ARMA Model

Auto-Regressive Moving Average (ARMA) Model functions. More...

Functions
int __stdcall	NDK_ARMA_GOF (double *pData, size_t nSize, double mean, double sigma, double *phis, size_t p, double *thetas, size_t q, WORD retType, double *retVal)
	Computes the goodness of fit measure (e.g., log-likelihood function (LLF), AIC, etc.) of the estimated ARMA model.
int __stdcall	NDK_ARMA_RESID (double *pData, size_t nSize, double mean, double sigma, double *phis, size_t p, double *thetas, size_t q, WORD retType)
	Returns the standardized residuals of a given ARMA model.
int __stdcall	NDK_ARMA_PARAM (double *pData, size_t nSize, double *mean, double *sigma, double *phis, size_t p, double *thetas, size_t q, MODEL_RETVAL_FUNC retType, size_t maxIter)
	Returns values for the quick guess, optimal (calibrated), or standard errors of the values of the model's parameters.
int __stdcall	NDK_ARMA_FORE (double *pData, size_t nSize, double mean, double sigma, double *phis, size_t p, double *thetas, size_t q, size_t nSteps, FORECAST_RETVAL_FUNC retType, double alpha, double *retVal)
	Calculates the out-of-sample arma-model forecast statistics.
int __stdcall	NDK_ARMA_FITTED (double *pData, size_t nSize, double mean, double sigma, double *phis, size_t p, double *thetas, size_t q, FIT_RETVAL_FUNC retType)
	Returns the fitted values (i.e. mean, volatility and residuals).
int __stdcall	NDK_ARMA_VALIDATE (double mean, double sigma, double *phis, size_t p, double *thetas, size_t q)
	Examines the model's parameters for stability constraints (e.g. stationarity, invertibility, causality, etc.).
int __stdcall	NDK_ARMA_SIM (double mean, double sigma, double *phis, size_t p, double *thetas, size_t q, double *pData, size_t nSize, UINT nSeed, double *retArray, size_t nSteps)
	Calculates the out-of-sample simulated values.

Detailed Description

The auto-regressive moving average (ARMA) is a stationary stochastic process made up of sums of auto-regressive and moving average components. Alternatively, in a simple formulation for an ARMA (p, q):

\[x_t -\phi_o - \phi_1 x_{t-1}-\phi_2 x_{t-2}-\cdots -\phi_p x_{t-p}=a_t + \theta_1 a_{t-1} + \theta_2 a_{t-2} + \cdots + \theta_q a_{t-q} \]

Where:

• \(x_t \) is the observed output at time \(t \).
• \(a_t \) is the innovation, shock, or error term at time \(t \).
• \(p \) is the order of the last lagged variables.
• \(q \) is the order of the last lagged innovation or shock.
• \(a_t \) time series observations are independent and identically distributed (i.e., $i.i.d$) and follow a Gaussian distribution (i.e., \(\Phi(0,\sigma^2) \)).

Using back-shift notations (i.e., \(L\)), we can express the ARMA process as follows:

\[ (1-\phi_1 L - \phi_2 L^2 -\cdots - \phi_p L^p) x_t - \phi_o= (1+\theta_1 L+\theta_2 L^2 + \cdots + \theta_q L^q)a_t \]

Assuming \(y_t\) is stationary with a long-run mean of \(\mu\), then taking the expectation from both sides, we can express \(\phi_o\) as follows:

\[ \phi_o = (1-\phi_1 -\phi_2 - \cdots - \phi_p)\mu \]

Thus, the ARMA (p, q) process can now be expressed as:

\[ (1-\phi_1 L - \phi_2 L^2 -\cdots - \phi_p L^p) (x_t - \mu)= (1+\theta_1 L+\theta_2 L^2 + \cdots + \theta_q L^q)a_t\]

\[z_t = x_t - \mu \]

\[(1-\phi_1 L - \phi_2 L^2 -\cdots - \phi_p L^p) z_t = (1+\theta_1 L+\theta_2 L^2 + \cdots + \theta_q L^q)a_t \]

In sum, \(z_t \) is the original signal after we subtract its long-run average.

