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(1234)
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.

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] -1.2070657494  0.2774292421  1.6977089295 -2.9459317186  0.7167390639
##   [6]  1.7418246382 -1.0346371303 -0.6203694070 -0.0009552807 -0.3362779159
##  [11]  0.2174936313 -0.2734866675 -0.0905836968  0.9739922420  1.3994628688
##  [16] -0.5327584083 -1.0425917434 -0.6406203177 -0.1660878101  3.2974211006
##  [21] -0.3299428431 -1.9938222918 -0.3247264458  0.9302201280 -0.6132272999
##  [26] -1.4464704762  1.5702698332 -0.3846350651  0.0494705563 -0.3156998472
##  [31]  1.4857599669 -0.3549429366 -1.1805813337  0.0678738017 -1.0029262223
##  [36] -0.2152270381 -0.7355358627  0.2315942247  1.5501272074  0.4564159184
##  [41]  1.8124315967 -1.3689027055 -1.2976053034  0.7027084857 -0.3688720953
##  [46] -0.4024044862 -0.1233084197 -0.2279500179  0.6413571509  0.4638728217
##  [51] -1.2929944025  0.4384465524  0.2075594995 -0.2221606049  0.9090090549
##  [56]  1.2369568341  1.5199808594 -1.7703139045  0.9265605147 -1.3361603449
##  [61]  0.1561661061  2.9810408136 -1.4483098967 -2.1851240662  0.2811049100
##  [66]  2.1819064982 -1.8448786626  0.7570196052  1.4655985611 -1.0160494268
##  [71] -0.9254351549 -0.6601095519 -0.1884721407  1.0681773835  2.0308705941
##  [76] -1.3704786975 -2.5715040985 -0.0752623509  1.3821150410  0.0137852464
##  [81] -0.2059233690  0.0913573752 -1.1976302805  0.4771300300  1.7431282568
##  [86]  0.4617881111 -0.2391854878 -1.0412961429 -0.3604993910 -0.8762511861
##  [91]  0.5396085954  0.9931762563  1.6357808981  0.1660100489 -1.9197671483
##  [96] -0.0471242772 -0.9794780965  1.1187166662  1.3024001287  1.2050282279
## [101] -1.0176733598 -1.9131981505  0.0071667625 -0.2440440958 -0.6646035704
## [106]  0.7988753840 -0.4674611859  0.4264976703  0.8254528817 -0.2950036539
## [111] -0.3880735286 -0.5394328340 -0.7325785595  1.6826229462  0.3485131839
## [116]  0.3126310853 -1.5901616139  0.1680571839  1.3521284522 -0.3969587475
## [121] -0.5847814021 -0.6895839534  2.4857599476  0.3963806461  0.2891105285
## [126] -1.0999245518 -1.7579581368 -1.3087601851  0.3480609963  1.3887143370
## [131]  1.3048296331 -0.2885069234 -1.8812625753  0.9858435726 -1.2144176785
## [136] -0.1070404714  0.8202831314 -1.6022873669  1.8890269386  0.1780530246
## [141] -0.6369225255  1.9024855095  0.2790372440 -1.2001901969  0.1751308106
## [146]  0.2588257141  1.1948900575 -0.6288122598 -0.6017986067 -0.0204931164
## [151] -0.8200220114  0.0399833371  1.8260394484 -1.5896620101 -0.5112497988
## [156]  1.8387821459 -0.8525293504 -1.7768127697 -0.3076578303  0.5898160974
## [161] -0.1694290979  0.3123768549  0.1982460944  0.1391007201  0.7289282408
## [166]  0.5716411677  1.1841188337 -1.1147558280 -1.3002858975  1.6405578743
## [171]  1.3084310586 -1.5210911518 -1.3725525729 -1.7151189124  2.3397509573
## [176] -0.3087461430 -1.6354872600  3.9958204643 -0.3082273696 -1.9535266651
## [181] -2.8599288639  1.0130523868  2.6552796841  0.0833305745  0.1611555545
## [186]  1.3704819868 -0.1717776632 -2.1666197414  0.2062098762 -0.1748461360
## [191] -0.8812123864 -2.5214805752  0.6754990831  2.5171285691  2.4466947968
## [196] -0.6592071826 -0.8808871658 -1.5144040602  0.1768898700 -1.5637574991
## [201]  1.2189290125  1.7496205755 -0.3843296882  0.2084666822 -0.0799207269
## [206]  0.2153498412  1.3512700642 -0.0809000264 -1.5177593547 -1.7949322526
## [211]  0.8442390338  0.1012110126  1.4027525041 -1.9643370385 -0.6690152740
## [216]  1.5525576525 -0.0710888128 -0.7986615246  0.0287812991  1.1151151966
## [221]  0.8261901817 -0.0755778202  1.4722952796 -0.1320234146 -1.5457794130
## [226] -1.4789525287  3.0104223313 -0.0943784956 -2.7067706227  0.0537709205
## [231]  0.0116020728 -0.2568421479 -0.6289856730  0.6550692122  0.8325241050
## [236] -1.2753134252 -2.9936314270  1.6849114742  2.0713589711  0.7353914336
## [241] -0.1962850397  0.0331805630 -1.2986128334  0.9022447903  1.0930819317
## [246] -0.4626095773  0.8252807706 -0.2894319446 -0.7117877804 -0.0803828866
## [251]  0.1710944535  0.7245253322 -0.2340175119 -0.1537518862 -1.6729672382
## [256] -0.3138672649  2.3069927290 -1.5923930762 -0.1385518663  1.0361177008
## [261]  0.6589611672 -0.9821997485 -0.2873058590 -1.6173792004  0.9487228100
## [266]  0.2559495915 -0.1751782408 -0.6923676150  1.0032294093  1.2252645564
## [271] -0.8022068888  1.5469089105  0.1233232786 -0.4145978134  1.2676143500
## [276] -2.0400543539 -0.1605770845  1.0538010704 -0.8859746054 -0.2704276920
## [281] -1.1357403683  0.6834648941 -0.2819442091  0.8119704665 -0.7163217300
## [286]  0.1557600832 -0.4874231776 -0.0987926101 -0.9466401476  0.8970355448
## [291]  1.8657851829 -0.2879902525 -2.4307942563 -0.0239905251  2.9639940437
## [296]  1.5197083888 -2.8925847444 -0.9825717830  1.2053288133  0.3977279966

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]  0.471173424 -1.745463687  0.141510314 -0.729712854 -0.114842790
##   [6] -0.472894305 -1.127948717 -1.443797742 -1.171879031 -1.723286222
##  [11] -1.461924092 -0.845074607  0.519525249  0.312187420  0.020572732
##  [16] -1.181770516 -1.508255550  1.534249256  0.598119283  1.012450498
##  [21] -0.556369299 -0.011041246 -0.774213194 -1.449939345 -0.420758391
##  [26] -1.662676381 -0.079747157 -1.556197355  0.718835126 -0.596243221
##  [31] -0.238298741 -1.070389923 -2.255260716 -2.120011486 -3.624543435
##  [36] -2.913580609 -2.138714925 -1.388210999  1.086560934 -0.768382742
##  [41] -0.413123964 -1.263954490 -1.619808057 -1.534624150 -2.091327966
##  [46] -2.276021753 -1.689013389 -1.457572736 -2.319068111 -1.602598402
##  [51] -2.425338748 -1.807763414 -1.233628102 -0.110845196  1.775654086
##  [56]  0.223607057  2.285258743 -0.979456751  1.157010823  2.116941328
##  [61]  1.378789117  0.845856907 -0.296314421  1.372262398 -0.432336811
##  [66]  1.978634754  1.193531021  1.688995021  0.939220832 -0.250827924
##  [71] -0.544575725  0.228395751  2.109671129  1.063681873 -0.209897795
##  [76] -1.371901203 -0.865591517 -0.647903476 -0.149656546 -0.431345529
##  [81] -1.546973491 -0.824704371 -0.042663743  0.933429312  1.339180190
##  [86]  0.235832192 -0.022688150 -1.512804573 -0.645926233 -0.482784254
##  [91]  1.776147116  1.837016456  0.928600262  0.758224871 -1.289737991
##  [96]  0.637690588  0.643433378  3.037205983  1.846720429  0.963761203
## [101]  0.124820225 -0.760911469 -0.987393604 -0.464896825 -1.325068067
## [106] -0.090126895 -0.115516360  0.190793420 -0.003795709 -0.758706669
## [111] -1.486955903  0.015925459 -0.303788132  1.349650149 -0.898414089
## [116]  0.169995644 -0.005795838  0.344405995  0.201997065 -0.874229340
## [121]  1.630564028  1.104622260  3.359306076  1.260043834  0.495139539
## [126] -1.717816055 -1.620260658 -0.936111272  0.722551062  0.794007194
## [131] -0.462634050  0.351585084 -2.085784191 -0.624664025 -1.452519789
## [136] -2.294204728 -0.048911894 -1.423796214 -0.031150775  0.887810276
## [141]  0.994311578  0.983326803  0.852394746  0.539717900  2.130822837
## [146]  1.180599067  1.614343853  0.715597066  0.065546716  0.155255589
## [151]  1.451449842 -0.161522938  0.754769796  0.885479176  0.383287177
## [156] -0.329952846 -1.431909381 -1.370070157 -1.525271048 -0.833655639
## [161] -1.027085507 -0.505202316  0.085183955  0.677625089  2.172292654
## [166]  0.977368521  0.658606072  1.301453560  2.100227737  1.607579227
## [171]  0.707237935 -1.929351923  0.482773840 -1.366418725 -0.612038329
## [176]  2.091711308  0.778269986  1.887009445 -2.604510182 -1.212633564
## [181] -0.703216214  0.744407256  1.663488769  2.596982415  2.509994691
## [186]  1.149145711  1.202150481 -0.221986412 -0.194929191 -3.190036735
## [191] -2.254792788 -1.541499299  1.889370282  1.660596411  2.121307573
## [196] -0.417402360  0.148419546 -2.592717585 -0.248475371 -0.356083018
## [201]  0.755357519  1.193000349  0.703282784  1.305574883  2.333657188
## [206]  2.305625709  1.583087270 -0.433965677 -0.008123389 -0.901681487
## [211]  1.584233701 -1.249824841 -0.162488303 -0.708540906 -0.232384260
## [216] -0.413640706 -0.638223437  0.143957003  0.964153779  1.396003084
## [221]  3.074671761  2.479018560  2.121198870  0.159412341  2.827857930
## [226]  1.449209496  1.338129934  0.319213246 -0.660388674 -0.292566289
## [231] -1.238021008 -0.421378522 -0.194203628 -0.879770820 -3.472672839
## [236] -2.194660779 -2.012323310  0.453156115  0.314555376  0.793617226
## [241] -0.896931516  0.520105724  0.344695528  0.965911715  1.889268102
## [246]  1.098368887  1.240516320  0.616470695  0.702766701  1.395722479
## [251]  1.138398305  1.480149118 -0.599779870 -0.427127769  0.646946451
## [256] -0.855414425  0.654688641 -0.226112090  1.292645476  0.284446275
## [261]  0.604556738 -1.909130933 -0.271530716 -1.589079651 -0.302115008
## [266] -1.683162950 -0.233358766  0.107894475  0.192979110  2.103113218
## [271]  1.217795463  2.311862960  2.831191653  0.737827135  1.777815630
## [276]  0.919360158  0.873633013  0.908532408 -0.887903437  0.256870201
## [281] -1.119996454  0.815395259 -0.906539838  0.483034891 -1.205622129
## [286] -0.392733748 -2.159077875 -0.640167479  0.105101596  0.654485299
## [291] -1.101664171 -1.217076868  0.348092030  2.099902382  0.542511208
## [296]  0.249165265 -0.498077915  0.240584445 -0.240119231 -0.993767765

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 o x o o o o o o o o o  o  o  o 
## 1 o x x o o o o o o o o  o  o  o 
## 2 x x o x o o o o o o o  o  o  o 
## 3 x x o x o o o o o o o  o  o  o 
## 4 x x o o x o o o o o o  o  o  o 
## 5 x x x x o o o o o o o  o  o  o 
## 6 o x x x o o o o o o o  o  o  o 
## 7 x x o x o x o 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 \(Yt\) dan \(Yt_{-1}\)

yt_ma <- MA[-1]
yt_ma

##   [1]  0.2774292421  1.6977089295 -2.9459317186  0.7167390639  1.7418246382
##   [6] -1.0346371303 -0.6203694070 -0.0009552807 -0.3362779159  0.2174936313
##  [11] -0.2734866675 -0.0905836968  0.9739922420  1.3994628688 -0.5327584083
##  [16] -1.0425917434 -0.6406203177 -0.1660878101  3.2974211006 -0.3299428431
##  [21] -1.9938222918 -0.3247264458  0.9302201280 -0.6132272999 -1.4464704762
##  [26]  1.5702698332 -0.3846350651  0.0494705563 -0.3156998472  1.4857599669
##  [31] -0.3549429366 -1.1805813337  0.0678738017 -1.0029262223 -0.2152270381
##  [36] -0.7355358627  0.2315942247  1.5501272074  0.4564159184  1.8124315967
##  [41] -1.3689027055 -1.2976053034  0.7027084857 -0.3688720953 -0.4024044862
##  [46] -0.1233084197 -0.2279500179  0.6413571509  0.4638728217 -1.2929944025
##  [51]  0.4384465524  0.2075594995 -0.2221606049  0.9090090549  1.2369568341
##  [56]  1.5199808594 -1.7703139045  0.9265605147 -1.3361603449  0.1561661061
##  [61]  2.9810408136 -1.4483098967 -2.1851240662  0.2811049100  2.1819064982
##  [66] -1.8448786626  0.7570196052  1.4655985611 -1.0160494268 -0.9254351549
##  [71] -0.6601095519 -0.1884721407  1.0681773835  2.0308705941 -1.3704786975
##  [76] -2.5715040985 -0.0752623509  1.3821150410  0.0137852464 -0.2059233690
##  [81]  0.0913573752 -1.1976302805  0.4771300300  1.7431282568  0.4617881111
##  [86] -0.2391854878 -1.0412961429 -0.3604993910 -0.8762511861  0.5396085954
##  [91]  0.9931762563  1.6357808981  0.1660100489 -1.9197671483 -0.0471242772
##  [96] -0.9794780965  1.1187166662  1.3024001287  1.2050282279 -1.0176733598
## [101] -1.9131981505  0.0071667625 -0.2440440958 -0.6646035704  0.7988753840
## [106] -0.4674611859  0.4264976703  0.8254528817 -0.2950036539 -0.3880735286
## [111] -0.5394328340 -0.7325785595  1.6826229462  0.3485131839  0.3126310853
## [116] -1.5901616139  0.1680571839  1.3521284522 -0.3969587475 -0.5847814021
## [121] -0.6895839534  2.4857599476  0.3963806461  0.2891105285 -1.0999245518
## [126] -1.7579581368 -1.3087601851  0.3480609963  1.3887143370  1.3048296331
## [131] -0.2885069234 -1.8812625753  0.9858435726 -1.2144176785 -0.1070404714
## [136]  0.8202831314 -1.6022873669  1.8890269386  0.1780530246 -0.6369225255
## [141]  1.9024855095  0.2790372440 -1.2001901969  0.1751308106  0.2588257141
## [146]  1.1948900575 -0.6288122598 -0.6017986067 -0.0204931164 -0.8200220114
## [151]  0.0399833371  1.8260394484 -1.5896620101 -0.5112497988  1.8387821459
## [156] -0.8525293504 -1.7768127697 -0.3076578303  0.5898160974 -0.1694290979
## [161]  0.3123768549  0.1982460944  0.1391007201  0.7289282408  0.5716411677
## [166]  1.1841188337 -1.1147558280 -1.3002858975  1.6405578743  1.3084310586
## [171] -1.5210911518 -1.3725525729 -1.7151189124  2.3397509573 -0.3087461430
## [176] -1.6354872600  3.9958204643 -0.3082273696 -1.9535266651 -2.8599288639
## [181]  1.0130523868  2.6552796841  0.0833305745  0.1611555545  1.3704819868
## [186] -0.1717776632 -2.1666197414  0.2062098762 -0.1748461360 -0.8812123864
## [191] -2.5214805752  0.6754990831  2.5171285691  2.4466947968 -0.6592071826
## [196] -0.8808871658 -1.5144040602  0.1768898700 -1.5637574991  1.2189290125
## [201]  1.7496205755 -0.3843296882  0.2084666822 -0.0799207269  0.2153498412
## [206]  1.3512700642 -0.0809000264 -1.5177593547 -1.7949322526  0.8442390338
## [211]  0.1012110126  1.4027525041 -1.9643370385 -0.6690152740  1.5525576525
## [216] -0.0710888128 -0.7986615246  0.0287812991  1.1151151966  0.8261901817
## [221] -0.0755778202  1.4722952796 -0.1320234146 -1.5457794130 -1.4789525287
## [226]  3.0104223313 -0.0943784956 -2.7067706227  0.0537709205  0.0116020728
## [231] -0.2568421479 -0.6289856730  0.6550692122  0.8325241050 -1.2753134252
## [236] -2.9936314270  1.6849114742  2.0713589711  0.7353914336 -0.1962850397
## [241]  0.0331805630 -1.2986128334  0.9022447903  1.0930819317 -0.4626095773
## [246]  0.8252807706 -0.2894319446 -0.7117877804 -0.0803828866  0.1710944535
## [251]  0.7245253322 -0.2340175119 -0.1537518862 -1.6729672382 -0.3138672649
## [256]  2.3069927290 -1.5923930762 -0.1385518663  1.0361177008  0.6589611672
## [261] -0.9821997485 -0.2873058590 -1.6173792004  0.9487228100  0.2559495915
## [266] -0.1751782408 -0.6923676150  1.0032294093  1.2252645564 -0.8022068888
## [271]  1.5469089105  0.1233232786 -0.4145978134  1.2676143500 -2.0400543539
## [276] -0.1605770845  1.0538010704 -0.8859746054 -0.2704276920 -1.1357403683
## [281]  0.6834648941 -0.2819442091  0.8119704665 -0.7163217300  0.1557600832
## [286] -0.4874231776 -0.0987926101 -0.9466401476  0.8970355448  1.8657851829
## [291] -0.2879902525 -2.4307942563 -0.0239905251  2.9639940437  1.5197083888
## [296] -2.8925847444 -0.9825717830  1.2053288133  0.3977279966

