Fits a univariate ARIMA (p, d, q) time series model with the inclusion of one or more regression variables.
Synopsis
#include <imsls.h>
float *imsls_f_regression_arima (int n_obs, float y[], int model[], ..., 0)
The type double function is imsls_d_regression_arima.
Required Arguments
int n_obs
(Input)
Number of observations.
float y[]
(Input)
Array of length n_obs containing the
observations.
int model[]
(Input)
Array of length 3 containing the model order parameters p,
d, q.
|
Element |
Description |
|
0 |
Order of the autoregressive part, p, where p ≥ 0. |
|
1 |
Order of the non-seasonal difference operator, d, where d ≥ 0. |
|
2 |
Order of the moving average part, q, where q ≥ 0. |
Return Value
Pointer to an array of length 1 + p + q with the estimated constant, autoregressive (AR), and moving average (MA) parameters.
Synopsis with Optional Arguments
#include <imsls.h>
float *imsls_f_regression_arima (int n_obs, float y[], int model[],
IMSLS_REGRESSION, int n_regressors, float x[],
IMSLS_REGRESSION_FORECASTS, float xlead[],
IMSLS_REGRESSION_INDICES, int n_indices, int indices[],
IMSLS_NO_TREND,
IMSLS_MAX_ITERATIONS, int max_iterations,
IMSLS_PRINT_LEVEL, int iprint,
IMSLS_FORECASTS, int n_predict, float **forecasts, float **forecast_variances,
IMSLS_FORECASTS_USER, int n_predict, float forecasts[], float forecast_variances[],
IMSLS_REGRESSION_COEF, float **coefficients,
IMSLS_REGRESSION_COEF_USER, float coefficients[],
IMSLS_SE_ARMA, float **arma_std_errors,
IMSLS_SE_ARMA_USER, float arma_std_errors[],
IMSLS_VAR_NOISE, float *avar,
IMSLS_SE_COEF, float **regcoef_std_errors,
IMSLS_SE_COEF_USER, float regcoef_std_errors[],
IMSLS_COEF_COVARIANCES, float **coef_covar,
IMSLS_COEF_COVARIANCES_USER, float coef_covar[],
IMSLS_AIC, float *aic,
IMSLS_LOG_LIKELIHOOD, float *log_likelihood,
IMSLS_RETURN_USER, float *constant, float ar[], float ma[],
0)
Optional Arguments
IMSLS_REGRESSION,
int n_regressors,
float x[]
(Input)
Array of length n_obs × n_regressors
containing the regression variables. Specific columns of x may be selected
using the optional argument IMSLS_REGRESSION_INDICES.
Default: n_regressors = 0 (No
regression variables are included.)
IMSLS_REGRESSION_FORECASTS,
float xlead[]
(Input)
Array of length n_predict × n_regressors
containing the regression variables to be used in obtaining forecasts.
Specific columns of xlead may be selected
using the optional argument IMSLS_REGRESSION_INDICES.
Note: If optional arguments IMSLS_FORECASTS and
IMSLS_REGRESSION
are present, then IMSLS_REGRESSION_FORECASTS
is required.
Default: Not used.
IMSLS_REGRESSION_INDICES,
int n_indices, int
indices[]
(Input)
Argument n_indices specifies
the length of array indices and the number
of regression variables to be included in the ARIMA fit. Argument indices contains the
indices of the regression variables in matrices x and xlead.
Default: All
regression variables in x and xlead will be
used.
IMSLS_NO_TREND, (Input)
If IMSLS_NO_TREND is
specified, the function will not include a trend variable. A trend variable has
the effect of fitting an intercept term in the regression. If the difference
operator model[1] =
d > 0, the effect of no trend on the model in the original,
undifferenced space is polynomial of order d.
Default: The function
will include a trend variable.
IMSLS_MAX_ITERATIONS,
int max_iterations
(Input)
Maximum number of iterations.
