -1.0604699200
## [101] -1.2323549937 0.3687490214 -0.5961504203 -1.1563333613 0.5823544504
## [106] -2.4651580155 -0.6721029368 -0.8781344511 -1.0874180244 -0.6465909279
## [111] -2.1501403282 -0.5079376822 -0.5742794875 1.0982944086 1.1648745588
## [116] 1.2916360209 1.1118011217 1.3652222306 -0.6982953603 -1.0079190711
## [121] -3.0997989464 -0.1045110079 -0.8670011155 -1.5091448797 -1.7253033946
## [126] -0.9522386083 -0.9511508590 0.4105727942 -1.0918847612 -0.1124743113
## [131] 1.3263703587 1.9758860667 1.1161327929 0.1466411658 -0.6167685819
## [136] -0.9129288077 -2.0152010753 -1.7921113090 -1.7171293502 -0.1367362503
## [141] -1.1334169855 -1.9648739708 -0.6322357996 0.2591906744 -1.2442644354
## [146] -0.4295216849 -0.6012956061 -0.2666046035 0.2020034393 1.3289238978
## [151] 1.4259272819 1.3620753210 -0.7713557220 0.1226569167 -0.9686600663
## [156] -1.4680204754 -0.3461845617 0.7262443519 0.7352360331 1.9391358235
## [161] -0.4077467286 0.2333486824 0.8605477752 0.9441682128 1.1733216766
## [166] -1.1097460174 -0.5085365235 0.1922280646 -0.2556189579 -0.3189819666
## [171] -1.5305582475 -0.1027979128 -0.3784047944 -0.4139003189 1.0054563402
## [176] 0.4183960863 1.1172863248 0.9426594865 1.6797464363 1.4467814506
## [181] 2.5816050898 1.4120843743 1.6424688459 0.3635698141 1.1028398582
## [186] 2.2070336998 1.6119036231 1.8016172551 0.2566905276 1.2341785606
## [191] -1.5131975128 -1.7985919608 -3.6919725607 -2.5458486583 -1.4960416420
## [196] -0.7403983695 -0.7571961238 -0.1603016025 1.3768511657 1.3398408079
## [201] 2.2041005425 2.3028273087 0.9421341274 0.8987884932 1.7019690895
## [206] -2.0023942272 -1.1132007272 1.7625894950 -1.5649450191 1.3306436751
## [211] -1.1944467094 -0.5425499034 -0.8368576901 0.5020182359 0.2858115497
## [216] -2.3089419094 -1.4337098680 -0.8857248322 -1.2520379937 -0.5374332169
## [221] -1.2016867375 0.1849641988 -0.6792216656 -0.2385181208 -1.5372115969
## [226] -1.9813836327 -0.7160887685 -0.6723286910 0.1354161862 -1.1705183538
## [231] 0.4013194768 -0.1330584065 2.0537282110 1.9588552681 2.5151632283
## [236] 1.4574347337 1.6344739973 3.0313260751 3.2015842648 1.8910906862
## [241] 3.5180968134 3.5352098463 1.5215641097 1.3033507636 1.8289626578
## [246] 1.0152030100 0.5580043396 0.9109794516 0.4161176122 0.7701732172
## [251] 1.0243457770 0.4629471262 2.3088735960 -0.2421535396 1.6679559758
## [256] 0.4085380936 0.8876103835 1.7824870110 0.0948950650 1.1315195004
## [261] -0.7467429074 -0.0813903669 -1.6947790487 -0.0972479954 -0.7765347037
## [266] -1.5830454318 -0.5887958482 -0.8224746430 -1.5181223592 0.1248683775
## [271] -0.6948623436 -0.6319318000 0.3098869089 0.4132612494 -0.1529102684
## [276] -0.4879447321 -1.1995455649 -0.3548623894 -0.4055887408 0.2360627975
## [281] -0.0390954987 -0.5567111742 -0.2687581847 -0.8115092727 -0.8891446997
## [286] 0.3183772907 1.0752590349 2.0633864637 1.4072009691 0.2273615401
## [291] 0.3642827445 0.7760965501 0.7933583066 0.6480641244 0.4218363672
## [296] 0.2091145687 0.6513462250 -1.4709084168 0.6524046217#Yt-1
yt_1_ar <- ar[-300]
yt_1_ar## [1] 2.0000000000 3.0000000000 1.5777489568 3.2941825568 1.5978135339
## [6] 1.7535972732 1.3450053906 -0.5581342021 0.3261690081 1.5254340049
## [11] 0.2396559060 0.9782860694 1.4280061994 0.9246097195 2.1477140618
## [16] 1.0824346015 0.8689107540 0.4528603017 2.4489628800 0.9163907073
## [21] -0.0461928397 -0.4874501497 -0.1321727478 1.3451599187 -0.7320328580
## [26] 0.3654710899 0.6776827513 1.7848402105 1.2970110168 0.1448946455
## [31] -0.8664208476 -1.0228538455 -0.3428852101 -1.0140721643 -0.2913230970
## [36] -0.7732908559 -0.9282881060 1.2179080447 -0.9365726180 -0.7440200072
## [41] -1.1757087785 0.8220666502 0.6268927577 -0.5324756731 -0.7748702931
## [46] -0.0087541161 0.1136969301 -0.2827849295 0.8811245850 0.7757645000
## [51] 1.8901856384 1.1570183945 1.1041794437 2.0521537420 0.9317553126
## [56] 0.5339543378 -0.4631607973 -0.1673712310 0.6211143001 -0.0172205828
## [61] -0.5990201049 -1.1776015215 -1.0067865409 -0.9837814718 0.3318411004
## [66] -0.7895722587 0.2265540591 0.8872056975 0.2891248820 3.2806598247
## [71] 1.7431179347 0.3638828245 0.6144478598 0.4329707110 1.2733000647
## [76] 2.3418840437 1.1900824926 1.9749275569 0.0520940933 -0.9455311177
## [81] 0.9803676582 0.1876985183 1.4013626213 0.4547553092 0.7264361466
## [86] -0.8278470766 1.3054175689 0.5000733545 1.4869953924 0.7602406157
## [91] -0.3403685081 -0.1586993606 0.3942567023 2.0561737540 2.4881506489
## [96] 2.1594135594 1.9683793594 1.8338159151 1.8356198584 0.0007075591
## [101] -1.0604699200 -1.2323549937 0.3687490214 -0.5961504203 -1.1563333613
## [106] 0.5823544504 -2.4651580155 -0.6721029368 -0.8781344511 -1.0874180244
## [111] -0.6465909279 -2.1501403282 -0.5079376822 -0.5742794875 1.0982944086
## [116] 1.1648745588 1.2916360209 1.1118011217 1.3652222306 -0.6982953603
## [121] -1.0079190711 -3.0997989464 -0.1045110079 -0.8670011155 -1.5091448797
## [126] -1.7253033946 -0.9522386083 -0.9511508590 0.4105727942 -1.0918847612
## [131] -0.1124743113 1.3263703587 1.9758860667 1.1161327929 0.1466411658
## [136] -0.6167685819 -0.9129288077 -2.0152010753 -1.7921113090 -1.7171293502
## [141] -0.1367362503 -1.1334169855 -1.9648739708 -0.6322357996 0.2591906744
## [146] -1.2442644354 -0.4295216849 -0.6012956061 -0.2666046035 0.2020034393
## [151] 1.3289238978 1.4259272819 1.3620753210 -0.7713557220 0.1226569167
## [156] -0.9686600663 -1.4680204754 -0.3461845617 0.7262443519 0.7352360331
## [161] 1.9391358235 -0.4077467286 0.2333486824 0.8605477752 0.9441682128
## [166] 1.1733216766 -1.1097460174 -0.5085365235 0.1922280646 -0.2556189579
## [171] -0.3189819666 -1.5305582475 -0.1027979128 -0.3784047944 -0.4139003189
## [176] 1.0054563402 0.4183960863 1.1172863248 0.9426594865 1.6797464363
## [181] 1.4467814506 2.5816050898 1.4120843743 1.6424688459 0.3635698141
## [186] 1.1028398582 2.2070336998 1.6119036231 1.8016172551 0.2566905276
## [191] 1.2341785606 -1.5131975128 -1.7985919608 -3.6919725607 -2.5458486583
## [196] -1.4960416420 -0.7403983695 -0.7571961238 -0.1603016025 1.3768511657
## [201] 1.3398408079 2.2041005425 2.3028273087 0.9421341274 0.8987884932
## [206] 1.7019690895 -2.0023942272 -1.1132007272 1.7625894950 -1.5649450191
## [211] 1.3306436751 -1.1944467094 -0.5425499034 -0.8368576901 0.5020182359
## [216] 0.2858115497 -2.3089419094 -1.4337098680 -0.8857248322 -1.2520379937
## [221] -0.5374332169 -1.2016867375 0.1849641988 -0.6792216656 -0.2385181208
## [226] -1.5372115969 -1.9813836327 -0.7160887685 -0.6723286910 0.1354161862
## [231] -1.1705183538 0.4013194768 -0.1330584065 2.0537282110 1.9588552681
## [236] 2.5151632283 1.4574347337 1.6344739973 3.0313260751 3.2015842648
## [241] 1.8910906862 3.5180968134 3.5352098463 1.5215641097 1.3033507636
## [246] 1.8289626578 1.0152030100 0.5580043396 0.9109794516 0.4161176122
## [251] 0.7701732172 1.0243457770 0.4629471262 2.3088735960 -0.2421535396
## [256] 1.6679559758 0.4085380936 0.8876103835 1.7824870110 0.0948950650
## [261] 1.1315195004 -0.7467429074 -0.0813903669 -1.6947790487 -0.0972479954
## [266] -0.7765347037 -1.5830454318 -0.5887958482 -0.8224746430 -1.5181223592
## [271] 0.1248683775 -0.6948623436 -0.6319318000 0.3098869089 0.4132612494
## [276] -0.1529102684 -0.4879447321 -1.1995455649 -0.3548623894 -0.4055887408
## [281] 0.2360627975 -0.0390954987 -0.5567111742 -0.2687581847 -0.8115092727
## [286] -0.8891446997 0.3183772907 1.0752590349 2.0633864637 1.4072009691
## [291] 0.2273615401 0.3642827445 0.7760965501 0.7933583066 0.6480641244
## [296] 0.4218363672 0.2091145687 0.6513462250 -1.4709084168cor(yt_ar,yt_1_ar)## [1] 0.5799812Korelasi antara \(Y_t\) dengan \(Y_{t-1}\) dari hasil simulasi sebesar 0.5799812
plot(y=yt_ar,x=yt_1_ar)
Berdasarkan scatterplot tersebut, terlihat bahwa cenderung terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-1}\).