# Yt-1
yt_1_ma <- MA[-300]
yt_1_ma

##   [1] -1.2070657494  0.2774292421  1.6977089295 -2.9459317186  0.7167390639
##   [6]  1.7418246382 -1.0346371303 -0.6203694070 -0.0009552807 -0.3362779159
##  [11]  0.2174936313 -0.2734866675 -0.0905836968  0.9739922420  1.3994628688
##  [16] -0.5327584083 -1.0425917434 -0.6406203177 -0.1660878101  3.2974211006
##  [21] -0.3299428431 -1.9938222918 -0.3247264458  0.9302201280 -0.6132272999
##  [26] -1.4464704762  1.5702698332 -0.3846350651  0.0494705563 -0.3156998472
##  [31]  1.4857599669 -0.3549429366 -1.1805813337  0.0678738017 -1.0029262223
##  [36] -0.2152270381 -0.7355358627  0.2315942247  1.5501272074  0.4564159184
##  [41]  1.8124315967 -1.3689027055 -1.2976053034  0.7027084857 -0.3688720953
##  [46] -0.4024044862 -0.1233084197 -0.2279500179  0.6413571509  0.4638728217
##  [51] -1.2929944025  0.4384465524  0.2075594995 -0.2221606049  0.9090090549
##  [56]  1.2369568341  1.5199808594 -1.7703139045  0.9265605147 -1.3361603449
##  [61]  0.1561661061  2.9810408136 -1.4483098967 -2.1851240662  0.2811049100
##  [66]  2.1819064982 -1.8448786626  0.7570196052  1.4655985611 -1.0160494268
##  [71] -0.9254351549 -0.6601095519 -0.1884721407  1.0681773835  2.0308705941
##  [76] -1.3704786975 -2.5715040985 -0.0752623509  1.3821150410  0.0137852464
##  [81] -0.2059233690  0.0913573752 -1.1976302805  0.4771300300  1.7431282568
##  [86]  0.4617881111 -0.2391854878 -1.0412961429 -0.3604993910 -0.8762511861
##  [91]  0.5396085954  0.9931762563  1.6357808981  0.1660100489 -1.9197671483
##  [96] -0.0471242772 -0.9794780965  1.1187166662  1.3024001287  1.2050282279
## [101] -1.0176733598 -1.9131981505  0.0071667625 -0.2440440958 -0.6646035704
## [106]  0.7988753840 -0.4674611859  0.4264976703  0.8254528817 -0.2950036539
## [111] -0.3880735286 -0.5394328340 -0.7325785595  1.6826229462  0.3485131839
## [116]  0.3126310853 -1.5901616139  0.1680571839  1.3521284522 -0.3969587475
## [121] -0.5847814021 -0.6895839534  2.4857599476  0.3963806461  0.2891105285
## [126] -1.0999245518 -1.7579581368 -1.3087601851  0.3480609963  1.3887143370
## [131]  1.3048296331 -0.2885069234 -1.8812625753  0.9858435726 -1.2144176785
## [136] -0.1070404714  0.8202831314 -1.6022873669  1.8890269386  0.1780530246
## [141] -0.6369225255  1.9024855095  0.2790372440 -1.2001901969  0.1751308106
## [146]  0.2588257141  1.1948900575 -0.6288122598 -0.6017986067 -0.0204931164
## [151] -0.8200220114  0.0399833371  1.8260394484 -1.5896620101 -0.5112497988
## [156]  1.8387821459 -0.8525293504 -1.7768127697 -0.3076578303  0.5898160974
## [161] -0.1694290979  0.3123768549  0.1982460944  0.1391007201  0.7289282408
## [166]  0.5716411677  1.1841188337 -1.1147558280 -1.3002858975  1.6405578743
## [171]  1.3084310586 -1.5210911518 -1.3725525729 -1.7151189124  2.3397509573
## [176] -0.3087461430 -1.6354872600  3.9958204643 -0.3082273696 -1.9535266651
## [181] -2.8599288639  1.0130523868  2.6552796841  0.0833305745  0.1611555545
## [186]  1.3704819868 -0.1717776632 -2.1666197414  0.2062098762 -0.1748461360
## [191] -0.8812123864 -2.5214805752  0.6754990831  2.5171285691  2.4466947968
## [196] -0.6592071826 -0.8808871658 -1.5144040602  0.1768898700 -1.5637574991
## [201]  1.2189290125  1.7496205755 -0.3843296882  0.2084666822 -0.0799207269
## [206]  0.2153498412  1.3512700642 -0.0809000264 -1.5177593547 -1.7949322526
## [211]  0.8442390338  0.1012110126  1.4027525041 -1.9643370385 -0.6690152740
## [216]  1.5525576525 -0.0710888128 -0.7986615246  0.0287812991  1.1151151966
## [221]  0.8261901817 -0.0755778202  1.4722952796 -0.1320234146 -1.5457794130
## [226] -1.4789525287  3.0104223313 -0.0943784956 -2.7067706227  0.0537709205
## [231]  0.0116020728 -0.2568421479 -0.6289856730  0.6550692122  0.8325241050
## [236] -1.2753134252 -2.9936314270  1.6849114742  2.0713589711  0.7353914336
## [241] -0.1962850397  0.0331805630 -1.2986128334  0.9022447903  1.0930819317
## [246] -0.4626095773  0.8252807706 -0.2894319446 -0.7117877804 -0.0803828866
## [251]  0.1710944535  0.7245253322 -0.2340175119 -0.1537518862 -1.6729672382
## [256] -0.3138672649  2.3069927290 -1.5923930762 -0.1385518663  1.0361177008
## [261]  0.6589611672 -0.9821997485 -0.2873058590 -1.6173792004  0.9487228100
## [266]  0.2559495915 -0.1751782408 -0.6923676150  1.0032294093  1.2252645564
## [271] -0.8022068888  1.5469089105  0.1233232786 -0.4145978134  1.2676143500
## [276] -2.0400543539 -0.1605770845  1.0538010704 -0.8859746054 -0.2704276920
## [281] -1.1357403683  0.6834648941 -0.2819442091  0.8119704665 -0.7163217300
## [286]  0.1557600832 -0.4874231776 -0.0987926101 -0.9466401476  0.8970355448
## [291]  1.8657851829 -0.2879902525 -2.4307942563 -0.0239905251  2.9639940437
## [296]  1.5197083888 -2.8925847444 -0.9825717830  1.2053288133

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] -0.0404004

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

\[ \rho_1=\frac{-\theta_1+\theta_1\theta_2}{1+(\theta_1)^2+\theta_2^2}=\frac{-(0.4)*(0.4*0.6)}{1+(0.4)^2+(0.6)^2}= -0.10\ \]

y <- (-0.4 +0.4*0.6)/ (1+0.4^2+0.6^2)
y

## [1] -0.1052632

Scatter Plot Yt dengan Yt-2

yt_ma2 <- MA[-c(1,2)]
yt_ma2

##   [1]  1.6977089295 -2.9459317186  0.7167390639  1.7418246382 -1.0346371303
##   [6] -0.6203694070 -0.0009552807 -0.3362779159  0.2174936313 -0.2734866675
##  [11] -0.0905836968  0.9739922420  1.3994628688 -0.5327584083 -1.0425917434
##  [16] -0.6406203177 -0.1660878101  3.2974211006 -0.3299428431 -1.9938222918
##  [21] -0.3247264458  0.9302201280 -0.6132272999 -1.4464704762  1.5702698332
##  [26] -0.3846350651  0.0494705563 -0.3156998472  1.4857599669 -0.3549429366
##  [31] -1.1805813337  0.0678738017 -1.0029262223 -0.2152270381 -0.7355358627
##  [36]  0.2315942247  1.5501272074  0.4564159184  1.8124315967 -1.3689027055
##  [41] -1.2976053034  0.7027084857 -0.3688720953 -0.4024044862 -0.1233084197
##  [46] -0.2279500179  0.6413571509  0.4638728217 -1.2929944025  0.4384465524
##  [51]  0.2075594995 -0.2221606049  0.9090090549  1.2369568341  1.5199808594
##  [56] -1.7703139045  0.9265605147 -1.3361603449  0.1561661061  2.9810408136
##  [61] -1.4483098967 -2.1851240662  0.2811049100  2.1819064982 -1.8448786626
##  [66]  0.7570196052  1.4655985611 -1.0160494268 -0.9254351549 -0.6601095519
##  [71] -0.1884721407  1.0681773835  2.0308705941 -1.3704786975 -2.5715040985
##  [76] -0.0752623509  1.3821150410  0.0137852464 -0.2059233690  0.0913573752
##  [81] -1.1976302805  0.4771300300  1.7431282568  0.4617881111 -0.2391854878
##  [86] -1.0412961429 -0.3604993910 -0.8762511861  0.5396085954  0.9931762563
##  [91]  1.6357808981  0.1660100489 -1.9197671483 -0.0471242772 -0.9794780965
##  [96]  1.1187166662  1.3024001287  1.2050282279 -1.0176733598 -1.9131981505
## [101]  0.0071667625 -0.2440440958 -0.6646035704  0.7988753840 -0.4674611859
## [106]  0.4264976703  0.8254528817 -0.2950036539 -0.3880735286 -0.5394328340
## [111] -0.7325785595  1.6826229462  0.3485131839  0.3126310853 -1.5901616139
## [116]  0.1680571839  1.3521284522 -0.3969587475 -0.5847814021 -0.6895839534
## [121]  2.4857599476  0.3963806461  0.2891105285 -1.0999245518 -1.7579581368
## [126] -1.3087601851  0.3480609963  1.3887143370  1.3048296331 -0.2885069234
## [131] -1.8812625753  0.9858435726 -1.2144176785 -0.1070404714  0.8202831314
## [136] -1.6022873669  1.8890269386  0.1780530246 -0.6369225255  1.9024855095
## [141]  0.2790372440 -1.2001901969  0.1751308106  0.2588257141  1.1948900575
## [146] -0.6288122598 -0.6017986067 -0.0204931164 -0.8200220114  0.0399833371
## [151]  1.8260394484 -1.5896620101 -0.5112497988  1.8387821459 -0.8525293504
## [156] -1.7768127697 -0.3076578303  0.5898160974 -0.1694290979  0.3123768549
## [161]  0.1982460944  0.1391007201  0.7289282408  0.5716411677  1.1841188337
## [166] -1.1147558280 -1.3002858975  1.6405578743  1.3084310586 -1.5210911518
## [171] -1.3725525729 -1.7151189124  2.3397509573 -0.3087461430 -1.6354872600
## [176]  3.9958204643 -0.3082273696 -1.9535266651 -2.8599288639  1.0130523868
## [181]  2.6552796841  0.0833305745  0.1611555545  1.3704819868 -0.1717776632
## [186] -2.1666197414  0.2062098762 -0.1748461360 -0.8812123864 -2.5214805752
## [191]  0.6754990831  2.5171285691  2.4466947968 -0.6592071826 -0.8808871658
## [196] -1.5144040602  0.1768898700 -1.5637574991  1.2189290125  1.7496205755
## [201] -0.3843296882  0.2084666822 -0.0799207269  0.2153498412  1.3512700642
## [206] -0.0809000264 -1.5177593547 -1.7949322526  0.8442390338  0.1012110126
## [211]  1.4027525041 -1.9643370385 -0.6690152740  1.5525576525 -0.0710888128
## [216] -0.7986615246  0.0287812991  1.1151151966  0.8261901817 -0.0755778202
## [221]  1.4722952796 -0.1320234146 -1.5457794130 -1.4789525287  3.0104223313
## [226] -0.0943784956 -2.7067706227  0.0537709205  0.0116020728 -0.2568421479
## [231] -0.6289856730  0.6550692122  0.8325241050 -1.2753134252 -2.9936314270
## [236]  1.6849114742  2.0713589711  0.7353914336 -0.1962850397  0.0331805630
## [241] -1.2986128334  0.9022447903  1.0930819317 -0.4626095773  0.8252807706
## [246] -0.2894319446 -0.7117877804 -0.0803828866  0.1710944535  0.7245253322
## [251] -0.2340175119 -0.1537518862 -1.6729672382 -0.3138672649  2.3069927290
## [256] -1.5923930762 -0.1385518663  1.0361177008  0.6589611672 -0.9821997485
## [261] -0.2873058590 -1.6173792004  0.9487228100  0.2559495915 -0.1751782408
## [266] -0.6923676150  1.0032294093  1.2252645564 -0.8022068888  1.5469089105
## [271]  0.1233232786 -0.4145978134  1.2676143500 -2.0400543539 -0.1605770845
## [276]  1.0538010704 -0.8859746054 -0.2704276920 -1.1357403683  0.6834648941
## [281] -0.2819442091  0.8119704665 -0.7163217300  0.1557600832 -0.4874231776
## [286] -0.0987926101 -0.9466401476  0.8970355448  1.8657851829 -0.2879902525
## [291] -2.4307942563 -0.0239905251  2.9639940437  1.5197083888 -2.8925847444
## [296] -0.9825717830  1.2053288133  0.3977279966

