Chapter 8: Time Series and Forecasting > arma_forecast

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 Examples 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

φ(B)Zt = θ0 + θ(B)At

for t {0, ±1, ±2, ...}, where B is the backward shift operator, θ0 is the constant, and

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

1 ≤ lφ (1) ≤ lφ (2) ≤ … ≤ lφ (p)

1 ≤ lθ (1) ≤ lθ (2) ≤ … ≤ lθ (q)

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

lφ (i) = i, 1 ≤ i ≤ p

lθ (j) = j, 1 ≤ j ≤ q

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

where the following is true:

The 100(1 − α) percent probability limits for Zt+l are given by

where z(a/2) is the 100(1 − α/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_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 (ψ1 = 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>

#include <stdio.h>

 

int main()

{

    int p=2, q=1, i, n_observations=100, max_iterations=0;

    float w[176][2], z[100], *parameters, relative_error=0.0;

 

    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, relative_error,

        IMSLS_MAX_ITERATIONS, max_iterations,

        0);

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

        parameters[1], parameters[2]);

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

}

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 = 0 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, q=1, i, n_observations=100;

    float w[176][2], z[100], *parameters;

 

    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_LEAST_SQUARES,

        0);

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

        parameters[1], parameters[2]);

    printf("MA estimate is %11.4f.\n", parameters[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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