Remarks

1. The variance of the shocks is constant or time-invariant.
2. The order of an AR component process is determined solely by the last lagged auto-regressive variable with a non-zero coefficient (i.e., \(w_{t-p} \)).
3. The order of an MA component process is solely determined by the order of the last moving average variable with a non-zero coefficient (i.e., \(a_{t-q} \)).
4. In principle, you can have fewer parameters than the orders of the model. Consider the following ARMA (12, 2) process (Airline Model):

   \[(1-\phi_1 L -\phi_{12} L^{12} )(y_t - \mu) = (1+\theta L^2)a_t \]

Related Links

• Wikipedia - Autoregressive moving average model

References

• D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics by D. S. G. Pollock; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906.
• James Douglas Hamilton; Time Series Analysis by James Douglas Hamilton, Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896.
• Tsay, Ruey S.; Analysis of Financial Time Series by Ruey S. Tsay, John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0-471-690740.
• Box, Jenkins and Reinsel; Time Series Analysis: Forecasting and Control by George E. P. Box & Gwilym M. Jenkins & Gregory C. Reinsel ; John Wiley & SONS.; 4th edition(Jun 30, 2008), ISBN: 470272848.
• Walter Enders; Applied Econometric Time Series by Walter Enders; Wiley; 4th edition(Nov 03, 2014), ISBN: 1118808568.

Function Documentation

NDK_ARMA_FITTED()

int __stdcall NDK_ARMA_FITTED	(	double *	pData,
		size_t	nSize,
		double	mean,
		double	sigma,
		double *	phis,
		size_t	p,
		double *	thetas,
		size_t	q,
		FIT_RETVAL_FUNC	retType )

Returns the fitted values (i.e. mean, volatility and residuals).

Parameters

[in,out]	pData	(Required) is the univariate time series data (a one dimensional array).
[in]	nSize	(Required) iis the number of observations in pData.
[in]	mean	(Optional) is the ARMA model mean (i.e. mu).
[in]	sigma	(Optional) iis the standard deviation of the model's residuals/innovations.
[in]	phis	(Optional) are the parameters of the AR(p) component model (starting with the lowest lag).
[in]	p	(Optional) is the number of elements in phis (order of AR component).
[in]	thetas	(Optional) are the parameters of the MA(q) component model (starting with the lowest lag).
[in]	q	(Optional) is the number of elements in thetas (order of MA component).
[in]	retType	(Required) s a switch to select a output type ( see FIT_RETVAL_FUNC). Value Output Type 1 Fitted conditional mean 2 Fitted conditional volatility or standard deviation 3 Raw residuals (actual - fitted mean) 4 Standardized residuals - (actual - fitted mean)/fitted volatility.

Returns

status code of the function call: zero = success, positive = success with a warning, and negative = error occurred.

Return values

NDK_SUCCESS	success
NDK_INVALID_ARG	failed (see remarks)
NDK_LOW_LICENSE_ARG_LIMIT	failed (see remarks)
NDK_FAILED	failed

Remarks

1. The time series is homogeneous or equally spaced.
2. The data sample may include one or more observation with missing values (i.e., NaN) at either end (not intermediary).
3. The function removes any observations with missing values from the dataset before calculating.
4. If the value of the nSize is zero, the function will fail and return NDK_INVALID_ARG.
5. If the value of the retVal argument is NULL, the function will fails and return NDK_INVALID_ARG.
6. The long-run mean can take any value or be omitted, in which a zero value is assumed.
7. The residuals/innovations standard deviation ( \(\sigma\)) must be greater than zero.
8. For the input argument - (phis):
   • The input argument is optional and can be omitted, so no AR component is included.
   • The order of the parameters starts with the lowest lag.
   • One or more parameters may have missing values (i.e., NaN).
   • The order of the last value solely determines the order of the AR component model in the array with a numeric value (vs. missing or error).
9. For the input argument - (thetas):
   • The input argument is optional and can be omitted, so no MA component is included.
   • The order of the parameters starts with the lowest lag.
   • One or more values in the input argument can be missing (i.e., NaN).
   • The order of the last value solely determines the order of the MA component model in the array with a numeric value (vs. missing or error).
10. Missing parameter values reduce the model's actual number of overall parameters, thus improving the AIC, BIC, and HQC statistics.
11. In lite-license mode (e.g., expired license key), the size of the input data set is limited to a maximum of 25. If this limit is exceeded, the function will fail and return NDK_LOW_LICENSE_ARG_LIMIT.