Korelasi antara \(Y_t\) dengan \(Y_{t-2}\)
#Yt
yt2_ar <- ar[-c(1,2)]
yt2_ar## [1] 1.5777489568 3.2941825568 1.5978135339 1.7535972732 1.3450053906
## [6] -0.5581342021 0.3261690081 1.5254340049 0.2396559060 0.9782860694
## [11] 1.4280061994 0.9246097195 2.1477140618 1.0824346015 0.8689107540
## [16] 0.4528603017 2.4489628800 0.9163907073 -0.0461928397 -0.4874501497
## [21] -0.1321727478 1.3451599187 -0.7320328580 0.3654710899 0.6776827513
## [26] 1.7848402105 1.2970110168 0.1448946455 -0.8664208476 -1.0228538455
## [31] -0.3428852101 -1.0140721643 -0.2913230970 -0.7732908559 -0.9282881060
## [36] 1.2179080447 -0.9365726180 -0.7440200072 -1.1757087785 0.8220666502
## [41] 0.6268927577 -0.5324756731 -0.7748702931 -0.0087541161 0.1136969301
## [46] -0.2827849295 0.8811245850 0.7757645000 1.8901856384 1.1570183945
## [51] 1.1041794437 2.0521537420 0.9317553126 0.5339543378 -0.4631607973
## [56] -0.1673712310 0.6211143001 -0.0172205828 -0.5990201049 -1.1776015215
## [61] -1.0067865409 -0.9837814718 0.3318411004 -0.7895722587 0.2265540591
## [66] 0.8872056975 0.2891248820 3.2806598247 1.7431179347 0.3638828245
## [71] 0.6144478598 0.4329707110 1.2733000647 2.3418840437 1.1900824926
## [76] 1.9749275569 0.0520940933 -0.9455311177 0.9803676582 0.1876985183
## [81] 1.4013626213 0.4547553092 0.7264361466 -0.8278470766 1.3054175689
## [86] 0.5000733545 1.4869953924 0.7602406157 -0.3403685081 -0.1586993606
## [91] 0.3942567023 2.0561737540 2.4881506489 2.1594135594 1.9683793594
## [96] 1.8338159151 1.8356198584 0.0007075591 -1.0604699200 -1.2323549937
## [101] 0.3687490214 -0.5961504203 -1.1563333613 0.5823544504 -2.4651580155
## [106] -0.6721029368 -0.8781344511 -1.0874180244 -0.6465909279 -2.1501403282
## [111] -0.5079376822 -0.5742794875 1.0982944086 1.1648745588 1.2916360209
## [116] 1.1118011217 1.3652222306 -0.6982953603 -1.0079190711 -3.0997989464
## [121] -0.1045110079 -0.8670011155 -1.5091448797 -1.7253033946 -0.9522386083
## [126] -0.9511508590 0.4105727942 -1.0918847612 -0.1124743113 1.3263703587
## [131] 1.9758860667 1.1161327929 0.1466411658 -0.6167685819 -0.9129288077
## [136] -2.0152010753 -1.7921113090 -1.7171293502 -0.1367362503 -1.1334169855
## [141] -1.9648739708 -0.6322357996 0.2591906744 -1.2442644354 -0.4295216849
## [146] -0.6012956061 -0.2666046035 0.2020034393 1.3289238978 1.4259272819
## [151] 1.3620753210 -0.7713557220 0.1226569167 -0.9686600663 -1.4680204754
## [156] -0.3461845617 0.7262443519 0.7352360331 1.9391358235 -0.4077467286
## [161] 0.2333486824 0.8605477752 0.9441682128 1.1733216766 -1.1097460174
## [166] -0.5085365235 0.1922280646 -0.2556189579 -0.3189819666 -1.5305582475
## [171] -0.1027979128 -0.3784047944 -0.4139003189 1.0054563402 0.4183960863
## [176] 1.1172863248 0.9426594865 1.6797464363 1.4467814506 2.5816050898
## [181] 1.4120843743 1.6424688459 0.3635698141 1.1028398582 2.2070336998
## [186] 1.6119036231 1.8016172551 0.2566905276 1.2341785606 -1.5131975128
## [191] -1.7985919608 -3.6919725607 -2.5458486583 -1.4960416420 -0.7403983695
## [196] -0.7571961238 -0.1603016025 1.3768511657 1.3398408079 2.2041005425
## [201] 2.3028273087 0.9421341274 0.8987884932 1.7019690895 -2.0023942272
## [206] -1.1132007272 1.7625894950 -1.5649450191 1.3306436751 -1.1944467094
## [211] -0.5425499034 -0.8368576901 0.5020182359 0.2858115497 -2.3089419094
## [216] -1.4337098680 -0.8857248322 -1.2520379937 -0.5374332169 -1.2016867375
## [221] 0.1849641988 -0.6792216656 -0.2385181208 -1.5372115969 -1.9813836327
## [226] -0.7160887685 -0.6723286910 0.1354161862 -1.1705183538 0.4013194768
## [231] -0.1330584065 2.0537282110 1.9588552681 2.5151632283 1.4574347337
## [236] 1.6344739973 3.0313260751 3.2015842648 1.8910906862 3.5180968134
## [241] 3.5352098463 1.5215641097 1.3033507636 1.8289626578 1.0152030100
## [246] 0.5580043396 0.9109794516 0.4161176122 0.7701732172 1.0243457770
## [251] 0.4629471262 2.3088735960 -0.2421535396 1.6679559758 0.4085380936
## [256] 0.8876103835 1.7824870110 0.0948950650 1.1315195004 -0.7467429074
## [261] -0.0813903669 -1.6947790487 -0.0972479954 -0.7765347037 -1.5830454318
## [266] -0.5887958482 -0.8224746430 -1.5181223592 0.1248683775 -0.6948623436
## [271] -0.6319318000 0.3098869089 0.4132612494 -0.1529102684 -0.4879447321
## [276] -1.1995455649 -0.3548623894 -0.4055887408 0.2360627975 -0.0390954987
## [281] -0.5567111742 -0.2687581847 -0.8115092727 -0.8891446997 0.3183772907
## [286] 1.0752590349 2.0633864637 1.4072009691 0.2273615401 0.3642827445
## [291] 0.7760965501 0.7933583066 0.6480641244 0.4218363672 0.2091145687
## [296] 0.6513462250 -1.4709084168 0.6524046217#Yt-2
yt_2_ar <- ar[-c(299,300)]
yt_2_ar## [1] 2.0000000000 3.0000000000 1.5777489568 3.2941825568 1.5978135339
## [6] 1.7535972732 1.3450053906 -0.5581342021 0.3261690081 1.5254340049
## [11] 0.2396559060 0.9782860694 1.4280061994 0.9246097195 2.1477140618
## [16] 1.0824346015 0.8689107540 0.4528603017 2.4489628800 0.9163907073
## [21] -0.0461928397 -0.4874501497 -0.1321727478 1.3451599187 -0.7320328580
## [26] 0.3654710899 0.6776827513 1.7848402105 1.2970110168 0.1448946455
## [31] -0.8664208476 -1.0228538455 -0.3428852101 -1.0140721643 -0.2913230970
## [36] -0.7732908559 -0.9282881060 1.2179080447 -0.9365726180 -0.7440200072
## [41] -1.1757087785 0.8220666502 0.6268927577 -0.5324756731 -0.7748702931
## [46] -0.0087541161 0.1136969301 -0.2827849295 0.8811245850 0.7757645000
## [51] 1.8901856384 1.1570183945 1.1041794437 2.0521537420 0.9317553126
## [56] 0.5339543378 -0.4631607973 -0.1673712310 0.6211143001 -0.0172205828
## [61] -0.5990201049 -1.1776015215 -1.0067865409 -0.9837814718 0.3318411004
## [66] -0.7895722587 0.2265540591 0.8872056975 0.2891248820 3.2806598247
## [71] 1.7431179347 0.3638828245 0.6144478598 0.4329707110 1.2733000647
## [76] 2.3418840437 1.1900824926 1.9749275569 0.0520940933 -0.9455311177
## [81] 0.9803676582 0.1876985183 1.4013626213 0.4547553092 0.7264361466
## [86] -0.8278470766 1.3054175689 0.5000733545 1.4869953924 0.7602406157
## [91] -0.3403685081 -0.1586993606 0.3942567023 2.0561737540 2.4881506489
## [96] 2.1594135594 1.9683793594 1.8338159151 1.8356198584 0.0007075591
## [101] -1.0604699200 -1.2323549937 0.3687490214 -0.5961504203 -1.1563333613
## [106] 0.5823544504 -2.4651580155 -0.6721029368 -0.8781344511 -1.0874180244
## [111] -0.6465909279 -2.1501403282 -0.5079376822 -0.5742794875 1.0982944086
## [116] 1.1648745588 1.2916360209 1.1118011217 1.3652222306 -0.6982953603
## [121] -1.0079190711 -3.0997989464 -0.1045110079 -0.8670011155 -1.5091448797
## [126] -1.7253033946 -0.9522386083 -0.9511508590 0.4105727942 -1.0918847612
## [131] -0.1124743113 1.3263703587 1.9758860667 1.1161327929 0.1466411658
## [136] -0.6167685819 -0.9129288077 -2.0152010753 -1.7921113090 -1.7171293502
## [141] -0.1367362503 -1.1334169855 -1.9648739708 -0.6322357996 0.2591906744
## [146] -1.2442644354 -0.4295216849 -0.6012956061 -0.2666046035 0.2020034393
## [151] 1.3289238978 1.4259272819 1.3620753210 -0.7713557220 0.1226569167
## [156] -0.9686600663 -1.4680204754 -0.3461845617 0.7262443519 0.7352360331
## [161] 1.9391358235 -0.4077467286 0.2333486824 0.8605477752 0.9441682128
## [166] 1.1733216766 -1.1097460174 -0.5085365235 0.1922280646 -0.2556189579
## [171] -0.3189819666 -1.5305582475 -0.1027979128 -0.3784047944 -0.4139003189
## [176] 1.0054563402 0.4183960863 1.1172863248 0.9426594865 1.6797464363
## [181] 1.4467814506 2.5816050898 1.4120843743 1.6424688459 0.3635698141
## [186] 1.1028398582 2.2070336998 1.6119036231 1.8016172551 0.2566905276
## [191] 1.2341785606 -1.5131975128 -1.7985919608 -3.6919725607 -2.5458486583
## [196] -1.4960416420 -0.7403983695 -0.7571961238 -0.1603016025 1.3768511657
## [201] 1.3398408079 2.2041005425 2.3028273087 0.9421341274 0.8987884932
## [206] 1.7019690895 -2.0023942272 -1.1132007272 1.7625894950 -1.5649450191
## [211] 1.3306436751 -1.1944467094 -0.5425499034 -0.8368576901 0.5020182359
## [216] 0.2858115497 -2.3089419094 -1.4337098680 -0.8857248322 -1.2520379937
## [221] -0.5374332169 -1.2016867375 0.1849641988 -0.6792216656 -0.2385181208
## [226] -1.5372115969 -1.9813836327 -0.7160887685 -0.6723286910 0.1354161862
## [231] -1.1705183538 0.4013194768 -0.1330584065 2.0537282110 1.9588552681
## [236] 2.5151632283 1.4574347337 1.6344739973 3.0313260751 3.2015842648
## [241] 1.8910906862 3.5180968134 3.5352098463 1.5215641097 1.3033507636
## [246] 1.8289626578 1.0152030100 0.5580043396 0.9109794516 0.4161176122
## [251] 0.7701732172 1.0243457770 0.4629471262 2.3088735960 -0.2421535396
## [256] 1.6679559758 0.4085380936 0.8876103835 1.7824870110 0.0948950650
## [261] 1.1315195004 -0.7467429074 -0.0813903669 -1.6947790487 -0.0972479954
## [266] -0.7765347037 -1.5830454318 -0.5887958482 -0.8224746430 -1.5181223592
## [271] 0.1248683775 -0.6948623436 -0.6319318000 0.3098869089 0.4132612494
## [276] -0.1529102684 -0.4879447321 -1.1995455649 -0.3548623894 -0.4055887408
## [281] 0.2360627975 -0.0390954987 -0.5567111742 -0.2687581847 -0.8115092727
## [286] -0.8891446997 0.3183772907 1.0752590349 2.0633864637 1.4072009691
## [291] 0.2273615401 0.3642827445 0.7760965501 0.7933583066 0.6480641244
## [296] 0.4218363672 0.2091145687 0.6513462250cor(yt2_ar,yt_2_ar)## [1] 0.5100164Korelasi antara \(Y_t\) dengan \(Y_{t-2}\) dari hasil simulasi sebesar 0.5100164.
