Example 1: ARAutoUnivariate

Consider the Wolfer Sunspot Data (Anderson 1971, p. 660) consisting of the number of sunspots observed each year from 1749 through 1924. The data set for this example consists of the number of sunspots observed from 1770 through 1869. In this example, ARAutoUnivariate found the minimum AIC fit is an autoregressive model with 3 lags:

\tilde{W_t}=\phi_1\tilde{W_{t-1}}+ \phi_2\tilde{W_{t-2}}+\phi_3\tilde{W_{t-3}} + a_t\rm{,}
where
\tilde{W_t} := W_t - \mu
\mu is the sample mean of the time series \{W_t\}. Defining the overall constant \theta_0 by \theta_0 := \mu(1- \sum_{i=1}^3\phi_i), we obtain the following equivalent representation:
W_t=\theta_0+\phi_1W_{t-1}+\phi_2W_{t-2}+\phi_3W_{t-3} + a_t\rm{.}
The example computes estimates for \theta_0\rm{,}\,\phi_1\rm{,}\,\phi_2\rm{,}\,\phi_3 for each of the three parameter estimation methods available.

using System;
using Imsl.Stat;
using PrintMatrix = Imsl.Math.PrintMatrix;

public class ARAutoUnivariateEx1
{
    public static void  Main(String[] args)
    {
        double[] z = new double[]{100.8, 81.6, 66.5, 34.8, 30.6, 
                                     7, 19.8, 92.5, 154.4, 125.9, 
                                     84.8, 68.1, 38.5, 22.8, 10.2, 
                                     24.1, 82.9, 132, 130.9, 118.1, 
                                     89.9, 66.6, 60, 46.9, 41,  
                                     21.3, 16, 6.4, 4.1, 6.8, 
                                     14.5, 34, 45, 43.1, 47.5, 
                                     42.2, 28.1, 10.1, 8.1, 2.5, 
                                     0, 1.4, 5, 12.2, 13.9,  
                                     35.4, 45.8, 41.1, 30.4, 23.9, 
                                     15.7, 6.6, 4, 1.8, 8.5, 
                                     16.6, 36.3, 49.7, 62.5, 67, 
                                     71, 47.8, 27.5, 8.5, 13.2, 
                                     56.9, 121.5, 138.3, 103.2, 85.8, 
                                     63.2, 36.8, 24.2, 10.7, 15, 
                                     40.1, 61.5, 98.5, 124.3, 95.9,  
                                     66.5, 64.5, 54.2, 39, 20.6,  
                                     6.7, 4.3, 22.8, 54.8, 93.8, 
                                     95.7, 77.2, 59.1, 44, 47, 
                                     30.5, 16.3, 7.3, 37.3, 73.9};
        
        ARAutoUnivariate autoAR = new ARAutoUnivariate(20, z);
        Console.Out.WriteLine("");
        Console.Out.WriteLine("    Method of Moments ");
        Console.Out.WriteLine("");
        
        autoAR.EstimationMethod = 
            Imsl.Stat.ARAutoUnivariate.ParamEstimation.MethodOfMoments;
        autoAR.Compute();
        
        Console.Out.WriteLine("Order Selected: " + autoAR.Order);
        Console.Out.WriteLine("AIC = " + autoAR.AIC 
            + "       Variance = " + autoAR.InnovationVariance);
        Console.Out.WriteLine("Constant = " + autoAR.Constant);
        Console.Out.WriteLine("");
        new PrintMatrix("AR estimates are:  ").Print(autoAR.GetAR());
        
        
        /* try another method */
        Console.Out.WriteLine("");
        Console.Out.WriteLine("");
        Console.Out.WriteLine("    Least Squares ");
        Console.Out.WriteLine("");
        autoAR.EstimationMethod = 
            Imsl.Stat.ARAutoUnivariate.ParamEstimation.LeastSquares;
        autoAR.Compute();
        
        Console.Out.WriteLine("Order Selected: " + autoAR.Order);
        Console.Out.WriteLine("AIC = " + autoAR.AIC 
            + "       Variance = " + autoAR.InnovationVariance);
        Console.Out.WriteLine("Constant = " + autoAR.Constant);
        Console.Out.WriteLine();
        new PrintMatrix("AR estimates are:  ").Print(autoAR.GetAR());
        
        Console.Out.WriteLine("");
        Console.Out.WriteLine("");
        Console.Out.WriteLine("    Maximum Likelihood ");
        Console.Out.WriteLine("");
        autoAR.EstimationMethod = 
            Imsl.Stat.ARAutoUnivariate.ParamEstimation.MaximumLikelihood;
        autoAR.Compute();
        
        Console.Out.WriteLine("Order Selected: " + autoAR.Order);
        Console.Out.WriteLine("AIC = " + autoAR.AIC 
            + "       Variance = " + autoAR.InnovationVariance);
        Console.Out.WriteLine("Constant = " + autoAR.Constant);
        Console.Out.WriteLine();
        new PrintMatrix("AR estimates are:  ").Print(autoAR.GetAR());
    }
}

Output


    Method of Moments 

Order Selected: 3
AIC = 405.981241965022       Variance = 287.269907744347
Constant = 13.7097894010953

  AR estimates are:  
            0           
0   1.36757589345393    
1  -0.737673445646019   
2   0.0782508772709516  



    Least Squares 

Order Selected: 3
AIC = 405.981241965022       Variance = 221.995196605174
Constant = 11.6104946249815

  AR estimates are:  
           0           
0   1.55022677857954   
1  -1.00394753883711   
2   0.205447994348897  



    Maximum Likelihood 

Order Selected: 3
AIC = 405.981241965022       Variance = 218.848375884726
Constant = 11.4178509825012

  AR estimates are:  
           0           
0   1.55139136379628   
1  -0.998935931377056  
2   0.204487453682164
Link to C# source.