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| Time Series Analysis and Control Examples |
proc iml;
y = { 2.430 2.506 2.767 2.940 3.169 3.450 3.594 3.774 3.695 3.411
2.718 1.991 2.265 2.446 2.612 3.359 3.429 3.533 3.261 2.612
2.179 1.653 1.832 2.328 2.737 3.014 3.328 3.404 2.981 2.557
2.576 2.352 2.556 2.864 3.214 3.435 3.458 3.326 2.835 2.476
2.373 2.389 2.742 3.210 3.520 3.828 3.628 2.837 2.406 2.675
2.554 2.894 3.202 3.224 3.352 3.154 2.878 2.476 2.303 2.360
2.671 2.867 3.310 3.449 3.646 3.400 2.590 1.863 1.581 1.690
1.771 2.274 2.576 3.111 3.605 3.543 2.769 2.021 2.185 2.588
2.880 3.115 3.540 3.845 3.800 3.579 3.264 2.538 2.582 2.907
3.142 3.433 3.580 3.490 3.475 3.579 2.829 1.909 1.903 2.033
2.360 2.601 3.054 3.386 3.553 3.468 3.187 2.723 2.686 2.821
3.000 3.201 3.424 3.531 };
call tsunimar(arcoef,ev,nar,aic) data=y opt={-1 1} print=1
maxlag=20;
You can also invoke the TSUNIMAR call as follows:
call tsunimar(arcoef,ev,nar,aic,y,20,{-1 1},,1);
The optional arguments can be omitted.
In this example, the argument MISSING is omitted,
and thus the default option (MISSING=0) is used.
The summary table of the minimum AIC method
is displayed in Figure 10.1 and Figure 10.2.
The final estimates are given in Figure 10.3.
ORDER INNOVATION VARIANCE M V(M) AIC(M) 0 0.31607294 -108.26753229 1 0.11481982 -201.45277331 2 0.04847420 -280.51201122 3 0.04828185 -278.88576251 4 0.04656506 -280.28905616 5 0.04615922 -279.11190502 6 0.04511943 -279.25356641 7 0.04312403 -281.50543541 8 0.04201118 -281.96304075 9 0.04128036 -281.61262868 10 0.03829179 -286.67686828 11 0.03318558 -298.13013264 12 0.03255171 -297.94298716 13 0.03247784 -296.15655602 14 0.03237083 -294.46677874 15 0.03234790 -292.53337704 16 0.03187416 -291.92021487 17 0.03183282 -290.04220196 18 0.03126946 -289.72064823 19 0.03087893 -288.90203735 20 0.02998019 -289.67854830 |
AIC(M)-AICMIN (truncated at 40.0)
0 10 20 30 40
M AIC(M)-AICMIN +---------+---------+---------+---------+
0 189.862600 | .
1 96.677359 | .
2 17.618121 | * |
3 19.244370 | * |
4 17.841076 | * |
5 19.018228 | * |
6 18.876566 | * |
7 16.624697 | * |
8 16.167092 | * |
9 16.517504 | * |
10 11.453264 | * |
11 0 * |
12 0.187145 * |
13 1.973577 | * |
14 3.663354 | * |
15 5.596756 | * |
16 6.209918 | * |
17 8.087931 | * |
18 8.409484 | * |
19 9.228095 | * |
20 8.451584 | * |
+---------+---------+---------+---------+
|
The minimum AIC order is selected as 11. Then the coefficients are estimated as in Figure 10.3. Note that the first 20 observations are used as presample values.
..........................M A I C E......................... . . . . . . . M AR Coefficients: AR(M) . . . . 1 1.181322 . . 2 -0.551571 . . 3 0.231372 . . 4 -0.178040 . . 5 0.019874 . . 6 -0.062573 . . 7 0.028569 . . 8 -0.050710 . . 9 0.199896 . . 10 0.161819 . . 11 -0.339086 . . . . . . AIC = -298.1301326 . . Innovation Variance = 0.033186 . . . . . . INPUT DATA START = 21 FINISH = 114 . ................................................................ |
You can estimate the AR(11) model directly by specifying OPT={-1 0} and using the first 11 observations as presample values. The AR(11) estimates shown in Figure 10.4 are different from the minimum AIC estimates in Figure 10.3 because the samples are slightly different.
