Pembangkitan Autoregressive dan Moving Average

Pertemuan kali ini akan melakukan proses ARIMA pada kestasioneran data time series

White Noise

Dimulai dengan membangkitkan data normal secara acak sejumlah 300 amatan sebagai data time series atau selanjutnya dianggap sebagai white noise.

set.seed(022)
wn <- rnorm(300)
ts.plot(wn)

Proses MA

Pembangkitan Proses MA(2)

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

ma2 <- wn[c(1,2)]

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

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

Cara Manual

for(i in 3:300){
   ma2[i] <- wn[i] + 0.4 * wn[i - 1] + 0.6 * wn[i - 2]
}
ma2

##   [1] -0.51213909  2.48518368  1.69461617  2.18705524  0.51286216  1.95019739
##   [7]  0.55183493  0.92567992 -0.30458250  0.12295849 -0.76359900 -0.04326397
##  [13]  0.31746867  0.26236626 -0.38068643 -1.28972475  0.01696422  1.79427496
##  [19]  2.25466531 -0.03934915 -0.65941993 -1.20343665 -1.02273579 -1.32245717
##  [25] -1.36879944 -0.29094466  0.14932719 -0.46699287  0.61536812 -1.77161201
##  [31]  0.40851692 -1.98932414  0.45597468 -1.04056622 -0.84963565 -1.53719572
##  [37] -0.87811307  0.48821259 -0.71258380 -0.08174685 -0.98502127  0.34661694
##  [43]  0.83513443  2.55293230  2.22373682  4.80715515  1.74391643  2.22106598
##  [49] -1.14221957  0.80638847 -1.27234300  0.77512092 -1.87772179  0.16125141
##  [55]  0.70940870  0.76947952  2.43492516 -0.24558080  1.66606188 -0.39797241
##  [61] -0.28924583  0.92113170 -0.06084652  0.65960855 -1.03003881  1.28051571
##  [67]  0.31385605  3.19682981  0.64362210  0.48658780 -0.93878232 -0.79643670
##  [73] -0.81251249 -0.68711692 -0.07021466 -0.58409030 -0.80943286 -0.40162958
##  [79]  1.71585040  1.87894327  2.72495011  1.33405693  0.16731264  0.61585863
##  [85]  1.56099364 -0.46505846  0.98790359 -1.92031145 -1.41097852 -2.47228337
##  [91] -0.62731574 -0.49414584  0.81565050 -0.03420602  0.59808754 -0.02771084
##  [97] -0.11332519 -0.57273554 -0.86027451 -1.21195964 -2.32218162 -1.94782008
## [103] -2.75797991 -0.51111492 -1.03111035 -0.12377864 -0.22737764  0.65950839
## [109] -0.48855603 -1.04152789 -1.13497311 -2.43239523 -2.08147739 -2.09496848
## [115] -3.50688323 -0.15204977 -0.22848834  2.63243030  1.18514480  2.01984543
## [121] -0.21541223  0.00402204 -0.97448203 -2.32938582 -1.87820958 -2.24489449
## [127] -2.21447293  0.12418850 -0.23188798  0.26894530 -0.62175043 -1.41336239
## [133] -2.85053314 -1.83647657 -0.23357008  0.72947999  0.16089316  0.76266960
## [139] -1.56029198  1.72467347  1.48133706  1.88236058  1.20489138 -1.13818969
## [145]  0.67205924 -0.76007935  0.44307757 -1.92410526 -1.26914254 -2.04915158
## [151] -1.43951917 -1.17500483 -0.06159182  1.64089681  0.89377073  1.13282339
## [157]  0.09681851  0.60641061 -1.01248690 -0.36319451 -1.49241751 -0.90006927
## [163] -0.24595296 -0.53308768 -0.19081898 -1.16204827  0.30563941 -0.64412274
## [169]  0.26691704 -0.92578561 -2.36052150 -0.92586793 -2.24716513 -0.22161843
## [175]  0.56694098  0.93668836  1.33196156  1.43069291  0.64239672  0.80718664
## [181]  0.57798260  0.04365511 -1.52053082 -0.52480273 -0.05959395  0.42296866
## [187] -0.94442554  0.05114117 -2.36063447 -0.77057102 -1.57472649 -1.93722272
## [193] -0.58856574 -0.43094516 -0.12107176 -1.13843788 -0.70308776 -0.47799678
## [199]  0.13230400  1.98346450  1.45645045 -0.64974118 -2.79405229 -2.89401714
## [205] -3.34844004 -2.45636226  0.26755649 -0.30109847 -0.12751765 -1.04407458
## [211] -2.30582095 -1.91232839 -2.84844322  0.12873723 -1.15133726  1.12091602
## [217] -0.11214974  1.85410575  0.03370631  0.56150652 -0.12796025  0.05658039
## [223]  0.48839134 -0.66613102 -1.30921287 -0.25864632 -1.63073729 -1.47130498
## [229] -0.87194859  0.36178868  1.05622427  1.26694543  0.45773206  0.71547311
## [235] -0.42979240  0.77223392 -0.37485584  0.70602610 -1.97141008  0.24638354
## [241] -0.89795116  1.38192413  0.83296011  0.04617360 -0.74661608  0.62295224
## [247]  2.48066595  1.33040368  0.97397256  0.99129349 -0.58662631  0.25196781
## [253] -2.18015553 -1.47208532 -2.00022146  0.13962967 -0.89208760  0.14949430
## [259]  2.80764017  1.93827915  3.61260041  1.07060008 -0.10903104 -0.83893583
## [265]  0.49518896 -0.32588029  0.69164340 -1.02761925 -1.14942628 -0.55067762
## [271]  1.24307205 -0.02402530  0.16670812 -1.02183257  1.01032563 -1.23062030
## [277] -0.43354204 -1.15494633 -2.14357023 -0.94015261 -1.73238543 -0.05273733
## [283] -1.53358451 -0.80519399  0.81229638 -0.19420484 -2.83967351 -1.04872855
## [289] -1.63696555  0.36437532 -2.30197721 -0.49864001  0.09538158  1.36219564
## [295] -0.35520808  0.75408250 -1.13883831  0.88466059 -1.97819903 -0.28030135

Fungs arima.sim

ma2 <- arima.sim(list(order=c(0,0,2), ma=c(0.4,0.6)), n=300)
ma2

## Time Series:
## Start = 1 
## End = 300 
## Frequency = 1 
##   [1]  2.198327037  0.256737131  0.210730157  0.766625272 -0.397169430
##   [6] -0.068333544 -1.014767960 -0.975904305  0.629789396  0.468403379
##  [11] -0.400669879  0.564095027 -1.039854411  0.061265290  0.644982965
##  [16] -0.245623724  1.491307617  0.813271007  0.920169898  0.794900726
##  [21]  0.466668892 -0.104897671 -1.180752848 -2.399760755 -1.845489666
##  [26] -1.954119879  0.056563276  0.493584496 -0.142112886 -0.260158929
##  [31] -2.085228014 -0.390931945 -0.057532243 -0.053722045 -0.130638222
##  [36]  0.645828554 -0.729403047 -0.370333182 -1.365213079 -1.611270366
##  [41] -0.707932909 -1.243359864  1.122102688  1.504700871  1.341782096
##  [46]  1.369339270  1.465098836  0.858577991  1.012183090  1.162406161
##  [51]  1.324519721  0.323729116  0.691831293  0.531558114  0.295197663
##  [56]  0.214496804 -0.102968794 -1.107427756  0.238662118  0.847345056
##  [61]  2.085101981  1.947810142  1.840236049  0.480516747  1.286661184
##  [66]  0.594993731 -1.739663058  0.969222459 -1.135579441  0.478710184
##  [71] -0.388897066 -0.204203299 -0.828524114  0.067422725 -0.101845708
##  [76]  0.029667316  0.254586960  0.520535589  1.746539552 -0.418209468
##  [81] -0.118979390  0.789742672  0.554266006  1.877869072  0.636862976
##  [86] -0.217599479  0.599841515  1.052964971  0.879175286 -0.880863458
##  [91] -2.693742014 -2.453361699  0.450461861 -0.686116009 -0.292776674
##  [96] -1.711404078 -0.918867194  0.899264760  0.117476431  2.543298648
## [101]  1.271637813  1.733972038  0.967369051  0.797733764 -0.026292789
## [106] -0.655758393 -1.089258071 -1.452123678  0.745628357 -0.180943905
## [111]  1.286828780 -1.527137746 -0.095320542  0.121962532  1.039760680
## [116] -0.200590299  0.760517150 -0.820704443  0.996365958  0.523813694
## [121]  2.477629395  1.936557366 -0.016117520 -0.367271918 -1.145417868
## [126] -1.332848895 -0.266839653  1.931648087  1.085875724  1.592165412
## [131]  0.623456180  0.272808470  0.098785826 -0.058467414  0.004328934
## [136] -0.341216304 -0.015749552  0.421748439  0.464742599 -0.492375670
## [141] -0.165574790 -0.439115079  0.899881601  1.028406929  0.073780314
## [146]  1.678533047  0.643288370  1.528085989 -0.293125461 -0.759029164
## [151] -1.097717622 -1.425809623 -0.827426707  0.110895333  1.310977002
## [156]  1.046876271 -0.701151350 -0.978549101 -3.054940206 -1.104452160
## [161] -1.321758109 -0.524601197 -0.388542387  2.362527597  1.599266052
## [166]  2.948831776 -0.124550611 -1.808071605 -0.776527759  0.627662067
## [171]  0.072225004 -0.382121197 -2.481396068 -1.703595934  0.727094596
## [176]  0.474287493  1.479500634 -0.566219721  0.278997481 -0.104424448
## [181] -1.829061685  1.423517561 -1.725159314  0.902429260 -0.984368492
## [186]  1.008183827  0.376070885 -0.198408292 -0.544321138 -0.844645942
## [191]  0.393989112  0.238929445  1.040084464  1.274098254  1.264141841
## [196]  0.429732159 -0.246493705  0.647582086  1.187549690 -0.738388893
## [201] -0.246977145 -1.659287019  0.942494047 -0.142656146  1.149978531
## [206] -1.042906088 -0.475675475  0.084695120  1.419077447  0.100289619
## [211]  0.928940628 -1.042369965 -2.029310892  0.692955872  0.572933560
## [216]  1.201031862 -0.435280227  0.072123662 -0.471911440  0.652403962
## [221]  0.174759067 -0.420422146 -0.122413686 -0.960531207  0.520175609
## [226] -0.246642012  1.479510072  1.920407048  0.049253120  2.410948668
## [231]  0.198517791  1.500187751 -0.497778326 -1.767346982 -2.191771939
## [236] -1.845532064 -1.534676414 -1.763197429  0.508517712  1.128180311
## [241] -0.597018811  1.358534075 -0.661014482  3.281067236  2.355457362
## [246]  1.712265522  1.383918601  0.130772983 -0.539085221  0.379084556
## [251] -2.257783390 -0.707537226 -0.114232581 -0.436963894  1.254925659
## [256] -0.232280148 -0.925141483  0.099354323 -0.781351321  0.971635478
## [261] -1.278289659 -0.566604140 -1.000779197 -0.066723937 -1.714517070
## [266] -0.224608222 -0.734668117 -0.649281904 -0.007208191  0.985330974
## [271] -0.947047578  0.119922554  2.018230565  0.399153678  2.286864719
## [276]  0.215612449  0.902373923 -0.989261018 -1.224407831 -2.440544588
## [281] -1.946771300  2.045475757 -0.839304227  0.354817940 -1.354455023
## [286] -2.157948379  0.237542950  0.265575809  3.228135690  2.010682381
## [291]  2.307858407  1.461384175 -0.245664295  0.725549412 -0.135603218
## [296] -0.359210423 -0.457191111 -0.481849244  1.572387199  0.455075045

Karakteristik MA(2)

Plot Time Series

ts.plot(ma2)

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

Plot ACF

acf(ma2,lag.max = 20)

Berdasarkan plot AFC tersebut, terlihat bahwa plot ACF cuts off setelah lag kedua maka ini menandakan MA(2)

Plot PACF

pacf(ma2)

Berdasarkan plot PACF tersebut, terlihat bahwa plot PACF cenderung tails off dan membentuk gelombang sinus yang artinya menandakan MA(2) tadi atau disini didapat AR(0)

Plot EACF

TSA::eacf(ma2)

## AR/MA
##   0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x o x o x x x o o o  o  o  o 
## 1 x x x x o o o o o o o  o  o  o 
## 2 x x o x o o o o o o o  o  o  o 
## 3 x x o x o o o o o o o  o  o  o 
## 4 x o x x o x o o o o o  o  o  o 
## 5 x o x x x x o o o o o  o  o  o 
## 6 x x o x o x o o o o o  o  o  o 
## 7 x x o x x x o x 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 Peubah Lag

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

#Yt
yt_ma2 <- ma2[-1]
yt_ma2

##   [1]  0.256737131  0.210730157  0.766625272 -0.397169430 -0.068333544
##   [6] -1.014767960 -0.975904305  0.629789396  0.468403379 -0.400669879
##  [11]  0.564095027 -1.039854411  0.061265290  0.644982965 -0.245623724
##  [16]  1.491307617  0.813271007  0.920169898  0.794900726  0.466668892
##  [21] -0.104897671 -1.180752848 -2.399760755 -1.845489666 -1.954119879
##  [26]  0.056563276  0.493584496 -0.142112886 -0.260158929 -2.085228014
##  [31] -0.390931945 -0.057532243 -0.053722045 -0.130638222  0.645828554
##  [36] -0.729403047 -0.370333182 -1.365213079 -1.611270366 -0.707932909
##  [41] -1.243359864  1.122102688  1.504700871  1.341782096  1.369339270
##  [46]  1.465098836  0.858577991  1.012183090  1.162406161  1.324519721
##  [51]  0.323729116  0.691831293  0.531558114  0.295197663  0.214496804
##  [56] -0.102968794 -1.107427756  0.238662118  0.847345056  2.085101981
##  [61]  1.947810142  1.840236049  0.480516747  1.286661184  0.594993731
##  [66] -1.739663058  0.969222459 -1.135579441  0.478710184 -0.388897066
##  [71] -0.204203299 -0.828524114  0.067422725 -0.101845708  0.029667316
##  [76]  0.254586960  0.520535589  1.746539552 -0.418209468 -0.118979390
##  [81]  0.789742672  0.554266006  1.877869072  0.636862976 -0.217599479
##  [86]  0.599841515  1.052964971  0.879175286 -0.880863458 -2.693742014
##  [91] -2.453361699  0.450461861 -0.686116009 -0.292776674 -1.711404078
##  [96] -0.918867194  0.899264760  0.117476431  2.543298648  1.271637813
## [101]  1.733972038  0.967369051  0.797733764 -0.026292789 -0.655758393
## [106] -1.089258071 -1.452123678  0.745628357 -0.180943905  1.286828780
## [111] -1.527137746 -0.095320542  0.121962532  1.039760680 -0.200590299
## [116]  0.760517150 -0.820704443  0.996365958  0.523813694  2.477629395
## [121]  1.936557366 -0.016117520 -0.367271918 -1.145417868 -1.332848895
## [126] -0.266839653  1.931648087  1.085875724  1.592165412  0.623456180
## [131]  0.272808470  0.098785826 -0.058467414  0.004328934 -0.341216304
## [136] -0.015749552  0.421748439  0.464742599 -0.492375670 -0.165574790
## [141] -0.439115079  0.899881601  1.028406929  0.073780314  1.678533047
## [146]  0.643288370  1.528085989 -0.293125461 -0.759029164 -1.097717622
## [151] -1.425809623 -0.827426707  0.110895333  1.310977002  1.046876271
## [156] -0.701151350 -0.978549101 -3.054940206 -1.104452160 -1.321758109
## [161] -0.524601197 -0.388542387  2.362527597  1.599266052  2.948831776
## [166] -0.124550611 -1.808071605 -0.776527759  0.627662067  0.072225004
## [171] -0.382121197 -2.481396068 -1.703595934  0.727094596  0.474287493
## [176]  1.479500634 -0.566219721  0.278997481 -0.104424448 -1.829061685
## [181]  1.423517561 -1.725159314  0.902429260 -0.984368492  1.008183827
## [186]  0.376070885 -0.198408292 -0.544321138 -0.844645942  0.393989112
## [191]  0.238929445  1.040084464  1.274098254  1.264141841  0.429732159
## [196] -0.246493705  0.647582086  1.187549690 -0.738388893 -0.246977145
## [201] -1.659287019  0.942494047 -0.142656146  1.149978531 -1.042906088
## [206] -0.475675475  0.084695120  1.419077447  0.100289619  0.928940628
## [211] -1.042369965 -2.029310892  0.692955872  0.572933560  1.201031862
## [216] -0.435280227  0.072123662 -0.471911440  0.652403962  0.174759067
## [221] -0.420422146 -0.122413686 -0.960531207  0.520175609 -0.246642012
## [226]  1.479510072  1.920407048  0.049253120  2.410948668  0.198517791
## [231]  1.500187751 -0.497778326 -1.767346982 -2.191771939 -1.845532064
## [236] -1.534676414 -1.763197429  0.508517712  1.128180311 -0.597018811
## [241]  1.358534075 -0.661014482  3.281067236  2.355457362  1.712265522
## [246]  1.383918601  0.130772983 -0.539085221  0.379084556 -2.257783390
## [251] -0.707537226 -0.114232581 -0.436963894  1.254925659 -0.232280148
## [256] -0.925141483  0.099354323 -0.781351321  0.971635478 -1.278289659
## [261] -0.566604140 -1.000779197 -0.066723937 -1.714517070 -0.224608222
## [266] -0.734668117 -0.649281904 -0.007208191  0.985330974 -0.947047578
## [271]  0.119922554  2.018230565  0.399153678  2.286864719  0.215612449
## [276]  0.902373923 -0.989261018 -1.224407831 -2.440544588 -1.946771300
## [281]  2.045475757 -0.839304227  0.354817940 -1.354455023 -2.157948379
## [286]  0.237542950  0.265575809  3.228135690  2.010682381  2.307858407
## [291]  1.461384175 -0.245664295  0.725549412 -0.135603218 -0.359210423
## [296] -0.457191111 -0.481849244  1.572387199  0.455075045

