NAG Library Function Document

nag_estimate_agarchI (g13fac)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{yt}\left[\mathbf{num}\right]$ – const doubleInput

On entry: the sequence of observations, $y_{\mathit{t}}$, for $\mathit{t}=1,2,\dots ,T$.

2:    $\mathbf{x}\left[\mathbf{num}\times \mathbf{tdx}\right]$ – const doubleInput

Note: the $i$th element of the $j$th vector $X$ is stored in $\mathbf{x}\left[\left(i-1\right)\times \mathbf{tdx}+j-1\right]$.

On entry: row $t$ of x must contain the time dependent exogenous vector $x_t$, where $x_{\mathit{t}}^{\mathrm{T}}=\left(x_{\mathit{t}}^1,\dots ,{x_{\mathit{t}}}^k\right)$, for $\mathit{t}=1,2,\dots ,T$.

3:    $\mathbf{tdx}$ – IntegerInput

On entry: the stride separating matrix column elements in the array x.

Constraint: $\mathbf{tdx}\ge \mathbf{nreg}$.

4:    $\mathbf{num}$ – IntegerInput

On entry: the number of terms in the sequence, $T$.

Constraint: $\mathbf{num}\ge \mathit{npar}$.

5:    $\mathbf{p}$ – IntegerInput

On entry: the GARCH$\left(p,q\right)$ parameter $p$.

Constraint: $\mathbf{p}\ge 0$.

6:    $\mathbf{q}$ – IntegerInput

On entry: the GARCH$\left(p,q\right)$ parameter $q$.

Constraint: $\mathbf{q}\ge 1$.

7:    $\mathbf{nreg}$ – IntegerInput

On entry: $k$, the number of refression coefficients.

Constraint: $\mathbf{nreg}\ge 0$.

8:    $\mathbf{mn}$ – IntegerInput

On entry: if $\mathbf{mn}=1$, the mean term $b_0$ will be included in the model.

Constraint: $\mathbf{mn}=0$ or $1$.

9:    $\mathbf{isym}$ – IntegerInput

On entry: if $\mathbf{isym}=1$, the asymmetry term $\gamma$ will be included in the model.

Constraint: $\mathbf{isym}=0$ or $1$.

10:  $\mathbf{theta}\left[\mathit{npar}\right]$ – doubleInput/Output

On entry: the initial parameter estimates for the vector $\theta$.

The first element contains the coefficient ${\alpha }_o$, the next q elements contain the autoregressive coefficients ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.

The next p elements are the moving average coefficients ${\beta }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,p$.

The next element contains the asymmetry parameter $\gamma$.

If $\mathbf{est\_opt}=\mathrm{Nag\_Garch\_Est\_Initial\_False}$, (when $\mathbf{mn}=1$) the next term contains an initial estimate of the mean term $b_o$ and the remaining nreg elements are taken as initial estimates of the linear regression coefficients $b_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,k$.

On exit: the estimated values $\hat{\theta }$ for the vector $\theta$.

The first element contains the coefficient ${\alpha }_o$, the next q elements contain the coefficients ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.

The next p elements are the coefficients ${\beta }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,p$.

The next element contains the estimate for the asymmetry parameter $\gamma$.

If $\mathbf{mn}=1$, the next element contains an estimate for the mean term $b_o$.

The final nreg elements are the estimated linear regression coefficients $b_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,k$.

11:  $\mathbf{se}\left[\mathit{npar}\right]$ – doubleOutput

On exit: the standard errors for $\hat{\theta }$.

The first element contains the standard error for ${\alpha }_o$.

The next q elements contain the standard errors for ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.

The next p elements are the standard errors for ${\beta }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,p$.

If $\mathbf{isym}=1$, the next element contains the standard error for $\gamma$.

If $\mathbf{mn}=1$, the next element contains the standard error for $b_o$.

The final nreg elements are the standard errors for $b_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,k$.

12:  $\mathbf{sc}\left[\mathit{npar}\right]$ – doubleOutput

On exit: the scores for $\hat{\theta }$.

The first element contains the score for ${\alpha }_o$.

The next q elements contain the score for ${\alpha }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$.

The next p elements are the scores for ${\beta }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,p$.

If $\mathbf{isym}=1$, the next element contains the score for $\gamma$.

If $\mathbf{mn}=1$, the next element contains the score for $b_o$.

The final nreg elements are the scores for $b_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,k$.

13:  $\mathbf{covar}\left[\mathit{npar}\times \mathbf{tdc}\right]$ – doubleOutput

Note: the $\left(i,j\right)$th element of the matrix is stored in $\mathbf{covar}\left[\left(i-1\right)\times \mathbf{tdc}+j-1\right]$.

On exit: the covariance matrix of the parameter estimates $\hat{\theta }$, that is the inverse of the Fisher Information Matrix.

14:  $\mathbf{tdc}$ – IntegerInput

On entry: the stride separating matrix column elements in the array covar.

Constraint: $\mathbf{tdc}\ge \mathit{npar}$.

15:  $\mathbf{hp}$ – double *Input/Output

On entry: if $\mathbf{est\_opt}=\mathrm{Nag\_Garch\_Est\_Initial\_False}$, hp is the value to be used for the pre-observed conditional variance.