Requirements

Requirement	Value
Target Platform	Windows
Header	SFSDK.h (include Windows.h)
Library	SFSDK.lib
DLL	SFSDK.dll

Since

v1.63

See also

• NDK_ARMA_RESID
• NDK_ARMA_PARAM
• NDK_ARMA_FORE
• NDK_ARMA_GOF
• NDK_ARMA_VALIDATE
• NDK_ARMA_SIM

NDK_ARMA_FORE()

int __stdcall NDK_ARMA_FORE	(	double *	pData,
		size_t	nSize,
		double	mean,
		double	sigma,
		double *	phis,
		size_t	p,
		double *	thetas,
		size_t	q,
		size_t	nSteps,
		FORECAST_RETVAL_FUNC	retType,
		double	alpha,
		double *	retVal )

Calculates the out-of-sample conditional forecast (i.e. mean, error, and confidence interval).

Parameters

[in]	pData	(Required) is the univariate time series data.
[in]	nSize	(Required) iis the number of observations in pData.
[in]	mean	(Optional) is the ARMA model mean (i.e. mu).
[in]	sigma	(Optional) iis the standard deviation of the model's residuals/innovations.
[in]	phis	(Optional) are the parameters of the AR(p) component model (starting with the lowest lag).
[in]	p	(Optional) is the number of elements in phis (order of AR component).
[in]	thetas	(Optional) are the parameters of the MA(q) component model (starting with the lowest lag).
[in]	q	(Optional) is the number of elements in thetas (order of MA component).
[in]	nSteps	(Optional) is the forecast time/horizon (expressed in terms of steps beyond end of the time series).
[in]	retType	(Required) is a switch to select the type of value returned (FORECAST_MEAN, FORECAST_STDEV , ..). (see FORECAST_RETVAL_FUNC). Value Return 1 Mean Value (default) 2 Forecast standard error (aka local volatility) 2 Volatility term structure. 3 Lower limit of the forecast confidence interval. 4 Upper limit of the forecast confidence interval.
[in]	alpha	(Required) is the statistical significance or confidence level (i.e. alpha). If missing or omitted, an alpha value of 5% is assumed.
[out]	retVal	(Required) is a pointer to a variable to hold the calculated forecast statistics.

Returns

status code of the function call: zero = success, positive = success with a warning, and negative = error occurred.

Return values

NDK_SUCCESS	success
NDK_INVALID_ARG	failed (see remarks)
NDK_LOW_LICENSE_ARG_LIMIT	failed (see remarks)
NDK_FAILED	failed

Remarks

1. The time series is homogeneous or equally spaced.
2. The data sample may include one or more observation with missing values (i.e., NaN) at either end (not intermediary).
3. The function removes any observations with missing values from the dataset before calculating.
4. If the value of the nSize is zero, the function will fail and return NDK_INVALID_ARG.
5. If the value of the retVal argument is NULL, the function will fails and return NDK_INVALID_ARG.
6. The long-run mean can take any value or be omitted, in which a zero value is assumed.
7. The residuals/innovations standard deviation ( \(\sigma\)) must be greater than zero.
8. For the input argument - (phis):
   • The input argument is optional and can be omitted, so no AR component is included.
   • The order of the parameters starts with the lowest lag.
   • One or more parameters may have missing values (i.e., NaN).
   • The order of the last value solely determines the order of the AR component model in the array with a numeric value (vs. missing or error).
9. For the input argument - (thetas):
   • The input argument is optional and can be omitted, so no MA component is included.
   • The order of the parameters starts with the lowest lag.
   • One or more values in the input argument can be missing (i.e., NaN).
   • The order of the last value solely determines the order of the MA component model in the array with a numeric value (vs. missing or error).
10. Missing parameter values reduce the model's actual number of overall parameters, thus improving the AIC, BIC, and HQC statistics.
11. In lite-license mode (e.g., expired license key), the size of the input data set is limited to a maximum of 25. If this limit is exceeded, the function will fail and return NDK_LOW_LICENSE_ARG_LIMIT.

