Question

Would you please help with r codes for the following

1. Independently generate a, a, and a, from a uniform distribution in the range (-1, 1) conditioned on la, + a2 + al < 1. 2. Generate a series of size 100+ S from the AR process as represented in Equation (1) of order p 3 with ao-0.0 and a,, a, and a, obtained from Step 1. Initialize the starting value, yo 0.0. Discard the first 100 observations to minimize the effect of initial value.

Where S is the sample size S=50,100,200

Do at least half of the steps please

Thanks

Answer 1

The error term is generated from standard normal distribution.  In our effort to minimize the initial effect, we simulate 100 + S observations and discard the first 100 observations, leaving the last S observations for order estimation.

R Code:

#Generate a1,a2 and a3 from uniform distribution
a1=2;a2=2;a3=2;
while (abs(a1+a2+a3)>1) {
a1 = runif(1, min = -1, max = 1)
a2 = runif(1, min = -1, max = 1)
a3 = runif(1, min = -1, max = 1)
}
a0=0;
# For S = 200 and p = 3
S = 50;
n = 100+S;
set.seed(1)
e=rnorm(n)
Y=rep(0,n);
y[0]=0
for(t in 4:n)
Y[t]=a0+a1*Y[t-1]+a2*Y[t-2]+a3*Y[t-3]+e[t]
n1 = length(Z)
Y = Y[101:n1]
# For S = 100 and p = 3
S = 100;
n = 100+S;
set.seed(1)
e=rnorm(n)
Y=rep(0,n);
y[0]=0
for(t in 4:n)
Y[t]=a0+a1*Y[t-1]+a2*Y[t-2]+a3*Y[t-3]+e[t]
n1 = length(Z)
Y = Y[101:n1]
# For S = 200 and p = 3
S = 200;
n = 100+S;
set.seed(1)
e=rnorm(n)
Y=rep(0,n);
y[0]=0
for(t in 4:n)
Y[t]=a0+a1*Y[t-1]+a2*Y[t-2]+a3*Y[t-3]+e[t]
n1 = length(Z)
Y = Y[101:n1]

Output:

For S = 50;

> a1
[1] 0.3474245
> a2
[1] -0.8102843
> a3
[1] -0.01480776
> Y
[1] 0.32287902 0.63377260 -0.86103374 -1.71366112
[5] -1.19969171 0.88768788 0.29132765 -0.30755738
[9] 1.43315518 1.84968667 0.46679287 -0.56464294
[13] -1.49396639 0.34129495 2.44164714 0.27898169
[17] -1.56981237 -0.65019564 2.39204148 -0.02491549
[21] -1.49440536 -0.79131151 0.79223157 0.59987858
[25] -0.61055845 -0.99622302 -0.24799895 1.49272956
[29] -0.54207748 -3.22714727 -0.01697972 3.06409058
[33] 1.92582934 -0.23230906 -2.30705395 1.08616817
[37] 1.09617732 -0.59148355 -2.51673652 -0.38319946
[41] 3.75548892 2.04820648 -0.73754200 -3.01567692
[45] 0.27380253 2.54486835 0.31072559 -0.58419050
[49] 1.13762026 -0.16611497 -1.03561485 -1.01774617
>

For S = 100

> Y
[1] 0.32287902 -0.34649406 -1.30231071 -0.01844825
[5] 0.39937869 1.94027383 1.06746897 -0.29704824
[9] -0.61270091 1.69419549 0.45372905 -1.66771542
[13] 0.46014034 0.85377255 -0.25890863 -1.18137125
[17] -0.53328300 0.49669156 1.11635543 -0.18404593
[21] -1.48181968 0.96081722 1.32265251 -0.47662736
[25] -1.35173485 0.60965869 1.24059323 -0.08060243
[29] -1.72392462 -0.87626339 1.15378768 0.54750950
[33] -0.20020614 -2.04867395 -0.25108509 0.03928999
[37] -0.05353925 -0.57499881 -0.80906321 0.12872016
[41] -1.20555324 0.66542511 -0.45885268 -1.14427902
[45] -1.15152297 -0.21690036 2.96181523 1.23920512
[49] -3.25247089 -3.81855978 1.74060222 3.72844877
[53] -0.37655238 -4.10706353 -2.66445015 1.33257748
[57] 3.68277733 -0.38209181 -4.52100399 0.55365254
[61] 4.28640924 0.86888632 -2.12105300 -0.61800010
[65] 0.87183827 3.04116422 0.10426217 -3.86538898
[69] -1.61684513 2.77632686 4.63988422 -0.50786052
[73] -3.52018083 -0.95734477 2.19326166 1.55511414
[77] -0.43506503 0.63153093 1.57630034 1.25027645
[81] -2.08354975 -0.77640143 1.61993779 -0.24448543
[85] -0.86503085 -0.28517243 2.07005262 0.19698484
[89] -2.03488276 -1.82334411 0.83533831 2.19978788
[93] -0.61734991 -1.17893234 -1.15001772 -0.48311678
[97] 2.22260975 0.16483266 -1.32455027 -1.00773041
>

