MPDW Sunnah

Pembangkitan Model MA, AR , ARMA

Pendahuluan

Pembangkitan White Noise Manual

Pembangkitan data berpola AR, MA, ARMA, dan banyak proses deret waktu lainnya diawali pembangkitan white noise. White noise merupakan sederet nilai dari peubah bebas stokastik identik. Oleh karena itu, white noise memiliki dua karakteristik penting:

  1. White noise tidak memiliki autokorelasi (karena saling bebas)
  2. Nilai harapan dan ragam white noise sama (karena berasal dari peubah acak bebas stokastik identik)

White noise dibangkitkan dari suatu peubah acak, umumnya peubah acak normal. Sesuai Perintah Soal, Saya akan membangkitkan 300 data.

set.seed(1000)
wn <- rnorm(300)
ts.plot(wn)

Pengecekan Autokorelasi

par(mfrow = c(1, 2)) 
acf(wn)
acf(wn, xlim = c(1, 20))

Dari Plot ACF, kita dapat melihat bahwa tidak ada korelasi / Lag yang terjadi yang melewati treshold. Disini observasi Pertama memiliki korelasi 1 karena dia dibandingkan dengan dirinya sendiri. Sehingga terbukti bahwa White noise yang dibangkitkan saling bebas.

Proses MA (2)

Moving Average sering kali dimodelkan sebagai berikut :

\[ y_{t} = c + e_t + \theta_{1}e_{t-1} + \theta_{2}e_{t-2} + \dots + \theta_{q}e_{t-q} = c+{e_t+\sum_{i=1}^p \theta_ie_{t-i}} \]

Dengan teta sebagai parameter dan e(t) sebagai white noise. Proses MA yang paling sederhana dilakukan dengan c atau nilai konstata = 0 , sehingga kita hanya melihat dan memperhitungkan White Noise. Dapat dilihat juga bahwa semakin banyak Ordonya, parameternya juga akan semakin bertambah mengikuti banyak ordo. Di sini nilai pertama white-noise akan dimasukkan sebagai nilai awal

set.seed(1000)
MA<- wn[c(1,2)]

Model Awal MA 2

\[ y_{t} = e_t + \theta_{1}e_{t-1} + \theta_{2}e_{t-2} \]

Masukkan Parameter \(\theta = 0.4\) dan \(\theta_2 = 0.6\)

for(i in 3:300){
   MA[i] <- wn[i] + 0.4 * wn[i - 1] + 0.6 * wn[i - 2] 
}
MA
##   [1] -0.44577826 -1.20585657 -0.70868327 -0.06767501 -0.50612320 -0.31647800
##   [7] -1.10199622  0.29810996 -0.01612608 -0.94866959 -1.54277830 -1.77133049
##  [13] -0.68987099 -0.40501306 -1.31156126 -0.43688233 -0.57852292  0.18899784
##  [19] -1.94356544 -0.59052094  1.52738206 -0.03109488  1.94548393  0.13006108
##  [25]  0.06675213  0.66397432 -1.93067461 -0.01669873 -0.37535389  1.64629124
##  [31]  0.61350611  1.34740948 -0.55353702 -0.32806790 -2.37612100 -0.79628142
##  [37] -1.35018440  1.02470206 -0.53828940 -1.01044693 -1.50162293  0.39579770
##  [43]  0.43977545  0.87753599 -0.12211094  1.35119513  0.68562875  1.07775550
##  [49] -1.19725312 -1.45484973 -0.96242331 -1.57491888  0.11079219 -1.49406108
##  [55] -0.05935708  0.39613028 -1.34850402  0.16778021  1.60447473  0.34777818
##  [61]  1.73198614 -0.69126931  1.40486375  1.45486038  0.72395281  0.22501188
##  [67] -0.98868211  1.21320500 -0.65334267  0.62999129 -2.05597220  0.08637725
##  [73] -0.34914498  1.89962484  0.56747748  0.30866343 -0.11153565  1.18631691
##  [79]  1.38768631  1.46375627  1.74036823  0.91673139  0.44260315  1.38901363
##  [85]  0.75057681  0.87191580  0.13180904 -2.42278732 -2.01439044  0.24611790
##  [91]  2.16728848  0.10234277  0.87664742 -1.12847075  0.10877448  0.32972035
##  [97]  0.56852966  1.28682478  0.21072349  2.36124980 -0.79406702  0.82661485
## [103] -0.86966677  0.91227492  0.40463187  0.76029060  0.54404496 -1.95589619
## [109]  0.36951233  0.27761944  1.84791413  0.73294821  0.10710414 -0.39502378
## [115]  1.62593060  0.15355169  1.17882571 -1.67703461  1.19673587 -0.01203474
## [121]  1.63717637  0.04728279  0.67084187  1.05780720  1.02657146  0.09835070
## [127] -0.90782728 -2.07059918 -0.81052589 -1.38664207 -0.64712070  0.18519904
## [133]  1.05073898  1.07363754  0.68864587 -0.40803864  0.90942018 -1.34437854
## [139] -1.31269769 -1.34972634 -0.53836408  0.65260978 -0.46232595  0.86674610
## [145] -1.31177522 -0.28788810  0.37244183  1.98029900  1.44844334  1.74624395
## [151] -0.64799155  0.84311765 -0.15144313  1.85528113  0.56640193 -1.03069885
## [157] -1.74424582 -1.18186877 -0.34399281 -0.84038721 -0.16701157  0.35121131
## [163]  2.47776799  1.05521102  0.64779663 -0.65705778  0.02750283  0.44386427
## [169] -0.19473509  0.03135829 -0.01225984 -0.62009120 -0.26702734 -1.34782478
## [175] -1.73369644 -2.20912569  0.64499434  1.26582125  1.86395001  0.63997837
## [181]  1.74842900  1.58863333  2.91747539  2.32705371  1.84448785  0.91659279
## [187] -0.19940894  0.64371526  1.11939192  0.54328682  0.14330464 -1.54131156
## [193] -1.39316055 -0.63277642  0.14010499  2.04117620  1.50237215  1.37261913
## [199]  0.74965735 -0.70809069  0.75168089  0.53883827 -0.83911947 -0.46790706
## [205] -1.76217493 -0.48000621 -0.27026455 -0.07790059  1.01556987 -0.30730168
## [211]  0.14444835  0.35050864  0.07542359 -1.63684593 -1.62591008 -3.15471712
## [217] -1.12739561 -1.50082720 -0.63486310 -0.93093075 -1.95046702 -1.68458535
## [223] -2.88580060 -0.45712345  0.64497177  0.96803353 -0.68574300 -0.08515739
## [229]  0.14273694  0.11908040 -0.56226445 -1.48733860  1.15551347 -1.63441038
## [235] -0.18682777 -1.07076885  0.16027792 -0.55666626  1.97240645 -0.63972394
## [241]  0.08810334 -0.15464880 -0.22721179  1.12945564  0.21254355  0.51639743
## [247]  0.50003393  0.72917293  0.58973785 -0.74601676  0.23221516 -0.51994522
## [253]  0.89096541  1.07799261  0.35396950  1.00819703 -0.59640806  0.34614609
## [259] -2.08610867  0.47313663 -1.30907082 -0.96282098 -1.28078396  1.13183130
## [265]  2.49932827  2.12991056  1.34273334  0.88271205  2.07724452  1.40901120
## [271] -0.26478648  0.33976713  0.31461353  0.31593828  0.11471051 -1.06773824
## [277] -0.64589379 -0.94739004 -0.89693970  0.45765802  0.48307679  1.81114656
## [283] -0.13469192  0.11083465  0.53008787  2.10832233  3.05308078  0.51795759
## [289]  2.55284338 -0.20453782  0.05204305 -1.09443875 -1.33505161 -2.28069968
## [295] -2.48469173 -1.84353691 -1.26365963 -0.16919897 -0.47997904 -0.56742791

Selain menggunakan cara di atas, pembangkitan proses MA(2) dapat dilakukan dengan fungsi arima.sim() sebagai berikut.

ARIMA.sim

MA2 <- arima.sim(list(order=c(0,0,2), ma=c(0.4, 0.6)), n=300)
MA2
## Time Series:
## Start = 1 
## End = 300 
## Frequency = 1 
##   [1] -0.70868327 -0.06767501 -0.50612320 -0.31647800 -1.10199622  0.29810996
##   [7] -0.01612608 -0.94866959 -1.54277830 -1.77133049 -0.68987099 -0.40501306
##  [13] -1.31156126 -0.43688233 -0.57852292  0.18899784 -1.94356544 -0.59052094
##  [19]  1.52738206 -0.03109488  1.94548393  0.13006108  0.06675213  0.66397432
##  [25] -1.93067461 -0.01669873 -0.37535389  1.64629124  0.61350611  1.34740948
##  [31] -0.55353702 -0.32806790 -2.37612100 -0.79628142 -1.35018440  1.02470206
##  [37] -0.53828940 -1.01044693 -1.50162293  0.39579770  0.43977545  0.87753599
##  [43] -0.12211094  1.35119513  0.68562875  1.07775550 -1.19725312 -1.45484973
##  [49] -0.96242331 -1.57491888  0.11079219 -1.49406108 -0.05935708  0.39613028
##  [55] -1.34850402  0.16778021  1.60447473  0.34777818  1.73198614 -0.69126931
##  [61]  1.40486375  1.45486038  0.72395281  0.22501188 -0.98868211  1.21320500
##  [67] -0.65334267  0.62999129 -2.05597220  0.08637725 -0.34914498  1.89962484
##  [73]  0.56747748  0.30866343 -0.11153565  1.18631691  1.38768631  1.46375627
##  [79]  1.74036823  0.91673139  0.44260315  1.38901363  0.75057681  0.87191580
##  [85]  0.13180904 -2.42278732 -2.01439044  0.24611790  2.16728848  0.10234277
##  [91]  0.87664742 -1.12847075  0.10877448  0.32972035  0.56852966  1.28682478
##  [97]  0.21072349  2.36124980 -0.79406702  0.82661485 -0.86966677  0.91227492
## [103]  0.40463187  0.76029060  0.54404496 -1.95589619  0.36951233  0.27761944
## [109]  1.84791413  0.73294821  0.10710414 -0.39502378  1.62593060  0.15355169
## [115]  1.17882571 -1.67703461  1.19673587 -0.01203474  1.63717637  0.04728279
## [121]  0.67084187  1.05780720  1.02657146  0.09835070 -0.90782728 -2.07059918
## [127] -0.81052589 -1.38664207 -0.64712070  0.18519904  1.05073898  1.07363754
## [133]  0.68864587 -0.40803864  0.90942018 -1.34437854 -1.31269769 -1.34972634
## [139] -0.53836408  0.65260978 -0.46232595  0.86674610 -1.31177522 -0.28788810
## [145]  0.37244183  1.98029900  1.44844334  1.74624395 -0.64799155  0.84311765
## [151] -0.15144313  1.85528113  0.56640193 -1.03069885 -1.74424582 -1.18186877
## [157] -0.34399281 -0.84038721 -0.16701157  0.35121131  2.47776799  1.05521102
## [163]  0.64779663 -0.65705778  0.02750283  0.44386427 -0.19473509  0.03135829
## [169] -0.01225984 -0.62009120 -0.26702734 -1.34782478 -1.73369644 -2.20912569
## [175]  0.64499434  1.26582125  1.86395001  0.63997837  1.74842900  1.58863333
## [181]  2.91747539  2.32705371  1.84448785  0.91659279 -0.19940894  0.64371526
## [187]  1.11939192  0.54328682  0.14330464 -1.54131156 -1.39316055 -0.63277642
## [193]  0.14010499  2.04117620  1.50237215  1.37261913  0.74965735 -0.70809069
## [199]  0.75168089  0.53883827 -0.83911947 -0.46790706 -1.76217493 -0.48000621
## [205] -0.27026455 -0.07790059  1.01556987 -0.30730168  0.14444835  0.35050864
## [211]  0.07542359 -1.63684593 -1.62591008 -3.15471712 -1.12739561 -1.50082720
## [217] -0.63486310 -0.93093075 -1.95046702 -1.68458535 -2.88580060 -0.45712345
## [223]  0.64497177  0.96803353 -0.68574300 -0.08515739  0.14273694  0.11908040
## [229] -0.56226445 -1.48733860  1.15551347 -1.63441038 -0.18682777 -1.07076885
## [235]  0.16027792 -0.55666626  1.97240645 -0.63972394  0.08810334 -0.15464880
## [241] -0.22721179  1.12945564  0.21254355  0.51639743  0.50003393  0.72917293
## [247]  0.58973785 -0.74601676  0.23221516 -0.51994522  0.89096541  1.07799261
## [253]  0.35396950  1.00819703 -0.59640806  0.34614609 -2.08610867  0.47313663
## [259] -1.30907082 -0.96282098 -1.28078396  1.13183130  2.49932827  2.12991056
## [265]  1.34273334  0.88271205  2.07724452  1.40901120 -0.26478648  0.33976713
## [271]  0.31461353  0.31593828  0.11471051 -1.06773824 -0.64589379 -0.94739004
## [277] -0.89693970  0.45765802  0.48307679  1.81114656 -0.13469192  0.11083465
## [283]  0.53008787  2.10832233  3.05308078  0.51795759  2.55284338 -0.20453782
## [289]  0.05204305 -1.09443875 -1.33505161 -2.28069968 -2.48469173 -1.84353691
## [295] -1.26365963 -0.16919897 -0.47997904 -0.56742791  0.24418800 -1.08960646

Plot

Time Series Plot

ts.plot(MA)

Dapat dilihat dari Plot time series dengan data bangkitan MA, data MA 2 cukup stasioner dalam rataan, dimana data berpusat pada sekitar nilai tengah.

Plot ACF

acf(MA,lag.max=20)

Berdasarkan plot AFC tersebut, terlihat bahwa plot ACF cuts off di lag kedua

Plot PACF

pacf(MA)

Berdasarkan plot PACF tersebut, terlihat bahwa plot PACF cenderung tails off dan membentuk gelombang sinus

TSA::eacf(MA)
## AR/MA
##   0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x o o o o o x o o o  o  o  o 
## 1 x x o o o o o o o o o  o  o  o 
## 2 x x x x o o o o o o o  o  o  o 
## 3 x x o x o o o o o o o  o  o  o 
## 4 x x o x o o o o o o o  o  o  o 
## 5 x x x x o x o o o o o  o  o  o 
## 6 x x x o x x x o o o o  o  o  o 
## 7 x o x o x o x o o o o  o  o  o

Dapat dilihat dari PLOT EACF bahwa Segita 0 dapat dibentuk paling awal di AR (0) dan MA (2) sesuai pada bangkitan Model yang kita buat. Berbicara dengan kestasioneran Model, Model ini dapat dikatakan stasioner dilihat dari PLOT ACF yang secara teoritis ACF MA memiliki kecenderungan untuk cut-off sesuai ordo yang dipakai, sehingga hal ini sejalan dengan teori dimana Plotnya tadi cut-off di ordo / lag 2, dapat dilihat juga Plot PACF yang membentuk pola tailsoff dan sinus yang mendukung bahwa model ini Stasioner.

Scatter Plot antar Lag

Kita akan membuat scatter plot antar lag

Scatter Plot \(Yt\) dan \(Yt_{-1}\)

