arma_forecast

Chapter 8: Time Series and Forecasting

arma_forecast

Computes forecasts and their associated probability limits for an ARMA model.

Synopsis

#include <imsls.h>

float *imsls_f_arma_forecast (Imsls_f_arma *arma_info, int n_predict, ..., 0)

The type double function is imsls_d_arma_forecast.

Required Arguments

Imsls_f_arma *arma_info (Input)
Pointer to a structure of type Imsls_f_arma that is passed from the imsls_f_arma function.

int n_predict (Input)
Maximum lead time for forecasts. Argument n_predict must be greater than 0.

Return Value

Pointer to an array of length n_predict (backward_origin + 3) containing the forecasts up to n_predict steps ahead and the information necessary to obtain pairwise confidence intervals. More information is given in the description of argument IMSLS_RETURN_USER.

Synopsis with Optional Arguments

#include <imsls.h>

float *imsls_f_arma_forecast (Imsls_f_arma *arma_info, int n_predict,
IMSLS_CONFIDENCE, float confidence,
IMSLS_BACKWARD_ORIGIN, int backward_origin,
IMSLS_ONE_STEP_FORECAST, float **forecast,
IMSLS_ONE_STEP_FORECAST_USER, float forecast[],
IMSLS_RETURN_USER, float forecasts[],
0)

Optional Arguments

IMSLS_CONFIDENCE, float confidence (Input)
Value in the exclusive interval (0, 100) used to specify the confidence percent probability limits of the forecasts. Typical choices for confidence are 90.0, 95.0, and 99.0.
Default: confidence = 95.0.

IMSLS_BACKWARD_ORIGIN, int backward_origin (Input)
If specified, the maximum backward origin. Argument backward_origin must be greater than or equal to 0 and less than or equal to n_observations - max (maxar, maxma), where maxar = max (ar_lags [i]), maxma = max (ma_lags [j]), and n_observations = the number of observations in the series, as input in function imsls_f_arma. n_predict forecasts beginning at origins n_observations - backward_origin +1 through n_observations are generated.
Default: backward_origin = 0.

IMSLS_ONE_STEP_FORECAST, float **forecast (Output)
Address of a pointer to an internally allocated array of length backward_origin + n_predict containing forecasts. The first backward_origin forecasts are one-step ahead forecasts for the last backward_origin values in the series. The next n_predict values in the returned series are forecasts for the next values beyond the series.

IMSLS_ONE_STEP_FORECAST_USER, float forecast[] (Output)
Storage for array forecast is provided by the user. See IMSLS_ONE_STEP_FORECAST.

IMSLS_RETURN_USER, float forecasts[] (Output)
If specified, a user-specified array of length
n_predict (backward_origin + 3) as defined below.

Column	Content
J	forecasts for lead times l = 1, ..., n_predict at origins n_observations - backward_origin + 1+ j, where j = 0, ..., backward_origin
backward_origin + 1	deviations from each forecast that give the confidence percent probability limits
backward_origin + 2	psi weights of the infinite order moving average form of the model

Also see Example for additional explanation of how to interpret this output.

Description

The Box-Jenkins forecasts and their associated probability limits for a nonseasonal ARMA model are computed given a sample of n = n_observations {Zt} for
t = 1, 2, ..., n, where n_observations = the number of observations in the series, as input in function imsls_f_arma.

Suppose the time series {Zt} is generated by a nonseasonal ARMA model of the form

f(B)Zt = q0 + q(B)At

for t {0, ±1, ±2, ...}, where B is the backward shift operator, q0 is the constant, and

with p autoregressive and q moving average parameters. Without loss of generality, the following is assumed:

1 £ lf (1) £ lf (2) £ … £ lf (p)

1 £ lq (1) £ lq (2) £ … £ lq (q)

so that the nonseasonal ARMA model is of order (p, q), where p = lq(p) and q = lq(q). Note that the usual hierarchical model assumes the following:

lf (i) = i, 1 £ i £ p

lq (j) = j, 1 £ j £ q

The Box-Jenkins forecast at origin t for lead time l of Zt+1 is defined in terms of the difference equation

where the following is true:

The 100(1 - a) percent probability limits for Zt+l are given by

where z(1-a/2) is the 100(1 - a/2) percentile of the standard normal distribution

(returned from imsls_f_arma) and

are the parameters of the random shock form of the difference equation. Note that the forecasts are computed for lead times l = 1, 2, ..., L at origins
t = (n - b), (n - b + 1), ..., n, where L = n_predict and b = backward_origin.