For S = 200

> Y
[1] 0.32287902 -0.34649406 -1.30231071 -0.01844825
[5] 0.39937869 1.94027383 1.06746897 -0.29704824
[9] -0.61270091 1.69419549 0.45372905 -1.66771542
[13] 0.46014034 0.85377255 -0.25890863 -1.18137125
[17] -0.53328300 0.49669156 1.11635543 -0.18404593
[21] -1.48181968 0.96081722 1.32265251 -0.47662736
[25] -1.35173485 0.60965869 1.24059323 -0.08060243
[29] -1.72392462 -0.87626339 1.15378768 0.54750950
[33] -0.20020614 -2.04867395 -0.25108509 0.03928999
[37] -0.05353925 -0.57499881 -0.80906321 0.12872016
[41] -1.20555324 0.66542511 -0.45885268 -1.14427902
[45] -1.15152297 -0.21690036 2.96181523 1.23920512
[49] -3.25247089 -3.81855978 1.74060222 3.72844877
[53] -0.37655238 -4.10706353 -2.66445015 1.33257748
[57] 3.68277733 -0.38209181 -4.52100399 0.55365254
[61] 4.28640924 0.86888632 -2.12105300 -0.61800010
[65] 0.87183827 3.04116422 0.10426217 -3.86538898
[69] -1.61684513 2.77632686 4.63988422 -0.50786052
[73] -3.52018083 -0.95734477 2.19326166 1.55511414
[77] -0.43506503 0.63153093 1.57630034 1.25027645
[81] -2.08354975 -0.77640143 1.61993779 -0.24448543
[85] -0.86503085 -0.28517243 2.07005262 0.19698484
[89] -2.03488276 -1.82334411 0.83533831 2.19978788
[93] -0.61734991 -1.17893234 -1.15001772 -0.48311678
[97] 2.22260975 0.16483266 -1.32455027 -1.00773041
[101] 1.13011312 2.91766397 1.69946560 -2.12134359
[105] -4.44249654 2.64795571 5.21812826 0.27441926
[109] -4.18543719 -1.24364126 2.79088323 2.45999567
[113] -1.78857736 -4.02622598 1.00185797 5.15668747
[117] 0.73064843 -5.19266273 -1.83020869 3.51614535
[121] 1.04825756 -2.45565386 -2.38490743 0.80471163
[125] 1.09181989 1.56573679 -0.68375725 -3.12792674
[129] -0.35867213 2.68319898 0.28332625 -4.95932905
[133] -2.63278092 3.67008612 3.42209199 -1.84408780
[137] -2.90707361 -0.75288528 3.21806916 1.76579029
[141] -1.27561293 -0.88751592 0.92259739 0.17985447
[145] 0.49102635 -1.98896523 -1.63634010 0.78018098
[149] 1.46028571 0.91986503 -0.73899379 -0.61655266
[153] 0.30131363 0.36754516 0.58822505 1.04831428
[157] -2.52095822 -1.16124590 1.99844883 1.24731134
[161] -0.21775696 -1.50516061 -0.64928513 1.85466472
[165] 2.91237679 -0.21130538 -2.88291325 -2.06261623
[169] 1.29147195 1.22285464 -0.84999967 -0.91091522
[173] -0.49769768 3.22694141 1.69389530 -0.88866221
[177] -4.01818749 0.03997075 1.96667494 1.63018735
[181] -0.62966056 -1.97632531 1.12369950 1.29987824
[185] -1.01025591 -2.42197019 -0.71028410 2.67585890
[189] 1.97475699 -0.46645060 -2.19191305 -0.03643806
[193] 2.01448525 -0.66439337 -0.08416340 0.61372474
[197] 1.05685645 0.82626922 -0.62894162 -1.20948777
>

I could only do step 1 and step 2 because you haven't mentioned any thing about the selection criteria. Please post step 3 onwards as separte and mention the six selection criteria for finding the order of AR.

Also, please like the solution if it helps you thank you.