plot(y=yt2_ar,x=yt_2_ar)
Berdasarkan scatterplot tersebut, terlihat bahwa cenderung terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-2}\).
Korelasi antara \(Y_t\) dengan \(Y_{t-3}\)
#Yt
yt3_ar <- ar[-c(1,2,3)]
yt3_ar## [1] 3.2941825568 1.5978135339 1.7535972732 1.3450053906 -0.5581342021
## [6] 0.3261690081 1.5254340049 0.2396559060 0.9782860694 1.4280061994
## [11] 0.9246097195 2.1477140618 1.0824346015 0.8689107540 0.4528603017
## [16] 2.4489628800 0.9163907073 -0.0461928397 -0.4874501497 -0.1321727478
## [21] 1.3451599187 -0.7320328580 0.3654710899 0.6776827513 1.7848402105
## [26] 1.2970110168 0.1448946455 -0.8664208476 -1.0228538455 -0.3428852101
## [31] -1.0140721643 -0.2913230970 -0.7732908559 -0.9282881060 1.2179080447
## [36] -0.9365726180 -0.7440200072 -1.1757087785 0.8220666502 0.6268927577
## [41] -0.5324756731 -0.7748702931 -0.0087541161 0.1136969301 -0.2827849295
## [46] 0.8811245850 0.7757645000 1.8901856384 1.1570183945 1.1041794437
## [51] 2.0521537420 0.9317553126 0.5339543378 -0.4631607973 -0.1673712310
## [56] 0.6211143001 -0.0172205828 -0.5990201049 -1.1776015215 -1.0067865409
## [61] -0.9837814718 0.3318411004 -0.7895722587 0.2265540591 0.8872056975
## [66] 0.2891248820 3.2806598247 1.7431179347 0.3638828245 0.6144478598
## [71] 0.4329707110 1.2733000647 2.3418840437 1.1900824926 1.9749275569
## [76] 0.0520940933 -0.9455311177 0.9803676582 0.1876985183 1.4013626213
## [81] 0.4547553092 0.7264361466 -0.8278470766 1.3054175689 0.5000733545
## [86] 1.4869953924 0.7602406157 -0.3403685081 -0.1586993606 0.3942567023
## [91] 2.0561737540 2.4881506489 2.1594135594 1.9683793594 1.8338159151
## [96] 1.8356198584 0.0007075591 -1.0604699200 -1.2323549937 0.3687490214
## [101] -0.5961504203 -1.1563333613 0.5823544504 -2.4651580155 -0.6721029368
## [106] -0.8781344511 -1.0874180244 -0.6465909279 -2.1501403282 -0.5079376822
## [111] -0.5742794875 1.0982944086 1.1648745588 1.2916360209 1.1118011217
## [116] 1.3652222306 -0.6982953603 -1.0079190711 -3.0997989464 -0.1045110079
## [121] -0.8670011155 -1.5091448797 -1.7253033946 -0.9522386083 -0.9511508590
## [126] 0.4105727942 -1.0918847612 -0.1124743113 1.3263703587 1.9758860667
## [131] 1.1161327929 0.1466411658 -0.6167685819 -0.9129288077 -2.0152010753
## [136] -1.7921113090 -1.7171293502 -0.1367362503 -1.1334169855 -1.9648739708
## [141] -0.6322357996 0.2591906744 -1.2442644354 -0.4295216849 -0.6012956061
## [146] -0.2666046035 0.2020034393 1.3289238978 1.4259272819 1.3620753210
## [151] -0.7713557220 0.1226569167 -0.9686600663 -1.4680204754 -0.3461845617
## [156] 0.7262443519 0.7352360331 1.9391358235 -0.4077467286 0.2333486824
## [161] 0.8605477752 0.9441682128 1.1733216766 -1.1097460174 -0.5085365235
## [166] 0.1922280646 -0.2556189579 -0.3189819666 -1.5305582475 -0.1027979128
## [171] -0.3784047944 -0.4139003189 1.0054563402 0.4183960863 1.1172863248
## [176] 0.9426594865 1.6797464363 1.4467814506 2.5816050898 1.4120843743
## [181] 1.6424688459 0.3635698141 1.1028398582 2.2070336998 1.6119036231
## [186] 1.8016172551 0.2566905276 1.2341785606 -1.5131975128 -1.7985919608
## [191] -3.6919725607 -2.5458486583 -1.4960416420 -0.7403983695 -0.7571961238
## [196] -0.1603016025 1.3768511657 1.3398408079 2.2041005425 2.3028273087
## [201] 0.9421341274 0.8987884932 1.7019690895 -2.0023942272 -1.1132007272
## [206] 1.7625894950 -1.5649450191 1.3306436751 -1.1944467094 -0.5425499034
## [211] -0.8368576901 0.5020182359 0.2858115497 -2.3089419094 -1.4337098680
## [216] -0.8857248322 -1.2520379937 -0.5374332169 -1.2016867375 0.1849641988
## [221] -0.6792216656 -0.2385181208 -1.5372115969 -1.9813836327 -0.7160887685
## [226] -0.6723286910 0.1354161862 -1.1705183538 0.4013194768 -0.1330584065
## [231] 2.0537282110 1.9588552681 2.5151632283 1.4574347337 1.6344739973
## [236] 3.0313260751 3.2015842648 1.8910906862 3.5180968134 3.5352098463
## [241] 1.5215641097 1.3033507636 1.8289626578 1.0152030100 0.5580043396
## [246] 0.9109794516 0.4161176122 0.7701732172 1.0243457770 0.4629471262
## [251] 2.3088735960 -0.2421535396 1.6679559758 0.4085380936 0.8876103835
## [256] 1.7824870110 0.0948950650 1.1315195004 -0.7467429074 -0.0813903669
## [261] -1.6947790487 -0.0972479954 -0.7765347037 -1.5830454318 -0.5887958482
## [266] -0.8224746430 -1.5181223592 0.1248683775 -0.6948623436 -0.6319318000
## [271] 0.3098869089 0.4132612494 -0.1529102684 -0.4879447321 -1.1995455649
## [276] -0.3548623894 -0.4055887408 0.2360627975 -0.0390954987 -0.5567111742
## [281] -0.2687581847 -0.8115092727 -0.8891446997 0.3183772907 1.0752590349
## [286] 2.0633864637 1.4072009691 0.2273615401 0.3642827445 0.7760965501
## [291] 0.7933583066 0.6480641244 0.4218363672 0.2091145687 0.6513462250
## [296] -1.4709084168 0.6524046217#Yt-2
yt_3_ar <- ar[-c(298,299,300)]
yt_3_ar## [1] 2.0000000000 3.0000000000 1.5777489568 3.2941825568 1.5978135339
## [6] 1.7535972732 1.3450053906 -0.5581342021 0.3261690081 1.5254340049
## [11] 0.2396559060 0.9782860694 1.4280061994 0.9246097195 2.1477140618
## [16] 1.0824346015 0.8689107540 0.4528603017 2.4489628800 0.9163907073
## [21] -0.0461928397 -0.4874501497 -0.1321727478 1.3451599187 -0.7320328580
## [26] 0.3654710899 0.6776827513 1.7848402105 1.2970110168 0.1448946455
## [31] -0.8664208476 -1.0228538455 -0.3428852101 -1.0140721643 -0.2913230970
## [36] -0.7732908559 -0.9282881060 1.2179080447 -0.9365726180 -0.7440200072
## [41] -1.1757087785 0.8220666502 0.6268927577 -0.5324756731 -0.7748702931
## [46] -0.0087541161 0.1136969301 -0.2827849295 0.8811245850 0.7757645000
## [51] 1.8901856384 1.1570183945 1.1041794437 2.0521537420 0.9317553126
## [56] 0.5339543378 -0.4631607973 -0.1673712310 0.6211143001 -0.0172205828
## [61] -0.5990201049 -1.1776015215 -1.0067865409 -0.9837814718 0.3318411004
## [66] -0.7895722587 0.2265540591 0.8872056975 0.2891248820 3.2806598247
## [71] 1.7431179347 0.3638828245 0.6144478598 0.4329707110 1.2733000647
## [76] 2.3418840437 1.1900824926 1.9749275569 0.0520940933 -0.9455311177
## [81] 0.9803676582 0.1876985183 1.4013626213 0.4547553092 0.7264361466
## [86] -0.8278470766 1.3054175689 0.5000733545 1.4869953924 0.7602406157
## [91] -0.3403685081 -0.1586993606 0.3942567023 2.0561737540 2.4881506489
## [96] 2.1594135594 1.9683793594 1.8338159151 1.8356198584 0.0007075591
## [101] -1.0604699200 -1.2323549937 0.3687490214 -0.5961504203 -1.1563333613
## [106] 0.5823544504 -2.4651580155 -0.6721029368 -0.8781344511 -1.0874180244
## [111] -0.6465909279 -2.1501403282 -0.5079376822 -0.5742794875 1.0982944086
## [116] 1.1648745588 1.2916360209 1.1118011217 1.3652222306 -0.6982953603
## [121] -1.0079190711 -3.0997989464 -0.1045110079 -0.8670011155 -1.5091448797
## [126] -1.7253033946 -0.9522386083 -0.9511508590 0.4105727942 -1.0918847612
## [131] -0.1124743113 1.3263703587 1.9758860667 1.1161327929 0.1466411658
## [136] -0.6167685819 -0.9129288077 -2.0152010753 -1.7921113090 -1.7171293502
## [141] -0.1367362503 -1.1334169855 -1.9648739708 -0.6322357996 0.2591906744
## [146] -1.2442644354 -0.4295216849 -0.6012956061 -0.2666046035 0.2020034393
## [151] 1.3289238978 1.4259272819 1.3620753210 -0.7713557220 0.1226569167
## [156] -0.9686600663 -1.4680204754 -0.3461845617 0.7262443519 0.7352360331
## [161] 1.9391358235 -0.4077467286 0.2333486824 0.8605477752 0.9441682128
## [166] 1.1733216766 -1.1097460174 -0.5085365235 0.1922280646 -0.2556189579
## [171] -0.3189819666 -1.5305582475 -0.1027979128 -0.3784047944 -0.4139003189
## [176] 1.0054563402 0.4183960863 1.1172863248 0.9426594865 1.6797464363
## [181] 1.4467814506 2.5816050898 1.4120843743 1.6424688459 0.3635698141
## [186] 1.1028398582 2.2070336998 1.6119036231 1.8016172551 0.2566905276
## [191] 1.2341785606 -1.5131975128 -1.7985919608 -3.6919725607 -2.5458486583
## [196] -1.4960416420 -0.7403983695 -0.7571961238 -0.1603016025 1.3768511657
## [201] 1.3398408079 2.2041005425 2.3028273087 0.9421341274 0.8987884932
## [206] 1.7019690895 -2.0023942272 -1.1132007272 1.7625894950 -1.5649450191
## [211] 