yt_ma <- MA[-1]
yt_ma
##   [1] -1.20585657 -0.70868327 -0.06767501 -0.50612320 -0.31647800 -1.10199622
##   [7]  0.29810996 -0.01612608 -0.94866959 -1.54277830 -1.77133049 -0.68987099
##  [13] -0.40501306 -1.31156126 -0.43688233 -0.57852292  0.18899784 -1.94356544
##  [19] -0.59052094  1.52738206 -0.03109488  1.94548393  0.13006108  0.06675213
##  [25]  0.66397432 -1.93067461 -0.01669873 -0.37535389  1.64629124  0.61350611
##  [31]  1.34740948 -0.55353702 -0.32806790 -2.37612100 -0.79628142 -1.35018440
##  [37]  1.02470206 -0.53828940 -1.01044693 -1.50162293  0.39579770  0.43977545
##  [43]  0.87753599 -0.12211094  1.35119513  0.68562875  1.07775550 -1.19725312
##  [49] -1.45484973 -0.96242331 -1.57491888  0.11079219 -1.49406108 -0.05935708
##  [55]  0.39613028 -1.34850402  0.16778021  1.60447473  0.34777818  1.73198614
##  [61] -0.69126931  1.40486375  1.45486038  0.72395281  0.22501188 -0.98868211
##  [67]  1.21320500 -0.65334267  0.62999129 -2.05597220  0.08637725 -0.34914498
##  [73]  1.89962484  0.56747748  0.30866343 -0.11153565  1.18631691  1.38768631
##  [79]  1.46375627  1.74036823  0.91673139  0.44260315  1.38901363  0.75057681
##  [85]  0.87191580  0.13180904 -2.42278732 -2.01439044  0.24611790  2.16728848
##  [91]  0.10234277  0.87664742 -1.12847075  0.10877448  0.32972035  0.56852966
##  [97]  1.28682478  0.21072349  2.36124980 -0.79406702  0.82661485 -0.86966677
## [103]  0.91227492  0.40463187  0.76029060  0.54404496 -1.95589619  0.36951233
## [109]  0.27761944  1.84791413  0.73294821  0.10710414 -0.39502378  1.62593060
## [115]  0.15355169  1.17882571 -1.67703461  1.19673587 -0.01203474  1.63717637
## [121]  0.04728279  0.67084187  1.05780720  1.02657146  0.09835070 -0.90782728
## [127] -2.07059918 -0.81052589 -1.38664207 -0.64712070  0.18519904  1.05073898
## [133]  1.07363754  0.68864587 -0.40803864  0.90942018 -1.34437854 -1.31269769
## [139] -1.34972634 -0.53836408  0.65260978 -0.46232595  0.86674610 -1.31177522
## [145] -0.28788810  0.37244183  1.98029900  1.44844334  1.74624395 -0.64799155
## [151]  0.84311765 -0.15144313  1.85528113  0.56640193 -1.03069885 -1.74424582
## [157] -1.18186877 -0.34399281 -0.84038721 -0.16701157  0.35121131  2.47776799
## [163]  1.05521102  0.64779663 -0.65705778  0.02750283  0.44386427 -0.19473509
## [169]  0.03135829 -0.01225984 -0.62009120 -0.26702734 -1.34782478 -1.73369644
## [175] -2.20912569  0.64499434  1.26582125  1.86395001  0.63997837  1.74842900
## [181]  1.58863333  2.91747539  2.32705371  1.84448785  0.91659279 -0.19940894
## [187]  0.64371526  1.11939192  0.54328682  0.14330464 -1.54131156 -1.39316055
## [193] -0.63277642  0.14010499  2.04117620  1.50237215  1.37261913  0.74965735
## [199] -0.70809069  0.75168089  0.53883827 -0.83911947 -0.46790706 -1.76217493
## [205] -0.48000621 -0.27026455 -0.07790059  1.01556987 -0.30730168  0.14444835
## [211]  0.35050864  0.07542359 -1.63684593 -1.62591008 -3.15471712 -1.12739561
## [217] -1.50082720 -0.63486310 -0.93093075 -1.95046702 -1.68458535 -2.88580060
## [223] -0.45712345  0.64497177  0.96803353 -0.68574300 -0.08515739  0.14273694
## [229]  0.11908040 -0.56226445 -1.48733860  1.15551347 -1.63441038 -0.18682777
## [235] -1.07076885  0.16027792 -0.55666626  1.97240645 -0.63972394  0.08810334
## [241] -0.15464880 -0.22721179  1.12945564  0.21254355  0.51639743  0.50003393
## [247]  0.72917293  0.58973785 -0.74601676  0.23221516 -0.51994522  0.89096541
## [253]  1.07799261  0.35396950  1.00819703 -0.59640806  0.34614609 -2.08610867
## [259]  0.47313663 -1.30907082 -0.96282098 -1.28078396  1.13183130  2.49932827
## [265]  2.12991056  1.34273334  0.88271205  2.07724452  1.40901120 -0.26478648
## [271]  0.33976713  0.31461353  0.31593828  0.11471051 -1.06773824 -0.64589379
## [277] -0.94739004 -0.89693970  0.45765802  0.48307679  1.81114656 -0.13469192
## [283]  0.11083465  0.53008787  2.10832233  3.05308078  0.51795759  2.55284338
## [289] -0.20453782  0.05204305 -1.09443875 -1.33505161 -2.28069968 -2.48469173
## [295] -1.84353691 -1.26365963 -0.16919897 -0.47997904 -0.56742791
# Yt-1
yt_1_ma <- MA[-300]
yt_1_ma
##   [1] -0.44577826 -1.20585657 -0.70868327 -0.06767501 -0.50612320 -0.31647800
##   [7] -1.10199622  0.29810996 -0.01612608 -0.94866959 -1.54277830 -1.77133049
##  [13] -0.68987099 -0.40501306 -1.31156126 -0.43688233 -0.57852292  0.18899784
##  [19] -1.94356544 -0.59052094  1.52738206 -0.03109488  1.94548393  0.13006108
##  [25]  0.06675213  0.66397432 -1.93067461 -0.01669873 -0.37535389  1.64629124
##  [31]  0.61350611  1.34740948 -0.55353702 -0.32806790 -2.37612100 -0.79628142
##  [37] -1.35018440  1.02470206 -0.53828940 -1.01044693 -1.50162293  0.39579770
##  [43]  0.43977545  0.87753599 -0.12211094  1.35119513  0.68562875  1.07775550
##  [49] -1.19725312 -1.45484973 -0.96242331 -1.57491888  0.11079219 -1.49406108
##  [55] -0.05935708  0.39613028 -1.34850402  0.16778021  1.60447473  0.34777818
##  [61]  1.73198614 -0.69126931  1.40486375  1.45486038  0.72395281  0.22501188
##  [67] -0.98868211  1.21320500 -0.65334267  0.62999129 -2.05597220  0.08637725
##  [73] -0.34914498  1.89962484  0.56747748  0.30866343 -0.11153565  1.18631691
##  [79]  1.38768631  1.46375627  1.74036823  0.91673139  0.44260315  1.38901363
##  [85]  0.75057681  0.87191580  0.13180904 -2.42278732 -2.01439044  0.24611790
##  [91]  2.16728848  0.10234277  0.87664742 -1.12847075  0.10877448  0.32972035
##  [97]  0.56852966  1.28682478  0.21072349  2.36124980 -0.79406702  0.82661485
## [103] -0.86966677  0.91227492  0.40463187  0.76029060  0.54404496 -1.95589619
## [109]  0.36951233  0.27761944  1.84791413  0.73294821  0.10710414 -0.39502378
## [115]  1.62593060  0.15355169  1.17882571 -1.67703461  1.19673587 -0.01203474
## [121]  1.63717637  0.04728279  0.67084187  1.05780720  1.02657146  0.09835070
## [127] -0.90782728 -2.07059918 -0.81052589 -1.38664207 -0.64712070  0.18519904
## [133]  1.05073898  1.07363754  0.68864587 -0.40803864  0.90942018 -1.34437854
## [139] -1.31269769 -1.34972634 -0.53836408  0.65260978 -0.46232595  0.86674610
## [145] -1.31177522 -0.28788810  0.37244183  1.98029900  1.44844334  1.74624395
## [151] -0.64799155  0.84311765 -0.15144313  1.85528113  0.56640193 -1.03069885
## [157] -1.74424582 -1.18186877 -0.34399281 -0.84038721 -0.16701157  0.35121131
## [163]  2.47776799  1.05521102  0.64779663 -0.65705778  0.02750283  0.44386427
## [169] -0.19473509  0.03135829 -0.01225984 -0.62009120 -0.26702734 -1.34782478
## [175] -1.73369644 -2.20912569  0.64499434  1.26582125  1.86395001  0.63997837
## [181]  1.74842900  1.58863333  2.91747539  2.32705371  1.84448785  0.91659279
## [187] -0.19940894  0.64371526  1.11939192  0.54328682  0.14330464 -1.54131156
## [193] -1.39316055 -0.63277642  0.14010499  2.04117620  1.50237215  1.37261913
## [199]  0.74965735 -0.70809069  0.75168089  0.53883827 -0.83911947 -0.46790706
## [205] -1.76217493 -0.48000621 -0.27026455 -0.07790059  1.01556987 -0.30730168
## [211]  0.14444835  0.35050864  0.07542359 -1.63684593 -1.62591008 -3.15471712
## [217] -1.12739561 -1.50082720 -0.63486310 -0.93093075 -1.95046702 -1.68458535
## [223] -2.88580060 -0.45712345  0.64497177  0.96803353 -0.68574300 -0.08515739
## [229]  0.14273694  0.11908040 -0.56226445 -1.48733860  1.15551347 -1.63441038
## [235] -0.18682777 -1.07076885  0.16027792 -0.55666626  1.97240645 -0.63972394
## [241]  0.08810334 -0.15464880 -0.22721179  1.12945564  0.21254355  0.51639743
## [247]  0.50003393  0.72917293  0.58973785 -0.74601676  0.23221516 -0.51994522
## [253]  0.89096541  1.07799261  0.35396950  1.00819703 -0.59640806  0.34614609
## [259] -2.08610867  0.47313663 -1.30907082 -0.96282098 -1.28078396  1.13183130
## [265]  2.49932827  2.12991056  1.34273334  0.88271205  2.07724452  1.40901120
## [271] -0.26478648  0.33976713  0.31461353  0.31593828  0.11471051 -1.06773824
## [277] -0.64589379 -0.94739004 -0.89693970  0.45765802  0.48307679  1.81114656
## [283] -0.13469192  0.11083465  0.53008787  2.10832233  3.05308078  0.51795759
## [289]  2.55284338 -0.20453782  0.05204305 -1.09443875 -1.33505161 -2.28069968
## [295] -2.48469173 -1.84353691 -1.26365963 -0.16919897 -0.47997904
plot(y=yt_ma,x=yt_1_ma)

Dapat dilihat bahwa Terdapat korelasi positif antara Yt dengan Yt-1, Mari kita hitung korelasinya:

cor(yt_ma,yt_1_ma)
## [1] 0.393322

Didapat Korelasinya Positif, akan dicek dengan Korelasi secara teoritis, dan ternyata hasilnya tidak terlalu jauh dan hampir mirip

\[ \rho_1=\frac{-\theta_1+\theta_1\theta_2}{1+(\theta_1)^2+\theta_2^2}=\frac{-(-0.4)*(-0.4*-0.6)}{1+(-0.4)^2+(-0.6)^2}= 0.42 \]

Scatter Plot Yt dengan Yt-2

yt_ma2 <- MA[-c(1,2)]
yt_ma2
##   [1] -0.70868327 -0.06767501 -0.50612320 -0.31647800 -1.10199622  0.29810996
##   [7] -0.01612608 -0.94866959 -1.54277830 -1.77133049 -0.68987099 -0.40501306
##  [13] -1.31156126 -0.43688233 -0.57852292  0.18899784 -1.94356544 -0.59052094
##  [19]  1.52738206 -0.03109488  1.94548393  0.13006108  0.06675213  0.66397432
##  [25] -1.93067461 -0.01669873 -0.37535389  1.64629124  0.61350611  1.34740948
##  [31] -0.55353702 -0.32806790 -2.37612100 -0.79628142 -1.35018440  1.02470206
##  [37] -0.53828940 -1.01044693 -1.50162293  0.39579770  0.43977545  0.87753599
##  [43] -0.12211094  1.35119513  0.68562875  1.07775550 -1.19725312 -1.45484973
##  [49] -0.96242331 -1.57491888  0.11079219 -1.49406108 -0.05935708  0.39613028
##  [55] -1.34850402  0.16778021  1.60447473  0.34777818  1.73198614 -0.69126931
##  [61]  1.40486375  1.45486038  0.72395281  0.22501188 -0.98868211  1.21320500
##  [67] -0.65334267  0.62999129 -2.05597220  0.08637725 -0.34914498  1.89962484
##  [73]  0.56747748  0.30866343 -0.11153565  1.18631691  1.38768631  1.46375627
##  [79]  1.74036823  0.91673139  0.44260315  1.38901363  0.75057681  0.87191580
##  [85]  0.13180904 -2.42278732 -2.01439044  0.24611790  2.16728848  0.10234277
##  [91]  0.87664742 -1.12847075  0.10877448  0.32972035  0.56852966  1.28682478
##  [97]  0.21072349  2.36124980 -0.79406702  0.82661485 -0.86966677  0.91227492
## [103]  0.40463187  0.76029060  0.54404496 -1.95589619  0.36951233  0.27761944
## [109]  1.84791413  0.73294821  0.10710414 -0.39502378  1.62593060  0.15355169
## [115]  1.17882571 -1.67703461  1.19673587 -0.01203474  1.63717637  0.04728279
## [121]  0.67084187  1.05780720  1.02657146  0.09835070 -0.90782728 -2.07059918
## [127] -0.81052589 -1.38664207 -0.64712070  0.18519904  1.05073898  1.07363754
## [133]  0.68864587 -0.40803864  0.90942018 -1.34437854 -1.31269769 -1.34972634
## [139] -0.53836408  0.65260978 -0.46232595  0.86674610 -1.31177522 -0.28788810
## [145]  0.37244183  1.98029900  1.44844334  1.74624395 -0.64799155  0.84311765
## [151] -0.15144313  1.85528113  0.56640193 -1.03069885 -1.74424582 -1.18186877
## [157] -0.34399281 -0.84038721 -0.16701157  0.35121131  2.47776799  1.05521102
## [163]  0.64779663 -0.65705778  0.02750283  0.44386427 -0.19473509  0.03135829
## [169] -0.01225984 -0.62009120 -0.26702734 -1.34782478 -1.73369644 -2.20912569
## [175]  0.64499434  1.26582125  1.86395001  0.63997837  1.74842900  1.58863333
## [181]  2.91747539  2.32705371  1.84448785  0.91659279 -0.19940894  0.64371526
## [187]  1.11939192  0.54328682  0.14330464 -1.54131156 -1.39316055 -0.63277642
## [193]  0.14010499  2.04117620  1.50237215  1.37261913  0.74965735 -0.70809069
## [199]  0.75168089  0.53883827 -0.83911947 -0.46790706 -1.76217493 -0.48000621
## [205] -0.27026455 -0.07790059  1.01556987 -0.30730168  0.14444835  0.35050864
## [211]  0.07542359 -1.63684593 -1.62591008 -3.15471712 -1.12739561 -1.50082720
## [217] -0.63486310 -0.93093075 -1.95046702 -1.68458535 -2.88580060 -0.45712345
## [223]  0.64497177  0.96803353 -0.68574300 -0.08515739  0.14273694  0.11908040
## [229] -0.56226445 -1.48733860  1.15551347 -1.63441038 -0.18682777 -1.07076885
## [235]  0.16027792 -0.55666626  1.97240645 -0.63972394  0.08810334 -0.15464880
## [241] -0.22721179  1.12945564  0.21254355  0.51639743  0.50003393  0.72917293
## [247]  0.58973785 -0.74601676  0.23221516 -0.51994522  0.89096541  1.07799261
## [253]  0.35396950  1.00819703 -0.59640806  0.34614609 -2.08610867  0.47313663
## [259] -1.30907082 -0.96282098 -1.28078396  1.13183130  2.49932827  2.12991056
## [265]  1.34273334  0.88271205  2.07724452  1.40901120 -0.26478648  0.33976713
## [271]  0.31461353  0.31593828  0.11471051 -1.06773824 -0.64589379 -0.94739004
## [277] -0.89693970  0.45765802  0.48307679  1.81114656 -0.13469192  0.11083465
## [283]  0.53008787  2.10832233  3.05308078  0.51795759  2.55284338 -0.20453782
## [289]  0.05204305 -1.09443875 -1.33505161 -2.28069968 -2.48469173 -1.84353691
## [295] -1.26365963 -0.16919897 -0.47997904 -0.56742791
yt_2_ma <- MA[-c(299,300)]
yt_2_ma
##   [1] -0.44577826 -1.20585657 -0.70868327 -0.06767501 -0.50612320 -0.31647800
##   [7] -1.10199622  0.29810996 -0.01612608 -0.94866959 -1.54277830 -1.77133049
##  [13] -0.68987099 -0.40501306 -1.31156126 -0.43688233 -0.57852292  0.18899784
##  [19] -1.94356544 -0.59052094  1.52738206 -0.03109488  1.94548393  0.13006108
##  [25]  0.06675213  0.66397432 -1.93067461 -0.01669873 -0.37535389  1.64629124
##  [31]  0.61350611  1.34740948 -0.55353702 -0.32806790 -2.37612100 -0.79628142
##  [37] -1.35018440  1.02470206 -0.53828940 -1.01044693 -1.50162293  0.39579770
##  [43]  0.43977545  0.87753599 -0.12211094  1.35119513  0.68562875  1.07775550
##  [49] -1.19725312 -1.45484973 -0.96242331 -1.57491888  0.11079219 -1.49406108
##  [55] -0.05935708  0.39613028 -1.34850402  0.16778021  1.60447473  0.34777818
##  [61]  1.73198614 -0.69126931  1.40486375  1.45486038  0.72395281  0.22501188
##  [67] -0.98868211  1.21320500 -0.65334267  0.62999129 -2.05597220  0.08637725
##  [73] -0.34914498  1.89962484  0.56747748  0.30866343 -0.11153565  1.18631691
##  [79]  1.38768631  1.46375627  1.74036823  0.91673139  0.44260315  1.38901363
##  [85]  0.75057681  0.87191580  0.13180904 -2.42278732 -2.01439044  0.24611790
##  [91]  2.16728848  0.10234277  0.87664742 -1.12847075  0.10877448  0.32972035
##  [97]  0.56852966  1.28682478  0.21072349  2.36124980 -0.79406702  0.82661485
## [103] -0.86966677  0.91227492  0.40463187  0.76029060  0.54404496 -1.95589619
## [109]  0.36951233  0.27761944  1.84791413  0.73294821  0.10710414 -0.39502378
## [115]  1.62593060  0.15355169  1.17882571 -1.67703461  1.19673587 -0.01203474
## [121]  1.63717637  0.04728279  0.67084187  1.05780720  1.02657146  0.09835070
## [127] -0.90782728 -2.07059918 -0.81052589 -1.38664207 -0.64712070  0.18519904
## [133]  1.05073898  1.07363754  0.68864587 -0.40803864  0.90942018 -1.34437854
## [139] -1.31269769 -1.34972634 -0.53836408  0.65260978 -0.46232595  0.86674610
## [145] -1.31177522 -0.28788810  0.37244183  1.98029900  1.44844334  1.74624395
## [151] -0.64799155  0.84311765 -0.15144313  1.85528113  0.56640193 -1.03069885
## [157] -1.74424582 -1.18186877 -0.34399281 -0.84038721 -0.16701157  0.35121131
## [163]  2.47776799  1.05521102  0.64779663 -0.65705778  0.02750283  0.44386427
## [169] -0.19473509  0.03135829 -0.01225984 -0.62009120 -0.26702734 -1.34782478
## [175] -1.73369644 -2.20912569  0.64499434  1.26582125  1.86395001  0.63997837
## [181]  1.74842900  1.58863333  2.91747539  2.32705371  1.84448785  0.91659279
## [187] -0.19940894  0.64371526  1.11939192  0.54328682  0.14330464 -1.54131156
## [193] -1.39316055 -0.63277642  0.14010499  2.04117620  1.50237215  1.37261913
## [199]  0.74965735 -0.70809069  0.75168089  0.53883827 -0.83911947 -0.46790706
## [205] -1.76217493 -0.48000621 -0.27026455 -0.07790059  1.01556987 -0.30730168
## [211]  0.14444835  0.35050864  0.07542359 -1.63684593 -1.62591008 -3.15471712
## [217] -1.12739561 -1.50082720 -0.63486310 -0.93093075 -1.95046702 -1.68458535
## [223] -2.88580060 -0.45712345  0.64497177  0.96803353 -0.68574300 -0.08515739
## [229]  0.14273694  0.11908040 -0.56226445 -1.48733860  1.15551347 -1.63441038
## [235] -0.18682777 -1.07076885  0.16027792 -0.55666626  1.97240645 -0.63972394
## [241]  0.08810334 -0.15464880 -0.22721179  1.12945564  0.21254355  0.51639743
## [247]  0.50003393  0.72917293  0.58973785 -0.74601676  0.23221516 -0.51994522
## [253]  0.89096541  1.07799261  0.35396950  1.00819703 -0.59640806  0.34614609
## [259] -2.08610867  0.47313663 -1.30907082 -0.96282098 -1.28078396  1.13183130
## [265]  2.49932827  2.12991056  1.34273334  0.88271205  2.07724452  1.40901120
## [271] -0.26478648  0.33976713  0.31461353  0.31593828  0.11471051 -1.06773824
## [277] -0.64589379 -0.94739004 -0.89693970  0.45765802  0.48307679  1.81114656
## [283] -0.13469192  0.11083465  0.53008787  2.10832233  3.05308078  0.51795759
## [289]  2.55284338 -0.20453782  0.05204305 -1.09443875 -1.33505161 -2.28069968
## [295] -2.48469173 -1.84353691 -1.26365963 -0.16919897
plot(y=yt_ma2,x=yt_2_ma)

Didapat Plot yang cukup positif namun dapat dilihat bahwa sebaran data mulai melebar, sehingga kita dapat mengasumsikan korelasinya berkurang namun tetap positif.

cor(yt_ma2,yt_2_ma)
## [1] 0.3670525

.