Requirements

Requirement	Value
Target Platform	Windows
Header	SFSDK.h (include Windows.h)
Library	SFSDK.lib
DLL	SFSDK.dll

Since

v1.63

See also

• NDK_ARMA_RESID
• NDK_ARMA_PARAM
• NDK_ARMA_GOF
• NDK_ARMA_FITTED
• NDK_ARMA_VALIDATE
• NDK_ARMA_SIM

NDK_ARMA_GOF()

int __stdcall NDK_ARMA_GOF	(	double *	pData,
		size_t	nSize,
		double	mean,
		double	sigma,
		double *	phis,
		size_t	p,
		double *	thetas,
		size_t	q,
		WORD	retType,
		double *	retVal )

Computes the log-likelihood (LLF), Akaike Information Criterion (AIC) or other goodness of fit functions of the ARMA model.

Parameters

[in]	pData	(Required) is the univariate time series data.
[in]	nSize	(Required) iis the number of observations in pData.
[in]	mean	(Optional) is the ARMA model mean (i.e. mu).
[in]	sigma	(Optional) iis the standard deviation of the model's residuals/innovations.
[in]	phis	(Optional) are the parameters of the AR(p) component model (starting with the lowest lag).
[in]	p	(Optional) is the number of elements in phis (order of AR component).
[in]	thetas	(Optional) are the parameters of the MA(q) component model (starting with the lowest lag).
[in]	q	(Optional) is the number of elements in thetas (order of MA component).
[in]	retType	(Required) is a switch to select the return output (1 = Mean Value (default), 2 = Std Error, 3 = Test Score, 4 = P-Value, 5 = Upper Value, 6 = Lower Value). ( see GOODNESS_OF_FIT_FUNC) Value Return 1 Mean Value (default) 2 Standard Error. 3 Upper Limit. 4 Lower Limit.
[out]	retVal	(Required) is a pointer to a variable to hold the calculated test statistics.

Returns

status code of the function call: zero = success, positive = success with a warning, and negative = error occurred.

Return values

NDK_SUCCESS	success
NDK_INVALID_ARG	failed (see remarks)
NDK_LOW_LICENSE_ARG_LIMIT	failed (see remarks)
NDK_FAILED	failed

Remarks

1. The time series is homogeneous or equally spaced.
2. The data sample may include one or more observation with missing values (i.e., NaN) at either end (not intermediary).
3. The function removes any observations with missing values from the dataset before calculating.
4. If the value of the nSize is zero, the function will fail and return NDK_INVALID_ARG.
5. If the value of the retVal argument is NULL, the function will fails and return NDK_INVALID_ARG.
6. The long-run mean can take any value or be omitted, in which a zero value is assumed.
7. The residuals/innovations standard deviation ( \(\sigma\)) must be greater than zero.
8. For the input argument - (phis):
   • The input argument is optional and can be omitted, so no AR component is included.
   • The order of the parameters starts with the lowest lag.
   • One or more parameters may have missing values (i.e., NaN).
   • The order of the last value solely determines the order of the AR component model in the array with a numeric value (vs. missing or error).
9. For the input argument - (thetas):
   • The input argument is optional and can be omitted, so no MA component is included.
   • The order of the parameters starts with the lowest lag.
   • One or more values in the input argument can be missing (i.e., NaN).
   • The order of the last value solely determines the order of the MA component model in the array with a numeric value (vs. missing or error).
10. Missing parameter values reduce the model's actual number of overall parameters, thus improving the AIC, BIC, and HQC statistics.
11. In lite-license mode (e.g., expired license key), the size of the input data set is limited to a maximum of 25. If this limit is exceeded, the function will fail and return NDK_LOW_LICENSE_ARG_LIMIT.

Requirements

Requirement	Value
Target Platform	Windows
Header	SFSDK.h (include Windows.h)
Library	SFSDK.lib
DLL	SFSDK.dll

Since

v1.63

See also

• NDK_ARMA_RESID
• NDK_ARMA_PARAM
• NDK_ARMA_FORE
• NDK_ARMA_FITTED
• NDK_ARMA_VALIDATE
• NDK_ARMA_SIM

NDK_ARMA_PARAM()

int __stdcall NDK_ARMA_PARAM	(	double *	pData,
		size_t	nSize,
		double *	mean,
		double *	sigma,
		double *	phis,
		size_t	p,
		double *	thetas,
		size_t	q,
		MODEL_RETVAL_FUNC	retType,
		size_t	maxIter )

Returns values for the quick guess, optimal (calibrated), or standard errors of the values of the model's parameters.