1.3306436751 -1.1944467094 -0.5425499034 -0.8368576901 0.5020182359
## [216] 0.2858115497 -2.3089419094 -1.4337098680 -0.8857248322 -1.2520379937
## [221] -0.5374332169 -1.2016867375 0.1849641988 -0.6792216656 -0.2385181208
## [226] -1.5372115969 -1.9813836327 -0.7160887685 -0.6723286910 0.1354161862
## [231] -1.1705183538 0.4013194768 -0.1330584065 2.0537282110 1.9588552681
## [236] 2.5151632283 1.4574347337 1.6344739973 3.0313260751 3.2015842648
## [241] 1.8910906862 3.5180968134 3.5352098463 1.5215641097 1.3033507636
## [246] 1.8289626578 1.0152030100 0.5580043396 0.9109794516 0.4161176122
## [251] 0.7701732172 1.0243457770 0.4629471262 2.3088735960 -0.2421535396
## [256] 1.6679559758 0.4085380936 0.8876103835 1.7824870110 0.0948950650
## [261] 1.1315195004 -0.7467429074 -0.0813903669 -1.6947790487 -0.0972479954
## [266] -0.7765347037 -1.5830454318 -0.5887958482 -0.8224746430 -1.5181223592
## [271] 0.1248683775 -0.6948623436 -0.6319318000 0.3098869089 0.4132612494
## [276] -0.1529102684 -0.4879447321 -1.1995455649 -0.3548623894 -0.4055887408
## [281] 0.2360627975 -0.0390954987 -0.5567111742 -0.2687581847 -0.8115092727
## [286] -0.8891446997 0.3183772907 1.0752590349 2.0633864637 1.4072009691
## [291] 0.2273615401 0.3642827445 0.7760965501 0.7933583066 0.6480641244
## [296] 0.4218363672 0.2091145687cor(yt3_ar,yt_3_ar)## [1] 0.3544178Korelasi antara \(Y_t\) dengan \(Y_{t-2}\) dari hasil simulasi sebesar 0.3544178.
plot(y=yt3_ar,x=yt_3_ar)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif lemah antara \(Y_t\) dengan \(Y_{t-3}\).
Fungsi pembangkitan ARMA
Setelah mengetahui cara membangkitkan data berpola AR, MA, bagaimana cara melakukan pembangkitan data berpola tersebut yang lebih kompleks? Apakah dapat dibuat suatu fungsi yang fleksibel yang memungkinan pembangkitan dengan berapapun jumlah koefisien?
Pertama, lihat kembali bentuk umum data berpola ARMA.
\[ y_{t} = c + \sum_{i=1}^p \phi_{i}y_{t-i} + \sum_{j=1}^q e_{t-j}+ e_{t} \]
Jika koefisien dan white noise/nilai deret waktu sebelumnya dapat diekstrak dalam bentuk vektor, dapat dilakukan perkalian matriks untuk mencari nilai bagian AR dan MA:
set.seed(1002)
coefs <- c(0.5, 0.2)
e <- c(1, 2)
coefs %*% e## [,1]
## [1,] 0.9Atau, dapat dilakukan perkalian elementwise yang dijumlahkan:
coefs * e## [1] 0.5 0.4sum(coefs * e)## [1] 0.9Dari prinsip ini, dapat dibuat fungsi umum untuk membangkitkan data ARMA. Input dari fungsi adalah jumlah data yang hendak dibangkitkan, koefisien MA, dan koefisien AR
arma.sim <- function(n, macoef, arcoef){
manum <- length(macoef)
arnum <- length(arcoef)
stopifnot(manum < n & arnum < n)
wn <- rnorm(n, sd = 0.5)
init <- max(manum, arnum)
arma <- wn[1:init]
for(i in {init+1}:n){
mastart <- i - manum
maend <- i-1
arstart <- i - arnum
arend <- i-1
arma[i] <- sum(arcoef * arma[arstart:arend]) + sum(macoef * wn[mastart:maend]) + wn[i]
}
return(arma)
}Terlihat bahwa komponen \(\sum_{i=1}^q
y_{t-1}\) disimulasikan melalui
sum(arcoef * arma[arstart:arend]). Jadi, koefisien
dikalikan dengan data \(y\) dari \(t-q\) di mana q adalah jumlah koefisien AR,
sampai data \(t-1\). Lalu komponen
\(\sum_{j=1}^q e_{t-j}\) disimulasikan
melalui sum(macoef * wn[mastart:maend]). Koefisien
dikalikan dengan white noise \(e\) dari \(t-p\), p jumlah koefisien MA, sampai \(t-1\).
# beberapa contoh pembangkitan melalui fungsi
ma2 <- arma.sim(300, c(0.4, 0.6), 0)
ar2 <- arma.sim(300, 0, c(0.5, 0.2))
par(mfrow = c(2, 2))
acf(ma2)
pacf(ma2)
acf(ar2)
pacf(ar2)
#contoh untuk ARMA
arma22 <- arma.sim(300, c(0.5, 0.2), c(0.4,0.6))
arma22 |> arima(c(2,0,2))##
## Call:
## arima(x = arma22, order = c(2, 0, 2))
##
## Coefficients:
## ar1 ar2 ma1 ma2 intercept
## 0.4759 0.4968 0.2355 0.4966 -1.2354
## s.e. 0.0781 0.0781 0.0710 0.0594 1.4949
##
## sigma^2 estimated as 0.221: log likelihood = -201.48, aic = 414.96set.seed(1002)
n = length(wn)
phi1 = 0.5
theta1 = 0.4
phi2=0.2
theta2=0.6
y.arma=c(1:n)
for (i in 3:n){y.arma[i] = phi1*y.arma[i-1] + theta1*wn[i-1] + phi2*y.arma[i-2] + theta2*wn[i-2]+wn[i]}
y.arma## [1] 1.00000000 2.00000000 0.90862746 2.80149851 1.78641982 2.74661744
## [7] 1.77868181 0.09427805 0.19575431 0.77638931 0.63038217 1.67290811
## [13] 1.72183406 1.89884364 3.23413555 2.38938496 2.50901726 1.38775726
## [19] 3.10409024 2.13158481 1.76221492 0.02292257 -0.37086605 0.98762505
## [25] -0.28256989 0.87266609 0.37924810 2.27107672 2.41441645 1.73220918
## [31] -0.03208144 -1.28387673 -1.27293993 -1.76574716 -0.90329953 -1.49873333
## [37] -1.41275656 0.38234517 -1.00659047 -0.38806296 -2.03538136 -0.09472111
## [43] 0.25022383 0.21146781 -0.61176578 -0.63821880 -0.35475065 -0.24257674
## [49] 0.83621497 0.95853285 2.72915817 2.37854523 2.70109352 3.18803300
## [55] 2.41512177 2.13794664 0.30947255 -0.03226415 0.27626841 0.13080170
## [61] -0.23324029 -1.42754232 -1.83723952 -2.09305724 -0.66574359 -1.24710484
## [67] 0.10982971 0.50408389 0.77993954 3.92863315 3.22885676 3.02952587
## [73] 1.80587173 0.89707953 1.81515705 3.11098649 2.89081614 3.85609097
## [79] 1.55611461 0.26026305 0.63341166 0.01252691 2.06466262 1.12791947
## [85] 1.74915584 -0.26441943 1.41014043 0.52553214 2.47027528 1.65508279
## [91] 0.85592497 0.16129761 0.12655585 2.11865682 3.54717417 4.38837807
## [97] 4.32503517 3.91681579 3.75017384 1.83524505 0.04118502 -1.65611843
## [103] -0.76047493 -1.18806381 -1.17354412 -0.23786915 -2.92601625 -1.30875347
## [109] -2.62607044 -1.84193357 -1.60843881 -3.06122751 -1.75594837 -2.06753876
## [115] 0.56382000 1.25962463 2.41656249 2.32738027 2.58492429 0.51487420
## [121] -0.46810388 -3.92194379 -1.94918203 -2.76868489 -1.91865193 -2.84916202
## [127] -2.54784689 -2.36722834 -0.54123071 -1.49834616 -0.30288454 0.62624978
## [133] 2.43894962 2.70230943 1.77862592 0.11156756 -1.07165154 -2.75043375
## [139] -3.14594902 -3.64309452 -1.89885478 -2.21838910 -2.50028252 -2.09823558
## [145] -1.17262803 -1.51992965 -0.77171305 -1.51966294 -0.76483586 -0.26541577
## [151] 1.24976251 2.07869890 2.72980057 0.62903078 0.63135982 -1.38241073
## [157] -1.78189035 -1.51458879 -0.29304176 0.81802304 2.66897685 0.80904922
## [163] 1.23373149 0.70923921 1.42839653 2.06731763 -0.07391642 -0.24844192
## [169] -0.67703416 -0.48384965 -0.30589271 -1.81152241 -0.90641039 -1.33785891
## [175] -0.62694098 0.61285334 0.57223843 1.88791856 1.64061167 2.72718203
## [181] 2.68427572 4.16816553 3.31279528 3.75626565 1.86780798 2.23374909
## [187] 2.86631153 3.15642102 3.77059892 1.94447960 2.41782512 -0.86551177
## [193] -1.66336383 -5.31932785 -5.10179286 -4.72956464 -2.86632422 -1.95098046
## [199] -0.90741907 0.85841285 1.79440031 3.56614757 3.98837201 3.18572538
## [205] 2.65733853 2.62676496 -0.78233350 -0.89297696 0.11587267 -1.52782966
## [211] 1.76221936 -1.60115625 -0.22194238 -1.77054568 -0.15825478 -0.01549577
## [217] -1.89340635 -2.18579970 -2.84457393 -2.46655385 -1.56968331 -2.16788282
## [223] -0.61817043 -1.32624803 -0.39922827 -2.04015184 -2.73937914 -2.43096918
## [229] -2.14759438 -0.56316855 -1.51974909 0.01436185 -0.67484163 2.24129653
## [235] 2.70051151 4.53094226 3.63881319 3.72654583 4.55957651 5.39479909
## [241] 4.99052004 6.19548365 6.07710298 5.04650614 4.03310232 3.26324143
## [247] 2.52879853 2.06146314 1.74330299 1.11531200 1.48320793 1.58208563
## [253] 1.33478937 3.10865991 0.95916417 2.95641872 0.93042836 2.05179921
## [259] 2.38265402 1.34045610 2.23896973 -0.23719807 0.29882417 -2.17538094
## [265] -0.82399383 -1.83230133 -1.95200811 -1.68793484 -2.00782024 -2.20038973
## [271] -0.97586535 -1.55578841 -0.83495571 -0.35980322 0.15805693 0.19832638
## [277] -0.30115209 -1.48646962 -1.12744745 -1.26726104 -0.13909013 -0.18802362
## [283] -0.43071170 -0.51489995 -1.25303925 -1.37500332 -0.52418615 0.66912313
## [289] 2.68451645 2.87771098 2.02827381 1.29954794 1.05822657 1.32236657
## [295] 1.43106538 1.15707700 0.76668759 0.98809387 -1.08490119 0.45484899Pembangkitan ARMA(p,q) juga dapat dilakukan dengan fungsi
arima.sim sebagai berikut.