Default: max_iterations = 50
IMSLS_PRINT_LEVEL,
int iprint
(Input)
Printing option.
|
iprint |
Action |
|
0 |
No printing |
|
1 |
Prints final results only. |
|
2 |
Prints intermediate and final results. |
Default: iprint = 0
IMSLS_FORECASTS,
int n_predict,
float
**forecasts, float **forecast_variances
(Output)
Addresses of pointers to internally allocated arrays of length n_predict containing
the forecasts and forecast variances for time points
t = n+1, n+2, …, n+n_predict, where
n = n_obs.
IMSLS_FORECASTS_USER,
int n_predict,
float forecasts[],
float forecast_variances[]
(Output)
Storage arrays forecast and forecast_variance are
provided by user. See IMSLS_FORECASTS.
IMSLS_REGRESSION_COEF,
float
**coefficients (Output)
Address of a pointer to an
internally allocated array of length n_regressors+t
containing the estimated regression coefficients, where
t = 0 if IMSLS_NO_TREND is
specified, otherwise t = 1.
IMSLS_REGRESSION_COEF_USER,
float coefficients[]
(Output)
Storage array coefficients is
provided by user. See IMSLS_REGRESSION_COEF.
IMSLS_SE_ARMA, float **arma_std_errors
(Output)
Address of a pointer to an internally allocated array of length
p+q containing the standard errors of the ARMA parameter estimates,
where p = model[0] and
q = model[2].
IMSLS_SE_ARMA_USER,
float
arma_std_errors[] (Output)
Storage array arma_std_errors is
provided by user. See IMSLS_SE_ARMA.
IMSLS_VAR_NOISE,
float *avar
(Output)
White noise variance estimate.
Note: If model[0]+model[2]= 0 and n_regressors > 0,
avar is the mean
squared regression residual.
IMSLS_SE_COEF,
float
**regcoef_std_error (Output)
Address of a pointer
to an internally allocated array of length n_regressors+ t containing
the standard errors of the ARMA parameter estimates, where
t = 0 if IMSLS_NO_TREND is
specified, otherwise t = 1.
IMSLS_SE_COEF_USER,
float regcoef_std_errors[]
(Output)
Storage array reg_std_errors is
provided by user. See IMSLS_SE_COEF.
IMSLS_COEF_COVARIANCES,
float
**coef_covar (Output)
Address of a pointer to an
internally allocated array of length
(n_regressors+t) × (n_regressors+t)
containing the variances and covariances of the regression coefficients,
where t = 0 if IMSLS_NO_TREND is
specified, otherwise t = 1.
IMSLS_COEF_COVARIANCES_USER,
float coef_covar[]
(Output)
Storage array coef_covar is provided
by user. See IMSLS_COEF_COVARIANCES.
IMSLS_AIC, float *aic
(Output)
Akaike’s Information Criterion for the fitted ARMA model.
IMSLS_LOG_LIKELIHOOD,
float *log_likelihood
(Output)
Value of –2(ln(likelihood)) for fitted model.
IMSLS_RETURN_USER,
float *constant,
float ar[], float
ma[]
(Output)
If specified, constant is the
constant parameter estimate, ar is an array of
length p
containing the final autoregressive parameter estimates, and ma is an array of
length q
containing the final moving average parameter estimates.
Description
Function imsls_f_regression_arima fits an ARIMA(p, d, q) to a univariate time series with the possible inclusion of one or more regression variables.
Suppose 
, 
, is a time series such that the
d-th difference is stationary. Further, suppose 
is a series of uncorrelated,
mean 0 random variables with variance 
.
The
Auto-Regressive Integrated Moving Average (ARIMA) model for
can be expressed as
where
B is the backshift operator, 
and
.
The notation
for this model is ARIMA(p, d, q) where p is the
order of the autoregressive polynomial 
, d is the order of the
differencing needed to make stationary, and
q is the order of the moving-average polynomial 
.
The ARIMA
model can be extended to include 
regression
variables
, by using the residuals (of the
multiple regression of on ) in place of in the above ARIMA model.
Equivalently,
where
is the differenced residual series.
To estimate the (p + q + K) parameters of the specified regression ARIMA model, imsls_f_regression_arima uses the iterative generalized least squares method (IGLS) as described in Otto, Bell, and Burman (1987).