#Yt-1
yt_1_ma2 <- ma2[-300]
yt_1_ma2

##   [1]  2.198327037  0.256737131  0.210730157  0.766625272 -0.397169430
##   [6] -0.068333544 -1.014767960 -0.975904305  0.629789396  0.468403379
##  [11] -0.400669879  0.564095027 -1.039854411  0.061265290  0.644982965
##  [16] -0.245623724  1.491307617  0.813271007  0.920169898  0.794900726
##  [21]  0.466668892 -0.104897671 -1.180752848 -2.399760755 -1.845489666
##  [26] -1.954119879  0.056563276  0.493584496 -0.142112886 -0.260158929
##  [31] -2.085228014 -0.390931945 -0.057532243 -0.053722045 -0.130638222
##  [36]  0.645828554 -0.729403047 -0.370333182 -1.365213079 -1.611270366
##  [41] -0.707932909 -1.243359864  1.122102688  1.504700871  1.341782096
##  [46]  1.369339270  1.465098836  0.858577991  1.012183090  1.162406161
##  [51]  1.324519721  0.323729116  0.691831293  0.531558114  0.295197663
##  [56]  0.214496804 -0.102968794 -1.107427756  0.238662118  0.847345056
##  [61]  2.085101981  1.947810142  1.840236049  0.480516747  1.286661184
##  [66]  0.594993731 -1.739663058  0.969222459 -1.135579441  0.478710184
##  [71] -0.388897066 -0.204203299 -0.828524114  0.067422725 -0.101845708
##  [76]  0.029667316  0.254586960  0.520535589  1.746539552 -0.418209468
##  [81] -0.118979390  0.789742672  0.554266006  1.877869072  0.636862976
##  [86] -0.217599479  0.599841515  1.052964971  0.879175286 -0.880863458
##  [91] -2.693742014 -2.453361699  0.450461861 -0.686116009 -0.292776674
##  [96] -1.711404078 -0.918867194  0.899264760  0.117476431  2.543298648
## [101]  1.271637813  1.733972038  0.967369051  0.797733764 -0.026292789
## [106] -0.655758393 -1.089258071 -1.452123678  0.745628357 -0.180943905
## [111]  1.286828780 -1.527137746 -0.095320542  0.121962532  1.039760680
## [116] -0.200590299  0.760517150 -0.820704443  0.996365958  0.523813694
## [121]  2.477629395  1.936557366 -0.016117520 -0.367271918 -1.145417868
## [126] -1.332848895 -0.266839653  1.931648087  1.085875724  1.592165412
## [131]  0.623456180  0.272808470  0.098785826 -0.058467414  0.004328934
## [136] -0.341216304 -0.015749552  0.421748439  0.464742599 -0.492375670
## [141] -0.165574790 -0.439115079  0.899881601  1.028406929  0.073780314
## [146]  1.678533047  0.643288370  1.528085989 -0.293125461 -0.759029164
## [151] -1.097717622 -1.425809623 -0.827426707  0.110895333  1.310977002
## [156]  1.046876271 -0.701151350 -0.978549101 -3.054940206 -1.104452160
## [161] -1.321758109 -0.524601197 -0.388542387  2.362527597  1.599266052
## [166]  2.948831776 -0.124550611 -1.808071605 -0.776527759  0.627662067
## [171]  0.072225004 -0.382121197 -2.481396068 -1.703595934  0.727094596
## [176]  0.474287493  1.479500634 -0.566219721  0.278997481 -0.104424448
## [181] -1.829061685  1.423517561 -1.725159314  0.902429260 -0.984368492
## [186]  1.008183827  0.376070885 -0.198408292 -0.544321138 -0.844645942
## [191]  0.393989112  0.238929445  1.040084464  1.274098254  1.264141841
## [196]  0.429732159 -0.246493705  0.647582086  1.187549690 -0.738388893
## [201] -0.246977145 -1.659287019  0.942494047 -0.142656146  1.149978531
## [206] -1.042906088 -0.475675475  0.084695120  1.419077447  0.100289619
## [211]  0.928940628 -1.042369965 -2.029310892  0.692955872  0.572933560
## [216]  1.201031862 -0.435280227  0.072123662 -0.471911440  0.652403962
## [221]  0.174759067 -0.420422146 -0.122413686 -0.960531207  0.520175609
## [226] -0.246642012  1.479510072  1.920407048  0.049253120  2.410948668
## [231]  0.198517791  1.500187751 -0.497778326 -1.767346982 -2.191771939
## [236] -1.845532064 -1.534676414 -1.763197429  0.508517712  1.128180311
## [241] -0.597018811  1.358534075 -0.661014482  3.281067236  2.355457362
## [246]  1.712265522  1.383918601  0.130772983 -0.539085221  0.379084556
## [251] -2.257783390 -0.707537226 -0.114232581 -0.436963894  1.254925659
## [256] -0.232280148 -0.925141483  0.099354323 -0.781351321  0.971635478
## [261] -1.278289659 -0.566604140 -1.000779197 -0.066723937 -1.714517070
## [266] -0.224608222 -0.734668117 -0.649281904 -0.007208191  0.985330974
## [271] -0.947047578  0.119922554  2.018230565  0.399153678  2.286864719
## [276]  0.215612449  0.902373923 -0.989261018 -1.224407831 -2.440544588
## [281] -1.946771300  2.045475757 -0.839304227  0.354817940 -1.354455023
## [286] -2.157948379  0.237542950  0.265575809  3.228135690  2.010682381
## [291]  2.307858407  1.461384175 -0.245664295  0.725549412 -0.135603218
## [296] -0.359210423 -0.457191111 -0.481849244  1.572387199

plot(y=yt_ma2,x=yt_1_ma2)

Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif yang rendah antara \(Y_t\) dengan \(Y_{t-1}\).

cor(yt_ma2,yt_1_ma2)

## [1] 0.3551089

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

#Yt
yt_ma2.2 <- ma2[-c(1,2)]
yt_ma2.2

##   [1]  0.210730157  0.766625272 -0.397169430 -0.068333544 -1.014767960
##   [6] -0.975904305  0.629789396  0.468403379 -0.400669879  0.564095027
##  [11] -1.039854411  0.061265290  0.644982965 -0.245623724  1.491307617
##  [16]  0.813271007  0.920169898  0.794900726  0.466668892 -0.104897671
##  [21] -1.180752848 -2.399760755 -1.845489666 -1.954119879  0.056563276
##  [26]  0.493584496 -0.142112886 -0.260158929 -2.085228014 -0.390931945
##  [31] -0.057532243 -0.053722045 -0.130638222  0.645828554 -0.729403047
##  [36] -0.370333182 -1.365213079 -1.611270366 -0.707932909 -1.243359864
##  [41]  1.122102688  1.504700871  1.341782096  1.369339270  1.465098836
##  [46]  0.858577991  1.012183090  1.162406161  1.324519721  0.323729116
##  [51]  0.691831293  0.531558114  0.295197663  0.214496804 -0.102968794
##  [56] -1.107427756  0.238662118  0.847345056  2.085101981  1.947810142
##  [61]  1.840236049  0.480516747  1.286661184  0.594993731 -1.739663058
##  [66]  0.969222459 -1.135579441  0.478710184 -0.388897066 -0.204203299
##  [71] -0.828524114  0.067422725 -0.101845708  0.029667316  0.254586960
##  [76]  0.520535589  1.746539552 -0.418209468 -0.118979390  0.789742672
##  [81]  0.554266006  1.877869072  0.636862976 -0.217599479  0.599841515
##  [86]  1.052964971  0.879175286 -0.880863458 -2.693742014 -2.453361699
##  [91]  0.450461861 -0.686116009 -0.292776674 -1.711404078 -0.918867194
##  [96]  0.899264760  0.117476431  2.543298648  1.271637813  1.733972038
## [101]  0.967369051  0.797733764 -0.026292789 -0.655758393 -1.089258071
## [106] -1.452123678  0.745628357 -0.180943905  1.286828780 -1.527137746
## [111] -0.095320542  0.121962532  1.039760680 -0.200590299  0.760517150
## [116] -0.820704443  0.996365958  0.523813694  2.477629395  1.936557366
## [121] -0.016117520 -0.367271918 -1.145417868 -1.332848895 -0.266839653
## [126]  1.931648087  1.085875724  1.592165412  0.623456180  0.272808470
## [131]  0.098785826 -0.058467414  0.004328934 -0.341216304 -0.015749552
## [136]  0.421748439  0.464742599 -0.492375670 -0.165574790 -0.439115079
## [141]  0.899881601  1.028406929  0.073780314  1.678533047  0.643288370
## [146]  1.528085989 -0.293125461 -0.759029164 -1.097717622 -1.425809623
## [151] -0.827426707  0.110895333  1.310977002  1.046876271 -0.701151350
## [156] -0.978549101 -3.054940206 -1.104452160 -1.321758109 -0.524601197
## [161] -0.388542387  2.362527597  1.599266052  2.948831776 -0.124550611
## [166] -1.808071605 -0.776527759  0.627662067  0.072225004 -0.382121197
## [171] -2.481396068 -1.703595934  0.727094596  0.474287493  1.479500634
## [176] -0.566219721  0.278997481 -0.104424448 -1.829061685  1.423517561
## [181] -1.725159314  0.902429260 -0.984368492  1.008183827  0.376070885
## [186] -0.198408292 -0.544321138 -0.844645942  0.393989112  0.238929445
## [191]  1.040084464  1.274098254  1.264141841  0.429732159 -0.246493705
## [196]  0.647582086  1.187549690 -0.738388893 -0.246977145 -1.659287019
## [201]  0.942494047 -0.142656146  1.149978531 -1.042906088 -0.475675475
## [206]  0.084695120  1.419077447  0.100289619  0.928940628 -1.042369965
## [211] -2.029310892  0.692955872  0.572933560  1.201031862 -0.435280227
## [216]  0.072123662 -0.471911440  0.652403962  0.174759067 -0.420422146
## [221] -0.122413686 -0.960531207  0.520175609 -0.246642012  1.479510072
## [226]  1.920407048  0.049253120  2.410948668  0.198517791  1.500187751
## [231] -0.497778326 -1.767346982 -2.191771939 -1.845532064 -1.534676414
## [236] -1.763197429  0.508517712  1.128180311 -0.597018811  1.358534075
## [241] -0.661014482  3.281067236  2.355457362  1.712265522  1.383918601
## [246]  0.130772983 -0.539085221  0.379084556 -2.257783390 -0.707537226
## [251] -0.114232581 -0.436963894  1.254925659 -0.232280148 -0.925141483
## [256]  0.099354323 -0.781351321  0.971635478 -1.278289659 -0.566604140
## [261] -1.000779197 -0.066723937 -1.714517070 -0.224608222 -0.734668117
## [266] -0.649281904 -0.007208191  0.985330974 -0.947047578  0.119922554
## [271]  2.018230565  0.399153678  2.286864719  0.215612449  0.902373923
## [276] -0.989261018 -1.224407831 -2.440544588 -1.946771300  2.045475757
## [281] -0.839304227  0.354817940 -1.354455023 -2.157948379  0.237542950
## [286]  0.265575809  3.228135690  2.010682381  2.307858407  1.461384175
## [291] -0.245664295  0.725549412 -0.135603218 -0.359210423 -0.457191111
## [296] -0.481849244  1.572387199  0.455075045

#Yt-2
yt_2_ma2 <- ma2[-c(299,300)]
yt_2_ma2

##   [1]  2.198327037  0.256737131  0.210730157  0.766625272 -0.397169430
##   [6] -0.068333544 -1.014767960 -0.975904305  0.629789396  0.468403379
##  [11] -0.400669879  0.564095027 -1.039854411  0.061265290  0.644982965
##  [16] -0.245623724  1.491307617  0.813271007  0.920169898  0.794900726
##  [21]  0.466668892 -0.104897671 -1.180752848 -2.399760755 -1.845489666
##  [26] -1.954119879  0.056563276  0.493584496 -0.142112886 -0.260158929
##  [31] -2.085228014 -0.390931945 -0.057532243 -0.053722045 -0.130638222
##  [36]  0.645828554 -0.729403047 -0.370333182 -1.365213079 -1.611270366
##  [41] -0.707932909 -1.243359864  1.122102688  1.504700871  1.341782096
##  [46]  1.369339270  1.465098836  0.858577991  1.012183090  1.162406161
##  [51]  1.324519721  0.323729116  0.691831293  0.531558114  0.295197663
##  [56]  0.214496804 -0.102968794 -1.107427756  0.238662118  0.847345056
##  [61]  2.085101981  1.947810142  1.840236049  0.480516747  1.286661184
##  [66]  0.594993731 -1.739663058  0.969222459 -1.135579441  0.478710184
##  [71] -0.388897066 -0.204203299 -0.828524114  0.067422725 -0.101845708
##  [76]  0.029667316  0.254586960  0.520535589  1.746539552 -0.418209468
##  [81] -0.118979390  0.789742672  0.554266006  1.877869072  0.636862976
##  [86] -0.217599479  0.599841515  1.052964971  0.879175286 -0.880863458
##  [91] -2.693742014 -2.453361699  0.450461861 -0.686116009 -0.292776674
##  [96] -1.711404078 -0.918867194  0.899264760  0.117476431  2.543298648
## [101]  1.271637813  1.733972038  0.967369051  0.797733764 -0.026292789
## [106] -0.655758393 -1.089258071 -1.452123678  0.745628357 -0.180943905
## [111]  1.286828780 -1.527137746 -0.095320542  0.121962532  1.039760680
## [116] -0.200590299  0.760517150 -0.820704443  0.996365958  0.523813694
## [121]  2.477629395  1.936557366 -0.016117520 -0.367271918 -1.145417868
## [126] -1.332848895 -0.266839653  1.931648087  1.085875724  1.592165412
## [131]  0.623456180  0.272808470  0.098785826 -0.058467414  0.004328934
## [136] -0.341216304 -0.015749552  0.421748439  0.464742599 -0.492375670
## [141] -0.165574790 -0.439115079  0.899881601  1.028406929  0.073780314
## [146]  1.678533047  0.643288370  1.528085989 -0.293125461 -0.759029164
## [151] -1.097717622 -1.425809623 -0.827426707  0.110895333  1.310977002
## [156]  1.046876271 -0.701151350 -0.978549101 -3.054940206 -1.104452160
## [161] -1.321758109 -0.524601197 -0.388542387  2.362527597  1.599266052
## [166]  2.948831776 -0.124550611 -1.808071605 -0.776527759  0.627662067
## [171]  0.072225004 -0.382121197 -2.481396068 -1.703595934  0.727094596
## [176]  0.474287493  1.479500634 -0.566219721  0.278997481 -0.104424448
## [181] -1.829061685  1.423517561 -1.725159314  0.902429260 -0.984368492
## [186]  1.008183827  0.376070885 -0.198408292 -0.544321138 -0.844645942
## [191]  0.393989112  0.238929445  1.040084464  1.274098254  1.264141841
## [196]  0.429732159 -0.246493705  0.647582086  1.187549690 -0.738388893
## [201] -0.246977145 -1.659287019  0.942494047 -0.142656146  1.149978531
## [206] -1.042906088 -0.475675475  0.084695120  1.419077447  0.100289619
## [211]  0.928940628 -1.042369965 -2.029310892  0.692955872  0.572933560
## [216]  1.201031862 -0.435280227  0.072123662 -0.471911440  0.652403962
## [221]  0.174759067 -0.420422146 -0.122413686 -0.960531207  0.520175609
## [226] -0.246642012  1.479510072  1.920407048  0.049253120  2.410948668
## [231]  0.198517791  1.500187751 -0.497778326 -1.767346982 -2.191771939
## [236] -1.845532064 -1.534676414 -1.763197429  0.508517712  1.128180311
## [241] -0.597018811  1.358534075 -0.661014482  3.281067236  2.355457362
## [246]  1.712265522  1.383918601  0.130772983 -0.539085221  0.379084556
## [251] -2.257783390 -0.707537226 -0.114232581 -0.436963894  1.254925659
## [256] -0.232280148 -0.925141483  0.099354323 -0.781351321  0.971635478
## [261] -1.278289659 -0.566604140 -1.000779197 -0.066723937 -1.714517070
## [266] -0.224608222 -0.734668117 -0.649281904 -0.007208191  0.985330974
## [271] -0.947047578  0.119922554  2.018230565  0.399153678  2.286864719
## [276]  0.215612449  0.902373923 -0.989261018 -1.224407831 -2.440544588
## [281] -1.946771300  2.045475757 -0.839304227  0.354817940 -1.354455023
## [286] -2.157948379  0.237542950  0.265575809  3.228135690  2.010682381
## [291]  2.307858407  1.461384175 -0.245664295  0.725549412 -0.135603218
## [296] -0.359210423 -0.457191111 -0.481849244

plot(y=yt_ma2.2,x=yt_2_ma2)