The Box-Jenkins forecasts minimize the mean-square error

Also, the forecasts can be easily updated according to the following equation:

This approach and others are discussed in Chapter 5: “Forecasting” of Box and Jenkins (1976).

Examples

Example 1

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. Function imsls_f_arma_forecast computes forecasts and 95-percent probability limits for the forecasts for an ARMA(2, 1) model fit using function imsls_f_max_arma with the method of moments option. With backward_origin = 3, columns zero through three of forecasts provide forecasts starting with 1867, 1868, 1869, and 1870, respectively. Note that the values in the first row are the one-step ahead forecasts for 1867, 1868, 1869, and 1870; the values in the second row are the two-step ahead forecasts for 1868, 1869, 1870, and 1871; etc. Column four gives the deviations for computing probability limits, and column five gives the psi weights, which can be used to update forecasts when more data is available. For example, the forecast for the 102nd observation (year 1871) given the data through the 100th observation (year 1869) is 77.21; and 95-percent probability limits are given by 77.2156.30. After observation 101 ( Z101 for year 1870) is available, the forecast can be updated by using

with the psi weight (y1 = 1.37) and the one-step-ahead forecast error for observation 101 (Z101 - 83.72) to give the following:

77.21 + 1.37 (Z101 - 83.72)

Since this updated forecast is one step ahead, the 95-percent probability limits are now given by the forecast 33.22.

#include <imsls.h>

int main()

{

int p = 2;

int q = 1;

int i;

int n_observations = 100;

int max_iterations = 0;

int n_predict = 12;

int backward_origin = 3;

float w[176][2];

float z[100];

float *parameters;

float rel_error = 0.0;

float *forecasts;

Imsls_f_arma *arma_info;

char *col_labels[] = {

"Lead Time",

"Forecast From 1866",

"Forecast From 1867",

"Forecast From 1868",

"Forecast From 1869",

"Dev. for Prob. Limits",

"Psi"};

imsls_f_data_sets(2, IMSLS_X_COL_DIM,

2, IMSLS_RETURN_USER, w,

0);

for (i=0; i<n_observations; i++) z[i] = w[21+i][1];

parameters = imsls_f_arma(n_observations, &z[0], p, q,

IMSLS_RELATIVE_ERROR,

rel_error,

IMSLS_MAX_ITERATIONS,

max_iterations,

IMSLS_ARMA_INFO,

&arma_info,

0);

printf("Method of Moments initial estimates:\n");

printf("AR estimates are %11.4f and %11.4f.\n",

parameters[1], parameters[2]);

printf("MA estimate is %11.4f.\n", parameters[3]);

forecasts = imsls_f_arma_forecast(arma_info, n_predict,

IMSLS_BACKWARD_ORIGIN,

backward_origin,

0);

imsls_f_write_matrix("* * * Forecast Table * * *\n",

n_predict, backward_origin+3,

forecasts,

IMSLS_COL_LABELS, col_labels,

IMSLS_WRITE_FORMAT, "%11.4f",

0);

}

Output

Method of Moments initial estimates:

AR estimates are 1.2443 and -0.5751.

MA estimate is -0.1241.