Scatter Plot Yt dengan Yt-3

yt_ma3 <- MA[-c(1,2,3)]
yt_ma3
##   [1] -0.06767501 -0.50612320 -0.31647800 -1.10199622  0.29810996 -0.01612608
##   [7] -0.94866959 -1.54277830 -1.77133049 -0.68987099 -0.40501306 -1.31156126
##  [13] -0.43688233 -0.57852292  0.18899784 -1.94356544 -0.59052094  1.52738206
##  [19] -0.03109488  1.94548393  0.13006108  0.06675213  0.66397432 -1.93067461
##  [25] -0.01669873 -0.37535389  1.64629124  0.61350611  1.34740948 -0.55353702
##  [31] -0.32806790 -2.37612100 -0.79628142 -1.35018440  1.02470206 -0.53828940
##  [37] -1.01044693 -1.50162293  0.39579770  0.43977545  0.87753599 -0.12211094
##  [43]  1.35119513  0.68562875  1.07775550 -1.19725312 -1.45484973 -0.96242331
##  [49] -1.57491888  0.11079219 -1.49406108 -0.05935708  0.39613028 -1.34850402
##  [55]  0.16778021  1.60447473  0.34777818  1.73198614 -0.69126931  1.40486375
##  [61]  1.45486038  0.72395281  0.22501188 -0.98868211  1.21320500 -0.65334267
##  [67]  0.62999129 -2.05597220  0.08637725 -0.34914498  1.89962484  0.56747748
##  [73]  0.30866343 -0.11153565  1.18631691  1.38768631  1.46375627  1.74036823
##  [79]  0.91673139  0.44260315  1.38901363  0.75057681  0.87191580  0.13180904
##  [85] -2.42278732 -2.01439044  0.24611790  2.16728848  0.10234277  0.87664742
##  [91] -1.12847075  0.10877448  0.32972035  0.56852966  1.28682478  0.21072349
##  [97]  2.36124980 -0.79406702  0.82661485 -0.86966677  0.91227492  0.40463187
## [103]  0.76029060  0.54404496 -1.95589619  0.36951233  0.27761944  1.84791413
## [109]  0.73294821  0.10710414 -0.39502378  1.62593060  0.15355169  1.17882571
## [115] -1.67703461  1.19673587 -0.01203474  1.63717637  0.04728279  0.67084187
## [121]  1.05780720  1.02657146  0.09835070 -0.90782728 -2.07059918 -0.81052589
## [127] -1.38664207 -0.64712070  0.18519904  1.05073898  1.07363754  0.68864587
## [133] -0.40803864  0.90942018 -1.34437854 -1.31269769 -1.34972634 -0.53836408
## [139]  0.65260978 -0.46232595  0.86674610 -1.31177522 -0.28788810  0.37244183
## [145]  1.98029900  1.44844334  1.74624395 -0.64799155  0.84311765 -0.15144313
## [151]  1.85528113  0.56640193 -1.03069885 -1.74424582 -1.18186877 -0.34399281
## [157] -0.84038721 -0.16701157  0.35121131  2.47776799  1.05521102  0.64779663
## [163] -0.65705778  0.02750283  0.44386427 -0.19473509  0.03135829 -0.01225984
## [169] -0.62009120 -0.26702734 -1.34782478 -1.73369644 -2.20912569  0.64499434
## [175]  1.26582125  1.86395001  0.63997837  1.74842900  1.58863333  2.91747539
## [181]  2.32705371  1.84448785  0.91659279 -0.19940894  0.64371526  1.11939192
## [187]  0.54328682  0.14330464 -1.54131156 -1.39316055 -0.63277642  0.14010499
## [193]  2.04117620  1.50237215  1.37261913  0.74965735 -0.70809069  0.75168089
## [199]  0.53883827 -0.83911947 -0.46790706 -1.76217493 -0.48000621 -0.27026455
## [205] -0.07790059  1.01556987 -0.30730168  0.14444835  0.35050864  0.07542359
## [211] -1.63684593 -1.62591008 -3.15471712 -1.12739561 -1.50082720 -0.63486310
## [217] -0.93093075 -1.95046702 -1.68458535 -2.88580060 -0.45712345  0.64497177
## [223]  0.96803353 -0.68574300 -0.08515739  0.14273694  0.11908040 -0.56226445
## [229] -1.48733860  1.15551347 -1.63441038 -0.18682777 -1.07076885  0.16027792
## [235] -0.55666626  1.97240645 -0.63972394  0.08810334 -0.15464880 -0.22721179
## [241]  1.12945564  0.21254355  0.51639743  0.50003393  0.72917293  0.58973785
## [247] -0.74601676  0.23221516 -0.51994522  0.89096541  1.07799261  0.35396950
## [253]  1.00819703 -0.59640806  0.34614609 -2.08610867  0.47313663 -1.30907082
## [259] -0.96282098 -1.28078396  1.13183130  2.49932827  2.12991056  1.34273334
## [265]  0.88271205  2.07724452  1.40901120 -0.26478648  0.33976713  0.31461353
## [271]  0.31593828  0.11471051 -1.06773824 -0.64589379 -0.94739004 -0.89693970
## [277]  0.45765802  0.48307679  1.81114656 -0.13469192  0.11083465  0.53008787
## [283]  2.10832233  3.05308078  0.51795759  2.55284338 -0.20453782  0.05204305
## [289] -1.09443875 -1.33505161 -2.28069968 -2.48469173 -1.84353691 -1.26365963
## [295] -0.16919897 -0.47997904 -0.56742791
yt_3_ma <- MA[-c(288,299,300)]
yt_3_ma
##   [1] -0.44577826 -1.20585657 -0.70868327 -0.06767501 -0.50612320 -0.31647800
##   [7] -1.10199622  0.29810996 -0.01612608 -0.94866959 -1.54277830 -1.77133049
##  [13] -0.68987099 -0.40501306 -1.31156126 -0.43688233 -0.57852292  0.18899784
##  [19] -1.94356544 -0.59052094  1.52738206 -0.03109488  1.94548393  0.13006108
##  [25]  0.06675213  0.66397432 -1.93067461 -0.01669873 -0.37535389  1.64629124
##  [31]  0.61350611  1.34740948 -0.55353702 -0.32806790 -2.37612100 -0.79628142
##  [37] -1.35018440  1.02470206 -0.53828940 -1.01044693 -1.50162293  0.39579770
##  [43]  0.43977545  0.87753599 -0.12211094  1.35119513  0.68562875  1.07775550
##  [49] -1.19725312 -1.45484973 -0.96242331 -1.57491888  0.11079219 -1.49406108
##  [55] -0.05935708  0.39613028 -1.34850402  0.16778021  1.60447473  0.34777818
##  [61]  1.73198614 -0.69126931  1.40486375  1.45486038  0.72395281  0.22501188
##  [67] -0.98868211  1.21320500 -0.65334267  0.62999129 -2.05597220  0.08637725
##  [73] -0.34914498  1.89962484  0.56747748  0.30866343 -0.11153565  1.18631691
##  [79]  1.38768631  1.46375627  1.74036823  0.91673139  0.44260315  1.38901363
##  [85]  0.75057681  0.87191580  0.13180904 -2.42278732 -2.01439044  0.24611790
##  [91]  2.16728848  0.10234277  0.87664742 -1.12847075  0.10877448  0.32972035
##  [97]  0.56852966  1.28682478  0.21072349  2.36124980 -0.79406702  0.82661485
## [103] -0.86966677  0.91227492  0.40463187  0.76029060  0.54404496 -1.95589619
## [109]  0.36951233  0.27761944  1.84791413  0.73294821  0.10710414 -0.39502378
## [115]  1.62593060  0.15355169  1.17882571 -1.67703461  1.19673587 -0.01203474
## [121]  1.63717637  0.04728279  0.67084187  1.05780720  1.02657146  0.09835070
## [127] -0.90782728 -2.07059918 -0.81052589 -1.38664207 -0.64712070  0.18519904
## [133]  1.05073898  1.07363754  0.68864587 -0.40803864  0.90942018 -1.34437854
## [139] -1.31269769 -1.34972634 -0.53836408  0.65260978 -0.46232595  0.86674610
## [145] -1.31177522 -0.28788810  0.37244183  1.98029900  1.44844334  1.74624395
## [151] -0.64799155  0.84311765 -0.15144313  1.85528113  0.56640193 -1.03069885
## [157] -1.74424582 -1.18186877 -0.34399281 -0.84038721 -0.16701157  0.35121131
## [163]  2.47776799  1.05521102  0.64779663 -0.65705778  0.02750283  0.44386427
## [169] -0.19473509  0.03135829 -0.01225984 -0.62009120 -0.26702734 -1.34782478
## [175] -1.73369644 -2.20912569  0.64499434  1.26582125  1.86395001  0.63997837
## [181]  1.74842900  1.58863333  2.91747539  2.32705371  1.84448785  0.91659279
## [187] -0.19940894  0.64371526  1.11939192  0.54328682  0.14330464 -1.54131156
## [193] -1.39316055 -0.63277642  0.14010499  2.04117620  1.50237215  1.37261913
## [199]  0.74965735 -0.70809069  0.75168089  0.53883827 -0.83911947 -0.46790706
## [205] -1.76217493 -0.48000621 -0.27026455 -0.07790059  1.01556987 -0.30730168
## [211]  0.14444835  0.35050864  0.07542359 -1.63684593 -1.62591008 -3.15471712
## [217] -1.12739561 -1.50082720 -0.63486310 -0.93093075 -1.95046702 -1.68458535
## [223] -2.88580060 -0.45712345  0.64497177  0.96803353 -0.68574300 -0.08515739
## [229]  0.14273694  0.11908040 -0.56226445 -1.48733860  1.15551347 -1.63441038
## [235] -0.18682777 -1.07076885  0.16027792 -0.55666626  1.97240645 -0.63972394
## [241]  0.08810334 -0.15464880 -0.22721179  1.12945564  0.21254355  0.51639743
## [247]  0.50003393  0.72917293  0.58973785 -0.74601676  0.23221516 -0.51994522
## [253]  0.89096541  1.07799261  0.35396950  1.00819703 -0.59640806  0.34614609
## [259] -2.08610867  0.47313663 -1.30907082 -0.96282098 -1.28078396  1.13183130
## [265]  2.49932827  2.12991056  1.34273334  0.88271205  2.07724452  1.40901120
## [271] -0.26478648  0.33976713  0.31461353  0.31593828  0.11471051 -1.06773824
## [277] -0.64589379 -0.94739004 -0.89693970  0.45765802  0.48307679  1.81114656
## [283] -0.13469192  0.11083465  0.53008787  2.10832233  3.05308078  2.55284338
## [289] -0.20453782  0.05204305 -1.09443875 -1.33505161 -2.28069968 -2.48469173
## [295] -1.84353691 -1.26365963 -0.16919897
plot(y=yt_ma3,x=yt_3_ma)

Didapat bahwa Plotnya mulai tidak menunjukkan adanya korelasi, secara teoritis , ketika kita menggunakan MA, Korelasi antara data dengan lag nya adalah sebanyak ordo, sehingga maksimum korelasi didapat di Ordo ke 2 ( Yt-2), dimana setelahnya ( Yt-3- Yt- k) tidak memiliki korelasi lagi

cor(y=yt_ma3,x=yt_3_ma)
## [1] 0.01210676

Dapat dilihat korelasinya sangat kecil dan mendekati 0 , sesuai dengan teori.

AR (2)

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. Dari Namanya sendiri autoregressive, dimana model ini memasukkan datanya sendiri ke dalam model. Pada Proses ini saya kan membangkitkan AR dengan Parameter 0.5 dan 0.2

\[ Y_t = e_t+0.5Y_{t-1}+0.2Y_{t-2}\]

wn <- rnorm(300)
set.seed(1500)

n<-length(wn)
n
## [1] 300
AR2 <- c(1:n) 
for (i in 3:n) {AR2[i]<-wn[i]+0.5*AR2[i-1]+0.2*AR2[i-2]}
AR2
##   [1]  1.00000000  2.00000000  2.22829705  0.96342943 -0.83459800 -1.30977273
##   [7] -1.69116559 -2.06791968 -1.36226114  0.19935900 -2.83219908 -1.22953224
##  [13] -1.15696942 -1.10490322 -1.38815026 -1.02666951 -1.98441460  0.12797058
##  [19] -0.78461749  0.31966119  1.39463079 -0.68895363  0.54826050  1.03295021
##  [25]  1.21937676  0.92726811  2.31269524  0.87851204 -0.91323817  1.84055862
##  [31]  1.19839555  0.26298741  1.22718319  0.14915794  0.90062837  0.40988371
##  [37]  1.40075079  1.63847534  0.58291758 -0.11269877  1.05017990  0.33092866
##  [43]  1.26980182  1.32114793  1.31002494  1.61012923  0.20380564  0.01559433
##  [49] -0.35846030 -1.61645351 -1.88622372 -0.56753362 -1.11930708 -0.57757744
##  [55]  1.05875830 -0.32975147 -0.69243787  0.12234080 -0.44399779 -0.47791673
##  [61] -1.00595951 -0.70272917 -1.73284282 -1.87489811 -1.99739779 -0.67999222
##  [67] -1.40491500 -0.08661475 -1.63649151  0.56548162 -1.89175278  0.65130406
##  [73]  1.98730967  2.10913685  2.19134921  2.94955709  1.53015472  0.35958709
##  [79]  1.15274956  1.54755015  2.43366261  1.56296223  0.05240549  2.07012303
##  [85]  0.34684784 -0.02706414 -0.09282581  1.17577278  2.23440678 -0.35793326
##  [91]  0.35049753 -1.16595823  0.28016902 -0.45522901 -0.39505734 -0.32639048
##  [97] -0.45966721  0.95818256 -0.26974804 -0.11857567 -0.16351374 -0.03631854
## [103]  0.47421634  0.65679220  0.38008067 -0.04461793  0.31727276  0.08744776
## [109]  0.13282747  0.29770279 -1.22023662 -0.66620413 -1.41672801 -1.14458121
## [115] -0.32387564 -0.87977905 -1.32386070 -1.18866160 -3.28687723 -3.07171376
## [121] -1.95651275 -2.62420688 -1.92029929 -1.29311722 -0.47459632 -1.06800330
## [127] -1.95995460 -1.91907457 -2.28343431 -0.61699283 -0.17265374 -0.13885990
## [133] -0.03576901  0.47761468  0.58534000  1.52053097  1.54033733  0.70783853
## [139]  0.12979412  0.47993955 -0.83617009  0.34579748 -0.74513644 -0.85649971
## [145]  0.91418283  0.44398067  2.00073140  0.31096191  2.16986317  1.69049187
## [151]  1.62192741  1.48375370  0.99470255  1.21177426  2.74048439  1.14093509
## [157]  1.54106887 -0.38074928  0.20709682  1.24686567  1.31653193  0.43552731
## [163]  0.03964804  0.60094975 -1.42115682  0.81867691 -1.33085765 -0.50242787
## [169] -0.65337301 -0.13036892 -0.23041918 -0.09966288  0.17773845 -0.02694753
## [175] -1.32549596 -0.81467137 -0.40008991 -0.76489860 -0.10715127 -1.08273497
## [181] -2.00268276 -0.10322204 -0.11966949  0.42609087  0.38549336  1.98880780
## [187]  2.78496227  2.26926745  1.59025698  0.70780493  0.35791903  0.54593790
## [193]  0.23107502 -1.71215076 -0.69833802 -0.01018600 -0.53531560 -1.02419932
## [199]  0.07254835  0.22048379  0.03012599 -1.88983009 -1.60448047 -1.16812932
## [205] -0.48402176 -1.19401772 -0.26578599 -0.38771542 -0.74623319 -2.04519290
## [211]  0.42814968  0.32925489  0.07208237 -0.30612031  0.85946660  1.61025961
## [217] -0.56066816 -0.39796748 -0.73519381 -0.27122522  0.13278797  0.65846145
## [223] -1.64867419 -1.62095619 -0.79187070 -0.02616839 -1.11914836  0.96266368
## [229]  0.64720446  0.88235160  0.99143429  1.44351551 -0.27251586  2.03873009
## [235]  0.60494257  0.81888096  0.15757627 -0.49973077 -0.22799864  1.22677387
## [241]  1.85574243  1.67422689 -0.34271673 -1.97193999 -1.03122207 -2.29629011
## [247] -0.79099247 -1.11178672 -0.68482757 -1.00957863 -0.33494977 -1.94964218
## [253] -1.57066222 -2.36054965 -2.92063265 -2.01414978 -0.89897709 -1.67223414
## [259] -0.75465345 -1.02989409 -0.29556686 -1.48309057 -1.01720291 -1.15225473
## [265]  1.13589925  0.78893380  1.32276755  1.25222785  1.73592544  1.54372302
## [271]  1.89700337  1.97756705  1.98212667  0.23630643 -1.05273399 -0.26645006
## [277] -0.57371204  0.47345054  1.13640287  2.64372559  2.64086711  0.41382075
## [283]  0.17944150  2.00669742  1.08171639  1.24620935  2.18745208  1.11164575
## [289]  1.66880970  1.19497686 -0.79636106 -0.62117128 -0.35045873 -3.66117717
## [295] -3.00339262 -0.92138089 -0.96039000 -2.23836978 -1.46228256 -1.28420522