call tsunimar(arcoef,ev,nar,aic,y,11,{-1 0},,1);
..........................M A I C E......................... . . . . . . . M AR Coefficients: AR(M) . . . . 1 1.149416 . . 2 -0.533719 . . 3 0.276312 . . 4 -0.326420 . . 5 0.169336 . . 6 -0.164108 . . 7 0.073123 . . 8 -0.030428 . . 9 0.151227 . . 10 0.192808 . . 11 -0.340200 . . . . . . AIC = -318.7984105 . . Innovation Variance = 0.036563 . . . . . . INPUT DATA START = 12 FINISH = 114 . ................................................................ |
The minimum AIC procedure can also be applied to the vector autoregressive (VAR) model using the TSMULMAR subroutine. See the section "Multivariate Time Series Analysis" for details. Three variables are used as input. The maximum lag is specified as 10.
data one;
input invest income consum @@;
datalines;
. . . data lines omitted . . .
;
proc iml;
use one;
read all into y var{invest income consum};
mdel = 1;
maice = 2;
misw = 0; /* instantaneous modeling ? */
opt = mdel || maice || misw;
maxlag = 10;
miss = 0;
print = 1;
call tsmulmar(arcoef,ev,nar,aic,y,maxlag,opt,miss,print);
The VAR(3) model minimizes the AIC and was
selected as an appropriate model (see Figure 10.5).
However, AICs of the VAR(4) and VAR(5)
models show little difference from VAR(3).
You can also choose VAR(4) or VAR(5) as an
appropriate model in the context of minimum AIC
since this AIC difference is much less than 1.
ORDER INNOVATION VARIANCE
M LOG(|V(M)|) AIC(M)
0 25.98001095 2136.36089828
1 15.70406486 1311.73331883
2 15.48896746 1312.09533158
3 15.18567834 1305.22562428
4 14.96865183 1305.42944974
5 14.74838535 1305.36759889
6 14.60269347 1311.42086432
7 14.54981887 1325.08514729
8 14.38596333 1329.64899297
9 14.16383772 1329.43469312
10 13.85377849 1322.00983656
AIC(M)-AICMIN (truncated at 40.0)
0 10 20 30 40
M AIC(M)-AICMIN +---------+---------+---------+---------+
0 831.135274 | .
1 6.507695 | * |
2 6.869707 | * |
3 0 * |
4 0.203825 * |
5 0.141975 * |
6 6.195240 | * |
7 19.859523 | * |
8 24.423369 | * |
9 24.209069 | * |
10 16.784212 | * |
+---------+---------+---------+---------+
|
The TSMULMAR subroutine estimates the instantaneous response model with diagonal error variance. See the section "Multivariate Time Series Analysis" for details on the instantaneous response model. Therefore, it is possible to select the minimum AIC model independently for each equation. The best model is selected by specifying MAXLAG=5.
call tsmulmar(arcoef,ev,nar,aic) data=y maxlag=5
opt={1 1 0} print=1;
----- INNOVATION VARIANCE MATRIX -----
256.643750 29.803549 76.846777
29.803549 228.973407 119.603867
76.846777 119.603867 134.217637
----- AR-COEFFICIENTS -----
LAG VAR = 1 VAR = 2 VAR = 3
1 0.825740 0.251480 0
0.095892 1.005709 0
0.032098 0.354435 0.469893
2 0.044719 -0.201035 0
0.005193 -0.023346 0
0.116986 -0.060196 0.048332
3 0.186783 0 0
0.021691 0 0
-0.117786 0 0.350037
4 0.154111 0 0
0.017897 0 0
0.046145 0 -0.191437
5 -0.389644 0 0
-0.045249 0 0
-0.116671 0 0
AIC = 1347.619775
|
You can print the intermediate results of the minimum AIC procedure using the PRINT=2 option.
Note that the AIC value depends on the MAXLAG=lag option and the number of parameters estimated. The minimum AIC VAR estimation procedure (MAICE=2) uses the following AIC formula:
The following code estimates the instantaneous response model. The results are shown in Figure 10.7.
call tsmulmar(arcoef,ev,nar,aic) data=y maxlag=3
opt={1 0 0};
print aic nar;
print arcoef;
The following code estimates the VAR model. The results are shown in Figure 10.8.
call tsmulmar(arcoef,ev,nar,aic) data=y maxlag=3
opt={1 2 0};
print aic nar;
print arcoef;
| |||||||||||||||||||||||||||||||||||||
The AIC computed from the instantaneous response model is greater than that obtained from the VAR model estimation by 6. There is a discrepancy between Figure 10.8 and Figure 10.5 because different observations are used for estimation.