yt_2_ma <- MA[-c(299,300)]
yt_2_ma

##   [1] -1.2070657494  0.2774292421  1.6977089295 -2.9459317186  0.7167390639
##   [6]  1.7418246382 -1.0346371303 -0.6203694070 -0.0009552807 -0.3362779159
##  [11]  0.2174936313 -0.2734866675 -0.0905836968  0.9739922420  1.3994628688
##  [16] -0.5327584083 -1.0425917434 -0.6406203177 -0.1660878101  3.2974211006
##  [21] -0.3299428431 -1.9938222918 -0.3247264458  0.9302201280 -0.6132272999
##  [26] -1.4464704762  1.5702698332 -0.3846350651  0.0494705563 -0.3156998472
##  [31]  1.4857599669 -0.3549429366 -1.1805813337  0.0678738017 -1.0029262223
##  [36] -0.2152270381 -0.7355358627  0.2315942247  1.5501272074  0.4564159184
##  [41]  1.8124315967 -1.3689027055 -1.2976053034  0.7027084857 -0.3688720953
##  [46] -0.4024044862 -0.1233084197 -0.2279500179  0.6413571509  0.4638728217
##  [51] -1.2929944025  0.4384465524  0.2075594995 -0.2221606049  0.9090090549
##  [56]  1.2369568341  1.5199808594 -1.7703139045  0.9265605147 -1.3361603449
##  [61]  0.1561661061  2.9810408136 -1.4483098967 -2.1851240662  0.2811049100
##  [66]  2.1819064982 -1.8448786626  0.7570196052  1.4655985611 -1.0160494268
##  [71] -0.9254351549 -0.6601095519 -0.1884721407  1.0681773835  2.0308705941
##  [76] -1.3704786975 -2.5715040985 -0.0752623509  1.3821150410  0.0137852464
##  [81] -0.2059233690  0.0913573752 -1.1976302805  0.4771300300  1.7431282568
##  [86]  0.4617881111 -0.2391854878 -1.0412961429 -0.3604993910 -0.8762511861
##  [91]  0.5396085954  0.9931762563  1.6357808981  0.1660100489 -1.9197671483
##  [96] -0.0471242772 -0.9794780965  1.1187166662  1.3024001287  1.2050282279
## [101] -1.0176733598 -1.9131981505  0.0071667625 -0.2440440958 -0.6646035704
## [106]  0.7988753840 -0.4674611859  0.4264976703  0.8254528817 -0.2950036539
## [111] -0.3880735286 -0.5394328340 -0.7325785595  1.6826229462  0.3485131839
## [116]  0.3126310853 -1.5901616139  0.1680571839  1.3521284522 -0.3969587475
## [121] -0.5847814021 -0.6895839534  2.4857599476  0.3963806461  0.2891105285
## [126] -1.0999245518 -1.7579581368 -1.3087601851  0.3480609963  1.3887143370
## [131]  1.3048296331 -0.2885069234 -1.8812625753  0.9858435726 -1.2144176785
## [136] -0.1070404714  0.8202831314 -1.6022873669  1.8890269386  0.1780530246
## [141] -0.6369225255  1.9024855095  0.2790372440 -1.2001901969  0.1751308106
## [146]  0.2588257141  1.1948900575 -0.6288122598 -0.6017986067 -0.0204931164
## [151] -0.8200220114  0.0399833371  1.8260394484 -1.5896620101 -0.5112497988
## [156]  1.8387821459 -0.8525293504 -1.7768127697 -0.3076578303  0.5898160974
## [161] -0.1694290979  0.3123768549  0.1982460944  0.1391007201  0.7289282408
## [166]  0.5716411677  1.1841188337 -1.1147558280 -1.3002858975  1.6405578743
## [171]  1.3084310586 -1.5210911518 -1.3725525729 -1.7151189124  2.3397509573
## [176] -0.3087461430 -1.6354872600  3.9958204643 -0.3082273696 -1.9535266651
## [181] -2.8599288639  1.0130523868  2.6552796841  0.0833305745  0.1611555545
## [186]  1.3704819868 -0.1717776632 -2.1666197414  0.2062098762 -0.1748461360
## [191] -0.8812123864 -2.5214805752  0.6754990831  2.5171285691  2.4466947968
## [196] -0.6592071826 -0.8808871658 -1.5144040602  0.1768898700 -1.5637574991
## [201]  1.2189290125  1.7496205755 -0.3843296882  0.2084666822 -0.0799207269
## [206]  0.2153498412  1.3512700642 -0.0809000264 -1.5177593547 -1.7949322526
## [211]  0.8442390338  0.1012110126  1.4027525041 -1.9643370385 -0.6690152740
## [216]  1.5525576525 -0.0710888128 -0.7986615246  0.0287812991  1.1151151966
## [221]  0.8261901817 -0.0755778202  1.4722952796 -0.1320234146 -1.5457794130
## [226] -1.4789525287  3.0104223313 -0.0943784956 -2.7067706227  0.0537709205
## [231]  0.0116020728 -0.2568421479 -0.6289856730  0.6550692122  0.8325241050
## [236] -1.2753134252 -2.9936314270  1.6849114742  2.0713589711  0.7353914336
## [241] -0.1962850397  0.0331805630 -1.2986128334  0.9022447903  1.0930819317
## [246] -0.4626095773  0.8252807706 -0.2894319446 -0.7117877804 -0.0803828866
## [251]  0.1710944535  0.7245253322 -0.2340175119 -0.1537518862 -1.6729672382
## [256] -0.3138672649  2.3069927290 -1.5923930762 -0.1385518663  1.0361177008
## [261]  0.6589611672 -0.9821997485 -0.2873058590 -1.6173792004  0.9487228100
## [266]  0.2559495915 -0.1751782408 -0.6923676150  1.0032294093  1.2252645564
## [271] -0.8022068888  1.5469089105  0.1233232786 -0.4145978134  1.2676143500
## [276] -2.0400543539 -0.1605770845  1.0538010704 -0.8859746054 -0.2704276920
## [281] -1.1357403683  0.6834648941 -0.2819442091  0.8119704665 -0.7163217300
## [286]  0.1557600832 -0.4874231776 -0.0987926101 -0.9466401476  0.8970355448
## [291]  1.8657851829 -0.2879902525 -2.4307942563 -0.0239905251  2.9639940437
## [296]  1.5197083888 -2.8925847444 -0.9825717830

plot(y=yt_ma2,x=yt_2_ma)

Didapat Plot yang negatif namun dapat dilihat bahwa sebaran data mulai memadat, sehingga kita dapat mengasumsikan korelasinya meningkat di ranah korelasi negatif. Korelasinya pun juga cukup mirip dengan korelasi teoritisnya.

cor(yt_ma2,yt_2_ma)

## [1] -0.4229229

\[ \rho_2=\frac{-\theta_2}{1+(\theta_1)^2+\theta_2^2}=\frac{-0.6}{1+(0.4)^2+(0.6)^2}= -0.394\ \]

z <- -0.6/ (1+0.4^2+0.6^2)
z

## [1] -0.3947368

Scatter Plot Yt dengan Yt-3

yt_ma3 <- MA[-c(1,2,3)]
yt_ma3

##   [1] -2.9459317186  0.7167390639  1.7418246382 -1.0346371303 -0.6203694070
##   [6] -0.0009552807 -0.3362779159  0.2174936313 -0.2734866675 -0.0905836968
##  [11]  0.9739922420  1.3994628688 -0.5327584083 -1.0425917434 -0.6406203177
##  [16] -0.1660878101  3.2974211006 -0.3299428431 -1.9938222918 -0.3247264458
##  [21]  0.9302201280 -0.6132272999 -1.4464704762  1.5702698332 -0.3846350651
##  [26]  0.0494705563 -0.3156998472  1.4857599669 -0.3549429366 -1.1805813337
##  [31]  0.0678738017 -1.0029262223 -0.2152270381 -0.7355358627  0.2315942247
##  [36]  1.5501272074  0.4564159184  1.8124315967 -1.3689027055 -1.2976053034
##  [41]  0.7027084857 -0.3688720953 -0.4024044862 -0.1233084197 -0.2279500179
##  [46]  0.6413571509  0.4638728217 -1.2929944025  0.4384465524  0.2075594995
##  [51] -0.2221606049  0.9090090549  1.2369568341  1.5199808594 -1.7703139045
##  [56]  0.9265605147 -1.3361603449  0.1561661061  2.9810408136 -1.4483098967
##  [61] -2.1851240662  0.2811049100  2.1819064982 -1.8448786626  0.7570196052
##  [66]  1.4655985611 -1.0160494268 -0.9254351549 -0.6601095519 -0.1884721407
##  [71]  1.0681773835  2.0308705941 -1.3704786975 -2.5715040985 -0.0752623509
##  [76]  1.3821150410  0.0137852464 -0.2059233690  0.0913573752 -1.1976302805
##  [81]  0.4771300300  1.7431282568  0.4617881111 -0.2391854878 -1.0412961429
##  [86] -0.3604993910 -0.8762511861  0.5396085954  0.9931762563  1.6357808981
##  [91]  0.1660100489 -1.9197671483 -0.0471242772 -0.9794780965  1.1187166662
##  [96]  1.3024001287  1.2050282279 -1.0176733598 -1.9131981505  0.0071667625
## [101] -0.2440440958 -0.6646035704  0.7988753840 -0.4674611859  0.4264976703
## [106]  0.8254528817 -0.2950036539 -0.3880735286 -0.5394328340 -0.7325785595
## [111]  1.6826229462  0.3485131839  0.3126310853 -1.5901616139  0.1680571839
## [116]  1.3521284522 -0.3969587475 -0.5847814021 -0.6895839534  2.4857599476
## [121]  0.3963806461  0.2891105285 -1.0999245518 -1.7579581368 -1.3087601851
## [126]  0.3480609963  1.3887143370  1.3048296331 -0.2885069234 -1.8812625753
## [131]  0.9858435726 -1.2144176785 -0.1070404714  0.8202831314 -1.6022873669
## [136]  1.8890269386  0.1780530246 -0.6369225255  1.9024855095  0.2790372440
## [141] -1.2001901969  0.1751308106  0.2588257141  1.1948900575 -0.6288122598
## [146] -0.6017986067 -0.0204931164 -0.8200220114  0.0399833371  1.8260394484
## [151] -1.5896620101 -0.5112497988  1.8387821459 -0.8525293504 -1.7768127697
## [156] -0.3076578303  0.5898160974 -0.1694290979  0.3123768549  0.1982460944
## [161]  0.1391007201  0.7289282408  0.5716411677  1.1841188337 -1.1147558280
## [166] -1.3002858975  1.6405578743  1.3084310586 -1.5210911518 -1.3725525729
## [171] -1.7151189124  2.3397509573 -0.3087461430 -1.6354872600  3.9958204643
## [176] -0.3082273696 -1.9535266651 -2.8599288639  1.0130523868  2.6552796841
## [181]  0.0833305745  0.1611555545  1.3704819868 -0.1717776632 -2.1666197414
## [186]  0.2062098762 -0.1748461360 -0.8812123864 -2.5214805752  0.6754990831
## [191]  2.5171285691  2.4466947968 -0.6592071826 -0.8808871658 -1.5144040602
## [196]  0.1768898700 -1.5637574991  1.2189290125  1.7496205755 -0.3843296882
## [201]  0.2084666822 -0.0799207269  0.2153498412  1.3512700642 -0.0809000264
## [206] -1.5177593547 -1.7949322526  0.8442390338  0.1012110126  1.4027525041
## [211] -1.9643370385 -0.6690152740  1.5525576525 -0.0710888128 -0.7986615246
## [216]  0.0287812991  1.1151151966  0.8261901817 -0.0755778202  1.4722952796
## [221] -0.1320234146 -1.5457794130 -1.4789525287  3.0104223313 -0.0943784956
## [226] -2.7067706227  0.0537709205  0.0116020728 -0.2568421479 -0.6289856730
## [231]  0.6550692122  0.8325241050 -1.2753134252 -2.9936314270  1.6849114742
## [236]  2.0713589711  0.7353914336 -0.1962850397  0.0331805630 -1.2986128334
## [241]  0.9022447903  1.0930819317 -0.4626095773  0.8252807706 -0.2894319446
## [246] -0.7117877804 -0.0803828866  0.1710944535  0.7245253322 -0.2340175119
## [251] -0.1537518862 -1.6729672382 -0.3138672649  2.3069927290 -1.5923930762
## [256] -0.1385518663  1.0361177008  0.6589611672 -0.9821997485 -0.2873058590
## [261] -1.6173792004  0.9487228100  0.2559495915 -0.1751782408 -0.6923676150
## [266]  1.0032294093  1.2252645564 -0.8022068888  1.5469089105  0.1233232786
## [271] -0.4145978134  1.2676143500 -2.0400543539 -0.1605770845  1.0538010704
## [276] -0.8859746054 -0.2704276920 -1.1357403683  0.6834648941 -0.2819442091
## [281]  0.8119704665 -0.7163217300  0.1557600832 -0.4874231776 -0.0987926101
## [286] -0.9466401476  0.8970355448  1.8657851829 -0.2879902525 -2.4307942563
## [291] -0.0239905251  2.9639940437  1.5197083888 -2.8925847444 -0.9825717830
## [296]  1.2053288133  0.3977279966