Pembangkitan dengan Arima.sim

ar_2 <- arima.sim(list(order=c(2,0,0), ar=c(0.5, 0.2)), n=300)
ar_2
## Time Series:
## Start = 1 
## End = 300 
## Frequency = 1 
##   [1]  1.00344679  0.90455781  1.09224883 -0.79887807 -2.28854782  0.75455134
##   [7]  1.13158068 -0.24531298 -2.22512085  1.11083074 -0.12478820  0.95840748
##  [13] -0.44450854 -0.74830296  1.09600719 -0.32902232  0.45858621  0.28035888
##  [19]  0.30364041 -0.25071350  1.56779514 -0.06692580 -0.06117411 -0.86928409
##  [25]  0.25979018 -1.28776020 -0.41069961  0.71433215 -0.39924056 -1.21927926
##  [31] -1.29378108 -1.70136700 -3.53555157 -1.13618283 -0.67477367 -0.08069306
##  [37] -0.67573062 -0.69220493  1.74285556  2.95324798  2.86115862  2.43470163
##  [43]  2.80402641  1.75974159  1.32152557  1.40497022 -0.16862939  0.57263976
##  [49]  0.74319200 -1.17649377 -0.23732538  1.04590991 -0.62835702 -0.21375307
##  [55] -0.43113102 -0.43759912 -1.45828699 -0.05352537 -0.77993611  1.07457422
##  [61]  0.12048632  2.21711913  1.52047914  2.73485163  1.38715319  1.29064517
##  [67]  0.47465564 -0.09806862  0.61867720  0.97635296  0.90115283  0.27557040
##  [73]  0.22412088 -0.22244154 -0.55087225 -0.67870209 -1.18320181 -0.48067388
##  [79] -0.54675528  0.11170776 -0.79284699  0.62304279  0.38388071 -0.05648456
##  [85]  0.89051592  1.38428621  0.48045511  0.61194698 -0.40249969 -2.37517992
##  [91] -1.71363250 -0.78296320 -1.76211278 -1.14591287 -2.49460160 -0.94839311
##  [97] -2.33221759 -0.08963746  0.79801605  0.20425300 -1.54680739 -0.32345510
## [103] -1.13930381 -1.78898938 -0.57405098 -1.92631670 -0.96306201 -1.47577189
## [109] -2.53160841 -1.48721430 -2.60109309 -4.33449596 -3.95665212 -3.04729349
## [115] -0.58750802 -0.46030306 -1.59689043 -0.52615535 -0.07147138  0.66217500
## [121] -0.80693697 -0.21916563  1.32953144 -0.06885138  0.93513828 -0.23332939
## [127]  0.26722264  1.12178714  0.94516747  2.71604461  1.42765749  2.19514444
## [133]  2.40334079  1.51803408  3.09797187  3.00391558  2.68088810  1.95828252
## [139]  2.19017552  1.45512220  0.46115143 -0.52865224 -1.59884973 -1.71252927
## [145] -0.18584484  0.61017955  0.92032978 -0.81931800 -0.57314454  0.11806296
## [151] -1.40600441 -1.21120938 -0.88654044 -0.24072441  0.05957926 -0.24251755
## [157] -0.18946911  1.26633097  1.33181758  0.81920103 -0.49346609  0.63730276
## [163]  0.47518060  0.59981886  0.99061541 -0.41318480 -0.72502402 -0.67086961
## [169]  1.12006771  1.09263376 -0.25382352 -0.09847768  0.13155688 -0.17578373
## [175] -1.67204469 -0.43085178 -0.17020486 -2.16123998 -0.26288333 -0.72777303
## [181] -0.18098709  0.04480527 -0.78388729 -2.23185898 -2.38648809 -1.88900244
## [187] -1.33217976 -0.50430659 -0.82593163 -1.05291159 -2.48455284 -1.43708908
## [193] -2.39841658 -1.02403704 -1.44483772 -0.82289431 -1.94833886 -3.98829221
## [199] -2.54726906 -3.19305119 -1.96246693 -2.07522267  0.21612516  0.65686696
## [205]  2.23694982  0.75931236  1.59987890  2.25369756  0.23380910  0.69095299
## [211]  0.39317909  1.47225038  1.63052298  0.41031510 -1.86359332 -0.46305749
## [217]  0.11485758 -1.36507916 -0.65101839 -1.23844971  0.23651130  1.97862222
## [223]  2.18962161  1.10442350  2.06713665  0.73797760  0.97433533  1.03678928
## [229]  0.83786629  1.31034342  1.64912441  2.56069760  1.72851680  2.55321133
## [235]  0.85707158 -0.21024114 -0.12170056  0.94130409  1.37952450  1.43176754
## [241]  0.35329842  1.39192315 -0.14529577 -1.71989420 -1.47526212 -2.33536106
## [247]  0.25858404 -1.48983667 -0.12952009 -0.36720688  0.84554511  1.35791203
## [253]  1.06965423  1.65689066  0.76509876  0.40632801  1.50771912  0.90111052
## [259]  2.54010229  0.96977653  1.56161344 -0.19072063  0.84319839 -0.10377355
## [265] -1.20779170 -0.41124391 -0.39677005  1.31130897  2.07069716  1.06033093
## [271]  1.79772902  2.04388920  0.84820657 -0.10436302 -0.20391984 -2.20718039
## [277] -0.85519800 -2.35928475 -2.39340055 -0.95386668  0.08102064 -0.59737715
## [283] -0.84691007  0.29209167 -0.49184917  0.46334715 -0.74923014 -1.27027129
## [289]  0.96910389  0.43495566 -0.96742813 -0.38777138 -0.48415872  0.82949202
## [295]  0.79428010 -0.58386735  0.85919738 -0.97994224 -0.28581998  0.88165783

Plot

Time Series AR

ts.plot(AR2)

Berdasarkan plot time series tersebut terlihat bahwa data cenderung stasioner pada rataan, namun sepertinya tidak terlalu stasioner dalam Ragam ,mengingat ada fluktuasi data disana.

Plot PACF

acf(AR2,lag.max = 20)

Dapat dilihat bahwa Pola dari Model AR 2 cenderung tails.off dan kemudian membentuk grafik sinus. Hal ini sesuai dengan teori dimana Plot AR akan tails.off

Plot PACF

pacf(AR2)

Berdasarkan plot PACF tersebut, terlihat bahwa plot PACF cuts off pada lag kedua, sejalan dengan teori yang ada

Plot EACF

TSA::eacf(AR2)
## AR/MA
##   0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x x x x x o o o  o  x  x 
## 1 x o o o o o o o o o o  o  o  o 
## 2 x o o o o o o o o o o  o  o  o 
## 3 o o o o o o o o o o o  o  o  o 
## 4 o x o o o o o o o o o  o  o  o 
## 5 x x x o o o o o o o o  o  o  o 
## 6 x x o o o o o o o o o  o  o  o 
## 7 x x o x o o o o o o o  o  o  o

Dapat dilihat bahwa kita dapat membentuk segitiga nol setidaknya pada AR 2 dan MA 0 , sesuai dengan model yang kita bangkitkan.

Scatter Plot antar Lag

Scatter Plot Yt dengan Yt-1

yt_ar <- AR2[-1]
yt_ar
##   [1]  2.00000000  2.22829705  0.96342943 -0.83459800 -1.30977273 -1.69116559
##   [7] -2.06791968 -1.36226114  0.19935900 -2.83219908 -1.22953224 -1.15696942
##  [13] -1.10490322 -1.38815026 -1.02666951 -1.98441460  0.12797058 -0.78461749
##  [19]  0.31966119  1.39463079 -0.68895363  0.54826050  1.03295021  1.21937676
##  [25]  0.92726811  2.31269524  0.87851204 -0.91323817  1.84055862  1.19839555
##  [31]  0.26298741  1.22718319  0.14915794  0.90062837  0.40988371  1.40075079
##  [37]  1.63847534  0.58291758 -0.11269877  1.05017990  0.33092866  1.26980182
##  [43]  1.32114793  1.31002494  1.61012923  0.20380564  0.01559433 -0.35846030
##  [49] -1.61645351 -1.88622372 -0.56753362 -1.11930708 -0.57757744  1.05875830
##  [55] -0.32975147 -0.69243787  0.12234080 -0.44399779 -0.47791673 -1.00595951
##  [61] -0.70272917 -1.73284282 -1.87489811 -1.99739779 -0.67999222 -1.40491500
##  [67] -0.08661475 -1.63649151  0.56548162 -1.89175278  0.65130406  1.98730967
##  [73]  2.10913685  2.19134921  2.94955709  1.53015472  0.35958709  1.15274956
##  [79]  1.54755015  2.43366261  1.56296223  0.05240549  2.07012303  0.34684784
##  [85] -0.02706414 -0.09282581  1.17577278  2.23440678 -0.35793326  0.35049753
##  [91] -1.16595823  0.28016902 -0.45522901 -0.39505734 -0.32639048 -0.45966721
##  [97]  0.95818256 -0.26974804 -0.11857567 -0.16351374 -0.03631854  0.47421634
## [103]  0.65679220  0.38008067 -0.04461793  0.31727276  0.08744776  0.13282747
## [109]  0.29770279 -1.22023662 -0.66620413 -1.41672801 -1.14458121 -0.32387564
## [115] -0.87977905 -1.32386070 -1.18866160 -3.28687723 -3.07171376 -1.95651275
## [121] -2.62420688 -1.92029929 -1.29311722 -0.47459632 -1.06800330 -1.95995460
## [127] -1.91907457 -2.28343431 -0.61699283 -0.17265374 -0.13885990 -0.03576901
## [133]  0.47761468  0.58534000  1.52053097  1.54033733  0.70783853  0.12979412
## [139]  0.47993955 -0.83617009  0.34579748 -0.74513644 -0.85649971  0.91418283
## [145]  0.44398067  2.00073140  0.31096191  2.16986317  1.69049187  1.62192741
## [151]  1.48375370  0.99470255  1.21177426  2.74048439  1.14093509  1.54106887
## [157] -0.38074928  0.20709682  1.24686567  1.31653193  0.43552731  0.03964804
## [163]  0.60094975 -1.42115682  0.81867691 -1.33085765 -0.50242787 -0.65337301
## [169] -0.13036892 -0.23041918 -0.09966288  0.17773845 -0.02694753 -1.32549596
## [175] -0.81467137 -0.40008991 -0.76489860 -0.10715127 -1.08273497 -2.00268276
## [181] -0.10322204 -0.11966949  0.42609087  0.38549336  1.98880780  2.78496227
## [187]  2.26926745  1.59025698  0.70780493  0.35791903  0.54593790  0.23107502
## [193] -1.71215076 -0.69833802 -0.01018600 -0.53531560 -1.02419932  0.07254835
## [199]  0.22048379  0.03012599 -1.88983009 -1.60448047 -1.16812932 -0.48402176
## [205] -1.19401772 -0.26578599 -0.38771542 -0.74623319 -2.04519290  0.42814968
## [211]  0.32925489  0.07208237 -0.30612031  0.85946660  1.61025961 -0.56066816
## [217] -0.39796748 -0.73519381 -0.27122522  0.13278797  0.65846145 -1.64867419
## [223] -1.62095619 -0.79187070 -0.02616839 -1.11914836  0.96266368  0.64720446
## [229]  0.88235160  0.99143429  1.44351551 -0.27251586  2.03873009  0.60494257
## [235]  0.81888096  0.15757627 -0.49973077 -0.22799864  1.22677387  1.85574243
## [241]  1.67422689 -0.34271673 -1.97193999 -1.03122207 -2.29629011 -0.79099247
## [247] -1.11178672 -0.68482757 -1.00957863 -0.33494977 -1.94964218 -1.57066222
## [253] -2.36054965 -2.92063265 -2.01414978 -0.89897709 -1.67223414 -0.75465345
## [259] -1.02989409 -0.29556686 -1.48309057 -1.01720291 -1.15225473  1.13589925
## [265]  0.78893380  1.32276755  1.25222785  1.73592544  1.54372302  1.89700337
## [271]  1.97756705  1.98212667  0.23630643 -1.05273399 -0.26645006 -0.57371204
## [277]  0.47345054  1.13640287  2.64372559  2.64086711  0.41382075  0.17944150
## [283]  2.00669742  1.08171639  1.24620935  2.18745208  1.11164575  1.66880970
## [289]  1.19497686 -0.79636106 -0.62117128 -0.35045873 -3.66117717 -3.00339262
## [295] -0.92138089 -0.96039000 -2.23836978 -1.46228256 -1.28420522
yt_1_ar <- AR2[-300]
yt_1_ar
##   [1]  1.00000000  2.00000000  2.22829705  0.96342943 -0.83459800 -1.30977273
##   [7] -1.69116559 -2.06791968 -1.36226114  0.19935900 -2.83219908 -1.22953224
##  [13] -1.15696942 -1.10490322 -1.38815026 -1.02666951 -1.98441460  0.12797058
##  [19] -0.78461749  0.31966119  1.39463079 -0.68895363  0.54826050  1.03295021
##  [25]  1.21937676  0.92726811  2.31269524  0.87851204 -0.91323817  1.84055862
##  [31]  1.19839555  0.26298741  1.22718319  0.14915794  0.90062837  0.40988371
##  [37]  1.40075079  1.63847534  0.58291758 -0.11269877  1.05017990  0.33092866
##  [43]  1.26980182  1.32114793  1.31002494  1.61012923  0.20380564  0.01559433
##  [49] -0.35846030 -1.61645351 -1.88622372 -0.56753362 -1.11930708 -0.57757744
##  [55]  1.05875830 -0.32975147 -0.69243787  0.12234080 -0.44399779 -0.47791673
##  [61] -1.00595951 -0.70272917 -1.73284282 -1.87489811 -1.99739779 -0.67999222
##  [67] -1.40491500 -0.08661475 -1.63649151  0.56548162 -1.89175278  0.65130406
##  [73]  1.98730967  2.10913685  2.19134921  2.94955709  1.53015472  0.35958709
##  [79]  1.15274956  1.54755015  2.43366261  1.56296223  0.05240549  2.07012303
##  [85]  0.34684784 -0.02706414 -0.09282581  1.17577278  2.23440678 -0.35793326
##  [91]  0.35049753 -1.16595823  0.28016902 -0.45522901 -0.39505734 -0.32639048
##  [97] -0.45966721  0.95818256 -0.26974804 -0.11857567 -0.16351374 -0.03631854
## [103]  0.47421634  0.65679220  0.38008067 -0.04461793  0.31727276  0.08744776
## [109]  0.13282747  0.29770279 -1.22023662 -0.66620413 -1.41672801 -1.14458121
## [115] -0.32387564 -0.87977905 -1.32386070 -1.18866160 -3.28687723 -3.07171376
## [121] -1.95651275 -2.62420688 -1.92029929 -1.29311722 -0.47459632 -1.06800330
## [127] -1.95995460 -1.91907457 -2.28343431 -0.61699283 -0.17265374 -0.13885990
## [133] -0.03576901  0.47761468  0.58534000  1.52053097  1.54033733  0.70783853
## [139]  0.12979412  0.47993955 -0.83617009  0.34579748 -0.74513644 -0.85649971
## [145]  0.91418283  0.44398067  2.00073140  0.31096191  2.16986317  1.69049187
## [151]  1.62192741  1.48375370  0.99470255  1.21177426  2.74048439  1.14093509
## [157]  1.54106887 -0.38074928  0.20709682  1.24686567  1.31653193  0.43552731
## [163]  0.03964804  0.60094975 -1.42115682  0.81867691 -1.33085765 -0.50242787
## [169] -0.65337301 -0.13036892 -0.23041918 -0.09966288  0.17773845 -0.02694753
## [175] -1.32549596 -0.81467137 -0.40008991 -0.76489860 -0.10715127 -1.08273497
## [181] -2.00268276 -0.10322204 -0.11966949  0.42609087  0.38549336  1.98880780
## [187]  2.78496227  2.26926745  1.59025698  0.70780493  0.35791903  0.54593790
## [193]  0.23107502 -1.71215076 -0.69833802 -0.01018600 -0.53531560 -1.02419932
## [199]  0.07254835  0.22048379  0.03012599 -1.88983009 -1.60448047 -1.16812932
## [205] -0.48402176 -1.19401772 -0.26578599 -0.38771542 -0.74623319 -2.04519290
## [211]  0.42814968  0.32925489  0.07208237 -0.30612031  0.85946660  1.61025961
## [217] -0.56066816 -0.39796748 -0.73519381 -0.27122522  0.13278797  0.65846145
## [223] -1.64867419 -1.62095619 -0.79187070 -0.02616839 -1.11914836  0.96266368
## [229]  0.64720446  0.88235160  0.99143429  1.44351551 -0.27251586  2.03873009
## [235]  0.60494257  0.81888096  0.15757627 -0.49973077 -0.22799864  1.22677387
## [241]  1.85574243  1.67422689 -0.34271673 -1.97193999 -1.03122207 -2.29629011
## [247] -0.79099247 -1.11178672 -0.68482757 -1.00957863 -0.33494977 -1.94964218
## [253] -1.57066222 -2.36054965 -2.92063265 -2.01414978 -0.89897709 -1.67223414
## [259] -0.75465345 -1.02989409 -0.29556686 -1.48309057 -1.01720291 -1.15225473
## [265]  1.13589925  0.78893380  1.32276755  1.25222785  1.73592544  1.54372302
## [271]  1.89700337  1.97756705  1.98212667  0.23630643 -1.05273399 -0.26645006
## [277] -0.57371204  0.47345054  1.13640287  2.64372559  2.64086711  0.41382075
## [283]  0.17944150  2.00669742  1.08171639  1.24620935  2.18745208  1.11164575
## [289]  1.66880970  1.19497686 -0.79636106 -0.62117128 -0.35045873 -3.66117717
## [295] -3.00339262 -0.92138089 -0.96039000 -2.23836978 -1.46228256
plot(y=yt_ar,x=yt_1_ar)

Dari Scatter Plot ini menunjukkan adanya hubungan positif , dimana hal ini sesuai dengan teori yang ada

cor(yt_ar,yt_1_ar)
## [1] 0.6539931

\[ \rho_1=\frac{\phi_1}{1-\phi_2}=\frac{0.5}{1-0.2}= 0.625\]