First, the locally stationary model is estimated. The whole series (1000 observations) is divided into three blocks of size 300 and one block of size 90, and the minimum AIC procedure is applied to each block of the data set. See the "Nonstationary Time Series" section for more details.
data one;
input y @@;
datalines;
. . . data lines omitted . . .
;
proc iml;
use one;
read all var{y};
mdel = -1;
lspan = 300; /* local span of data */
maice = 1;
opt = mdel || lspan || maice;
call tsmlocar(arcoef,ev,nar,aic,first,last)
data=y maxlag=10 opt=opt print=2;
Estimation results are displayed with the graphs
of power spectrum (log10(fYY(g))), where
fYY(g) is a rational spectral density function.
See the "Spectral Analysis" section.
As the first block and the second block do not have any
sizable difference, the pooled model (AIC=45.892) is selected
instead of the moving model (AIC=46.957) in Figure 10.10.
However, you can notice a slight change in
the shape of the spectrum of the third block
of the data (observations 611 through 910).
See Figure 10.11 and Figure 10.13 for comparison.
The moving model is selected since the AIC (106.830) of the
moving model is smaller than that of the pooled model (108.867).
INITIAL LOCAL MODEL: N_CURR = 300
NAR_CURR = 8
AIC = 37.583203
..........................CURRENT MODEL.........................
. .
. .
. .
. M AR Coefficients: AR(M) .
. .
. 1 1.605717 .
. 2 -1.245350 .
. 3 1.014847 .
. 4 -0.931554 .
. 5 0.394230 .
. 6 -0.004344 .
. 7 0.111608 .
. 8 -0.124992 .
. .
. .
. AIC = 37.5832030 .
. Innovation Variance = 1.067455 .
. .
. .
. INPUT DATA START = 11 FINISH = 310 .
................................................................
|
--- THE FOLLOWING TWO MODELS ARE COMPARED ---
MOVING MODEL: (N_PREV = 300, N_CURR = 300)
NAR_CURR = 7
AIC = 46.957398
CONSTANT MODEL: N_POOLED = 600
NAR_POOLED = 8
AIC = 45.892350
***** CONSTANT MODEL ADOPTED *****
..........................CURRENT MODEL.........................
. .
. .
. .
. M AR Coefficients: AR(M) .
. .
. 1 1.593890 .
. 2 -1.262379 .
. 3 1.013733 .
. 4 -0.926052 .
. 5 0.314480 .
. 6 0.193973 .
. 7 -0.058043 .
. 8 -0.078508 .
. .
. .
. AIC = 45.8923501 .
. Innovation Variance = 1.047585 .
. .
. .
. INPUT DATA START = 11 FINISH = 610 .
................................................................
|
POWER SPECTRAL DENSITY
20.00+
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-20.0+-----------+-----------+-----------+-----------+-----------+
0.0 0.1 0.2 0.3 0.4 0.5
FREQUENCY
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--- THE FOLLOWING TWO MODELS ARE COMPARED ---
MOVING MODEL: (N_PREV = 600, N_CURR = 300)
NAR_CURR = 7
AIC = 106.829869
CONSTANT MODEL: N_POOLED = 900
NAR_POOLED = 8
AIC = 108.867091
*************************************
***** *****
***** NEW MODEL ADOPTED *****
***** *****
*************************************
..........................CURRENT MODEL.........................
. .
. .
. .
. M AR Coefficients: AR(M) .
. .
. 1 1.648544 .
. 2 -1.201812 .
. 3 0.674933 .
. 4 -0.567576 .
. 5 -0.018924 .
. 6 0.516627 .
. 7 -0.283410 .
. .
. .
. AIC = 60.9375188 .
. Innovation Variance = 1.161592 .
. .
. .
. INPUT DATA START = 611 FINISH = 910 .
................................................................
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POWER SPECTRAL DENSITY
20.00+ X
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-20.0+-----------+-----------+-----------+-----------+-----------+
0.0 0.1 0.2 0.3 0.4 0.5
FREQUENCY
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Finally, the moving model is selected since there is a structural change in the last block of data (observations 911 through 1000). The final estimates are stored in variables ARCOEF, EV, NAR, AIC, FIRST, and LAST. The final estimates and spectrum are given in Figure 10.14 and Figure 10.15, respectively. The power spectrum of the final model (Figure 10.15) is significantly different from that of the first and second blocks (see Figure 10.11).