* * * Forecast Table * * *

Lead Time Forecast From Forecast From Forecast From Forecast From

1866 1867 1868 1869

1 18.2833 16.6151 55.1893 83.7196

2 28.9182 32.0189 62.7606 77.2092

3 41.0101 45.8275 61.8922 63.4608

4 49.9387 54.1496 56.4571 50.0987

5 54.0937 56.5623 50.1939 41.3803

6 54.1282 54.7780 45.5268 38.2174

7 51.7815 51.1701 43.3221 39.2965

8 48.8417 47.7072 43.2631 42.4582

9 46.5335 45.4736 44.4577 45.7715

10 45.3524 44.6861 45.9781 48.0758

11 45.2103 44.9909 47.1827 49.0371

12 45.7128 45.8230 47.8072 48.9080

Lead Time Dev. for Prob. Psi

Limits

1 33.2179 1.3684

2 56.2980 1.1274

3 67.6168 0.6158

4 70.6432 0.1178

5 70.7515 -0.2076

6 71.0869 -0.3261

7 71.9074 -0.2863

8 72.5337 -0.1687

9 72.7498 -0.0452

10 72.7653 0.0407

11 72.7779 0.0767

12 72.8225 0.0720

Example 2

Using the same data as in example 1, option IMSLS_ONE_STEP_FORECAST is used to compute the one-step ahead forecasts with backward_origin = 10 and n_predict = 5. This obtains the one-step ahead forecasts for the last 10 observations in the series, i.e. years 1860-1869, plus the next 5 years. The upper 90% confidence limits are computed for these forecasts using the deviations in column backward_origin +1 of forecasts.

#include <imsls.h>

#include <stdio.h>

int main()

{

int p = 2;

int q = 1;

int i;

int n_observations = 100;

int max_iterations = 0;

int n_predict = 5;

int backward_origin = 10;

int year=1860;

int devindex;

float w[176][2];

float z[100];

float *parameters;

float rel_error = 0.0;

float *forecasts;

float *one_step_forecast;

float confidence=90.;

Imsls_f_arma *arma_info;

imsls_f_data_sets(2, IMSLS_X_COL_DIM, 2,

IMSLS_RETURN_USER, w,

0);

for (i=0; i<n_observations; i++) z[i] = w[21+i][1];

parameters = imsls_f_arma(n_observations, &z[0], p, q,

IMSLS_RELATIVE_ERROR, rel_error,

IMSLS_MAX_ITERATIONS, max_iterations,

IMSLS_ARMA_INFO, &arma_info,

0);

/* get one-step ahead forecasts */

forecasts = imsls_f_arma_forecast(arma_info, n_predict,

IMSLS_BACKWARD_ORIGIN, backward_origin,

IMSLS_ONE_STEP_FORECAST, &one_step_forecast,

IMSLS_CONFIDENCE, confidence,

0);

devindex = backward_origin+1; /* forecasts index for deviations */

printf (" ARMA ONE-STEP AHEAD FORECASTS\n");

printf ("Year Observed Forecast Residual UCL(90\x25) \n\n");

for (i=0; i<backward_origin; i++)

printf ("%d %7.3f %7.3f %7.3f %7.3f\n", year+i,

z[n_observations-backward_origin+i],

one_step_forecast[i],

z[n_observations-backward_origin+i]-one_step_forecast[i],

one_step_forecast[i]+forecasts[devindex]);

for (i=backward_origin; i<backward_origin+n_predict; i++)

printf ("%d - %7.3f - %7.3f\n",

year+i, one_step_forecast[i],

one_step_forecast[i]+

forecasts[devindex+(i-backward_origin)*(backward_origin+3)] );

}

Output

ARMA ONE-STEP AHEAD FORECASTS

Year Observed Forecast Residual UCL(90%)

1860 95.700 100.737 -5.037 128.615

1861 77.200 81.295 -4.095 109.173

1862 59.100 57.067 2.033 84.944

1863 44.000 44.426 -0.426 72.303

1864 47.000 36.353 10.647 64.230

1865 30.500 47.396 -16.896 75.274

1866 16.300 28.558 -12.258 56.436

1867 7.300 19.804 -12.504 47.682

1868 37.300 16.804 20.496 44.681

1869 73.900 55.213 18.687 83.090

1870 - 83.723 - 111.600

1871 - 77.213 - 124.460

1872 - 63.464 - 120.210

1873 - 50.100 - 109.386

1874 - 41.380 - 100.757

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