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

cor(yt_ma2.2,yt_2_ma2)

## [1] 0.2844754

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

#Yt
yt_ma2.3 <- ma2[-c(1,2,3)]
yt_ma2.3

##   [1]  0.766625272 -0.397169430 -0.068333544 -1.014767960 -0.975904305
##   [6]  0.629789396  0.468403379 -0.400669879  0.564095027 -1.039854411
##  [11]  0.061265290  0.644982965 -0.245623724  1.491307617  0.813271007
##  [16]  0.920169898  0.794900726  0.466668892 -0.104897671 -1.180752848
##  [21] -2.399760755 -1.845489666 -1.954119879  0.056563276  0.493584496
##  [26] -0.142112886 -0.260158929 -2.085228014 -0.390931945 -0.057532243
##  [31] -0.053722045 -0.130638222  0.645828554 -0.729403047 -0.370333182
##  [36] -1.365213079 -1.611270366 -0.707932909 -1.243359864  1.122102688
##  [41]  1.504700871  1.341782096  1.369339270  1.465098836  0.858577991
##  [46]  1.012183090  1.162406161  1.324519721  0.323729116  0.691831293
##  [51]  0.531558114  0.295197663  0.214496804 -0.102968794 -1.107427756
##  [56]  0.238662118  0.847345056  2.085101981  1.947810142  1.840236049
##  [61]  0.480516747  1.286661184  0.594993731 -1.739663058  0.969222459
##  [66] -1.135579441  0.478710184 -0.388897066 -0.204203299 -0.828524114
##  [71]  0.067422725 -0.101845708  0.029667316  0.254586960  0.520535589
##  [76]  1.746539552 -0.418209468 -0.118979390  0.789742672  0.554266006
##  [81]  1.877869072  0.636862976 -0.217599479  0.599841515  1.052964971
##  [86]  0.879175286 -0.880863458 -2.693742014 -2.453361699  0.450461861
##  [91] -0.686116009 -0.292776674 -1.711404078 -0.918867194  0.899264760
##  [96]  0.117476431  2.543298648  1.271637813  1.733972038  0.967369051
## [101]  0.797733764 -0.026292789 -0.655758393 -1.089258071 -1.452123678
## [106]  0.745628357 -0.180943905  1.286828780 -1.527137746 -0.095320542
## [111]  0.121962532  1.039760680 -0.200590299  0.760517150 -0.820704443
## [116]  0.996365958  0.523813694  2.477629395  1.936557366 -0.016117520
## [121] -0.367271918 -1.145417868 -1.332848895 -0.266839653  1.931648087
## [126]  1.085875724  1.592165412  0.623456180  0.272808470  0.098785826
## [131] -0.058467414  0.004328934 -0.341216304 -0.015749552  0.421748439
## [136]  0.464742599 -0.492375670 -0.165574790 -0.439115079  0.899881601
## [141]  1.028406929  0.073780314  1.678533047  0.643288370  1.528085989
## [146] -0.293125461 -0.759029164 -1.097717622 -1.425809623 -0.827426707
## [151]  0.110895333  1.310977002  1.046876271 -0.701151350 -0.978549101
## [156] -3.054940206 -1.104452160 -1.321758109 -0.524601197 -0.388542387
## [161]  2.362527597  1.599266052  2.948831776 -0.124550611 -1.808071605
## [166] -0.776527759  0.627662067  0.072225004 -0.382121197 -2.481396068
## [171] -1.703595934  0.727094596  0.474287493  1.479500634 -0.566219721
## [176]  0.278997481 -0.104424448 -1.829061685  1.423517561 -1.725159314
## [181]  0.902429260 -0.984368492  1.008183827  0.376070885 -0.198408292
## [186] -0.544321138 -0.844645942  0.393989112  0.238929445  1.040084464
## [191]  1.274098254  1.264141841  0.429732159 -0.246493705  0.647582086
## [196]  1.187549690 -0.738388893 -0.246977145 -1.659287019  0.942494047
## [201] -0.142656146  1.149978531 -1.042906088 -0.475675475  0.084695120
## [206]  1.419077447  0.100289619  0.928940628 -1.042369965 -2.029310892
## [211]  0.692955872  0.572933560  1.201031862 -0.435280227  0.072123662
## [216] -0.471911440  0.652403962  0.174759067 -0.420422146 -0.122413686
## [221] -0.960531207  0.520175609 -0.246642012  1.479510072  1.920407048
## [226]  0.049253120  2.410948668  0.198517791  1.500187751 -0.497778326
## [231] -1.767346982 -2.191771939 -1.845532064 -1.534676414 -1.763197429
## [236]  0.508517712  1.128180311 -0.597018811  1.358534075 -0.661014482
## [241]  3.281067236  2.355457362  1.712265522  1.383918601  0.130772983
## [246] -0.539085221  0.379084556 -2.257783390 -0.707537226 -0.114232581
## [251] -0.436963894  1.254925659 -0.232280148 -0.925141483  0.099354323
## [256] -0.781351321  0.971635478 -1.278289659 -0.566604140 -1.000779197
## [261] -0.066723937 -1.714517070 -0.224608222 -0.734668117 -0.649281904
## [266] -0.007208191  0.985330974 -0.947047578  0.119922554  2.018230565
## [271]  0.399153678  2.286864719  0.215612449  0.902373923 -0.989261018
## [276] -1.224407831 -2.440544588 -1.946771300  2.045475757 -0.839304227
## [281]  0.354817940 -1.354455023 -2.157948379  0.237542950  0.265575809
## [286]  3.228135690  2.010682381  2.307858407  1.461384175 -0.245664295
## [291]  0.725549412 -0.135603218 -0.359210423 -0.457191111 -0.481849244
## [296]  1.572387199  0.455075045

#Yt-2
yt_3_ma2 <- ma2[-c(298,299,300)]
yt_3_ma2

##   [1]  2.198327037  0.256737131  0.210730157  0.766625272 -0.397169430
##   [6] -0.068333544 -1.014767960 -0.975904305  0.629789396  0.468403379
##  [11] -0.400669879  0.564095027 -1.039854411  0.061265290  0.644982965
##  [16] -0.245623724  1.491307617  0.813271007  0.920169898  0.794900726
##  [21]  0.466668892 -0.104897671 -1.180752848 -2.399760755 -1.845489666
##  [26] -1.954119879  0.056563276  0.493584496 -0.142112886 -0.260158929
##  [31] -2.085228014 -0.390931945 -0.057532243 -0.053722045 -0.130638222
##  [36]  0.645828554 -0.729403047 -0.370333182 -1.365213079 -1.611270366
##  [41] -0.707932909 -1.243359864  1.122102688  1.504700871  1.341782096
##  [46]  1.369339270  1.465098836  0.858577991  1.012183090  1.162406161
##  [51]  1.324519721  0.323729116  0.691831293  0.531558114  0.295197663
##  [56]  0.214496804 -0.102968794 -1.107427756  0.238662118  0.847345056
##  [61]  2.085101981  1.947810142  1.840236049  0.480516747  1.286661184
##  [66]  0.594993731 -1.739663058  0.969222459 -1.135579441  0.478710184
##  [71] -0.388897066 -0.204203299 -0.828524114  0.067422725 -0.101845708
##  [76]  0.029667316  0.254586960  0.520535589  1.746539552 -0.418209468
##  [81] -0.118979390  0.789742672  0.554266006  1.877869072  0.636862976
##  [86] -0.217599479  0.599841515  1.052964971  0.879175286 -0.880863458
##  [91] -2.693742014 -2.453361699  0.450461861 -0.686116009 -0.292776674
##  [96] -1.711404078 -0.918867194  0.899264760  0.117476431  2.543298648
## [101]  1.271637813  1.733972038  0.967369051  0.797733764 -0.026292789
## [106] -0.655758393 -1.089258071 -1.452123678  0.745628357 -0.180943905
## [111]  1.286828780 -1.527137746 -0.095320542  0.121962532  1.039760680
## [116] -0.200590299  0.760517150 -0.820704443  0.996365958  0.523813694
## [121]  2.477629395  1.936557366 -0.016117520 -0.367271918 -1.145417868
## [126] -1.332848895 -0.266839653  1.931648087  1.085875724  1.592165412
## [131]  0.623456180  0.272808470  0.098785826 -0.058467414  0.004328934
## [136] -0.341216304 -0.015749552  0.421748439  0.464742599 -0.492375670
## [141] -0.165574790 -0.439115079  0.899881601  1.028406929  0.073780314
## [146]  1.678533047  0.643288370  1.528085989 -0.293125461 -0.759029164
## [151] -1.097717622 -1.425809623 -0.827426707  0.110895333  1.310977002
## [156]  1.046876271 -0.701151350 -0.978549101 -3.054940206 -1.104452160
## [161] -1.321758109 -0.524601197 -0.388542387  2.362527597  1.599266052
## [166]  2.948831776 -0.124550611 -1.808071605 -0.776527759  0.627662067
## [171]  0.072225004 -0.382121197 -2.481396068 -1.703595934  0.727094596
## [176]  0.474287493  1.479500634 -0.566219721  0.278997481 -0.104424448
## [181] -1.829061685  1.423517561 -1.725159314  0.902429260 -0.984368492
## [186]  1.008183827  0.376070885 -0.198408292 -0.544321138 -0.844645942
## [191]  0.393989112  0.238929445  1.040084464  1.274098254  1.264141841
## [196]  0.429732159 -0.246493705  0.647582086  1.187549690 -0.738388893
## [201] -0.246977145 -1.659287019  0.942494047 -0.142656146  1.149978531
## [206] -1.042906088 -0.475675475  0.084695120  1.419077447  0.100289619
## [211]  0.928940628 -1.042369965 -2.029310892  0.692955872  0.572933560
## [216]  1.201031862 -0.435280227  0.072123662 -0.471911440  0.652403962
## [221]  0.174759067 -0.420422146 -0.122413686 -0.960531207  0.520175609
## [226] -0.246642012  1.479510072  1.920407048  0.049253120  2.410948668
## [231]  0.198517791  1.500187751 -0.497778326 -1.767346982 -2.191771939
## [236] -1.845532064 -1.534676414 -1.763197429  0.508517712  1.128180311
## [241] -0.597018811  1.358534075 -0.661014482  3.281067236  2.355457362
## [246]  1.712265522  1.383918601  0.130772983 -0.539085221  0.379084556
## [251] -2.257783390 -0.707537226 -0.114232581 -0.436963894  1.254925659
## [256] -0.232280148 -0.925141483  0.099354323 -0.781351321  0.971635478
## [261] -1.278289659 -0.566604140 -1.000779197 -0.066723937 -1.714517070
## [266] -0.224608222 -0.734668117 -0.649281904 -0.007208191  0.985330974
## [271] -0.947047578  0.119922554  2.018230565  0.399153678  2.286864719
## [276]  0.215612449  0.902373923 -0.989261018 -1.224407831 -2.440544588
## [281] -1.946771300  2.045475757 -0.839304227  0.354817940 -1.354455023
## [286] -2.157948379  0.237542950  0.265575809  3.228135690  2.010682381
## [291]  2.307858407  1.461384175 -0.245664295  0.725549412 -0.135603218
## [296] -0.359210423 -0.457191111

plot(y=yt_ma2.3,x=yt_3_ma2)

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

cor(yt_ma2.3,yt_3_ma2)

## [1] -0.07238159

Perbandingan korelasi \(Yt_t\) dengan \(Y_{t-1}\), \(Yt_t\) dengan \(Y_{t-2}\), & \(Yt_t\) dengan \(Y_{t-3}\)

par(mfrow=c(2,2))
plot(y=yt_ma2,x=yt_1_ma2)
plot(y=yt_ma2.2,x=yt_2_ma2)
plot(y=yt_ma2.3,x=yt_3_ma2)

Proses AR

Pembangkitan Proses AR(2)

Akan dicoba membangkitkan proses AR paling sederhana, yaitu AR(2) dengan \(\phi_1 = 0.5\) dan \(\phi_2 = 0.2\) sebanyak 300 observasi dan \(c=0\).

Nilai-nilai selanjutnya dapat dicari melalui loop. Bentuk loop dapat dilihat dari rumus AR(1) yang hendak dibangkitkan:

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

Cara Manual

n<-length(wn)
n

## [1] 300

ar2 <- c(2:n) 
for (i in 3:n) {ar2[i]<-wn[i]+0.5*ar2[i-1]+0.2*ar2[i-2]}
ar2

##   [1]  2.000000000  3.000000000  2.907826150  2.346727647  1.545969693
##   [6]  3.100422766  1.793378916  1.354009060  0.835819633  0.989273362
##  [11] -0.102106675  0.228763239  0.836988560  0.380224734 -0.435384438
##  [16] -1.063800903  0.242585039  1.911474527  1.940805284 -0.263037324
##  [21] -0.318414195 -0.215787651 -0.847689268 -1.516630439 -1.471133641
##  [26] -0.482748378 -0.282763200 -1.139745950  0.197965741 -1.689246072
##  [31] -0.267035884 -1.739721043 -0.282747871 -1.025079962 -1.588732372
##  [36] -1.807263685 -1.164553092  0.006482120 -1.356433058 -0.878088399
##  [41] -0.938826664 -0.086314752  0.517822725  2.159852781  2.190698747
##  [46]  4.780669330  2.666726401  2.623252612  0.666298611  1.935404159
##  [51]  0.104780858  0.966501963 -0.986618351  0.141354406  1.300712261
##  [56]  0.612345128  2.171337697  0.360317128  1.656602674  0.594214169
##  [61] -0.163663917  1.458669550  0.542346923  0.431222747 -0.560731164
##  [66]  1.519330217  0.806881578  2.812320816  1.273535612  0.540411820
##  [71]  0.026059441 -0.080491087 -0.467592718 -0.643025971 -0.068458190
##  [76] -0.649675088 -1.161169728 -0.490988759  1.643893949  1.622324911
##  [81]  2.232487717  1.798601774  0.714426011  1.370605202  2.506536157
##  [86] -0.001211787  1.093036278 -0.693809105 -1.398642830 -2.058177464
##  [91] -0.685888029 -0.765840518 -0.073700522 -0.396026720  0.199899017
##  [96] -0.048422659 -0.317481477 -0.566360740 -0.847827015 -1.309923872
## [101] -2.536923460 -2.329665208 -3.083072048 -1.474710865 -1.751670617
## [106] -1.455130728 -0.952915360  0.012642864 -1.059858826 -1.686765874
## [111] -1.201223486 -2.616393612 -2.871026147 -2.517669198 -4.322821311
## [116] -1.485755230 -0.813737038  0.903415238  0.354908183  1.387142498
## [121]  0.097969733 -0.020334548 -0.426406305 -2.164273941 -2.005326477
## [126] -2.177046888 -2.904705174 -0.752609571 -0.794124898 -1.024978864
## [131] -1.199979596 -1.720445184 -3.267368694 -2.398114109 -0.617674478
## [136] -0.300718165 -1.049011363  0.195422098 -1.519341335  1.098944418
## [141]  1.843478807  1.293020521  1.200775593 -0.444320956  1.100037277
## [146]  0.050245753  0.203345801 -1.549124178 -1.313626254 -1.787388470
## [151] -1.919798455 -1.694579862 -0.683932883  0.967409465  0.252982087
## [156]  0.501395629  0.381905305  0.756544345 -0.792325342 -0.388444418
## [161] -1.039481391 -1.136626836 -0.394407282 -0.786819301 -0.747275423
## [166] -1.365681938 -0.027793910 -0.752148724 -0.411371236 -0.990936507
## [171] -2.666492684 -1.240908818 -2.263877900 -1.332004058  0.094991680
## [176]  0.203405679  0.555476907  1.321815820  0.752176082  0.853530991
## [181]  1.081847724  0.425578490 -1.279739607 -0.224366532  0.465458873
## [186]  0.079156222 -1.268429726  0.058417345 -2.015172365 -1.456276275
## [191] -1.447332621 -2.549456146 -1.349250497 -0.780718527 -1.071766766
## [196] -1.908120775 -1.138129582 -0.711143181 -0.564882311  1.407741167
## [201]  1.303391069 -1.100942769 -2.697669018 -2.279377940 -3.298795357
## [206] -3.487593562 -0.611327692 -1.191780117 -1.845571116 -1.641091419
## [211] -2.627051638 -2.691145909 -3.437230758 -0.871967255 -1.888969477
## [216] -0.522672671 -0.530438357  1.083151981 -0.177182768  0.062870554
## [221]  0.261708505  0.132825915  0.451961107 -0.540506394 -1.371776865
## [226] -0.100038395 -1.517542849 -2.189187558 -0.989986591  0.111956639
## [231]  0.251420835  0.630798774  0.394272905  0.737715360 -0.165001414
## [236]  0.833710264  0.069160056  0.572025356 -1.631029824  0.095201535
## [241] -0.336556195  0.778079915  0.818532349 -0.144056319 -0.669444553
## [246]  0.989230935  2.756960439  1.136577846  0.831812670  2.013511870
## [251]  0.211097180  0.322853315 -1.325137608 -1.347331395 -1.721911157
## [256] -0.227905059 -1.241498313 -0.745080526  2.688198523  1.856997146
## [261]  2.828550023  1.914161434  0.544917832  0.130412337  1.466053973
## [266]  0.231410788  0.536520509 -0.447499366 -1.037617741 -0.333292431
## [271]  1.311599053 -0.274201571 -0.374197097 -0.545999417  1.083760401
## [276] -1.188143463 -1.021491897 -0.673157810 -2.328030899 -1.569068272
## [281] -1.802065072 -0.884560861 -2.137237236 -1.715078465  0.515855049
## [286] -0.717901099 -3.922832917 -1.307216123 -1.194020393 -1.070408132
## [291] -3.137696025 -1.208921670  0.051991083  0.288405620 -1.172679492
## [296]  0.453819437 -0.743087402  0.308578431 -1.766993864 -0.746631302

