### Example 2 (Logging and Forecasting): ARAutoUnivariate

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.

```
import java.text.*;
import com.imsl.stat.*;
import com.imsl.math.*;
import java.util.logging.*;

public class ARAutoUnivariateEx2 {

public static void main(String args[]) throws Exception {
/* THE CANDIAN LYNX DATA AS USED IN TIMSAC 1821-1934 */
double[] y = {
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;
double timsacAR[], mmAR[], mleAR[], lsAR[];
double forecasts[], residuals[];
double timsacConstant, mmConstant, mleConstant, lsConstant;
double timsacVar, timsacEquivalentVar, mmVar, mleVar, lsVar;
int maxlag = 20;
String[] colLabels = {"TIMSAC", "Method of Moments", "Least Squares",
"Maximum Likelihood"};
String[] colLabels2 = {"Observed", "Forecast", "Residual"};
PrintMatrixFormat pmf = new PrintMatrixFormat();
PrintMatrix pm = new PrintMatrix();
NumberFormat nf = NumberFormat.getNumberInstance();
pm.setColumnSpacing(4);
nf.setMinimumFractionDigits(4);
pmf.setNumberFormat(nf);
pmf.setColumnLabels(colLabels);
System.out.println("Automatic Selection of Minimum AIC AR Model");
System.out.println("");

ARAutoUnivariate autoAR = new ARAutoUnivariate(maxlag, y);

// logging to console
Logger logger = autoAR.getLogger();
ConsoleHandler ch = new ConsoleHandler();
ch.setLevel(Level.ALL);         // default ConsoleHandler Level is INFO
logger.setLevel(Level.FINE);
ch.setFormatter(new com.imsl.IMSLFormatter());

autoAR.compute();
int orderSelected = autoAR.getOrder();
System.out.println("Minimum AIC Selected=" + autoAR.getAIC()
+ " with an optimum lag of k= " + autoAR.getOrder());
System.out.println("");

timsacAR = autoAR.getTimsacAR();
timsacConstant = autoAR.getTimsacConstant();
timsacVar = autoAR.getTimsacVariance();
lsAR = autoAR.getAR();
lsConstant = autoAR.getConstant();
lsVar = autoAR.getInnovationVariance();

autoAR.setEstimationMethod(ARAutoUnivariate.METHOD_OF_MOMENTS);
autoAR.compute();
mmAR = autoAR.getAR();
mmConstant = autoAR.getConstant();
mmVar = autoAR.getInnovationVariance();

autoAR.setEstimationMethod(ARAutoUnivariate.MAX_LIKELIHOOD);
autoAR.compute();
mleAR = autoAR.getAR();
mleConstant = autoAR.getConstant();
mleVar = autoAR.getInnovationVariance();

printOutput = new double[orderSelected + 1][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.getTimsacConstant(), autoAR.getTimsacAR(),
new double[0], autoAR.getTimsacVariance());
armaLS.setBackwardOrigin(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][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;
nf.setMinimumFractionDigits(5);
pmf.setNumberFormat(nf);
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.setBackwardOrigin(10);
forecasts = autoAR.getForecast(15);
residuals = autoAR.getResiduals();
printOutput = new double[15][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;
}
nf.setMaximumFractionDigits(3);
pmf.setFirstRowNumber(105);
pmf.setNumberFormat(nf);
pmf.setColumnLabels(colLabels2);
pm.setTitle("Maximum Likelihood Forecasts of Last 10 Observations & "
+ "the Next 5");
pm.print(pmf, printOutput);
}
}
```

#### Output

```Automatic Selection of Minimum AIC AR Model

ORDER SELECTED = 11
AIC = -122.36968839314528

Minimum AIC Selected=-296.13013263562374 with an optimum lag of k= 11

ORDER SELECTED = 11
AIC = -122.36968839314528

ORDER SELECTED = 11
AIC = -122.36968839314528

Comparison of AR Estimates
TIMSAC     Method of Moments    Least Squares    Maximum Likelihood
0     1.0427          1.1679             1.1144             1.1187
1     1.1813          1.1381             1.1481             1.1664
2    -0.5516         -0.5061            -0.5331            -0.5420
3     0.2314          0.2098             0.2757             0.2624
4    -0.1780         -0.2672            -0.3263            -0.3051
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            -0.0309
9     0.1999          0.1366             0.1509             0.1379
10     0.1618          0.1828             0.1935             0.1994
11    -0.3391         -0.3101            -0.3414            -0.3375

Comparison of Equivalent Innovation Variances
TIMSAC     Method of Moments    Least Squares    Maximum Likelihood
0    0.03769         0.04274            0.03687            0.03618

Maximum Likelihood Forecasts of Last 10 Observations & the Next 5
Observed    Forecast    Residual
105     3.553       3.439        0.114
106     3.468       3.480       -0.012
107     3.187       2.924        0.263
108     2.723       2.703        0.020
109     2.686       2.556        0.130
110     2.821       2.785        0.036
111     3.000       2.949        0.051
112     3.201       3.186        0.015
113     3.424       3.385        0.039
114     3.531       3.527        0.004
115     ?           3.446        ?
116     ?           3.195        ?
117     ?           2.829        ?
118     ?           2.492        ?
119     ?           2.414        ?

```