Korelasi Yt dengan Yt-2

yt_ar2 <- AR2[-c(1,2)]
yt_ar2
##   [1]  2.22829705  0.96342943 -0.83459800 -1.30977273 -1.69116559 -2.06791968
##   [7] -1.36226114  0.19935900 -2.83219908 -1.22953224 -1.15696942 -1.10490322
##  [13] -1.38815026 -1.02666951 -1.98441460  0.12797058 -0.78461749  0.31966119
##  [19]  1.39463079 -0.68895363  0.54826050  1.03295021  1.21937676  0.92726811
##  [25]  2.31269524  0.87851204 -0.91323817  1.84055862  1.19839555  0.26298741
##  [31]  1.22718319  0.14915794  0.90062837  0.40988371  1.40075079  1.63847534
##  [37]  0.58291758 -0.11269877  1.05017990  0.33092866  1.26980182  1.32114793
##  [43]  1.31002494  1.61012923  0.20380564  0.01559433 -0.35846030 -1.61645351
##  [49] -1.88622372 -0.56753362 -1.11930708 -0.57757744  1.05875830 -0.32975147
##  [55] -0.69243787  0.12234080 -0.44399779 -0.47791673 -1.00595951 -0.70272917
##  [61] -1.73284282 -1.87489811 -1.99739779 -0.67999222 -1.40491500 -0.08661475
##  [67] -1.63649151  0.56548162 -1.89175278  0.65130406  1.98730967  2.10913685
##  [73]  2.19134921  2.94955709  1.53015472  0.35958709  1.15274956  1.54755015
##  [79]  2.43366261  1.56296223  0.05240549  2.07012303  0.34684784 -0.02706414
##  [85] -0.09282581  1.17577278  2.23440678 -0.35793326  0.35049753 -1.16595823
##  [91]  0.28016902 -0.45522901 -0.39505734 -0.32639048 -0.45966721  0.95818256
##  [97] -0.26974804 -0.11857567 -0.16351374 -0.03631854  0.47421634  0.65679220
## [103]  0.38008067 -0.04461793  0.31727276  0.08744776  0.13282747  0.29770279
## [109] -1.22023662 -0.66620413 -1.41672801 -1.14458121 -0.32387564 -0.87977905
## [115] -1.32386070 -1.18866160 -3.28687723 -3.07171376 -1.95651275 -2.62420688
## [121] -1.92029929 -1.29311722 -0.47459632 -1.06800330 -1.95995460 -1.91907457
## [127] -2.28343431 -0.61699283 -0.17265374 -0.13885990 -0.03576901  0.47761468
## [133]  0.58534000  1.52053097  1.54033733  0.70783853  0.12979412  0.47993955
## [139] -0.83617009  0.34579748 -0.74513644 -0.85649971  0.91418283  0.44398067
## [145]  2.00073140  0.31096191  2.16986317  1.69049187  1.62192741  1.48375370
## [151]  0.99470255  1.21177426  2.74048439  1.14093509  1.54106887 -0.38074928
## [157]  0.20709682  1.24686567  1.31653193  0.43552731  0.03964804  0.60094975
## [163] -1.42115682  0.81867691 -1.33085765 -0.50242787 -0.65337301 -0.13036892
## [169] -0.23041918 -0.09966288  0.17773845 -0.02694753 -1.32549596 -0.81467137
## [175] -0.40008991 -0.76489860 -0.10715127 -1.08273497 -2.00268276 -0.10322204
## [181] -0.11966949  0.42609087  0.38549336  1.98880780  2.78496227  2.26926745
## [187]  1.59025698  0.70780493  0.35791903  0.54593790  0.23107502 -1.71215076
## [193] -0.69833802 -0.01018600 -0.53531560 -1.02419932  0.07254835  0.22048379
## [199]  0.03012599 -1.88983009 -1.60448047 -1.16812932 -0.48402176 -1.19401772
## [205] -0.26578599 -0.38771542 -0.74623319 -2.04519290  0.42814968  0.32925489
## [211]  0.07208237 -0.30612031  0.85946660  1.61025961 -0.56066816 -0.39796748
## [217] -0.73519381 -0.27122522  0.13278797  0.65846145 -1.64867419 -1.62095619
## [223] -0.79187070 -0.02616839 -1.11914836  0.96266368  0.64720446  0.88235160
## [229]  0.99143429  1.44351551 -0.27251586  2.03873009  0.60494257  0.81888096
## [235]  0.15757627 -0.49973077 -0.22799864  1.22677387  1.85574243  1.67422689
## [241] -0.34271673 -1.97193999 -1.03122207 -2.29629011 -0.79099247 -1.11178672
## [247] -0.68482757 -1.00957863 -0.33494977 -1.94964218 -1.57066222 -2.36054965
## [253] -2.92063265 -2.01414978 -0.89897709 -1.67223414 -0.75465345 -1.02989409
## [259] -0.29556686 -1.48309057 -1.01720291 -1.15225473  1.13589925  0.78893380
## [265]  1.32276755  1.25222785  1.73592544  1.54372302  1.89700337  1.97756705
## [271]  1.98212667  0.23630643 -1.05273399 -0.26645006 -0.57371204  0.47345054
## [277]  1.13640287  2.64372559  2.64086711  0.41382075  0.17944150  2.00669742
## [283]  1.08171639  1.24620935  2.18745208  1.11164575  1.66880970  1.19497686
## [289] -0.79636106 -0.62117128 -0.35045873 -3.66117717 -3.00339262 -0.92138089
## [295] -0.96039000 -2.23836978 -1.46228256 -1.28420522
yt_2_ar <- AR2[-c(299,300)]
yt_2_ar
##   [1]  1.00000000  2.00000000  2.22829705  0.96342943 -0.83459800 -1.30977273
##   [7] -1.69116559 -2.06791968 -1.36226114  0.19935900 -2.83219908 -1.22953224
##  [13] -1.15696942 -1.10490322 -1.38815026 -1.02666951 -1.98441460  0.12797058
##  [19] -0.78461749  0.31966119  1.39463079 -0.68895363  0.54826050  1.03295021
##  [25]  1.21937676  0.92726811  2.31269524  0.87851204 -0.91323817  1.84055862
##  [31]  1.19839555  0.26298741  1.22718319  0.14915794  0.90062837  0.40988371
##  [37]  1.40075079  1.63847534  0.58291758 -0.11269877  1.05017990  0.33092866
##  [43]  1.26980182  1.32114793  1.31002494  1.61012923  0.20380564  0.01559433
##  [49] -0.35846030 -1.61645351 -1.88622372 -0.56753362 -1.11930708 -0.57757744
##  [55]  1.05875830 -0.32975147 -0.69243787  0.12234080 -0.44399779 -0.47791673
##  [61] -1.00595951 -0.70272917 -1.73284282 -1.87489811 -1.99739779 -0.67999222
##  [67] -1.40491500 -0.08661475 -1.63649151  0.56548162 -1.89175278  0.65130406
##  [73]  1.98730967  2.10913685  2.19134921  2.94955709  1.53015472  0.35958709
##  [79]  1.15274956  1.54755015  2.43366261  1.56296223  0.05240549  2.07012303
##  [85]  0.34684784 -0.02706414 -0.09282581  1.17577278  2.23440678 -0.35793326
##  [91]  0.35049753 -1.16595823  0.28016902 -0.45522901 -0.39505734 -0.32639048
##  [97] -0.45966721  0.95818256 -0.26974804 -0.11857567 -0.16351374 -0.03631854
## [103]  0.47421634  0.65679220  0.38008067 -0.04461793  0.31727276  0.08744776
## [109]  0.13282747  0.29770279 -1.22023662 -0.66620413 -1.41672801 -1.14458121
## [115] -0.32387564 -0.87977905 -1.32386070 -1.18866160 -3.28687723 -3.07171376
## [121] -1.95651275 -2.62420688 -1.92029929 -1.29311722 -0.47459632 -1.06800330
## [127] -1.95995460 -1.91907457 -2.28343431 -0.61699283 -0.17265374 -0.13885990
## [133] -0.03576901  0.47761468  0.58534000  1.52053097  1.54033733  0.70783853
## [139]  0.12979412  0.47993955 -0.83617009  0.34579748 -0.74513644 -0.85649971
## [145]  0.91418283  0.44398067  2.00073140  0.31096191  2.16986317  1.69049187
## [151]  1.62192741  1.48375370  0.99470255  1.21177426  2.74048439  1.14093509
## [157]  1.54106887 -0.38074928  0.20709682  1.24686567  1.31653193  0.43552731
## [163]  0.03964804  0.60094975 -1.42115682  0.81867691 -1.33085765 -0.50242787
## [169] -0.65337301 -0.13036892 -0.23041918 -0.09966288  0.17773845 -0.02694753
## [175] -1.32549596 -0.81467137 -0.40008991 -0.76489860 -0.10715127 -1.08273497
## [181] -2.00268276 -0.10322204 -0.11966949  0.42609087  0.38549336  1.98880780
## [187]  2.78496227  2.26926745  1.59025698  0.70780493  0.35791903  0.54593790
## [193]  0.23107502 -1.71215076 -0.69833802 -0.01018600 -0.53531560 -1.02419932
## [199]  0.07254835  0.22048379  0.03012599 -1.88983009 -1.60448047 -1.16812932
## [205] -0.48402176 -1.19401772 -0.26578599 -0.38771542 -0.74623319 -2.04519290
## [211]  0.42814968  0.32925489  0.07208237 -0.30612031  0.85946660  1.61025961
## [217] -0.56066816 -0.39796748 -0.73519381 -0.27122522  0.13278797  0.65846145
## [223] -1.64867419 -1.62095619 -0.79187070 -0.02616839 -1.11914836  0.96266368
## [229]  0.64720446  0.88235160  0.99143429  1.44351551 -0.27251586  2.03873009
## [235]  0.60494257  0.81888096  0.15757627 -0.49973077 -0.22799864  1.22677387
## [241]  1.85574243  1.67422689 -0.34271673 -1.97193999 -1.03122207 -2.29629011
## [247] -0.79099247 -1.11178672 -0.68482757 -1.00957863 -0.33494977 -1.94964218
## [253] -1.57066222 -2.36054965 -2.92063265 -2.01414978 -0.89897709 -1.67223414
## [259] -0.75465345 -1.02989409 -0.29556686 -1.48309057 -1.01720291 -1.15225473
## [265]  1.13589925  0.78893380  1.32276755  1.25222785  1.73592544  1.54372302
## [271]  1.89700337  1.97756705  1.98212667  0.23630643 -1.05273399 -0.26645006
## [277] -0.57371204  0.47345054  1.13640287  2.64372559  2.64086711  0.41382075
## [283]  0.17944150  2.00669742  1.08171639  1.24620935  2.18745208  1.11164575
## [289]  1.66880970  1.19497686 -0.79636106 -0.62117128 -0.35045873 -3.66117717
## [295] -3.00339262 -0.92138089 -0.96039000 -2.23836978
plot(y=yt_ar2,x=yt_2_ar)

Didapatkan hasil plot korelasi yang positif, hal ini juga sesuai dengan teori, kemudian kita akan mencari korelasinya:

cor(yt_2_ar,yt_ar2)
## [1] 0.546988

\[ \rho_2=\frac{\phi_2(1-\phi_2)+\phi_1^2}{1-\phi_2}=\frac{0.2*(1-0.2)+0.5^2}{1-0.2}= 0.5125\]

x <- 0.2*0.8+0.5^2 
x
## [1] 0.41
y <- x / 0.8
y
## [1] 0.5125

Scatter Plot Yt dengan Yt-3

#Yt
yt_ar2 <- AR2[-c(1,2,3)]
yt_ar2
##   [1]  0.96342943 -0.83459800 -1.30977273 -1.69116559 -2.06791968 -1.36226114
##   [7]  0.19935900 -2.83219908 -1.22953224 -1.15696942 -1.10490322 -1.38815026
##  [13] -1.02666951 -1.98441460  0.12797058 -0.78461749  0.31966119  1.39463079
##  [19] -0.68895363  0.54826050  1.03295021  1.21937676  0.92726811  2.31269524
##  [25]  0.87851204 -0.91323817  1.84055862  1.19839555  0.26298741  1.22718319
##  [31]  0.14915794  0.90062837  0.40988371  1.40075079  1.63847534  0.58291758
##  [37] -0.11269877  1.05017990  0.33092866  1.26980182  1.32114793  1.31002494
##  [43]  1.61012923  0.20380564  0.01559433 -0.35846030 -1.61645351 -1.88622372
##  [49] -0.56753362 -1.11930708 -0.57757744  1.05875830 -0.32975147 -0.69243787
##  [55]  0.12234080 -0.44399779 -0.47791673 -1.00595951 -0.70272917 -1.73284282
##  [61] -1.87489811 -1.99739779 -0.67999222 -1.40491500 -0.08661475 -1.63649151
##  [67]  0.56548162 -1.89175278  0.65130406  1.98730967  2.10913685  2.19134921
##  [73]  2.94955709  1.53015472  0.35958709  1.15274956  1.54755015  2.43366261
##  [79]  1.56296223  0.05240549  2.07012303  0.34684784 -0.02706414 -0.09282581
##  [85]  1.17577278  2.23440678 -0.35793326  0.35049753 -1.16595823  0.28016902
##  [91] -0.45522901 -0.39505734 -0.32639048 -0.45966721  0.95818256 -0.26974804
##  [97] -0.11857567 -0.16351374 -0.03631854  0.47421634  0.65679220  0.38008067
## [103] -0.04461793  0.31727276  0.08744776  0.13282747  0.29770279 -1.22023662
## [109] -0.66620413 -1.41672801 -1.14458121 -0.32387564 -0.87977905 -1.32386070
## [115] -1.18866160 -3.28687723 -3.07171376 -1.95651275 -2.62420688 -1.92029929
## [121] -1.29311722 -0.47459632 -1.06800330 -1.95995460 -1.91907457 -2.28343431
## [127] -0.61699283 -0.17265374 -0.13885990 -0.03576901  0.47761468  0.58534000
## [133]  1.52053097  1.54033733  0.70783853  0.12979412  0.47993955 -0.83617009
## [139]  0.34579748 -0.74513644 -0.85649971  0.91418283  0.44398067  2.00073140
## [145]  0.31096191  2.16986317  1.69049187  1.62192741  1.48375370  0.99470255
## [151]  1.21177426  2.74048439  1.14093509  1.54106887 -0.38074928  0.20709682
## [157]  1.24686567  1.31653193  0.43552731  0.03964804  0.60094975 -1.42115682
## [163]  0.81867691 -1.33085765 -0.50242787 -0.65337301 -0.13036892 -0.23041918
## [169] -0.09966288  0.17773845 -0.02694753 -1.32549596 -0.81467137 -0.40008991
## [175] -0.76489860 -0.10715127 -1.08273497 -2.00268276 -0.10322204 -0.11966949
## [181]  0.42609087  0.38549336  1.98880780  2.78496227  2.26926745  1.59025698
## [187]  0.70780493  0.35791903  0.54593790  0.23107502 -1.71215076 -0.69833802
## [193] -0.01018600 -0.53531560 -1.02419932  0.07254835  0.22048379  0.03012599
## [199] -1.88983009 -1.60448047 -1.16812932 -0.48402176 -1.19401772 -0.26578599
## [205] -0.38771542 -0.74623319 -2.04519290  0.42814968  0.32925489  0.07208237
## [211] -0.30612031  0.85946660  1.61025961 -0.56066816 -0.39796748 -0.73519381
## [217] -0.27122522  0.13278797  0.65846145 -1.64867419 -1.62095619 -0.79187070
## [223] -0.02616839 -1.11914836  0.96266368  0.64720446  0.88235160  0.99143429
## [229]  1.44351551 -0.27251586  2.03873009  0.60494257  0.81888096  0.15757627
## [235] -0.49973077 -0.22799864  1.22677387  1.85574243  1.67422689 -0.34271673
## [241] -1.97193999 -1.03122207 -2.29629011 -0.79099247 -1.11178672 -0.68482757
## [247] -1.00957863 -0.33494977 -1.94964218 -1.57066222 -2.36054965 -2.92063265
## [253] -2.01414978 -0.89897709 -1.67223414 -0.75465345 -1.02989409 -0.29556686
## [259] -1.48309057 -1.01720291 -1.15225473  1.13589925  0.78893380  1.32276755
## [265]  1.25222785  1.73592544  1.54372302  1.89700337  1.97756705  1.98212667
## [271]  0.23630643 -1.05273399 -0.26645006 -0.57371204  0.47345054  1.13640287
## [277]  2.64372559  2.64086711  0.41382075  0.17944150  2.00669742  1.08171639
## [283]  1.24620935  2.18745208  1.11164575  1.66880970  1.19497686 -0.79636106
## [289] -0.62117128 -0.35045873 -3.66117717 -3.00339262 -0.92138089 -0.96039000
## [295] -2.23836978 -1.46228256 -1.28420522
#Yt-3
yt_3_ar <- AR2[-c(288,299,300)]
yt_3_ar
##   [1]  1.00000000  2.00000000  2.22829705  0.96342943 -0.83459800 -1.30977273
##   [7] -1.69116559 -2.06791968 -1.36226114  0.19935900 -2.83219908 -1.22953224
##  [13] -1.15696942 -1.10490322 -1.38815026 -1.02666951 -1.98441460  0.12797058
##  [19] -0.78461749  0.31966119  1.39463079 -0.68895363  0.54826050  1.03295021
##  [25]  1.21937676  0.92726811  2.31269524  0.87851204 -0.91323817  1.84055862
##  [31]  1.19839555  0.26298741  1.22718319  0.14915794  0.90062837  0.40988371
##  [37]  1.40075079  1.63847534  0.58291758 -0.11269877  1.05017990  0.33092866
##  [43]  1.26980182  1.32114793  1.31002494  1.61012923  0.20380564  0.01559433
##  [49] -0.35846030 -1.61645351 -1.88622372 -0.56753362 -1.11930708 -0.57757744
##  [55]  1.05875830 -0.32975147 -0.69243787  0.12234080 -0.44399779 -0.47791673
##  [61] -1.00595951 -0.70272917 -1.73284282 -1.87489811 -1.99739779 -0.67999222
##  [67] -1.40491500 -0.08661475 -1.63649151  0.56548162 -1.89175278  0.65130406
##  [73]  1.98730967  2.10913685  2.19134921  2.94955709  1.53015472  0.35958709
##  [79]  1.15274956  1.54755015  2.43366261  1.56296223  0.05240549  2.07012303
##  [85]  0.34684784 -0.02706414 -0.09282581  1.17577278  2.23440678 -0.35793326
##  [91]  0.35049753 -1.16595823  0.28016902 -0.45522901 -0.39505734 -0.32639048
##  [97] -0.45966721  0.95818256 -0.26974804 -0.11857567 -0.16351374 -0.03631854
## [103]  0.47421634  0.65679220  0.38008067 -0.04461793  0.31727276  0.08744776
## [109]  0.13282747  0.29770279 -1.22023662 -0.66620413 -1.41672801 -1.14458121
## [115] -0.32387564 -0.87977905 -1.32386070 -1.18866160 -3.28687723 -3.07171376
## [121] -1.95651275 -2.62420688 -1.92029929 -1.29311722 -0.47459632 -1.06800330
## [127] -1.95995460 -1.91907457 -2.28343431 -0.61699283 -0.17265374 -0.13885990
## [133] -0.03576901  0.47761468  0.58534000  1.52053097  1.54033733  0.70783853
## [139]  0.12979412  0.47993955 -0.83617009  0.34579748 -0.74513644 -0.85649971
## [145]  0.91418283  0.44398067  2.00073140  0.31096191  2.16986317  1.69049187
## [151]  1.62192741  1.48375370  0.99470255  1.21177426  2.74048439  1.14093509
## [157]  1.54106887 -0.38074928  0.20709682  1.24686567  1.31653193  0.43552731
## [163]  0.03964804  0.60094975 -1.42115682  0.81867691 -1.33085765 -0.50242787
## [169] -0.65337301 -0.13036892 -0.23041918 -0.09966288  0.17773845 -0.02694753
## [175] -1.32549596 -0.81467137 -0.40008991 -0.76489860 -0.10715127 -1.08273497
## [181] -2.00268276 -0.10322204 -0.11966949  0.42609087  0.38549336  1.98880780
## [187]  2.78496227  2.26926745  1.59025698  0.70780493  0.35791903  0.54593790
## [193]  0.23107502 -1.71215076 -0.69833802 -0.01018600 -0.53531560 -1.02419932
## [199]  0.07254835  0.22048379  0.03012599 -1.88983009 -1.60448047 -1.16812932
## [205] -0.48402176 -1.19401772 -0.26578599 -0.38771542 -0.74623319 -2.04519290
## [211]  0.42814968  0.32925489  0.07208237 -0.30612031  0.85946660  1.61025961
## [217] -0.56066816 -0.39796748 -0.73519381 -0.27122522  0.13278797  0.65846145
## [223] -1.64867419 -1.62095619 -0.79187070 -0.02616839 -1.11914836  0.96266368
## [229]  0.64720446  0.88235160  0.99143429  1.44351551 -0.27251586  2.03873009
## [235]  0.60494257  0.81888096  0.15757627 -0.49973077 -0.22799864  1.22677387
## [241]  1.85574243  1.67422689 -0.34271673 -1.97193999 -1.03122207 -2.29629011
## [247] -0.79099247 -1.11178672 -0.68482757 -1.00957863 -0.33494977 -1.94964218
## [253] -1.57066222 -2.36054965 -2.92063265 -2.01414978 -0.89897709 -1.67223414
## [259] -0.75465345 -1.02989409 -0.29556686 -1.48309057 -1.01720291 -1.15225473
## [265]  1.13589925  0.78893380  1.32276755  1.25222785  1.73592544  1.54372302
## [271]  1.89700337  1.97756705  1.98212667  0.23630643 -1.05273399 -0.26645006
## [277] -0.57371204  0.47345054  1.13640287  2.64372559  2.64086711  0.41382075
## [283]  0.17944150  2.00669742  1.08171639  1.24620935  2.18745208  1.66880970
## [289]  1.19497686 -0.79636106 -0.62117128 -0.35045873 -3.66117717 -3.00339262
## [295] -0.92138089 -0.96039000 -2.23836978
plot(y=yt_ar2,x=yt_3_ar)

