Latihan Pertemuan 4

White noise dibangkitkan dari suatu peubah acak, umumnya peubah acak normal.

wn <- rnorm(300)
ts.plot(wn)

Dapat terlihat bahwa white noise tidak memiliki autokorelasi dari ACF. Perhatikan bahwa lag ke-0 adalah korelasi observasi ke-t dengan dirinya sendiri. Nilai korelasi tersebut pasti 1. Sebagai alternatif, lag pertama di plot ACF dapat ditetapkan sebagai 1 (alih-alih 0) dengan menambahkan argumen xlim(1, lag akhir). Plot tersebut dapat disandingkan bersamaan dengan membuat matriks \(1 \times 2\) dengan par(mfrow = c(1,2)).

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

Pembangkitan Proses MA(2)

Akan dicoba membangkitkan proses MA(2) dengan \(\theta_1 = 0.4\) dan \(\theta_2 = 0.6\) sebanyak 300 observasi dan \(c=0\). Karena diperlukan satu nilai awal untuk \(e_{t-1}\), masukkan nilai pertama white noise sebagai nilai awal tersebut.

set.seed(1234)
ma <- c(1:300)
ma[1] <- wn[1]
ma[2] <- wn[2]

Nilai-nilai selanjutnya dapat dicari melalui loop. Bentuk loop dapat dilihat dari rumus MA(2) yang hendak dibangkitkan:

\[ y_t = e_t+0.4e_{t-1}+0.6e_{t-2} \]

for(i in 3:300){
   ma[i] <- wn[i] - (0.4 * wn[i - 1]) - (0.6 * wn[i - 2])
}
ma
##   [1] -0.609325752  0.502173207  1.323136707  1.079503321 -2.122439921
##   [6] -1.131846578 -0.389904575  0.451381434  0.708937815 -1.091752095
##  [11]  0.799177873  0.077535614 -0.009155793  1.319737391 -0.125327190
##  [16] -0.580322546  1.793504738 -1.226599547 -0.668477511 -0.047155572
##  [21] -0.571094332  0.274900221 -0.610338781 -0.503210943  0.872912386
##  [26] -1.013015216  1.071734605  1.147441586 -1.812674773 -0.025083717
##  [31]  0.882457695 -0.875739742 -1.589637562  3.956536049 -1.389613106
##  [36]  0.180854458  0.084747791  0.088612569 -0.940318747  0.446912489
##  [41]  0.459646270 -0.727774292  0.899828878 -0.668955219 -0.234251080
##  [46]  1.459587250 -2.377160112  1.074650446  1.688515442  0.067814434
##  [51] -2.182606367  0.739774454  0.765206031 -1.781682969 -0.583105734
##  [56]  2.191822617  0.200969495 -1.210370120 -0.473210294 -0.161900111
##  [61] -0.390731569  1.557389094  0.066821413 -1.224590674  0.508869533
##  [66]  2.069115302 -2.581734299 -0.733724221  0.895010778 -0.320407624
##  [71]  1.007225429  0.112838572 -0.692704368  0.115207291  0.777638270
##  [76] -0.400146629 -0.948404358  0.090426511  0.353971097  0.385632981
##  [81]  0.806727344 -0.317006326 -0.758772719  0.422364211 -0.049843236
##  [86]  0.822121923 -0.357781372 -0.196275531 -0.555167242 -1.248176098
##  [91] -0.919388186 -0.757205912  3.768394440  1.266214499 -0.380397255
##  [96] -2.146678447 -0.806942306  2.513644888 -1.324084170  2.177336623
## [101]  0.473948856 -0.704161991 -2.388686520 -1.010053911  2.071560302
## [106]  1.333566894 -0.452748520 -0.599741846 -0.540512967  0.208640298
## [111]  0.670645762 -0.982381022  0.928570349  1.263054209 -0.268230846
## [116] -2.498826005  0.830162887 -0.543319031  0.021475172  0.248075449
## [121]  0.271402760 -0.306230345  0.476561939 -0.147135228  0.937772530
## [126]  1.408472663 -1.910648211  0.448089780 -0.222431757 -1.178059809
## [131] -1.061808176  0.767243619 -0.918462911 -1.352353860  3.398975004
## [136]  0.675696100 -1.021696793 -0.877903157 -0.601703670  0.644934774
## [141]  0.999281742 -0.836724265 -0.562056533 -0.819662741  0.905597982
## [146]  2.140009144 -1.216948269 -3.018431223  1.101657207  2.709279620
## [151]  1.783888273  0.038989322 -1.573173923  0.130682298 -0.921570071
## [156] -0.017717936  0.226605976  0.532044676  0.842996351 -0.011268027
## [161] -0.941857522  0.091587682  0.402494991 -0.524290310  0.101037676
## [166] -1.826747222 -0.177684385  0.647260387 -0.916733730  1.534085022
## [171] -0.235307473 -0.321397851 -1.495058895  1.223320787  0.581443656
## [176] -0.713407096 -0.994210998  0.697981936 -0.398863048  2.210188564
## [181]  0.499821508  0.905535266 -0.601564613 -1.972218582  1.695019359
## [186] -0.218926393 -1.388909919 -0.465994651  0.753487716  0.254019347
## [191]  0.129562207 -0.843145359 -0.650182387  1.955206691  0.113539734
## [196] -0.610227429 -0.040618954 -1.147589009  0.067961014  1.242906024
## [201] -0.309644771 -2.041944378  0.847917199  2.276426937  0.239973551
## [206]  0.973251914 -1.869374009 -0.076984941  0.837448410 -1.413387035
## [211] -0.795598079  0.757232926  1.591654704 -0.759050399  0.127052524
## [216]  0.923138305 -1.234916649  0.069552303 -0.712905002  0.618874636
## [221] -0.163172706 -1.353011775 -0.007502246  1.417652156  2.173758006
## [226] -2.305927204 -1.062261658  2.108581831 -0.034984677 -2.165162074
## [231]  0.425192872  0.679827828 -0.361058768 -1.531222808 -0.279071055
## [236]  1.319782357  0.184171287 -0.317995691 -1.320759530  1.417665407
## [241]  0.944959945  0.077224051  0.924784958 -2.483893468 -0.316660865
## [246]  0.577761193 -0.352558506  0.274810241  2.805198635  0.495346270
## [251] -0.959647752 -0.915048797 -0.956877375  0.658261429  0.466647403
## [256] -0.815329007 -1.254052514  1.099970570  0.930170142 -1.167730478
## [261]  1.101864240  0.648978966 -2.382078808  1.236911765  1.206909143
## [266] -0.961777183  0.711098534  0.470077048 -0.032043030 -1.133625460
## [271]  1.553758088 -0.999973152 -1.768344423  1.749021065  0.953349755
## [276] -2.422380883 -0.399658591  0.411523639 -0.225397818  0.490785171
## [281]  0.332498153  0.386919352 -0.814413416 -0.668199367  1.142543518
## [286]  0.873097506  0.776812850 -1.366610991 -0.748030104  2.404996411
## [291] -1.832425823 -1.080545696  0.142481729  1.941707796  0.132696274
## [296]  0.214057643  0.010037719 -1.380630136 -0.931869642 -0.720948791

Selain menggunakan cara di atas, pembangkitan proses MA(1) dapat dilakukan dengan fungsi arima.sim() sebagai berikut.

ma1 <- arima.sim(list(order=c(0,0,2), ma=c(0.4,0.6)), n=300)
ma1
## 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

Karakteristik MA(2)

Plot Time Series

ts.plot(ma)

Berdasarkan plot time series, terlihat bahwa data MA(2) yang dibangkitkan stasioner dalam rataan

Plot ACF

acf(ma,lag.max = 20)

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

Plot PACF

pacf(ma)

Berdasarkan plot PACF tersebut, terlihat bahwa plot PACF cenderung tails off

Plot EACF

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

Berdasarkan pola segitiga nol pada plot EACF, terlihat bahwa segitiga nol berada pada ordo AR(0) dan ordo MA(2)

Scatterplot Antar Lag

Korelasi antara \(Y_t\) dengan \(Y_{t-1}\)

#Yt
yt_ma <- ma[-1]
yt_ma
##   [1]  0.502173207  1.323136707  1.079503321 -2.122439921 -1.131846578
##   [6] -0.389904575  0.451381434  0.708937815 -1.091752095  0.799177873
##  [11]  0.077535614 -0.009155793  1.319737391 -0.125327190 -0.580322546
##  [16]  1.793504738 -1.226599547 -0.668477511 -0.047155572 -0.571094332
##  [21]  0.274900221 -0.610338781 -0.503210943  0.872912386 -1.013015216
##  [26]  1.071734605  1.147441586 -1.812674773 -0.025083717  0.882457695
##  [31] -0.875739742 -1.589637562  3.956536049 -1.389613106  0.180854458
##  [36]  0.084747791  0.088612569 -0.940318747  0.446912489  0.459646270
##  [41] -0.727774292  0.899828878 -0.668955219 -0.234251080  1.459587250
##  [46] -2.377160112  1.074650446  1.688515442  0.067814434 -2.182606367
##  [51]  0.739774454  0.765206031 -1.781682969 -0.583105734  2.191822617
##  [56]  0.200969495 -1.210370120 -0.473210294 -0.161900111 -0.390731569
##  [61]  1.557389094  0.066821413 -1.224590674  0.508869533  2.069115302
##  [66] -2.581734299 -0.733724221  0.895010778 -0.320407624  1.007225429
##  [71]  0.112838572 -0.692704368  0.115207291  0.777638270 -0.400146629
##  [76] -0.948404358  0.090426511  0.353971097  0.385632981  0.806727344
##  [81] -0.317006326 -0.758772719  0.422364211 -0.049843236  0.822121923
##  [86] -0.357781372 -0.196275531 -0.555167242 -1.248176098 -0.919388186
##  [91] -0.757205912  3.768394440  1.266214499 -0.380397255 -2.146678447
##  [96] -0.806942306  2.513644888 -1.324084170  2.177336623  0.473948856
## [101] -0.704161991 -2.388686520 -1.010053911  2.071560302  1.333566894
## [106] -0.452748520 -0.599741846 -0.540512967  0.208640298  0.670645762
## [111] -0.982381022  0.928570349  1.263054209 -0.268230846 -2.498826005
## [116]  0.830162887 -0.543319031  0.021475172  0.248075449  0.271402760
## [121] -0.306230345  0.476561939 -0.147135228  0.937772530  1.408472663
## [126] -1.910648211  0.448089780 -0.222431757 -1.178059809 -1.061808176
## [131]  0.767243619 -0.918462911 -1.352353860  3.398975004  0.675696100
## [136] -1.021696793 -0.877903157 -0.601703670  0.644934774  0.999281742
## [141] -0.836724265 -0.562056533 -0.819662741  0.905597982  2.140009144
## [146] -1.216948269 -3.018431223  1.101657207  2.709279620  1.783888273
## [151]  0.038989322 -1.573173923  0.130682298 -0.921570071 -0.017717936
## [156]  0.226605976  0.532044676  0.842996351 -0.011268027 -0.941857522
## [161]  0.091587682  0.402494991 -0.524290310  0.101037676 -1.826747222
## [166] -0.177684385  0.647260387 -0.916733730  1.534085022 -0.235307473
## [171] -0.321397851 -1.495058895  1.223320787  0.581443656 -0.713407096
## [176] -0.994210998  0.697981936 -0.398863048  2.210188564  0.499821508
## [181]  0.905535266 -0.601564613 -1.972218582  1.695019359 -0.218926393
## [186] -1.388909919 -0.465994651  0.753487716  0.254019347  0.129562207
## [191] -0.843145359 -0.650182387  1.955206691  0.113539734 -0.610227429
## [196] -0.040618954 -1.147589009  0.067961014  1.242906024 -0.309644771
## [201] -2.041944378  0.847917199  2.276426937  0.239973551  0.973251914
## [206] -1.869374009 -0.076984941  0.837448410 -1.413387035 -0.795598079
## [211]  0.757232926  1.591654704 -0.759050399  0.127052524  0.923138305
## [216] -1.234916649  0.069552303 -0.712905002  0.618874636 -0.163172706
## [221] -1.353011775 -0.007502246  1.417652156  2.173758006 -2.305927204
## [226] -1.062261658  2.108581831 -0.034984677 -2.165162074  0.425192872
## [231]  0.679827828 -0.361058768 -1.531222808 -0.279071055  1.319782357
## [236]  0.184171287 -0.317995691 -1.320759530  1.417665407  0.944959945
## [241]  0.077224051  0.924784958 -2.483893468 -0.316660865  0.577761193
## [246] -0.352558506  0.274810241  2.805198635  0.495346270 -0.959647752
## [251] -0.915048797 -0.956877375  0.658261429  0.466647403 -0.815329007
## [256] -1.254052514  1.099970570  0.930170142 -1.167730478  1.101864240
## [261]  0.648978966 -2.382078808  1.236911765  1.206909143 -0.961777183
## [266]  0.711098534  0.470077048 -0.032043030 -1.133625460  1.553758088
## [271] -0.999973152 -1.768344423  1.749021065  0.953349755 -2.422380883
## [276] -0.399658591  0.411523639 -0.225397818  0.490785171  0.332498153
## [281]  0.386919352 -0.814413416 -0.668199367  1.142543518  0.873097506
## [286]  0.776812850 -1.366610991 -0.748030104  2.404996411 -1.832425823
## [291] -1.080545696  0.142481729  1.941707796  0.132696274  0.214057643
## [296]  0.010037719 -1.380630136 -0.931869642 -0.720948791
#Yt-1
yt_1_ma <- ma[-300]
yt_1_ma
##   [1] -0.609325752  0.502173207  1.323136707  1.079503321 -2.122439921
##   [6] -1.131846578 -0.389904575  0.451381434  0.708937815 -1.091752095
##  [11]  0.799177873  0.077535614 -0.009155793  1.319737391 -0.125327190
##  [16] -0.580322546  1.793504738 -1.226599547 -0.668477511 -0.047155572
##  [21] -0.571094332  0.274900221 -0.610338781 -0.503210943  0.872912386
##  [26] -1.013015216  1.071734605  1.147441586 -1.812674773 -0.025083717
##  [31]  0.882457695 -0.875739742 -1.589637562  3.956536049 -1.389613106
##  [36]  0.180854458  0.084747791  0.088612569 -0.940318747  0.446912489
##  [41]  0.459646270 -0.727774292  0.899828878 -0.668955219 -0.234251080
##  [46]  1.459587250 -2.377160112  1.074650446  1.688515442  0.067814434
##  [51] -2.182606367  0.739774454  0.765206031 -1.781682969 -0.583105734
##  [56]  2.191822617  0.200969495 -1.210370120 -0.473210294 -0.161900111
##  [61] -0.390731569  1.557389094  0.066821413 -1.224590674  0.508869533
##  [66]  2.069115302 -2.581734299 -0.733724221  0.895010778 -0.320407624
##  [71]  1.007225429  0.112838572 -0.692704368  0.115207291  0.777638270
##  [76] -0.400146629 -0.948404358  0.090426511  0.353971097  0.385632981
##  [81]  0.806727344 -0.317006326 -0.758772719  0.422364211 -0.049843236
##  [86]  0.822121923 -0.357781372 -0.196275531 -0.555167242 -1.248176098
##  [91] -0.919388186 -0.757205912  3.768394440  1.266214499 -0.380397255
##  [96] -2.146678447 -0.806942306  2.513644888 -1.324084170  2.177336623
## [101]  0.473948856 -0.704161991 -2.388686520 -1.010053911  2.071560302
## [106]  1.333566894 -0.452748520 -0.599741846 -0.540512967  0.208640298
## [111]  0.670645762 -0.982381022  0.928570349  1.263054209 -0.268230846
## [116] -2.498826005  0.830162887 -0.543319031  0.021475172  0.248075449
## [121]  0.271402760 -0.306230345  0.476561939 -0.147135228  0.937772530
## [126]  1.408472663 -1.910648211  0.448089780 -0.222431757 -1.178059809
## [131] -1.061808176  0.767243619 -0.918462911 -1.352353860  3.398975004
## [136]  0.675696100 -1.021696793 -0.877903157 -0.601703670  0.644934774
## [141]  0.999281742 -0.836724265 -0.562056533 -0.819662741  0.905597982
## [146]  2.140009144 -1.216948269 -3.018431223  1.101657207  2.709279620
## [151]  1.783888273  0.038989322 -1.573173923  0.130682298 -0.921570071
## [156] -0.017717936  0.226605976  0.532044676  0.842996351 -0.011268027
## [161] -0.941857522  0.091587682  0.402494991 -0.524290310  0.101037676
## [166] -1.826747222 -0.177684385  0.647260387 -0.916733730  1.534085022
## [171] -0.235307473 -0.321397851 -1.495058895  1.223320787  0.581443656
## [176] -0.713407096 -0.994210998  0.697981936 -0.398863048  2.210188564
## [181]  0.499821508  0.905535266 -0.601564613 -1.972218582  1.695019359
## [186] -0.218926393 -1.388909919 -0.465994651  0.753487716  0.254019347
## [191]  0.129562207 -0.843145359 -0.650182387  1.955206691  0.113539734
## [196] -0.610227429 -0.040618954 -1.147589009  0.067961014  1.242906024
## [201] -0.309644771 -2.041944378  0.847917199  2.276426937  0.239973551
## [206]  0.973251914 -1.869374009 -0.076984941  0.837448410 -1.413387035
## [211] -0.795598079  0.757232926  1.591654704 -0.759050399  0.127052524
## [216]  0.923138305 -1.234916649  0.069552303 -0.712905002  0.618874636
## [221] -0.163172706 -1.353011775 -0.007502246  1.417652156  2.173758006
## [226] -2.305927204 -1.062261658  2.108581831 -0.034984677 -2.165162074
## [231]  0.425192872  0.679827828 -0.361058768 -1.531222808 -0.279071055
## [236]  1.319782357  0.184171287 -0.317995691 -1.320759530  1.417665407
## [241]  0.944959945  0.077224051  0.924784958 -2.483893468 -0.316660865
## [246]  0.577761193 -0.352558506  0.274810241  2.805198635  0.495346270
## [251] -0.959647752 -0.915048797 -0.956877375  0.658261429  0.466647403
## [256] -0.815329007 -1.254052514  1.099970570  0.930170142 -1.167730478
## [261]  1.101864240  0.648978966 -2.382078808  1.236911765  1.206909143
## [266] -0.961777183  0.711098534  0.470077048 -0.032043030 -1.133625460
## [271]  1.553758088 -0.999973152 -1.768344423  1.749021065  0.953349755
## [276] -2.422380883 -0.399658591  0.411523639 -0.225397818  0.490785171
## [281]  0.332498153  0.386919352 -0.814413416 -0.668199367  1.142543518
## [286]  0.873097506  0.776812850 -1.366610991 -0.748030104  2.404996411
## [291] -1.832425823 -1.080545696  0.142481729  1.941707796  0.132696274
## [296]  0.214057643  0.010037719 -1.380630136 -0.931869642
plot(y=yt_ma,x=yt_1_ma)