Parameters

[in]	pData	(Required) is the univariate time series data.
[in]	nSize	(Required) iis the number of observations in pData.
[in,out]	mean	(Optional) is the ARMA model mean (i.e. mu).
[in,out]	sigma	(Optional) iis the standard deviation of the model's residuals/innovations.
[in,out]	phis	(Optional) are the parameters of the AR(p) component model (starting with the lowest lag).
[in]	p	(Optional) is the number of elements in phis (order of AR component).
[in,out]	thetas	(Optional) are the parameters of the MA(q) component model (starting with the lowest lag).
[in]	q	(Optional) is the number of elements in thetas (order of MA component).
[in]	retType	(Required) is a switch to select the type of value returned: 1= Quick Guess, 2=Calibrated, 3= Std. Errors ( see MODEL_RETVAL_FUNC). Value Type 1 Quick guess (non-optimal) of parameters values 2 Run a calibration process to find optimal values for the model's parameters 3 Compute the standard error of the parameters' values
[in]	maxIter	(Required) is the maximum number of iterations used to calibrate the model. If none (i.e., 0), the default maximum of 100 is assumed.

Returns

status code of the function call: zero = success, positive = success with a warning, and negative = error occurred.

Return values

NDK_SUCCESS	success
NDK_INVALID_ARG	failed (see remarks)
NDK_LOW_LICENSE_ARG_LIMIT	failed (see remarks)
NDK_FAILED	failed

Remarks

1. The time series is homogeneous or equally spaced.
2. The data sample may include one or more observation with missing values (i.e., NaN) at either end (not intermediary).
3. The function removes any observations with missing values from the dataset before calculating.
4. If the value of the nSize is zero, the function will fail and return NDK_INVALID_ARG.
5. If the value of the retVal argument is NULL, the function will fails and return NDK_INVALID_ARG.
6. The long-run mean can take any value or be omitted, in which a zero value is assumed.
7. The residuals/innovations standard deviation ( \(\sigma\)) must be greater than zero.
8. For the input argument - (phis):
   • The input argument is optional and can be omitted, so no AR component is included.
   • The order of the parameters starts with the lowest lag.
   • One or more parameters may have missing values (i.e., NaN).
   • The order of the last value solely determines the order of the AR component model in the array with a numeric value (vs. missing or error).
9. For the input argument - (thetas):
   • The input argument is optional and can be omitted, so no MA component is included.
   • The order of the parameters starts with the lowest lag.
   • One or more values in the input argument can be missing (i.e., NaN).
   • The order of the last value solely determines the order of the MA component model in the array with a numeric value (vs. missing or error).
10. Missing parameter values reduce the model's actual number of overall parameters, thus improving the AIC, BIC, and HQC statistics.
11. In lite-license mode (e.g., expired license key), the size of the input data set is limited to a maximum of 25. If this limit is exceeded, the function will fail and return NDK_LOW_LICENSE_ARG_LIMIT.

Requirements

Requirement	Value
Target Platform	Windows
Header	SFSDK.h (include Windows.h)
Library	SFSDK.lib
DLL	SFSDK.dll

Since

v1.63

See also

• NDK_ARMA_RESID
• NDK_ARMA_GOF
• NDK_ARMA_FORE
• NDK_ARMA_FITTED
• NDK_ARMA_VALIDATE
• NDK_ARMA_SIM

NDK_ARMA_RESID()

int __stdcall NDK_ARMA_RESID	(	double *	pData,
		size_t	nSize,
		double	mean,
		double	sigma,
		double *	phis,
		size_t	p,
		double *	thetas,
		size_t	q,
		WORD	retType )

Returns the standardized residuals of a given ARMA model

Parameters

[in,out]	pData	(Required) is the univariate time series data.
[in]	nSize	(Required) iis the number of observations in pData.
[in]	mean	(Optional) is the ARMA model mean (i.e. mu).
[in]	sigma	(Optional) iis the standard deviation of the model's residuals/innovations.
[in]	phis	(Optional) are the parameters of the AR(p) component model (starting with the lowest lag).
[in]	p	(Optional) is the number of elements in phis (order of AR component).
[in]	thetas	(Optional) are the parameters of the MA(q) component model (starting with the lowest lag).
[in]	q	(Optional) is the number of elements in thetas (order of MA component).
[in]	retType	(Required) is a switch to select a residuals-type:raw or standardized. see RESID_RETVAL_FUNC Value Residuals Type 0 Standardized residuals 1 Raw Residuals

Returns

status code of the function call: zero = success, positive = success with a warning, and negative = error occurred.