The IGLS method iterates between two steps, one step to estimate the regression parameters via generalized least squares (GLS) and the second step to estimate the ARMA parameters. In particular, at iteration m, the first step finds
by solving the GLS problem with weight matrix
, 
where
That is, 
minimizes 
, where 
, 
is an N by K
matrix with i-th column, 
, and 
is an N by N weight
matrix defined using the theoretical autocovariances of the series
. The series
is modeled as an ARMA(p,q) with
parameters 
and 
. At iteration m, the second
step is then to obtain new estimates of 
and 
for the updated series,
. To find the estimates and , imsls_f_regression_arima
uses the exact likelihood method as described in Akaike, Kitagawa, Arahata and
Tada (1979) and used in function, imsls_f_max_arma.
Remarks
When forecasts are requested (n_predict > 0), imsls_f_regression_arima requires that future values of the independent variables be provided in optional argument IMSLS_REGRESSION_FORECASTS. In effect, imsls_f_regression_arima assumes the future X’s are known without error, which is valid for any deterministic function of time such as a seasonal indicator. Also, in economics, certain factors that are considered to be leading indicators are treated as deterministic for the purpose of predicting changes in the economy. Users may consider using a more general transfer function model if this is an unreasonable assumption. Function imsls_f_regression_arima calculates forecast variances using the asymptotic result found in Fuller (1996), Theorem 2.9.4. To obtain the standard errors of the ARMA parameters, imsls_f_regression_arima calls function imsls_f_arma for the final w series.
Examples
Example 1
The data set consists of annual mileage per passenger vehicle and annual US population (in 1000’s) spanning the years 1980 to 2006 (U.S. Energy Information Administration, 2008). Consider modeling the annual mileage using US population as a regression variable.
#include <imsls.h>
int main()
{
int nobs= 24, model[3] = {1, 0, 0};
int indices[1] = {0}, n_predict=5;
float avar, llike, *result;
float *regcoef, *regstderr, *coefcovar, *armastderr;
float *fcst, *fcst_var;
float y[29] = {
9062.0, 8813.0, 8873.0, 9050.0, 9118.0,
9248.0, 9419.0, 9464.0, 9720.0, 9972.0,
10157.0, 10504.0, 10571.0, 10857.0, 10804.0,
10992.0, 11203.0, 11330.0, 11581.0, 11754.0,
11848.0, 11976.0, 11831.0, 12202.0, 12325.0,
12460.0, 12510.0, 12485.0, 12293.0
};
float regX[29][2] = {
{22722.4681, 9062.0},
{22946.5714, 8813.0},
{23166.4458, 8873.0},
{23379.1990, 9050.0},
{23582.4902, 9118.0},
{23792.3795, 9248.0},
{24013.2887, 9419.0},
{24228.8918, 9464.0},
{24449.8982, 9720.0},
{24681.923, 9972.0},
{24962.2814, 10157.0},
{25298.0941, 10504.0},
{25651.4224, 10571.0},
{25991.8588, 10857.0},
{26312.5820999999, 10804.0},
{26627.8393, 10992.0},
{26939.4284, 11203.0},
{27264.6925, 11330.0},
{27585.4104, 11581.0},
{27904.0168, 11754.0},
{28217.1936, 11848.0},
{28503.9803, 11976.0},
{28772.6647, 11831.0},
{29021.0914, 12202.0},
{29289.2127, 12325.0},
{29556.0549, 12460.0},
{29836.2973, 12510.0},
{30129.0332, 12485.0},
{30405.9724, 12293.0}
};
result = imsls_f_regression_arima (nobs, y, model,
IMSLS_REGRESSION,2, regX,
IMSLS_REGRESSION_FORECASTS, ®X[24][0],
IMSLS_FORECASTS, n_predict, &fcst, &fcst_var,
IMSLS_REGRESSION_INDICES, 1, indices,
IMSLS_VAR_NOISE, &avar,
IMSLS_LOG_LIKELIHOOD, &llike,
IMSLS_REGRESSION_COEF, ®coef,
IMSLS_SE_COEF, ®stderr,
IMSLS_COEF_COVARIANCES, &coefcovar,
IMSLS_SE_ARMA, &armastderr,
IMSLS_PRINT_LEVEL, 1,
0);
}
Output
Final results for regression ARIMA model (p,d,q) = 1, 0, 0s
Final AR parameter estimates/ std errors
0.73000 0.13498
-2*ln(maximum log likelihood) = 231.835464
White noise variance = 15427.915039
Regression estimates:
COEFFICIENTS Regression STD Errors
0 -3483.13306 687.21167
1 0.54244 0.02666
Forecasts with standard deviation
T Y fcst Y fcst std dev
24 12360.51563 124.20916
25 12514.80664 153.78410
26 12673.78906 167.42434
27 12837.66895 174.25776
28 12991.60547 177.79208
Example 2
The data set consists of simulated weekly observations containing a strong annual seasonality. The seasonal variables are constructed and sent into regression_arima as regression variables.