cor(yt_ar2,yt_3_ar)
## [1] 0.4305581

Didapat Nilai Korelasi yang positif sesuai dengan Grafik, nilai ini juga hampir mirip dengan perhitungan teoritisnya yaitu bila dihitung dengan :

\[ \rho_3=\phi_1*\rho2+\phi_2*\rho_1= 0.5*0.5125+0.2*0.625= 0.38125\]

Pengecekan

x <- 0.5*0.5125+0.2*0.625
x
## [1] 0.38125

ARMA (2,2)

Pola Data ARMA

\[ y_{t} = c + \sum_{i=1}^p \phi_{i}y_{t-i} + \sum_{j=1}^q e_{t-j}+ e_{t} \]

set.seed(1500)
n = length(wn)
phi1 = 0.5
phi2 = 0.2 
theta1 = 0.4
theta2 = 0.6 
y.arma <- numeric(n)
for (i in 3:n) {
  y.arma[i] <- phi1 * y.arma[i-1] + phi2 * y.arma[i-2] + 
               theta1 * wn[i-1] + theta2 * wn[i-2] + wn[i]
}
y.arma
##   [1]  0.0000000000  0.0000000000  1.0070355131  0.2910283464 -1.0183602565
##   [6] -2.5713543828 -3.8499559890 -4.3984708317 -3.8650635080 -2.0904090937
##  [11] -3.9540551059 -2.5357403043 -3.5714222653 -2.4756593463 -2.6540818075
##  [16] -2.3438156556 -3.3034023400 -1.3393006745 -1.9679158312  0.0491769000
##  [21]  1.0262473815  0.0412727756  1.0946507347  0.8275943183  1.9529078365
##  [26]  2.0282287056  3.4102273607  2.3561383659  0.8228772370  2.0001547761
##  [31]  1.3849868837  1.8453930386  2.0504337544  0.7970752463  1.6960309079
##  [36]  0.8591948681  2.1047497041  2.4444530905  2.0785654743  1.1034065490
##  [41]  1.3547389456  0.6832959756  2.0322161333  2.0275762310  2.6003273742
##  [46]  2.9267991261  1.6338503128  1.0631773685 -0.2299519560 -1.7504907667
##  [51] -2.7478887295 -2.2919008716 -2.4780590718 -1.3658237289  0.1561405734
##  [56] -0.2527965210 -0.1890849377 -0.3524863461 -0.8105250463 -0.5821120192
##  [61] -1.4635253715 -1.3918633911 -2.6175104801 -2.9896729578 -3.7870628901
##  [66] -2.6038903271 -2.8753506569 -1.0565761578 -2.5140864637 -0.1410838769
##  [71] -2.6474550673  0.2338918937  1.1127796103  3.2948431393  4.2273897406
##  [76]  5.0915788739  4.0247870727  2.7413832260  2.2146772265  2.2244022237
##  [81]  3.7443324038  3.4649573580  2.1377879477  3.0288625658  1.2063403453
##  [86]  1.3537488180  0.1044572328  1.1224039706  2.6490204070  1.2412931237
##  [91]  1.5479683017 -1.2405191747  0.0240842435 -1.0427363484 -0.4090475338
##  [96] -0.7575508283 -0.8272578083  0.5784813882 -0.1622753406  0.3484346563
## [101] -0.3727928320 -0.1728694326  0.3615806785  0.8246876130  0.9273273565
## [106]  0.5014896554  0.5274739863  0.1875861014  0.3581702276  0.4033024359
## [111] -1.0214590170 -0.9756771042 -2.4153516323 -2.1109948906 -1.6317449264
## [116] -1.6960780251 -1.8700977002 -2.2460733024 -4.5566582894 -5.0996616117
## [121] -5.1573245905 -5.2498402332 -4.1438896933 -3.6357610644 -2.1440227889
## [126] -2.0337121617 -2.6719137142 -3.3438583860 -4.2270368965 -2.6818112986
## [131] -1.7895114602 -0.5781170983 -0.1949052111  0.3799911360  0.7549244703
## [136]  2.0412357783  2.4997537236  2.2362920406  1.3371319301  0.9565603120
## [141] -0.5663177994  0.2992931688 -1.1085194992 -0.9470758005  0.1245010832
## [146]  0.2957539715  2.7268333690  1.3776428700  3.4946867738  2.7450142842
## [151]  3.6000420629  3.1468197893  2.5613604803  2.4999074973  3.8220156234
## [156]  2.9641933997  3.6417335448  0.9202393226  0.9794384353  1.1012548299
## [161]  1.9395362934  1.7102594891  1.0037781237  0.8781253553 -1.1569880963
## [166]  0.6107840321 -1.8560809790 -0.5435647814 -1.6528587461 -0.6931748399
## [171] -0.6745905484 -0.2700519030 -0.0003782134 -0.0156498803 -1.2296319056
## [176] -1.3610382729 -1.5212560354 -1.4137373862 -0.6531646511 -1.5845346407
## [181] -2.5000675109 -1.5539361346 -1.3625679657  0.3162898500  0.4841280112
## [186]  2.3986596660  3.8117814084  4.5765370434  4.1689413221  2.7054681908
## [191]  1.5951951830  1.1137884636  0.6642015910 -1.2921580133 -1.2445533131
## [196] -1.3168116596 -0.9583928156 -1.2444371606 -0.6583207408 -0.3650164600
## [201]  0.1618485183 -1.7454894136 -2.3423369099 -2.9438195607 -1.9139617705
## [206] -2.0885040167 -1.0338061379 -1.2104404488 -1.0607909523 -2.5763154278
## [211] -0.8376673893 -0.7266009768  0.4606741405 -0.0797344283  0.7802678976
## [216]  1.7703740599  0.5991156398  0.3439210212 -1.2307817029 -0.8040832323
## [221] -0.4168184100  0.5488415007 -1.3056168347 -1.8853489989 -2.4294576903
## [226] -1.3154903878 -1.6047381370  0.4993032976  0.3607809119  1.7188315881
## [231]  1.7326976017  2.3695001883  0.8997509204  2.7958330588  1.2569250892
## [236]  2.2840960422  0.8480941899  0.0546283080 -0.3333451895  0.8357359520
## [241]  2.2096527922  3.1525881865  1.4404194837 -1.1044905458 -2.0256281043
## [246] -3.8919429335 -2.3282417599 -2.8059577794 -1.6041377423 -1.9505816925
## [251] -1.1496777608 -2.6893692628 -2.5514889502 -4.1585998491 -4.8072498472
## [256] -4.5987326380 -3.4570165931 -3.2403148488 -1.9629333596 -2.3350959519
## [261] -1.1603165610 -2.2192537621 -1.7877792538 -2.4489902341  0.0646756104
## [266]  0.5519406673  2.3198806174  2.2546951485  3.0304771019  2.9894299055
## [271]  3.5560478426  3.6626022116  3.9113555091  2.2156973230  0.2310645782
## [276] -0.5457597965 -1.3119324615  0.0840956912  0.9815558609  3.3823570677
## [281]  4.3801990716  3.0564029481  1.9294900662  2.3267664708  1.9920602633
## [286]  2.8829143584  3.3349656545  2.7343521942  3.4259392475  2.5294881904
## [291]  0.6829154986 -0.2227295916 -1.0767438788 -4.1740634290 -4.6781387204
## [296] -4.3194442344 -3.1309779247 -3.1753543129 -2.9338644682 -3.2121401154

Arma Dengan Fungsi Arma.sim

armasim <- arima.sim(list(order=c(2,0,2), ar = c(0.5,0.2), ma = c(0.4, 0.6)), n=300)
armasim
## Time Series:
## Start = 1 
## End = 300 
## Frequency = 1 
##   [1]  2.05563944  0.18037452 -1.95304068 -0.64041642  0.06010345  0.65992120
##   [7] -1.64439590  0.07351970 -1.01558552  1.57494711 -0.13605165 -0.35108718
##  [13]  0.52996160 -0.33961592  0.98457039  0.26637142  0.69092917  0.03895303
##  [19]  1.64969021  0.40976128  0.85273046 -0.93391088 -0.12462919 -1.70541555
##  [25] -0.76993032 -0.22260438 -0.35992790 -0.95037653 -2.02103737 -2.95044718
##  [31] -4.99236716 -3.57122377 -3.25057782 -1.03231229 -1.11287210 -1.01091305
##  [37]  1.06053518  3.23506722  5.08817113  5.35111385  5.49460222  4.34217312
##  [43]  3.70783804  2.98942540  1.18627404  1.34817014  0.87107027 -0.53563312
##  [49] -0.26200770  0.24508349 -0.35238829  0.16245006 -0.89364646 -0.73830337
##  [55] -1.89200524 -0.89939964 -1.67631845  0.73048456  0.08235434  2.91005819
##  [61]  2.47961858  4.67331476  3.39338132  3.48641743  1.82320563  0.86618074
##  [67]  0.86424314  1.16498267  1.66290033  1.22184331  0.87504074  0.03254905
##  [73] -0.50537634 -1.03251591 -1.78520600 -1.36117586 -1.44894592 -0.39539868
##  [79] -1.07621706  0.37292865  0.15738963  0.47089339  1.09825052  1.70660184
##  [85]  1.56847915  1.63470075  0.13055217 -2.16901161 -2.90520428 -2.89352415
##  [91] -3.10347756 -2.32053589 -4.01023442 -2.63378147 -4.20833579 -1.59156036
##  [97] -0.63716949  0.46967694 -0.98629656 -0.81962625 -2.19677028 -2.43878396
## [103] -1.97322901 -3.22933072 -2.07801928 -3.01678672 -3.69975438 -3.38532080
## [109] -4.71494386 -6.26726177 -7.25110636 -7.23065191 -4.18041669 -2.52368237
## [115] -2.13351647 -1.44109336 -1.24006778  0.31789323 -0.58494981 -0.14463542
## [121]  0.75770300  0.33146181  1.70531659  0.09941510  0.73497385  1.08867856
## [127]  1.55421591  3.76718388  3.08117582  4.39583420  4.13799306  3.79645706
## [133]  5.14718998  5.15392478  5.74123746  4.83298711  4.58202139  3.50616192
## [139]  2.35730563  0.52888165 -1.53361977 -2.66926050 -1.83016639 -0.49167595
## [145]  1.05289469 -0.08507836 -0.34867387 -0.60278566 -1.70266595 -1.70277337
## [151] -2.21462684 -1.32206621 -0.56863476 -0.36312049 -0.25072857  1.04503279
## [157]  1.72466850  2.11172664  0.63330487  0.93143694  0.43402205  1.17227276
## [163]  1.51565131  0.34295267 -0.29592870 -1.20879010  0.41670545  1.13813907
## [169]  0.85527061  0.45557316 -0.06012830 -0.18224759 -1.66342405 -1.20513989
## [175] -1.34577238 -2.48783299 -1.22950223 -2.12967035 -0.62982630 -0.46425338
## [181] -0.87455744 -2.51853074 -3.74956406 -4.18271307 -3.51967359 -2.17057996
## [187] -1.82696212 -1.68586819 -3.40127645 -3.06265716 -4.46398391 -2.84565712
## [193] -3.29350249 -2.01525162 -3.14439922 -5.26136433 -5.31158926 -6.60493414
## [199] -4.76804884 -4.77604015 -1.79144406 -0.50181658  2.62937170  2.04821246
## [205]  3.24577373  3.34923653  2.09521546  2.13669517  0.80984575  2.04409381
## [211]  2.45533059  1.94587452 -0.72115349 -0.96230576 -1.18852141 -1.59697062
## [217] -1.12813550 -2.31790456 -0.64947961  1.33015692  3.12297728  3.16744548
## [223]  3.82267902  2.22748636  2.50980836  1.86930997  1.83718320  2.26756350
## [229]  2.67598155  4.00655342  3.74227049  4.78103661  2.91546619  1.66451429
## [235]  0.30844593  0.76647919  1.68302580  2.54835979  1.75372013  2.39230304
## [241]  0.62345254 -0.94285862 -2.25039727 -3.95740243 -1.56071766 -2.78761969
## [247] -0.57030433 -1.31291692  0.62095031  1.47580595  2.12014611  2.89949958
## [253]  2.06964756  1.70650191  2.12930958  1.74799497  3.80517797  2.52648375
## [259]  3.47358542  1.01579066  1.70387820  0.11907342 -0.74338209 -0.95662472
## [265] -1.28594263  0.90585460  2.35715871  2.67539517  3.46427969  3.39917937
## [271]  2.74439967  1.46125313  0.26325889 -2.35136614 -1.86042206 -4.02567218
## [277] -3.85023325 -3.32679775 -1.73656636 -1.13728890 -1.03724854 -0.40509865
## [283] -0.88315854  0.44186248 -0.85900078 -1.29195506  0.01145730  0.06043444
## [289] -0.21198353 -0.51376924 -1.21972415  0.40316571  0.83558168  0.23153991
## [295]  1.10221851 -0.98658370 -0.16227845  0.17936449  0.38045863 -0.79086414

Karakteristik ARMA

Plot Time Series

par(mfrow = c(1, 2))
ts.plot(y.arma)
ts.plot(armasim)

Dapat dilihat dari Plot bahwa Model ARMA 2,2 baik bangkitan secara manual ataupun simulasi stasioner dalam rataan namun belum tentu stasioner secara ragam karena adanya data yang fluktuatif.

Plot ACF

par(mfrow = c(1, 2))
acf(y.arma)
acf(armasim)

Dapat dilihat bahwa Baik ARMA bangkitan secara manual ataupun simulasi memiliki pola yang Tails off sesuai dengan teori yang ada .

PACF

par(mfrow = c(1, 2))
pacf(y.arma)
pacf(armasim)

Berdasarkan plot PACF tersebut, terlihat bahwa model ARMA(2,2) manual maupun hasil simulasi memiliki plot PACF yang tails off, sesuai dengan teori

Plot EACF

TSA::eacf(y.arma)
## AR/MA
##   0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x x x x x o o o  o  x  x 
## 1 o x x o o o o o o o o  o  o  o 
## 2 x x o o o x 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 o o o o o o  o  o  o 
## 5 x x o x o o o o o o o  o  o  o 
## 6 x x x x o x o o o o o  o  o  o 
## 7 o x x x o o o o o o o  o  o  o

Pola Segitiga 0 dapat dilihat dari Model Sesuai dengan Ordo AR (2) dan MA (2) pada ARMA Manual

TSA::eacf(armasim)
## AR/MA
##   0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x x x x o o o o  o  o  o 
## 1 o x o o o o o o o o o  o  o  o 
## 2 o x x o o o o o o o o  o  o  o 
## 3 x x x x o o o o x o o  o  o  o 
## 4 x x o o o o o o o o o  o  o  o 
## 5 x x x o o o o o o o o  o  o  o 
## 6 x x o o o x x o o o o  o  o  o 
## 7 x x x o o x x o x o o  o  o  o

Didapat Pula Hasil yang sama dengan menggunakan ARMA simulasi, namun perlu dicatat bahwa simulasi sewaktu-waktu tidak memberikan hasil yang sama yaitu Ordo AR 2 MA 2, karena pada dasaranya data simulasi tidak dibangkitkan dengan asumsi kenormalan dan data yang digunakan adalah random.