Return values

NDK_SUCCESS	success
NDK_INVALID_ARG	failed (see remarks)
NDK_LOW_LICENSE_ARG_LIMIT	failed (see remarks)
NDK_FAILED	failed

Remarks

1. The time series is homogeneous or equally spaced.
2. The data sample may include one or more observation with missing values (i.e., NaN) at either end (not intermediary).
3. The function removes any observations with missing values from the dataset before calculating.
4. If the value of the nSize is zero, the function will fail and return NDK_INVALID_ARG.
5. If the value of the retVal argument is NULL, the function will fails and return NDK_INVALID_ARG.
6. The long-run mean can take any value or be omitted, in which a zero value is assumed.
7. The residuals/innovations standard deviation ( \(\sigma\)) must be greater than zero.
8. For the input argument - (phis):
   • The input argument is optional and can be omitted, so no AR component is included.
   • The order of the parameters starts with the lowest lag.
   • One or more parameters may have missing values (i.e., NaN).
   • The order of the last value solely determines the order of the AR component model in the array with a numeric value (vs. missing or error).
9. For the input argument - (thetas):
   • The input argument is optional and can be omitted, so no MA component is included.
   • The order of the parameters starts with the lowest lag.
   • One or more values in the input argument can be missing (i.e., NaN).
   • The order of the last value solely determines the order of the MA component model in the array with a numeric value (vs. missing or error).
10. Missing parameter values reduce the model's actual number of overall parameters, thus improving the AIC, BIC, and HQC statistics.
11. In lite-license mode (e.g., expired license key), the size of the input data set is limited to a maximum of 25. If this limit is exceeded, the function will fail and return NDK_LOW_LICENSE_ARG_LIMIT.

Requirements

Requirement	Value
Target Platform	Windows
Header	SFSDK.h (include Windows.h)
Library	SFSDK.lib
DLL	SFSDK.dll

Since

v1.63

See also

• NDK_ARMA_GOF
• NDK_ARMA_PARAM
• NDK_ARMA_FORE
• NDK_ARMA_FITTED
• NDK_ARMA_VALIDATE
• NDK_ARMA_SIM

NDK_ARMA_SIM()

int __stdcall NDK_ARMA_SIM	(	double	mean,
		double	sigma,
		double *	phis,
		size_t	p,
		double *	thetas,
		size_t	q,
		double *	pData,
		size_t	nSize,
		UINT	nSeed,
		double *	retArray,
		size_t	nSteps )

Calculates the out-of-sample simulated values.

Parameters

[in]	mean	(Optional) is the ARMA model mean (i.e. mu).
[in]	sigma	(Optional) is the standard deviation of the model's residuals/innovations.
[in]	phis	(Optional) are the parameters of the AR(p) component model (starting with the lowest lag).
[in]	p	(Optional) is the number of elements in phis (order of AR component).
[in]	thetas	(Optional) are the parameters of the MA(q) component model (starting with the lowest lag).
[in]	q	(Optional) is the number of elements in thetas (order of MA component).
[in]	pData	(Required) The values of the latest (most recent) observations.
[in]	nSize	(Required) is the number of observations in pData.
[in]	nSeed	(Required) is an unsigned integer to initialize the psuedorandom number generator.
[out]	retArray	(Required) is the output array to hold nSteps future simulations.
[in]	nSteps	(Required) is the number of future steps to simulate for.

Returns

status code of the function call: zero = success, positive = success with a warning, and negative = error occurred.