Fungsi arima.sim

ar2 <- arima.sim(list(order=c(2,0,0), ar=c(0.5,0.2)), n=300)
ar2

## Time Series:
## Start = 1 
## End = 300 
## Frequency = 1 
##   [1]  0.6739349283  0.9076604902  1.9914390098  1.3539009856  0.7500803953
##   [6]  0.6347400983 -0.5643707523 -1.0794914635  1.2246041100  0.3656149571
##  [11]  0.1857602467 -0.7673915289 -1.8335976688 -1.8829328493 -1.4433673123
##  [16] -1.4790464669 -1.8536960966 -1.5851897819 -2.1437247548 -1.9844423449
##  [21] -1.0330610046 -2.2980781084 -2.9842466251 -3.3707774768 -0.8982790943
##  [26] -1.4552530286 -2.5228140398 -0.8510301656 -2.2195202673 -0.8986074949
##  [31]  0.4663785665  0.3443671675  0.4262749985 -0.5742331071  0.8626312354
##  [36] -0.4097095049 -1.2144767072 -2.7951759136 -1.1817580234 -1.8117649944
##  [41] -0.0139355621 -0.7830298786 -0.6416627279  0.2225530770 -0.2039478885
##  [46] -0.0364417078 -0.4115592998 -1.1910944922 -0.5967967203 -0.0034814070
##  [51]  0.7924631503 -0.1024529552  1.5028360454  1.0620822310 -0.2922992306
##  [56]  0.6113033486 -0.0144757220  1.6150566869  3.7767688014  2.5134898117
##  [61]  1.8193010018  0.1569636619  1.3350307851 -0.9956683306  2.1212439241
##  [66]  1.0313883413  2.1848368098 -0.6065105044  0.1147979087  1.4604409542
##  [71]  0.2652752680 -1.0412773436  0.7607271039  0.9718998970  0.1903257190
##  [76] -1.6199991451  0.3806078701  1.0559650057  1.5925550202  1.7566389339
##  [81]  0.3731780321  1.9722895156  0.7627719616  1.2050112214  0.2974420475
##  [86]  1.2583585482  0.8719330070  1.2373387010  0.9259432046 -0.1915627581
##  [91] -0.6116261568  1.3543231037  0.1747968619  0.4195683628  1.3404394562
##  [96] -0.5103517966 -0.0462849381 -1.0861057882 -2.1422139738 -1.1313819924
## [101] -0.1571743436 -0.9180747818 -0.6827462016 -0.8011491283  0.3806699098
## [106] -0.8833573051  0.2077090325  1.6787548699 -1.8079601237 -1.8917120196
## [111] -0.8200005120  2.1980305565  1.5719072909 -0.4609206306 -1.5056406632
## [116] -1.3343878176  0.3847065565  0.9467178487  1.2303091062  1.2274515121
## [121] -1.0783635414  0.0596422586  1.5534556714  0.8887706150  0.8663748299
## [126]  0.7380488856  1.2857207894  0.1814609400  0.1587468460 -0.0318320930
## [131]  1.1943975924  0.8552626428 -0.4643699283 -0.9181340681 -1.4869637093
## [136] -0.5275327491 -0.7448665434 -0.6726628486 -0.5121633064 -0.4235790854
## [141]  1.1897437284  2.5653362021  1.6202724074  0.4645847308 -0.1884631954
## [146] -2.0619181630 -1.9779126040  1.2626105014 -0.2636438881  0.1677600226
## [151] -1.3987534614 -0.1700578886  0.1651567670 -0.1167698027  0.6319715705
## [156] -0.0508086453  1.0748884367  2.0475088218  2.6595516762  1.9410196426
## [161]  0.3507416549 -1.0229581560 -0.9627220475 -0.9558728878 -1.0393676911
## [166] -1.9550187981 -2.9164942870 -1.7742639436  0.5267534108 -0.0149290657
## [171]  0.0702683067  0.0973893875  0.3412274679 -0.8472093157 -1.5335072170
## [176]  0.7554907588  0.8215478224  1.0662516127  1.4823625861  0.8704265583
## [181]  0.2613680744 -0.4170795539  1.1240563030  0.5581099764  0.7028000385
## [186] -0.3251612526 -2.0864949312  0.0964940224 -0.5214771959 -1.3362166924
## [191]  0.0008050617  1.3105193327  0.7705957552  1.4823770007  1.4532891037
## [196]  0.5804803934 -0.5072554461 -1.4768463593 -0.2942161653 -1.1886723609
## [201]  0.6073939384  0.9233355120 -0.9241903290 -1.1436196873 -0.0917360432
## [206]  0.1839338479 -0.0258868808 -0.1067211637 -1.4843174034  0.4177741749
## [211]  0.8960822615 -0.1965641103  1.6232954649  2.6063387803  4.0819833612
## [216]  1.8242641635  1.4707511490  2.1549538259  0.8551305920  1.1022077285
## [221]  1.4269923383  0.1610602021  0.4727421748 -0.3711539387 -0.1053673799
## [226]  0.3306832168  0.5463109874 -1.2396974924 -1.7914899288 -0.6759388895
## [231] -0.1578182824  1.1696463827  0.4391326921  2.1922767669 -1.2329125179
## [236]  0.3258349194 -0.5735369315 -0.3938937500 -1.3439792714 -2.2749956569
## [241] -1.6804699388 -0.6931987131 -0.6736489288 -0.8497905416 -1.4424589548
## [246] -0.0409603050 -0.2111770427  1.4563833264  1.0763492003  0.8671101832
## [251]  0.2895226535 -0.0478217598  0.5132098935 -1.0332105062 -0.1983999479
## [256] -0.9161540574 -0.1271787157 -1.6833500423 -2.0348818148 -1.7658902032
## [261] -0.7711289638 -2.9100819781 -2.1051746191 -2.4144982458 -1.6005752678
## [266] -0.1413717208 -1.2919575659 -0.4823718960 -0.5962667358  0.3349125563
## [271]  0.6290593161  0.3206883276 -2.1065267755 -1.4393735797 -2.0167968679
## [276] -0.3642024544  0.0632780462 -1.3000727765  0.8658758776  1.7858568757
## [281]  1.0726273597  1.2565145808 -0.6363211813  0.3893826410 -2.0205005996
## [286]  0.9016953303  1.7793220987  1.9245656319  1.4929276273  1.6095793669
## [291]  1.7416497174  0.2686250167 -0.4034650728 -0.1825269381  1.5310584870
## [296] -1.2127173873 -0.4820041350 -2.4901715257 -1.4144498140 -1.3541903826

Karakteristik AR(2)

Plot Time Series

ts.plot(ar2)

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

Plot ACF

acf(ar2,lag.max = 20)

Berdasarkan plot AFC tersebut, terlihat bahwa plot ACF tells off penanda model adalah AR

Plot PACF

pacf(ar2)

Berdasarkan plot PACF tersebut, terlihat bahwa plot PACF cuts off pada lag dua, menandakan AR(2)

Plot EACF

TSA::eacf(ar2)

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

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

Scatterplot Antar Peubah Lag

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

#Yt
yt_ar2 <- ar2[-1]
yt_ar2

##   [1]  0.9076604902  1.9914390098  1.3539009856  0.7500803953  0.6347400983
##   [6] -0.5643707523 -1.0794914635  1.2246041100  0.3656149571  0.1857602467
##  [11] -0.7673915289 -1.8335976688 -1.8829328493 -1.4433673123 -1.4790464669
##  [16] -1.8536960966 -1.5851897819 -2.1437247548 -1.9844423449 -1.0330610046
##  [21] -2.2980781084 -2.9842466251 -3.3707774768 -0.8982790943 -1.4552530286
##  [26] -2.5228140398 -0.8510301656 -2.2195202673 -0.8986074949  0.4663785665
##  [31]  0.3443671675  0.4262749985 -0.5742331071  0.8626312354 -0.4097095049
##  [36] -1.2144767072 -2.7951759136 -1.1817580234 -1.8117649944 -0.0139355621
##  [41] -0.7830298786 -0.6416627279  0.2225530770 -0.2039478885 -0.0364417078
##  [46] -0.4115592998 -1.1910944922 -0.5967967203 -0.0034814070  0.7924631503
##  [51] -0.1024529552  1.5028360454  1.0620822310 -0.2922992306  0.6113033486
##  [56] -0.0144757220  1.6150566869  3.7767688014  2.5134898117  1.8193010018
##  [61]  0.1569636619  1.3350307851 -0.9956683306  2.1212439241  1.0313883413
##  [66]  2.1848368098 -0.6065105044  0.1147979087  1.4604409542  0.2652752680
##  [71] -1.0412773436  0.7607271039  0.9718998970  0.1903257190 -1.6199991451
##  [76]  0.3806078701  1.0559650057  1.5925550202  1.7566389339  0.3731780321
##  [81]  1.9722895156  0.7627719616  1.2050112214  0.2974420475  1.2583585482
##  [86]  0.8719330070  1.2373387010  0.9259432046 -0.1915627581 -0.6116261568
##  [91]  1.3543231037  0.1747968619  0.4195683628  1.3404394562 -0.5103517966
##  [96] -0.0462849381 -1.0861057882 -2.1422139738 -1.1313819924 -0.1571743436
## [101] -0.9180747818 -0.6827462016 -0.8011491283  0.3806699098 -0.8833573051
## [106]  0.2077090325  1.6787548699 -1.8079601237 -1.8917120196 -0.8200005120
## [111]  2.1980305565  1.5719072909 -0.4609206306 -1.5056406632 -1.3343878176
## [116]  0.3847065565  0.9467178487  1.2303091062  1.2274515121 -1.0783635414
## [121]  0.0596422586  1.5534556714  0.8887706150  0.8663748299  0.7380488856
## [126]  1.2857207894  0.1814609400  0.1587468460 -0.0318320930  1.1943975924
## [131]  0.8552626428 -0.4643699283 -0.9181340681 -1.4869637093 -0.5275327491
## [136] -0.7448665434 -0.6726628486 -0.5121633064 -0.4235790854  1.1897437284
## [141]  2.5653362021  1.6202724074  0.4645847308 -0.1884631954 -2.0619181630
## [146] -1.9779126040  1.2626105014 -0.2636438881  0.1677600226 -1.3987534614
## [151] -0.1700578886  0.1651567670 -0.1167698027  0.6319715705 -0.0508086453
## [156]  1.0748884367  2.0475088218  2.6595516762  1.9410196426  0.3507416549
## [161] -1.0229581560 -0.9627220475 -0.9558728878 -1.0393676911 -1.9550187981
## [166] -2.9164942870 -1.7742639436  0.5267534108 -0.0149290657  0.0702683067
## [171]  0.0973893875  0.3412274679 -0.8472093157 -1.5335072170  0.7554907588
## [176]  0.8215478224  1.0662516127  1.4823625861  0.8704265583  0.2613680744
## [181] -0.4170795539  1.1240563030  0.5581099764  0.7028000385 -0.3251612526
## [186] -2.0864949312  0.0964940224 -0.5214771959 -1.3362166924  0.0008050617
## [191]  1.3105193327  0.7705957552  1.4823770007  1.4532891037  0.5804803934
## [196] -0.5072554461 -1.4768463593 -0.2942161653 -1.1886723609  0.6073939384
## [201]  0.9233355120 -0.9241903290 -1.1436196873 -0.0917360432  0.1839338479
## [206] -0.0258868808 -0.1067211637 -1.4843174034  0.4177741749  0.8960822615
## [211] -0.1965641103  1.6232954649  2.6063387803  4.0819833612  1.8242641635
## [216]  1.4707511490  2.1549538259  0.8551305920  1.1022077285  1.4269923383
## [221]  0.1610602021  0.4727421748 -0.3711539387 -0.1053673799  0.3306832168
## [226]  0.5463109874 -1.2396974924 -1.7914899288 -0.6759388895 -0.1578182824
## [231]  1.1696463827  0.4391326921  2.1922767669 -1.2329125179  0.3258349194
## [236] -0.5735369315 -0.3938937500 -1.3439792714 -2.2749956569 -1.6804699388
## [241] -0.6931987131 -0.6736489288 -0.8497905416 -1.4424589548 -0.0409603050
## [246] -0.2111770427  1.4563833264  1.0763492003  0.8671101832  0.2895226535
## [251] -0.0478217598  0.5132098935 -1.0332105062 -0.1983999479 -0.9161540574
## [256] -0.1271787157 -1.6833500423 -2.0348818148 -1.7658902032 -0.7711289638
## [261] -2.9100819781 -2.1051746191 -2.4144982458 -1.6005752678 -0.1413717208
## [266] -1.2919575659 -0.4823718960 -0.5962667358  0.3349125563  0.6290593161
## [271]  0.3206883276 -2.1065267755 -1.4393735797 -2.0167968679 -0.3642024544
## [276]  0.0632780462 -1.3000727765  0.8658758776  1.7858568757  1.0726273597
## [281]  1.2565145808 -0.6363211813  0.3893826410 -2.0205005996  0.9016953303
## [286]  1.7793220987  1.9245656319  1.4929276273  1.6095793669  1.7416497174
## [291]  0.2686250167 -0.4034650728 -0.1825269381  1.5310584870 -1.2127173873
## [296] -0.4820041350 -2.4901715257 -1.4144498140 -1.3541903826