Scatter Plot antar Lag

Yt dengan Yt-1

yt_arma <- y.arma[-1]
yt_arma
##   [1]  0.0000000000  1.0070355131  0.2910283464 -1.0183602565 -2.5713543828
##   [6] -3.8499559890 -4.3984708317 -3.8650635080 -2.0904090937 -3.9540551059
##  [11] -2.5357403043 -3.5714222653 -2.4756593463 -2.6540818075 -2.3438156556
##  [16] -3.3034023400 -1.3393006745 -1.9679158312  0.0491769000  1.0262473815
##  [21]  0.0412727756  1.0946507347  0.8275943183  1.9529078365  2.0282287056
##  [26]  3.4102273607  2.3561383659  0.8228772370  2.0001547761  1.3849868837
##  [31]  1.8453930386  2.0504337544  0.7970752463  1.6960309079  0.8591948681
##  [36]  2.1047497041  2.4444530905  2.0785654743  1.1034065490  1.3547389456
##  [41]  0.6832959756  2.0322161333  2.0275762310  2.6003273742  2.9267991261
##  [46]  1.6338503128  1.0631773685 -0.2299519560 -1.7504907667 -2.7478887295
##  [51] -2.2919008716 -2.4780590718 -1.3658237289  0.1561405734 -0.2527965210
##  [56] -0.1890849377 -0.3524863461 -0.8105250463 -0.5821120192 -1.4635253715
##  [61] -1.3918633911 -2.6175104801 -2.9896729578 -3.7870628901 -2.6038903271
##  [66] -2.8753506569 -1.0565761578 -2.5140864637 -0.1410838769 -2.6474550673
##  [71]  0.2338918937  1.1127796103  3.2948431393  4.2273897406  5.0915788739
##  [76]  4.0247870727  2.7413832260  2.2146772265  2.2244022237  3.7443324038
##  [81]  3.4649573580  2.1377879477  3.0288625658  1.2063403453  1.3537488180
##  [86]  0.1044572328  1.1224039706  2.6490204070  1.2412931237  1.5479683017
##  [91] -1.2405191747  0.0240842435 -1.0427363484 -0.4090475338 -0.7575508283
##  [96] -0.8272578083  0.5784813882 -0.1622753406  0.3484346563 -0.3727928320
## [101] -0.1728694326  0.3615806785  0.8246876130  0.9273273565  0.5014896554
## [106]  0.5274739863  0.1875861014  0.3581702276  0.4033024359 -1.0214590170
## [111] -0.9756771042 -2.4153516323 -2.1109948906 -1.6317449264 -1.6960780251
## [116] -1.8700977002 -2.2460733024 -4.5566582894 -5.0996616117 -5.1573245905
## [121] -5.2498402332 -4.1438896933 -3.6357610644 -2.1440227889 -2.0337121617
## [126] -2.6719137142 -3.3438583860 -4.2270368965 -2.6818112986 -1.7895114602
## [131] -0.5781170983 -0.1949052111  0.3799911360  0.7549244703  2.0412357783
## [136]  2.4997537236  2.2362920406  1.3371319301  0.9565603120 -0.5663177994
## [141]  0.2992931688 -1.1085194992 -0.9470758005  0.1245010832  0.2957539715
## [146]  2.7268333690  1.3776428700  3.4946867738  2.7450142842  3.6000420629
## [151]  3.1468197893  2.5613604803  2.4999074973  3.8220156234  2.9641933997
## [156]  3.6417335448  0.9202393226  0.9794384353  1.1012548299  1.9395362934
## [161]  1.7102594891  1.0037781237  0.8781253553 -1.1569880963  0.6107840321
## [166] -1.8560809790 -0.5435647814 -1.6528587461 -0.6931748399 -0.6745905484
## [171] -0.2700519030 -0.0003782134 -0.0156498803 -1.2296319056 -1.3610382729
## [176] -1.5212560354 -1.4137373862 -0.6531646511 -1.5845346407 -2.5000675109
## [181] -1.5539361346 -1.3625679657  0.3162898500  0.4841280112  2.3986596660
## [186]  3.8117814084  4.5765370434  4.1689413221  2.7054681908  1.5951951830
## [191]  1.1137884636  0.6642015910 -1.2921580133 -1.2445533131 -1.3168116596
## [196] -0.9583928156 -1.2444371606 -0.6583207408 -0.3650164600  0.1618485183
## [201] -1.7454894136 -2.3423369099 -2.9438195607 -1.9139617705 -2.0885040167
## [206] -1.0338061379 -1.2104404488 -1.0607909523 -2.5763154278 -0.8376673893
## [211] -0.7266009768  0.4606741405 -0.0797344283  0.7802678976  1.7703740599
## [216]  0.5991156398  0.3439210212 -1.2307817029 -0.8040832323 -0.4168184100
## [221]  0.5488415007 -1.3056168347 -1.8853489989 -2.4294576903 -1.3154903878
## [226] -1.6047381370  0.4993032976  0.3607809119  1.7188315881  1.7326976017
## [231]  2.3695001883  0.8997509204  2.7958330588  1.2569250892  2.2840960422
## [236]  0.8480941899  0.0546283080 -0.3333451895  0.8357359520  2.2096527922
## [241]  3.1525881865  1.4404194837 -1.1044905458 -2.0256281043 -3.8919429335
## [246] -2.3282417599 -2.8059577794 -1.6041377423 -1.9505816925 -1.1496777608
## [251] -2.6893692628 -2.5514889502 -4.1585998491 -4.8072498472 -4.5987326380
## [256] -3.4570165931 -3.2403148488 -1.9629333596 -2.3350959519 -1.1603165610
## [261] -2.2192537621 -1.7877792538 -2.4489902341  0.0646756104  0.5519406673
## [266]  2.3198806174  2.2546951485  3.0304771019  2.9894299055  3.5560478426
## [271]  3.6626022116  3.9113555091  2.2156973230  0.2310645782 -0.5457597965
## [276] -1.3119324615  0.0840956912  0.9815558609  3.3823570677  4.3801990716
## [281]  3.0564029481  1.9294900662  2.3267664708  1.9920602633  2.8829143584
## [286]  3.3349656545  2.7343521942  3.4259392475  2.5294881904  0.6829154986
## [291] -0.2227295916 -1.0767438788 -4.1740634290 -4.6781387204 -4.3194442344
## [296] -3.1309779247 -3.1753543129 -2.9338644682 -3.2121401154
yt_1_arma <- y.arma[-300]
yt_1_arma
##   [1]  0.0000000000  0.0000000000  1.0070355131  0.2910283464 -1.0183602565
##   [6] -2.5713543828 -3.8499559890 -4.3984708317 -3.8650635080 -2.0904090937
##  [11] -3.9540551059 -2.5357403043 -3.5714222653 -2.4756593463 -2.6540818075
##  [16] -2.3438156556 -3.3034023400 -1.3393006745 -1.9679158312  0.0491769000
##  [21]  1.0262473815  0.0412727756  1.0946507347  0.8275943183  1.9529078365
##  [26]  2.0282287056  3.4102273607  2.3561383659  0.8228772370  2.0001547761
##  [31]  1.3849868837  1.8453930386  2.0504337544  0.7970752463  1.6960309079
##  [36]  0.8591948681  2.1047497041  2.4444530905  2.0785654743  1.1034065490
##  [41]  1.3547389456  0.6832959756  2.0322161333  2.0275762310  2.6003273742
##  [46]  2.9267991261  1.6338503128  1.0631773685 -0.2299519560 -1.7504907667
##  [51] -2.7478887295 -2.2919008716 -2.4780590718 -1.3658237289  0.1561405734
##  [56] -0.2527965210 -0.1890849377 -0.3524863461 -0.8105250463 -0.5821120192
##  [61] -1.4635253715 -1.3918633911 -2.6175104801 -2.9896729578 -3.7870628901
##  [66] -2.6038903271 -2.8753506569 -1.0565761578 -2.5140864637 -0.1410838769
##  [71] -2.6474550673  0.2338918937  1.1127796103  3.2948431393  4.2273897406
##  [76]  5.0915788739  4.0247870727  2.7413832260  2.2146772265  2.2244022237
##  [81]  3.7443324038  3.4649573580  2.1377879477  3.0288625658  1.2063403453
##  [86]  1.3537488180  0.1044572328  1.1224039706  2.6490204070  1.2412931237
##  [91]  1.5479683017 -1.2405191747  0.0240842435 -1.0427363484 -0.4090475338
##  [96] -0.7575508283 -0.8272578083  0.5784813882 -0.1622753406  0.3484346563
## [101] -0.3727928320 -0.1728694326  0.3615806785  0.8246876130  0.9273273565
## [106]  0.5014896554  0.5274739863  0.1875861014  0.3581702276  0.4033024359
## [111] -1.0214590170 -0.9756771042 -2.4153516323 -2.1109948906 -1.6317449264
## [116] -1.6960780251 -1.8700977002 -2.2460733024 -4.5566582894 -5.0996616117
## [121] -5.1573245905 -5.2498402332 -4.1438896933 -3.6357610644 -2.1440227889
## [126] -2.0337121617 -2.6719137142 -3.3438583860 -4.2270368965 -2.6818112986
## [131] -1.7895114602 -0.5781170983 -0.1949052111  0.3799911360  0.7549244703
## [136]  2.0412357783  2.4997537236  2.2362920406  1.3371319301  0.9565603120
## [141] -0.5663177994  0.2992931688 -1.1085194992 -0.9470758005  0.1245010832
## [146]  0.2957539715  2.7268333690  1.3776428700  3.4946867738  2.7450142842
## [151]  3.6000420629  3.1468197893  2.5613604803  2.4999074973  3.8220156234
## [156]  2.9641933997  3.6417335448  0.9202393226  0.9794384353  1.1012548299
## [161]  1.9395362934  1.7102594891  1.0037781237  0.8781253553 -1.1569880963
## [166]  0.6107840321 -1.8560809790 -0.5435647814 -1.6528587461 -0.6931748399
## [171] -0.6745905484 -0.2700519030 -0.0003782134 -0.0156498803 -1.2296319056
## [176] -1.3610382729 -1.5212560354 -1.4137373862 -0.6531646511 -1.5845346407
## [181] -2.5000675109 -1.5539361346 -1.3625679657  0.3162898500  0.4841280112
## [186]  2.3986596660  3.8117814084  4.5765370434  4.1689413221  2.7054681908
## [191]  1.5951951830  1.1137884636  0.6642015910 -1.2921580133 -1.2445533131
## [196] -1.3168116596 -0.9583928156 -1.2444371606 -0.6583207408 -0.3650164600
## [201]  0.1618485183 -1.7454894136 -2.3423369099 -2.9438195607 -1.9139617705
## [206] -2.0885040167 -1.0338061379 -1.2104404488 -1.0607909523 -2.5763154278
## [211] -0.8376673893 -0.7266009768  0.4606741405 -0.0797344283  0.7802678976
## [216]  1.7703740599  0.5991156398  0.3439210212 -1.2307817029 -0.8040832323
## [221] -0.4168184100  0.5488415007 -1.3056168347 -1.8853489989 -2.4294576903
## [226] -1.3154903878 -1.6047381370  0.4993032976  0.3607809119  1.7188315881
## [231]  1.7326976017  2.3695001883  0.8997509204  2.7958330588  1.2569250892
## [236]  2.2840960422  0.8480941899  0.0546283080 -0.3333451895  0.8357359520
## [241]  2.2096527922  3.1525881865  1.4404194837 -1.1044905458 -2.0256281043
## [246] -3.8919429335 -2.3282417599 -2.8059577794 -1.6041377423 -1.9505816925
## [251] -1.1496777608 -2.6893692628 -2.5514889502 -4.1585998491 -4.8072498472
## [256] -4.5987326380 -3.4570165931 -3.2403148488 -1.9629333596 -2.3350959519
## [261] -1.1603165610 -2.2192537621 -1.7877792538 -2.4489902341  0.0646756104
## [266]  0.5519406673  2.3198806174  2.2546951485  3.0304771019  2.9894299055
## [271]  3.5560478426  3.6626022116  3.9113555091  2.2156973230  0.2310645782
## [276] -0.5457597965 -1.3119324615  0.0840956912  0.9815558609  3.3823570677
## [281]  4.3801990716  3.0564029481  1.9294900662  2.3267664708  1.9920602633
## [286]  2.8829143584  3.3349656545  2.7343521942  3.4259392475  2.5294881904
## [291]  0.6829154986 -0.2227295916 -1.0767438788 -4.1740634290 -4.6781387204
## [296] -4.3194442344 -3.1309779247 -3.1753543129 -2.9338644682
plot(y=yt_arma,x=yt_1_arma)

Didapatkan dari Plot korelasinya positif, mari kita lihat berapa korelasinya:

cor(yt_arma,yt_1_arma)
## [1] 0.8779016

Korelasi Antara Yt dan Yt-2

yt_arma <- y.arma[-c(1,2)]
yt_arma
##   [1]  1.0070355131  0.2910283464 -1.0183602565 -2.5713543828 -3.8499559890
##   [6] -4.3984708317 -3.8650635080 -2.0904090937 -3.9540551059 -2.5357403043
##  [11] -3.5714222653 -2.4756593463 -2.6540818075 -2.3438156556 -3.3034023400
##  [16] -1.3393006745 -1.9679158312  0.0491769000  1.0262473815  0.0412727756
##  [21]  1.0946507347  0.8275943183  1.9529078365  2.0282287056  3.4102273607
##  [26]  2.3561383659  0.8228772370  2.0001547761  1.3849868837  1.8453930386
##  [31]  2.0504337544  0.7970752463  1.6960309079  0.8591948681  2.1047497041
##  [36]  2.4444530905  2.0785654743  1.1034065490  1.3547389456  0.6832959756
##  [41]  2.0322161333  2.0275762310  2.6003273742  2.9267991261  1.6338503128
##  [46]  1.0631773685 -0.2299519560 -1.7504907667 -2.7478887295 -2.2919008716
##  [51] -2.4780590718 -1.3658237289  0.1561405734 -0.2527965210 -0.1890849377
##  [56] -0.3524863461 -0.8105250463 -0.5821120192 -1.4635253715 -1.3918633911
##  [61] -2.6175104801 -2.9896729578 -3.7870628901 -2.6038903271 -2.8753506569
##  [66] -1.0565761578 -2.5140864637 -0.1410838769 -2.6474550673  0.2338918937
##  [71]  1.1127796103  3.2948431393  4.2273897406  5.0915788739  4.0247870727
##  [76]  2.7413832260  2.2146772265  2.2244022237  3.7443324038  3.4649573580
##  [81]  2.1377879477  3.0288625658  1.2063403453  1.3537488180  0.1044572328
##  [86]  1.1224039706  2.6490204070  1.2412931237  1.5479683017 -1.2405191747
##  [91]  0.0240842435 -1.0427363484 -0.4090475338 -0.7575508283 -0.8272578083
##  [96]  0.5784813882 -0.1622753406  0.3484346563 -0.3727928320 -0.1728694326
## [101]  0.3615806785  0.8246876130  0.9273273565  0.5014896554  0.5274739863
## [106]  0.1875861014  0.3581702276  0.4033024359 -1.0214590170 -0.9756771042
## [111] -2.4153516323 -2.1109948906 -1.6317449264 -1.6960780251 -1.8700977002
## [116] -2.2460733024 -4.5566582894 -5.0996616117 -5.1573245905 -5.2498402332
## [121] -4.1438896933 -3.6357610644 -2.1440227889 -2.0337121617 -2.6719137142
## [126] -3.3438583860 -4.2270368965 -2.6818112986 -1.7895114602 -0.5781170983
## [131] -0.1949052111  0.3799911360  0.7549244703  2.0412357783  2.4997537236
## [136]  2.2362920406  1.3371319301  0.9565603120 -0.5663177994  0.2992931688
## [141] -1.1085194992 -0.9470758005  0.1245010832  0.2957539715  2.7268333690
## [146]  1.3776428700  3.4946867738  2.7450142842  3.6000420629  3.1468197893
## [151]  2.5613604803  2.4999074973  3.8220156234  2.9641933997  3.6417335448
## [156]  0.9202393226  0.9794384353  1.1012548299  1.9395362934  1.7102594891
## [161]  1.0037781237  0.8781253553 -1.1569880963  0.6107840321 -1.8560809790
## [166] -0.5435647814 -1.6528587461 -0.6931748399 -0.6745905484 -0.2700519030
## [171] -0.0003782134 -0.0156498803 -1.2296319056 -1.3610382729 -1.5212560354
## [176] -1.4137373862 -0.6531646511 -1.5845346407 -2.5000675109 -1.5539361346
## [181] -1.3625679657  0.3162898500  0.4841280112  2.3986596660  3.8117814084
## [186]  4.5765370434  4.1689413221  2.7054681908  1.5951951830  1.1137884636
## [191]  0.6642015910 -1.2921580133 -1.2445533131 -1.3168116596 -0.9583928156
## [196] -1.2444371606 -0.6583207408 -0.3650164600  0.1618485183 -1.7454894136
## [201] -2.3423369099 -2.9438195607 -1.9139617705 -2.0885040167 -1.0338061379
## [206] -1.2104404488 -1.0607909523 -2.5763154278 -0.8376673893 -0.7266009768
## [211]  0.4606741405 -0.0797344283  0.7802678976  1.7703740599  0.5991156398
## [216]  0.3439210212 -1.2307817029 -0.8040832323 -0.4168184100  0.5488415007
## [221] -1.3056168347 -1.8853489989 -2.4294576903 -1.3154903878 -1.6047381370
## [226]  0.4993032976  0.3607809119  1.7188315881  1.7326976017  2.3695001883
## [231]  0.8997509204  2.7958330588  1.2569250892  2.2840960422  0.8480941899
## [236]  0.0546283080 -0.3333451895  0.8357359520  2.2096527922  3.1525881865
## [241]  1.4404194837 -1.1044905458 -2.0256281043 -3.8919429335 -2.3282417599
## [246] -2.8059577794 -1.6041377423 -1.9505816925 -1.1496777608 -2.6893692628
## [251] -2.5514889502 -4.1585998491 -4.8072498472 -4.5987326380 -3.4570165931
## [256] -3.2403148488 -1.9629333596 -2.3350959519 -1.1603165610 -2.2192537621
## [261] -1.7877792538 -2.4489902341  0.0646756104  0.5519406673  2.3198806174
## [266]  2.2546951485  3.0304771019  2.9894299055  3.5560478426  3.6626022116
## [271]  3.9113555091  2.2156973230  0.2310645782 -0.5457597965 -1.3119324615
## [276]  0.0840956912  0.9815558609  3.3823570677  4.3801990716  3.0564029481
## [281]  1.9294900662  2.3267664708  1.9920602633  2.8829143584  3.3349656545
## [286]  2.7343521942  3.4259392475  2.5294881904  0.6829154986 -0.2227295916
## [291] -1.0767438788 -4.1740634290 -4.6781387204 -4.3194442344 -3.1309779247
## [296] -3.1753543129 -2.9338644682 -3.2121401154
yt_2_arma <- y.arma[-c(299,300)]
yt_2_arma
##   [1]  0.0000000000  0.0000000000  1.0070355131  0.2910283464 -1.0183602565
##   [6] -2.5713543828 -3.8499559890 -4.3984708317 -3.8650635080 -2.0904090937
##  [11] -3.9540551059 -2.5357403043 -3.5714222653 -2.4756593463 -2.6540818075
##  [16] -2.3438156556 -3.3034023400 -1.3393006745 -1.9679158312  0.0491769000
##  [21]  1.0262473815  0.0412727756  1.0946507347  0.8275943183  1.9529078365
##  [26]  2.0282287056  3.4102273607  2.3561383659  0.8228772370  2.0001547761
##  [31]  1.3849868837  1.8453930386  2.0504337544  0.7970752463  1.6960309079
##  [36]  0.8591948681  2.1047497041  2.4444530905  2.0785654743  1.1034065490
##  [41]  1.3547389456  0.6832959756  2.0322161333  2.0275762310  2.6003273742
##  [46]  2.9267991261  1.6338503128  1.0631773685 -0.2299519560 -1.7504907667
##  [51] -2.7478887295 -2.2919008716 -2.4780590718 -1.3658237289  0.1561405734
##  [56] -0.2527965210 -0.1890849377 -0.3524863461 -0.8105250463 -0.5821120192
##  [61] -1.4635253715 -1.3918633911 -2.6175104801 -2.9896729578 -3.7870628901
##  [66] -2.6038903271 -2.8753506569 -1.0565761578 -2.5140864637 -0.1410838769
##  [71] -2.6474550673  0.2338918937  1.1127796103  3.2948431393  4.2273897406
##  [76]  5.0915788739  4.0247870727  2.7413832260  2.2146772265  2.2244022237
##  [81]  3.7443324038  3.4649573580  2.1377879477  3.0288625658  1.2063403453
##  [86]  1.3537488180  0.1044572328  1.1224039706  2.6490204070  1.2412931237
##  [91]  1.5479683017 -1.2405191747  0.0240842435 -1.0427363484 -0.4090475338
##  [96] -0.7575508283 -0.8272578083  0.5784813882 -0.1622753406  0.3484346563
## [101] -0.3727928320 -0.1728694326  0.3615806785  0.8246876130  0.9273273565
## [106]  0.5014896554  0.5274739863  0.1875861014  0.3581702276  0.4033024359
## [111] -1.0214590170 -0.9756771042 -2.4153516323 -2.1109948906 -1.6317449264
## [116] -1.6960780251 -1.8700977002 -2.2460733024 -4.5566582894 -5.0996616117
## [121] -5.1573245905 -5.2498402332 -4.1438896933 -3.6357610644 -2.1440227889
## [126] -2.0337121617 -2.6719137142 -3.3438583860 -4.2270368965 -2.6818112986
## [131] -1.7895114602 -0.5781170983 -0.1949052111  0.3799911360  0.7549244703
## [136]  2.0412357783  2.4997537236  2.2362920406  1.3371319301  0.9565603120
## [141] -0.5663177994  0.2992931688 -1.1085194992 -0.9470758005  0.1245010832
## [146]  0.2957539715  2.7268333690  1.3776428700  3.4946867738  2.7450142842
## [151]  3.6000420629  3.1468197893  2.5613604803  2.4999074973  3.8220156234
## [156]  2.9641933997  3.6417335448  0.9202393226  0.9794384353  1.1012548299
## [161]  1.9395362934  1.7102594891  1.0037781237  0.8781253553 -1.1569880963
## [166]  0.6107840321 -1.8560809790 -0.5435647814 -1.6528587461 -0.6931748399
## [171] -0.6745905484 -0.2700519030 -0.0003782134 -0.0156498803 -1.2296319056
## [176] -1.3610382729 -1.5212560354 -1.4137373862 -0.6531646511 -1.5845346407
## [181] -2.5000675109 -1.5539361346 -1.3625679657  0.3162898500  0.4841280112
## [186]  2.3986596660  3.8117814084  4.5765370434  4.1689413221  2.7054681908
## [191]  1.5951951830  1.1137884636  0.6642015910 -1.2921580133 -1.2445533131
## [196] -1.3168116596 -0.9583928156 -1.2444371606 -0.6583207408 -0.3650164600
## [201]  0.1618485183 -1.7454894136 -2.3423369099 -2.9438195607 -1.9139617705
## [206] -2.0885040167 -1.0338061379 -1.2104404488 -1.0607909523 -2.5763154278
## [211] -0.8376673893 -0.7266009768  0.4606741405 -0.0797344283  0.7802678976
## [216]  1.7703740599  0.5991156398  0.3439210212 -1.2307817029 -0.8040832323
## [221] -0.4168184100  0.5488415007 -1.3056168347 -1.8853489989 -2.4294576903
## [226] -1.3154903878 -1.6047381370  0.4993032976  0.3607809119  1.7188315881
## [231]  1.7326976017  2.3695001883  0.8997509204  2.7958330588  1.2569250892
## [236]  2.2840960422  0.8480941899  0.0546283080 -0.3333451895  0.8357359520
## [241]  2.2096527922  3.1525881865  1.4404194837 -1.1044905458 -2.0256281043
## [246] -3.8919429335 -2.3282417599 -2.8059577794 -1.6041377423 -1.9505816925
## [251] -1.1496777608 -2.6893692628 -2.5514889502 -4.1585998491 -4.8072498472
## [256] -4.5987326380 -3.4570165931 -3.2403148488 -1.9629333596 -2.3350959519
## [261] -1.1603165610 -2.2192537621 -1.7877792538 -2.4489902341  0.0646756104
## [266]  0.5519406673  2.3198806174  2.2546951485  3.0304771019  2.9894299055
## [271]  3.5560478426  3.6626022116  3.9113555091  2.2156973230  0.2310645782
## [276] -0.5457597965 -1.3119324615  0.0840956912  0.9815558609  3.3823570677
## [281]  4.3801990716  3.0564029481  1.9294900662  2.3267664708  1.9920602633
## [286]  2.8829143584  3.3349656545  2.7343521942  3.4259392475  2.5294881904
## [291]  0.6829154986 -0.2227295916 -1.0767438788 -4.1740634290 -4.6781387204
## [296] -4.3194442344 -3.1309779247 -3.1753543129
plot(y=yt_arma,x=yt_2_arma)

