Using the Canadian Lynx data included in TIMSAC-78, ARAutoUnivariate
is used to find the minimum AIC autoregressive model using a maximum number of lags of maxlag
=20.
This example compares the three different methods for estimating the autoregressive coefficients, and it illustrates the relationship between these estimates and those calculated within the TIMSAC UNIMAR code. As illustrated, the UNIMAR code estimates the coefficients and innovation variance using only the last N-maxlag values in the time series. The other estimation methods use all N-k
values, where k
is the number of lags with minimum AIC selected by ARAutoUnivariate
.
This example also illustrates how to generate forecasts for the observed series values and beyond by setting the backward orgin for the forecasts.
using System;
using Imsl.Stat;
using Imsl.Math;
using PrintMatrix = Imsl.Math.PrintMatrix;
public class ARAutoUnivariateEx2
{
public static void Main(String[] args)
{
/* THE CANDIAN LYNX DATA AS USED IN TIMSAC 1821-1934 */
double[] y = new double[]{
0.24300e01, 0.25060e01, 0.27670e01, 0.29400e01, 0.31690e01,
0.34500e01, 0.35940e01, 0.37740e01, 0.36950e01, 0.34110e01,
0.27180e01, 0.19910e01, 0.22650e01, 0.24460e01, 0.26120e01,
0.33590e01, 0.34290e01, 0.35330e01, 0.32610e01, 0.26120e01,
0.21790e01, 0.16530e01, 0.18320e01, 0.23280e01, 0.27370e01,
0.30140e01, 0.33280e01, 0.34040e01, 0.29810e01, 0.25570e01,
0.25760e01, 0.23520e01, 0.25560e01, 0.28640e01, 0.32140e01,
0.34350e01, 0.34580e01, 0.33260e01, 0.28350e01, 0.24760e01,
0.23730e01, 0.23890e01, 0.27420e01, 0.32100e01, 0.35200e01,
0.38280e01, 0.36280e01, 0.28370e01, 0.24060e01, 0.26750e01,
0.25540e01, 0.28940e01, 0.32020e01, 0.32240e01, 0.33520e01,
0.31540e01, 0.28780e01, 0.24760e01, 0.23030e01, 0.23600e01,
0.26710e01, 0.28670e01, 0.33100e01, 0.34490e01, 0.36460e01,
0.34000e01, 0.25900e01, 0.18630e01, 0.15810e01, 0.16900e01,
0.17710e01, 0.22740e01, 0.25760e01, 0.31110e01, 0.36050e01,
0.35430e01, 0.27690e01, 0.20210e01, 0.21850e01, 0.25880e01,
0.28800e01, 0.31150e01, 0.35400e01, 0.38450e01, 0.38000e01,
0.35790e01, 0.32640e01, 0.25380e01, 0.25820e01, 0.29070e01,
0.31420e01, 0.34330e01, 0.35800e01, 0.34900e01, 0.34750e01,
0.35790e01, 0.28290e01, 0.19090e01, 0.19030e01, 0.20330e01,
0.23600e01, 0.26010e01, 0.30540e01, 0.33860e01, 0.35530e01,
0.34680e01, 0.31870e01, 0.27230e01, 0.26860e01, 0.28210e01,
0.30000e01, 0.32010e01, 0.34240e01, 0.35310e01};
double[][] printOutput = null;
double[] timsacAR, mmAR, mleAR, lsAR;
double[] forecasts, residuals;
double timsacConstant, mmConstant, mleConstant, lsConstant;
double timsacVar, timsacEquivalentVar, mmVar, mleVar, lsVar;
int maxlag = 20;
String[] colLabels = new String[]{"TIMSAC",
"Method of Moments",
"Least Squares",
"Maximum Likelihood"};
String[] colLabels2 = new String[]{"Observed", "Forecast", "Residual"};
PrintMatrixFormat pmf = new PrintMatrixFormat();
PrintMatrix pm = new PrintMatrix();
pmf.SetColumnLabels(colLabels);
pmf.NumberFormat = "0.0000";
Console.Out.WriteLine
("Automatic Selection of Minimum AIC AR Model");
Console.Out.WriteLine("");
ARAutoUnivariate autoAR = new ARAutoUnivariate(maxlag, y);
autoAR.Compute();
int orderSelected = autoAR.Order;
Console.Out.WriteLine("Minimum AIC Selected=" + autoAR.AIC
+ " with an optimum lag of k= " + autoAR.Order);
Console.Out.WriteLine("");
timsacAR = autoAR.GetTimsacAR();
timsacConstant = autoAR.TimsacConstant;
timsacVar = autoAR.TimsacVariance;
lsAR = autoAR.GetAR();
lsConstant = autoAR.Constant;
lsVar = autoAR.InnovationVariance;
autoAR.EstimationMethod =
Imsl.Stat.ARAutoUnivariate.ParamEstimation.MethodOfMoments;
autoAR.Compute();
mmAR = autoAR.GetAR();
mmConstant = autoAR.Constant;
mmVar = autoAR.InnovationVariance;
autoAR.EstimationMethod =
Imsl.Stat.ARAutoUnivariate.ParamEstimation.MaximumLikelihood;
autoAR.Compute();
mleAR = autoAR.GetAR();
mleConstant = autoAR.Constant;
mleVar = autoAR.InnovationVariance;
printOutput = new double[orderSelected + 1][];