#Yt-1
yt_1_ar2 <- ar2[-300]
yt_1_ar2

##   [1]  0.6739349283  0.9076604902  1.9914390098  1.3539009856  0.7500803953
##   [6]  0.6347400983 -0.5643707523 -1.0794914635  1.2246041100  0.3656149571
##  [11]  0.1857602467 -0.7673915289 -1.8335976688 -1.8829328493 -1.4433673123
##  [16] -1.4790464669 -1.8536960966 -1.5851897819 -2.1437247548 -1.9844423449
##  [21] -1.0330610046 -2.2980781084 -2.9842466251 -3.3707774768 -0.8982790943
##  [26] -1.4552530286 -2.5228140398 -0.8510301656 -2.2195202673 -0.8986074949
##  [31]  0.4663785665  0.3443671675  0.4262749985 -0.5742331071  0.8626312354
##  [36] -0.4097095049 -1.2144767072 -2.7951759136 -1.1817580234 -1.8117649944
##  [41] -0.0139355621 -0.7830298786 -0.6416627279  0.2225530770 -0.2039478885
##  [46] -0.0364417078 -0.4115592998 -1.1910944922 -0.5967967203 -0.0034814070
##  [51]  0.7924631503 -0.1024529552  1.5028360454  1.0620822310 -0.2922992306
##  [56]  0.6113033486 -0.0144757220  1.6150566869  3.7767688014  2.5134898117
##  [61]  1.8193010018  0.1569636619  1.3350307851 -0.9956683306  2.1212439241
##  [66]  1.0313883413  2.1848368098 -0.6065105044  0.1147979087  1.4604409542
##  [71]  0.2652752680 -1.0412773436  0.7607271039  0.9718998970  0.1903257190
##  [76] -1.6199991451  0.3806078701  1.0559650057  1.5925550202  1.7566389339
##  [81]  0.3731780321  1.9722895156  0.7627719616  1.2050112214  0.2974420475
##  [86]  1.2583585482  0.8719330070  1.2373387010  0.9259432046 -0.1915627581
##  [91] -0.6116261568  1.3543231037  0.1747968619  0.4195683628  1.3404394562
##  [96] -0.5103517966 -0.0462849381 -1.0861057882 -2.1422139738 -1.1313819924
## [101] -0.1571743436 -0.9180747818 -0.6827462016 -0.8011491283  0.3806699098
## [106] -0.8833573051  0.2077090325  1.6787548699 -1.8079601237 -1.8917120196
## [111] -0.8200005120  2.1980305565  1.5719072909 -0.4609206306 -1.5056406632
## [116] -1.3343878176  0.3847065565  0.9467178487  1.2303091062  1.2274515121
## [121] -1.0783635414  0.0596422586  1.5534556714  0.8887706150  0.8663748299
## [126]  0.7380488856  1.2857207894  0.1814609400  0.1587468460 -0.0318320930
## [131]  1.1943975924  0.8552626428 -0.4643699283 -0.9181340681 -1.4869637093
## [136] -0.5275327491 -0.7448665434 -0.6726628486 -0.5121633064 -0.4235790854
## [141]  1.1897437284  2.5653362021  1.6202724074  0.4645847308 -0.1884631954
## [146] -2.0619181630 -1.9779126040  1.2626105014 -0.2636438881  0.1677600226
## [151] -1.3987534614 -0.1700578886  0.1651567670 -0.1167698027  0.6319715705
## [156] -0.0508086453  1.0748884367  2.0475088218  2.6595516762  1.9410196426
## [161]  0.3507416549 -1.0229581560 -0.9627220475 -0.9558728878 -1.0393676911
## [166] -1.9550187981 -2.9164942870 -1.7742639436  0.5267534108 -0.0149290657
## [171]  0.0702683067  0.0973893875  0.3412274679 -0.8472093157 -1.5335072170
## [176]  0.7554907588  0.8215478224  1.0662516127  1.4823625861  0.8704265583
## [181]  0.2613680744 -0.4170795539  1.1240563030  0.5581099764  0.7028000385
## [186] -0.3251612526 -2.0864949312  0.0964940224 -0.5214771959 -1.3362166924
## [191]  0.0008050617  1.3105193327  0.7705957552  1.4823770007  1.4532891037
## [196]  0.5804803934 -0.5072554461 -1.4768463593 -0.2942161653 -1.1886723609
## [201]  0.6073939384  0.9233355120 -0.9241903290 -1.1436196873 -0.0917360432
## [206]  0.1839338479 -0.0258868808 -0.1067211637 -1.4843174034  0.4177741749
## [211]  0.8960822615 -0.1965641103  1.6232954649  2.6063387803  4.0819833612
## [216]  1.8242641635  1.4707511490  2.1549538259  0.8551305920  1.1022077285
## [221]  1.4269923383  0.1610602021  0.4727421748 -0.3711539387 -0.1053673799
## [226]  0.3306832168  0.5463109874 -1.2396974924 -1.7914899288 -0.6759388895
## [231] -0.1578182824  1.1696463827  0.4391326921  2.1922767669 -1.2329125179
## [236]  0.3258349194 -0.5735369315 -0.3938937500 -1.3439792714 -2.2749956569
## [241] -1.6804699388 -0.6931987131 -0.6736489288 -0.8497905416 -1.4424589548
## [246] -0.0409603050 -0.2111770427  1.4563833264  1.0763492003  0.8671101832
## [251]  0.2895226535 -0.0478217598  0.5132098935 -1.0332105062 -0.1983999479
## [256] -0.9161540574 -0.1271787157 -1.6833500423 -2.0348818148 -1.7658902032
## [261] -0.7711289638 -2.9100819781 -2.1051746191 -2.4144982458 -1.6005752678
## [266] -0.1413717208 -1.2919575659 -0.4823718960 -0.5962667358  0.3349125563
## [271]  0.6290593161  0.3206883276 -2.1065267755 -1.4393735797 -2.0167968679
## [276] -0.3642024544  0.0632780462 -1.3000727765  0.8658758776  1.7858568757
## [281]  1.0726273597  1.2565145808 -0.6363211813  0.3893826410 -2.0205005996
## [286]  0.9016953303  1.7793220987  1.9245656319  1.4929276273  1.6095793669
## [291]  1.7416497174  0.2686250167 -0.4034650728 -0.1825269381  1.5310584870
## [296] -1.2127173873 -0.4820041350 -2.4901715257 -1.4144498140

plot(y=yt_ar2,x=yt_1_ar2)

Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-1}\).

cor(yt_ar2,yt_1_ar2)

## [1] 0.5745402

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

#Yt
yt_ar2.2 <- ar2[-c(1,2)]
yt_ar2.2

##   [1]  1.9914390098  1.3539009856  0.7500803953  0.6347400983 -0.5643707523
##   [6] -1.0794914635  1.2246041100  0.3656149571  0.1857602467 -0.7673915289
##  [11] -1.8335976688 -1.8829328493 -1.4433673123 -1.4790464669 -1.8536960966
##  [16] -1.5851897819 -2.1437247548 -1.9844423449 -1.0330610046 -2.2980781084
##  [21] -2.9842466251 -3.3707774768 -0.8982790943 -1.4552530286 -2.5228140398
##  [26] -0.8510301656 -2.2195202673 -0.8986074949  0.4663785665  0.3443671675
##  [31]  0.4262749985 -0.5742331071  0.8626312354 -0.4097095049 -1.2144767072
##  [36] -2.7951759136 -1.1817580234 -1.8117649944 -0.0139355621 -0.7830298786
##  [41] -0.6416627279  0.2225530770 -0.2039478885 -0.0364417078 -0.4115592998
##  [46] -1.1910944922 -0.5967967203 -0.0034814070  0.7924631503 -0.1024529552
##  [51]  1.5028360454  1.0620822310 -0.2922992306  0.6113033486 -0.0144757220
##  [56]  1.6150566869  3.7767688014  2.5134898117  1.8193010018  0.1569636619
##  [61]  1.3350307851 -0.9956683306  2.1212439241  1.0313883413  2.1848368098
##  [66] -0.6065105044  0.1147979087  1.4604409542  0.2652752680 -1.0412773436
##  [71]  0.7607271039  0.9718998970  0.1903257190 -1.6199991451  0.3806078701
##  [76]  1.0559650057  1.5925550202  1.7566389339  0.3731780321  1.9722895156
##  [81]  0.7627719616  1.2050112214  0.2974420475  1.2583585482  0.8719330070
##  [86]  1.2373387010  0.9259432046 -0.1915627581 -0.6116261568  1.3543231037
##  [91]  0.1747968619  0.4195683628  1.3404394562 -0.5103517966 -0.0462849381
##  [96] -1.0861057882 -2.1422139738 -1.1313819924 -0.1571743436 -0.9180747818
## [101] -0.6827462016 -0.8011491283  0.3806699098 -0.8833573051  0.2077090325
## [106]  1.6787548699 -1.8079601237 -1.8917120196 -0.8200005120  2.1980305565
## [111]  1.5719072909 -0.4609206306 -1.5056406632 -1.3343878176  0.3847065565
## [116]  0.9467178487  1.2303091062  1.2274515121 -1.0783635414  0.0596422586
## [121]  1.5534556714  0.8887706150  0.8663748299  0.7380488856  1.2857207894
## [126]  0.1814609400  0.1587468460 -0.0318320930  1.1943975924  0.8552626428
## [131] -0.4643699283 -0.9181340681 -1.4869637093 -0.5275327491 -0.7448665434
## [136] -0.6726628486 -0.5121633064 -0.4235790854  1.1897437284  2.5653362021
## [141]  1.6202724074  0.4645847308 -0.1884631954 -2.0619181630 -1.9779126040
## [146]  1.2626105014 -0.2636438881  0.1677600226 -1.3987534614 -0.1700578886
## [151]  0.1651567670 -0.1167698027  0.6319715705 -0.0508086453  1.0748884367
## [156]  2.0475088218  2.6595516762  1.9410196426  0.3507416549 -1.0229581560
## [161] -0.9627220475 -0.9558728878 -1.0393676911 -1.9550187981 -2.9164942870
## [166] -1.7742639436  0.5267534108 -0.0149290657  0.0702683067  0.0973893875
## [171]  0.3412274679 -0.8472093157 -1.5335072170  0.7554907588  0.8215478224
## [176]  1.0662516127  1.4823625861  0.8704265583  0.2613680744 -0.4170795539
## [181]  1.1240563030  0.5581099764  0.7028000385 -0.3251612526 -2.0864949312
## [186]  0.0964940224 -0.5214771959 -1.3362166924  0.0008050617  1.3105193327
## [191]  0.7705957552  1.4823770007  1.4532891037  0.5804803934 -0.5072554461
## [196] -1.4768463593 -0.2942161653 -1.1886723609  0.6073939384  0.9233355120
## [201] -0.9241903290 -1.1436196873 -0.0917360432  0.1839338479 -0.0258868808
## [206] -0.1067211637 -1.4843174034  0.4177741749  0.8960822615 -0.1965641103
## [211]  1.6232954649  2.6063387803  4.0819833612  1.8242641635  1.4707511490
## [216]  2.1549538259  0.8551305920  1.1022077285  1.4269923383  0.1610602021
## [221]  0.4727421748 -0.3711539387 -0.1053673799  0.3306832168  0.5463109874
## [226] -1.2396974924 -1.7914899288 -0.6759388895 -0.1578182824  1.1696463827
## [231]  0.4391326921  2.1922767669 -1.2329125179  0.3258349194 -0.5735369315
## [236] -0.3938937500 -1.3439792714 -2.2749956569 -1.6804699388 -0.6931987131
## [241] -0.6736489288 -0.8497905416 -1.4424589548 -0.0409603050 -0.2111770427
## [246]  1.4563833264  1.0763492003  0.8671101832  0.2895226535 -0.0478217598
## [251]  0.5132098935 -1.0332105062 -0.1983999479 -0.9161540574 -0.1271787157
## [256] -1.6833500423 -2.0348818148 -1.7658902032 -0.7711289638 -2.9100819781
## [261] -2.1051746191 -2.4144982458 -1.6005752678 -0.1413717208 -1.2919575659
## [266] -0.4823718960 -0.5962667358  0.3349125563  0.6290593161  0.3206883276
## [271] -2.1065267755 -1.4393735797 -2.0167968679 -0.3642024544  0.0632780462
## [276] -1.3000727765  0.8658758776  1.7858568757  1.0726273597  1.2565145808
## [281] -0.6363211813  0.3893826410 -2.0205005996  0.9016953303  1.7793220987
## [286]  1.9245656319  1.4929276273  1.6095793669  1.7416497174  0.2686250167
## [291] -0.4034650728 -0.1825269381  1.5310584870 -1.2127173873 -0.4820041350
## [296] -2.4901715257 -1.4144498140 -1.3541903826

#Yt-2
yt_2_ar2 <- ar2[-c(299,300)]
yt_2_ar2

##   [1]  0.6739349283  0.9076604902  1.9914390098  1.3539009856  0.7500803953
##   [6]  0.6347400983 -0.5643707523 -1.0794914635  1.2246041100  0.3656149571
##  [11]  0.1857602467 -0.7673915289 -1.8335976688 -1.8829328493 -1.4433673123
##  [16] -1.4790464669 -1.8536960966 -1.5851897819 -2.1437247548 -1.9844423449
##  [21] -1.0330610046 -2.2980781084 -2.9842466251 -3.3707774768 -0.8982790943
##  [26] -1.4552530286 -2.5228140398 -0.8510301656 -2.2195202673 -0.8986074949
##  [31]  0.4663785665  0.3443671675  0.4262749985 -0.5742331071  0.8626312354
##  [36] -0.4097095049 -1.2144767072 -2.7951759136 -1.1817580234 -1.8117649944
##  [41] -0.0139355621 -0.7830298786 -0.6416627279  0.2225530770 -0.2039478885
##  [46] -0.0364417078 -0.4115592998 -1.1910944922 -0.5967967203 -0.0034814070
##  [51]  0.7924631503 -0.1024529552  1.5028360454  1.0620822310 -0.2922992306
##  [56]  0.6113033486 -0.0144757220  1.6150566869  3.7767688014  2.5134898117
##  [61]  1.8193010018  0.1569636619  1.3350307851 -0.9956683306  2.1212439241
##  [66]  1.0313883413  2.1848368098 -0.6065105044  0.1147979087  1.4604409542
##  [71]  0.2652752680 -1.0412773436  0.7607271039  0.9718998970  0.1903257190
##  [76] -1.6199991451  0.3806078701  1.0559650057  1.5925550202  1.7566389339
##  [81]  0.3731780321  1.9722895156  0.7627719616  1.2050112214  0.2974420475
##  [86]  1.2583585482  0.8719330070  1.2373387010  0.9259432046 -0.1915627581
##  [91] -0.6116261568  1.3543231037  0.1747968619  0.4195683628  1.3404394562
##  [96] -0.5103517966 -0.0462849381 -1.0861057882 -2.1422139738 -1.1313819924
## [101] -0.1571743436 -0.9180747818 -0.6827462016 -0.8011491283  0.3806699098
## [106] -0.8833573051  0.2077090325  1.6787548699 -1.8079601237 -1.8917120196
## [111] -0.8200005120  2.1980305565  1.5719072909 -0.4609206306 -1.5056406632
## [116] -1.3343878176  0.3847065565  0.9467178487  1.2303091062  1.2274515121
## [121] -1.0783635414  0.0596422586  1.5534556714  0.8887706150  0.8663748299
## [126]  0.7380488856  1.2857207894  0.1814609400  0.1587468460 -0.0318320930
## [131]  1.1943975924  0.8552626428 -0.4643699283 -0.9181340681 -1.4869637093
## [136] -0.5275327491 -0.7448665434 -0.6726628486 -0.5121633064 -0.4235790854
## [141]  1.1897437284  2.5653362021  1.6202724074  0.4645847308 -0.1884631954
## [146] -2.0619181630 -1.9779126040  1.2626105014 -0.2636438881  0.1677600226
## [151] -1.3987534614 -0.1700578886  0.1651567670 -0.1167698027  0.6319715705
## [156] -0.0508086453  1.0748884367  2.0475088218  2.6595516762  1.9410196426
## [161]  0.3507416549 -1.0229581560 -0.9627220475 -0.9558728878 -1.0393676911
## [166] -1.9550187981 -2.9164942870 -1.7742639436  0.5267534108 -0.0149290657
## [171]  0.0702683067  0.0973893875  0.3412274679 -0.8472093157 -1.5335072170
## [176]  0.7554907588  0.8215478224  1.0662516127  1.4823625861  0.8704265583
## [181]  0.2613680744 -0.4170795539  1.1240563030  0.5581099764  0.7028000385
## [186] -0.3251612526 -2.0864949312  0.0964940224 -0.5214771959 -1.3362166924
## [191]  0.0008050617  1.3105193327  0.7705957552  1.4823770007  1.4532891037
## [196]  0.5804803934 -0.5072554461 -1.4768463593 -0.2942161653 -1.1886723609
## [201]  0.6073939384  0.9233355120 -0.9241903290 -1.1436196873 -0.0917360432
## [206]  0.1839338479 -0.0258868808 -0.1067211637 -1.4843174034  0.4177741749
## [211]  0.8960822615 -0.1965641103  1.6232954649  2.6063387803  4.0819833612
## [216]  1.8242641635  1.4707511490  2.1549538259  0.8551305920  1.1022077285
## [221]  1.4269923383  0.1610602021  0.4727421748 -0.3711539387 -0.1053673799
## [226]  0.3306832168  0.5463109874 -1.2396974924 -1.7914899288 -0.6759388895
## [231] -0.1578182824  1.1696463827  0.4391326921  2.1922767669 -1.2329125179
## [236]  0.3258349194 -0.5735369315 -0.3938937500 -1.3439792714 -2.2749956569
## [241] -1.6804699388 -0.6931987131 -0.6736489288 -0.8497905416 -1.4424589548
## [246] -0.0409603050 -0.2111770427  1.4563833264  1.0763492003  0.8671101832
## [251]  0.2895226535 -0.0478217598  0.5132098935 -1.0332105062 -0.1983999479
## [256] -0.9161540574 -0.1271787157 -1.6833500423 -2.0348818148 -1.7658902032
## [261] -0.7711289638 -2.9100819781 -2.1051746191 -2.4144982458 -1.6005752678
## [266] -0.1413717208 -1.2919575659 -0.4823718960 -0.5962667358  0.3349125563
## [271]  0.6290593161  0.3206883276 -2.1065267755 -1.4393735797 -2.0167968679
## [276] -0.3642024544  0.0632780462 -1.3000727765  0.8658758776  1.7858568757
## [281]  1.0726273597  1.2565145808 -0.6363211813  0.3893826410 -2.0205005996
## [286]  0.9016953303  1.7793220987  1.9245656319  1.4929276273  1.6095793669
## [291]  1.7416497174  0.2686250167 -0.4034650728 -0.1825269381  1.5310584870
## [296] -1.2127173873 -0.4820041350 -2.4901715257

plot(y=yt_ar2.2,x=yt_2_ar2)

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

cor(yt_ar2.2,yt_2_ar2)