#include <imsls.h>
#include <math.h>
int main()
{
int nobs=100, n_predict=4, n_regressors=2;
int i, model[3] = {2,0,0};
float PI, *coefcovar, *regcoef, *regstderr, *result;
float *armastderr, *fcst, *fcstvar;
float avar, llike;
float x[104][2];
float y[104] = {
32.27778, 32.63300, 33.13768, 34.4517,
34.63824, 37.31262, 37.35704, 37.03092,
36.39894, 35.75541, 35.10829, 34.70107,
34.69592, 32.75326, 30.85370, 31.10936,
29.47493, 29.14361, 28.50466, 30.09714,
28.49403, 27.23268, 23.49674, 22.71225,
21.42798, 18.68601, 17.40035, 16.06832,
15.31862, 14.75179, 13.40089, 13.01101,
12.44863, 11.27890, 11.51770, 14.31982,
14.67036, 14.76331, 15.35644, 17.04353,
18.39931, 18.21919, 18.72777, 19.61794,
22.31733, 23.79600, 25.41326, 25.60497,
27.93579, 29.21765, 29.60981, 28.46994,
28.780810, 30.96402, 35.49537, 35.75124,
36.18933, 37.2627, 35.02454, 33.57089,
35.00683, 34.83886, 34.19827, 33.73966,
34.49709, 34.07127, 32.74709, 31.97856,
31.3029, 30.21916, 27.46015, 26.78431,
25.32815, 23.97863, 21.83837, 21.00647,
20.58846, 19.94578, 17.38271, 17.12572,
16.71847, 17.45425, 16.15050, 13.07448,
12.54188, 12.42137, 13.51771, 14.84232,
14.28870, 13.39561, 15.48938, 16.47175,
17.62758, 16.57677, 18.20737, 20.8491,
20.15616, 20.93857, 23.73973, 25.30449,
26.51106, 29.43261, 32.02672, 32.18846
};
/*
* The data are simulated weekly observations
* with an annual seasonal cycle
*/
PI = imsls_f_constant("PI",0);
for (i=0; i<nobs+n_predict;i++)
{
x[i][0] = sin(2*PI*i/ 52.0);
x[i][1] = cos(2*PI*i/ 52.0);
}
result = imsls_f_regression_arima (nobs, y, model,
IMSLS_REGRESSION,2, x,
IMSLS_REGRESSION_FORECASTS, &x[100][0],
IMSLS_FORECASTS, n_predict, &fcst, &fcstvar,
IMSLS_VAR_NOISE, &avar,
IMSLS_LOG_LIKELIHOOD, &llike,
IMSLS_REGRESSION_COEF, ®coef,
IMSLS_SE_COEF, ®stderr,
IMSLS_COEF_COVARIANCES, &coefcovar,
IMSLS_SE_ARMA, &armastderr,
IMSLS_PRINT_LEVEL, 1,
0);
}
Output
Final AR parameter estimates/ std errors
0.71855 0.09838
-0.25989 0.09828
-2*ln(maximum log likelihood) = -13.621020
White noise variance = 0.868007
Regression estimates:
COEFFICIENTS Regression STD Errors
0 24.81011 0.17177
1 8.91971 0.24042
2 6.84814 0.24709
Forecasts with standard deviation
T Y fcst Y fcst std dev
100 26.74492 0.93167
101 28.07804 1.14725
102 29.33707 1.35615
103 30.53160 1.52323