--- THE FOLLOWING TWO MODELS ARE COMPARED ---
MOVING MODEL: (N_PREV = 300, N_CURR = 90)
NAR_CURR = 6
AIC = 139.579012
CONSTANT MODEL: N_POOLED = 390
NAR_POOLED = 9
AIC = 167.783711
*************************************
***** *****
***** NEW MODEL ADOPTED *****
***** *****
*************************************
..........................CURRENT MODEL.........................
. .
. .
. .
. M AR Coefficients: AR(M) .
. .
. 1 1.181022 .
. 2 -0.321178 .
. 3 -0.113001 .
. 4 -0.137846 .
. 5 -0.141799 .
. 6 0.260728 .
. .
. .
. AIC = 78.6414932 .
. Innovation Variance = 2.050818 .
. .
. .
. INPUT DATA START = 911 FINISH = 1000 .
................................................................
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POWER SPECTRAL DENSITY
30.00+
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-10.0+-----------+-----------+-----------+-----------+-----------+
0.0 0.1 0.2 0.3 0.4 0.5
FREQUENCY
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The multivariate analysis for locally stationary data is a straightforward extension of the univariate analysis. The bivariate locally stationary VAR models are estimated. The selected model is the VAR(7) process with some zero coefficients over the last block of data. There seems to be a structural difference between observations from 11 to 610 and those from 611 to 896.
proc iml;
rudder = {. . . data lines omitted . . .};
yawing = {. . . data lines omitted . . .};
y = rudder` || yawing`;
c = {0.01795 0.02419};
/*-- calibration of data --*/
y = y # (c @ j(n,1,1));
mdel = -1;
lspan = 300; /* local span of data */
maice = 1;
call tsmlomar(arcoef,ev,nar,aic,first,last) data=y maxlag=10
opt = (mdel || lspan || maice) print=1;
--- THE FOLLOWING TWO MODELS ARE COMPARED ---
MOVING MODEL: (N_PREV = 600, N_CURR = 286)
NAR_CURR = 7
AIC = -823.845234
CONSTANT MODEL: N_POOLED = 886
NAR_POOLED = 10
AIC = -716.818588
*************************************
***** *****
***** NEW MODEL ADOPTED *****
***** *****
*************************************
..........................CURRENT MODEL.........................
. .
. .
. .
. M AR Coefficients .
. .
. 1 0.932904 -0.130964 .
. -0.024401 0.599483 .
. 2 0.163141 0.266876 .
. -0.135605 0.377923 .
. 3 -0.322283 0.178194 .
. 0.188603 -0.081245 .
. 4 0.166094 -0.304755 .
. -0.084626 -0.180638 .
. 5 0 0 .
. 0 -0.036958 .
. 6 0 0 .
. 0 0.034578 .
. 7 0 0 .
. 0 0.268414 .
. .
. .
. AIC = -114.6911872 .
. .
. Innovation Variance .
. .
. 1.069929 0.145558 .
. 0.145558 0.563985 .
. .
. .
. INPUT DATA START = 611 FINISH = 896 .
................................................................
|
Consider the time series decomposition
![H& = & [ 1 & 0 & 1 & 0
] \
F& = & [ 2 & -1 & 0 & 0 \ 1 & 0 & 0 & 0 \ 0 & 0 & ...
..._1^2 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & \sigma_2^2 & 0 \ 0 & 0 & 0 & 0
]
)](images/i10eq10.gif)
, 
, 
, and 
.
proc iml;
y = { 116.8 120.1 123.2 130.2 131.4 125.6 124.5 134.3
135.2 151.8 146.4 139.0 127.8 147.0 165.9 165.5
179.4 190.0 189.8 190.9 203.6 183.5 169.3 144.2
141.5 154.3 169.5 193.0 203.2 192.9 209.4 227.2
263.7 297.8 337.1 361.3 355.2 312.6 309.9 323.7
324.1 355.3 383.4 395.1 412.8 406.0 438.0 446.1
452.5 447.3 475.9 487.7 497.2 529.8 551.0 581.1
617.8 658.1 675.2 706.6 724.7 };
y = y`; /*-- convert to column vector --*/
mdel = 0;
trade = 0;
tvreg = 0;
year = 0;
period= 0;
log = 0;
maxit = 100;
update = .; /* use default update method */
line = .; /* use default line search method */
sigmax = 0; /* no upper bound for variances */
back = 100;
opt = mdel || trade || year || period || log || maxit ||
update || line || sigmax || back;
call tsdecomp(cmp,coef,aic) data=y order=2 sorder=0 nar=2
npred=5 opt=opt icmp={1 3} print=1;
The estimated parameters are printed
when you specify the PRINT= option.