yt_3_ma <- MA[-c(288,299,300)]
yt_3_ma

##   [1] -1.2070657494  0.2774292421  1.6977089295 -2.9459317186  0.7167390639
##   [6]  1.7418246382 -1.0346371303 -0.6203694070 -0.0009552807 -0.3362779159
##  [11]  0.2174936313 -0.2734866675 -0.0905836968  0.9739922420  1.3994628688
##  [16] -0.5327584083 -1.0425917434 -0.6406203177 -0.1660878101  3.2974211006
##  [21] -0.3299428431 -1.9938222918 -0.3247264458  0.9302201280 -0.6132272999
##  [26] -1.4464704762  1.5702698332 -0.3846350651  0.0494705563 -0.3156998472
##  [31]  1.4857599669 -0.3549429366 -1.1805813337  0.0678738017 -1.0029262223
##  [36] -0.2152270381 -0.7355358627  0.2315942247  1.5501272074  0.4564159184
##  [41]  1.8124315967 -1.3689027055 -1.2976053034  0.7027084857 -0.3688720953
##  [46] -0.4024044862 -0.1233084197 -0.2279500179  0.6413571509  0.4638728217
##  [51] -1.2929944025  0.4384465524  0.2075594995 -0.2221606049  0.9090090549
##  [56]  1.2369568341  1.5199808594 -1.7703139045  0.9265605147 -1.3361603449
##  [61]  0.1561661061  2.9810408136 -1.4483098967 -2.1851240662  0.2811049100
##  [66]  2.1819064982 -1.8448786626  0.7570196052  1.4655985611 -1.0160494268
##  [71] -0.9254351549 -0.6601095519 -0.1884721407  1.0681773835  2.0308705941
##  [76] -1.3704786975 -2.5715040985 -0.0752623509  1.3821150410  0.0137852464
##  [81] -0.2059233690  0.0913573752 -1.1976302805  0.4771300300  1.7431282568
##  [86]  0.4617881111 -0.2391854878 -1.0412961429 -0.3604993910 -0.8762511861
##  [91]  0.5396085954  0.9931762563  1.6357808981  0.1660100489 -1.9197671483
##  [96] -0.0471242772 -0.9794780965  1.1187166662  1.3024001287  1.2050282279
## [101] -1.0176733598 -1.9131981505  0.0071667625 -0.2440440958 -0.6646035704
## [106]  0.7988753840 -0.4674611859  0.4264976703  0.8254528817 -0.2950036539
## [111] -0.3880735286 -0.5394328340 -0.7325785595  1.6826229462  0.3485131839
## [116]  0.3126310853 -1.5901616139  0.1680571839  1.3521284522 -0.3969587475
## [121] -0.5847814021 -0.6895839534  2.4857599476  0.3963806461  0.2891105285
## [126] -1.0999245518 -1.7579581368 -1.3087601851  0.3480609963  1.3887143370
## [131]  1.3048296331 -0.2885069234 -1.8812625753  0.9858435726 -1.2144176785
## [136] -0.1070404714  0.8202831314 -1.6022873669  1.8890269386  0.1780530246
## [141] -0.6369225255  1.9024855095  0.2790372440 -1.2001901969  0.1751308106
## [146]  0.2588257141  1.1948900575 -0.6288122598 -0.6017986067 -0.0204931164
## [151] -0.8200220114  0.0399833371  1.8260394484 -1.5896620101 -0.5112497988
## [156]  1.8387821459 -0.8525293504 -1.7768127697 -0.3076578303  0.5898160974
## [161] -0.1694290979  0.3123768549  0.1982460944  0.1391007201  0.7289282408
## [166]  0.5716411677  1.1841188337 -1.1147558280 -1.3002858975  1.6405578743
## [171]  1.3084310586 -1.5210911518 -1.3725525729 -1.7151189124  2.3397509573
## [176] -0.3087461430 -1.6354872600  3.9958204643 -0.3082273696 -1.9535266651
## [181] -2.8599288639  1.0130523868  2.6552796841  0.0833305745  0.1611555545
## [186]  1.3704819868 -0.1717776632 -2.1666197414  0.2062098762 -0.1748461360
## [191] -0.8812123864 -2.5214805752  0.6754990831  2.5171285691  2.4466947968
## [196] -0.6592071826 -0.8808871658 -1.5144040602  0.1768898700 -1.5637574991
## [201]  1.2189290125  1.7496205755 -0.3843296882  0.2084666822 -0.0799207269
## [206]  0.2153498412  1.3512700642 -0.0809000264 -1.5177593547 -1.7949322526
## [211]  0.8442390338  0.1012110126  1.4027525041 -1.9643370385 -0.6690152740
## [216]  1.5525576525 -0.0710888128 -0.7986615246  0.0287812991  1.1151151966
## [221]  0.8261901817 -0.0755778202  1.4722952796 -0.1320234146 -1.5457794130
## [226] -1.4789525287  3.0104223313 -0.0943784956 -2.7067706227  0.0537709205
## [231]  0.0116020728 -0.2568421479 -0.6289856730  0.6550692122  0.8325241050
## [236] -1.2753134252 -2.9936314270  1.6849114742  2.0713589711  0.7353914336
## [241] -0.1962850397  0.0331805630 -1.2986128334  0.9022447903  1.0930819317
## [246] -0.4626095773  0.8252807706 -0.2894319446 -0.7117877804 -0.0803828866
## [251]  0.1710944535  0.7245253322 -0.2340175119 -0.1537518862 -1.6729672382
## [256] -0.3138672649  2.3069927290 -1.5923930762 -0.1385518663  1.0361177008
## [261]  0.6589611672 -0.9821997485 -0.2873058590 -1.6173792004  0.9487228100
## [266]  0.2559495915 -0.1751782408 -0.6923676150  1.0032294093  1.2252645564
## [271] -0.8022068888  1.5469089105  0.1233232786 -0.4145978134  1.2676143500
## [276] -2.0400543539 -0.1605770845  1.0538010704 -0.8859746054 -0.2704276920
## [281] -1.1357403683  0.6834648941 -0.2819442091  0.8119704665 -0.7163217300
## [286]  0.1557600832 -0.4874231776 -0.9466401476  0.8970355448  1.8657851829
## [291] -0.2879902525 -2.4307942563 -0.0239905251  2.9639940437  1.5197083888
## [296] -2.8925847444 -0.9825717830

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] -0.1229898

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

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 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 o o o o o o o  o  o  o 
## 1 x o o x o o o o o o o  o  o  o 
## 2 x o o x o x o o o o o  o  o  o 
## 3 x o o o o o o o o o o  o  o  o 
## 4 o o x o o o o o o o o  o  o  o 
## 5 o x o x o o o o o o o  o  o  o 
## 6 x x o x o o o o o o o  o  o  o 
## 7 o o x 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 Yt dengan Yt-1

yt_ar <- AR2[-1]
yt_ar

##   [1]  2.00000000  1.22362661  0.36278509 -0.07825636  1.64782034  0.36129909
##   [7]  1.27339037  2.18067369  1.78867982  0.90875278  0.77211073  0.07552595
##  [13]  1.41990224  0.57550274  2.12171520  0.61434562  0.08439860  0.30820058
##  [19]  0.19516866 -0.34522708 -1.71497662 -0.89646730 -1.50780567  0.14941466
##  [25] -1.17953926  0.56659603 -0.60165285  0.10496286  0.83085358 -0.08232299
##  [31]  0.67944777  0.23528562 -0.88168057 -0.66386277  1.11152238  0.20885746
##  [37] -0.49104926 -0.25777606  0.10304373  0.95529127  1.64221426  1.11268778
##  [43]  2.04936199  0.48295860 -1.69316171 -1.22167256 -1.46532411 -3.29303272
##  [49] -1.37710942 -2.13093640 -1.56694407 -2.79676230 -1.16424576  0.74975168
##  [55] -0.73605042 -0.33063379  1.63618609  1.68578262  3.08318795  1.87351644
##  [61]  1.40113576  0.56563951  1.99762061 -0.17390032  0.61988818  0.22884549
##  [67]  2.49024218  0.68285646 -0.66981150  0.03429731 -0.15646234 -0.91049678
##  [73] -0.35424977 -0.63474899 -1.06698309 -0.15960541 -0.62486163 -2.17933215
##  [79] -3.86637965 -2.94963843 -0.79390827 -0.14875244  0.98189570  1.44370281
##  [85]  1.23399457 -0.60132478  0.15170631  1.55281629 -2.58931413 -1.76544609
##  [91] -0.29812123  0.02659519  0.74306775  0.83395242  1.10442287  0.73364504
##  [97] -0.32878204 -1.24447710 -0.65184209 -0.99620958 -1.52783762 -0.54571941
## [103] -0.42498249  1.14164792 -0.63567504 -0.60729602 -0.50573010 -1.78211434
## [109] -1.27690906 -1.70304857 -3.25454510 -2.25171943 -2.31084090 -0.47275414
## [115] -1.30261417 -0.18834632 -0.21206670 -1.38056283 -0.35855078 -0.56147408
## [121] -0.18638521  0.45254743  0.30295959  1.82714845  1.08971958 -0.34370956
## [127]  0.78795392 -0.89766657 -2.21013663 -2.09453802 -2.23397357 -2.14340031
## [133]  0.08826470  1.25105173  1.36926990 -0.33898963  0.12750572  0.48307617
## [139]  0.13001173  0.33336658 -0.75210828 -1.59700113  0.45884352  0.17739422
## [145] -0.22793183  0.24678415  2.14028668  1.48216713  2.58054554  2.95425051
## [151]  1.58605822  2.14642094  0.73922129 -0.67476095 -1.39120201 -0.97927367
## [157]  1.02918519  0.42354658 -0.38522765  0.12247046  0.68216489 -0.92962519
## [163] -1.38305984 -2.81906245 -2.95570378 -2.92910319 -2.34760812 -0.34953064
## [169]  0.24435449  0.32524543 -0.35514785 -0.15273178  0.40021679 -0.33447128
## [175]  0.97187785  1.51649823 -0.21387994 -0.55055164  0.88868693 -1.35298674
## [181] -0.07904609 -0.07662163  3.14178116 -1.17411409 -0.79949661  0.03945755
## [187]  1.53415596  1.61465657 -0.31420896  2.40521501 -0.61750051 -0.94072334
## [193] -0.63700216  1.71894116  1.23838673  1.69337765  2.82244377  2.73467738
## [199]  0.70708957  1.61020648  0.83730116  2.52329977  1.18566544 -0.42961435
## [205]  0.51416028  0.52566093  0.34803618 -0.77740005 -1.15800745 -0.74710945
## [211]  0.43295393 -0.29740264 -0.93356001 -3.23046356 -1.30769394 -0.87259936
## [217]  0.61135239  1.59787897  0.77815390  2.00716647  1.16052756  0.30528700
## [223] -0.46344759 -0.51760749 -0.12401382 -0.19292981  0.07961012  0.20016177
## [229]  0.83060226  0.18215171  1.89254377  1.42039611 -0.10482904 -0.04498510
## [235]  0.47242469 -0.18384001  0.34755998  0.57140785  1.65869361  0.29193246
## [241]  0.46082409 -0.27889392 -1.17441501 -2.49195660 -1.28533225 -0.53949222
## [247]  0.11234560 -1.44171846 -1.82806796 -2.34353187 -2.22605103 -1.41153407
## [253] -0.55649381 -1.37558637 -3.66343878 -1.15301318 -0.69390958 -1.27842813
## [259] -1.85165732 -1.48150548 -1.00297764 -0.07458915  1.15220499 -0.28769873
## [265]  0.37731448  1.46529352 -0.09323250  0.02367904 -1.77827501 -0.59697944
## [271] -1.29533649 -1.69200177 -1.88504565 -0.83641981 -2.95768804 -2.07163888
## [277] -2.64107471 -2.69698416 -1.40207239 -0.88867876 -0.83177421 -0.05610560
## [283] -0.79651254 -2.43277504 -0.22126875 -0.82788238 -0.82328371 -0.41970040
## [289]  0.39142528  0.53785953  1.27448022  0.79094661  0.14608371 -0.06921823
## [295]  0.81752360 -0.45435064 -1.01728070 -0.28603176  0.26600129

yt_1_ar <- AR2[-300]
yt_1_ar

##   [1]  1.00000000  2.00000000  1.22362661  0.36278509 -0.07825636  1.64782034
##   [7]  0.36129909  1.27339037  2.18067369  1.78867982  0.90875278  0.77211073
##  [13]  0.07552595  1.41990224  0.57550274  2.12171520  0.61434562  0.08439860
##  [19]  0.30820058  0.19516866 -0.34522708 -1.71497662 -0.89646730 -1.50780567
##  [25]  0.14941466 -1.17953926  0.56659603 -0.60165285  0.10496286  0.83085358
##  [31] -0.08232299  0.67944777  0.23528562 -0.88168057 -0.66386277  1.11152238
##  [37]  0.20885746 -0.49104926 -0.25777606  0.10304373  0.95529127  1.64221426
##  [43]  1.11268778  2.04936199  0.48295860 -1.69316171 -1.22167256 -1.46532411
##  [49] -3.29303272 -1.37710942 -2.13093640 -1.56694407 -2.79676230 -1.16424576
##  [55]  0.74975168 -0.73605042 -0.33063379  1.63618609  1.68578262  3.08318795
##  [61]  1.87351644  1.40113576  0.56563951  1.99762061 -0.17390032  0.61988818
##  [67]  0.22884549  2.49024218  0.68285646 -0.66981150  0.03429731 -0.15646234
##  [73] -0.91049678 -0.35424977 -0.63474899 -1.06698309 -0.15960541 -0.62486163
##  [79] -2.17933215 -3.86637965 -2.94963843 -0.79390827 -0.14875244  0.98189570
##  [85]  1.44370281  1.23399457 -0.60132478  0.15170631  1.55281629 -2.58931413
##  [91] -1.76544609 -0.29812123  0.02659519  0.74306775  0.83395242  1.10442287
##  [97]  0.73364504 -0.32878204 -1.24447710 -0.65184209 -0.99620958 -1.52783762
## [103] -0.54571941 -0.42498249  1.14164792 -0.63567504 -0.60729602 -0.50573010
## [109] -1.78211434 -1.27690906 -1.70304857 -3.25454510 -2.25171943 -2.31084090
## [115] -0.47275414 -1.30261417 -0.18834632 -0.21206670 -1.38056283 -0.35855078
## [121] -0.56147408 -0.18638521  0.45254743  0.30295959  1.82714845  1.08971958
## [127] -0.34370956  0.78795392 -0.89766657 -2.21013663 -2.09453802 -2.23397357
## [133] -2.14340031  0.08826470  1.25105173  1.36926990 -0.33898963  0.12750572
## [139]  0.48307617  0.13001173  0.33336658 -0.75210828 -1.59700113  0.45884352
## [145]  0.17739422 -0.22793183  0.24678415  2.14028668  1.48216713  2.58054554
## [151]  2.95425051  1.58605822  2.14642094  0.73922129 -0.67476095 -1.39120201
## [157] -0.97927367  1.02918519  0.42354658 -0.38522765  0.12247046  0.68216489
## [163] -0.92962519 -1.38305984 -2.81906245 -2.95570378 -2.92910319 -2.34760812
## [169] -0.34953064  0.24435449  0.32524543 -0.35514785 -0.15273178  0.40021679
## [175] -0.33447128  0.97187785  1.51649823 -0.21387994 -0.55055164  0.88868693
## [181] -1.35298674 -0.07904609 -0.07662163  3.14178116 -1.17411409 -0.79949661
## [187]  0.03945755  1.53415596  1.61465657 -0.31420896  2.40521501 -0.61750051
## [193] -0.94072334 -0.63700216  1.71894116  1.23838673  1.69337765  2.82244377
## [199]  2.73467738  0.70708957  1.61020648  0.83730116  2.52329977  1.18566544
## [205] -0.42961435  0.51416028  0.52566093  0.34803618 -0.77740005 -1.15800745
## [211] -0.74710945  0.43295393 -0.29740264 -0.93356001 -3.23046356 -1.30769394
## [217] -0.87259936  0.61135239  1.59787897  0.77815390  2.00716647  1.16052756
## [223]  0.30528700 -0.46344759 -0.51760749 -0.12401382 -0.19292981  0.07961012
## [229]  0.20016177  0.83060226  0.18215171  1.89254377  1.42039611 -0.10482904
## [235] -0.04498510  0.47242469 -0.18384001  0.34755998  0.57140785  1.65869361
## [241]  0.29193246  0.46082409 -0.27889392 -1.17441501 -2.49195660 -1.28533225
## [247] -0.53949222  0.11234560 -1.44171846 -1.82806796 -2.34353187 -2.22605103
## [253] -1.41153407 -0.55649381 -1.37558637 -3.66343878 -1.15301318 -0.69390958
## [259] -1.27842813 -1.85165732 -1.48150548 -1.00297764 -0.07458915  1.15220499
## [265] -0.28769873  0.37731448  1.46529352 -0.09323250  0.02367904 -1.77827501
## [271] -0.59697944 -1.29533649 -1.69200177 -1.88504565 -0.83641981 -2.95768804
## [277] -2.07163888 -2.64107471 -2.69698416 -1.40207239 -0.88867876 -0.83177421
## [283] -0.05610560 -0.79651254 -2.43277504 -0.22126875 -0.82788238 -0.82328371
## [289] -0.41970040  0.39142528  0.53785953  1.27448022  0.79094661  0.14608371
## [295] -0.06921823  0.81752360 -0.45435064 -1.01728070 -0.28603176