Return values

NDK_SUCCESS	success
NDK_INVALID_ARG	failed (see remarks)
NDK_LOW_LICENSE_ARG_LIMIT	failed (see remarks)
NDK_FAILED	failed

Remarks

1. The time series is homogeneous or equally spaced.
2. The data sample may include one or more observation with missing values (i.e., NaN) at either end (not intermediary).
3. The function removes any observations with missing values from the dataset before calculating.
4. The input data argument (i.e. latest observations) is optional. If omitted, an array of zeroes is assumed.
5. The function returns an array of one simulation path starting from the end of the input data.
6. The long-run mean can take any value or be omitted, in which a zero value is assumed.
7. The residuals/innovations standard deviation ( \(\sigma\)) must be greater than zero.
8. For the input argument - (phis):
   • The input argument is optional and can be omitted, so no AR component is included.
   • The order of the parameters starts with the lowest lag.
   • One or more parameters may have missing values (i.e., NaN).
   • The order of the last value solely determines the order of the AR component model in the array with a numeric value (vs. missing or error).
9. For the input argument - (thetas):
   • The input argument is optional and can be omitted, so no MA component is included.
   • The order of the parameters starts with the lowest lag.
   • One or more values in the input argument can be missing (i.e., NaN).
   • The order of the last value solely determines the order of the MA component model in the array with a numeric value (vs. missing or error).
10. Missing parameter values reduce the model's actual number of overall parameters, thus improving the AIC, BIC, and HQC statistics.
11. In lite-license mode (e.g., expired license key), the size of the input data set is limited to a maximum of 25. If this limit is exceeded, the function will fail and return NDK_LOW_LICENSE_ARG_LIMIT.

Requirements

Requirement	Value
Target Platform	Windows
Header	SFSDK.h (include Windows.h)
Library	SFSDK.lib
DLL	SFSDK.dll

Since

v1.63

See also

• NDK_ARMA_RESID
• NDK_ARMA_PARAM
• NDK_ARMA_FORE
• NDK_ARMA_FITTED
• NDK_ARMA_VALIDATE
• NDK_ARMA_GOF

NDK_ARMA_VALIDATE()

int __stdcall NDK_ARMA_VALIDATE	(	double	mean,
		double	sigma,
		double *	phis,
		size_t	p,
		double *	thetas,
		size_t	q )

Examines the model's parameters for stability constraints (e.g. stationarity, invertibility, causality, etc.).

Parameters

[in]	mean	(Optional) is the ARMA model mean (i.e. mu).
[in]	sigma	(Optional) iis the standard deviation of the model's residuals/innovations.
[in]	phis	(Optional) are the parameters of the AR(p) component model (starting with the lowest lag).
[in]	p	(Optional) is the number of elements in phis (order of AR component).
[in]	thetas	(Optional) are the parameters of the MA(q) component model (starting with the lowest lag).
[in]	q	(Optional) is the number of elements in thetas (order of MA component).

Returns

status code of the function call: zero = success, positive = success with a warning, and negative = error occurred.

Return values

NDK_TRUE	model is stable
NDK_FALSE	model is instable
NDK_INVALID_ARG	failed (see remarks)
NDK_FAILED	failed

Remarks

1. The long-run mean can take any value or be omitted, in which a zero value is assumed.
2. The residuals/innovations standard deviation ( \(\sigma\)) must be greater than zero.
3. For the input argument - (phis):
   • The input argument is optional and can be omitted, so no AR component is included.
   • The order of the parameters starts with the lowest lag.
   • One or more parameters may have missing values (i.e., NaN).
   • The order of the last value solely determines the order of the AR component model in the array with a numeric value (vs. missing or error).
4. For the input argument - (thetas):
   • The input argument is optional and can be omitted, so no MA component is included.
   • The order of the parameters starts with the lowest lag.
   • One or more values in the input argument can be missing (i.e., NaN).
   • The order of the last value solely determines the order of the MA component model in the array with a numeric value (vs. missing or error).
5. Missing parameter values reduce the model's actual number of overall parameters, thus improving the AIC, BIC, and HQC statistics.

Requirements

Requirement	Value
Target Platform	Windows
Header	SFSDK.h (include Windows.h)
Library	SFSDK.lib
DLL	SFSDK.dll

Since

v1.63

See also

• NDK_ARMA_RESID
• NDK_ARMA_PARAM
• NDK_ARMA_FORE
• NDK_ARMA_FITTED
• NDK_ARMA_GOF
• NDK_ARMA_SIM