If $\mathbf{est\_opt}=\mathrm{Nag\_Garch\_Est\_Initial\_True}$, hp is not referenced.

On exit: if $\mathbf{est\_opt}=\mathrm{Nag\_Garch\_Est\_Initial\_True}$, hp is the estimated value of the pre-observed of the conditional variance.

16:  $\mathbf{et}\left[\mathbf{num}\right]$ – doubleOutput

On exit: the estimated residuals, ${\epsilon }_{\mathit{t}}$, for $\mathit{t}=1,2,\dots ,T$.

17:  $\mathbf{ht}\left[\mathbf{num}\right]$ – doubleOutput

On exit: the estimated conditional variances, $h_{\mathit{t}}$, for $\mathit{t}=1,2,\dots ,T$.

18:  $\mathbf{lgf}$ – double *Output

On exit: the value of the log-likelihood function at $\hat{\theta }$.

19:  $\mathbf{stat\_opt}$ – Nag_Garch_Stationary_TypeInput

On entry: if $\mathbf{stat\_opt}=\mathrm{Nag\_Garch\_Stationary\_True}$, Stationary conditions are enforced.

If $\mathbf{stat\_opt}=\mathrm{Nag\_Garch\_Stationary\_False}$, Stationary conditions are not enforced.

Constraint: $\mathbf{stat\_opt}=\mathrm{Nag\_Garch\_Stationary\_True}$ or $\mathrm{Nag\_Garch\_Stationary\_False}$.

20:  $\mathbf{est\_opt}$ – Nag_Garch_Est_Initial_TypeInput

On entry: if $\mathbf{est\_opt}=\mathrm{Nag\_Garch\_Est\_Initial\_True}$, the function provides initial parameter estimates of the regression terms $\left(b_0,b^{\mathrm{T}}\right)$.

If $\mathbf{est\_opt}=\mathrm{Nag\_Garch\_Est\_Initial\_False}$, you must supply the initial estimations of the regression parameters $\left(b_0,b^{\mathrm{T}}\right)$.

Constraint: $\mathbf{est\_opt}=\mathrm{Nag\_Garch\_Est\_Initial\_True}$ or $\mathrm{Nag\_Garch\_Est\_Initial\_False}$.

21:  $\mathbf{max\_iter}$ – IntegerInput

On entry: the maximum number of iterations to be used by the optimization function when estimating the GARCH$\left(p,q\right)$ parameters. If max_iter is set to 0, the standard errors, score vector and variance-covariance are calculated for the input value of $\theta$ in theta; however the value of $\theta$ is not updated.

Constraint: $\mathbf{max\_iter}\ge 0$.

22:  $\mathbf{tol}$ – doubleInput

On entry: the tolerance to be used by the optimization function when estimating the GARCH$\left(p,q\right)$ parameters.

23:  $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_2_INT_ARG_LT

On entry, $\mathbf{num}=〈\mathit{\text{value}}〉$ while $1+\mathbf{q}+\mathbf{p}+\mathbf{isym}+\mathbf{mn}+\mathbf{nreg}=〈\mathit{\text{value}}〉$. These parameters must satisfy $\mathbf{num}\ge 1+\mathbf{q}+\mathbf{p}+\mathbf{isym}+\mathbf{mn}+\mathbf{nreg}$.

On entry, $\mathbf{tdc}=〈\mathit{\text{value}}〉$ while $1+\mathbf{q}+\mathbf{p}+\mathbf{isym}+\mathbf{mn}+\mathbf{nreg}=〈\mathit{\text{value}}〉$. These parameters must satisfy $\mathbf{tdc}\ge 1+\mathbf{q}+\mathbf{p}+\mathbf{isym}+\mathbf{mn}+\mathbf{nreg}$.

On entry, $\mathbf{tdx}=〈\mathit{\text{value}}〉$ while $\mathbf{nreg}=〈\mathit{\text{value}}〉$. These parameters must satisfy $\mathbf{tdx}\ge \mathbf{nreg}$.

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument est_opt had an illegal value.

On entry, argument stat_opt had an illegal value.

NE_INT_ARG_LT

On entry, max_iter must not be less than 0: $\mathbf{max\_iter}=〈\mathit{\text{value}}〉$.

On entry, $\mathbf{nreg}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nreg}\ge 0$.

On entry, $\mathbf{p}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{p}\ge 0$.

On entry, $\mathbf{q}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{q}\ge 1$.

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

NE_INVALID_INT_RANGE_2

Value $〈\mathit{\text{value}}〉$ given to isym is not valid. Correct range is 0 to 1.

Value $〈\mathit{\text{value}}〉$ given to mn is not valid. Correct range is 0 to 1.

NE_MAT_NOT_FULL_RANK

Matrix $X$ does not give a model of full rank.

NE_MAT_NOT_POS_DEF

Attempt to invert the second derivative matrix needed in the calculation of the covariance matrix of the parameter estimates has failed. The matrix is not positive definite, possibly due to rounding errors.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (g13face.c)

10.2

Program Data

10.3

Program Results

Program Results (g13face.r)