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] 0.6242197

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

Korelasi Yt dengan Yt-2

yt_ar2 <- AR2[-c(1,2)]
yt_ar2

##   [1]  1.22362661  0.36278509 -0.07825636  1.64782034  0.36129909  1.27339037
##   [7]  2.18067369  1.78867982  0.90875278  0.77211073  0.07552595  1.41990224
##  [13]  0.57550274  2.12171520  0.61434562  0.08439860  0.30820058  0.19516866
##  [19] -0.34522708 -1.71497662 -0.89646730 -1.50780567  0.14941466 -1.17953926
##  [25]  0.56659603 -0.60165285  0.10496286  0.83085358 -0.08232299  0.67944777
##  [31]  0.23528562 -0.88168057 -0.66386277  1.11152238  0.20885746 -0.49104926
##  [37] -0.25777606  0.10304373  0.95529127  1.64221426  1.11268778  2.04936199
##  [43]  0.48295860 -1.69316171 -1.22167256 -1.46532411 -3.29303272 -1.37710942
##  [49] -2.13093640 -1.56694407 -2.79676230 -1.16424576  0.74975168 -0.73605042
##  [55] -0.33063379  1.63618609  1.68578262  3.08318795  1.87351644  1.40113576
##  [61]  0.56563951  1.99762061 -0.17390032  0.61988818  0.22884549  2.49024218
##  [67]  0.68285646 -0.66981150  0.03429731 -0.15646234 -0.91049678 -0.35424977
##  [73] -0.63474899 -1.06698309 -0.15960541 -0.62486163 -2.17933215 -3.86637965
##  [79] -2.94963843 -0.79390827 -0.14875244  0.98189570  1.44370281  1.23399457
##  [85] -0.60132478  0.15170631  1.55281629 -2.58931413 -1.76544609 -0.29812123
##  [91]  0.02659519  0.74306775  0.83395242  1.10442287  0.73364504 -0.32878204
##  [97] -1.24447710 -0.65184209 -0.99620958 -1.52783762 -0.54571941 -0.42498249
## [103]  1.14164792 -0.63567504 -0.60729602 -0.50573010 -1.78211434 -1.27690906
## [109] -1.70304857 -3.25454510 -2.25171943 -2.31084090 -0.47275414 -1.30261417
## [115] -0.18834632 -0.21206670 -1.38056283 -0.35855078 -0.56147408 -0.18638521
## [121]  0.45254743  0.30295959  1.82714845  1.08971958 -0.34370956  0.78795392
## [127] -0.89766657 -2.21013663 -2.09453802 -2.23397357 -2.14340031  0.08826470
## [133]  1.25105173  1.36926990 -0.33898963  0.12750572  0.48307617  0.13001173
## [139]  0.33336658 -0.75210828 -1.59700113  0.45884352  0.17739422 -0.22793183
## [145]  0.24678415  2.14028668  1.48216713  2.58054554  2.95425051  1.58605822
## [151]  2.14642094  0.73922129 -0.67476095 -1.39120201 -0.97927367  1.02918519
## [157]  0.42354658 -0.38522765  0.12247046  0.68216489 -0.92962519 -1.38305984
## [163] -2.81906245 -2.95570378 -2.92910319 -2.34760812 -0.34953064  0.24435449
## [169]  0.32524543 -0.35514785 -0.15273178  0.40021679 -0.33447128  0.97187785
## [175]  1.51649823 -0.21387994 -0.55055164  0.88868693 -1.35298674 -0.07904609
## [181] -0.07662163  3.14178116 -1.17411409 -0.79949661  0.03945755  1.53415596
## [187]  1.61465657 -0.31420896  2.40521501 -0.61750051 -0.94072334 -0.63700216
## [193]  1.71894116  1.23838673  1.69337765  2.82244377  2.73467738  0.70708957
## [199]  1.61020648  0.83730116  2.52329977  1.18566544 -0.42961435  0.51416028
## [205]  0.52566093  0.34803618 -0.77740005 -1.15800745 -0.74710945  0.43295393
## [211] -0.29740264 -0.93356001 -3.23046356 -1.30769394 -0.87259936  0.61135239
## [217]  1.59787897  0.77815390  2.00716647  1.16052756  0.30528700 -0.46344759
## [223] -0.51760749 -0.12401382 -0.19292981  0.07961012  0.20016177  0.83060226
## [229]  0.18215171  1.89254377  1.42039611 -0.10482904 -0.04498510  0.47242469
## [235] -0.18384001  0.34755998  0.57140785  1.65869361  0.29193246  0.46082409
## [241] -0.27889392 -1.17441501 -2.49195660 -1.28533225 -0.53949222  0.11234560
## [247] -1.44171846 -1.82806796 -2.34353187 -2.22605103 -1.41153407 -0.55649381
## [253] -1.37558637 -3.66343878 -1.15301318 -0.69390958 -1.27842813 -1.85165732
## [259] -1.48150548 -1.00297764 -0.07458915  1.15220499 -0.28769873  0.37731448
## [265]  1.46529352 -0.09323250  0.02367904 -1.77827501 -0.59697944 -1.29533649
## [271] -1.69200177 -1.88504565 -0.83641981 -2.95768804 -2.07163888 -2.64107471
## [277] -2.69698416 -1.40207239 -0.88867876 -0.83177421 -0.05610560 -0.79651254
## [283] -2.43277504 -0.22126875 -0.82788238 -0.82328371 -0.41970040  0.39142528
## [289]  0.53785953  1.27448022  0.79094661  0.14608371 -0.06921823  0.81752360
## [295] -0.45435064 -1.01728070 -0.28603176  0.26600129

yt_2_ar <- AR2[-c(299,300)]
yt_2_ar

##   [1]  1.00000000  2.00000000  1.22362661  0.36278509 -0.07825636  1.64782034
##   [7]  0.36129909  1.27339037  2.18067369  1.78867982  0.90875278  0.77211073
##  [13]  0.07552595  1.41990224  0.57550274  2.12171520  0.61434562  0.08439860
##  [19]  0.30820058  0.19516866 -0.34522708 -1.71497662 -0.89646730 -1.50780567
##  [25]  0.14941466 -1.17953926  0.56659603 -0.60165285  0.10496286  0.83085358
##  [31] -0.08232299  0.67944777  0.23528562 -0.88168057 -0.66386277  1.11152238
##  [37]  0.20885746 -0.49104926 -0.25777606  0.10304373  0.95529127  1.64221426
##  [43]  1.11268778  2.04936199  0.48295860 -1.69316171 -1.22167256 -1.46532411
##  [49] -3.29303272 -1.37710942 -2.13093640 -1.56694407 -2.79676230 -1.16424576
##  [55]  0.74975168 -0.73605042 -0.33063379  1.63618609  1.68578262  3.08318795
##  [61]  1.87351644  1.40113576  0.56563951  1.99762061 -0.17390032  0.61988818
##  [67]  0.22884549  2.49024218  0.68285646 -0.66981150  0.03429731 -0.15646234
##  [73] -0.91049678 -0.35424977 -0.63474899 -1.06698309 -0.15960541 -0.62486163
##  [79] -2.17933215 -3.86637965 -2.94963843 -0.79390827 -0.14875244  0.98189570
##  [85]  1.44370281  1.23399457 -0.60132478  0.15170631  1.55281629 -2.58931413
##  [91] -1.76544609 -0.29812123  0.02659519  0.74306775  0.83395242  1.10442287
##  [97]  0.73364504 -0.32878204 -1.24447710 -0.65184209 -0.99620958 -1.52783762
## [103] -0.54571941 -0.42498249  1.14164792 -0.63567504 -0.60729602 -0.50573010
## [109] -1.78211434 -1.27690906 -1.70304857 -3.25454510 -2.25171943 -2.31084090
## [115] -0.47275414 -1.30261417 -0.18834632 -0.21206670 -1.38056283 -0.35855078
## [121] -0.56147408 -0.18638521  0.45254743  0.30295959  1.82714845  1.08971958
## [127] -0.34370956  0.78795392 -0.89766657 -2.21013663 -2.09453802 -2.23397357
## [133] -2.14340031  0.08826470  1.25105173  1.36926990 -0.33898963  0.12750572
## [139]  0.48307617  0.13001173  0.33336658 -0.75210828 -1.59700113  0.45884352
## [145]  0.17739422 -0.22793183  0.24678415  2.14028668  1.48216713  2.58054554
## [151]  2.95425051  1.58605822  2.14642094  0.73922129 -0.67476095 -1.39120201
## [157] -0.97927367  1.02918519  0.42354658 -0.38522765  0.12247046  0.68216489
## [163] -0.92962519 -1.38305984 -2.81906245 -2.95570378 -2.92910319 -2.34760812
## [169] -0.34953064  0.24435449  0.32524543 -0.35514785 -0.15273178  0.40021679
## [175] -0.33447128  0.97187785  1.51649823 -0.21387994 -0.55055164  0.88868693
## [181] -1.35298674 -0.07904609 -0.07662163  3.14178116 -1.17411409 -0.79949661
## [187]  0.03945755  1.53415596  1.61465657 -0.31420896  2.40521501 -0.61750051
## [193] -0.94072334 -0.63700216  1.71894116  1.23838673  1.69337765  2.82244377
## [199]  2.73467738  0.70708957  1.61020648  0.83730116  2.52329977  1.18566544
## [205] -0.42961435  0.51416028  0.52566093  0.34803618 -0.77740005 -1.15800745
## [211] -0.74710945  0.43295393 -0.29740264 -0.93356001 -3.23046356 -1.30769394
## [217] -0.87259936  0.61135239  1.59787897  0.77815390  2.00716647  1.16052756
## [223]  0.30528700 -0.46344759 -0.51760749 -0.12401382 -0.19292981  0.07961012
## [229]  0.20016177  0.83060226  0.18215171  1.89254377  1.42039611 -0.10482904
## [235] -0.04498510  0.47242469 -0.18384001  0.34755998  0.57140785  1.65869361
## [241]  0.29193246  0.46082409 -0.27889392 -1.17441501 -2.49195660 -1.28533225
## [247] -0.53949222  0.11234560 -1.44171846 -1.82806796 -2.34353187 -2.22605103
## [253] -1.41153407 -0.55649381 -1.37558637 -3.66343878 -1.15301318 -0.69390958
## [259] -1.27842813 -1.85165732 -1.48150548 -1.00297764 -0.07458915  1.15220499
## [265] -0.28769873  0.37731448  1.46529352 -0.09323250  0.02367904 -1.77827501
## [271] -0.59697944 -1.29533649 -1.69200177 -1.88504565 -0.83641981 -2.95768804
## [277] -2.07163888 -2.64107471 -2.69698416 -1.40207239 -0.88867876 -0.83177421
## [283] -0.05610560 -0.79651254 -2.43277504 -0.22126875 -0.82788238 -0.82328371
## [289] -0.41970040  0.39142528  0.53785953  1.27448022  0.79094661  0.14608371
## [295] -0.06921823  0.81752360 -0.45435064 -1.01728070

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] 0.4451309

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

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]  0.36278509 -0.07825636  1.64782034  0.36129909  1.27339037  2.18067369
##   [7]  1.78867982  0.90875278  0.77211073  0.07552595  1.41990224  0.57550274
##  [13]  2.12171520  0.61434562  0.08439860  0.30820058  0.19516866 -0.34522708
##  [19] -1.71497662 -0.89646730 -1.50780567  0.14941466 -1.17953926  0.56659603
##  [25] -0.60165285  0.10496286  0.83085358 -0.08232299  0.67944777  0.23528562
##  [31] -0.88168057 -0.66386277  1.11152238  0.20885746 -0.49104926 -0.25777606
##  [37]  0.10304373  0.95529127  1.64221426  1.11268778  2.04936199  0.48295860
##  [43] -1.69316171 -1.22167256 -1.46532411 -3.29303272 -1.37710942 -2.13093640
##  [49] -1.56694407 -2.79676230 -1.16424576  0.74975168 -0.73605042 -0.33063379
##  [55]  1.63618609  1.68578262  3.08318795  1.87351644  1.40113576  0.56563951
##  [61]  1.99762061 -0.17390032  0.61988818  0.22884549  2.49024218  0.68285646
##  [67] -0.66981150  0.03429731 -0.15646234 -0.91049678 -0.35424977 -0.63474899
##  [73] -1.06698309 -0.15960541 -0.62486163 -2.17933215 -3.86637965 -2.94963843
##  [79] -0.79390827 -0.14875244  0.98189570  1.44370281  1.23399457 -0.60132478
##  [85]  0.15170631  1.55281629 -2.58931413 -1.76544609 -0.29812123  0.02659519
##  [91]  0.74306775  0.83395242  1.10442287  0.73364504 -0.32878204 -1.24447710
##  [97] -0.65184209 -0.99620958 -1.52783762 -0.54571941 -0.42498249  1.14164792
## [103] -0.63567504 -0.60729602 -0.50573010 -1.78211434 -1.27690906 -1.70304857
## [109] -3.25454510 -2.25171943 -2.31084090 -0.47275414 -1.30261417 -0.18834632
## [115] -0.21206670 -1.38056283 -0.35855078 -0.56147408 -0.18638521  0.45254743
## [121]  0.30295959  1.82714845  1.08971958 -0.34370956  0.78795392 -0.89766657
## [127] -2.21013663 -2.09453802 -2.23397357 -2.14340031  0.08826470  1.25105173
## [133]  1.36926990 -0.33898963  0.12750572  0.48307617  0.13001173  0.33336658
## [139] -0.75210828 -1.59700113  0.45884352  0.17739422 -0.22793183  0.24678415
## [145]  2.14028668  1.48216713  2.58054554  2.95425051  1.58605822  2.14642094
## [151]  0.73922129 -0.67476095 -1.39120201 -0.97927367  1.02918519  0.42354658
## [157] -0.38522765  0.12247046  0.68216489 -0.92962519 -1.38305984 -2.81906245
## [163] -2.95570378 -2.92910319 -2.34760812 -0.34953064  0.24435449  0.32524543
## [169] -0.35514785 -0.15273178  0.40021679 -0.33447128  0.97187785  1.51649823
## [175] -0.21387994 -0.55055164  0.88868693 -1.35298674 -0.07904609 -0.07662163
## [181]  3.14178116 -1.17411409 -0.79949661  0.03945755  1.53415596  1.61465657
## [187] -0.31420896  2.40521501 -0.61750051 -0.94072334 -0.63700216  1.71894116
## [193]  1.23838673  1.69337765  2.82244377  2.73467738  0.70708957  1.61020648
## [199]  0.83730116  2.52329977  1.18566544 -0.42961435  0.51416028  0.52566093
## [205]  0.34803618 -0.77740005 -1.15800745 -0.74710945  0.43295393 -0.29740264
## [211] -0.93356001 -3.23046356 -1.30769394 -0.87259936  0.61135239  1.59787897
## [217]  0.77815390  2.00716647  1.16052756  0.30528700 -0.46344759 -0.51760749
## [223] -0.12401382 -0.19292981  0.07961012  0.20016177  0.83060226  0.18215171
## [229]  1.89254377  1.42039611 -0.10482904 -0.04498510  0.47242469 -0.18384001
## [235]  0.34755998  0.57140785  1.65869361  0.29193246  0.46082409 -0.27889392
## [241] -1.17441501 -2.49195660 -1.28533225 -0.53949222  0.11234560 -1.44171846
## [247] -1.82806796 -2.34353187 -2.22605103 -1.41153407 -0.55649381 -1.37558637
## [253] -3.66343878 -1.15301318 -0.69390958 -1.27842813 -1.85165732 -1.48150548
## [259] -1.00297764 -0.07458915  1.15220499 -0.28769873  0.37731448  1.46529352
## [265] -0.09323250  0.02367904 -1.77827501 -0.59697944 -1.29533649 -1.69200177
## [271] -1.88504565 -0.83641981 -2.95768804 -2.07163888 -2.64107471 -2.69698416
## [277] -1.40207239 -0.88867876 -0.83177421 -0.05610560 -0.79651254 -2.43277504
## [283] -0.22126875 -0.82788238 -0.82328371 -0.41970040  0.39142528  0.53785953
## [289]  1.27448022  0.79094661  0.14608371 -0.06921823  0.81752360 -0.45435064
## [295] -1.01728070 -0.28603176  0.26600129