Berdasarkan scatterplot tersebut, terlihat bahwa cenderung terdapat hubungan negatif antara \(Y_t\) dengan \(Y_{t-1}\). Hal ini sesuai dengan teori yang ada

cor(yt_ma,yt_1_ma)
## [1] -0.1090929

Korelasi antara \(Y_t\) dengan \(Y_{t-1}\) dari hasil simulasi mendekati perhitungan teoritis yaitu

\[ \rho_1=\frac{-\theta_1+(\theta_1)(\theta_2)}{1+(\theta_1)^2+(\theta_2)^2}=\frac{-0.4+(0.4)(0.6)}{1+(0.4)^2+(0.6)^2}=\frac{-0.16}{1.52}=-0.105 \]

Korelasi antara \(Y_t\) dengan \(Y_{t-2}\)

#Yt
yt_ma2 <- ma[-c(1,2)]
yt_ma2
##   [1]  1.323136707  1.079503321 -2.122439921 -1.131846578 -0.389904575
##   [6]  0.451381434  0.708937815 -1.091752095  0.799177873  0.077535614
##  [11] -0.009155793  1.319737391 -0.125327190 -0.580322546  1.793504738
##  [16] -1.226599547 -0.668477511 -0.047155572 -0.571094332  0.274900221
##  [21] -0.610338781 -0.503210943  0.872912386 -1.013015216  1.071734605
##  [26]  1.147441586 -1.812674773 -0.025083717  0.882457695 -0.875739742
##  [31] -1.589637562  3.956536049 -1.389613106  0.180854458  0.084747791
##  [36]  0.088612569 -0.940318747  0.446912489  0.459646270 -0.727774292
##  [41]  0.899828878 -0.668955219 -0.234251080  1.459587250 -2.377160112
##  [46]  1.074650446  1.688515442  0.067814434 -2.182606367  0.739774454
##  [51]  0.765206031 -1.781682969 -0.583105734  2.191822617  0.200969495
##  [56] -1.210370120 -0.473210294 -0.161900111 -0.390731569  1.557389094
##  [61]  0.066821413 -1.224590674  0.508869533  2.069115302 -2.581734299
##  [66] -0.733724221  0.895010778 -0.320407624  1.007225429  0.112838572
##  [71] -0.692704368  0.115207291  0.777638270 -0.400146629 -0.948404358
##  [76]  0.090426511  0.353971097  0.385632981  0.806727344 -0.317006326
##  [81] -0.758772719  0.422364211 -0.049843236  0.822121923 -0.357781372
##  [86] -0.196275531 -0.555167242 -1.248176098 -0.919388186 -0.757205912
##  [91]  3.768394440  1.266214499 -0.380397255 -2.146678447 -0.806942306
##  [96]  2.513644888 -1.324084170  2.177336623  0.473948856 -0.704161991
## [101] -2.388686520 -1.010053911  2.071560302  1.333566894 -0.452748520
## [106] -0.599741846 -0.540512967  0.208640298  0.670645762 -0.982381022
## [111]  0.928570349  1.263054209 -0.268230846 -2.498826005  0.830162887
## [116] -0.543319031  0.021475172  0.248075449  0.271402760 -0.306230345
## [121]  0.476561939 -0.147135228  0.937772530  1.408472663 -1.910648211
## [126]  0.448089780 -0.222431757 -1.178059809 -1.061808176  0.767243619
## [131] -0.918462911 -1.352353860  3.398975004  0.675696100 -1.021696793
## [136] -0.877903157 -0.601703670  0.644934774  0.999281742 -0.836724265
## [141] -0.562056533 -0.819662741  0.905597982  2.140009144 -1.216948269
## [146] -3.018431223  1.101657207  2.709279620  1.783888273  0.038989322
## [151] -1.573173923  0.130682298 -0.921570071 -0.017717936  0.226605976
## [156]  0.532044676  0.842996351 -0.011268027 -0.941857522  0.091587682
## [161]  0.402494991 -0.524290310  0.101037676 -1.826747222 -0.177684385
## [166]  0.647260387 -0.916733730  1.534085022 -0.235307473 -0.321397851
## [171] -1.495058895  1.223320787  0.581443656 -0.713407096 -0.994210998
## [176]  0.697981936 -0.398863048  2.210188564  0.499821508  0.905535266
## [181] -0.601564613 -1.972218582  1.695019359 -0.218926393 -1.388909919
## [186] -0.465994651  0.753487716  0.254019347  0.129562207 -0.843145359
## [191] -0.650182387  1.955206691  0.113539734 -0.610227429 -0.040618954
## [196] -1.147589009  0.067961014  1.242906024 -0.309644771 -2.041944378
## [201]  0.847917199  2.276426937  0.239973551  0.973251914 -1.869374009
## [206] -0.076984941  0.837448410 -1.413387035 -0.795598079  0.757232926
## [211]  1.591654704 -0.759050399  0.127052524  0.923138305 -1.234916649
## [216]  0.069552303 -0.712905002  0.618874636 -0.163172706 -1.353011775
## [221] -0.007502246  1.417652156  2.173758006 -2.305927204 -1.062261658
## [226]  2.108581831 -0.034984677 -2.165162074  0.425192872  0.679827828
## [231] -0.361058768 -1.531222808 -0.279071055  1.319782357  0.184171287
## [236] -0.317995691 -1.320759530  1.417665407  0.944959945  0.077224051
## [241]  0.924784958 -2.483893468 -0.316660865  0.577761193 -0.352558506
## [246]  0.274810241  2.805198635  0.495346270 -0.959647752 -0.915048797
## [251] -0.956877375  0.658261429  0.466647403 -0.815329007 -1.254052514
## [256]  1.099970570  0.930170142 -1.167730478  1.101864240  0.648978966
## [261] -2.382078808  1.236911765  1.206909143 -0.961777183  0.711098534
## [266]  0.470077048 -0.032043030 -1.133625460  1.553758088 -0.999973152
## [271] -1.768344423  1.749021065  0.953349755 -2.422380883 -0.399658591
## [276]  0.411523639 -0.225397818  0.490785171  0.332498153  0.386919352
## [281] -0.814413416 -0.668199367  1.142543518  0.873097506  0.776812850
## [286] -1.366610991 -0.748030104  2.404996411 -1.832425823 -1.080545696
## [291]  0.142481729  1.941707796  0.132696274  0.214057643  0.010037719
## [296] -1.380630136 -0.931869642 -0.720948791
#Yt-2
yt_2_ma <- ma[-c(299,300)]
yt_2_ma
##   [1] -0.609325752  0.502173207  1.323136707  1.079503321 -2.122439921
##   [6] -1.131846578 -0.389904575  0.451381434  0.708937815 -1.091752095
##  [11]  0.799177873  0.077535614 -0.009155793  1.319737391 -0.125327190
##  [16] -0.580322546  1.793504738 -1.226599547 -0.668477511 -0.047155572
##  [21] -0.571094332  0.274900221 -0.610338781 -0.503210943  0.872912386
##  [26] -1.013015216  1.071734605  1.147441586 -1.812674773 -0.025083717
##  [31]  0.882457695 -0.875739742 -1.589637562  3.956536049 -1.389613106
##  [36]  0.180854458  0.084747791  0.088612569 -0.940318747  0.446912489
##  [41]  0.459646270 -0.727774292  0.899828878 -0.668955219 -0.234251080
##  [46]  1.459587250 -2.377160112  1.074650446  1.688515442  0.067814434
##  [51] -2.182606367  0.739774454  0.765206031 -1.781682969 -0.583105734
##  [56]  2.191822617  0.200969495 -1.210370120 -0.473210294 -0.161900111
##  [61] -0.390731569  1.557389094  0.066821413 -1.224590674  0.508869533
##  [66]  2.069115302 -2.581734299 -0.733724221  0.895010778 -0.320407624
##  [71]  1.007225429  0.112838572 -0.692704368  0.115207291  0.777638270
##  [76] -0.400146629 -0.948404358  0.090426511  0.353971097  0.385632981
##  [81]  0.806727344 -0.317006326 -0.758772719  0.422364211 -0.049843236
##  [86]  0.822121923 -0.357781372 -0.196275531 -0.555167242 -1.248176098
##  [91] -0.919388186 -0.757205912  3.768394440  1.266214499 -0.380397255
##  [96] -2.146678447 -0.806942306  2.513644888 -1.324084170  2.177336623
## [101]  0.473948856 -0.704161991 -2.388686520 -1.010053911  2.071560302
## [106]  1.333566894 -0.452748520 -0.599741846 -0.540512967  0.208640298
## [111]  0.670645762 -0.982381022  0.928570349  1.263054209 -0.268230846
## [116] -2.498826005  0.830162887 -0.543319031  0.021475172  0.248075449
## [121]  0.271402760 -0.306230345  0.476561939 -0.147135228  0.937772530
## [126]  1.408472663 -1.910648211  0.448089780 -0.222431757 -1.178059809
## [131] -1.061808176  0.767243619 -0.918462911 -1.352353860  3.398975004
## [136]  0.675696100 -1.021696793 -0.877903157 -0.601703670  0.644934774
## [141]  0.999281742 -0.836724265 -0.562056533 -0.819662741  0.905597982
## [146]  2.140009144 -1.216948269 -3.018431223  1.101657207  2.709279620
## [151]  1.783888273  0.038989322 -1.573173923  0.130682298 -0.921570071
## [156] -0.017717936  0.226605976  0.532044676  0.842996351 -0.011268027
## [161] -0.941857522  0.091587682  0.402494991 -0.524290310  0.101037676
## [166] -1.826747222 -0.177684385  0.647260387 -0.916733730  1.534085022
## [171] -0.235307473 -0.321397851 -1.495058895  1.223320787  0.581443656
## [176] -0.713407096 -0.994210998  0.697981936 -0.398863048  2.210188564
## [181]  0.499821508  0.905535266 -0.601564613 -1.972218582  1.695019359
## [186] -0.218926393 -1.388909919 -0.465994651  0.753487716  0.254019347
## [191]  0.129562207 -0.843145359 -0.650182387  1.955206691  0.113539734
## [196] -0.610227429 -0.040618954 -1.147589009  0.067961014  1.242906024
## [201] -0.309644771 -2.041944378  0.847917199  2.276426937  0.239973551
## [206]  0.973251914 -1.869374009 -0.076984941  0.837448410 -1.413387035
## [211] -0.795598079  0.757232926  1.591654704 -0.759050399  0.127052524
## [216]  0.923138305 -1.234916649  0.069552303 -0.712905002  0.618874636
## [221] -0.163172706 -1.353011775 -0.007502246  1.417652156  2.173758006
## [226] -2.305927204 -1.062261658  2.108581831 -0.034984677 -2.165162074
## [231]  0.425192872  0.679827828 -0.361058768 -1.531222808 -0.279071055
## [236]  1.319782357  0.184171287 -0.317995691 -1.320759530  1.417665407
## [241]  0.944959945  0.077224051  0.924784958 -2.483893468 -0.316660865
## [246]  0.577761193 -0.352558506  0.274810241  2.805198635  0.495346270
## [251] -0.959647752 -0.915048797 -0.956877375  0.658261429  0.466647403
## [256] -0.815329007 -1.254052514  1.099970570  0.930170142 -1.167730478
## [261]  1.101864240  0.648978966 -2.382078808  1.236911765  1.206909143
## [266] -0.961777183  0.711098534  0.470077048 -0.032043030 -1.133625460
## [271]  1.553758088 -0.999973152 -1.768344423  1.749021065  0.953349755
## [276] -2.422380883 -0.399658591  0.411523639 -0.225397818  0.490785171
## [281]  0.332498153  0.386919352 -0.814413416 -0.668199367  1.142543518
## [286]  0.873097506  0.776812850 -1.366610991 -0.748030104  2.404996411
## [291] -1.832425823 -1.080545696  0.142481729  1.941707796  0.132696274
## [296]  0.214057643  0.010037719 -1.380630136
plot(y=yt_ma2,x=yt_2_ma)

Berdasarkan scatterplot tersebut, terlihat bahwa cenderung terdapat hubungan negatif antara \(Y_t\) dengan \(Y_{t-2}\).

cor(yt_ma2,yt_2_ma)
## [1] -0.4105339

Korelasi antara \(Y_t\) dengan \(Y_{t-2}\) hasil simulasi mendekati hasil perhitungan teoritis yaitu

\[ \rho_2=\frac{-\theta_2}{1+(\theta_1)^2+(\theta_2)^2}=\frac{-0.6}{1+(0.4)^2+(0.6)^2}=\frac{-0.6}{1.52}=-0.394 \]

Korelasi antara \(Y_t\) dengan \(Y_{t-3}\)