In Figure 10.17, the estimated variances are
printed under the title of TAU2(I), showing that
and 
.
The AR coefficient estimates are
and 
.
These estimates are also stored in the output matrix COEF.
<<< Final Estimates >>>
--- PARAMETER VECTOR ---
1.607426E-01 6.283837E+00 8.761628E-01 -5.94879E-01
--- GRADIENT ---
3.385021E-04 5.760929E-06 3.029534E-04 -1.18396E-04
LIKELIHOOD = -249.937193 SIG2 = 18.135052
AIC = 509.874385
I TAU2(I) AR(I) PARCOR(I)
1 2.915075 1.397374 0.876163
2 113.957712 -0.594879 -0.594879
|
The trend and stationary AR components are estimated using the smoothing method, and out-of-sample forecasts are computed using a Kalman filter prediction algorithm. The trend and AR components are stored in the matrix CMP since the ICMP={1 3} option is specified. The last 10 observations of the original series Y and the last 15 observations of two components are shown in Figure 10.18. Note that the first column of CMP is the trend component and the second column is the AR component. The last 5 observations of the CMP matrix are out-of-sample forecasts.
|
|
.
The details of the adjustment procedure are
given in the section "Bayesian Seasonal Adjustment".
The monthly labor force series (1972-1978) are analyzed. You do not need to specify the options vector if you want to use the default options. However, you should change OPT[2] when the data frequency is not monthly (OPT[2]=12). The NPRED= option produces the multistep forecasts for the trend and seasonal components. The stochastic constraints are specified as ORDER=2 and SORDER=1.
is shown in Figure 10.21.
proc iml;
y =
{ 5447 5412 5215 4697 4344 5426
5173 4857 4658 4470 4268 4116
4675 4845 4512 4174 3799 4847
4550 4208 4165 3763 4056 4058
5008 5140 4755 4301 4144 5380
5260 4885 5202 5044 5685 6106
8180 8309 8359 7820 7623 8569
8209 7696 7522 7244 7231 7195
8174 8033 7525 6890 6304 7655
7577 7322 7026 6833 7095 7022
7848 8109 7556 6568 6151 7453
6941 6757 6437 6221 6346 5880 };
y = y`;
call tsbaysea(trend,season,series,adj,abic)
data=y order=2 sorder=1 npred=12 print=2;
print trend season series adj abic;
Seasonal Component
576.866752 612.796066 324.020037 -198.760111
-572.556158 493.248873 218.901469 -126.976886
-223.927593 -440.622170 -345.477541 -339.527540
567.417780 649.108143 315.457702 -195.764740
-567.242588 503.917031 226.829019 -142.216380
-209.010499 -511.275202 -344.187789 -365.761124
647.626707 686.576003 324.601881 -242.421270
-582.439797 516.512576 248.795247 -160.227108
-212.583209 -538.237178 -364.306967 -416.965872
749.318446 705.520212 361.245687 -273.971547
-617.748290 506.336574 239.146930 -132.685481
-254.706508 -510.461942 -348.035057 -391.992877
711.125340 748.595903 367.983922 -290.532690
-700.824658 519.764643 242.638512 -73.786428
-288.809493 -509.321443 -302.485088 -397.322723
650.134120 800.460271 395.841362 -340.552541
-719.314201 553.049123 201.955997 -54.527951
-295.332122 -487.701411 -266.216231 -440.347213
650.770701 800.937334 396.198661 -340.285229
-719.114602 553.197644 202.065816 -54.447682
-295.274714 -487.662081 -266.191701 -440.335439
*** Last 12 Values Are Forecasted ***
|