set.seed(1002)
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] 2.26962428 2.41330919 1.73136506 -0.03272495 -1.28436730 -1.27331392
## [7] -1.76603227 -0.90351688 -1.49889903 -1.41288288 0.38224887 -1.00666389
## [13] -0.38811892 -2.03542402 -0.09475364 0.25019903 0.21144890 -0.61178019
## [19] -0.63822978 -0.35475902 -0.24258312 0.83621010 0.95852914 2.72915534
## [25] 2.37854308 2.70109188 3.18803175 2.41512081 2.13794591 0.30947200
## [31] -0.03226457 0.27626809 0.13080145 -0.23324048 -1.42754246 -1.83723963
## [37] -2.09305732 -0.66574366 -1.24710489 0.10982967 0.50408386 0.77993951
## [43] 3.92863313 3.22885675 3.02952586 1.80587172 0.89707953 1.81515705
## [49] 3.11098648 2.89081614 3.85609097 1.55611461 0.26026305 0.63341166
## [55] 0.01252691 2.06466262 1.12791947 1.74915584 -0.26441943 1.41014043
## [61] 0.52553214 2.47027528 1.65508279 0.85592497 0.16129761 0.12655585
## [67] 2.11865682 3.54717417 4.38837807 4.32503517 3.91681579 3.75017384
## [73] 1.83524505 0.04118502 -1.65611843 -0.76047493 -1.18806381 -1.17354412
## [79] -0.23786915 -2.92601625 -1.30875347 -2.62607044 -1.84193357 -1.60843881
## [85] -3.06122751 -1.75594837 -2.06753876 0.56382000 1.25962463 2.41656249
## [91] 2.32738027 2.58492429 0.51487420 -0.46810388 -3.92194379 -1.94918203
## [97] -2.76868489 -1.91865193 -2.84916202 -2.54784689 -2.36722834 -0.54123071
## [103] -1.49834616 -0.30288454 0.62624978 2.43894962 2.70230943 1.77862592
## [109] 0.11156756 -1.07165154 -2.75043375 -3.14594902 -3.64309452 -1.89885478
## [115] -2.21838910 -2.50028252 -2.09823558 -1.17262803 -1.51992965 -0.77171305
## [121] -1.51966294 -0.76483586 -0.26541577 1.24976251 2.07869890 2.72980057
## [127] 0.62903078 0.63135982 -1.38241073 -1.78189035 -1.51458879 -0.29304176
## [133] 0.81802304 2.66897685 0.80904922 1.23373149 0.70923921 1.42839653
## [139] 2.06731763 -0.07391642 -0.24844192 -0.67703416 -0.48384965 -0.30589271
## [145] -1.81152241 -0.90641039 -1.33785891 -0.62694098 0.61285334 0.57223843
## [151] 1.88791856 1.64061167 2.72718203 2.68427572 4.16816553 3.31279528
## [157] 3.75626565 1.86780798 2.23374909 2.86631153 3.15642102 3.77059892
## [163] 1.94447960 2.41782512 -0.86551177 -1.66336383 -5.31932785 -5.10179286
## [169] -4.72956464 -2.86632422 -1.95098046 -0.90741907 0.85841285 1.79440031
## [175] 3.56614757 3.98837201 3.18572538 2.65733853 2.62676496 -0.78233350
## [181] -0.89297696 0.11587267 -1.52782966 1.76221936 -1.60115625 -0.22194238
## [187] -1.77054568 -0.15825478 -0.01549577 -1.89340635 -2.18579970 -2.84457393
## [193] -2.46655385 -1.56968331 -2.16788282 -0.61817043 -1.32624803 -0.39922827
## [199] -2.04015184 -2.73937914 -2.43096918 -2.14759438 -0.56316855 -1.51974909
## [205] 0.01436185 -0.67484163 2.24129653 2.70051151 4.53094226 3.63881319
## [211] 3.72654583 4.55957651 5.39479909 4.99052004 6.19548365 6.07710298
## [217] 5.04650614 4.03310232 3.26324143 2.52879853 2.06146314 1.74330299
## [223] 1.11531200 1.48320793 1.58208563 1.33478937 3.10865991 0.95916417
## [229] 2.95641872 0.93042836 2.05179921 2.38265402 1.34045610 2.23896973
## [235] -0.23719807 0.29882417 -2.17538094 -0.82399383 -1.83230133 -1.95200811
## [241] -1.68793484 -2.00782024 -2.20038973 -0.97586535 -1.55578841 -0.83495571
## [247] -0.35980322 0.15805693 0.19832638 -0.30115209 -1.48646962 -1.12744745
## [253] -1.26726104 -0.13909013 -0.18802362 -0.43071170 -0.51489995 -1.25303925
## [259] -1.37500332 -0.52418615 0.66912313 2.68451645 2.87771098 2.02827381
## [265] 1.29954794 1.05822657 1.32236657 1.43106538 1.15707700 0.76668759
## [271] 0.98809387 -1.08490119 0.45484899 -1.00423828 1.70169503 1.07265235
## [277] 0.84904993 -0.55717178 -0.28437368 -1.61833190 -0.46212695 -0.85734611
## [283] 1.32334789 1.03077209 1.81284188 1.60898453 2.49065180 3.57380175
## [289] 4.45900946 3.91123613 2.04864728 1.69284019 -0.37815901 1.50104762
## [295] 0.36947578 1.42342958 0.06245159 1.99546624 1.08846122 1.36826417Karakteristik 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(2, 2))Berdasarkan plot ACF tersebut, terlihat bahwa model ARMA(2,2) hasil simulasi memiliki plot ACF yang tails off, sesuai dengan teori yang ada
Plot PACF
par(mfrow = c(1, 2))
pacf(y.arma)
pacf(arma22)
par(mfrow = c(2, 2))Berdasarkan plot PACF tersebut, terlihat bahwa model ARMA(2,2) hasil simulasi memiliki plot PACF yang cuts off, tidak sesuai dengan teori
Plot EACF
TSA::eacf(y.arma)## AR/MA
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x x x o o o o o o o o
## 1 x x o o x o o o o o o o o o
## 2 x x x o x o o o o o o o o o
## 3 o x x x x o o o o o o o o o
## 4 x x x o x x o o o o o o o o
## 5 x x x x x x o o o o o o o o
## 6 x x o x o o o o o o o o o o
## 7 x x o x o o o x 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 o o o o o o o o o
## 1 x x o o x o o o o o o o o o
## 2 x x x o o o o o o o o o o o
## 3 o x x x x o o o o o o o o o
## 4 x x x o x o o o o o o o o o
## 5 x x x o x o o o o o o o o o
## 6 x x x x x o o o o o o o o o
## 7 x x x o x x o o o o o o o oBerdasarkan pola segitiga nol pada plot EACF, tidak 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
yt1_arma <- arma22[-1]
yt1_arma## [1] 2.41330919 1.73136506 -0.03272495 -1.28436730 -1.27331392 -1.76603227
## [7] -0.90351688 -1.49889903 -1.41288288 0.38224887 -1.00666389 -0.38811892
## [13] -2.03542402 -0.09475364 0.25019903 0.21144890 -0.61178019 -0.63822978
## [19] -0.35475902 -0.24258312 0.83621010 0.95852914 2.72915534 2.37854308
## [25] 2.70109188 3.18803175 2.41512081 2.13794591 0.30947200 -0.03226457
## [31] 0.27626809 0.13080145 -0.23324048 -1.42754246 -1.83723963 -2.09305732
## [37] -0.66574366 -1.24710489 0.10982967 0.50408386 0.77993951 3.92863313
## [43] 3.22885675 3.02952586 1.80587172 0.89707953 1.81515705 3.11098648
## [49] 2.89081614 3.85609097 1.55611461 0.26026305 0.63341166 0.01252691
## [55] 2.06466262 1.12791947 1.74915584 -0.26441943 1.41014043 0.52553214
## [61] 2.47027528 1.65508279 0.85592497 0.16129761 0.12655585 2.11865682
## [67] 3.54717417 4.38837807 4.32503517 3.91681579 3.75017384 1.83524505
## [73] 0.04118502 -1.65611843 -0.76047493 -1.18806381 -1.17354412 -0.23786915
## [79] -2.92601625 -1.30875347 -2.62607044 -1.84193357 -1.60843881 -3.06122751
## [85] -1.75594837 -2.06753876 0.56382000 1.25962463 2.41656249 2.32738027
## [91] 2.58492429 0.51487420 -0.46810388 -3.92194379 -1.94918203 -2.76868489
## [97] -1.91865193 -2.84916202 -2.54784689 -2.36722834 -0.54123071 -1.49834616
## [103] -0.30288454 0.62624978 2.43894962 2.70230943 1.77862592 0.11156756
## [109] -1.07165154 -2.75043375 -3.14594902 -3.64309452 -1.89885478 -2.21838910
## [115] -2.50028252 -2.09823558 -1.17262803 -1.51992965 -0.77171305 -1.51966294
## [121] -0.76483586 -0.26541577 1.24976251 2.07869890 2.72980057 0.62903078
## [127] 0.63135982 -1.38241073 -1.78189035 -1.51458879 -0.29304176 0.81802304
## [133] 2.66897685 0.80904922 1.23373149 0.70923921 1.42839653 2.06731763
## [139] -0.07391642 -0.24844192 -0.67703416 -0.48384965 -0.30589271 -1.81152241
## [145] -0.90641039 -1.33785891 -0.62694098 0.61285334 0.57223843 1.88791856
## [151] 1.64061167 2.72718203 2.68427572 4.16816553 3.31279528 3.75626565
## [157] 1.86780798 2.23374909 2.86631153 3.15642102 3.77059892 1.94447960
## [163] 2.41782512 -0.86551177 -1.66336383 -5.31932785 -5.10179286 -4.72956464
## [169] -2.86632422 -1.95098046 -0.90741907 0.85841285 1.79440031 3.56614757
## [175] 3.98837201 3.18572538 2.65733853 2.62676496 -0.78233350 -0.89297696
## [181] 0.11587267 -1.52782966 1.76221936 -1.60115625 -0.22194238 -1.77054568
## [187] -0.15825478 -0.01549577 -1.89340635 -2.18579970 -2.84457393 -2.46655385
## [193] -1.56968331 -2.16788282 -0.61817043 -1.32624803 -0.39922827 -2.04015184
## [199] -2.73937914 -2.43096918 -2.14759438 -0.56316855 -1.51974909 0.01436185
## [205] -0.67484163 2.24129653 2.70051151 4.53094226 3.63881319 3.72654583
## [211] 4.55957651 5.39479909 4.99052004 6.19548365 6.07710298 5.04650614
## [217] 4.03310232 3.26324143 2.52879853 2.06146314 1.74330299 1.11531200
## [223] 1.48320793 1.58208563 1.33478937 3.10865991 0.95916417 2.95641872
## [229] 0.93042836 2.05179921 2.38265402 1.34045610 2.23896973 -0.23719807
## [235] 0.29882417 -2.17538094 -0.82399383 -1.83230133 -1.95200811 -1.68793484
## [241] -2.00782024 -2.20038973 -0.97586535 -1.55578841 -0.83495571 -0.35980322
## [247] 0.15805693 0.19832638 -0.30115209 -1.48646962 -1.12744745 -1.26726104
## [253] -0.13909013 -0.18802362 -0.43071170 -0.51489995 -1.25303925 -1.37500332
## [259] -0.52418615 0.66912313 2.68451645 2.87771098 2.02827381 1.29954794
## [265] 1.05822657 1.32236657 1.43106538 1.15707700 0.76668759 0.98809387
## [271] -1.08490119 0.45484899 -1.00423828 1.70169503 1.07265235 0.84904993
## [277] -0.55717178 -0.28437368 -1.61833190 -0.46212695 -0.85734611 1.32334789
## [283] 1.03077209 1.81284188 1.60898453 2.49065180 3.57380175 4.45900946
## [289] 3.91123613 2.04864728 1.69284019 -0.37815901 1.50104762 0.36947578
## [295] 1.42342958 0.06245159 1.99546624 1.08846122 1.36826417#Yt-1
yt_1_arma <- arma22[-300]
yt_1_arma## [1] 2.26962428 2.41330919 1.73136506 -0.03272495 -1.28436730 -1.27331392
## [7] -1.76603227 -0.90351688 -1.49889903 -1.41288288 0.38224887 -1.00666389
## [13] -0.38811892 -2.03542402 -0.09475364 0.25019903 0.21144890 -0.61178019
## [19] -0.63822978 -0.35475902 -0.24258312 0.83621010 0.95852914 2.72915534
## [25] 2.37854308 2.70109188 3.18803175 2.41512081 2.13794591 0.30947200
## [31] -0.03226457 0.27626809 0.13080145 -0.23324048 -1.42754246 -1.83723963
## [37] -2.09305732 -0.66574366 -1.24710489 0.10982967 0.50408386 0.77993951
## [43] 3.92863313 3.22885675 3.02952586 1.80587172 0.89707953 1.81515705
## [49] 3.11098648 2.89081614 3.85609097 1.55611461 0.26026305 0.63341166
## [55] 0.01252691 2.06466262 1.12791947 1.74915584 -0.26441943 1.41014043
## [61] 0.52553214 2.47027528 1.65508279 0.85592497 0.16129761 0.12655585
## [67] 2.11865682 3.54717417 4.38837807 4.32503517 3.91681579 3.75017384
## [73] 1.83524505 0.04118502 -1.65611843 -0.76047493 -1.18806381 -1.17354412
## [79] -0.23786915 -2.92601625 -1.30875347 -2.62607044 -1.84193357 -1.60843881
## [85] -3.06122751 -1.75594837 -2.06753876 0.56382000 1.25962463 2.41656249
## [91] 2.32738027 2.58492429 0.51487420 -0.46810388 -3.92194379 -1.94918203
## [97] -2.76868489 -1.91865193 -2.84916202 -2.54784689 -2.36722834 -0.54123071
## [103] -1.49834616 -0.30288454 0.62624978 2.43894962 2.70230943 1.77862592
## [109] 0.11156756 -1.07165154 -2.75043375 -3.14594902 -3.64309452 -1.89885478
## [115] -2.21838910 -2.50028252 -2.09823558 -1.17262803 -1.51992965 -0.77171305
## [121] -1.51966294 -0.76483586 -0.26541577 1.24976251 2.07869890 2.72980057
## [127] 0.62903078 0.63135982 -1.38241073 -1.78189035 -1.51458879 -0.29304176
## [133] 0.81802304 2.66897685 0.80904922 1.23373149 0.70923921 1.42839653
## [139] 2.06731763 -0.07391642 -0.24844192 -0.67703416 -0.48384965 -0.30589271
## [145] -1.81152241 -0.90641039 -1.33785891 -0.62694098 0.61285334 0.57223843
## [151] 1.88791856 1.64061167 2.72718203 2.68427572 4.16816553 3.31279528
## [157] 3.75626565 1.86780798 2.23374909 2.86631153 3.15642102 3.77059892
## [163] 1.94447960 2.41782512 -0.86551177 -1.66336383 -5.31932785 -5.10179286
## [169] -4.72956464 -2.86632422 -1.95098046 -0.90741907 0.85841285 1.79440031
## [175] 3.56614757 3.98837201 3.18572538 2.65733853 2.62676496 -0.78233350
## [181] -0.89297696 0.11587267 -1.52782966 1.76221936 -1.60115625 -0.22194238
## [187] -1.77054568 -0.15825478 -0.01549577 -1.89340635 -2.18579970 -2.84457393
## [193] -2.46655385 -1.56968331 -2.16788282 -0.61817043 -1.32624803 -0.39922827
## [199] -2.04015184 -2.73937914 -2.43096918 -2.14759438 -0.56316855 -1.51974909
## [205] 0.01436185 -0.67484163 2.24129653 2.70051151 4.53094226 3.63881319
## [211] 3.72654583 4.55957651 5.39479909 4.99052004 6.19548365 6.07710298
## [217] 5.04650614 4.03310232 3.26324143 2.52879853 2.06146314 1.74330299
## [223] 1.11531200 1.48320793 1.58208563 1.33478937 3.10865991 0.95916417
## [229] 2.95641872 0.93042836 2.05179921 2.38265402 1.34045610 2.23896973
## [235] -0.23719807 0.29882417 -2.17538094 -0.82399383 -1.83230133 -1.95200811
## [241] -1.68793484 -2.00782024 -2.20038973 -0.97586535 -1.55578841 -0.83495571
## [247] -0.35980322 0.15805693 0.19832638 -0.30115209 -1.48646962 -1.12744745
## [253] -1.26726104 -0.13909013 -0.18802362 -0.43071170 -0.51489995 -1.25303925
## [259] -1.37500332 -0.52418615 0.66912313 2.68451645 2.87771098 2.02827381
## [265] 1.29954794 1.05822657 1.32236657 1.43106538 1.15707700 0.76668759
## [271] 0.98809387 -1.08490119 0.45484899 -1.00423828 1.70169503 1.07265235
## [277] 0.84904993 -0.55717178 -0.28437368 -1.61833190 -0.46212695 -0.85734611
## [283] 1.32334789 1.03077209 1.81284188 1.60898453 2.49065180 3.57380175
## [289] 4.45900946 3.91123613 2.04864728 1.69284019 -0.37815901 1.50104762
## [295] 0.36947578 1.42342958 0.06245159 1.99546624 1.08846122plot(y=yt1_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(yt1_arma,yt_1_arma)## [1] 0.8277724Korelasi antara \(Y_t\) dengan \(Y_{t-1}\) dari hasil simulasi sebesar 0.8277724
Korelasi antara \(Y_t\) dengan \(Y_{t-2}\)
#Yt
yt2_arma <- arma22[-c(1,2)]
yt2_arma## [1] 1.73136506 -0.03272495 -1.28436730 -1.27331392 -1.76603227 -0.90351688
## [7] -1.49889903 -1.41288288 0.38224887 -1.00666389 -0.38811892 -2.03542402
## [13] -0.09475364 0.25019903 0.21144890 -0.61178019 -0.63822978 -0.35475902
## [19] -0.24258312 0.83621010 0.95852914 2.72915534 2.37854308 2.70109188
## [25] 3.18803175 2.41512081 2.13794591 0.30947200 -0.03226457 0.27626809
## [31] 0.13080145 -0.23324048 -1.42754246 -1.83723963 -2.09305732 -0.66574366
## [37] -1.24710489 0.10982967 0.50408386 0.77993951 3.92863313 3.22885675
## [43] 3.02952586 1.80587172 0.89707953 1.81515705 3.11098648 2.89081614
## [49] 3.85609097 1.55611461 0.26026305 0.63341166 0.01252691 2.06466262
## [55] 1.12791947 1.74915584 -0.26441943 1.41014043 0.52553214 2.47027528
## [61] 1.65508279 0.85592497 0.16129761 0.12655585 2.11865682 3.54717417
## [67] 4.38837807 4.32503517 3.91681579 3.75017384 1.83524505 0.04118502
## [73] -1.65611843 -0.76047493 -1.18806381 -1.17354412 -0.23786915 -2.92601625
## [79] -1.30875347 -2.62607044 -1.84193357 -1.60843881 -3.06122751 -1.75594837
## [85] -2.06753876 0.56382000 1.25962463 2.41656249 2.32738027 2.58492429
## [91] 0.51487420 -0.46810388 -3.92194379 -1.94918203 -2.76868489 -1.91865193
## [97] -2.84916202 -2.54784689 -2.36722834 -0.54123071 -1.49834616 -0.30288454
## [103] 0.62624978 2.43894962 2.70230943 1.77862592 0.11156756 -1.07165154
## [109] -2.75043375 -3.14594902 -3.64309452 -1.89885478 -2.21838910 -2.50028252
## [115] -2.09823558 -1.17262803 -1.51992965 -0.77171305 -1.51966294 -0.76483586
## [121] -0.26541577 1.24976251 2.07869890 2.72980057 0.62903078 0.63135982
## [127] -1.38241073 -1.78189035 -1.51458879 -0.29304176 0.81802304 2.66897685
## [133] 0.80904922 1.23373149 0.70923921 1.42839653 2.06731763 -0.07391642
## [139] -0.24844192 -0.67703416 -0.48384965 -0.30589271 -1.81152241 -0.90641039
## [145] -1.33785891 -0.62694098 0.61285334 0.57223843 1.88791856 1.64061167
## [151] 2.72718203 2.68427572 4.16816553 3.31279528 3.75626565 1.86780798
## [157] 2.23374909 2.86631153 3.15642102 3.77059892 1.94447960 2.41782512
## [163] -0.86551177 -1.66336383 -5.31932785 -5.10179286 -4.72956464 -2.86632422
## [169] -1.95098046 -0.90741907 0.85841285 1.79440031 3.56614757 3.98837201
## [175] 3.18572538 2.65733853 2.62676496 -0.78233350 -0.89297696 0.11587267
## [181] -1.52782966 1.76221936 -1.60115625 -0.22194238 -1.77054568 -0.15825478
## [187] -0.01549577 -1.89340635 -2.18579970 -2.84457393 -2.46655385 -1.56968331
## [193] -2.16788282 -0.61817043 -1.32624803 -0.39922827 -2.04015184 -2.73937914
## [199] -2.43096918 -2.14759438 -0.56316855 -1.51974909 0.01436185 -0.67484163
## [205] 2.24129653 2.70051151 4.53094226 3.63881319 3.72654583 4.55957651
## [211] 5.39479909 4.99052004 6.19548365 6.07710298 5.04650614 4.03310232
## [217] 3.26324143 2.52879853 2.06146314 1.74330299 1.11531200 1.48320793
## [223] 1.58208563 1.33478937 3.10865991 0.95916417 2.95641872 0.93042836
## [229] 2.05179921 2.38265402 1.34045610 2.23896973 -0.23719807 0.29882417
## [235] -2.17538094 -0.82399383 -1.83230133 -1.95200811 -1.68793484 -2.00782024
## [241] -2.20038973 -0.97586535 -1.55578841 -0.83495571 -0.35980322 0.15805693
## [247] 0.19832638 -0.30115209 -1.48646962 -1.12744745 -1.26726104 -0.13909013
## [253] -0.18802362 -0.43071170 -0.51489995 -1.25303925 -1.37500332 -0.52418615
## [259] 0.66912313 2.68451645 2.87771098 2.02827381 1.29954794 1.05822657
## [265] 1.32236657 1.43106538 1.15707700 0.76668759 0.98809387 -1.08490119
## [271] 0.45484899 -1.00423828 1.70169503 1.07265235 0.84904993 -0.55717178
## [277] -0.28437368 -1.61833190 -0.46212695 -0.85734611 1.32334789 1.03077209
## [283] 1.81284188 1.60898453 2.49065180 3.57380175 4.45900946 3.91123613
## [289] 2.04864728 1.69284019 -0.37815901 1.50104762 0.36947578 1.42342958
## [295] 0.06245159 1.99546624 1.08846122 1.36826417#Yt-1
yt_2_arma <- arma22[-c(299,300)]
yt_2_arma## [1] 2.26962428 2.41330919 1.73136506 -0.03272495 -1.28436730 -1.27331392
## [7] -1.76603227 -0.90351688 -1.49889903 -1.41288288 0.38224887 -1.00666389
## [13] -0.38811892 -2.03542402 -0.09475364 0.25019903 0.21144890 -0.61178019
## [19] -0.63822978 -0.35475902 -0.24258312 0.83621010 0.95852914 2.72915534
## [25] 2.37854308 2.70109188 3.18803175 2.41512081 2.13794591 0.30947200
## [31] -0.03226457 0.27626809 0.13080145 -0.23324048 -1.42754246 -1.83723963
## [37] -2.09305732 -0.66574366 -1.24710489 0.10982967 0.50408386 0.77993951
## [43] 3.92863313 3.22885675 3.02952586 1.80587172 0.89707953 1.81515705
## [49] 3.11098648 2.89081614 3.85609097 1.55611461 0.26026305 0.63341166
## [55] 0.01252691 2.06466262 1.12791947 1.74915584 -0.26441943 1.41014043
## [61] 0.52553214 2.47027528 1.65508279 0.85592497 0.16129761 0.12655585
## [67] 2.11865682 3.54717417 4.38837807 4.32503517 3.91681579 3.75017384
## [73] 1.83524505 0.04118502 -1.65611843 -0.76047493 -1.18806381 -1.17354412
## [79] -0.23786915 -2.92601625 -1.30875347 -2.62607044 -1.84193357 -1.60843881
## [85] -3.06122751 -1.75594837 -2.06753876 0.56382000 1.25962463 2.41656249
## [91] 2.32738027 2.58492429 0.51487420 -0.46810388 -3.92194379 -1.94918203
## [97] -2.76868489 -1.91865193 -2.84916202 -2.54784689 -2.36722834 -0.54123071
## [103] -1.49834616 -0.30288454 0.62624978 2.43894962 2.70230943 1.77862592
## [109] 0.11156756 -1.07165154 -2.75043375 -3.14594902 -3.64309452 -1.89885478
## [115] -2.21838910 -2.50028252 -2.09823558 -1.17262803 -1.51992965 -0.77171305
## [121] -1.51966294 -0.76483586 -0.26541577 1.24976251 2.07869890 2.72980057
## [127] 0.62903078 0.63135982 -1.38241073 -1.78189035 -1.51458879 -0.29304176
## [133] 0.81802304 2.66897685 0.80904922 1.23373149 0.70923921 1.42839653
## [139] 2.06731763 -0.07391642 -0.24844192 -0.67703416 -0.48384965 -0.30589271
## [145] -1.81152241 -0.90641039 -1.33785891 -0.62694098 0.61285334 0.57223843
## [151] 1.88791856 1.64061167 2.72718203 2.68427572 4.16816553 3.31279528
## [157] 3.75626565 1.86780798 2.23374909 2.86631153 3.15642102 3.77059892
## [163] 1.94447960 2.41782512 -0.86551177 -1.66336383 -5.31932785 -5.10179286
## [169] -4.72956464 -2.86632422 -1.95098046 -0.90741907 0.85841285 1.79440031
## [175] 3.56614757 3.98837201 3.18572538 2.65733853 2.62676496 -0.78233350
## [181] -0.89297696 0.11587267 -1.52782966 1.76221936 -1.60115625 -0.22194238
## [187] -1.77054568 -0.15825478 -0.01549577 -1.89340635 -2.18579970 -2.84457393
## [193] -2.46655385 -1.56968331 -2.16788282 -0.61817043 -1.32624803 -0.39922827
## [199] -2.04015184 -2.73937914 -2.43096918 -2.14759438 -0.56316855 -1.51974909
## [205] 0.01436185 -0.67484163 2.24129653 2.70051151 4.53094226 3.63881319
## [211] 3.72654583 4.55957651 5.39479909 4.99052004 6.19548365 6.07710298
## [217] 5.04650614 4.03310232 3.26324143 2.52879853 2.06146314 1.74330299
## [223] 1.11531200 1.48320793 1.58208563 1.33478937 3.10865991 0.95916417
## [229] 2.95641872 0.93042836 2.05179921 2.38265402 1.34045610 2.23896973
## [235] -0.23719807 0.29882417 -2.17538094 -0.82399383 -1.83230133 -1.95200811
## [241] -1.68793484 -2.00782024 -2.20038973 -0.97586535 -1.55578841 -0.83495571
## [247] -0.35980322 0.15805693 0.19832638 -0.30115209 -1.48646962 -1.12744745
## [253] -1.26726104 -0.13909013 -0.18802362 -0.43071170 -0.51489995 -1.25303925
## [259] -1.37500332 -0.52418615 0.66912313 2.68451645 2.87771098 2.02827381
## [265] 1.29954794 1.05822657 1.32236657 1.43106538 1.15707700 0.76668759
## [271] 0.98809387 -1.08490119 0.45484899 -1.00423828 1.70169503 1.07265235
## [277] 0.84904993 -0.55717178 -0.28437368 -1.61833190 -0.46212695 -0.85734611
## [283] 1.32334789 1.03077209 1.81284188 1.60898453 2.49065180 3.57380175
## [289] 4.45900946 3.91123613 2.04864728 1.69284019 -0.37815901 1.50104762
## [295] 0.36947578 1.42342958 0.06245159 1.99546624plot(y=yt2_arma,x=yt_2_arma)
Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-2}\). Hal ini sesuai dengan teori yang ada
cor(yt2_arma,yt_2_arma)## [1] 0.7273334Korelasi antara \(Y_t\) dengan \(Y_{t-2}\) dari hasil simulasi sebesar 0.7273334
Korelasi antara \(Y_t\) dengan \(Y_{t-3}\)
#Yt
yt3_arma <- arma22[-c(1,2,3)]
yt3_arma## [1] -0.03272495 -1.28436730 -1.27331392 -1.76603227 -0.90351688 -1.49889903