#Yt 
yt_ma3 <- ma[-c(1,2,3)] 
yt_ma3 
##   [1]  1.079503321 -2.122439921 -1.131846578 -0.389904575  0.451381434
##   [6]  0.708937815 -1.091752095  0.799177873  0.077535614 -0.009155793
##  [11]  1.319737391 -0.125327190 -0.580322546  1.793504738 -1.226599547
##  [16] -0.668477511 -0.047155572 -0.571094332  0.274900221 -0.610338781
##  [21] -0.503210943  0.872912386 -1.013015216  1.071734605  1.147441586
##  [26] -1.812674773 -0.025083717  0.882457695 -0.875739742 -1.589637562
##  [31]  3.956536049 -1.389613106  0.180854458  0.084747791  0.088612569
##  [36] -0.940318747  0.446912489  0.459646270 -0.727774292  0.899828878
##  [41] -0.668955219 -0.234251080  1.459587250 -2.377160112  1.074650446
##  [46]  1.688515442  0.067814434 -2.182606367  0.739774454  0.765206031
##  [51] -1.781682969 -0.583105734  2.191822617  0.200969495 -1.210370120
##  [56] -0.473210294 -0.161900111 -0.390731569  1.557389094  0.066821413
##  [61] -1.224590674  0.508869533  2.069115302 -2.581734299 -0.733724221
##  [66]  0.895010778 -0.320407624  1.007225429  0.112838572 -0.692704368
##  [71]  0.115207291  0.777638270 -0.400146629 -0.948404358  0.090426511
##  [76]  0.353971097  0.385632981  0.806727344 -0.317006326 -0.758772719
##  [81]  0.422364211 -0.049843236  0.822121923 -0.357781372 -0.196275531
##  [86] -0.555167242 -1.248176098 -0.919388186 -0.757205912  3.768394440
##  [91]  1.266214499 -0.380397255 -2.146678447 -0.806942306  2.513644888
##  [96] -1.324084170  2.177336623  0.473948856 -0.704161991 -2.388686520
## [101] -1.010053911  2.071560302  1.333566894 -0.452748520 -0.599741846
## [106] -0.540512967  0.208640298  0.670645762 -0.982381022  0.928570349
## [111]  1.263054209 -0.268230846 -2.498826005  0.830162887 -0.543319031
## [116]  0.021475172  0.248075449  0.271402760 -0.306230345  0.476561939
## [121] -0.147135228  0.937772530  1.408472663 -1.910648211  0.448089780
## [126] -0.222431757 -1.178059809 -1.061808176  0.767243619 -0.918462911
## [131] -1.352353860  3.398975004  0.675696100 -1.021696793 -0.877903157
## [136] -0.601703670  0.644934774  0.999281742 -0.836724265 -0.562056533
## [141] -0.819662741  0.905597982  2.140009144 -1.216948269 -3.018431223
## [146]  1.101657207  2.709279620  1.783888273  0.038989322 -1.573173923
## [151]  0.130682298 -0.921570071 -0.017717936  0.226605976  0.532044676
## [156]  0.842996351 -0.011268027 -0.941857522  0.091587682  0.402494991
## [161] -0.524290310  0.101037676 -1.826747222 -0.177684385  0.647260387
## [166] -0.916733730  1.534085022 -0.235307473 -0.321397851 -1.495058895
## [171]  1.223320787  0.581443656 -0.713407096 -0.994210998  0.697981936
## [176] -0.398863048  2.210188564  0.499821508  0.905535266 -0.601564613
## [181] -1.972218582  1.695019359 -0.218926393 -1.388909919 -0.465994651
## [186]  0.753487716  0.254019347  0.129562207 -0.843145359 -0.650182387
## [191]  1.955206691  0.113539734 -0.610227429 -0.040618954 -1.147589009
## [196]  0.067961014  1.242906024 -0.309644771 -2.041944378  0.847917199
## [201]  2.276426937  0.239973551  0.973251914 -1.869374009 -0.076984941
## [206]  0.837448410 -1.413387035 -0.795598079  0.757232926  1.591654704
## [211] -0.759050399  0.127052524  0.923138305 -1.234916649  0.069552303
## [216] -0.712905002  0.618874636 -0.163172706 -1.353011775 -0.007502246
## [221]  1.417652156  2.173758006 -2.305927204 -1.062261658  2.108581831
## [226] -0.034984677 -2.165162074  0.425192872  0.679827828 -0.361058768
## [231] -1.531222808 -0.279071055  1.319782357  0.184171287 -0.317995691
## [236] -1.320759530  1.417665407  0.944959945  0.077224051  0.924784958
## [241] -2.483893468 -0.316660865  0.577761193 -0.352558506  0.274810241
## [246]  2.805198635  0.495346270 -0.959647752 -0.915048797 -0.956877375
## [251]  0.658261429  0.466647403 -0.815329007 -1.254052514  1.099970570
## [256]  0.930170142 -1.167730478  1.101864240  0.648978966 -2.382078808
## [261]  1.236911765  1.206909143 -0.961777183  0.711098534  0.470077048
## [266] -0.032043030 -1.133625460  1.553758088 -0.999973152 -1.768344423
## [271]  1.749021065  0.953349755 -2.422380883 -0.399658591  0.411523639
## [276] -0.225397818  0.490785171  0.332498153  0.386919352 -0.814413416
## [281] -0.668199367  1.142543518  0.873097506  0.776812850 -1.366610991
## [286] -0.748030104  2.404996411 -1.832425823 -1.080545696  0.142481729
## [291]  1.941707796  0.132696274  0.214057643  0.010037719 -1.380630136
## [296] -0.931869642 -0.720948791
#Yt-3 
yt_3_ma <- ma[-c(298,299,300)] 
yt_3_ma
##   [1] -0.609325752  0.502173207  1.323136707  1.079503321 -2.122439921
##   [6] -1.131846578 -0.389904575  0.451381434  0.708937815 -1.091752095
##  [11]  0.799177873  0.077535614 -0.009155793  1.319737391 -0.125327190
##  [16] -0.580322546  1.793504738 -1.226599547 -0.668477511 -0.047155572
##  [21] -0.571094332  0.274900221 -0.610338781 -0.503210943  0.872912386
##  [26] -1.013015216  1.071734605  1.147441586 -1.812674773 -0.025083717
##  [31]  0.882457695 -0.875739742 -1.589637562  3.956536049 -1.389613106
##  [36]  0.180854458  0.084747791  0.088612569 -0.940318747  0.446912489
##  [41]  0.459646270 -0.727774292  0.899828878 -0.668955219 -0.234251080
##  [46]  1.459587250 -2.377160112  1.074650446  1.688515442  0.067814434
##  [51] -2.182606367  0.739774454  0.765206031 -1.781682969 -0.583105734
##  [56]  2.191822617  0.200969495 -1.210370120 -0.473210294 -0.161900111
##  [61] -0.390731569  1.557389094  0.066821413 -1.224590674  0.508869533
##  [66]  2.069115302 -2.581734299 -0.733724221  0.895010778 -0.320407624
##  [71]  1.007225429  0.112838572 -0.692704368  0.115207291  0.777638270
##  [76] -0.400146629 -0.948404358  0.090426511  0.353971097  0.385632981
##  [81]  0.806727344 -0.317006326 -0.758772719  0.422364211 -0.049843236
##  [86]  0.822121923 -0.357781372 -0.196275531 -0.555167242 -1.248176098
##  [91] -0.919388186 -0.757205912  3.768394440  1.266214499 -0.380397255
##  [96] -2.146678447 -0.806942306  2.513644888 -1.324084170  2.177336623
## [101]  0.473948856 -0.704161991 -2.388686520 -1.010053911  2.071560302
## [106]  1.333566894 -0.452748520 -0.599741846 -0.540512967  0.208640298
## [111]  0.670645762 -0.982381022  0.928570349  1.263054209 -0.268230846
## [116] -2.498826005  0.830162887 -0.543319031  0.021475172  0.248075449
## [121]  0.271402760 -0.306230345  0.476561939 -0.147135228  0.937772530
## [126]  1.408472663 -1.910648211  0.448089780 -0.222431757 -1.178059809
## [131] -1.061808176  0.767243619 -0.918462911 -1.352353860  3.398975004
## [136]  0.675696100 -1.021696793 -0.877903157 -0.601703670  0.644934774
## [141]  0.999281742 -0.836724265 -0.562056533 -0.819662741  0.905597982
## [146]  2.140009144 -1.216948269 -3.018431223  1.101657207  2.709279620
## [151]  1.783888273  0.038989322 -1.573173923  0.130682298 -0.921570071
## [156] -0.017717936  0.226605976  0.532044676  0.842996351 -0.011268027
## [161] -0.941857522  0.091587682  0.402494991 -0.524290310  0.101037676
## [166] -1.826747222 -0.177684385  0.647260387 -0.916733730  1.534085022
## [171] -0.235307473 -0.321397851 -1.495058895  1.223320787  0.581443656
## [176] -0.713407096 -0.994210998  0.697981936 -0.398863048  2.210188564
## [181]  0.499821508  0.905535266 -0.601564613 -1.972218582  1.695019359
## [186] -0.218926393 -1.388909919 -0.465994651  0.753487716  0.254019347
## [191]  0.129562207 -0.843145359 -0.650182387  1.955206691  0.113539734
## [196] -0.610227429 -0.040618954 -1.147589009  0.067961014  1.242906024
## [201] -0.309644771 -2.041944378  0.847917199  2.276426937  0.239973551
## [206]  0.973251914 -1.869374009 -0.076984941  0.837448410 -1.413387035
## [211] -0.795598079  0.757232926  1.591654704 -0.759050399  0.127052524
## [216]  0.923138305 -1.234916649  0.069552303 -0.712905002  0.618874636
## [221] -0.163172706 -1.353011775 -0.007502246  1.417652156  2.173758006
## [226] -2.305927204 -1.062261658  2.108581831 -0.034984677 -2.165162074
## [231]  0.425192872  0.679827828 -0.361058768 -1.531222808 -0.279071055
## [236]  1.319782357  0.184171287 -0.317995691 -1.320759530  1.417665407
## [241]  0.944959945  0.077224051  0.924784958 -2.483893468 -0.316660865
## [246]  0.577761193 -0.352558506  0.274810241  2.805198635  0.495346270
## [251] -0.959647752 -0.915048797 -0.956877375  0.658261429  0.466647403
## [256] -0.815329007 -1.254052514  1.099970570  0.930170142 -1.167730478
## [261]  1.101864240  0.648978966 -2.382078808  1.236911765  1.206909143
## [266] -0.961777183  0.711098534  0.470077048 -0.032043030 -1.133625460
## [271]  1.553758088 -0.999973152 -1.768344423  1.749021065  0.953349755
## [276] -2.422380883 -0.399658591  0.411523639 -0.225397818  0.490785171
## [281]  0.332498153  0.386919352 -0.814413416 -0.668199367  1.142543518
## [286]  0.873097506  0.776812850 -1.366610991 -0.748030104  2.404996411
## [291] -1.832425823 -1.080545696  0.142481729  1.941707796  0.132696274
## [296]  0.214057643  0.010037719
plot(y=yt_ma3,x=yt_3_ma)

Berdasarkan scatterplot tersebut, terlihat bahwa tidak terdapat hubungan antara \(Y_t\) dengan \(Y_{t-3}\).

cor(yt_ma3,yt_3_ma)
## [1] 0.008488369

Korelasi antara \(Y_t\) dengan \(Y_{t-2}\) hasil simulasi mendekati teori yang ada yaitu 0.

Autokorelasi

autokol <- acf(ma, plot = FALSE)
autokol$acf
## , , 1
## 
##                [,1]
##  [1,]  1.0000000000
##  [2,] -0.1089713766
##  [3,] -0.4095202645
##  [4,]  0.0083978605
##  [5,]  0.0370528887
##  [6,] -0.0382706706
##  [7,]  0.0133933364
##  [8,]  0.0435135530
##  [9,] -0.0740714401
## [10,]  0.0700875763
## [11,] -0.0206042122
## [12,] -0.0942738107
## [13,]  0.0861870337
## [14,]  0.0496839668
## [15,] -0.0518314182
## [16,]  0.0383804618
## [17,]  0.0097059755
## [18,] -0.1126433551
## [19,] -0.0077606690
## [20,]  0.0657356053
## [21,] -0.0124355562
## [22,] -0.0002973591
## [23,]  0.0037911821
## [24,] -0.0340173757
## [25,]  0.0199116308
autokol <- acf(ma1, plot = FALSE)
autokol$acf
## , , 1
## 
##               [,1]
##  [1,]  1.000000000
##  [2,]  0.527945553
##  [3,]  0.435325276
##  [4,]  0.082036325
##  [5,]  0.070764433
##  [6,]  0.085724277
##  [7,]  0.060349271
##  [8,]  0.083104675
##  [9,]  0.005286790
## [10,] -0.019363790
## [11,] -0.080664802
## [12,] -0.067262880
## [13,] -0.078636116
## [14,] -0.119619909
## [15,] -0.124603683
## [16,] -0.124080368
## [17,] -0.001507892
## [18,]  0.053031442
## [19,]  0.123031556
## [20,]  0.105720468
## [21,]  0.053745284
## [22,]  0.074779882
## [23,]  0.058009167
## [24,]  0.153448258
## [25,]  0.131836632

Pembangkitan Proses AR(2)

Akan dicoba membangkitkan proses AR dengan \(\phi_1 = 0.5\) dan \(\phi_2 = 0.2\) sebanyak 300 observasi dan \(c=0\).

set.seed(1234)

Nilai-nilai selanjutnya dapat dicari melalui loop. Bentuk loop dapat dilihat dari rumus AR(2) yang hendak dibangkitkan:

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

n<-length(wn)
n
## [1] 300
ar <- c(1:n) 
for (i in 3:n) {ar[i]<-wn[i]+0.5*ar[i-1]+0.2*ar[i-2]}
ar
##   [1]  1.000000000  2.000000000  2.358410538  3.423376729  1.493645458
##   [6]  1.130264367 -0.060371791  0.096814960  0.151110289 -1.010354302
##  [11] -0.049019895 -0.641834578 -0.250417450  0.849131015  0.738419780
##  [16]  0.765912646  2.633258296  1.220385269  1.630167029  1.059678245
##  [21]  0.580978909  0.667678483 -0.259138578 -0.683761948 -0.221391806
##  [26] -1.604173769 -0.213930702  0.158580013 -1.162158133 -0.702083510
##  [31] -0.481301065 -1.307570317 -2.648980777  1.055055585 -1.474818698
##  [36]  0.650072114 -0.298121954  0.644170514 -0.609487616  0.320153631
##  [41]  0.173081808 -0.225603654  0.752114808 -0.231603607  0.073534131
##  [46]  1.128074615 -1.320016887  0.563329480  0.966009393  1.641458040
##  [51] -0.181351671  1.126749667  0.930794547 -0.395979407 -0.787413145
##  [56]  0.756651254  0.448284565 -0.006188735 -0.402847315 -0.789322221
##  [61] -1.394272386 -0.017224758 -0.436960179 -1.003647517 -0.482731246
##  [66]  1.200580510 -1.357029310 -0.930826718 -1.155243699 -1.547021507
##  [71] -0.561682476 -0.770196750 -0.996393512 -0.844582997 -0.220245872
##  [76] -0.634064568 -1.210701233 -1.194599948 -1.180215555 -0.857166771
##  [81] -0.073619954 -0.305730442 -0.610752850 -0.179916024 -0.453207405
##  [86]  0.575059380  0.029506058  0.369124286 -0.369389426 -1.439371437
##  [91] -2.580264091 -3.846993389 -0.650770188 -0.474449776  0.573625013
##  [96] -1.206153923 -1.289922491  0.467978761 -1.287353291  1.934414935
## [101]  1.419469093  2.167045953 -0.167258987 -0.631887475  0.408692430
## [106]  1.125770497  1.065848703  0.955507659  0.482117267  0.675735467
## [111]  1.077083288  0.094569737  1.345299536  2.040185728  2.209929536
## [116]  0.191659822  1.391907928 -0.260201514  0.284416289 -0.203994990
## [121]  0.190304667 -0.334207147  0.333346448 -0.095484088  1.156006876
## [126]  2.305021606  0.853757739  2.171660187  1.229688020  0.630624987
## [131] -0.684114675 -0.197965266 -1.894300513 -2.991714903 -0.272828154
## [136] -0.621288154 -0.380385299 -1.130342332 -1.578414071 -1.234742297
## [141] -0.583858446 -1.367603817 -1.484603274 -2.606332693 -1.741110943
## [146] -0.262528619 -1.329331775 -3.397964910 -2.445416088 -0.993724297
## [151]  0.873033231  1.565498555  1.030659879  1.785069610  0.603735067
## [156]  1.017174810  0.702298708  1.330790721  2.003112588  2.201073249
## [161]  1.650999635  1.977253632  2.095841696  1.656810744  1.900196026
## [166] -0.156174632 -0.059213025 -0.420627894 -1.499503812 -0.026603240
## [171] -0.992008514 -0.609880661 -2.449106909 -0.966648204 -1.407208199
## [176] -1.556037093 -2.577750155 -1.904878559 -2.899744764 -0.376239349
## [181] -0.545456894  1.519368889  0.929533576  0.028405312  1.766398935
## [186]  0.852328810  0.315687323 -0.345119007  0.096440184 -0.088496531
## [191]  0.201042183 -0.710538302 -1.146985689  0.430820144  0.058965943
## [196]  0.222453165  0.168895088 -0.936217007 -0.764905356 -0.098113514
## [201] -0.521395095 -2.167057995 -1.286200275  0.028515650  0.379966920
## [206]  2.081134596  0.375135366  1.361508338  1.451459958  0.317544340
## [211] -0.201321611  0.051634295  1.222490559  0.410568573  1.234596136
## [216]  1.809873443  0.832012207  1.385872595  0.197684212  1.094964533
## [221]  0.314432850 -0.654468202 -0.847672547  0.011185715  1.886187597
## [226] -0.200995186 -0.013974633  1.257313976  0.938248780 -0.536920002
## [231]  0.028811783 -0.123805783 -0.363757155 -1.879405795 -2.145201779
## [236] -1.585458308 -1.772037198 -1.823398782 -3.165142382 -1.661372874
## [241] -1.543825070 -0.887480278  0.210896443 -2.040556564 -1.504122393
## [246] -1.973925914 -2.281461343 -2.146427219  0.435123866  0.702927839
## [251]  1.023479327  0.520063498 -0.194060073  0.322371977  0.319904489
## [256] -0.322656573 -1.451714878 -0.560415107 -0.361011423 -1.238519380
## [261]  0.157753470  0.252279615 -1.546416337  0.085180837  0.240925424
## [266] -0.136471145  0.886031001  1.083848900  1.497987565  0.544557445
## [271]  2.424466026  0.809483596  0.028180091  1.273429321  1.517804510
## [276] -0.400145697 -0.336391765 -0.860893600 -1.232118293 -0.958811098
## [281] -0.902196232 -0.428831732 -1.229477301 -1.574136077 -0.740645726
## [286] -0.219304111  0.880760700 -0.235167122 -0.259011044  1.722413625
## [291] -0.453987964 -0.329042437 -1.049485095  0.765571541  0.371534285
## [296]  1.446070852  1.369443083  0.486459536 -0.266481248 -1.362823066