Adjusted = Data - Seasonal - Trading_Day_Comp - OCF 4870.133248 4799.203934 4890.979963 4895.760111 4916.556158 4932.751127 4954.098531 4983.976886 4881.927593 4910.622170 4613.477541 4455.527540 4107.582220 4195.891857 4196.542298 4369.764740 4366.242588 4343.082969 4323.170981 4350.216380 4374.010499 4274.275202 4400.187789 4423.761124 4360.373293 4453.423997 4430.398119 4543.421270 4726.439797 4863.487424 5011.204753 5045.227108 5414.583209 5582.237178 6049.306967 6522.965872 7430.681554 7603.479788 7997.754313 8093.971547 8240.748290 8062.663426 7969.853070 7828.685481 7776.706508 7754.461942 7579.035057 7586.992877 7462.874660 7284.404097 7157.016078 7180.532690 7004.824658 7135.235357 7334.361488 7395.786428 7314.809493 7342.321443 7397.485088 7419.322723 7197.865880 7308.539729 7160.158638 6908.552541 6870.314201 6899.950877 6739.044003 6811.527951 6732.332122 6708.701411 6612.216231 6320.347213 |
I Rational 0.0 10.0 20.0 30.0 40.0 50.0 60.0
Spectrum +---------+---------+---------+---------+---------+---------+
0 1.366798E+00 |* ===>X
1 1.571261E+00 |*
2 2.414836E+00 | *
3 5.151906E+00 | *
4 1.634887E+01 | *
5 8.085674E+01 | *
6 3.805530E+02 | *
7 8.082536E+02 | *
8 6.366350E+02 | *
9 3.479435E+02 | *
10 3.872650E+02 | * ===>X
11 1.264805E+03 | *
12 1.726138E+04 | *
13 1.559041E+03 | *
14 1.276516E+03 | *
15 3.861089E+03 | *
16 9.593184E+03 | *
17 3.662145E+03 | *
18 5.499783E+03 | *
19 4.443303E+03 | *
20 1.238135E+03 | * ===>X
21 8.392131E+02 | *
22 1.258933E+03 | *
23 2.932003E+03 | *
24 1.857923E+03 | *
25 1.171437E+03 | *
26 1.611958E+03 | *
27 4.822498E+03 | *
28 4.464961E+03 | *
29 1.951547E+03 | *
30 1.653182E+03 | * ===>X
31 2.308152E+03 | *
32 5.475758E+03 | *
33 2.349584E+04 | *
34 5.266969E+03 | *
35 2.058667E+03 | *
36 2.215595E+03 | *
37 8.181540E+03 | *
38 3.077329E+03 | *
39 7.577961E+02 | *
40 5.057636E+02 | * ===>X
41 7.312090E+02 | *
42 3.131377E+03 | * ===>T
43 8.173276E+03 | *
44 1.958359E+03 | *
45 2.216458E+03 | *
46 4.215465E+03 | *
47 9.659340E+02 | *
48 3.758466E+02 | *
49 2.849326E+02 | *
50 3.617848E+02 | * ===>X
51 7.659839E+02 | *
52 3.191969E+03 | *
53 1.768107E+04 | *
54 5.281385E+03 | *
55 2.959704E+03 | *
56 3.783522E+03 | *
57 1.896625E+04 | *
58 1.041753E+04 | *
59 2.038940E+03 | *
60 1.347568E+03 | * ===>X
X: If peaks (troughs) appear at these frequencies, try lower (higher) values
of rigid and watch ABIC
T: If a peaks appears here try trading-day adjustment
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The VARMA(p,q) model is written
proc iml;
c = { 264 235 239 239 275 277 274 334 334 306
308 309 295 271 277 221 223 227 215 223
241 250 270 303 311 307 322 335 335 334
309 262 228 191 188 215 215 249 291 296 };
f = { 690 690 688 690 694 702 702 702 700 702
702 694 708 702 702 708 700 700 702 694
698 694 700 702 700 702 708 708 710 704
704 700 700 694 702 694 710 710 710 708 };
t = { 1152 1288 1288 1288 1368 1456 1656 1496 1744 1464
1560 1376 1336 1336 1296 1296 1280 1264 1280 1272
1344 1328 1352 1480 1472 1600 1512 1456 1368 1280
1224 1112 1112 1048 1176 1064 1168 1280 1336 1248 };
p = { 254.14 253.12 251.85 250.41 249.09 249.19 249.52 250.19
248.74 248.41 249.95 250.64 250.87 250.94 250.96 251.33
251.18 251.05 251.00 250.99 250.79 250.44 250.12 250.19
249.77 250.27 250.74 250.90 252.21 253.68 254.47 254.80
254.92 254.96 254.96 254.96 254.96 254.54 253.21 252.08 };
y = c` || f` || t` || p`;
ar = { .82028 -.97167 .079386 -5.4382,
-.39983 .94448 .027938 -1.7477,
-.42278 -2.3314 1.4682 -70.996,
.031038 -.019231 -.0004904 1.3677,
-.029811 .89262 -.047579 4.7873,