#Yt-3
yt_3_ar <- AR2[-c(288,299,300)]
yt_3_ar

##   [1]  1.00000000  2.00000000  1.22362661  0.36278509 -0.07825636  1.64782034
##   [7]  0.36129909  1.27339037  2.18067369  1.78867982  0.90875278  0.77211073
##  [13]  0.07552595  1.41990224  0.57550274  2.12171520  0.61434562  0.08439860
##  [19]  0.30820058  0.19516866 -0.34522708 -1.71497662 -0.89646730 -1.50780567
##  [25]  0.14941466 -1.17953926  0.56659603 -0.60165285  0.10496286  0.83085358
##  [31] -0.08232299  0.67944777  0.23528562 -0.88168057 -0.66386277  1.11152238
##  [37]  0.20885746 -0.49104926 -0.25777606  0.10304373  0.95529127  1.64221426
##  [43]  1.11268778  2.04936199  0.48295860 -1.69316171 -1.22167256 -1.46532411
##  [49] -3.29303272 -1.37710942 -2.13093640 -1.56694407 -2.79676230 -1.16424576
##  [55]  0.74975168 -0.73605042 -0.33063379  1.63618609  1.68578262  3.08318795
##  [61]  1.87351644  1.40113576  0.56563951  1.99762061 -0.17390032  0.61988818
##  [67]  0.22884549  2.49024218  0.68285646 -0.66981150  0.03429731 -0.15646234
##  [73] -0.91049678 -0.35424977 -0.63474899 -1.06698309 -0.15960541 -0.62486163
##  [79] -2.17933215 -3.86637965 -2.94963843 -0.79390827 -0.14875244  0.98189570
##  [85]  1.44370281  1.23399457 -0.60132478  0.15170631  1.55281629 -2.58931413
##  [91] -1.76544609 -0.29812123  0.02659519  0.74306775  0.83395242  1.10442287
##  [97]  0.73364504 -0.32878204 -1.24447710 -0.65184209 -0.99620958 -1.52783762
## [103] -0.54571941 -0.42498249  1.14164792 -0.63567504 -0.60729602 -0.50573010
## [109] -1.78211434 -1.27690906 -1.70304857 -3.25454510 -2.25171943 -2.31084090
## [115] -0.47275414 -1.30261417 -0.18834632 -0.21206670 -1.38056283 -0.35855078
## [121] -0.56147408 -0.18638521  0.45254743  0.30295959  1.82714845  1.08971958
## [127] -0.34370956  0.78795392 -0.89766657 -2.21013663 -2.09453802 -2.23397357
## [133] -2.14340031  0.08826470  1.25105173  1.36926990 -0.33898963  0.12750572
## [139]  0.48307617  0.13001173  0.33336658 -0.75210828 -1.59700113  0.45884352
## [145]  0.17739422 -0.22793183  0.24678415  2.14028668  1.48216713  2.58054554
## [151]  2.95425051  1.58605822  2.14642094  0.73922129 -0.67476095 -1.39120201
## [157] -0.97927367  1.02918519  0.42354658 -0.38522765  0.12247046  0.68216489
## [163] -0.92962519 -1.38305984 -2.81906245 -2.95570378 -2.92910319 -2.34760812
## [169] -0.34953064  0.24435449  0.32524543 -0.35514785 -0.15273178  0.40021679
## [175] -0.33447128  0.97187785  1.51649823 -0.21387994 -0.55055164  0.88868693
## [181] -1.35298674 -0.07904609 -0.07662163  3.14178116 -1.17411409 -0.79949661
## [187]  0.03945755  1.53415596  1.61465657 -0.31420896  2.40521501 -0.61750051
## [193] -0.94072334 -0.63700216  1.71894116  1.23838673  1.69337765  2.82244377
## [199]  2.73467738  0.70708957  1.61020648  0.83730116  2.52329977  1.18566544
## [205] -0.42961435  0.51416028  0.52566093  0.34803618 -0.77740005 -1.15800745
## [211] -0.74710945  0.43295393 -0.29740264 -0.93356001 -3.23046356 -1.30769394
## [217] -0.87259936  0.61135239  1.59787897  0.77815390  2.00716647  1.16052756
## [223]  0.30528700 -0.46344759 -0.51760749 -0.12401382 -0.19292981  0.07961012
## [229]  0.20016177  0.83060226  0.18215171  1.89254377  1.42039611 -0.10482904
## [235] -0.04498510  0.47242469 -0.18384001  0.34755998  0.57140785  1.65869361
## [241]  0.29193246  0.46082409 -0.27889392 -1.17441501 -2.49195660 -1.28533225
## [247] -0.53949222  0.11234560 -1.44171846 -1.82806796 -2.34353187 -2.22605103
## [253] -1.41153407 -0.55649381 -1.37558637 -3.66343878 -1.15301318 -0.69390958
## [259] -1.27842813 -1.85165732 -1.48150548 -1.00297764 -0.07458915  1.15220499
## [265] -0.28769873  0.37731448  1.46529352 -0.09323250  0.02367904 -1.77827501
## [271] -0.59697944 -1.29533649 -1.69200177 -1.88504565 -0.83641981 -2.95768804
## [277] -2.07163888 -2.64107471 -2.69698416 -1.40207239 -0.88867876 -0.83177421
## [283] -0.05610560 -0.79651254 -2.43277504 -0.22126875 -0.82788238 -0.41970040
## [289]  0.39142528  0.53785953  1.27448022  0.79094661  0.14608371 -0.06921823
## [295]  0.81752360 -0.45435064 -1.01728070

plot(y=yt_ar2,x=yt_3_ar)

cor(yt_ar2,yt_3_ar)

## [1] 0.3164807

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

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

x <- 0.5*0.5125+0.2*0.625
x

## [1] 0.38125

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] -2.75189059 -1.74814288 -2.98064692 -1.12111691 -1.75293106 -1.33898765
##   [7] -2.09046996 -3.56829323 -4.32225209 -6.49932813 -7.02769509 -6.95242810
##  [13] -6.26996407 -3.43890672 -3.74182891 -2.97181976 -3.49823015 -3.96328709
##  [19] -4.21591372 -4.99194225 -5.61517562 -5.49498965 -5.32810268 -6.08211738
##  [25] -5.70927763 -6.49640104 -6.19781946 -5.63181804 -4.16631811 -1.43386858
##  [31] -1.32659085  1.33518960 -0.57718012  1.13545868  2.56923465  2.89049818
##  [37]  2.80495292  1.68426168  2.77538382  1.29220743  3.17981524  3.04188012
##  [43]  3.84589813  3.47054592  2.25362466  1.27634579  1.31729358  3.02358708
##  [49]  2.83893413  1.81428668  0.10302896 -0.45121970 -0.85290753 -0.66635425
##  [55] -0.93510416 -2.14779642 -2.08562341 -1.51503473 -0.24121274  0.91556687
##  [61]  0.64537308  0.48311177 -1.14217407 -1.12039092 -1.27141453  0.91636167
##  [67]  2.04091439  2.13232979  2.23257264  0.25301429  1.21071226  1.29939237
##  [73]  3.92904462  4.07112121  3.78513073  2.83160983  1.41191959  0.28488816
##  [79] -0.04006883 -1.28812485 -0.74220308 -0.74424287 -0.32976863 -0.31752860
##  [85] -0.98342470 -2.04217397 -1.20184647 -1.31314616  0.45270778 -0.93468943
##  [91] -0.20680752 -0.29613748  0.15497575  0.22025744 -0.73310547  1.30806278
##  [97]  1.61203256  4.42693491  3.79591780  3.27848542  0.68061022 -0.62425847
## [103] -1.11211846  0.04164014  0.59240357 -0.15810424  0.39101368 -1.92189820
## [109] -1.50741039 -2.59060462 -3.89098912 -2.51252738 -3.45825773 -2.26278511
## [115] -0.93523383  0.07413764  0.83334886  1.28389670  1.34833602  3.06177019
## [121]  2.98115137  3.71727357  3.17046413  2.39423349  1.98646516  2.92352912
## [127]  1.69753466  2.18824295  2.31910758  1.98048956  1.12411345 -0.47375475
## [133] -1.38212484 -2.31108442 -2.26562282 -2.62211380 -2.26938378 -1.57393069
## [139] -0.56321701  1.57589801  1.65267412  1.80012273  2.53204975  3.72627716
## [145]  3.97712776  3.44105725  0.58660225  1.46428641 -0.51695507 -0.57765858
## [151]  1.69949100  1.51248377  2.98314953 -0.81043866 -1.02122299 -1.37591544
## [157] -0.14779506  1.31440815  3.22462748  4.38519006  3.98666624  4.07252161
## [163]  2.61160764  1.92537895 -1.70502573 -2.72222986 -3.24361938 -0.27688538
## [169]  0.87342985  2.50264542  1.00860632  1.15325179 -1.81437043 -0.92501023
## [175] -1.18146222 -0.02037563  0.94652009  1.17246770  2.08111275  3.60870710
## [181]  4.52620181  4.56792960  2.75523948  2.28308227  0.79090755  2.43630393
## [187]  0.12650863  0.38802680 -0.48922578 -0.39939179 -0.71118176 -1.07369267
## [193] -0.53512569  0.48185240  1.52990415  3.93599432  4.75299655  5.28489601
## [199]  3.75245965  5.76106696  5.08023491  5.03046078  3.85049062  2.27094879
## [205]  1.61300623  0.02267186 -0.08744134 -0.23338993 -1.01395405 -4.02632785
## [211] -4.41061552 -5.02289664 -2.94041531 -2.16023161 -0.87458164 -1.76626866
## [217] -0.53794493 -0.27753067  0.71955739  2.19354067  2.33905070  2.84874980
## [223]  2.50865574  2.52684453  3.16087589  3.22420516  3.72442687  1.90727460
## [229]  1.27139490  1.66409882  0.23091397  1.10296539  0.37155340  1.69901525
## [235]  1.20826458  1.54849208 -0.89323198 -0.40844829 -1.97195019 -1.36977976
## [241] -2.76244287 -1.88853615 -1.38886217 -0.87915921  1.38576118  1.73484421
## [247]  3.45643730  4.90637915  3.88230417  4.70024354  4.04594276  3.83665310
## [253]  3.63604751  1.69745094  1.83280517  0.13589632  1.24990445 -0.25440835
## [259]  0.60581161 -0.95359799 -0.74837042 -2.72398269 -2.15183291 -1.51561139
## [265] -0.53368698 -1.67162994 -2.15962923 -1.06604857  1.13495225  0.89677762
## [271]  0.92454452  0.14354987  0.49726828  0.03722489 -0.87570167 -1.33912012
## [277] -0.47863083 -0.18724669 -0.22304216 -0.89877996  0.52922646 -0.01897051
## [283]  1.64938777  2.32971329  2.98499263  2.59721428  1.95311423  0.73468623
## [289]  1.76477012  1.07548761  3.11749005  1.74249123  1.55297139  0.67230168
## [295]  0.33991628 -0.10447830 -1.75292012 -1.80251894 -2.90523171 -0.99909951

Karakteristik ARMA

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

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 . Untuk Model ARMA Manual / Bangkitan, Bila grafik di jadikan ke satu sumbu, maka polanya tetap akan taiils off.

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 o o x x o o o o o  o  o  o 
## 1 x x o o o x o o o o o  x  o  o 
## 2 x x o o o x o o o o o  o  o  o 
## 3 x x x x o x o o o o o  o  o  o 
## 4 x o o x o o o o o o o  o  o  o 
## 5 x x o x x o o o o o o  o  o  o 
## 6 x x x o x o x o o o o  o  o  o 
## 7 x x x x x o x o o o o  o  o  o

Pola Segitiga 0 dapat dilihat dari Model Sesuai dengan Ordo AR (1) 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  x  o 
## 1 o x o o o o x o o o o  o  o  o 
## 2 x x o o o o o o o o o  o  o  o 
## 3 x x x x o o o o o o o  o  o  o 
## 4 x x x x o o o o o o o  o  o  o 
## 5 x x x x x x x o o o o  o  o  o 
## 6 o o x o x x o o o o o  o  o  o 
## 7 x o x x x x o x o o o  o  o  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.