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

ar1 <- arima.sim(list(order=c(2,0,0), ar=c(0.5,0.2)), n=300)
ar1
## Time Series:
## Start = 1 
## End = 300 
## Frequency = 1 
##   [1] -1.7066325607 -0.4040571790 -1.5670108246 -0.8794551485 -1.6890783403
##   [6]  0.0818673463 -0.7724750738 -1.0793041051 -1.1954051279 -2.4426568541
##  [11] -2.6280287147 -3.9825853771 -3.8578916236 -3.0197567460 -2.7473542381
##  [16] -0.5281322030 -1.8821796728 -1.9020809107 -1.6080993919 -2.1788059544
##  [21] -2.3795371734 -2.7328479702 -3.0943173054 -2.6175563658 -2.4244916012
##  [26] -3.5417883306 -2.8378684102 -3.2361814957 -3.2006264394 -2.4098590424
##  [31] -1.2819989901  0.5248461693 -0.7673301373  1.3272137941 -0.6476676781
##  [36]  0.5981973841  2.7185562271  1.4441572006  0.5961562661  0.5793048174
##  [41]  2.1859681101  0.0702372818  1.8401394423  2.2636819683  1.8363416698
##  [46]  1.3778000670  0.6006996293  0.2093858952  0.8731194410  2.5487077608
##  [51]  1.2955793561 -0.2331697165 -0.5810507641 -0.0788975632 -0.4727180494
##  [56] -0.4299284950 -0.4795019342 -1.6980385520 -1.1187068333 -0.0487288699
##  [61]  0.4495029101  0.7650030321  0.0696701226 -0.0037581027 -1.1824729064
##  [66] -0.6451468925 -0.3038720265  1.4249986155  1.6532381548  0.6160353575
##  [71]  0.9942156067 -0.5142931687  0.8199001639  1.2800082016  2.9251012390
##  [76]  2.1330757942  1.1768396714  1.0810284882  0.2734043960 -0.4730906915
##  [81] -0.0148751868 -0.9983203579 -0.3339498287 -0.0116707249 -0.1247304451
##  [86] -0.2606339862 -0.8043328337 -1.5640604455 -0.0936225868 -0.3372608564
##  [91]  0.6437856717 -0.9898471868 -0.1971400453  0.3766268470  0.1226090380
##  [96] -0.0547622802 -0.7847659792  1.6548265421  1.4209615284  2.8656543750
## [101]  1.7970791343  0.8402611432 -0.7337417216 -0.8349184633 -0.3379060434
## [106]  0.6777536331  0.5240457429 -0.7743747148  0.3863361197 -1.6118078179
## [111] -1.0944889332 -1.1857243590 -2.7600060135 -0.6970903562 -1.5234179757
## [116] -1.2351637091  0.4728824429  0.6260828907  0.2991862369  0.7885724750
## [121]  0.8533952922  2.2472685885  1.5702067563  1.7408297190  1.5320081858
## [126]  0.7369323892  0.7724872949  2.1723747704  0.3650923700  0.7387811377
## [131]  1.8045397037  0.8154049925 -0.2847723710 -0.8490887926 -0.8716259002
## [136] -1.4529807817 -1.1614549630 -1.2857433442 -1.0582134624 -0.3791993022
## [141]  0.2233907846  1.7140612756  0.8330151412  0.4384799130  1.8568487018
## [146]  2.7204497313  1.7748386436  1.0988519489 -0.9178417144  1.1721119313
## [151] -0.4350948113 -1.1068878138  2.4033030170  1.2152952540  1.0550496203
## [156] -1.9616356620 -0.8695984954  0.1489053548  0.3144018938  1.0993041795
## [161]  2.5962646692  2.6871016845  1.3540667607  1.9186338957  1.0317140262
## [166]  0.3615130033 -2.4686593481 -1.9516739257 -0.9817541973  1.2868206557
## [171]  0.9477541027  1.3514513862 -0.1006266965  0.3826316371 -1.9070470626
## [176] -0.3917703833  0.1194741744  0.1668969261  0.8080768135  0.7490988206
## [181]  1.2966271350  2.6405969578  2.6919867472  1.9067767226  0.3773367460
## [186]  0.9880815399  0.1692728818  1.7757458515 -0.6853534377 -0.4032793370
## [191]  0.0832980172 -0.1907433953 -0.6848632095 -0.6853013528  0.1499127816
## [196]  0.8330681005  1.1067292384  2.9934617597  2.8915743002  2.3321892304
## [201]  1.0846393814  3.9278976675  2.8582922105  1.5304052946  1.5233531725
## [206]  0.7433643447  0.4016485886 -0.5840061774 -0.0948280260  0.1549449902
## [211] -1.0190352329 -3.7116807513 -2.3145220749 -1.8700793574 -0.8036703196
## [216] -0.7167158631 -0.1056931007 -1.2939618975  0.0430556884  0.4816241945
## [221]  0.5010743040  1.7041364270  1.3567515456  1.2835673280  1.1811778773
## [226]  1.2842329813  1.9384759716  1.6782749786  1.8900312993  0.1442970914
## [231]  0.0796572885  1.5456576523 -0.4351434667  0.3496281845  0.4927882044
## [236]  1.2921230609  0.3957424348  0.6149212691 -1.3766459451 -0.2267426716
## [241] -1.0552655544 -0.8116279358 -1.8046323613 -0.6797064460 -0.0342001796
## [246] -0.4576552685  1.5893433939  1.3737000138  1.9533512590  3.3008186339
## [251]  1.3899659595  2.1637659794  2.3464567956  1.5998107976  1.5882491158
## [256]  0.1022648147  0.8389497782 -0.2610424796  0.8509515784 -0.4381634907
## [261]  0.2705060574 -0.7989023226 -0.5911131289 -2.0081960401 -0.9938866129
## [266]  0.0868608749  0.0279006379 -1.7349067195 -1.4824069288  0.5678582287
## [271]  1.7972531142 -0.1628385652 -0.0886719186  0.2767217766  0.4397827255
## [276] -0.3047212710 -1.0176827923 -0.7492142373  0.4316645378  0.0896160364
## [281] -0.5178872938 -0.7453946617  1.1381167058 -0.0269803907  0.9773099069
## [286]  1.9549775615  1.6166156658  0.7775814752  0.6721122445 -0.0007075509
## [291]  1.3617857908  0.5311978199  2.0879394521  0.5885967547  0.0647690171
## [296]  0.2932360185  0.1837604607 -0.3539240905 -1.7216067631 -0.9015217784

Karakteristik AR(2)

Plot Time Series

ts.plot(ar)

Berdasarkan plot time series tersebut terlihat bahwa data cenderung stasioner pada rataan

Plot ACF

acf(ar)

Berdasarkan plot ACF tersebut terlihat bahwa plot ACF cenderung tails off

Plot PACF

pacf(ar)

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

Plot EACF

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

Berdasarkan pola segitiga nol pada plot EACF, terlihat bahwa segitiga nol berada pada ordo AR(1) dan ordo MA(2)

Scatterplot Antar Lag

Korelasi antara \(Y_t\) dengan \(Y_{t-1}\)

#Yt
yt_ar <- ar[-1]
yt_ar
##   [1]  2.000000000  2.358410538  3.423376729  1.493645458  1.130264367
##   [6] -0.060371791  0.096814960  0.151110289 -1.010354302 -0.049019895
##  [11] -0.641834578 -0.250417450  0.849131015  0.738419780  0.765912646
##  [16]  2.633258296  1.220385269  1.630167029  1.059678245  0.580978909
##  [21]  0.667678483 -0.259138578 -0.683761948 -0.221391806 -1.604173769
##  [26] -0.213930702  0.158580013 -1.162158133 -0.702083510 -0.481301065
##  [31] -1.307570317 -2.648980777  1.055055585 -1.474818698  0.650072114
##  [36] -0.298121954  0.644170514 -0.609487616  0.320153631  0.173081808
##  [41] -0.225603654  0.752114808 -0.231603607  0.073534131  1.128074615
##  [46] -1.320016887  0.563329480  0.966009393  1.641458040 -0.181351671
##  [51]  1.126749667  0.930794547 -0.395979407 -0.787413145  0.756651254
##  [56]  0.448284565 -0.006188735 -0.402847315 -0.789322221 -1.394272386
##  [61] -0.017224758 -0.436960179 -1.003647517 -0.482731246  1.200580510
##  [66] -1.357029310 -0.930826718 -1.155243699 -1.547021507 -0.561682476
##  [71] -0.770196750 -0.996393512 -0.844582997 -0.220245872 -0.634064568
##  [76] -1.210701233 -1.194599948 -1.180215555 -0.857166771 -0.073619954
##  [81] -0.305730442 -0.610752850 -0.179916024 -0.453207405  0.575059380
##  [86]  0.029506058  0.369124286 -0.369389426 -1.439371437 -2.580264091
##  [91] -3.846993389 -0.650770188 -0.474449776  0.573625013 -1.206153923
##  [96] -1.289922491  0.467978761 -1.287353291  1.934414935  1.419469093
## [101]  2.167045953 -0.167258987 -0.631887475  0.408692430  1.125770497
## [106]  1.065848703  0.955507659  0.482117267  0.675735467  1.077083288
## [111]  0.094569737  1.345299536  2.040185728  2.209929536  0.191659822
## [116]  1.391907928 -0.260201514  0.284416289 -0.203994990  0.190304667
## [121] -0.334207147  0.333346448 -0.095484088  1.156006876  2.305021606
## [126]  0.853757739  2.171660187  1.229688020  0.630624987 -0.684114675
## [131] -0.197965266 -1.894300513 -2.991714903 -0.272828154 -0.621288154
## [136] -0.380385299 -1.130342332 -1.578414071 -1.234742297 -0.583858446
## [141] -1.367603817 -1.484603274 -2.606332693 -1.741110943 -0.262528619
## [146] -1.329331775 -3.397964910 -2.445416088 -0.993724297  0.873033231
## [151]  1.565498555  1.030659879  1.785069610  0.603735067  1.017174810
## [156]  0.702298708  1.330790721  2.003112588  2.201073249  1.650999635
## [161]  1.977253632  2.095841696  1.656810744  1.900196026 -0.156174632
## [166] -0.059213025 -0.420627894 -1.499503812 -0.026603240 -0.992008514
## [171] -0.609880661 -2.449106909 -0.966648204 -1.407208199 -1.556037093
## [176] -2.577750155 -1.904878559 -2.899744764 -0.376239349 -0.545456894
## [181]  1.519368889  0.929533576  0.028405312  1.766398935  0.852328810
## [186]  0.315687323 -0.345119007  0.096440184 -0.088496531  0.201042183
## [191] -0.710538302 -1.146985689  0.430820144  0.058965943  0.222453165
## [196]  0.168895088 -0.936217007 -0.764905356 -0.098113514 -0.521395095
## [201] -2.167057995 -1.286200275  0.028515650  0.379966920  2.081134596
## [206]  0.375135366  1.361508338  1.451459958  0.317544340 -0.201321611
## [211]  0.051634295  1.222490559  0.410568573  1.234596136  1.809873443
## [216]  0.832012207  1.385872595  0.197684212  1.094964533  0.314432850
## [221] -0.654468202 -0.847672547  0.011185715  1.886187597 -0.200995186
## [226] -0.013974633  1.257313976  0.938248780 -0.536920002  0.028811783
## [231] -0.123805783 -0.363757155 -1.879405795 -2.145201779 -1.585458308
## [236] -1.772037198 -1.823398782 -3.165142382 -1.661372874 -1.543825070
## [241] -0.887480278  0.210896443 -2.040556564 -1.504122393 -1.973925914
## [246] -2.281461343 -2.146427219  0.435123866  0.702927839  1.023479327
## [251]  0.520063498 -0.194060073  0.322371977  0.319904489 -0.322656573
## [256] -1.451714878 -0.560415107 -0.361011423 -1.238519380  0.157753470
## [261]  0.252279615 -1.546416337  0.085180837  0.240925424 -0.136471145
## [266]  0.886031001  1.083848900  1.497987565  0.544557445  2.424466026
## [271]  0.809483596  0.028180091  1.273429321  1.517804510 -0.400145697
## [276] -0.336391765 -0.860893600 -1.232118293 -0.958811098 -0.902196232
## [281] -0.428831732 -1.229477301 -1.574136077 -0.740645726 -0.219304111
## [286]  0.880760700 -0.235167122 -0.259011044  1.722413625 -0.453987964
## [291] -0.329042437 -1.049485095  0.765571541  0.371534285  1.446070852
## [296]  1.369443083  0.486459536 -0.266481248 -1.362823066
#Yt-1
yt_1_ar <- ar[-300]
yt_1_ar
##   [1]  1.000000000  2.000000000  2.358410538  3.423376729  1.493645458
##   [6]  1.130264367 -0.060371791  0.096814960  0.151110289 -1.010354302
##  [11] -0.049019895 -0.641834578 -0.250417450  0.849131015  0.738419780
##  [16]  0.765912646  2.633258296  1.220385269  1.630167029  1.059678245
##  [21]  0.580978909  0.667678483 -0.259138578 -0.683761948 -0.221391806
##  [26] -1.604173769 -0.213930702  0.158580013 -1.162158133 -0.702083510
##  [31] -0.481301065 -1.307570317 -2.648980777  1.055055585 -1.474818698
##  [36]  0.650072114 -0.298121954  0.644170514 -0.609487616  0.320153631
##  [41]  0.173081808 -0.225603654  0.752114808 -0.231603607  0.073534131
##  [46]  1.128074615 -1.320016887  0.563329480  0.966009393  1.641458040
##  [51] -0.181351671  1.126749667  0.930794547 -0.395979407 -0.787413145
##  [56]  0.756651254  0.448284565 -0.006188735 -0.402847315 -0.789322221
##  [61] -1.394272386 -0.017224758 -0.436960179 -1.003647517 -0.482731246
##  [66]  1.200580510 -1.357029310 -0.930826718 -1.155243699 -1.547021507
##  [71] -0.561682476 -0.770196750 -0.996393512 -0.844582997 -0.220245872
##  [76] -0.634064568 -1.210701233 -1.194599948 -1.180215555 -0.857166771
##  [81] -0.073619954 -0.305730442 -0.610752850 -0.179916024 -0.453207405
##  [86]  0.575059380  0.029506058  0.369124286 -0.369389426 -1.439371437
##  [91] -2.580264091 -3.846993389 -0.650770188 -0.474449776  0.573625013
##  [96] -1.206153923 -1.289922491  0.467978761 -1.287353291  1.934414935
## [101]  1.419469093  2.167045953 -0.167258987 -0.631887475  0.408692430
## [106]  1.125770497  1.065848703  0.955507659  0.482117267  0.675735467
## [111]  1.077083288  0.094569737  1.345299536  2.040185728  2.209929536
## [116]  0.191659822  1.391907928 -0.260201514  0.284416289 -0.203994990
## [121]  0.190304667 -0.334207147  0.333346448 -0.095484088  1.156006876
## [126]  2.305021606  0.853757739  2.171660187  1.229688020  0.630624987
## [131] -0.684114675 -0.197965266 -1.894300513 -2.991714903 -0.272828154
## [136] -0.621288154 -0.380385299 -1.130342332 -1.578414071 -1.234742297
## [141] -0.583858446 -1.367603817 -1.484603274 -2.606332693 -1.741110943
## [146] -0.262528619 -1.329331775 -3.397964910 -2.445416088 -0.993724297
## [151]  0.873033231  1.565498555  1.030659879  1.785069610  0.603735067
## [156]  1.017174810  0.702298708  1.330790721  2.003112588  2.201073249
## [161]  1.650999635  1.977253632  2.095841696  1.656810744  1.900196026
## [166] -0.156174632 -0.059213025 -0.420627894 -1.499503812 -0.026603240
## [171] -0.992008514 -0.609880661 -2.449106909 -0.966648204 -1.407208199
## [176] -1.556037093 -2.577750155 -1.904878559 -2.899744764 -0.376239349
## [181] -0.545456894  1.519368889  0.929533576  0.028405312  1.766398935
## [186]  0.852328810  0.315687323 -0.345119007  0.096440184 -0.088496531
## [191]  0.201042183 -0.710538302 -1.146985689  0.430820144  0.058965943
## [196]  0.222453165  0.168895088 -0.936217007 -0.764905356 -0.098113514
## [201] -0.521395095 -2.167057995 -1.286200275  0.028515650  0.379966920
## [206]  2.081134596  0.375135366  1.361508338  1.451459958  0.317544340
## [211] -0.201321611  0.051634295  1.222490559  0.410568573  1.234596136
## [216]  1.809873443  0.832012207  1.385872595  0.197684212  1.094964533
## [221]  0.314432850 -0.654468202 -0.847672547  0.011185715  1.886187597
## [226] -0.200995186 -0.013974633  1.257313976  0.938248780 -0.536920002
## [231]  0.028811783 -0.123805783 -0.363757155 -1.879405795 -2.145201779
## [236] -1.585458308 -1.772037198 -1.823398782 -3.165142382 -1.661372874
## [241] -1.543825070 -0.887480278  0.210896443 -2.040556564 -1.504122393
## [246] -1.973925914 -2.281461343 -2.146427219  0.435123866  0.702927839
## [251]  1.023479327  0.520063498 -0.194060073  0.322371977  0.319904489
## [256] -0.322656573 -1.451714878 -0.560415107 -0.361011423 -1.238519380
## [261]  0.157753470  0.252279615 -1.546416337  0.085180837  0.240925424
## [266] -0.136471145  0.886031001  1.083848900  1.497987565  0.544557445
## [271]  2.424466026  0.809483596  0.028180091  1.273429321  1.517804510
## [276] -0.400145697 -0.336391765 -0.860893600 -1.232118293 -0.958811098
## [281] -0.902196232 -0.428831732 -1.229477301 -1.574136077 -0.740645726
## [286] -0.219304111  0.880760700 -0.235167122 -0.259011044  1.722413625
## [291] -0.453987964 -0.329042437 -1.049485095  0.765571541  0.371534285
## [296]  1.446070852  1.369443083  0.486459536 -0.266481248
plot(y=yt_ar,x=yt_1_ar)

Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-1}\). Hal ini sesuai dengan teori yang ada

cor(yt_ar,yt_1_ar)
## [1] 0.6047724

Korelasi antara \(Y_t\) dengan \(Y_{t-1}\) dari hasil simulasi mendekati perhitungan teoritis yaitu \(\rho_1=\frac{\phi_1}{1-\phi_2}=\frac{0.5}{1-0.2}=0.625\)

Korelasi antara \(Y_t\) dengan \(Y_{t-2}\)