.31476 .0061959 -.012221 1.4921,
.3813 2.7182 -.52993 67.711,
-.020818 .01764 .00037981 -.38154 };
ma = { .083035 -1.0509 .055898 -3.9778,
-.40452 .36876 .026369 -.81146,
.062379 -2.6506 .80784 -76.952,
.03273 -.031555 -.00019776 -.025205 };
coef = ar // ma;
ev = { 188.55 6.8082 42.385 .042942,
6.8082 32.169 37.995 -.062341,
42.385 37.995 5138.8 -.10757,
.042942 -.062341 -.10757 .34313 };
nar = 2; nma = 1;
call tspred(forecast,impulse,mse,y,coef,nar,nma,ev,
5,nrow(y),-1);
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The first 40 forecasts are one-step predictions. The last observation is the five-step forecast values of variables C and F. You can construct the confidence interval for these forecasts using the mean square error matrix, MSE. See the "Multivariate Time Series Analysis" section for more details on impulse response functions and the mean square error matrix.
The TSROOT call computes the polynomial roots of the AR and MA equations. When the AR(p) process is written
are the MA coefficients.
For example, the best AR model is selected and estimated by
the TSUNIMAR subroutine (see Figure 10.23).
You can obtain the roots of the preceding
equation by calling the TSROOT call.
Since the TSROOT call can handle the complex AR
or MA coefficients, note that you should add zero
imaginary coefficients for the second column
of the MATIN matrix for real coefficients.
proc iml;
y = { 2.430 2.506 2.767 2.940 3.169 3.450 3.594 3.774 3.695 3.411
2.718 1.991 2.265 2.446 2.612 3.359 3.429 3.533 3.261 2.612
2.179 1.653 1.832 2.328 2.737 3.014 3.328 3.404 2.981 2.557
2.576 2.352 2.556 2.864 3.214 3.435 3.458 3.326 2.835 2.476
2.373 2.389 2.742 3.210 3.520 3.828 3.628 2.837 2.406 2.675
2.554 2.894 3.202 3.224 3.352 3.154 2.878 2.476 2.303 2.360
2.671 2.867 3.310 3.449 3.646 3.400 2.590 1.863 1.581 1.690
1.771 2.274 2.576 3.111 3.605 3.543 2.769 2.021 2.185 2.588
2.880 3.115 3.540 3.845 3.800 3.579 3.264 2.538 2.582 2.907
3.142 3.433 3.580 3.490 3.475 3.579 2.829 1.909 1.903 2.033
2.360 2.601 3.054 3.386 3.553 3.468 3.187 2.723 2.686 2.821
3.000 3.201 3.424 3.531 };
call tsunimar(ar,v,nar,aic) data=y maxlag=5
opt=({-1 1}) print=1;
/*-- set up complex coefficient matrix --*/
ar_cx = ar || j(nrow(ar),1,0);
call tsroot(root) matin=ar_cx nar=nar
nma=0 print=1;
In Figure 10.24, the roots and their
lengths from the origin are shown.
The roots are also stored in the matrix ROOT.
All roots are within the unit circle, while the mod values
of the fourth and fifth roots appear to be sizable (0.9194).
..........................M A I C E......................... . . . . . . . M AR Coefficients: AR(M) . . . . 1 1.300307 . . 2 -0.723280 . . 3 0.242193 . . 4 -0.378757 . . 5 0.137727 . . . . . . AIC = -318.6137704 . . Innovation Variance = 0.049055 . . . . . . INPUT DATA START = 6 FINISH = 114 . ................................................................ |
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The TSROOT call can also recover the polynomial coefficients if the roots are given as an input. You should specify the QCOEF=1 option when you want to compute the polynomial coefficients instead of polynomial roots. You can compare the result with the preceding output of the TSUNIMAR call.
call tsroot(ar_cx) matin=root nar=nar qcoef=1
nma=0 print=1;
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