Yt dengan Yt-1

yt_arma <- y.arma[-1]
yt_arma

##   [1]  0.000000000 -0.272639519 -1.400683554 -1.013808584  1.418517113
##   [6] -0.283595036  0.115231693  1.435520708  0.137878065 -1.126175869
##  [11] -0.673024142 -0.784993441  0.921528532 -0.041506854  1.036726819
##  [16] -0.581811711 -1.436022758 -0.095427141  0.020288013 -0.608947701
##  [21] -1.694545637 -0.003766299 -0.120557458  1.290169793 -0.334810413
##  [26]  0.948619090 -0.120677374  0.005582779  1.149796418 -0.477690729
##  [31]  0.213827780  0.012872066 -1.383485003 -0.452378325  1.906063316
##  [36]  0.162556635 -1.241512939 -0.186676377  0.500779481  1.068736190
##  [41]  1.198269057 -0.117374555  0.618956894 -1.004399949 -3.115963169
##  [46] -0.834183669  0.039241455 -1.973899907  0.819297849  0.395726792
##  [51]  0.111695981 -0.891422961  0.894625505  2.893507289 -0.337403693
##  [56] -0.486064666  2.210069824  1.229688432  1.427163230 -0.371228324
##  [61] -1.198183595 -1.118924665  0.930683344 -1.312332280 -0.509124065
##  [66]  0.085230411  2.026771077 -0.450547708 -2.437099397 -0.107491966
##  [71]  0.231705636 -0.868490226  0.103826347  0.053248987 -0.600533633
##  [76]  0.648037217  0.079170382 -1.833624250 -2.619729810 -0.095487277
##  [81]  2.705774888  1.938593924  1.517741645  1.140195990  0.067376023
##  [86] -1.961144294 -0.348160518  1.852928634 -3.301464430 -1.661410213
##  [91]  1.961645683  1.205111333  0.911302414  0.520768206  0.325001257
##  [96] -0.208495557 -1.284893781 -1.553151312  0.043217974  0.011213518
## [101] -0.738248530  0.663141388  0.710007843  1.639072559 -0.837344710
## [106] -1.038014754  0.118593324 -1.215444685 -0.260625260 -0.123016349
## [111] -1.807180236  0.071927759  0.542573933  1.802613875  0.272992022
## [116]  0.616351834  0.644840330 -1.182728358  0.320914370  0.410283930
## [121]  0.253334891  0.863985959  0.233771747  1.434436152  0.177084448
## [126] -1.875886459  0.271605991 -1.006622400 -2.323842355 -0.671883426
## [131] -0.070076378  0.006911929  2.286008961  2.501786040  0.815890391
## [136] -1.637328628 -0.558460368  0.635467657 -0.139722175 -0.008483813
## [141] -0.963461947 -1.496177766  1.548908938  0.952057493 -0.574195627
## [146]  0.231520348  2.178332119  0.477981964  0.703506684  1.032732018
## [151] -1.143969314 -0.260552652 -1.070982016 -2.258302027 -1.564830408
## [156] -0.017936301  2.255615866  0.599436706 -1.172157392  0.022433568
## [161]  0.864313297 -1.275973425 -1.420508694 -1.708063396 -0.998242903
## [166] -0.055384210  0.597455432  2.346974518  1.792731612  0.437222026
## [171] -0.631858717 -0.205819896  0.674398208 -0.402918932  0.865536287
## [176]  1.328429864 -1.403605944 -1.374898597  1.237235548 -1.378130534
## [181] -0.071063550  0.766788850  3.219857471 -2.384853572 -2.214919675
## [186]  1.063724642  1.998070909  0.977319662 -1.880565162  1.562104644
## [191] -1.391061141 -2.136852141  0.109787480  2.538176030  0.933011569
## [196]  0.166658262  1.402060666  0.589673285 -2.080247644 -0.313435778
## [201] -0.231035172  1.222255420 -0.326035167 -2.417860387 -0.025393239
## [206]  0.577765429 -0.170724368 -1.232011078 -1.055869141  0.182533557
## [211]  1.426602183 -0.022318540 -1.074371308 -2.678597970  0.544627484
## [216]  1.588756353  1.745008502  1.876897625 -0.227809127  0.737177529
## [221] -0.109231364 -1.363223907 -1.281878921 -0.515400658  0.361097728
## [226]  0.167240212  0.231190338  0.284075609  0.702771485 -0.270186256
## [231]  1.321321723  0.554087579 -1.808513746 -0.855291153  0.553316157
## [236] -0.345818822  0.137641172  0.542687862  1.221594475 -0.714389691
## [241] -0.651165058 -0.638383031 -1.339351898 -1.854854245  0.416099399
## [246]  1.469814641  1.099341837 -1.162961370 -1.318787940 -0.747273607
## [251] -0.191797501  0.885005463  1.343750431 -0.306068404 -2.779307949
## [256]  1.137714152  1.965358966 -0.309056384 -0.923940323  0.026214322
## [261]  0.700618943  1.215505192  1.783827238 -0.703827238 -0.198929025
## [266]  1.486986975 -0.905738593 -0.818204073 -1.731807125  0.100123132
## [271]  0.010420296 -0.815679513 -0.431043052  0.932799518 -1.492092722
## [276] -0.386711780 -0.037806332 -0.397570947  1.261366094  1.290340693
## [281]  0.364940725  0.809811339 -0.275005773 -2.080506659  1.229748794
## [286]  0.720290142 -0.359369508  0.406342510  1.053275666  0.633109654
## [291]  0.824481240 -0.041561196 -0.934983070 -0.602219681  0.757560673
## [296] -0.739829142 -1.326054601  0.393490903  0.990782410

yt_1_arma <- y.arma[-300]
yt_1_arma

##   [1]  0.000000000  0.000000000 -0.272639519 -1.400683554 -1.013808584
##   [6]  1.418517113 -0.283595036  0.115231693  1.435520708  0.137878065
##  [11] -1.126175869 -0.673024142 -0.784993441  0.921528532 -0.041506854
##  [16]  1.036726819 -0.581811711 -1.436022758 -0.095427141  0.020288013
##  [21] -0.608947701 -1.694545637 -0.003766299 -0.120557458  1.290169793
##  [26] -0.334810413  0.948619090 -0.120677374  0.005582779  1.149796418
##  [31] -0.477690729  0.213827780  0.012872066 -1.383485003 -0.452378325
##  [36]  1.906063316  0.162556635 -1.241512939 -0.186676377  0.500779481
##  [41]  1.068736190  1.198269057 -0.117374555  0.618956894 -1.004399949
##  [46] -3.115963169 -0.834183669  0.039241455 -1.973899907  0.819297849
##  [51]  0.395726792  0.111695981 -0.891422961  0.894625505  2.893507289
##  [56] -0.337403693 -0.486064666  2.210069824  1.229688432  1.427163230
##  [61] -0.371228324 -1.198183595 -1.118924665  0.930683344 -1.312332280
##  [66] -0.509124065  0.085230411  2.026771077 -0.450547708 -2.437099397
##  [71] -0.107491966  0.231705636 -0.868490226  0.103826347  0.053248987
##  [76] -0.600533633  0.648037217  0.079170382 -1.833624250 -2.619729810
##  [81] -0.095487277  2.705774888  1.938593924  1.517741645  1.140195990
##  [86]  0.067376023 -1.961144294 -0.348160518  1.852928634 -3.301464430
##  [91] -1.661410213  1.961645683  1.205111333  0.911302414  0.520768206
##  [96]  0.325001257 -0.208495557 -1.284893781 -1.553151312  0.043217974
## [101]  0.011213518 -0.738248530  0.663141388  0.710007843  1.639072559
## [106] -0.837344710 -1.038014754  0.118593324 -1.215444685 -0.260625260
## [111] -0.123016349 -1.807180236  0.071927759  0.542573933  1.802613875
## [116]  0.272992022  0.616351834  0.644840330 -1.182728358  0.320914370
## [121]  0.410283930  0.253334891  0.863985959  0.233771747  1.434436152
## [126]  0.177084448 -1.875886459  0.271605991 -1.006622400 -2.323842355
## [131] -0.671883426 -0.070076378  0.006911929  2.286008961  2.501786040
## [136]  0.815890391 -1.637328628 -0.558460368  0.635467657 -0.139722175
## [141] -0.008483813 -0.963461947 -1.496177766  1.548908938  0.952057493
## [146] -0.574195627  0.231520348  2.178332119  0.477981964  0.703506684
## [151]  1.032732018 -1.143969314 -0.260552652 -1.070982016 -2.258302027
## [156] -1.564830408 -0.017936301  2.255615866  0.599436706 -1.172157392
## [161]  0.022433568  0.864313297 -1.275973425 -1.420508694 -1.708063396
## [166] -0.998242903 -0.055384210  0.597455432  2.346974518  1.792731612
## [171]  0.437222026 -0.631858717 -0.205819896  0.674398208 -0.402918932
## [176]  0.865536287  1.328429864 -1.403605944 -1.374898597  1.237235548
## [181] -1.378130534 -0.071063550  0.766788850  3.219857471 -2.384853572
## [186] -2.214919675  1.063724642  1.998070909  0.977319662 -1.880565162
## [191]  1.562104644 -1.391061141 -2.136852141  0.109787480  2.538176030
## [196]  0.933011569  0.166658262  1.402060666  0.589673285 -2.080247644
## [201] -0.313435778 -0.231035172  1.222255420 -0.326035167 -2.417860387
## [206] -0.025393239  0.577765429 -0.170724368 -1.232011078 -1.055869141
## [211]  0.182533557  1.426602183 -0.022318540 -1.074371308 -2.678597970
## [216]  0.544627484  1.588756353  1.745008502  1.876897625 -0.227809127
## [221]  0.737177529 -0.109231364 -1.363223907 -1.281878921 -0.515400658
## [226]  0.361097728  0.167240212  0.231190338  0.284075609  0.702771485
## [231] -0.270186256  1.321321723  0.554087579 -1.808513746 -0.855291153
## [236]  0.553316157 -0.345818822  0.137641172  0.542687862  1.221594475
## [241] -0.714389691 -0.651165058 -0.638383031 -1.339351898 -1.854854245
## [246]  0.416099399  1.469814641  1.099341837 -1.162961370 -1.318787940
## [251] -0.747273607 -0.191797501  0.885005463  1.343750431 -0.306068404
## [256] -2.779307949  1.137714152  1.965358966 -0.309056384 -0.923940323
## [261]  0.026214322  0.700618943  1.215505192  1.783827238 -0.703827238
## [266] -0.198929025  1.486986975 -0.905738593 -0.818204073 -1.731807125
## [271]  0.100123132  0.010420296 -0.815679513 -0.431043052  0.932799518
## [276] -1.492092722 -0.386711780 -0.037806332 -0.397570947  1.261366094
## [281]  1.290340693  0.364940725  0.809811339 -0.275005773 -2.080506659
## [286]  1.229748794  0.720290142 -0.359369508  0.406342510  1.053275666
## [291]  0.633109654  0.824481240 -0.041561196 -0.934983070 -0.602219681
## [296]  0.757560673 -0.739829142 -1.326054601  0.393490903

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] 0.1947432

Korelasi Antara Yt dan Yt-2

yt_arma <- y.arma[-c(1,2)]
yt_arma

##   [1] -0.272639519 -1.400683554 -1.013808584  1.418517113 -0.283595036
##   [6]  0.115231693  1.435520708  0.137878065 -1.126175869 -0.673024142
##  [11] -0.784993441  0.921528532 -0.041506854  1.036726819 -0.581811711
##  [16] -1.436022758 -0.095427141  0.020288013 -0.608947701 -1.694545637
##  [21] -0.003766299 -0.120557458  1.290169793 -0.334810413  0.948619090
##  [26] -0.120677374  0.005582779  1.149796418 -0.477690729  0.213827780
##  [31]  0.012872066 -1.383485003 -0.452378325  1.906063316  0.162556635
##  [36] -1.241512939 -0.186676377  0.500779481  1.068736190  1.198269057
##  [41] -0.117374555  0.618956894 -1.004399949 -3.115963169 -0.834183669
##  [46]  0.039241455 -1.973899907  0.819297849  0.395726792  0.111695981
##  [51] -0.891422961  0.894625505  2.893507289 -0.337403693 -0.486064666
##  [56]  2.210069824  1.229688432  1.427163230 -0.371228324 -1.198183595
##  [61] -1.118924665  0.930683344 -1.312332280 -0.509124065  0.085230411
##  [66]  2.026771077 -0.450547708 -2.437099397 -0.107491966  0.231705636
##  [71] -0.868490226  0.103826347  0.053248987 -0.600533633  0.648037217
##  [76]  0.079170382 -1.833624250 -2.619729810 -0.095487277  2.705774888
##  [81]  1.938593924  1.517741645  1.140195990  0.067376023 -1.961144294
##  [86] -0.348160518  1.852928634 -3.301464430 -1.661410213  1.961645683
##  [91]  1.205111333  0.911302414  0.520768206  0.325001257 -0.208495557
##  [96] -1.284893781 -1.553151312  0.043217974  0.011213518 -0.738248530
## [101]  0.663141388  0.710007843  1.639072559 -0.837344710 -1.038014754
## [106]  0.118593324 -1.215444685 -0.260625260 -0.123016349 -1.807180236
## [111]  0.071927759  0.542573933  1.802613875  0.272992022  0.616351834
## [116]  0.644840330 -1.182728358  0.320914370  0.410283930  0.253334891
## [121]  0.863985959  0.233771747  1.434436152  0.177084448 -1.875886459
## [126]  0.271605991 -1.006622400 -2.323842355 -0.671883426 -0.070076378
## [131]  0.006911929  2.286008961  2.501786040  0.815890391 -1.637328628
## [136] -0.558460368  0.635467657 -0.139722175 -0.008483813 -0.963461947
## [141] -1.496177766  1.548908938  0.952057493 -0.574195627  0.231520348
## [146]  2.178332119  0.477981964  0.703506684  1.032732018 -1.143969314
## [151] -0.260552652 -1.070982016 -2.258302027 -1.564830408 -0.017936301
## [156]  2.255615866  0.599436706 -1.172157392  0.022433568  0.864313297
## [161] -1.275973425 -1.420508694 -1.708063396 -0.998242903 -0.055384210
## [166]  0.597455432  2.346974518  1.792731612  0.437222026 -0.631858717
## [171] -0.205819896  0.674398208 -0.402918932  0.865536287  1.328429864
## [176] -1.403605944 -1.374898597  1.237235548 -1.378130534 -0.071063550
## [181]  0.766788850  3.219857471 -2.384853572 -2.214919675  1.063724642
## [186]  1.998070909  0.977319662 -1.880565162  1.562104644 -1.391061141
## [191] -2.136852141  0.109787480  2.538176030  0.933011569  0.166658262
## [196]  1.402060666  0.589673285 -2.080247644 -0.313435778 -0.231035172
## [201]  1.222255420 -0.326035167 -2.417860387 -0.025393239  0.577765429
## [206] -0.170724368 -1.232011078 -1.055869141  0.182533557  1.426602183
## [211] -0.022318540 -1.074371308 -2.678597970  0.544627484  1.588756353
## [216]  1.745008502  1.876897625 -0.227809127  0.737177529 -0.109231364
## [221] -1.363223907 -1.281878921 -0.515400658  0.361097728  0.167240212
## [226]  0.231190338  0.284075609  0.702771485 -0.270186256  1.321321723
## [231]  0.554087579 -1.808513746 -0.855291153  0.553316157 -0.345818822
## [236]  0.137641172  0.542687862  1.221594475 -0.714389691 -0.651165058
## [241] -0.638383031 -1.339351898 -1.854854245  0.416099399  1.469814641
## [246]  1.099341837 -1.162961370 -1.318787940 -0.747273607 -0.191797501
## [251]  0.885005463  1.343750431 -0.306068404 -2.779307949  1.137714152
## [256]  1.965358966 -0.309056384 -0.923940323  0.026214322  0.700618943
## [261]  1.215505192  1.783827238 -0.703827238 -0.198929025  1.486986975
## [266] -0.905738593 -0.818204073 -1.731807125  0.100123132  0.010420296
## [271] -0.815679513 -0.431043052  0.932799518 -1.492092722 -0.386711780
## [276] -0.037806332 -0.397570947  1.261366094  1.290340693  0.364940725
## [281]  0.809811339 -0.275005773 -2.080506659  1.229748794  0.720290142
## [286] -0.359369508  0.406342510  1.053275666  0.633109654  0.824481240
## [291] -0.041561196 -0.934983070 -0.602219681  0.757560673 -0.739829142
## [296] -1.326054601  0.393490903  0.990782410