#Yt
yt_ar2 <- ar[-c(1,2)]
yt_ar2
##   [1]  2.358410538  3.423376729  1.493645458  1.130264367 -0.060371791
##   [6]  0.096814960  0.151110289 -1.010354302 -0.049019895 -0.641834578
##  [11] -0.250417450  0.849131015  0.738419780  0.765912646  2.633258296
##  [16]  1.220385269  1.630167029  1.059678245  0.580978909  0.667678483
##  [21] -0.259138578 -0.683761948 -0.221391806 -1.604173769 -0.213930702
##  [26]  0.158580013 -1.162158133 -0.702083510 -0.481301065 -1.307570317
##  [31] -2.648980777  1.055055585 -1.474818698  0.650072114 -0.298121954
##  [36]  0.644170514 -0.609487616  0.320153631  0.173081808 -0.225603654
##  [41]  0.752114808 -0.231603607  0.073534131  1.128074615 -1.320016887
##  [46]  0.563329480  0.966009393  1.641458040 -0.181351671  1.126749667
##  [51]  0.930794547 -0.395979407 -0.787413145  0.756651254  0.448284565
##  [56] -0.006188735 -0.402847315 -0.789322221 -1.394272386 -0.017224758
##  [61] -0.436960179 -1.003647517 -0.482731246  1.200580510 -1.357029310
##  [66] -0.930826718 -1.155243699 -1.547021507 -0.561682476 -0.770196750
##  [71] -0.996393512 -0.844582997 -0.220245872 -0.634064568 -1.210701233
##  [76] -1.194599948 -1.180215555 -0.857166771 -0.073619954 -0.305730442
##  [81] -0.610752850 -0.179916024 -0.453207405  0.575059380  0.029506058
##  [86]  0.369124286 -0.369389426 -1.439371437 -2.580264091 -3.846993389
##  [91] -0.650770188 -0.474449776  0.573625013 -1.206153923 -1.289922491
##  [96]  0.467978761 -1.287353291  1.934414935  1.419469093  2.167045953
## [101] -0.167258987 -0.631887475  0.408692430  1.125770497  1.065848703
## [106]  0.955507659  0.482117267  0.675735467  1.077083288  0.094569737
## [111]  1.345299536  2.040185728  2.209929536  0.191659822  1.391907928
## [116] -0.260201514  0.284416289 -0.203994990  0.190304667 -0.334207147
## [121]  0.333346448 -0.095484088  1.156006876  2.305021606  0.853757739
## [126]  2.171660187  1.229688020  0.630624987 -0.684114675 -0.197965266
## [131] -1.894300513 -2.991714903 -0.272828154 -0.621288154 -0.380385299
## [136] -1.130342332 -1.578414071 -1.234742297 -0.583858446 -1.367603817
## [141] -1.484603274 -2.606332693 -1.741110943 -0.262528619 -1.329331775
## [146] -3.397964910 -2.445416088 -0.993724297  0.873033231  1.565498555
## [151]  1.030659879  1.785069610  0.603735067  1.017174810  0.702298708
## [156]  1.330790721  2.003112588  2.201073249  1.650999635  1.977253632
## [161]  2.095841696  1.656810744  1.900196026 -0.156174632 -0.059213025
## [166] -0.420627894 -1.499503812 -0.026603240 -0.992008514 -0.609880661
## [171] -2.449106909 -0.966648204 -1.407208199 -1.556037093 -2.577750155
## [176] -1.904878559 -2.899744764 -0.376239349 -0.545456894  1.519368889
## [181]  0.929533576  0.028405312  1.766398935  0.852328810  0.315687323
## [186] -0.345119007  0.096440184 -0.088496531  0.201042183 -0.710538302
## [191] -1.146985689  0.430820144  0.058965943  0.222453165  0.168895088
## [196] -0.936217007 -0.764905356 -0.098113514 -0.521395095 -2.167057995
## [201] -1.286200275  0.028515650  0.379966920  2.081134596  0.375135366
## [206]  1.361508338  1.451459958  0.317544340 -0.201321611  0.051634295
## [211]  1.222490559  0.410568573  1.234596136  1.809873443  0.832012207
## [216]  1.385872595  0.197684212  1.094964533  0.314432850 -0.654468202
## [221] -0.847672547  0.011185715  1.886187597 -0.200995186 -0.013974633
## [226]  1.257313976  0.938248780 -0.536920002  0.028811783 -0.123805783
## [231] -0.363757155 -1.879405795 -2.145201779 -1.585458308 -1.772037198
## [236] -1.823398782 -3.165142382 -1.661372874 -1.543825070 -0.887480278
## [241]  0.210896443 -2.040556564 -1.504122393 -1.973925914 -2.281461343
## [246] -2.146427219  0.435123866  0.702927839  1.023479327  0.520063498
## [251] -0.194060073  0.322371977  0.319904489 -0.322656573 -1.451714878
## [256] -0.560415107 -0.361011423 -1.238519380  0.157753470  0.252279615
## [261] -1.546416337  0.085180837  0.240925424 -0.136471145  0.886031001
## [266]  1.083848900  1.497987565  0.544557445  2.424466026  0.809483596
## [271]  0.028180091  1.273429321  1.517804510 -0.400145697 -0.336391765
## [276] -0.860893600 -1.232118293 -0.958811098 -0.902196232 -0.428831732
## [281] -1.229477301 -1.574136077 -0.740645726 -0.219304111  0.880760700
## [286] -0.235167122 -0.259011044  1.722413625 -0.453987964 -0.329042437
## [291] -1.049485095  0.765571541  0.371534285  1.446070852  1.369443083
## [296]  0.486459536 -0.266481248 -1.362823066
#Yt-2
yt_2_ar <- ar[-c(299,300)]
yt_2_ar
##   [1]  1.000000000  2.000000000  2.358410538  3.423376729  1.493645458
##   [6]  1.130264367 -0.060371791  0.096814960  0.151110289 -1.010354302
##  [11] -0.049019895 -0.641834578 -0.250417450  0.849131015  0.738419780
##  [16]  0.765912646  2.633258296  1.220385269  1.630167029  1.059678245
##  [21]  0.580978909  0.667678483 -0.259138578 -0.683761948 -0.221391806
##  [26] -1.604173769 -0.213930702  0.158580013 -1.162158133 -0.702083510
##  [31] -0.481301065 -1.307570317 -2.648980777  1.055055585 -1.474818698
##  [36]  0.650072114 -0.298121954  0.644170514 -0.609487616  0.320153631
##  [41]  0.173081808 -0.225603654  0.752114808 -0.231603607  0.073534131
##  [46]  1.128074615 -1.320016887  0.563329480  0.966009393  1.641458040
##  [51] -0.181351671  1.126749667  0.930794547 -0.395979407 -0.787413145
##  [56]  0.756651254  0.448284565 -0.006188735 -0.402847315 -0.789322221
##  [61] -1.394272386 -0.017224758 -0.436960179 -1.003647517 -0.482731246
##  [66]  1.200580510 -1.357029310 -0.930826718 -1.155243699 -1.547021507
##  [71] -0.561682476 -0.770196750 -0.996393512 -0.844582997 -0.220245872
##  [76] -0.634064568 -1.210701233 -1.194599948 -1.180215555 -0.857166771
##  [81] -0.073619954 -0.305730442 -0.610752850 -0.179916024 -0.453207405
##  [86]  0.575059380  0.029506058  0.369124286 -0.369389426 -1.439371437
##  [91] -2.580264091 -3.846993389 -0.650770188 -0.474449776  0.573625013
##  [96] -1.206153923 -1.289922491  0.467978761 -1.287353291  1.934414935
## [101]  1.419469093  2.167045953 -0.167258987 -0.631887475  0.408692430
## [106]  1.125770497  1.065848703  0.955507659  0.482117267  0.675735467
## [111]  1.077083288  0.094569737  1.345299536  2.040185728  2.209929536
## [116]  0.191659822  1.391907928 -0.260201514  0.284416289 -0.203994990
## [121]  0.190304667 -0.334207147  0.333346448 -0.095484088  1.156006876
## [126]  2.305021606  0.853757739  2.171660187  1.229688020  0.630624987
## [131] -0.684114675 -0.197965266 -1.894300513 -2.991714903 -0.272828154
## [136] -0.621288154 -0.380385299 -1.130342332 -1.578414071 -1.234742297
## [141] -0.583858446 -1.367603817 -1.484603274 -2.606332693 -1.741110943
## [146] -0.262528619 -1.329331775 -3.397964910 -2.445416088 -0.993724297
## [151]  0.873033231  1.565498555  1.030659879  1.785069610  0.603735067
## [156]  1.017174810  0.702298708  1.330790721  2.003112588  2.201073249
## [161]  1.650999635  1.977253632  2.095841696  1.656810744  1.900196026
## [166] -0.156174632 -0.059213025 -0.420627894 -1.499503812 -0.026603240
## [171] -0.992008514 -0.609880661 -2.449106909 -0.966648204 -1.407208199
## [176] -1.556037093 -2.577750155 -1.904878559 -2.899744764 -0.376239349
## [181] -0.545456894  1.519368889  0.929533576  0.028405312  1.766398935
## [186]  0.852328810  0.315687323 -0.345119007  0.096440184 -0.088496531
## [191]  0.201042183 -0.710538302 -1.146985689  0.430820144  0.058965943
## [196]  0.222453165  0.168895088 -0.936217007 -0.764905356 -0.098113514
## [201] -0.521395095 -2.167057995 -1.286200275  0.028515650  0.379966920
## [206]  2.081134596  0.375135366  1.361508338  1.451459958  0.317544340
## [211] -0.201321611  0.051634295  1.222490559  0.410568573  1.234596136
## [216]  1.809873443  0.832012207  1.385872595  0.197684212  1.094964533
## [221]  0.314432850 -0.654468202 -0.847672547  0.011185715  1.886187597
## [226] -0.200995186 -0.013974633  1.257313976  0.938248780 -0.536920002
## [231]  0.028811783 -0.123805783 -0.363757155 -1.879405795 -2.145201779
## [236] -1.585458308 -1.772037198 -1.823398782 -3.165142382 -1.661372874
## [241] -1.543825070 -0.887480278  0.210896443 -2.040556564 -1.504122393
## [246] -1.973925914 -2.281461343 -2.146427219  0.435123866  0.702927839
## [251]  1.023479327  0.520063498 -0.194060073  0.322371977  0.319904489
## [256] -0.322656573 -1.451714878 -0.560415107 -0.361011423 -1.238519380
## [261]  0.157753470  0.252279615 -1.546416337  0.085180837  0.240925424
## [266] -0.136471145  0.886031001  1.083848900  1.497987565  0.544557445
## [271]  2.424466026  0.809483596  0.028180091  1.273429321  1.517804510
## [276] -0.400145697 -0.336391765 -0.860893600 -1.232118293 -0.958811098
## [281] -0.902196232 -0.428831732 -1.229477301 -1.574136077 -0.740645726
## [286] -0.219304111  0.880760700 -0.235167122 -0.259011044  1.722413625
## [291] -0.453987964 -0.329042437 -1.049485095  0.765571541  0.371534285
## [296]  1.446070852  1.369443083  0.486459536
plot(y=yt_ar2,x=yt_2_ar)

Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan yang cenderung positif antara \(Y_t\) dengan \(Y_{t-2}\). Hal ini sesuai dengan teori yang ada

cor(yt_ar2,yt_2_ar)
## [1] 0.4912454

Korelasi antara \(Y_t\) dengan \(Y_{t-2}\) dari hasil simulasi mendekati perhitungan teoritis yaitu \(\rho_2=\frac{\phi_2(1-\phi_2)+\phi_1^2}{1-\phi_2}=\frac{0.2(1-0.2)+0.5^2}{1-0.2}=\frac{0.41}{0.8}=0.5125\).

Korelasi antara \(Y_t\) dengan \(Y_{t-3}\)

#Yt 
yt_ar3 <- ar[-c(1,2,3)] 
yt_ar3 
##   [1]  3.423376729  1.493645458  1.130264367 -0.060371791  0.096814960
##   [6]  0.151110289 -1.010354302 -0.049019895 -0.641834578 -0.250417450
##  [11]  0.849131015  0.738419780  0.765912646  2.633258296  1.220385269
##  [16]  1.630167029  1.059678245  0.580978909  0.667678483 -0.259138578
##  [21] -0.683761948 -0.221391806 -1.604173769 -0.213930702  0.158580013
##  [26] -1.162158133 -0.702083510 -0.481301065 -1.307570317 -2.648980777
##  [31]  1.055055585 -1.474818698  0.650072114 -0.298121954  0.644170514
##  [36] -0.609487616  0.320153631  0.173081808 -0.225603654  0.752114808
##  [41] -0.231603607  0.073534131  1.128074615 -1.320016887  0.563329480
##  [46]  0.966009393  1.641458040 -0.181351671  1.126749667  0.930794547
##  [51] -0.395979407 -0.787413145  0.756651254  0.448284565 -0.006188735
##  [56] -0.402847315 -0.789322221 -1.394272386 -0.017224758 -0.436960179
##  [61] -1.003647517 -0.482731246  1.200580510 -1.357029310 -0.930826718
##  [66] -1.155243699 -1.547021507 -0.561682476 -0.770196750 -0.996393512
##  [71] -0.844582997 -0.220245872 -0.634064568 -1.210701233 -1.194599948
##  [76] -1.180215555 -0.857166771 -0.073619954 -0.305730442 -0.610752850
##  [81] -0.179916024 -0.453207405  0.575059380  0.029506058  0.369124286
##  [86] -0.369389426 -1.439371437 -2.580264091 -3.846993389 -0.650770188
##  [91] -0.474449776  0.573625013 -1.206153923 -1.289922491  0.467978761
##  [96] -1.287353291  1.934414935  1.419469093  2.167045953 -0.167258987
## [101] -0.631887475  0.408692430  1.125770497  1.065848703  0.955507659
## [106]  0.482117267  0.675735467  1.077083288  0.094569737  1.345299536
## [111]  2.040185728  2.209929536  0.191659822  1.391907928 -0.260201514
## [116]  0.284416289 -0.203994990  0.190304667 -0.334207147  0.333346448
## [121] -0.095484088  1.156006876  2.305021606  0.853757739  2.171660187
## [126]  1.229688020  0.630624987 -0.684114675 -0.197965266 -1.894300513
## [131] -2.991714903 -0.272828154 -0.621288154 -0.380385299 -1.130342332
## [136] -1.578414071 -1.234742297 -0.583858446 -1.367603817 -1.484603274
## [141] -2.606332693 -1.741110943 -0.262528619 -1.329331775 -3.397964910
## [146] -2.445416088 -0.993724297  0.873033231  1.565498555  1.030659879
## [151]  1.785069610  0.603735067  1.017174810  0.702298708  1.330790721
## [156]  2.003112588  2.201073249  1.650999635  1.977253632  2.095841696
## [161]  1.656810744  1.900196026 -0.156174632 -0.059213025 -0.420627894
## [166] -1.499503812 -0.026603240 -0.992008514 -0.609880661 -2.449106909
## [171] -0.966648204 -1.407208199 -1.556037093 -2.577750155 -1.904878559
## [176] -2.899744764 -0.376239349 -0.545456894  1.519368889  0.929533576
## [181]  0.028405312  1.766398935  0.852328810  0.315687323 -0.345119007
## [186]  0.096440184 -0.088496531  0.201042183 -0.710538302 -1.146985689
## [191]  0.430820144  0.058965943  0.222453165  0.168895088 -0.936217007
## [196] -0.764905356 -0.098113514 -0.521395095 -2.167057995 -1.286200275
## [201]  0.028515650  0.379966920  2.081134596  0.375135366  1.361508338
## [206]  1.451459958  0.317544340 -0.201321611  0.051634295  1.222490559
## [211]  0.410568573  1.234596136  1.809873443  0.832012207  1.385872595
## [216]  0.197684212  1.094964533  0.314432850 -0.654468202 -0.847672547
## [221]  0.011185715  1.886187597 -0.200995186 -0.013974633  1.257313976
## [226]  0.938248780 -0.536920002  0.028811783 -0.123805783 -0.363757155
## [231] -1.879405795 -2.145201779 -1.585458308 -1.772037198 -1.823398782
## [236] -3.165142382 -1.661372874 -1.543825070 -0.887480278  0.210896443
## [241] -2.040556564 -1.504122393 -1.973925914 -2.281461343 -2.146427219
## [246]  0.435123866  0.702927839  1.023479327  0.520063498 -0.194060073
## [251]  0.322371977  0.319904489 -0.322656573 -1.451714878 -0.560415107
## [256] -0.361011423 -1.238519380  0.157753470  0.252279615 -1.546416337
## [261]  0.085180837  0.240925424 -0.136471145  0.886031001  1.083848900
## [266]  1.497987565  0.544557445  2.424466026  0.809483596  0.028180091
## [271]  1.273429321  1.517804510 -0.400145697 -0.336391765 -0.860893600
## [276] -1.232118293 -0.958811098 -0.902196232 -0.428831732 -1.229477301
## [281] -1.574136077 -0.740645726 -0.219304111  0.880760700 -0.235167122
## [286] -0.259011044  1.722413625 -0.453987964 -0.329042437 -1.049485095
## [291]  0.765571541  0.371534285  1.446070852  1.369443083  0.486459536
## [296] -0.266481248 -1.362823066
#Yt-2 
yt_3_ar <- ar[-c(298,299,300)] 
yt_3_ar
##   [1]  1.000000000  2.000000000  2.358410538  3.423376729  1.493645458
##   [6]  1.130264367 -0.060371791  0.096814960  0.151110289 -1.010354302
##  [11] -0.049019895 -0.641834578 -0.250417450  0.849131015  0.738419780
##  [16]  0.765912646  2.633258296  1.220385269  1.630167029  1.059678245
##  [21]  0.580978909  0.667678483 -0.259138578 -0.683761948 -0.221391806
##  [26] -1.604173769 -0.213930702  0.158580013 -1.162158133 -0.702083510
##  [31] -0.481301065 -1.307570317 -2.648980777  1.055055585 -1.474818698
##  [36]  0.650072114 -0.298121954  0.644170514 -0.609487616  0.320153631
##  [41]  0.173081808 -0.225603654  0.752114808 -0.231603607  0.073534131
##  [46]  1.128074615 -1.320016887  0.563329480  0.966009393  1.641458040
##  [51] -0.181351671  1.126749667  0.930794547 -0.395979407 -0.787413145
##  [56]  0.756651254  0.448284565 -0.006188735 -0.402847315 -0.789322221
##  [61] -1.394272386 -0.017224758 -0.436960179 -1.003647517 -0.482731246
##  [66]  1.200580510 -1.357029310 -0.930826718 -1.155243699 -1.547021507
##  [71] -0.561682476 -0.770196750 -0.996393512 -0.844582997 -0.220245872
##  [76] -0.634064568 -1.210701233 -1.194599948 -1.180215555 -0.857166771
##  [81] -0.073619954 -0.305730442 -0.610752850 -0.179916024 -0.453207405
##  [86]  0.575059380  0.029506058  0.369124286 -0.369389426 -1.439371437
##  [91] -2.580264091 -3.846993389 -0.650770188 -0.474449776  0.573625013
##  [96] -1.206153923 -1.289922491  0.467978761 -1.287353291  1.934414935
## [101]  1.419469093  2.167045953 -0.167258987 -0.631887475  0.408692430
## [106]  1.125770497  1.065848703  0.955507659  0.482117267  0.675735467
## [111]  1.077083288  0.094569737  1.345299536  2.040185728  2.209929536
## [116]  0.191659822  1.391907928 -0.260201514  0.284416289 -0.203994990
## [121]  0.190304667 -0.334207147  0.333346448 -0.095484088  1.156006876
## [126]  2.305021606  0.853757739  2.171660187  1.229688020  0.630624987
## [131] -0.684114675 -0.197965266 -1.894300513 -2.991714903 -0.272828154
## [136] -0.621288154 -0.380385299 -1.130342332 -1.578414071 -1.234742297
## [141] -0.583858446 -1.367603817 -1.484603274 -2.606332693 -1.741110943
## [146] -0.262528619 -1.329331775 -3.397964910 -2.445416088 -0.993724297
## [151]  0.873033231  1.565498555  1.030659879  1.785069610  0.603735067
## [156]  1.017174810  0.702298708  1.330790721  2.003112588  2.201073249
## [161]  1.650999635  1.977253632  2.095841696  1.656810744  1.900196026
## [166] -0.156174632 -0.059213025 -0.420627894 -1.499503812 -0.026603240
## [171] -0.992008514 -0.609880661 -2.449106909 -0.966648204 -1.407208199
## [176] -1.556037093 -2.577750155 -1.904878559 -2.899744764 -0.376239349
## [181] -0.545456894  1.519368889  0.929533576  0.028405312  1.766398935
## [186]  0.852328810  0.315687323 -0.345119007  0.096440184 -0.088496531
## [191]  0.201042183 -0.710538302 -1.146985689  0.430820144  0.058965943
## [196]  0.222453165  0.168895088 -0.936217007 -0.764905356 -0.098113514
## [201] -0.521395095 -2.167057995 -1.286200275  0.028515650  0.379966920
## [206]  2.081134596  0.375135366  1.361508338  1.451459958  0.317544340
## [211] -0.201321611  0.051634295  1.222490559  0.410568573  1.234596136
## [216]  1.809873443  0.832012207  1.385872595  0.197684212  1.094964533
## [221]  0.314432850 -0.654468202 -0.847672547  0.011185715  1.886187597
## [226] -0.200995186 -0.013974633  1.257313976  0.938248780 -0.536920002
## [231]  0.028811783 -0.123805783 -0.363757155 -1.879405795 -2.145201779
## [236] -1.585458308 -1.772037198 -1.823398782 -3.165142382 -1.661372874
## [241] -1.543825070 -0.887480278  0.210896443 -2.040556564 -1.504122393
## [246] -1.973925914 -2.281461343 -2.146427219  0.435123866  0.702927839
## [251]  1.023479327  0.520063498 -0.194060073  0.322371977  0.319904489
## [256] -0.322656573 -1.451714878 -0.560415107 -0.361011423 -1.238519380
## [261]  0.157753470  0.252279615 -1.546416337  0.085180837  0.240925424
## [266] -0.136471145  0.886031001  1.083848900  1.497987565  0.544557445
## [271]  2.424466026  0.809483596  0.028180091  1.273429321  1.517804510
## [276] -0.400145697 -0.336391765 -0.860893600 -1.232118293 -0.958811098
## [281] -0.902196232 -0.428831732 -1.229477301 -1.574136077 -0.740645726
## [286] -0.219304111  0.880760700 -0.235167122 -0.259011044  1.722413625
## [291] -0.453987964 -0.329042437 -1.049485095  0.765571541  0.371534285
## [296]  1.446070852  1.369443083
plot(y=yt_ar3,x=yt_3_ar)

Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan yang cenderung positif antara \(Y_t\) dengan \(Y_{t-2}\). Hal ini sesuai dengan teori yang ada

cor(yt_ar3,yt_3_ar)
## [1] 0.3556605

Korelasi antara \(Y_t\) dengan \(Y_{t-2}\) dari hasil simulasi mendekati perhitungan teoritis yaitu \(\rho_3=\frac{\phi_1\phi_2(2-\phi_2)+\phi_1^3}{1-\phi_2}=\frac{0.16(2-0.2)+0.5^3}{1-0.2}=\frac{0.}{0.8}=0.413\).

Autokorelasi

autokol <- acf(ar, plot = FALSE)
autokol$acf
## , , 1
## 
##               [,1]
##  [1,]  1.000000000
##  [2,]  0.602787967
##  [3,]  0.487140835
##  [4,]  0.350091024
##  [5,]  0.265207538
##  [6,]  0.177674204
##  [7,]  0.122460994
##  [8,]  0.096548897
##  [9,]  0.008479435
## [10,]  0.016039713
## [11,] -0.039508167
## [12,] -0.062694173
## [13,] -0.038378650
## [14,] -0.067894182
## [15,] -0.115612274
## [16,] -0.132693820
## [17,] -0.170775813
## [18,] -0.219657305
## [19,] -0.194235909
## [20,] -0.161757517
## [21,] -0.153845140
## [22,] -0.128496578
## [23,] -0.110193567
## [24,] -0.099150552
## [25,] -0.068514529
autokol <- acf(ar1, plot = FALSE)
autokol$acf
## , , 1
## 
##               [,1]
##  [1,]  1.000000000
##  [2,]  0.713530667
##  [3,]  0.582836497
##  [4,]  0.442735461
##  [5,]  0.373365980
##  [6,]  0.291252230
##  [7,]  0.206842648
##  [8,]  0.168196392
##  [9,]  0.102666233
## [10,]  0.028535825
## [11,] -0.032694282
## [12,] -0.053286667
## [13,] -0.065098303
## [14,] -0.096725466
## [15,] -0.106877264
## [16,] -0.088976382
## [17,] -0.009359047
## [18,]  0.025396683
## [19,]  0.079782447
## [20,]  0.080132655
## [21,]  0.071232050
## [22,]  0.085637298
## [23,]  0.106245454
## [24,]  0.129906305
## [25,]  0.103902988

Fungsi pembangkitan ARMA(2,2)

Jika koefisien dan white noise/nilai deret waktu sebelumnya dapat diekstrak dalam bentuk vektor, dapat dilakukan perkalian matriks untuk mencari nilai bagian AR dan MA:

set.seed(30)
coefs <- c(0.5, 0.2)
e <- c(0.4,0.6)

coefs %*% e
##      [,1]
## [1,] 0.32

Atau, dapat dilakukan perkalian elementwise yang dijumlahkan:

coefs * e
## [1] 0.20 0.12
sum(coefs * e)
## [1] 0.32
set.seed(1234)
n = length(wn)
phi1 = 0.5
phi2 = 0.2
theta1 = 0.4
theta2 = 0.6
y.arma=c(1:n)
for (i in 3:n){y.arma[i] = phi1*y.arma[i-1] + phi2*y.arma[i-2] + wn[i] - theta1*wn[i-1] - theta2*wn[i-2]}
y.arma
##   [1]  1.000000000  2.000000000  2.523136707  2.741071674 -0.247276743
##   [6] -0.707270615 -0.792995231 -0.086570305  0.507053616 -0.855539348
##  [11]  0.472818922  0.142837205  0.156826594  1.426718129  0.619397194
##  [16]  0.014719677  1.924744015 -0.261283604 -0.414170510 -0.306497548
##  [21] -0.807177208 -0.189987893 -0.866768169 -0.974592606  0.212262449
##  [26] -1.101802513  0.563285838  1.208724002 -1.095655604 -0.331166718
##  [31]  0.497743215 -0.693101478 -1.836639659  2.899595924 -0.307143076
##  [36]  0.607202105  0.326920228  0.373513104 -0.688178149  0.177526035
##  [41]  0.410773658 -0.486882256  0.738542481 -0.397060430 -0.285072799
##  [46]  1.237638765 -1.815355289  0.414500555  1.532694662  0.917061876
##  [51] -1.417536496  0.214418581  0.588908023 -1.444345242 -1.187496751
##  [56]  1.309205193  0.618072741 -0.639492711 -0.669342101 -0.624469704
##  [61] -0.836834841  1.014077733  0.406493311 -0.818528472  0.180903959
##  [66]  1.995861587 -1.547622714 -1.108363260  0.031304605 -0.526427974
##  [71]  0.750272363  0.382689159 -0.351305316  0.016092465  0.715423439
##  [76] -0.039216416 -0.824927879 -0.329880711  0.024045165  0.331679422
##  [81]  0.977376088  0.238017603 -0.444288700  0.247823381 -0.014789285
##  [86]  0.864291957  0.071406750  0.012286235 -0.534742775 -1.513090239
##  [91] -1.782881861 -1.951264890  2.436185623  2.094054332  1.153867036
##  [96] -1.150934062 -1.151635930  1.707640111 -0.700591301  2.168568995
## [101]  1.418115094  0.438609355 -1.885758824 -1.865211452  0.761802812
## [106]  1.341426010  0.370325047 -0.146294121 -0.539595018 -0.090416035
## [111]  0.517518741 -0.741704859  0.661221668  1.445324072  0.586675523
## [116] -1.916423429 -0.010713723 -0.931960578 -0.446647862 -0.161640598
## [121]  0.101252889 -0.287932020  0.352846507 -0.028298379  0.994192642
## [126]  1.899909309 -0.761855029  0.447144128 -0.151230699 -1.164246333
## [131] -1.674177482 -0.302694388 -1.404645601 -2.115215538  2.060438114
## [136]  1.282872049  0.031826855 -0.605415319 -0.898045959  0.074828730
## [141]  0.857086916 -0.393215061 -0.587246680 -1.191929093  0.192184099
## [146]  1.997715374 -0.179653762 -2.708715029 -0.288631060  2.023221084
## [151]  2.737772603  1.812519841 -0.119359482  0.433506525 -0.728688704
## [156] -0.295360983 -0.066812256  0.439566351  1.049417075  0.601353781
## [161] -0.431297217 -0.003790171  0.314340462 -0.367878114 -0.020033289
## [166] -1.910339489 -1.136860788 -0.303237905 -1.295724840  0.825575021
## [171] -0.081664930 -0.197115311 -1.609949536  0.378922957  0.448915227
## [176] -0.413164891 -1.111010398  0.059843759 -0.591143248  1.926585692
## [181]  1.344885704  1.963295257  0.649060156 -1.255029452  1.197316664
## [186]  0.128726049 -1.085083561 -0.982791222  0.045075393  0.079998799
## [191]  0.178576685 -0.737857256 -0.983395678  1.315937401  0.574829299
## [196] -0.059625299  0.044534256 -1.137246941 -0.491755605  0.769578833
## [201] -0.023206476 -1.899631849 -0.106540020  1.843230557  1.140280826
## [206]  1.912038438 -0.685298625 -0.037226565  0.681775403 -1.079944646
## [211] -1.199215322 -0.058363664  1.322629808 -0.109408228  0.336874371
## [216]  1.069693845 -0.632694852 -0.032856354 -0.855872150  0.184367291
## [221] -0.242163491 -1.437220062 -0.774544976  0.742935655  2.390316839
## [226] -0.962181654 -1.065289117  1.383500941  0.443707970 -1.666607900
## [231] -0.319369484  0.186821506 -0.331521912 -1.659619463 -1.175185169
## [236]  0.400265880  0.149267193 -0.163308918 -1.372560551  0.698723348
## [241]  1.019809508  0.726873474  1.492183597 -1.592426974 -0.814437633
## [246] -0.147943018 -0.589417542 -0.049487134  2.662571560  1.816734623
## [251]  0.481233872 -0.311084936 -1.016173069  0.087957907  0.307391743
## [256] -0.644041554 -1.514594943  0.213864788  0.734183547 -0.757865747
## [261]  0.869768076  0.932289855 -1.741980265  0.552379603  1.134702892
## [266] -0.283949817  0.796064204  0.811319187  0.532829404 -0.704946921
## [271]  1.307850509 -0.487037282 -1.750292962  0.776467128  0.991524727
## [276] -1.771325094 -1.087016192 -0.486249476 -0.685925794  0.050572379
## [281]  0.220599183  0.507333420 -0.516626869 -0.825046117  0.626695086
## [286]  1.021435825  1.412869780 -0.455888936 -0.693400616  1.967118317
## [291] -0.987546788 -1.180895426 -0.645475342  1.382791040  0.694996726
## [296]  0.838114214  0.568094171 -0.928960208 -1.282730912 -1.548106289

Pembangkitan ARMA(p,q) juga dapat dilakukan dengan fungsi arima.sim sebagai berikut.

arma11 <- arima.sim(list(order=c(2,0,2), ar = c(0.5,0.2), ma = c(0.4,0.)), n=300)
arma11
## Time Series:
## Start = 1 
## End = 300 
## Frequency = 1 
##   [1] -1.727911058 -1.505708577 -2.040440421 -0.593443820 -0.739484055
##   [6] -1.388108061 -1.626984917 -2.920710764 -3.605009015 -5.033734014
##  [11] -5.450877862 -4.562876869 -3.955229091 -1.627052670 -2.093416371
##  [16] -2.654940443 -2.368922351 -2.822038541 -3.251054089 -3.684658673
##  [21] -4.187453317 -3.855280866 -3.471512301 -4.511583564 -4.254582669
##  [26] -4.371328042 -4.495098414 -3.690109143 -2.245942245  0.012046850
##  [31] -0.557391459  1.020281900 -0.116782038  0.339130406  2.957835252
##  [36]  2.531579746  1.173819188  0.817767355  2.417690061  0.944624544
##  [41]  1.868234369  2.999737756  2.741814465  2.112336741  1.151819661
##  [46]  0.449665751  0.956873802  2.897955539  2.315062462  0.285062027
##  [51] -0.674318650 -0.311317868 -0.504277074 -0.619015714 -0.651473332
##  [56] -1.889839325 -1.797922254 -0.496211603  0.430011362  0.944804196
##  [61]  0.375671336  0.024109946 -1.183976147 -1.118136055 -0.561930784
##  [66]  1.303449805  2.223237601  1.277330619  1.240629750 -0.116606926
##  [71]  0.614182896  1.607968267  3.437104520  3.303116290  2.030069989
##  [76]  1.551764357  0.705815791 -0.363728933 -0.204111463 -1.004270433
##  [81] -0.733277972 -0.145250656 -0.129398735 -0.310526164 -0.908586428
##  [86] -1.885793579 -0.719246765 -0.374709891  0.508881329 -0.732332918
##  [91] -0.593078920  0.297770829  0.273259777 -0.005718665 -0.806670891
##  [96]  1.340920150  2.082892145  3.434038986  2.943340884  1.559092797
## [101] -0.397637264 -1.128415152 -0.671873429  0.542591216  0.795147196
## [106] -0.564756418  0.076586234 -1.457273370 -1.739212060 -1.623519932
## [111] -3.234295757 -1.801092762 -1.802254118 -1.844530899 -0.021183041
## [116]  0.815235868  0.549619393  0.908246970  1.168824282  2.588626705
## [121]  2.469114192  2.368912422  2.228340073  1.349735664  1.067260251
## [126]  2.481369688  1.234042278  0.884818086  2.100052159  1.537220874
## [131]  0.041389626 -0.962997741 -1.211261417 -1.801631142 -1.742647276
## [136] -1.750325329 -1.572510800 -0.802484687  0.071711064  1.803417589
## [141]  1.518639651  0.771685970  2.032240667  3.463189212  2.863018536
## [146]  1.808787406 -0.478300935  0.804975246  0.033749961 -1.280925738
## [151]  1.960547891  2.176616461  1.541167722 -1.539615814 -1.654252760
## [156] -0.198934043  0.373964036  1.225064937  3.035986341  3.725607552
## [161]  2.428907434  2.460260600  1.799167584  0.774198614 -2.324054147
## [166] -2.939137665 -1.762423768  0.894118977  1.462482365  1.730553027
## [171]  0.439953858  0.342380958 -1.753994408 -1.154589208 -0.037233979
## [176]  0.214686596  0.874835584  1.072329546  1.596266663  3.159247812
## [181]  3.748225530  2.983571422  1.140047435  1.139016238  0.564505498
## [186]  1.843455004  0.024944903 -0.677420712 -0.078013718 -0.157424188
## [191] -0.761160568 -0.959246637 -0.124207760  0.893033213  1.439956479
## [196]  3.436153455  4.088959004  3.488818950  2.017515074  4.361753420
## [201]  4.429451277  2.673722179  2.135515290  1.352705614  0.698994326
## [206] -0.423346742 -0.328430497  0.117013780 -0.957057237 -4.119294844
## [211] -3.799194375 -2.795888187 -1.551702063 -1.038183991 -0.392379446
## [216] -1.336239138 -0.474529071  0.498846470  0.693723982  1.904566149
## [221]  2.038406116  1.826267946  1.694604808  1.756704132  2.452169164
## [226]  2.453665367  2.561341291  0.900309611  0.137376125  1.577520568
## [231]  0.183119594  0.175570798  0.632639478  1.489238343  0.912591659
## [236]  0.773218243 -1.130677437 -0.777401050 -1.145962623 -1.233734158
## [241] -2.129283536 -1.401559391 -0.306082758 -0.471335340  1.406281287
## [246]  2.009437371  2.502831264  4.082159138  2.710293413  2.719752363
## [251]  3.211963187  2.538393516  2.228173435  0.737564461  0.879855704
## [256]  0.074537432  0.746534587 -0.097782859  0.095240661 -0.690699900
## [261] -0.910674058 -2.244641292 -1.797165029 -0.310693770  0.062644988
## [266] -1.723746464 -2.176369617 -0.025104543  2.024396406  0.556062681
## [271] -0.153807345  0.241253009  0.550471436 -0.128808181 -1.139571301
## [276] -1.156287354  0.131978843  0.262281851 -0.482040879 -0.952549579
## [281]  0.839958841  0.428266292  0.966517751  2.345901524  2.398606690
## [286]  1.424227741  0.983144835  0.268137347  1.361502770  1.075912136
## [291]  2.300418580  1.423772536  0.300207719  0.319143625  0.301054868
## [296] -0.280419906 -1.863176399 -1.590164484 -1.872267652 -0.458186447

Karakteristik ARMA(2,2)

Plot Time Series

par(mfrow = c(1, 2))
ts.plot(y.arma)
ts.plot(arma11)

par(mfrow = c(1, 1))

Berdasarkan plot time series tersebut, terlihat bahwa model ARMA(1,1) cenderung stasioner dalam rataan

Plot ACF

par(mfrow = c(1, 2))
acf(y.arma)
acf(arma11)

par(mfrow = c(1, 1))

Berdasarkan plot ACF tersebut, terlihat bahwa model ARMA(1,1) hasil simulasi memiliki plot ACF yang tails off, sesuai dengan teori yang ada

Plot PACF

par(mfrow = c(1, 2))
pacf(y.arma)
pacf(arma11)

par(mfrow = c(1, 1))

Berdasarkan plot PACF tersebut, terlihat bahwa model ARMA(1,1) 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 o o o o o o o  o  o  o 
## 1 x x o o o o o o o o o  o  o  o 
## 2 o x x o o o o o o o o  o  o  o 
## 3 o x x x o o o o o o o  o  o  o 
## 4 x o o x o o o o o o o  o  o  o 
## 5 x o o x o o o o o o o  o  o  o 
## 6 x o o x o x o o o o o  o  o  o 
## 7 x x x o o o o o o o o  o  o  o
TSA::eacf(arma11)
## 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  x 
## 1 x o o o o o o o o o o  o  o  x 
## 2 x o o o o o o o o o o  o  o  o 
## 3 x x o o o o o o o o o  o  o  o 
## 4 x x o o o o o o o o o  o  o  o 
## 5 x x o x o o o o o o o  o  o  o 
## 6 x x x o o o o o o o o  o  o  o 
## 7 x x x o x x o o o o o  o  o  o