## [1] 0.3859049

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

#Yt
yt_ar2.3 <- ar2[-c(1,2,3)]
yt_ar2.3

##   [1]  1.3539009856  0.7500803953  0.6347400983 -0.5643707523 -1.0794914635
##   [6]  1.2246041100  0.3656149571  0.1857602467 -0.7673915289 -1.8335976688
##  [11] -1.8829328493 -1.4433673123 -1.4790464669 -1.8536960966 -1.5851897819
##  [16] -2.1437247548 -1.9844423449 -1.0330610046 -2.2980781084 -2.9842466251
##  [21] -3.3707774768 -0.8982790943 -1.4552530286 -2.5228140398 -0.8510301656
##  [26] -2.2195202673 -0.8986074949  0.4663785665  0.3443671675  0.4262749985
##  [31] -0.5742331071  0.8626312354 -0.4097095049 -1.2144767072 -2.7951759136
##  [36] -1.1817580234 -1.8117649944 -0.0139355621 -0.7830298786 -0.6416627279
##  [41]  0.2225530770 -0.2039478885 -0.0364417078 -0.4115592998 -1.1910944922
##  [46] -0.5967967203 -0.0034814070  0.7924631503 -0.1024529552  1.5028360454
##  [51]  1.0620822310 -0.2922992306  0.6113033486 -0.0144757220  1.6150566869
##  [56]  3.7767688014  2.5134898117  1.8193010018  0.1569636619  1.3350307851
##  [61] -0.9956683306  2.1212439241  1.0313883413  2.1848368098 -0.6065105044
##  [66]  0.1147979087  1.4604409542  0.2652752680 -1.0412773436  0.7607271039
##  [71]  0.9718998970  0.1903257190 -1.6199991451  0.3806078701  1.0559650057
##  [76]  1.5925550202  1.7566389339  0.3731780321  1.9722895156  0.7627719616
##  [81]  1.2050112214  0.2974420475  1.2583585482  0.8719330070  1.2373387010
##  [86]  0.9259432046 -0.1915627581 -0.6116261568  1.3543231037  0.1747968619
##  [91]  0.4195683628  1.3404394562 -0.5103517966 -0.0462849381 -1.0861057882
##  [96] -2.1422139738 -1.1313819924 -0.1571743436 -0.9180747818 -0.6827462016
## [101] -0.8011491283  0.3806699098 -0.8833573051  0.2077090325  1.6787548699
## [106] -1.8079601237 -1.8917120196 -0.8200005120  2.1980305565  1.5719072909
## [111] -0.4609206306 -1.5056406632 -1.3343878176  0.3847065565  0.9467178487
## [116]  1.2303091062  1.2274515121 -1.0783635414  0.0596422586  1.5534556714
## [121]  0.8887706150  0.8663748299  0.7380488856  1.2857207894  0.1814609400
## [126]  0.1587468460 -0.0318320930  1.1943975924  0.8552626428 -0.4643699283
## [131] -0.9181340681 -1.4869637093 -0.5275327491 -0.7448665434 -0.6726628486
## [136] -0.5121633064 -0.4235790854  1.1897437284  2.5653362021  1.6202724074
## [141]  0.4645847308 -0.1884631954 -2.0619181630 -1.9779126040  1.2626105014
## [146] -0.2636438881  0.1677600226 -1.3987534614 -0.1700578886  0.1651567670
## [151] -0.1167698027  0.6319715705 -0.0508086453  1.0748884367  2.0475088218
## [156]  2.6595516762  1.9410196426  0.3507416549 -1.0229581560 -0.9627220475
## [161] -0.9558728878 -1.0393676911 -1.9550187981 -2.9164942870 -1.7742639436
## [166]  0.5267534108 -0.0149290657  0.0702683067  0.0973893875  0.3412274679
## [171] -0.8472093157 -1.5335072170  0.7554907588  0.8215478224  1.0662516127
## [176]  1.4823625861  0.8704265583  0.2613680744 -0.4170795539  1.1240563030
## [181]  0.5581099764  0.7028000385 -0.3251612526 -2.0864949312  0.0964940224
## [186] -0.5214771959 -1.3362166924  0.0008050617  1.3105193327  0.7705957552
## [191]  1.4823770007  1.4532891037  0.5804803934 -0.5072554461 -1.4768463593
## [196] -0.2942161653 -1.1886723609  0.6073939384  0.9233355120 -0.9241903290
## [201] -1.1436196873 -0.0917360432  0.1839338479 -0.0258868808 -0.1067211637
## [206] -1.4843174034  0.4177741749  0.8960822615 -0.1965641103  1.6232954649
## [211]  2.6063387803  4.0819833612  1.8242641635  1.4707511490  2.1549538259
## [216]  0.8551305920  1.1022077285  1.4269923383  0.1610602021  0.4727421748
## [221] -0.3711539387 -0.1053673799  0.3306832168  0.5463109874 -1.2396974924
## [226] -1.7914899288 -0.6759388895 -0.1578182824  1.1696463827  0.4391326921
## [231]  2.1922767669 -1.2329125179  0.3258349194 -0.5735369315 -0.3938937500
## [236] -1.3439792714 -2.2749956569 -1.6804699388 -0.6931987131 -0.6736489288
## [241] -0.8497905416 -1.4424589548 -0.0409603050 -0.2111770427  1.4563833264
## [246]  1.0763492003  0.8671101832  0.2895226535 -0.0478217598  0.5132098935
## [251] -1.0332105062 -0.1983999479 -0.9161540574 -0.1271787157 -1.6833500423
## [256] -2.0348818148 -1.7658902032 -0.7711289638 -2.9100819781 -2.1051746191
## [261] -2.4144982458 -1.6005752678 -0.1413717208 -1.2919575659 -0.4823718960
## [266] -0.5962667358  0.3349125563  0.6290593161  0.3206883276 -2.1065267755
## [271] -1.4393735797 -2.0167968679 -0.3642024544  0.0632780462 -1.3000727765
## [276]  0.8658758776  1.7858568757  1.0726273597  1.2565145808 -0.6363211813
## [281]  0.3893826410 -2.0205005996  0.9016953303  1.7793220987  1.9245656319
## [286]  1.4929276273  1.6095793669  1.7416497174  0.2686250167 -0.4034650728
## [291] -0.1825269381  1.5310584870 -1.2127173873 -0.4820041350 -2.4901715257
## [296] -1.4144498140 -1.3541903826

#Yt-2
yt_3_ar2 <- ar2[-c(298,299,300)]
yt_3_ar2

##   [1]  0.6739349283  0.9076604902  1.9914390098  1.3539009856  0.7500803953
##   [6]  0.6347400983 -0.5643707523 -1.0794914635  1.2246041100  0.3656149571
##  [11]  0.1857602467 -0.7673915289 -1.8335976688 -1.8829328493 -1.4433673123
##  [16] -1.4790464669 -1.8536960966 -1.5851897819 -2.1437247548 -1.9844423449
##  [21] -1.0330610046 -2.2980781084 -2.9842466251 -3.3707774768 -0.8982790943
##  [26] -1.4552530286 -2.5228140398 -0.8510301656 -2.2195202673 -0.8986074949
##  [31]  0.4663785665  0.3443671675  0.4262749985 -0.5742331071  0.8626312354
##  [36] -0.4097095049 -1.2144767072 -2.7951759136 -1.1817580234 -1.8117649944
##  [41] -0.0139355621 -0.7830298786 -0.6416627279  0.2225530770 -0.2039478885
##  [46] -0.0364417078 -0.4115592998 -1.1910944922 -0.5967967203 -0.0034814070
##  [51]  0.7924631503 -0.1024529552  1.5028360454  1.0620822310 -0.2922992306
##  [56]  0.6113033486 -0.0144757220  1.6150566869  3.7767688014  2.5134898117
##  [61]  1.8193010018  0.1569636619  1.3350307851 -0.9956683306  2.1212439241
##  [66]  1.0313883413  2.1848368098 -0.6065105044  0.1147979087  1.4604409542
##  [71]  0.2652752680 -1.0412773436  0.7607271039  0.9718998970  0.1903257190
##  [76] -1.6199991451  0.3806078701  1.0559650057  1.5925550202  1.7566389339
##  [81]  0.3731780321  1.9722895156  0.7627719616  1.2050112214  0.2974420475
##  [86]  1.2583585482  0.8719330070  1.2373387010  0.9259432046 -0.1915627581
##  [91] -0.6116261568  1.3543231037  0.1747968619  0.4195683628  1.3404394562
##  [96] -0.5103517966 -0.0462849381 -1.0861057882 -2.1422139738 -1.1313819924
## [101] -0.1571743436 -0.9180747818 -0.6827462016 -0.8011491283  0.3806699098
## [106] -0.8833573051  0.2077090325  1.6787548699 -1.8079601237 -1.8917120196
## [111] -0.8200005120  2.1980305565  1.5719072909 -0.4609206306 -1.5056406632
## [116] -1.3343878176  0.3847065565  0.9467178487  1.2303091062  1.2274515121
## [121] -1.0783635414  0.0596422586  1.5534556714  0.8887706150  0.8663748299
## [126]  0.7380488856  1.2857207894  0.1814609400  0.1587468460 -0.0318320930
## [131]  1.1943975924  0.8552626428 -0.4643699283 -0.9181340681 -1.4869637093
## [136] -0.5275327491 -0.7448665434 -0.6726628486 -0.5121633064 -0.4235790854
## [141]  1.1897437284  2.5653362021  1.6202724074  0.4645847308 -0.1884631954
## [146] -2.0619181630 -1.9779126040  1.2626105014 -0.2636438881  0.1677600226
## [151] -1.3987534614 -0.1700578886  0.1651567670 -0.1167698027  0.6319715705
## [156] -0.0508086453  1.0748884367  2.0475088218  2.6595516762  1.9410196426
## [161]  0.3507416549 -1.0229581560 -0.9627220475 -0.9558728878 -1.0393676911
## [166] -1.9550187981 -2.9164942870 -1.7742639436  0.5267534108 -0.0149290657
## [171]  0.0702683067  0.0973893875  0.3412274679 -0.8472093157 -1.5335072170
## [176]  0.7554907588  0.8215478224  1.0662516127  1.4823625861  0.8704265583
## [181]  0.2613680744 -0.4170795539  1.1240563030  0.5581099764  0.7028000385
## [186] -0.3251612526 -2.0864949312  0.0964940224 -0.5214771959 -1.3362166924
## [191]  0.0008050617  1.3105193327  0.7705957552  1.4823770007  1.4532891037
## [196]  0.5804803934 -0.5072554461 -1.4768463593 -0.2942161653 -1.1886723609
## [201]  0.6073939384  0.9233355120 -0.9241903290 -1.1436196873 -0.0917360432
## [206]  0.1839338479 -0.0258868808 -0.1067211637 -1.4843174034  0.4177741749
## [211]  0.8960822615 -0.1965641103  1.6232954649  2.6063387803  4.0819833612
## [216]  1.8242641635  1.4707511490  2.1549538259  0.8551305920  1.1022077285
## [221]  1.4269923383  0.1610602021  0.4727421748 -0.3711539387 -0.1053673799
## [226]  0.3306832168  0.5463109874 -1.2396974924 -1.7914899288 -0.6759388895
## [231] -0.1578182824  1.1696463827  0.4391326921  2.1922767669 -1.2329125179
## [236]  0.3258349194 -0.5735369315 -0.3938937500 -1.3439792714 -2.2749956569
## [241] -1.6804699388 -0.6931987131 -0.6736489288 -0.8497905416 -1.4424589548
## [246] -0.0409603050 -0.2111770427  1.4563833264  1.0763492003  0.8671101832
## [251]  0.2895226535 -0.0478217598  0.5132098935 -1.0332105062 -0.1983999479
## [256] -0.9161540574 -0.1271787157 -1.6833500423 -2.0348818148 -1.7658902032
## [261] -0.7711289638 -2.9100819781 -2.1051746191 -2.4144982458 -1.6005752678
## [266] -0.1413717208 -1.2919575659 -0.4823718960 -0.5962667358  0.3349125563
## [271]  0.6290593161  0.3206883276 -2.1065267755 -1.4393735797 -2.0167968679
## [276] -0.3642024544  0.0632780462 -1.3000727765  0.8658758776  1.7858568757
## [281]  1.0726273597  1.2565145808 -0.6363211813  0.3893826410 -2.0205005996
## [286]  0.9016953303  1.7793220987  1.9245656319  1.4929276273  1.6095793669
## [291]  1.7416497174  0.2686250167 -0.4034650728 -0.1825269381  1.5310584870
## [296] -1.2127173873 -0.4820041350

plot(y=yt_ar2.3,x=yt_3_ar2)

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

cor(yt_ar2.3,yt_3_ar2)

## [1] 0.2264197

Perbandingan korelasi \(Yt_t\) dengan \(Y_{t-1}\), \(Yt_t\) dengan \(Y_{t-2}\), & \(Yt_t\) dengan \(Y_{t-3}\)

par(mfrow=c(2,2))
plot(y=yt_ar2,x=yt_1_ar2)
plot(y=yt_ar2.2,x=yt_2_ar2)
plot(y=yt_ar2.3,x=yt_3_ar2)

Proses ARMA

Pembangkitan Proses ARMA(2,2)

Akan dicoba membangkitkan proses AR paling sederhana, yaitu ARMA(2,2) dengan \(\theta_1 = 0.6\), \(\phi_1 = 0.5\), \(\theta_2 = 0.4\), dan \(\phi_2 = 0.2\) sebanyak 300 observasi dan \(c=0\).

Cara Manual

y.arma=c(1:n)
for (i in 3:n){y.arma[i] = 0.5*y.arma[i-1] + 0.2*y.arma[i-2] + 0.4*wn[i-1]+  0.6*wn[i-2]+wn[i]}

Fungsi arima.sim

arma22 <- arima.sim(list(order=c(2,0,2), ar = c(0.5,0.2), ma = c(0.5,0.2)), n=300)
arma22

## Time Series:
## Start = 1 
## End = 300 
## Frequency = 1 
##   [1] -1.74650961  1.84212574  1.68856917  2.13483906  2.96194453  2.27909247
##   [7]  2.00430099  0.71682006 -0.89497815  0.04864253  1.41826850  0.74792408
##  [13] -0.58442929 -2.59184200 -1.12155678 -1.78691629  0.45355922  1.17043171
##  [19]  1.85786553  1.68533085  0.97736618  0.42674828  1.71468402  1.88475263
##  [25]  0.31868216 -1.95677476 -3.23293413 -2.04796080 -3.50154620 -0.14850281
##  [31] -0.84241205 -0.09904779  0.98347628  0.40537181  0.80051058  1.93966047
##  [37]  1.94018792  1.49219033  2.34709325  3.92474462  3.91906833  3.13306971
##  [43]  2.16928793  2.06848733  0.38189044  0.10480999  0.93111620  2.58304860
##  [49]  1.72480305  2.08008778  0.44799092  1.44672387  1.19872106  3.34263037
##  [55]  2.26530231  1.35969662  2.26333305  3.17326337  2.94719719  4.16441731
##  [61]  3.57972014  2.82466277  2.61564224  2.11655419  0.32488963  0.66099253
##  [67]  0.52624370  0.06373925 -0.35784740  1.03242230  1.06911203 -0.41597984
##  [73] -0.24149670 -2.60049809 -2.41676378 -2.62799158 -1.24026800 -1.94581157
##  [79] -1.94284872 -2.05250578 -1.96678958 -3.31746059 -2.04650579 -1.36000718
##  [85]  1.98159061  2.64942025  2.30147448  1.64053611  0.42307363 -0.01501245
##  [91]  0.15777867  1.85095220  3.61362565  1.90010766  0.83161341 -1.08308679
##  [97] -0.24887354 -0.58370827 -2.09066260 -2.38615567 -2.13169180 -2.36131544
## [103] -2.02185719  0.04892401  0.73347202  0.71321593 -0.63186749 -1.48982580
## [109] -2.06916048 -0.34819431  0.42621972  0.27779544 -1.48483570 -0.95756852
## [115] -0.93823655 -0.05459029 -0.10154393  0.81700169  0.07924658  0.89822643
## [121]  0.13815886 -0.67354298 -0.17675454 -1.87794691 -1.80457518 -0.37965128
## [127] -1.16500218 -0.34621989 -0.63381243 -1.83254618 -0.60113922  0.56772770
## [133]  0.06589133 -0.17501360 -0.69941261  0.67758863 -0.38464271  0.49950799
## [139]  1.93783162  1.43189519  1.82933697  2.35166115  3.03047868  3.62535827
## [145]  2.86201068  3.30372623  3.31098178  2.70794432  2.89456806  3.41630044
## [151]  2.36547055  0.35395738 -1.33000072 -3.80871160 -3.42927264 -2.83624079
## [157] -1.51258949 -0.30838137 -0.08933384  0.30787902  1.12989924 -0.14712740
## [163]  0.02755425 -1.34329815 -3.01152132 -2.08722716 -2.03088646 -1.38604944
## [169] -1.07893635 -1.57224145 -2.98480078 -2.42021756  0.29194879  0.15008644
## [175]  1.06436719 -1.10227677 -1.29730754 -0.51245874 -0.79091573 -0.31909909
## [181] -0.91892791 -0.99975430  0.22264232  1.34152710  2.07065592 -0.50249779
## [187] -1.47261502 -1.48777266  0.31149646  2.32266628  1.38143745  0.69022919
## [193] -1.13246270 -0.97647298 -0.73009886  1.23398575  0.59147217 -0.06650082
## [199] -1.20349436 -0.41909478  0.81468480  0.83520111  0.52821900 -1.00379600
## [205]  1.26706191  1.89237274  0.66980320  0.51826936  0.95673508  3.14414567
## [211]  2.74180769  1.36573777 -1.47136605 -1.13929899 -1.26138103 -2.87300485
## [217] -1.78940053 -1.67765475 -2.66370958 -2.55056697 -3.24465555 -3.24295959
## [223] -2.03371710 -1.24220601 -0.70388318 -0.17096367 -0.70958630 -2.62660221
## [229] -0.68046853  1.48162800  2.43847799  2.99838010  2.34417736  2.90383247
## [235]  2.93031348  2.47971021  1.14925097  2.47363792  2.46084663  3.98794261
## [241]  4.28774275  3.32454839  2.27019712  2.89342765  3.64287055  3.62158905
## [247]  3.79227758  1.12375525  0.18394795 -2.04787990 -3.29806814 -2.56240795
## [253] -1.08471576 -2.10468381 -0.91726893 -0.54600782  0.23708751 -0.25954186
## [259] -0.96739098 -0.95444048 -0.82317391 -0.10198069  0.25261059  0.70834934
## [265]  1.62957416  1.18751345  1.70589425  0.92860639  0.72582207  0.67640692
## [271]  1.53456079  2.87957799  2.89335087  0.80434298 -0.73991953 -0.36374042
## [277]  0.13763276 -0.07270916  1.16174085  0.69941293  1.07993785 -0.54656439
## [283] -0.67044288 -0.32518707  1.32404394 -1.15413692 -2.37347758 -1.89669726
## [289] -1.56461104 -1.94321529 -1.64486354 -2.46361366 -3.58595094 -2.66204313
## [295] -0.80500899 -0.45592718 -1.42792139 -1.70735653 -0.16704081  0.31406658