## [7] -1.41288288 0.38224887 -1.00666389 -0.38811892 -2.03542402 -0.09475364
## [13] 0.25019903 0.21144890 -0.61178019 -0.63822978 -0.35475902 -0.24258312
## [19] 0.83621010 0.95852914 2.72915534 2.37854308 2.70109188 3.18803175
## [25] 2.41512081 2.13794591 0.30947200 -0.03226457 0.27626809 0.13080145
## [31] -0.23324048 -1.42754246 -1.83723963 -2.09305732 -0.66574366 -1.24710489
## [37] 0.10982967 0.50408386 0.77993951 3.92863313 3.22885675 3.02952586
## [43] 1.80587172 0.89707953 1.81515705 3.11098648 2.89081614 3.85609097
## [49] 1.55611461 0.26026305 0.63341166 0.01252691 2.06466262 1.12791947
## [55] 1.74915584 -0.26441943 1.41014043 0.52553214 2.47027528 1.65508279
## [61] 0.85592497 0.16129761 0.12655585 2.11865682 3.54717417 4.38837807
## [67] 4.32503517 3.91681579 3.75017384 1.83524505 0.04118502 -1.65611843
## [73] -0.76047493 -1.18806381 -1.17354412 -0.23786915 -2.92601625 -1.30875347
## [79] -2.62607044 -1.84193357 -1.60843881 -3.06122751 -1.75594837 -2.06753876
## [85] 0.56382000 1.25962463 2.41656249 2.32738027 2.58492429 0.51487420
## [91] -0.46810388 -3.92194379 -1.94918203 -2.76868489 -1.91865193 -2.84916202
## [97] -2.54784689 -2.36722834 -0.54123071 -1.49834616 -0.30288454 0.62624978
## [103] 2.43894962 2.70230943 1.77862592 0.11156756 -1.07165154 -2.75043375
## [109] -3.14594902 -3.64309452 -1.89885478 -2.21838910 -2.50028252 -2.09823558
## [115] -1.17262803 -1.51992965 -0.77171305 -1.51966294 -0.76483586 -0.26541577
## [121] 1.24976251 2.07869890 2.72980057 0.62903078 0.63135982 -1.38241073
## [127] -1.78189035 -1.51458879 -0.29304176 0.81802304 2.66897685 0.80904922
## [133] 1.23373149 0.70923921 1.42839653 2.06731763 -0.07391642 -0.24844192
## [139] -0.67703416 -0.48384965 -0.30589271 -1.81152241 -0.90641039 -1.33785891
## [145] -0.62694098 0.61285334 0.57223843 1.88791856 1.64061167 2.72718203
## [151] 2.68427572 4.16816553 3.31279528 3.75626565 1.86780798 2.23374909
## [157] 2.86631153 3.15642102 3.77059892 1.94447960 2.41782512 -0.86551177
## [163] -1.66336383 -5.31932785 -5.10179286 -4.72956464 -2.86632422 -1.95098046
## [169] -0.90741907 0.85841285 1.79440031 3.56614757 3.98837201 3.18572538
## [175] 2.65733853 2.62676496 -0.78233350 -0.89297696 0.11587267 -1.52782966
## [181] 1.76221936 -1.60115625 -0.22194238 -1.77054568 -0.15825478 -0.01549577
## [187] -1.89340635 -2.18579970 -2.84457393 -2.46655385 -1.56968331 -2.16788282
## [193] -0.61817043 -1.32624803 -0.39922827 -2.04015184 -2.73937914 -2.43096918
## [199] -2.14759438 -0.56316855 -1.51974909 0.01436185 -0.67484163 2.24129653
## [205] 2.70051151 4.53094226 3.63881319 3.72654583 4.55957651 5.39479909
## [211] 4.99052004 6.19548365 6.07710298 5.04650614 4.03310232 3.26324143
## [217] 2.52879853 2.06146314 1.74330299 1.11531200 1.48320793 1.58208563
## [223] 1.33478937 3.10865991 0.95916417 2.95641872 0.93042836 2.05179921
## [229] 2.38265402 1.34045610 2.23896973 -0.23719807 0.29882417 -2.17538094
## [235] -0.82399383 -1.83230133 -1.95200811 -1.68793484 -2.00782024 -2.20038973
## [241] -0.97586535 -1.55578841 -0.83495571 -0.35980322 0.15805693 0.19832638
## [247] -0.30115209 -1.48646962 -1.12744745 -1.26726104 -0.13909013 -0.18802362
## [253] -0.43071170 -0.51489995 -1.25303925 -1.37500332 -0.52418615 0.66912313
## [259] 2.68451645 2.87771098 2.02827381 1.29954794 1.05822657 1.32236657
## [265] 1.43106538 1.15707700 0.76668759 0.98809387 -1.08490119 0.45484899
## [271] -1.00423828 1.70169503 1.07265235 0.84904993 -0.55717178 -0.28437368
## [277] -1.61833190 -0.46212695 -0.85734611 1.32334789 1.03077209 1.81284188
## [283] 1.60898453 2.49065180 3.57380175 4.45900946 3.91123613 2.04864728
## [289] 1.69284019 -0.37815901 1.50104762 0.36947578 1.42342958 0.06245159
## [295] 1.99546624 1.08846122 1.36826417#Yt-1
yt_3_arma <- arma22[-c(298,299,300)]
yt_3_arma## [1] 2.26962428 2.41330919 1.73136506 -0.03272495 -1.28436730 -1.27331392
## [7] -1.76603227 -0.90351688 -1.49889903 -1.41288288 0.38224887 -1.00666389
## [13] -0.38811892 -2.03542402 -0.09475364 0.25019903 0.21144890 -0.61178019
## [19] -0.63822978 -0.35475902 -0.24258312 0.83621010 0.95852914 2.72915534
## [25] 2.37854308 2.70109188 3.18803175 2.41512081 2.13794591 0.30947200
## [31] -0.03226457 0.27626809 0.13080145 -0.23324048 -1.42754246 -1.83723963
## [37] -2.09305732 -0.66574366 -1.24710489 0.10982967 0.50408386 0.77993951
## [43] 3.92863313 3.22885675 3.02952586 1.80587172 0.89707953 1.81515705
## [49] 3.11098648 2.89081614 3.85609097 1.55611461 0.26026305 0.63341166
## [55] 0.01252691 2.06466262 1.12791947 1.74915584 -0.26441943 1.41014043
## [61] 0.52553214 2.47027528 1.65508279 0.85592497 0.16129761 0.12655585
## [67] 2.11865682 3.54717417 4.38837807 4.32503517 3.91681579 3.75017384
## [73] 1.83524505 0.04118502 -1.65611843 -0.76047493 -1.18806381 -1.17354412
## [79] -0.23786915 -2.92601625 -1.30875347 -2.62607044 -1.84193357 -1.60843881
## [85] -3.06122751 -1.75594837 -2.06753876 0.56382000 1.25962463 2.41656249
## [91] 2.32738027 2.58492429 0.51487420 -0.46810388 -3.92194379 -1.94918203
## [97] -2.76868489 -1.91865193 -2.84916202 -2.54784689 -2.36722834 -0.54123071
## [103] -1.49834616 -0.30288454 0.62624978 2.43894962 2.70230943 1.77862592
## [109] 0.11156756 -1.07165154 -2.75043375 -3.14594902 -3.64309452 -1.89885478
## [115] -2.21838910 -2.50028252 -2.09823558 -1.17262803 -1.51992965 -0.77171305
## [121] -1.51966294 -0.76483586 -0.26541577 1.24976251 2.07869890 2.72980057
## [127] 0.62903078 0.63135982 -1.38241073 -1.78189035 -1.51458879 -0.29304176
## [133] 0.81802304 2.66897685 0.80904922 1.23373149 0.70923921 1.42839653
## [139] 2.06731763 -0.07391642 -0.24844192 -0.67703416 -0.48384965 -0.30589271
## [145] -1.81152241 -0.90641039 -1.33785891 -0.62694098 0.61285334 0.57223843
## [151] 1.88791856 1.64061167 2.72718203 2.68427572 4.16816553 3.31279528
## [157] 3.75626565 1.86780798 2.23374909 2.86631153 3.15642102 3.77059892
## [163] 1.94447960 2.41782512 -0.86551177 -1.66336383 -5.31932785 -5.10179286
## [169] -4.72956464 -2.86632422 -1.95098046 -0.90741907 0.85841285 1.79440031
## [175] 3.56614757 3.98837201 3.18572538 2.65733853 2.62676496 -0.78233350
## [181] -0.89297696 0.11587267 -1.52782966 1.76221936 -1.60115625 -0.22194238
## [187] -1.77054568 -0.15825478 -0.01549577 -1.89340635 -2.18579970 -2.84457393
## [193] -2.46655385 -1.56968331 -2.16788282 -0.61817043 -1.32624803 -0.39922827
## [199] -2.04015184 -2.73937914 -2.43096918 -2.14759438 -0.56316855 -1.51974909
## [205] 0.01436185 -0.67484163 2.24129653 2.70051151 4.53094226 3.63881319
## [211] 3.72654583 4.55957651 5.39479909 4.99052004 6.19548365 6.07710298
## [217] 5.04650614 4.03310232 3.26324143 2.52879853 2.06146314 1.74330299
## [223] 1.11531200 1.48320793 1.58208563 1.33478937 3.10865991 0.95916417
## [229] 2.95641872 0.93042836 2.05179921 2.38265402 1.34045610 2.23896973
## [235] -0.23719807 0.29882417 -2.17538094 -0.82399383 -1.83230133 -1.95200811
## [241] -1.68793484 -2.00782024 -2.20038973 -0.97586535 -1.55578841 -0.83495571
## [247] -0.35980322 0.15805693 0.19832638 -0.30115209 -1.48646962 -1.12744745
## [253] -1.26726104 -0.13909013 -0.18802362 -0.43071170 -0.51489995 -1.25303925
## [259] -1.37500332 -0.52418615 0.66912313 2.68451645 2.87771098 2.02827381
## [265] 1.29954794 1.05822657 1.32236657 1.43106538 1.15707700 0.76668759
## [271] 0.98809387 -1.08490119 0.45484899 -1.00423828 1.70169503 1.07265235
## [277] 0.84904993 -0.55717178 -0.28437368 -1.61833190 -0.46212695 -0.85734611
## [283] 1.32334789 1.03077209 1.81284188 1.60898453 2.49065180 3.57380175
## [289] 4.45900946 3.91123613 2.04864728 1.69284019 -0.37815901 1.50104762
## [295] 0.36947578 1.42342958 0.06245159plot(y=yt3_arma,x=yt_3_arma)
Berdasarkan scatterplot tersebut, terlihat bahwa cenderung terdapat hubungan positif lemah antara \(Y_t\) dengan \(Y_{t-3}\). Hal ini sesuai dengan teori yang ada
cor(yt3_arma,yt_3_arma)## [1] 0.4929514Korelasi antara \(Y_t\) dengan \(Y_{t-3}\) dari hasil simulasi sebesar 0.4929514