Berdasarkan pola segitiga nol pada plot EACF, terlihat bahwa segitiga nol berada pada ordo AR(1) dan ordo MA(2)

Scatterplot Antar Lag

Korelasi antara \(Y_t\) dengan \(Y_{t-1}\)

#Yt
yt_arma <- arma11[-1]
yt_arma
##   [1] -1.505708577 -2.040440421 -0.593443820 -0.739484055 -1.388108061
##   [6] -1.626984917 -2.920710764 -3.605009015 -5.033734014 -5.450877862
##  [11] -4.562876869 -3.955229091 -1.627052670 -2.093416371 -2.654940443
##  [16] -2.368922351 -2.822038541 -3.251054089 -3.684658673 -4.187453317
##  [21] -3.855280866 -3.471512301 -4.511583564 -4.254582669 -4.371328042
##  [26] -4.495098414 -3.690109143 -2.245942245  0.012046850 -0.557391459
##  [31]  1.020281900 -0.116782038  0.339130406  2.957835252  2.531579746
##  [36]  1.173819188  0.817767355  2.417690061  0.944624544  1.868234369
##  [41]  2.999737756  2.741814465  2.112336741  1.151819661  0.449665751
##  [46]  0.956873802  2.897955539  2.315062462  0.285062027 -0.674318650
##  [51] -0.311317868 -0.504277074 -0.619015714 -0.651473332 -1.889839325
##  [56] -1.797922254 -0.496211603  0.430011362  0.944804196  0.375671336
##  [61]  0.024109946 -1.183976147 -1.118136055 -0.561930784  1.303449805
##  [66]  2.223237601  1.277330619  1.240629750 -0.116606926  0.614182896
##  [71]  1.607968267  3.437104520  3.303116290  2.030069989  1.551764357
##  [76]  0.705815791 -0.363728933 -0.204111463 -1.004270433 -0.733277972
##  [81] -0.145250656 -0.129398735 -0.310526164 -0.908586428 -1.885793579
##  [86] -0.719246765 -0.374709891  0.508881329 -0.732332918 -0.593078920
##  [91]  0.297770829  0.273259777 -0.005718665 -0.806670891  1.340920150
##  [96]  2.082892145  3.434038986  2.943340884  1.559092797 -0.397637264
## [101] -1.128415152 -0.671873429  0.542591216  0.795147196 -0.564756418
## [106]  0.076586234 -1.457273370 -1.739212060 -1.623519932 -3.234295757
## [111] -1.801092762 -1.802254118 -1.844530899 -0.021183041  0.815235868
## [116]  0.549619393  0.908246970  1.168824282  2.588626705  2.469114192
## [121]  2.368912422  2.228340073  1.349735664  1.067260251  2.481369688
## [126]  1.234042278  0.884818086  2.100052159  1.537220874  0.041389626
## [131] -0.962997741 -1.211261417 -1.801631142 -1.742647276 -1.750325329
## [136] -1.572510800 -0.802484687  0.071711064  1.803417589  1.518639651
## [141]  0.771685970  2.032240667  3.463189212  2.863018536  1.808787406
## [146] -0.478300935  0.804975246  0.033749961 -1.280925738  1.960547891
## [151]  2.176616461  1.541167722 -1.539615814 -1.654252760 -0.198934043
## [156]  0.373964036  1.225064937  3.035986341  3.725607552  2.428907434
## [161]  2.460260600  1.799167584  0.774198614 -2.324054147 -2.939137665
## [166] -1.762423768  0.894118977  1.462482365  1.730553027  0.439953858
## [171]  0.342380958 -1.753994408 -1.154589208 -0.037233979  0.214686596
## [176]  0.874835584  1.072329546  1.596266663  3.159247812  3.748225530
## [181]  2.983571422  1.140047435  1.139016238  0.564505498  1.843455004
## [186]  0.024944903 -0.677420712 -0.078013718 -0.157424188 -0.761160568
## [191] -0.959246637 -0.124207760  0.893033213  1.439956479  3.436153455
## [196]  4.088959004  3.488818950  2.017515074  4.361753420  4.429451277
## [201]  2.673722179  2.135515290  1.352705614  0.698994326 -0.423346742
## [206] -0.328430497  0.117013780 -0.957057237 -4.119294844 -3.799194375
## [211] -2.795888187 -1.551702063 -1.038183991 -0.392379446 -1.336239138
## [216] -0.474529071  0.498846470  0.693723982  1.904566149  2.038406116
## [221]  1.826267946  1.694604808  1.756704132  2.452169164  2.453665367
## [226]  2.561341291  0.900309611  0.137376125  1.577520568  0.183119594
## [231]  0.175570798  0.632639478  1.489238343  0.912591659  0.773218243
## [236] -1.130677437 -0.777401050 -1.145962623 -1.233734158 -2.129283536
## [241] -1.401559391 -0.306082758 -0.471335340  1.406281287  2.009437371
## [246]  2.502831264  4.082159138  2.710293413  2.719752363  3.211963187
## [251]  2.538393516  2.228173435  0.737564461  0.879855704  0.074537432
## [256]  0.746534587 -0.097782859  0.095240661 -0.690699900 -0.910674058
## [261] -2.244641292 -1.797165029 -0.310693770  0.062644988 -1.723746464
## [266] -2.176369617 -0.025104543  2.024396406  0.556062681 -0.153807345
## [271]  0.241253009  0.550471436 -0.128808181 -1.139571301 -1.156287354
## [276]  0.131978843  0.262281851 -0.482040879 -0.952549579  0.839958841
## [281]  0.428266292  0.966517751  2.345901524  2.398606690  1.424227741
## [286]  0.983144835  0.268137347  1.361502770  1.075912136  2.300418580
## [291]  1.423772536  0.300207719  0.319143625  0.301054868 -0.280419906
## [296] -1.863176399 -1.590164484 -1.872267652 -0.458186447
#Yt-1
yt_1_arma <- arma11[-300]
yt_1_arma
##   [1] -1.727911058 -1.505708577 -2.040440421 -0.593443820 -0.739484055
##   [6] -1.388108061 -1.626984917 -2.920710764 -3.605009015 -5.033734014
##  [11] -5.450877862 -4.562876869 -3.955229091 -1.627052670 -2.093416371
##  [16] -2.654940443 -2.368922351 -2.822038541 -3.251054089 -3.684658673
##  [21] -4.187453317 -3.855280866 -3.471512301 -4.511583564 -4.254582669
##  [26] -4.371328042 -4.495098414 -3.690109143 -2.245942245  0.012046850
##  [31] -0.557391459  1.020281900 -0.116782038  0.339130406  2.957835252
##  [36]  2.531579746  1.173819188  0.817767355  2.417690061  0.944624544
##  [41]  1.868234369  2.999737756  2.741814465  2.112336741  1.151819661
##  [46]  0.449665751  0.956873802  2.897955539  2.315062462  0.285062027
##  [51] -0.674318650 -0.311317868 -0.504277074 -0.619015714 -0.651473332
##  [56] -1.889839325 -1.797922254 -0.496211603  0.430011362  0.944804196
##  [61]  0.375671336  0.024109946 -1.183976147 -1.118136055 -0.561930784
##  [66]  1.303449805  2.223237601  1.277330619  1.240629750 -0.116606926
##  [71]  0.614182896  1.607968267  3.437104520  3.303116290  2.030069989
##  [76]  1.551764357  0.705815791 -0.363728933 -0.204111463 -1.004270433
##  [81] -0.733277972 -0.145250656 -0.129398735 -0.310526164 -0.908586428
##  [86] -1.885793579 -0.719246765 -0.374709891  0.508881329 -0.732332918
##  [91] -0.593078920  0.297770829  0.273259777 -0.005718665 -0.806670891
##  [96]  1.340920150  2.082892145  3.434038986  2.943340884  1.559092797
## [101] -0.397637264 -1.128415152 -0.671873429  0.542591216  0.795147196
## [106] -0.564756418  0.076586234 -1.457273370 -1.739212060 -1.623519932
## [111] -3.234295757 -1.801092762 -1.802254118 -1.844530899 -0.021183041
## [116]  0.815235868  0.549619393  0.908246970  1.168824282  2.588626705
## [121]  2.469114192  2.368912422  2.228340073  1.349735664  1.067260251
## [126]  2.481369688  1.234042278  0.884818086  2.100052159  1.537220874
## [131]  0.041389626 -0.962997741 -1.211261417 -1.801631142 -1.742647276
## [136] -1.750325329 -1.572510800 -0.802484687  0.071711064  1.803417589
## [141]  1.518639651  0.771685970  2.032240667  3.463189212  2.863018536
## [146]  1.808787406 -0.478300935  0.804975246  0.033749961 -1.280925738
## [151]  1.960547891  2.176616461  1.541167722 -1.539615814 -1.654252760
## [156] -0.198934043  0.373964036  1.225064937  3.035986341  3.725607552
## [161]  2.428907434  2.460260600  1.799167584  0.774198614 -2.324054147
## [166] -2.939137665 -1.762423768  0.894118977  1.462482365  1.730553027
## [171]  0.439953858  0.342380958 -1.753994408 -1.154589208 -0.037233979
## [176]  0.214686596  0.874835584  1.072329546  1.596266663  3.159247812
## [181]  3.748225530  2.983571422  1.140047435  1.139016238  0.564505498
## [186]  1.843455004  0.024944903 -0.677420712 -0.078013718 -0.157424188
## [191] -0.761160568 -0.959246637 -0.124207760  0.893033213  1.439956479
## [196]  3.436153455  4.088959004  3.488818950  2.017515074  4.361753420
## [201]  4.429451277  2.673722179  2.135515290  1.352705614  0.698994326
## [206] -0.423346742 -0.328430497  0.117013780 -0.957057237 -4.119294844
## [211] -3.799194375 -2.795888187 -1.551702063 -1.038183991 -0.392379446
## [216] -1.336239138 -0.474529071  0.498846470  0.693723982  1.904566149
## [221]  2.038406116  1.826267946  1.694604808  1.756704132  2.452169164
## [226]  2.453665367  2.561341291  0.900309611  0.137376125  1.577520568
## [231]  0.183119594  0.175570798  0.632639478  1.489238343  0.912591659
## [236]  0.773218243 -1.130677437 -0.777401050 -1.145962623 -1.233734158
## [241] -2.129283536 -1.401559391 -0.306082758 -0.471335340  1.406281287
## [246]  2.009437371  2.502831264  4.082159138  2.710293413  2.719752363
## [251]  3.211963187  2.538393516  2.228173435  0.737564461  0.879855704
## [256]  0.074537432  0.746534587 -0.097782859  0.095240661 -0.690699900
## [261] -0.910674058 -2.244641292 -1.797165029 -0.310693770  0.062644988
## [266] -1.723746464 -2.176369617 -0.025104543  2.024396406  0.556062681
## [271] -0.153807345  0.241253009  0.550471436 -0.128808181 -1.139571301
## [276] -1.156287354  0.131978843  0.262281851 -0.482040879 -0.952549579
## [281]  0.839958841  0.428266292  0.966517751  2.345901524  2.398606690
## [286]  1.424227741  0.983144835  0.268137347  1.361502770  1.075912136
## [291]  2.300418580  1.423772536  0.300207719  0.319143625  0.301054868
## [296] -0.280419906 -1.863176399 -1.590164484 -1.872267652
plot(y=yt_arma,x=yt_1_arma)

Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-1}\). Hal ini sesuai dengan teori yang ada

cor(yt_arma,yt_1_arma)
## [1] 0.8455292

Korelasi antara \(Y_t\) dengan \(Y_{t-2}\)

#Yt 
yt_arma2 <- arma11[-c(1,2)] 
yt_arma2 
##   [1] -2.040440421 -0.593443820 -0.739484055 -1.388108061 -1.626984917
##   [6] -2.920710764 -3.605009015 -5.033734014 -5.450877862 -4.562876869
##  [11] -3.955229091 -1.627052670 -2.093416371 -2.654940443 -2.368922351
##  [16] -2.822038541 -3.251054089 -3.684658673 -4.187453317 -3.855280866
##  [21] -3.471512301 -4.511583564 -4.254582669 -4.371328042 -4.495098414
##  [26] -3.690109143 -2.245942245  0.012046850 -0.557391459  1.020281900
##  [31] -0.116782038  0.339130406  2.957835252  2.531579746  1.173819188
##  [36]  0.817767355  2.417690061  0.944624544  1.868234369  2.999737756
##  [41]  2.741814465  2.112336741  1.151819661  0.449665751  0.956873802
##  [46]  2.897955539  2.315062462  0.285062027 -0.674318650 -0.311317868
##  [51] -0.504277074 -0.619015714 -0.651473332 -1.889839325 -1.797922254
##  [56] -0.496211603  0.430011362  0.944804196  0.375671336  0.024109946
##  [61] -1.183976147 -1.118136055 -0.561930784  1.303449805  2.223237601
##  [66]  1.277330619  1.240629750 -0.116606926  0.614182896  1.607968267
##  [71]  3.437104520  3.303116290  2.030069989  1.551764357  0.705815791
##  [76] -0.363728933 -0.204111463 -1.004270433 -0.733277972 -0.145250656
##  [81] -0.129398735 -0.310526164 -0.908586428 -1.885793579 -0.719246765
##  [86] -0.374709891  0.508881329 -0.732332918 -0.593078920  0.297770829
##  [91]  0.273259777 -0.005718665 -0.806670891  1.340920150  2.082892145
##  [96]  3.434038986  2.943340884  1.559092797 -0.397637264 -1.128415152
## [101] -0.671873429  0.542591216  0.795147196 -0.564756418  0.076586234
## [106] -1.457273370 -1.739212060 -1.623519932 -3.234295757 -1.801092762
## [111] -1.802254118 -1.844530899 -0.021183041  0.815235868  0.549619393
## [116]  0.908246970  1.168824282  2.588626705  2.469114192  2.368912422
## [121]  2.228340073  1.349735664  1.067260251  2.481369688  1.234042278
## [126]  0.884818086  2.100052159  1.537220874  0.041389626 -0.962997741
## [131] -1.211261417 -1.801631142 -1.742647276 -1.750325329 -1.572510800
## [136] -0.802484687  0.071711064  1.803417589  1.518639651  0.771685970
## [141]  2.032240667  3.463189212  2.863018536  1.808787406 -0.478300935
## [146]  0.804975246  0.033749961 -1.280925738  1.960547891  2.176616461
## [151]  1.541167722 -1.539615814 -1.654252760 -0.198934043  0.373964036
## [156]  1.225064937  3.035986341  3.725607552  2.428907434  2.460260600
## [161]  1.799167584  0.774198614 -2.324054147 -2.939137665 -1.762423768
## [166]  0.894118977  1.462482365  1.730553027  0.439953858  0.342380958
## [171] -1.753994408 -1.154589208 -0.037233979  0.214686596  0.874835584
## [176]  1.072329546  1.596266663  3.159247812  3.748225530  2.983571422
## [181]  1.140047435  1.139016238  0.564505498  1.843455004  0.024944903
## [186] -0.677420712 -0.078013718 -0.157424188 -0.761160568 -0.959246637
## [191] -0.124207760  0.893033213  1.439956479  3.436153455  4.088959004
## [196]  3.488818950  2.017515074  4.361753420  4.429451277  2.673722179
## [201]  2.135515290  1.352705614  0.698994326 -0.423346742 -0.328430497
## [206]  0.117013780 -0.957057237 -4.119294844 -3.799194375 -2.795888187
## [211] -1.551702063 -1.038183991 -0.392379446 -1.336239138 -0.474529071
## [216]  0.498846470  0.693723982  1.904566149  2.038406116  1.826267946
## [221]  1.694604808  1.756704132  2.452169164  2.453665367  2.561341291
## [226]  0.900309611  0.137376125  1.577520568  0.183119594  0.175570798
## [231]  0.632639478  1.489238343  0.912591659  0.773218243 -1.130677437
## [236] -0.777401050 -1.145962623 -1.233734158 -2.129283536 -1.401559391
## [241] -0.306082758 -0.471335340  1.406281287  2.009437371  2.502831264
## [246]  4.082159138  2.710293413  2.719752363  3.211963187  2.538393516
## [251]  2.228173435  0.737564461  0.879855704  0.074537432  0.746534587
## [256] -0.097782859  0.095240661 -0.690699900 -0.910674058 -2.244641292
## [261] -1.797165029 -0.310693770  0.062644988 -1.723746464 -2.176369617
## [266] -0.025104543  2.024396406  0.556062681 -0.153807345  0.241253009
## [271]  0.550471436 -0.128808181 -1.139571301 -1.156287354  0.131978843
## [276]  0.262281851 -0.482040879 -0.952549579  0.839958841  0.428266292
## [281]  0.966517751  2.345901524  2.398606690  1.424227741  0.983144835
## [286]  0.268137347  1.361502770  1.075912136  2.300418580  1.423772536
## [291]  0.300207719  0.319143625  0.301054868 -0.280419906 -1.863176399
## [296] -1.590164484 -1.872267652 -0.458186447
#Yt-2 
yt_2_arma <- arma11[-c(299,300)] 
yt_2_arma
##   [1] -1.727911058 -1.505708577 -2.040440421 -0.593443820 -0.739484055
##   [6] -1.388108061 -1.626984917 -2.920710764 -3.605009015 -5.033734014
##  [11] -5.450877862 -4.562876869 -3.955229091 -1.627052670 -2.093416371
##  [16] -2.654940443 -2.368922351 -2.822038541 -3.251054089 -3.684658673
##  [21] -4.187453317 -3.855280866 -3.471512301 -4.511583564 -4.254582669
##  [26] -4.371328042 -4.495098414 -3.690109143 -2.245942245  0.012046850
##  [31] -0.557391459  1.020281900 -0.116782038  0.339130406  2.957835252
##  [36]  2.531579746  1.173819188  0.817767355  2.417690061  0.944624544
##  [41]  1.868234369  2.999737756  2.741814465  2.112336741  1.151819661
##  [46]  0.449665751  0.956873802  2.897955539  2.315062462  0.285062027
##  [51] -0.674318650 -0.311317868 -0.504277074 -0.619015714 -0.651473332
##  [56] -1.889839325 -1.797922254 -0.496211603  0.430011362  0.944804196
##  [61]  0.375671336  0.024109946 -1.183976147 -1.118136055 -0.561930784
##  [66]  1.303449805  2.223237601  1.277330619  1.240629750 -0.116606926
##  [71]  0.614182896  1.607968267  3.437104520  3.303116290  2.030069989
##  [76]  1.551764357  0.705815791 -0.363728933 -0.204111463 -1.004270433
##  [81] -0.733277972 -0.145250656 -0.129398735 -0.310526164 -0.908586428
##  [86] -1.885793579 -0.719246765 -0.374709891  0.508881329 -0.732332918
##  [91] -0.593078920  0.297770829  0.273259777 -0.005718665 -0.806670891
##  [96]  1.340920150  2.082892145  3.434038986  2.943340884  1.559092797
## [101] -0.397637264 -1.128415152 -0.671873429  0.542591216  0.795147196
## [106] -0.564756418  0.076586234 -1.457273370 -1.739212060 -1.623519932
## [111] -3.234295757 -1.801092762 -1.802254118 -1.844530899 -0.021183041
## [116]  0.815235868  0.549619393  0.908246970  1.168824282  2.588626705
## [121]  2.469114192  2.368912422  2.228340073  1.349735664  1.067260251
## [126]  2.481369688  1.234042278  0.884818086  2.100052159  1.537220874
## [131]  0.041389626 -0.962997741 -1.211261417 -1.801631142 -1.742647276
## [136] -1.750325329 -1.572510800 -0.802484687  0.071711064  1.803417589
## [141]  1.518639651  0.771685970  2.032240667  3.463189212  2.863018536
## [146]  1.808787406 -0.478300935  0.804975246  0.033749961 -1.280925738
## [151]  1.960547891  2.176616461  1.541167722 -1.539615814 -1.654252760
## [156] -0.198934043  0.373964036  1.225064937  3.035986341  3.725607552
## [161]  2.428907434  2.460260600  1.799167584  0.774198614 -2.324054147
## [166] -2.939137665 -1.762423768  0.894118977  1.462482365  1.730553027
## [171]  0.439953858  0.342380958 -1.753994408 -1.154589208 -0.037233979
## [176]  0.214686596  0.874835584  1.072329546  1.596266663  3.159247812
## [181]  3.748225530  2.983571422  1.140047435  1.139016238  0.564505498
## [186]  1.843455004  0.024944903 -0.677420712 -0.078013718 -0.157424188
## [191] -0.761160568 -0.959246637 -0.124207760  0.893033213  1.439956479
## [196]  3.436153455  4.088959004  3.488818950  2.017515074  4.361753420
## [201]  4.429451277  2.673722179  2.135515290  1.352705614  0.698994326
## [206] -0.423346742 -0.328430497  0.117013780 -0.957057237 -4.119294844
## [211] -3.799194375 -2.795888187 -1.551702063 -1.038183991 -0.392379446
## [216] -1.336239138 -0.474529071  0.498846470  0.693723982  1.904566149
## [221]  2.038406116  1.826267946  1.694604808  1.756704132  2.452169164
## [226]  2.453665367  2.561341291  0.900309611  0.137376125  1.577520568
## [231]  0.183119594  0.175570798  0.632639478  1.489238343  0.912591659
## [236]  0.773218243 -1.130677437 -0.777401050 -1.145962623 -1.233734158
## [241] -2.129283536 -1.401559391 -0.306082758 -0.471335340  1.406281287
## [246]  2.009437371  2.502831264  4.082159138  2.710293413  2.719752363
## [251]  3.211963187  2.538393516  2.228173435  0.737564461  0.879855704
## [256]  0.074537432  0.746534587 -0.097782859  0.095240661 -0.690699900
## [261] -0.910674058 -2.244641292 -1.797165029 -0.310693770  0.062644988
## [266] -1.723746464 -2.176369617 -0.025104543  2.024396406  0.556062681
## [271] -0.153807345  0.241253009  0.550471436 -0.128808181 -1.139571301
## [276] -1.156287354  0.131978843  0.262281851 -0.482040879 -0.952549579
## [281]  0.839958841  0.428266292  0.966517751  2.345901524  2.398606690
## [286]  1.424227741  0.983144835  0.268137347  1.361502770  1.075912136
## [291]  2.300418580  1.423772536  0.300207719  0.319143625  0.301054868
## [296] -0.280419906 -1.863176399 -1.590164484
plot(y=yt_arma2,x=yt_2_arma)

Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-1}\). Hal ini sesuai dengan teori yang ada

cor(yt_arma2,yt_2_arma)
## [1] 0.6604866

Korelasi antara \(Y_t\) dengan \(Y_{t-3}\)

#Yt 
yt_arma3 <- arma11[-c(1,2,3)] 
yt_arma3 
##   [1] -0.593443820 -0.739484055 -1.388108061 -1.626984917 -2.920710764
##   [6] -3.605009015 -5.033734014 -5.450877862 -4.562876869 -3.955229091
##  [11] -1.627052670 -2.093416371 -2.654940443 -2.368922351 -2.822038541
##  [16] -3.251054089 -3.684658673 -4.187453317 -3.855280866 -3.471512301
##  [21] -4.511583564 -4.254582669 -4.371328042 -4.495098414 -3.690109143
##  [26] -2.245942245  0.012046850 -0.557391459  1.020281900 -0.116782038
##  [31]  0.339130406  2.957835252  2.531579746  1.173819188  0.817767355
##  [36]  2.417690061  0.944624544  1.868234369  2.999737756  2.741814465
##  [41]  2.112336741  1.151819661  0.449665751  0.956873802  2.897955539
##  [46]  2.315062462  0.285062027 -0.674318650 -0.311317868 -0.504277074
##  [51] -0.619015714 -0.651473332 -1.889839325 -1.797922254 -0.496211603
##  [56]  0.430011362  0.944804196  0.375671336  0.024109946 -1.183976147
##  [61] -1.118136055 -0.561930784  1.303449805  2.223237601  1.277330619
##  [66]  1.240629750 -0.116606926  0.614182896  1.607968267  3.437104520
##  [71]  3.303116290  2.030069989  1.551764357  0.705815791 -0.363728933
##  [76] -0.204111463 -1.004270433 -0.733277972 -0.145250656 -0.129398735
##  [81] -0.310526164 -0.908586428 -1.885793579 -0.719246765 -0.374709891
##  [86]  0.508881329 -0.732332918 -0.593078920  0.297770829  0.273259777
##  [91] -0.005718665 -0.806670891  1.340920150  2.082892145  3.434038986
##  [96]  2.943340884  1.559092797 -0.397637264 -1.128415152 -0.671873429
## [101]  0.542591216  0.795147196 -0.564756418  0.076586234 -1.457273370
## [106] -1.739212060 -1.623519932 -3.234295757 -1.801092762 -1.802254118
## [111] -1.844530899 -0.021183041  0.815235868  0.549619393  0.908246970
## [116]  1.168824282  2.588626705  2.469114192  2.368912422  2.228340073
## [121]  1.349735664  1.067260251  2.481369688  1.234042278  0.884818086
## [126]  2.100052159  1.537220874  0.041389626 -0.962997741 -1.211261417
## [131] -1.801631142 -1.742647276 -1.750325329 -1.572510800 -0.802484687
## [136]  0.071711064  1.803417589  1.518639651  0.771685970  2.032240667
## [141]  3.463189212  2.863018536  1.808787406 -0.478300935  0.804975246
## [146]  0.033749961 -1.280925738  1.960547891  2.176616461  1.541167722
## [151] -1.539615814 -1.654252760 -0.198934043  0.373964036  1.225064937
## [156]  3.035986341  3.725607552  2.428907434  2.460260600  1.799167584
## [161]  0.774198614 -2.324054147 -2.939137665 -1.762423768  0.894118977
## [166]  1.462482365  1.730553027  0.439953858  0.342380958 -1.753994408
## [171] -1.154589208 -0.037233979  0.214686596  0.874835584  1.072329546
## [176]  1.596266663  3.159247812  3.748225530  2.983571422  1.140047435
## [181]  1.139016238  0.564505498  1.843455004  0.024944903 -0.677420712
## [186] -0.078013718 -0.157424188 -0.761160568 -0.959246637 -0.124207760
## [191]  0.893033213  1.439956479  3.436153455  4.088959004  3.488818950
## [196]  2.017515074  4.361753420  4.429451277  2.673722179  2.135515290
## [201]  1.352705614  0.698994326 -0.423346742 -0.328430497  0.117013780
## [206] -0.957057237 -4.119294844 -3.799194375 -2.795888187 -1.551702063
## [211] -1.038183991 -0.392379446 -1.336239138 -0.474529071  0.498846470
## [216]  0.693723982  1.904566149  2.038406116  1.826267946  1.694604808
## [221]  1.756704132  2.452169164  2.453665367  2.561341291  0.900309611
## [226]  0.137376125  1.577520568  0.183119594  0.175570798  0.632639478
## [231]  1.489238343  0.912591659  0.773218243 -1.130677437 -0.777401050
## [236] -1.145962623 -1.233734158 -2.129283536 -1.401559391 -0.306082758
## [241] -0.471335340  1.406281287  2.009437371  2.502831264  4.082159138
## [246]  2.710293413  2.719752363  3.211963187  2.538393516  2.228173435
## [251]  0.737564461  0.879855704  0.074537432  0.746534587 -0.097782859
## [256]  0.095240661 -0.690699900 -0.910674058 -2.244641292 -1.797165029
## [261] -0.310693770  0.062644988 -1.723746464 -2.176369617 -0.025104543
## [266]  2.024396406  0.556062681 -0.153807345  0.241253009  0.550471436
## [271] -0.128808181 -1.139571301 -1.156287354  0.131978843  0.262281851
## [276] -0.482040879 -0.952549579  0.839958841  0.428266292  0.966517751
## [281]  2.345901524  2.398606690  1.424227741  0.983144835  0.268137347
## [286]  1.361502770  1.075912136  2.300418580  1.423772536  0.300207719
## [291]  0.319143625  0.301054868 -0.280419906 -1.863176399 -1.590164484
## [296] -1.872267652 -0.458186447
#Yt-1 
yt_3_arma <- arma11[-c(298,299,300)] 
yt_3_arma
##   [1] -1.727911058 -1.505708577 -2.040440421 -0.593443820 -0.739484055
##   [6] -1.388108061 -1.626984917 -2.920710764 -3.605009015 -5.033734014
##  [11] -5.450877862 -4.562876869 -3.955229091 -1.627052670 -2.093416371
##  [16] -2.654940443 -2.368922351 -2.822038541 -3.251054089 -3.684658673
##  [21] -4.187453317 -3.855280866 -3.471512301 -4.511583564 -4.254582669
##  [26] -4.371328042 -4.495098414 -3.690109143 -2.245942245  0.012046850
##  [31] -0.557391459  1.020281900 -0.116782038  0.339130406  2.957835252
##  [36]  2.531579746  1.173819188  0.817767355  2.417690061  0.944624544
##  [41]  1.868234369  2.999737756  2.741814465  2.112336741  1.151819661
##  [46]  0.449665751  0.956873802  2.897955539  2.315062462  0.285062027
##  [51] -0.674318650 -0.311317868 -0.504277074 -0.619015714 -0.651473332
##  [56] -1.889839325 -1.797922254 -0.496211603  0.430011362  0.944804196
##  [61]  0.375671336  0.024109946 -1.183976147 -1.118136055 -0.561930784
##  [66]  1.303449805  2.223237601  1.277330619  1.240629750 -0.116606926
##  [71]  0.614182896  1.607968267  3.437104520  3.303116290  2.030069989
##  [76]  1.551764357  0.705815791 -0.363728933 -0.204111463 -1.004270433
##  [81] -0.733277972 -0.145250656 -0.129398735 -0.310526164 -0.908586428
##  [86] -1.885793579 -0.719246765 -0.374709891  0.508881329 -0.732332918
##  [91] -0.593078920  0.297770829  0.273259777 -0.005718665 -0.806670891
##  [96]  1.340920150  2.082892145  3.434038986  2.943340884  1.559092797
## [101] -0.397637264 -1.128415152 -0.671873429  0.542591216  0.795147196
## [106] -0.564756418  0.076586234 -1.457273370 -1.739212060 -1.623519932
## [111] -3.234295757 -1.801092762 -1.802254118 -1.844530899 -0.021183041
## [116]  0.815235868  0.549619393  0.908246970  1.168824282  2.588626705
## [121]  2.469114192  2.368912422  2.228340073  1.349735664  1.067260251
## [126]  2.481369688  1.234042278  0.884818086  2.100052159  1.537220874
## [131]  0.041389626 -0.962997741 -1.211261417 -1.801631142 -1.742647276
## [136] -1.750325329 -1.572510800 -0.802484687  0.071711064  1.803417589
## [141]  1.518639651  0.771685970  2.032240667  3.463189212  2.863018536
## [146]  1.808787406 -0.478300935  0.804975246  0.033749961 -1.280925738
## [151]  1.960547891  2.176616461  1.541167722 -1.539615814 -1.654252760
## [156] -0.198934043  0.373964036  1.225064937  3.035986341  3.725607552
## [161]  2.428907434  2.460260600  1.799167584  0.774198614 -2.324054147
## [166] -2.939137665 -1.762423768  0.894118977  1.462482365  1.730553027
## [171]  0.439953858  0.342380958 -1.753994408 -1.154589208 -0.037233979
## [176]  0.214686596  0.874835584  1.072329546  1.596266663  3.159247812
## [181]  3.748225530  2.983571422  1.140047435  1.139016238  0.564505498
## [186]  1.843455004  0.024944903 -0.677420712 -0.078013718 -0.157424188
## [191] -0.761160568 -0.959246637 -0.124207760  0.893033213  1.439956479
## [196]  3.436153455  4.088959004  3.488818950  2.017515074  4.361753420
## [201]  4.429451277  2.673722179  2.135515290  1.352705614  0.698994326
## [206] -0.423346742 -0.328430497  0.117013780 -0.957057237 -4.119294844
## [211] -3.799194375 -2.795888187 -1.551702063 -1.038183991 -0.392379446
## [216] -1.336239138 -0.474529071  0.498846470  0.693723982  1.904566149
## [221]  2.038406116  1.826267946  1.694604808  1.756704132  2.452169164
## [226]  2.453665367  2.561341291  0.900309611  0.137376125  1.577520568
## [231]  0.183119594  0.175570798  0.632639478  1.489238343  0.912591659
## [236]  0.773218243 -1.130677437 -0.777401050 -1.145962623 -1.233734158
## [241] -2.129283536 -1.401559391 -0.306082758 -0.471335340  1.406281287
## [246]  2.009437371  2.502831264  4.082159138  2.710293413  2.719752363
## [251]  3.211963187  2.538393516  2.228173435  0.737564461  0.879855704
## [256]  0.074537432  0.746534587 -0.097782859  0.095240661 -0.690699900
## [261] -0.910674058 -2.244641292 -1.797165029 -0.310693770  0.062644988
## [266] -1.723746464 -2.176369617 -0.025104543  2.024396406  0.556062681
## [271] -0.153807345  0.241253009  0.550471436 -0.128808181 -1.139571301
## [276] -1.156287354  0.131978843  0.262281851 -0.482040879 -0.952549579
## [281]  0.839958841  0.428266292  0.966517751  2.345901524  2.398606690
## [286]  1.424227741  0.983144835  0.268137347  1.361502770  1.075912136
## [291]  2.300418580  1.423772536  0.300207719  0.319143625  0.301054868
## [296] -0.280419906 -1.863176399
plot(y=yt_arma3,x=yt_3_arma)

Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-1}\). Hal ini sesuai dengan teori yang ada

cor(yt_arma3,yt_3_arma)
## [1] 0.5189441

Autokorelasi

autokol <- acf(y.arma, plot = FALSE)
autokol$acf
## , , 1
## 
##               [,1]
##  [1,]  1.000000000
##  [2,]  0.161440270
##  [3,] -0.211160745
##  [4,] -0.066341449
##  [5,] -0.056203060
##  [6,] -0.074117401
##  [7,] -0.024138201
##  [8,]  0.015104911
##  [9,] -0.064262461
## [10,]  0.026677287
## [11,] -0.029350092
## [12,] -0.057251460
## [13,]  0.077517536
## [14,]  0.057889817
## [15,] -0.022902557
## [16,]  0.010420987
## [17,] -0.036238247
## [18,] -0.140074845
## [19,] -0.059134825
## [20,]  0.016210180
## [21,] -0.016645380
## [22,] -0.001920717
## [23,]  0.002287085
## [24,] -0.005060659
## [25,]  0.045479944
autokol <- acf(arma11, plot = FALSE)
autokol$acf
## , , 1
## 
##              [,1]
##  [1,]  1.00000000
##  [2,]  0.84392020
##  [3,]  0.65705241
##  [4,]  0.51430934
##  [5,]  0.41536250
##  [6,]  0.32669189
##  [7,]  0.24116024
##  [8,]  0.17744505
##  [9,]  0.10344714
## [10,]  0.02274274
## [11,] -0.04281499
## [12,] -0.07496889
## [13,] -0.09508799
## [14,] -0.12023368
## [15,] -0.12816703
## [16,] -0.09713769
## [17,] -0.03089063
## [18,]  0.02441983
## [19,]  0.06863941
## [20,]  0.08061106
## [21,]  0.07713181
## [22,]  0.08670884
## [23,]  0.10860147
## [24,]  0.12630603
## [25,]  0.11893181