Karakteristik ARMA(2,2)

Plot Time Series

ts.plot(y.arma)

Berdasarkan plot time series tersebut, terlihat bahwa model ARMA(2,2) cenderung stasioner dalam rataan

Plot ACF

acf(y.arma,lag.max = 20)

Berdasarkan plot ACF tersebut, terlihat bahwa model ARMA(2,2) hasil simulasi memiliki plot ACF yang tails off

Plot PACF

pacf(arma22)

Berdasarkan plot PACF tersebut, terlihat bahwa model ARMA(2,2) hasil simulasi memiliki plot PACF yang tails off

Plot EACF

TSA::eacf(arma22)

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

Scatterplot Antar Peubah Lag

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

#Yt
yt_arma22 <- arma22[-1]
yt_arma22

##   [1]  1.84212574  1.68856917  2.13483906  2.96194453  2.27909247  2.00430099
##   [7]  0.71682006 -0.89497815  0.04864253  1.41826850  0.74792408 -0.58442929
##  [13] -2.59184200 -1.12155678 -1.78691629  0.45355922  1.17043171  1.85786553
##  [19]  1.68533085  0.97736618  0.42674828  1.71468402  1.88475263  0.31868216
##  [25] -1.95677476 -3.23293413 -2.04796080 -3.50154620 -0.14850281 -0.84241205
##  [31] -0.09904779  0.98347628  0.40537181  0.80051058  1.93966047  1.94018792
##  [37]  1.49219033  2.34709325  3.92474462  3.91906833  3.13306971  2.16928793
##  [43]  2.06848733  0.38189044  0.10480999  0.93111620  2.58304860  1.72480305
##  [49]  2.08008778  0.44799092  1.44672387  1.19872106  3.34263037  2.26530231
##  [55]  1.35969662  2.26333305  3.17326337  2.94719719  4.16441731  3.57972014
##  [61]  2.82466277  2.61564224  2.11655419  0.32488963  0.66099253  0.52624370
##  [67]  0.06373925 -0.35784740  1.03242230  1.06911203 -0.41597984 -0.24149670
##  [73] -2.60049809 -2.41676378 -2.62799158 -1.24026800 -1.94581157 -1.94284872
##  [79] -2.05250578 -1.96678958 -3.31746059 -2.04650579 -1.36000718  1.98159061
##  [85]  2.64942025  2.30147448  1.64053611  0.42307363 -0.01501245  0.15777867
##  [91]  1.85095220  3.61362565  1.90010766  0.83161341 -1.08308679 -0.24887354
##  [97] -0.58370827 -2.09066260 -2.38615567 -2.13169180 -2.36131544 -2.02185719
## [103]  0.04892401  0.73347202  0.71321593 -0.63186749 -1.48982580 -2.06916048
## [109] -0.34819431  0.42621972  0.27779544 -1.48483570 -0.95756852 -0.93823655
## [115] -0.05459029 -0.10154393  0.81700169  0.07924658  0.89822643  0.13815886
## [121] -0.67354298 -0.17675454 -1.87794691 -1.80457518 -0.37965128 -1.16500218
## [127] -0.34621989 -0.63381243 -1.83254618 -0.60113922  0.56772770  0.06589133
## [133] -0.17501360 -0.69941261  0.67758863 -0.38464271  0.49950799  1.93783162
## [139]  1.43189519  1.82933697  2.35166115  3.03047868  3.62535827  2.86201068
## [145]  3.30372623  3.31098178  2.70794432  2.89456806  3.41630044  2.36547055
## [151]  0.35395738 -1.33000072 -3.80871160 -3.42927264 -2.83624079 -1.51258949
## [157] -0.30838137 -0.08933384  0.30787902  1.12989924 -0.14712740  0.02755425
## [163] -1.34329815 -3.01152132 -2.08722716 -2.03088646 -1.38604944 -1.07893635
## [169] -1.57224145 -2.98480078 -2.42021756  0.29194879  0.15008644  1.06436719
## [175] -1.10227677 -1.29730754 -0.51245874 -0.79091573 -0.31909909 -0.91892791
## [181] -0.99975430  0.22264232  1.34152710  2.07065592 -0.50249779 -1.47261502
## [187] -1.48777266  0.31149646  2.32266628  1.38143745  0.69022919 -1.13246270
## [193] -0.97647298 -0.73009886  1.23398575  0.59147217 -0.06650082 -1.20349436
## [199] -0.41909478  0.81468480  0.83520111  0.52821900 -1.00379600  1.26706191
## [205]  1.89237274  0.66980320  0.51826936  0.95673508  3.14414567  2.74180769
## [211]  1.36573777 -1.47136605 -1.13929899 -1.26138103 -2.87300485 -1.78940053
## [217] -1.67765475 -2.66370958 -2.55056697 -3.24465555 -3.24295959 -2.03371710
## [223] -1.24220601 -0.70388318 -0.17096367 -0.70958630 -2.62660221 -0.68046853
## [229]  1.48162800  2.43847799  2.99838010  2.34417736  2.90383247  2.93031348
## [235]  2.47971021  1.14925097  2.47363792  2.46084663  3.98794261  4.28774275
## [241]  3.32454839  2.27019712  2.89342765  3.64287055  3.62158905  3.79227758
## [247]  1.12375525  0.18394795 -2.04787990 -3.29806814 -2.56240795 -1.08471576
## [253] -2.10468381 -0.91726893 -0.54600782  0.23708751 -0.25954186 -0.96739098
## [259] -0.95444048 -0.82317391 -0.10198069  0.25261059  0.70834934  1.62957416
## [265]  1.18751345  1.70589425  0.92860639  0.72582207  0.67640692  1.53456079
## [271]  2.87957799  2.89335087  0.80434298 -0.73991953 -0.36374042  0.13763276
## [277] -0.07270916  1.16174085  0.69941293  1.07993785 -0.54656439 -0.67044288
## [283] -0.32518707  1.32404394 -1.15413692 -2.37347758 -1.89669726 -1.56461104
## [289] -1.94321529 -1.64486354 -2.46361366 -3.58595094 -2.66204313 -0.80500899
## [295] -0.45592718 -1.42792139 -1.70735653 -0.16704081  0.31406658

#Yt-1
yt_1_arma22 <- arma22[-300]
yt_1_arma22

##   [1] -1.74650961  1.84212574  1.68856917  2.13483906  2.96194453  2.27909247
##   [7]  2.00430099  0.71682006 -0.89497815  0.04864253  1.41826850  0.74792408
##  [13] -0.58442929 -2.59184200 -1.12155678 -1.78691629  0.45355922  1.17043171
##  [19]  1.85786553  1.68533085  0.97736618  0.42674828  1.71468402  1.88475263
##  [25]  0.31868216 -1.95677476 -3.23293413 -2.04796080 -3.50154620 -0.14850281
##  [31] -0.84241205 -0.09904779  0.98347628  0.40537181  0.80051058  1.93966047
##  [37]  1.94018792  1.49219033  2.34709325  3.92474462  3.91906833  3.13306971
##  [43]  2.16928793  2.06848733  0.38189044  0.10480999  0.93111620  2.58304860
##  [49]  1.72480305  2.08008778  0.44799092  1.44672387  1.19872106  3.34263037
##  [55]  2.26530231  1.35969662  2.26333305  3.17326337  2.94719719  4.16441731
##  [61]  3.57972014  2.82466277  2.61564224  2.11655419  0.32488963  0.66099253
##  [67]  0.52624370  0.06373925 -0.35784740  1.03242230  1.06911203 -0.41597984
##  [73] -0.24149670 -2.60049809 -2.41676378 -2.62799158 -1.24026800 -1.94581157
##  [79] -1.94284872 -2.05250578 -1.96678958 -3.31746059 -2.04650579 -1.36000718
##  [85]  1.98159061  2.64942025  2.30147448  1.64053611  0.42307363 -0.01501245
##  [91]  0.15777867  1.85095220  3.61362565  1.90010766  0.83161341 -1.08308679
##  [97] -0.24887354 -0.58370827 -2.09066260 -2.38615567 -2.13169180 -2.36131544
## [103] -2.02185719  0.04892401  0.73347202  0.71321593 -0.63186749 -1.48982580
## [109] -2.06916048 -0.34819431  0.42621972  0.27779544 -1.48483570 -0.95756852
## [115] -0.93823655 -0.05459029 -0.10154393  0.81700169  0.07924658  0.89822643
## [121]  0.13815886 -0.67354298 -0.17675454 -1.87794691 -1.80457518 -0.37965128
## [127] -1.16500218 -0.34621989 -0.63381243 -1.83254618 -0.60113922  0.56772770
## [133]  0.06589133 -0.17501360 -0.69941261  0.67758863 -0.38464271  0.49950799
## [139]  1.93783162  1.43189519  1.82933697  2.35166115  3.03047868  3.62535827
## [145]  2.86201068  3.30372623  3.31098178  2.70794432  2.89456806  3.41630044
## [151]  2.36547055  0.35395738 -1.33000072 -3.80871160 -3.42927264 -2.83624079
## [157] -1.51258949 -0.30838137 -0.08933384  0.30787902  1.12989924 -0.14712740
## [163]  0.02755425 -1.34329815 -3.01152132 -2.08722716 -2.03088646 -1.38604944
## [169] -1.07893635 -1.57224145 -2.98480078 -2.42021756  0.29194879  0.15008644
## [175]  1.06436719 -1.10227677 -1.29730754 -0.51245874 -0.79091573 -0.31909909
## [181] -0.91892791 -0.99975430  0.22264232  1.34152710  2.07065592 -0.50249779
## [187] -1.47261502 -1.48777266  0.31149646  2.32266628  1.38143745  0.69022919
## [193] -1.13246270 -0.97647298 -0.73009886  1.23398575  0.59147217 -0.06650082
## [199] -1.20349436 -0.41909478  0.81468480  0.83520111  0.52821900 -1.00379600
## [205]  1.26706191  1.89237274  0.66980320  0.51826936  0.95673508  3.14414567
## [211]  2.74180769  1.36573777 -1.47136605 -1.13929899 -1.26138103 -2.87300485
## [217] -1.78940053 -1.67765475 -2.66370958 -2.55056697 -3.24465555 -3.24295959
## [223] -2.03371710 -1.24220601 -0.70388318 -0.17096367 -0.70958630 -2.62660221
## [229] -0.68046853  1.48162800  2.43847799  2.99838010  2.34417736  2.90383247
## [235]  2.93031348  2.47971021  1.14925097  2.47363792  2.46084663  3.98794261
## [241]  4.28774275  3.32454839  2.27019712  2.89342765  3.64287055  3.62158905
## [247]  3.79227758  1.12375525  0.18394795 -2.04787990 -3.29806814 -2.56240795
## [253] -1.08471576 -2.10468381 -0.91726893 -0.54600782  0.23708751 -0.25954186
## [259] -0.96739098 -0.95444048 -0.82317391 -0.10198069  0.25261059  0.70834934
## [265]  1.62957416  1.18751345  1.70589425  0.92860639  0.72582207  0.67640692
## [271]  1.53456079  2.87957799  2.89335087  0.80434298 -0.73991953 -0.36374042
## [277]  0.13763276 -0.07270916  1.16174085  0.69941293  1.07993785 -0.54656439
## [283] -0.67044288 -0.32518707  1.32404394 -1.15413692 -2.37347758 -1.89669726
## [289] -1.56461104 -1.94321529 -1.64486354 -2.46361366 -3.58595094 -2.66204313
## [295] -0.80500899 -0.45592718 -1.42792139 -1.70735653 -0.16704081

plot(y=yt_arma22,x=yt_1_arma22)

Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-1}\).

cor(yt_arma22,yt_1_arma22)

## [1] 0.8123794

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

#Yt
yt_arma22.2 <- arma22[-c(1,2)]
yt_arma22.2

##   [1]  1.68856917  2.13483906  2.96194453  2.27909247  2.00430099  0.71682006
##   [7] -0.89497815  0.04864253  1.41826850  0.74792408 -0.58442929 -2.59184200
##  [13] -1.12155678 -1.78691629  0.45355922  1.17043171  1.85786553  1.68533085
##  [19]  0.97736618  0.42674828  1.71468402  1.88475263  0.31868216 -1.95677476
##  [25] -3.23293413 -2.04796080 -3.50154620 -0.14850281 -0.84241205 -0.09904779
##  [31]  0.98347628  0.40537181  0.80051058  1.93966047  1.94018792  1.49219033
##  [37]  2.34709325  3.92474462  3.91906833  3.13306971  2.16928793  2.06848733
##  [43]  0.38189044  0.10480999  0.93111620  2.58304860  1.72480305  2.08008778
##  [49]  0.44799092  1.44672387  1.19872106  3.34263037  2.26530231  1.35969662
##  [55]  2.26333305  3.17326337  2.94719719  4.16441731  3.57972014  2.82466277
##  [61]  2.61564224  2.11655419  0.32488963  0.66099253  0.52624370  0.06373925
##  [67] -0.35784740  1.03242230  1.06911203 -0.41597984 -0.24149670 -2.60049809
##  [73] -2.41676378 -2.62799158 -1.24026800 -1.94581157 -1.94284872 -2.05250578
##  [79] -1.96678958 -3.31746059 -2.04650579 -1.36000718  1.98159061  2.64942025
##  [85]  2.30147448  1.64053611  0.42307363 -0.01501245  0.15777867  1.85095220
##  [91]  3.61362565  1.90010766  0.83161341 -1.08308679 -0.24887354 -0.58370827
##  [97] -2.09066260 -2.38615567 -2.13169180 -2.36131544 -2.02185719  0.04892401
## [103]  0.73347202  0.71321593 -0.63186749 -1.48982580 -2.06916048 -0.34819431
## [109]  0.42621972  0.27779544 -1.48483570 -0.95756852 -0.93823655 -0.05459029
## [115] -0.10154393  0.81700169  0.07924658  0.89822643  0.13815886 -0.67354298
## [121] -0.17675454 -1.87794691 -1.80457518 -0.37965128 -1.16500218 -0.34621989
## [127] -0.63381243 -1.83254618 -0.60113922  0.56772770  0.06589133 -0.17501360
## [133] -0.69941261  0.67758863 -0.38464271  0.49950799  1.93783162  1.43189519
## [139]  1.82933697  2.35166115  3.03047868  3.62535827  2.86201068  3.30372623
## [145]  3.31098178  2.70794432  2.89456806  3.41630044  2.36547055  0.35395738
## [151] -1.33000072 -3.80871160 -3.42927264 -2.83624079 -1.51258949 -0.30838137
## [157] -0.08933384  0.30787902  1.12989924 -0.14712740  0.02755425 -1.34329815
## [163] -3.01152132 -2.08722716 -2.03088646 -1.38604944 -1.07893635 -1.57224145
## [169] -2.98480078 -2.42021756  0.29194879  0.15008644  1.06436719 -1.10227677
## [175] -1.29730754 -0.51245874 -0.79091573 -0.31909909 -0.91892791 -0.99975430
## [181]  0.22264232  1.34152710  2.07065592 -0.50249779 -1.47261502 -1.48777266
## [187]  0.31149646  2.32266628  1.38143745  0.69022919 -1.13246270 -0.97647298
## [193] -0.73009886  1.23398575  0.59147217 -0.06650082 -1.20349436 -0.41909478
## [199]  0.81468480  0.83520111  0.52821900 -1.00379600  1.26706191  1.89237274
## [205]  0.66980320  0.51826936  0.95673508  3.14414567  2.74180769  1.36573777
## [211] -1.47136605 -1.13929899 -1.26138103 -2.87300485 -1.78940053 -1.67765475
## [217] -2.66370958 -2.55056697 -3.24465555 -3.24295959 -2.03371710 -1.24220601
## [223] -0.70388318 -0.17096367 -0.70958630 -2.62660221 -0.68046853  1.48162800
## [229]  2.43847799  2.99838010  2.34417736  2.90383247  2.93031348  2.47971021
## [235]  1.14925097  2.47363792  2.46084663  3.98794261  4.28774275  3.32454839
## [241]  2.27019712  2.89342765  3.64287055  3.62158905  3.79227758  1.12375525
## [247]  0.18394795 -2.04787990 -3.29806814 -2.56240795 -1.08471576 -2.10468381
## [253] -0.91726893 -0.54600782  0.23708751 -0.25954186 -0.96739098 -0.95444048
## [259] -0.82317391 -0.10198069  0.25261059  0.70834934  1.62957416  1.18751345
## [265]  1.70589425  0.92860639  0.72582207  0.67640692  1.53456079  2.87957799
## [271]  2.89335087  0.80434298 -0.73991953 -0.36374042  0.13763276 -0.07270916
## [277]  1.16174085  0.69941293  1.07993785 -0.54656439 -0.67044288 -0.32518707
## [283]  1.32404394 -1.15413692 -2.37347758 -1.89669726 -1.56461104 -1.94321529
## [289] -1.64486354 -2.46361366 -3.58595094 -2.66204313 -0.80500899 -0.45592718
## [295] -1.42792139 -1.70735653 -0.16704081  0.31406658