Dari Scatterplot tersebut, dapat disimpulkan adanya hubungan positif antara Yt dengan Yt-2

cor(yt_arma,yt_2_arma)
## [1] 0.7851878

Scatter Plot Yt dengan Yt-3

yt_arma <- y.arma[-c(1,2,3)]
yt_arma
##   [1]  0.2910283464 -1.0183602565 -2.5713543828 -3.8499559890 -4.3984708317
##   [6] -3.8650635080 -2.0904090937 -3.9540551059 -2.5357403043 -3.5714222653
##  [11] -2.4756593463 -2.6540818075 -2.3438156556 -3.3034023400 -1.3393006745
##  [16] -1.9679158312  0.0491769000  1.0262473815  0.0412727756  1.0946507347
##  [21]  0.8275943183  1.9529078365  2.0282287056  3.4102273607  2.3561383659
##  [26]  0.8228772370  2.0001547761  1.3849868837  1.8453930386  2.0504337544
##  [31]  0.7970752463  1.6960309079  0.8591948681  2.1047497041  2.4444530905
##  [36]  2.0785654743  1.1034065490  1.3547389456  0.6832959756  2.0322161333
##  [41]  2.0275762310  2.6003273742  2.9267991261  1.6338503128  1.0631773685
##  [46] -0.2299519560 -1.7504907667 -2.7478887295 -2.2919008716 -2.4780590718
##  [51] -1.3658237289  0.1561405734 -0.2527965210 -0.1890849377 -0.3524863461
##  [56] -0.8105250463 -0.5821120192 -1.4635253715 -1.3918633911 -2.6175104801
##  [61] -2.9896729578 -3.7870628901 -2.6038903271 -2.8753506569 -1.0565761578
##  [66] -2.5140864637 -0.1410838769 -2.6474550673  0.2338918937  1.1127796103
##  [71]  3.2948431393  4.2273897406  5.0915788739  4.0247870727  2.7413832260
##  [76]  2.2146772265  2.2244022237  3.7443324038  3.4649573580  2.1377879477
##  [81]  3.0288625658  1.2063403453  1.3537488180  0.1044572328  1.1224039706
##  [86]  2.6490204070  1.2412931237  1.5479683017 -1.2405191747  0.0240842435
##  [91] -1.0427363484 -0.4090475338 -0.7575508283 -0.8272578083  0.5784813882
##  [96] -0.1622753406  0.3484346563 -0.3727928320 -0.1728694326  0.3615806785
## [101]  0.8246876130  0.9273273565  0.5014896554  0.5274739863  0.1875861014
## [106]  0.3581702276  0.4033024359 -1.0214590170 -0.9756771042 -2.4153516323
## [111] -2.1109948906 -1.6317449264 -1.6960780251 -1.8700977002 -2.2460733024
## [116] -4.5566582894 -5.0996616117 -5.1573245905 -5.2498402332 -4.1438896933
## [121] -3.6357610644 -2.1440227889 -2.0337121617 -2.6719137142 -3.3438583860
## [126] -4.2270368965 -2.6818112986 -1.7895114602 -0.5781170983 -0.1949052111
## [131]  0.3799911360  0.7549244703  2.0412357783  2.4997537236  2.2362920406
## [136]  1.3371319301  0.9565603120 -0.5663177994  0.2992931688 -1.1085194992
## [141] -0.9470758005  0.1245010832  0.2957539715  2.7268333690  1.3776428700
## [146]  3.4946867738  2.7450142842  3.6000420629  3.1468197893  2.5613604803
## [151]  2.4999074973  3.8220156234  2.9641933997  3.6417335448  0.9202393226
## [156]  0.9794384353  1.1012548299  1.9395362934  1.7102594891  1.0037781237
## [161]  0.8781253553 -1.1569880963  0.6107840321 -1.8560809790 -0.5435647814
## [166] -1.6528587461 -0.6931748399 -0.6745905484 -0.2700519030 -0.0003782134
## [171] -0.0156498803 -1.2296319056 -1.3610382729 -1.5212560354 -1.4137373862
## [176] -0.6531646511 -1.5845346407 -2.5000675109 -1.5539361346 -1.3625679657
## [181]  0.3162898500  0.4841280112  2.3986596660  3.8117814084  4.5765370434
## [186]  4.1689413221  2.7054681908  1.5951951830  1.1137884636  0.6642015910
## [191] -1.2921580133 -1.2445533131 -1.3168116596 -0.9583928156 -1.2444371606
## [196] -0.6583207408 -0.3650164600  0.1618485183 -1.7454894136 -2.3423369099
## [201] -2.9438195607 -1.9139617705 -2.0885040167 -1.0338061379 -1.2104404488
## [206] -1.0607909523 -2.5763154278 -0.8376673893 -0.7266009768  0.4606741405
## [211] -0.0797344283  0.7802678976  1.7703740599  0.5991156398  0.3439210212
## [216] -1.2307817029 -0.8040832323 -0.4168184100  0.5488415007 -1.3056168347
## [221] -1.8853489989 -2.4294576903 -1.3154903878 -1.6047381370  0.4993032976
## [226]  0.3607809119  1.7188315881  1.7326976017  2.3695001883  0.8997509204
## [231]  2.7958330588  1.2569250892  2.2840960422  0.8480941899  0.0546283080
## [236] -0.3333451895  0.8357359520  2.2096527922  3.1525881865  1.4404194837
## [241] -1.1044905458 -2.0256281043 -3.8919429335 -2.3282417599 -2.8059577794
## [246] -1.6041377423 -1.9505816925 -1.1496777608 -2.6893692628 -2.5514889502
## [251] -4.1585998491 -4.8072498472 -4.5987326380 -3.4570165931 -3.2403148488
## [256] -1.9629333596 -2.3350959519 -1.1603165610 -2.2192537621 -1.7877792538
## [261] -2.4489902341  0.0646756104  0.5519406673  2.3198806174  2.2546951485
## [266]  3.0304771019  2.9894299055  3.5560478426  3.6626022116  3.9113555091
## [271]  2.2156973230  0.2310645782 -0.5457597965 -1.3119324615  0.0840956912
## [276]  0.9815558609  3.3823570677  4.3801990716  3.0564029481  1.9294900662
## [281]  2.3267664708  1.9920602633  2.8829143584  3.3349656545  2.7343521942
## [286]  3.4259392475  2.5294881904  0.6829154986 -0.2227295916 -1.0767438788
## [291] -4.1740634290 -4.6781387204 -4.3194442344 -3.1309779247 -3.1753543129
## [296] -2.9338644682 -3.2121401154
yt_3_arma <- y.arma[c(-298,-299,-300)]
yt_3_arma
##   [1]  0.0000000000  0.0000000000  1.0070355131  0.2910283464 -1.0183602565
##   [6] -2.5713543828 -3.8499559890 -4.3984708317 -3.8650635080 -2.0904090937
##  [11] -3.9540551059 -2.5357403043 -3.5714222653 -2.4756593463 -2.6540818075
##  [16] -2.3438156556 -3.3034023400 -1.3393006745 -1.9679158312  0.0491769000
##  [21]  1.0262473815  0.0412727756  1.0946507347  0.8275943183  1.9529078365
##  [26]  2.0282287056  3.4102273607  2.3561383659  0.8228772370  2.0001547761
##  [31]  1.3849868837  1.8453930386  2.0504337544  0.7970752463  1.6960309079
##  [36]  0.8591948681  2.1047497041  2.4444530905  2.0785654743  1.1034065490
##  [41]  1.3547389456  0.6832959756  2.0322161333  2.0275762310  2.6003273742
##  [46]  2.9267991261  1.6338503128  1.0631773685 -0.2299519560 -1.7504907667
##  [51] -2.7478887295 -2.2919008716 -2.4780590718 -1.3658237289  0.1561405734
##  [56] -0.2527965210 -0.1890849377 -0.3524863461 -0.8105250463 -0.5821120192
##  [61] -1.4635253715 -1.3918633911 -2.6175104801 -2.9896729578 -3.7870628901
##  [66] -2.6038903271 -2.8753506569 -1.0565761578 -2.5140864637 -0.1410838769
##  [71] -2.6474550673  0.2338918937  1.1127796103  3.2948431393  4.2273897406
##  [76]  5.0915788739  4.0247870727  2.7413832260  2.2146772265  2.2244022237
##  [81]  3.7443324038  3.4649573580  2.1377879477  3.0288625658  1.2063403453
##  [86]  1.3537488180  0.1044572328  1.1224039706  2.6490204070  1.2412931237
##  [91]  1.5479683017 -1.2405191747  0.0240842435 -1.0427363484 -0.4090475338
##  [96] -0.7575508283 -0.8272578083  0.5784813882 -0.1622753406  0.3484346563
## [101] -0.3727928320 -0.1728694326  0.3615806785  0.8246876130  0.9273273565
## [106]  0.5014896554  0.5274739863  0.1875861014  0.3581702276  0.4033024359
## [111] -1.0214590170 -0.9756771042 -2.4153516323 -2.1109948906 -1.6317449264
## [116] -1.6960780251 -1.8700977002 -2.2460733024 -4.5566582894 -5.0996616117
## [121] -5.1573245905 -5.2498402332 -4.1438896933 -3.6357610644 -2.1440227889
## [126] -2.0337121617 -2.6719137142 -3.3438583860 -4.2270368965 -2.6818112986
## [131] -1.7895114602 -0.5781170983 -0.1949052111  0.3799911360  0.7549244703
## [136]  2.0412357783  2.4997537236  2.2362920406  1.3371319301  0.9565603120
## [141] -0.5663177994  0.2992931688 -1.1085194992 -0.9470758005  0.1245010832
## [146]  0.2957539715  2.7268333690  1.3776428700  3.4946867738  2.7450142842
## [151]  3.6000420629  3.1468197893  2.5613604803  2.4999074973  3.8220156234
## [156]  2.9641933997  3.6417335448  0.9202393226  0.9794384353  1.1012548299
## [161]  1.9395362934  1.7102594891  1.0037781237  0.8781253553 -1.1569880963
## [166]  0.6107840321 -1.8560809790 -0.5435647814 -1.6528587461 -0.6931748399
## [171] -0.6745905484 -0.2700519030 -0.0003782134 -0.0156498803 -1.2296319056
## [176] -1.3610382729 -1.5212560354 -1.4137373862 -0.6531646511 -1.5845346407
## [181] -2.5000675109 -1.5539361346 -1.3625679657  0.3162898500  0.4841280112
## [186]  2.3986596660  3.8117814084  4.5765370434  4.1689413221  2.7054681908
## [191]  1.5951951830  1.1137884636  0.6642015910 -1.2921580133 -1.2445533131
## [196] -1.3168116596 -0.9583928156 -1.2444371606 -0.6583207408 -0.3650164600
## [201]  0.1618485183 -1.7454894136 -2.3423369099 -2.9438195607 -1.9139617705
## [206] -2.0885040167 -1.0338061379 -1.2104404488 -1.0607909523 -2.5763154278
## [211] -0.8376673893 -0.7266009768  0.4606741405 -0.0797344283  0.7802678976
## [216]  1.7703740599  0.5991156398  0.3439210212 -1.2307817029 -0.8040832323
## [221] -0.4168184100  0.5488415007 -1.3056168347 -1.8853489989 -2.4294576903
## [226] -1.3154903878 -1.6047381370  0.4993032976  0.3607809119  1.7188315881
## [231]  1.7326976017  2.3695001883  0.8997509204  2.7958330588  1.2569250892
## [236]  2.2840960422  0.8480941899  0.0546283080 -0.3333451895  0.8357359520
## [241]  2.2096527922  3.1525881865  1.4404194837 -1.1044905458 -2.0256281043
## [246] -3.8919429335 -2.3282417599 -2.8059577794 -1.6041377423 -1.9505816925
## [251] -1.1496777608 -2.6893692628 -2.5514889502 -4.1585998491 -4.8072498472
## [256] -4.5987326380 -3.4570165931 -3.2403148488 -1.9629333596 -2.3350959519
## [261] -1.1603165610 -2.2192537621 -1.7877792538 -2.4489902341  0.0646756104
## [266]  0.5519406673  2.3198806174  2.2546951485  3.0304771019  2.9894299055
## [271]  3.5560478426  3.6626022116  3.9113555091  2.2156973230  0.2310645782
## [276] -0.5457597965 -1.3119324615  0.0840956912  0.9815558609  3.3823570677
## [281]  4.3801990716  3.0564029481  1.9294900662  2.3267664708  1.9920602633
## [286]  2.8829143584  3.3349656545  2.7343521942  3.4259392475  2.5294881904
## [291]  0.6829154986 -0.2227295916 -1.0767438788 -4.1740634290 -4.6781387204
## [296] -4.3194442344 -3.1309779247
plot(y=yt_arma,x=yt_3_arma)

cor(yt_arma,yt_3_arma)
## [1] 0.6069982