for (int i = 0; i < orderSelected + 1; i++)
{
printOutput[i] = new double[4];
}
printOutput[0][0] = timsacConstant;
for (int i = 0; i < orderSelected; i++)
printOutput[i + 1][0] = timsacAR[i];
printOutput[0][1] = mmConstant;
for (int i = 0; i < orderSelected; i++)
printOutput[i + 1][1] = mmAR[i];
printOutput[0][2] = lsConstant;
for (int i = 0; i < orderSelected; i++)
printOutput[i + 1][2] = lsAR[i];
printOutput[0][3] = mleConstant;
for (int i = 0; i < orderSelected; i++)
printOutput[i + 1][3] = mleAR[i];
pm.SetTitle("Comparison of AR Estimates");
pm.Print(pmf, printOutput);
/* calculation of equivalent innovation variance using TIMSAC
coefficients. The Timsac innovation variance is calculated using
only N-maxlag observations in the series. The following code
calculates the innovation variance using N-k observations in the
series with the Timsac coefficient. This illustrates that the
least squares Timsac coefficients will not have the least value for
the sum of squared residuals, which is calculated using all N-k
observations. */
ARMA armaLS = new ARMA(orderSelected, 0, y);
armaLS.SetARMAInfo(autoAR.TimsacConstant, autoAR.GetTimsacAR(),
new double[0], autoAR.TimsacVariance);
armaLS.BackwardOrigin = y.Length - orderSelected;
forecasts = armaLS.GetForecast(1);
double sumResiduals = 0.0;
for (int i = 0; i < y.Length - orderSelected; i++)
{
sumResiduals += (y[i + orderSelected] - forecasts[i]) *
(y[i + orderSelected] - forecasts[i]);
}
timsacEquivalentVar = sumResiduals / (y.Length - orderSelected - 1);
printOutput = new double[1][];
for (int i2 = 0; i2 < 1; i2++)
{
printOutput[i2] = new double[4];
}
printOutput[0][0] = timsacEquivalentVar;
/* the method of moments variance */
printOutput[0][1] = mmVar;
/* the least squares variance */
printOutput[0][2] = lsVar;
/* the maximum likelihood estimate of the variance */
printOutput[0][3] = mleVar;
pm.SetTitle("Comparison of Equivalent Innovation Variances");
pm.Print(pmf, printOutput);
/* FORECASTING - An example of forecasting using the maximum
* likelihood estimates for the minimum AIC AR model. In this example,
* forecasts are returned for the last 10 values in the series followed
* by the forecasts for the next 5 values.
*/
autoAR.BackwardOrigin = 10;
forecasts = autoAR.GetForecast(15);
residuals = autoAR.GetResiduals();
printOutput = new double[15][];
for (int i3 = 0; i3 < 15; i3++)
{
printOutput[i3] = new double[3];
}
for (int i = 0; i < 10; i++)
{
printOutput[i][0] = y[y.Length - 10 + i];
printOutput[i][1] = forecasts[i];
printOutput[i][2] = residuals[i];
}
for (int i = 10; i < 15; i++)
{
printOutput[i][0] = Double.NaN;
printOutput[i][1] = forecasts[i];
printOutput[i][2] = Double.NaN;
}
pmf.FirstRowNumber = 105;
pmf.SetColumnLabels(colLabels2);
pm.SetTitle("Maximum Likelihood Forecasts of Last 10 Values");
pm.Print(pmf, printOutput);
}
}
Automatic Selection of Minimum AIC AR Model
Minimum AIC Selected=-296.130132635624 with an optimum lag of k= 11
Comparison of AR Estimates
TIMSAC Method of Moments Least Squares Maximum Likelihood
0 1.0427 1.1679 1.1144 1.1189
1 1.1813 1.1381 1.1481 1.1664
2 -0.5516 -0.5061 -0.5331 -0.5419
3 0.2314 0.2098 0.2757 0.2624
4 -0.1780 -0.2672 -0.3263 -0.3052
5 0.0199 0.1112 0.1685 0.1519
6 -0.0626 -0.1246 -0.1643 -0.1460
7 0.0286 0.0693 0.0728 0.0581
8 -0.0507 -0.0419 -0.0305 -0.0310
9 0.1999 0.1366 0.1509 0.1380
10 0.1618 0.1828 0.1935 0.1995
11 -0.3391 -0.3101 -0.3414 -0.3376
Comparison of Equivalent Innovation Variances
TIMSAC Method of Moments Least Squares Maximum Likelihood
0 0.0377 0.0427 0.0369 0.0362
Maximum Likelihood Forecasts of Last 10 Values
Observed Forecast Residual
105 3.5530 3.4388 0.1142
106 3.4680 3.4801 -0.0121
107 3.1870 2.9243 0.2627
108 2.7230 2.7026 0.0204
109 2.6860 2.5558 0.1302
110 2.8210 2.7852 0.0358
111 3.0000 2.9493 0.0507
112 3.2010 3.1861 0.0149
113 3.4240 3.3855 0.0385
114 3.5310 3.5272 0.0038
115 NaN 3.4465 NaN
116 NaN 3.1947 NaN
117 NaN 2.8289 NaN
118 NaN 2.4917 NaN
119 NaN 2.4142 NaN
Link to C# source.