#Yt-2
yt_2_arma22 <- arma22[-c(299,300)]
yt_2_arma22

##   [1] -1.74650961  1.84212574  1.68856917  2.13483906  2.96194453  2.27909247
##   [7]  2.00430099  0.71682006 -0.89497815  0.04864253  1.41826850  0.74792408
##  [13] -0.58442929 -2.59184200 -1.12155678 -1.78691629  0.45355922  1.17043171
##  [19]  1.85786553  1.68533085  0.97736618  0.42674828  1.71468402  1.88475263
##  [25]  0.31868216 -1.95677476 -3.23293413 -2.04796080 -3.50154620 -0.14850281
##  [31] -0.84241205 -0.09904779  0.98347628  0.40537181  0.80051058  1.93966047
##  [37]  1.94018792  1.49219033  2.34709325  3.92474462  3.91906833  3.13306971
##  [43]  2.16928793  2.06848733  0.38189044  0.10480999  0.93111620  2.58304860
##  [49]  1.72480305  2.08008778  0.44799092  1.44672387  1.19872106  3.34263037
##  [55]  2.26530231  1.35969662  2.26333305  3.17326337  2.94719719  4.16441731
##  [61]  3.57972014  2.82466277  2.61564224  2.11655419  0.32488963  0.66099253
##  [67]  0.52624370  0.06373925 -0.35784740  1.03242230  1.06911203 -0.41597984
##  [73] -0.24149670 -2.60049809 -2.41676378 -2.62799158 -1.24026800 -1.94581157
##  [79] -1.94284872 -2.05250578 -1.96678958 -3.31746059 -2.04650579 -1.36000718
##  [85]  1.98159061  2.64942025  2.30147448  1.64053611  0.42307363 -0.01501245
##  [91]  0.15777867  1.85095220  3.61362565  1.90010766  0.83161341 -1.08308679
##  [97] -0.24887354 -0.58370827 -2.09066260 -2.38615567 -2.13169180 -2.36131544
## [103] -2.02185719  0.04892401  0.73347202  0.71321593 -0.63186749 -1.48982580
## [109] -2.06916048 -0.34819431  0.42621972  0.27779544 -1.48483570 -0.95756852
## [115] -0.93823655 -0.05459029 -0.10154393  0.81700169  0.07924658  0.89822643
## [121]  0.13815886 -0.67354298 -0.17675454 -1.87794691 -1.80457518 -0.37965128
## [127] -1.16500218 -0.34621989 -0.63381243 -1.83254618 -0.60113922  0.56772770
## [133]  0.06589133 -0.17501360 -0.69941261  0.67758863 -0.38464271  0.49950799
## [139]  1.93783162  1.43189519  1.82933697  2.35166115  3.03047868  3.62535827
## [145]  2.86201068  3.30372623  3.31098178  2.70794432  2.89456806  3.41630044
## [151]  2.36547055  0.35395738 -1.33000072 -3.80871160 -3.42927264 -2.83624079
## [157] -1.51258949 -0.30838137 -0.08933384  0.30787902  1.12989924 -0.14712740
## [163]  0.02755425 -1.34329815 -3.01152132 -2.08722716 -2.03088646 -1.38604944
## [169] -1.07893635 -1.57224145 -2.98480078 -2.42021756  0.29194879  0.15008644
## [175]  1.06436719 -1.10227677 -1.29730754 -0.51245874 -0.79091573 -0.31909909
## [181] -0.91892791 -0.99975430  0.22264232  1.34152710  2.07065592 -0.50249779
## [187] -1.47261502 -1.48777266  0.31149646  2.32266628  1.38143745  0.69022919
## [193] -1.13246270 -0.97647298 -0.73009886  1.23398575  0.59147217 -0.06650082
## [199] -1.20349436 -0.41909478  0.81468480  0.83520111  0.52821900 -1.00379600
## [205]  1.26706191  1.89237274  0.66980320  0.51826936  0.95673508  3.14414567
## [211]  2.74180769  1.36573777 -1.47136605 -1.13929899 -1.26138103 -2.87300485
## [217] -1.78940053 -1.67765475 -2.66370958 -2.55056697 -3.24465555 -3.24295959
## [223] -2.03371710 -1.24220601 -0.70388318 -0.17096367 -0.70958630 -2.62660221
## [229] -0.68046853  1.48162800  2.43847799  2.99838010  2.34417736  2.90383247
## [235]  2.93031348  2.47971021  1.14925097  2.47363792  2.46084663  3.98794261
## [241]  4.28774275  3.32454839  2.27019712  2.89342765  3.64287055  3.62158905
## [247]  3.79227758  1.12375525  0.18394795 -2.04787990 -3.29806814 -2.56240795
## [253] -1.08471576 -2.10468381 -0.91726893 -0.54600782  0.23708751 -0.25954186
## [259] -0.96739098 -0.95444048 -0.82317391 -0.10198069  0.25261059  0.70834934
## [265]  1.62957416  1.18751345  1.70589425  0.92860639  0.72582207  0.67640692
## [271]  1.53456079  2.87957799  2.89335087  0.80434298 -0.73991953 -0.36374042
## [277]  0.13763276 -0.07270916  1.16174085  0.69941293  1.07993785 -0.54656439
## [283] -0.67044288 -0.32518707  1.32404394 -1.15413692 -2.37347758 -1.89669726
## [289] -1.56461104 -1.94321529 -1.64486354 -2.46361366 -3.58595094 -2.66204313
## [295] -0.80500899 -0.45592718 -1.42792139 -1.70735653

plot(y=yt_arma22.2,x=yt_2_arma22)

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

cor(yt_arma22.2,yt_2_arma22)

## [1] 0.6070096

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

#Yt
yt_arma22.3 <- arma22[-c(1,2,3)]
yt_arma22.3

##   [1]  2.13483906  2.96194453  2.27909247  2.00430099  0.71682006 -0.89497815
##   [7]  0.04864253  1.41826850  0.74792408 -0.58442929 -2.59184200 -1.12155678
##  [13] -1.78691629  0.45355922  1.17043171  1.85786553  1.68533085  0.97736618
##  [19]  0.42674828  1.71468402  1.88475263  0.31868216 -1.95677476 -3.23293413
##  [25] -2.04796080 -3.50154620 -0.14850281 -0.84241205 -0.09904779  0.98347628
##  [31]  0.40537181  0.80051058  1.93966047  1.94018792  1.49219033  2.34709325
##  [37]  3.92474462  3.91906833  3.13306971  2.16928793  2.06848733  0.38189044
##  [43]  0.10480999  0.93111620  2.58304860  1.72480305  2.08008778  0.44799092
##  [49]  1.44672387  1.19872106  3.34263037  2.26530231  1.35969662  2.26333305
##  [55]  3.17326337  2.94719719  4.16441731  3.57972014  2.82466277  2.61564224
##  [61]  2.11655419  0.32488963  0.66099253  0.52624370  0.06373925 -0.35784740
##  [67]  1.03242230  1.06911203 -0.41597984 -0.24149670 -2.60049809 -2.41676378
##  [73] -2.62799158 -1.24026800 -1.94581157 -1.94284872 -2.05250578 -1.96678958
##  [79] -3.31746059 -2.04650579 -1.36000718  1.98159061  2.64942025  2.30147448
##  [85]  1.64053611  0.42307363 -0.01501245  0.15777867  1.85095220  3.61362565
##  [91]  1.90010766  0.83161341 -1.08308679 -0.24887354 -0.58370827 -2.09066260
##  [97] -2.38615567 -2.13169180 -2.36131544 -2.02185719  0.04892401  0.73347202
## [103]  0.71321593 -0.63186749 -1.48982580 -2.06916048 -0.34819431  0.42621972
## [109]  0.27779544 -1.48483570 -0.95756852 -0.93823655 -0.05459029 -0.10154393
## [115]  0.81700169  0.07924658  0.89822643  0.13815886 -0.67354298 -0.17675454
## [121] -1.87794691 -1.80457518 -0.37965128 -1.16500218 -0.34621989 -0.63381243
## [127] -1.83254618 -0.60113922  0.56772770  0.06589133 -0.17501360 -0.69941261
## [133]  0.67758863 -0.38464271  0.49950799  1.93783162  1.43189519  1.82933697
## [139]  2.35166115  3.03047868  3.62535827  2.86201068  3.30372623  3.31098178
## [145]  2.70794432  2.89456806  3.41630044  2.36547055  0.35395738 -1.33000072
## [151] -3.80871160 -3.42927264 -2.83624079 -1.51258949 -0.30838137 -0.08933384
## [157]  0.30787902  1.12989924 -0.14712740  0.02755425 -1.34329815 -3.01152132
## [163] -2.08722716 -2.03088646 -1.38604944 -1.07893635 -1.57224145 -2.98480078
## [169] -2.42021756  0.29194879  0.15008644  1.06436719 -1.10227677 -1.29730754
## [175] -0.51245874 -0.79091573 -0.31909909 -0.91892791 -0.99975430  0.22264232
## [181]  1.34152710  2.07065592 -0.50249779 -1.47261502 -1.48777266  0.31149646
## [187]  2.32266628  1.38143745  0.69022919 -1.13246270 -0.97647298 -0.73009886
## [193]  1.23398575  0.59147217 -0.06650082 -1.20349436 -0.41909478  0.81468480
## [199]  0.83520111  0.52821900 -1.00379600  1.26706191  1.89237274  0.66980320
## [205]  0.51826936  0.95673508  3.14414567  2.74180769  1.36573777 -1.47136605
## [211] -1.13929899 -1.26138103 -2.87300485 -1.78940053 -1.67765475 -2.66370958
## [217] -2.55056697 -3.24465555 -3.24295959 -2.03371710 -1.24220601 -0.70388318
## [223] -0.17096367 -0.70958630 -2.62660221 -0.68046853  1.48162800  2.43847799
## [229]  2.99838010  2.34417736  2.90383247  2.93031348  2.47971021  1.14925097
## [235]  2.47363792  2.46084663  3.98794261  4.28774275  3.32454839  2.27019712
## [241]  2.89342765  3.64287055  3.62158905  3.79227758  1.12375525  0.18394795
## [247] -2.04787990 -3.29806814 -2.56240795 -1.08471576 -2.10468381 -0.91726893
## [253] -0.54600782  0.23708751 -0.25954186 -0.96739098 -0.95444048 -0.82317391
## [259] -0.10198069  0.25261059  0.70834934  1.62957416  1.18751345  1.70589425
## [265]  0.92860639  0.72582207  0.67640692  1.53456079  2.87957799  2.89335087
## [271]  0.80434298 -0.73991953 -0.36374042  0.13763276 -0.07270916  1.16174085
## [277]  0.69941293  1.07993785 -0.54656439 -0.67044288 -0.32518707  1.32404394
## [283] -1.15413692 -2.37347758 -1.89669726 -1.56461104 -1.94321529 -1.64486354
## [289] -2.46361366 -3.58595094 -2.66204313 -0.80500899 -0.45592718 -1.42792139
## [295] -1.70735653 -0.16704081  0.31406658

#Yt-2
yt_3_arma22 <- arma22[-c(298,299,300)]
yt_3_arma22

##   [1] -1.74650961  1.84212574  1.68856917  2.13483906  2.96194453  2.27909247
##   [7]  2.00430099  0.71682006 -0.89497815  0.04864253  1.41826850  0.74792408
##  [13] -0.58442929 -2.59184200 -1.12155678 -1.78691629  0.45355922  1.17043171
##  [19]  1.85786553  1.68533085  0.97736618  0.42674828  1.71468402  1.88475263
##  [25]  0.31868216 -1.95677476 -3.23293413 -2.04796080 -3.50154620 -0.14850281
##  [31] -0.84241205 -0.09904779  0.98347628  0.40537181  0.80051058  1.93966047
##  [37]  1.94018792  1.49219033  2.34709325  3.92474462  3.91906833  3.13306971
##  [43]  2.16928793  2.06848733  0.38189044  0.10480999  0.93111620  2.58304860
##  [49]  1.72480305  2.08008778  0.44799092  1.44672387  1.19872106  3.34263037
##  [55]  2.26530231  1.35969662  2.26333305  3.17326337  2.94719719  4.16441731
##  [61]  3.57972014  2.82466277  2.61564224  2.11655419  0.32488963  0.66099253
##  [67]  0.52624370  0.06373925 -0.35784740  1.03242230  1.06911203 -0.41597984
##  [73] -0.24149670 -2.60049809 -2.41676378 -2.62799158 -1.24026800 -1.94581157
##  [79] -1.94284872 -2.05250578 -1.96678958 -3.31746059 -2.04650579 -1.36000718
##  [85]  1.98159061  2.64942025  2.30147448  1.64053611  0.42307363 -0.01501245
##  [91]  0.15777867  1.85095220  3.61362565  1.90010766  0.83161341 -1.08308679
##  [97] -0.24887354 -0.58370827 -2.09066260 -2.38615567 -2.13169180 -2.36131544
## [103] -2.02185719  0.04892401  0.73347202  0.71321593 -0.63186749 -1.48982580
## [109] -2.06916048 -0.34819431  0.42621972  0.27779544 -1.48483570 -0.95756852
## [115] -0.93823655 -0.05459029 -0.10154393  0.81700169  0.07924658  0.89822643
## [121]  0.13815886 -0.67354298 -0.17675454 -1.87794691 -1.80457518 -0.37965128
## [127] -1.16500218 -0.34621989 -0.63381243 -1.83254618 -0.60113922  0.56772770
## [133]  0.06589133 -0.17501360 -0.69941261  0.67758863 -0.38464271  0.49950799
## [139]  1.93783162  1.43189519  1.82933697  2.35166115  3.03047868  3.62535827
## [145]  2.86201068  3.30372623  3.31098178  2.70794432  2.89456806  3.41630044
## [151]  2.36547055  0.35395738 -1.33000072 -3.80871160 -3.42927264 -2.83624079
## [157] -1.51258949 -0.30838137 -0.08933384  0.30787902  1.12989924 -0.14712740
## [163]  0.02755425 -1.34329815 -3.01152132 -2.08722716 -2.03088646 -1.38604944
## [169] -1.07893635 -1.57224145 -2.98480078 -2.42021756  0.29194879  0.15008644
## [175]  1.06436719 -1.10227677 -1.29730754 -0.51245874 -0.79091573 -0.31909909
## [181] -0.91892791 -0.99975430  0.22264232  1.34152710  2.07065592 -0.50249779
## [187] -1.47261502 -1.48777266  0.31149646  2.32266628  1.38143745  0.69022919
## [193] -1.13246270 -0.97647298 -0.73009886  1.23398575  0.59147217 -0.06650082
## [199] -1.20349436 -0.41909478  0.81468480  0.83520111  0.52821900 -1.00379600
## [205]  1.26706191  1.89237274  0.66980320  0.51826936  0.95673508  3.14414567
## [211]  2.74180769  1.36573777 -1.47136605 -1.13929899 -1.26138103 -2.87300485
## [217] -1.78940053 -1.67765475 -2.66370958 -2.55056697 -3.24465555 -3.24295959
## [223] -2.03371710 -1.24220601 -0.70388318 -0.17096367 -0.70958630 -2.62660221
## [229] -0.68046853  1.48162800  2.43847799  2.99838010  2.34417736  2.90383247
## [235]  2.93031348  2.47971021  1.14925097  2.47363792  2.46084663  3.98794261
## [241]  4.28774275  3.32454839  2.27019712  2.89342765  3.64287055  3.62158905
## [247]  3.79227758  1.12375525  0.18394795 -2.04787990 -3.29806814 -2.56240795
## [253] -1.08471576 -2.10468381 -0.91726893 -0.54600782  0.23708751 -0.25954186
## [259] -0.96739098 -0.95444048 -0.82317391 -0.10198069  0.25261059  0.70834934
## [265]  1.62957416  1.18751345  1.70589425  0.92860639  0.72582207  0.67640692
## [271]  1.53456079  2.87957799  2.89335087  0.80434298 -0.73991953 -0.36374042
## [277]  0.13763276 -0.07270916  1.16174085  0.69941293  1.07993785 -0.54656439
## [283] -0.67044288 -0.32518707  1.32404394 -1.15413692 -2.37347758 -1.89669726
## [289] -1.56461104 -1.94321529 -1.64486354 -2.46361366 -3.58595094 -2.66204313
## [295] -0.80500899 -0.45592718 -1.42792139

plot(y=yt_arma22.3,x=yt_3_arma22)

Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dengan \(Y_{t-3}\). Hal ini sesuai dengan teori yang ada

cor(yt_arma22.3,yt_3_arma22)

## [1] 0.433211

Perbandingan korelasi \(Yt_t\) dengan \(Y_{t-1}\), \(Yt_t\) dengan \(Y_{t-2}\), & \(Yt_t\) dengan \(Y_{t-3}\)

par(mfrow=c(2,2))
plot(y=yt_arma22,x=yt_1_arma22)
plot(y=yt_arma22.2,x=yt_2_arma22)
plot(y=yt_arma22.3,x=yt_3_arma22)