yt_2_arma <- y.arma[-c(299,300)]
yt_2_arma

##   [1]  0.000000000  0.000000000 -0.272639519 -1.400683554 -1.013808584
##   [6]  1.418517113 -0.283595036  0.115231693  1.435520708  0.137878065
##  [11] -1.126175869 -0.673024142 -0.784993441  0.921528532 -0.041506854
##  [16]  1.036726819 -0.581811711 -1.436022758 -0.095427141  0.020288013
##  [21] -0.608947701 -1.694545637 -0.003766299 -0.120557458  1.290169793
##  [26] -0.334810413  0.948619090 -0.120677374  0.005582779  1.149796418
##  [31] -0.477690729  0.213827780  0.012872066 -1.383485003 -0.452378325
##  [36]  1.906063316  0.162556635 -1.241512939 -0.186676377  0.500779481
##  [41]  1.068736190  1.198269057 -0.117374555  0.618956894 -1.004399949
##  [46] -3.115963169 -0.834183669  0.039241455 -1.973899907  0.819297849
##  [51]  0.395726792  0.111695981 -0.891422961  0.894625505  2.893507289
##  [56] -0.337403693 -0.486064666  2.210069824  1.229688432  1.427163230
##  [61] -0.371228324 -1.198183595 -1.118924665  0.930683344 -1.312332280
##  [66] -0.509124065  0.085230411  2.026771077 -0.450547708 -2.437099397
##  [71] -0.107491966  0.231705636 -0.868490226  0.103826347  0.053248987
##  [76] -0.600533633  0.648037217  0.079170382 -1.833624250 -2.619729810
##  [81] -0.095487277  2.705774888  1.938593924  1.517741645  1.140195990
##  [86]  0.067376023 -1.961144294 -0.348160518  1.852928634 -3.301464430
##  [91] -1.661410213  1.961645683  1.205111333  0.911302414  0.520768206
##  [96]  0.325001257 -0.208495557 -1.284893781 -1.553151312  0.043217974
## [101]  0.011213518 -0.738248530  0.663141388  0.710007843  1.639072559
## [106] -0.837344710 -1.038014754  0.118593324 -1.215444685 -0.260625260
## [111] -0.123016349 -1.807180236  0.071927759  0.542573933  1.802613875
## [116]  0.272992022  0.616351834  0.644840330 -1.182728358  0.320914370
## [121]  0.410283930  0.253334891  0.863985959  0.233771747  1.434436152
## [126]  0.177084448 -1.875886459  0.271605991 -1.006622400 -2.323842355
## [131] -0.671883426 -0.070076378  0.006911929  2.286008961  2.501786040
## [136]  0.815890391 -1.637328628 -0.558460368  0.635467657 -0.139722175
## [141] -0.008483813 -0.963461947 -1.496177766  1.548908938  0.952057493
## [146] -0.574195627  0.231520348  2.178332119  0.477981964  0.703506684
## [151]  1.032732018 -1.143969314 -0.260552652 -1.070982016 -2.258302027
## [156] -1.564830408 -0.017936301  2.255615866  0.599436706 -1.172157392
## [161]  0.022433568  0.864313297 -1.275973425 -1.420508694 -1.708063396
## [166] -0.998242903 -0.055384210  0.597455432  2.346974518  1.792731612
## [171]  0.437222026 -0.631858717 -0.205819896  0.674398208 -0.402918932
## [176]  0.865536287  1.328429864 -1.403605944 -1.374898597  1.237235548
## [181] -1.378130534 -0.071063550  0.766788850  3.219857471 -2.384853572
## [186] -2.214919675  1.063724642  1.998070909  0.977319662 -1.880565162
## [191]  1.562104644 -1.391061141 -2.136852141  0.109787480  2.538176030
## [196]  0.933011569  0.166658262  1.402060666  0.589673285 -2.080247644
## [201] -0.313435778 -0.231035172  1.222255420 -0.326035167 -2.417860387
## [206] -0.025393239  0.577765429 -0.170724368 -1.232011078 -1.055869141
## [211]  0.182533557  1.426602183 -0.022318540 -1.074371308 -2.678597970
## [216]  0.544627484  1.588756353  1.745008502  1.876897625 -0.227809127
## [221]  0.737177529 -0.109231364 -1.363223907 -1.281878921 -0.515400658
## [226]  0.361097728  0.167240212  0.231190338  0.284075609  0.702771485
## [231] -0.270186256  1.321321723  0.554087579 -1.808513746 -0.855291153
## [236]  0.553316157 -0.345818822  0.137641172  0.542687862  1.221594475
## [241] -0.714389691 -0.651165058 -0.638383031 -1.339351898 -1.854854245
## [246]  0.416099399  1.469814641  1.099341837 -1.162961370 -1.318787940
## [251] -0.747273607 -0.191797501  0.885005463  1.343750431 -0.306068404
## [256] -2.779307949  1.137714152  1.965358966 -0.309056384 -0.923940323
## [261]  0.026214322  0.700618943  1.215505192  1.783827238 -0.703827238
## [266] -0.198929025  1.486986975 -0.905738593 -0.818204073 -1.731807125
## [271]  0.100123132  0.010420296 -0.815679513 -0.431043052  0.932799518
## [276] -1.492092722 -0.386711780 -0.037806332 -0.397570947  1.261366094
## [281]  1.290340693  0.364940725  0.809811339 -0.275005773 -2.080506659
## [286]  1.229748794  0.720290142 -0.359369508  0.406342510  1.053275666
## [291]  0.633109654  0.824481240 -0.041561196 -0.934983070 -0.602219681
## [296]  0.757560673 -0.739829142 -1.326054601

plot(y=yt_arma,x=yt_2_arma)

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

cor(yt_arma,yt_2_arma)

## [1] -0.2592442

Scatter Plot Yt dengan Yt-3

yt_arma <- y.arma[-c(1,2,3)]
yt_arma

##   [1] -1.400683554 -1.013808584  1.418517113 -0.283595036  0.115231693
##   [6]  1.435520708  0.137878065 -1.126175869 -0.673024142 -0.784993441
##  [11]  0.921528532 -0.041506854  1.036726819 -0.581811711 -1.436022758
##  [16] -0.095427141  0.020288013 -0.608947701 -1.694545637 -0.003766299
##  [21] -0.120557458  1.290169793 -0.334810413  0.948619090 -0.120677374
##  [26]  0.005582779  1.149796418 -0.477690729  0.213827780  0.012872066
##  [31] -1.383485003 -0.452378325  1.906063316  0.162556635 -1.241512939
##  [36] -0.186676377  0.500779481  1.068736190  1.198269057 -0.117374555
##  [41]  0.618956894 -1.004399949 -3.115963169 -0.834183669  0.039241455
##  [46] -1.973899907  0.819297849  0.395726792  0.111695981 -0.891422961
##  [51]  0.894625505  2.893507289 -0.337403693 -0.486064666  2.210069824
##  [56]  1.229688432  1.427163230 -0.371228324 -1.198183595 -1.118924665
##  [61]  0.930683344 -1.312332280 -0.509124065  0.085230411  2.026771077
##  [66] -0.450547708 -2.437099397 -0.107491966  0.231705636 -0.868490226
##  [71]  0.103826347  0.053248987 -0.600533633  0.648037217  0.079170382
##  [76] -1.833624250 -2.619729810 -0.095487277  2.705774888  1.938593924
##  [81]  1.517741645  1.140195990  0.067376023 -1.961144294 -0.348160518
##  [86]  1.852928634 -3.301464430 -1.661410213  1.961645683  1.205111333
##  [91]  0.911302414  0.520768206  0.325001257 -0.208495557 -1.284893781
##  [96] -1.553151312  0.043217974  0.011213518 -0.738248530  0.663141388
## [101]  0.710007843  1.639072559 -0.837344710 -1.038014754  0.118593324
## [106] -1.215444685 -0.260625260 -0.123016349 -1.807180236  0.071927759
## [111]  0.542573933  1.802613875  0.272992022  0.616351834  0.644840330
## [116] -1.182728358  0.320914370  0.410283930  0.253334891  0.863985959
## [121]  0.233771747  1.434436152  0.177084448 -1.875886459  0.271605991
## [126] -1.006622400 -2.323842355 -0.671883426 -0.070076378  0.006911929
## [131]  2.286008961  2.501786040  0.815890391 -1.637328628 -0.558460368
## [136]  0.635467657 -0.139722175 -0.008483813 -0.963461947 -1.496177766
## [141]  1.548908938  0.952057493 -0.574195627  0.231520348  2.178332119
## [146]  0.477981964  0.703506684  1.032732018 -1.143969314 -0.260552652
## [151] -1.070982016 -2.258302027 -1.564830408 -0.017936301  2.255615866
## [156]  0.599436706 -1.172157392  0.022433568  0.864313297 -1.275973425
## [161] -1.420508694 -1.708063396 -0.998242903 -0.055384210  0.597455432
## [166]  2.346974518  1.792731612  0.437222026 -0.631858717 -0.205819896
## [171]  0.674398208 -0.402918932  0.865536287  1.328429864 -1.403605944
## [176] -1.374898597  1.237235548 -1.378130534 -0.071063550  0.766788850
## [181]  3.219857471 -2.384853572 -2.214919675  1.063724642  1.998070909
## [186]  0.977319662 -1.880565162  1.562104644 -1.391061141 -2.136852141
## [191]  0.109787480  2.538176030  0.933011569  0.166658262  1.402060666
## [196]  0.589673285 -2.080247644 -0.313435778 -0.231035172  1.222255420
## [201] -0.326035167 -2.417860387 -0.025393239  0.577765429 -0.170724368
## [206] -1.232011078 -1.055869141  0.182533557  1.426602183 -0.022318540
## [211] -1.074371308 -2.678597970  0.544627484  1.588756353  1.745008502
## [216]  1.876897625 -0.227809127  0.737177529 -0.109231364 -1.363223907
## [221] -1.281878921 -0.515400658  0.361097728  0.167240212  0.231190338
## [226]  0.284075609  0.702771485 -0.270186256  1.321321723  0.554087579
## [231] -1.808513746 -0.855291153  0.553316157 -0.345818822  0.137641172
## [236]  0.542687862  1.221594475 -0.714389691 -0.651165058 -0.638383031
## [241] -1.339351898 -1.854854245  0.416099399  1.469814641  1.099341837
## [246] -1.162961370 -1.318787940 -0.747273607 -0.191797501  0.885005463
## [251]  1.343750431 -0.306068404 -2.779307949  1.137714152  1.965358966
## [256] -0.309056384 -0.923940323  0.026214322  0.700618943  1.215505192
## [261]  1.783827238 -0.703827238 -0.198929025  1.486986975 -0.905738593
## [266] -0.818204073 -1.731807125  0.100123132  0.010420296 -0.815679513
## [271] -0.431043052  0.932799518 -1.492092722 -0.386711780 -0.037806332
## [276] -0.397570947  1.261366094  1.290340693  0.364940725  0.809811339
## [281] -0.275005773 -2.080506659  1.229748794  0.720290142 -0.359369508
## [286]  0.406342510  1.053275666  0.633109654  0.824481240 -0.041561196
## [291] -0.934983070 -0.602219681  0.757560673 -0.739829142 -1.326054601
## [296]  0.393490903  0.990782410

yt_3_arma <- y.arma[c(-298,-299,-300)]
yt_3_arma

##   [1]  0.000000000  0.000000000 -0.272639519 -1.400683554 -1.013808584
##   [6]  1.418517113 -0.283595036  0.115231693  1.435520708  0.137878065
##  [11] -1.126175869 -0.673024142 -0.784993441  0.921528532 -0.041506854
##  [16]  1.036726819 -0.581811711 -1.436022758 -0.095427141  0.020288013
##  [21] -0.608947701 -1.694545637 -0.003766299 -0.120557458  1.290169793
##  [26] -0.334810413  0.948619090 -0.120677374  0.005582779  1.149796418
##  [31] -0.477690729  0.213827780  0.012872066 -1.383485003 -0.452378325
##  [36]  1.906063316  0.162556635 -1.241512939 -0.186676377  0.500779481
##  [41]  1.068736190  1.198269057 -0.117374555  0.618956894 -1.004399949
##  [46] -3.115963169 -0.834183669  0.039241455 -1.973899907  0.819297849
##  [51]  0.395726792  0.111695981 -0.891422961  0.894625505  2.893507289
##  [56] -0.337403693 -0.486064666  2.210069824  1.229688432  1.427163230
##  [61] -0.371228324 -1.198183595 -1.118924665  0.930683344 -1.312332280
##  [66] -0.509124065  0.085230411  2.026771077 -0.450547708 -2.437099397
##  [71] -0.107491966  0.231705636 -0.868490226  0.103826347  0.053248987
##  [76] -0.600533633  0.648037217  0.079170382 -1.833624250 -2.619729810
##  [81] -0.095487277  2.705774888  1.938593924  1.517741645  1.140195990
##  [86]  0.067376023 -1.961144294 -0.348160518  1.852928634 -3.301464430
##  [91] -1.661410213  1.961645683  1.205111333  0.911302414  0.520768206
##  [96]  0.325001257 -0.208495557 -1.284893781 -1.553151312  0.043217974
## [101]  0.011213518 -0.738248530  0.663141388  0.710007843  1.639072559
## [106] -0.837344710 -1.038014754  0.118593324 -1.215444685 -0.260625260
## [111] -0.123016349 -1.807180236  0.071927759  0.542573933  1.802613875
## [116]  0.272992022  0.616351834  0.644840330 -1.182728358  0.320914370
## [121]  0.410283930  0.253334891  0.863985959  0.233771747  1.434436152
## [126]  0.177084448 -1.875886459  0.271605991 -1.006622400 -2.323842355
## [131] -0.671883426 -0.070076378  0.006911929  2.286008961  2.501786040
## [136]  0.815890391 -1.637328628 -0.558460368  0.635467657 -0.139722175
## [141] -0.008483813 -0.963461947 -1.496177766  1.548908938  0.952057493
## [146] -0.574195627  0.231520348  2.178332119  0.477981964  0.703506684
## [151]  1.032732018 -1.143969314 -0.260552652 -1.070982016 -2.258302027
## [156] -1.564830408 -0.017936301  2.255615866  0.599436706 -1.172157392
## [161]  0.022433568  0.864313297 -1.275973425 -1.420508694 -1.708063396
## [166] -0.998242903 -0.055384210  0.597455432  2.346974518  1.792731612
## [171]  0.437222026 -0.631858717 -0.205819896  0.674398208 -0.402918932
## [176]  0.865536287  1.328429864 -1.403605944 -1.374898597  1.237235548
## [181] -1.378130534 -0.071063550  0.766788850  3.219857471 -2.384853572
## [186] -2.214919675  1.063724642  1.998070909  0.977319662 -1.880565162
## [191]  1.562104644 -1.391061141 -2.136852141  0.109787480  2.538176030
## [196]  0.933011569  0.166658262  1.402060666  0.589673285 -2.080247644
## [201] -0.313435778 -0.231035172  1.222255420 -0.326035167 -2.417860387
## [206] -0.025393239  0.577765429 -0.170724368 -1.232011078 -1.055869141
## [211]  0.182533557  1.426602183 -0.022318540 -1.074371308 -2.678597970
## [216]  0.544627484  1.588756353  1.745008502  1.876897625 -0.227809127
## [221]  0.737177529 -0.109231364 -1.363223907 -1.281878921 -0.515400658
## [226]  0.361097728  0.167240212  0.231190338  0.284075609  0.702771485
## [231] -0.270186256  1.321321723  0.554087579 -1.808513746 -0.855291153
## [236]  0.553316157 -0.345818822  0.137641172  0.542687862  1.221594475
## [241] -0.714389691 -0.651165058 -0.638383031 -1.339351898 -1.854854245
## [246]  0.416099399  1.469814641  1.099341837 -1.162961370 -1.318787940
## [251] -0.747273607 -0.191797501  0.885005463  1.343750431 -0.306068404
## [256] -2.779307949  1.137714152  1.965358966 -0.309056384 -0.923940323
## [261]  0.026214322  0.700618943  1.215505192  1.783827238 -0.703827238
## [266] -0.198929025  1.486986975 -0.905738593 -0.818204073 -1.731807125
## [271]  0.100123132  0.010420296 -0.815679513 -0.431043052  0.932799518
## [276] -1.492092722 -0.386711780 -0.037806332 -0.397570947  1.261366094
## [281]  1.290340693  0.364940725  0.809811339 -0.275005773 -2.080506659
## [286]  1.229748794  0.720290142 -0.359369508  0.406342510  1.053275666
## [291]  0.633109654  0.824481240 -0.041561196 -0.934983070 -0.602219681
## [296]  0.757560673 -0.739829142

plot(y=yt_arma,x=yt_3_arma)

Dapat dilihat bahwa sepertinya terdapat hubungan negatif antara yt dengan yt-3

cor(yt_arma,yt_3_arma)

## [1] -0.06776492