naginterfaces.library.tsa.uni_​garch_​exp_​estim

naginterfaces.library.tsa.uni_garch_exp_estim(dist, yt, x, ip, iq, mn, theta, pht, copts, maxit, tol, io_manager=None)

uni_garch_exp_estim estimates the parameters of a univariate regression-exponential $\text{GARCH}(p,q)$ process (see Engle and Ng (1993)).

For full information please refer to the NAG Library document for g13fg

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g13/g13fgf.html

Parameters

diststr, length 1

The type of distribution to use for $e_t$.

$dist=\text{'N'}$

A Normal distribution is used.

$dist=\text{'T'}$

A Student’s $t$-distribution is used.

ytfloat, array-like, shape $\left(\text{num}\right)$

The sequence of observations, $y_{\text{t}}$, for $\text{t}=1,2,\dots ,T$.

xfloat, array-like, shape $(\text{num},\text{nreg})$

Row $\text{t}$ of $x$ must contain the time dependent exogenous vector $x_{\text{t}}$, where $x_{\text{t}}^T=(x_{\text{t}}^1,\dots ,x_{\text{t}}^k)$, for $\text{t}=1,2,\dots ,T$.

ipint

The number of coefficients, ${\beta }_{\text{i}}$, for $\text{i}=1,2,\dots ,p$.

iqint

The number of coefficients, ${\alpha }_{\text{i}}$, for $\text{i}=1,2,\dots ,q$.

mnint

If $mn=1$, the mean term $b_0$ will be included in the model.

thetafloat, array-like, shape $\left(\text{npar}\right)$

The initial parameter estimates for the vector $\theta$.

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

The next $iq$ elements contain the coefficients ${\varphi }_{\text{i}}$, for $\text{i}=1,2,\dots ,q$.

The next $ip$ elements must contain the moving average coefficients ${\beta }_{\text{i}}$, for $\text{i}=1,2,\dots ,p$.

If $dist=\text{'T'}$, the next element must contain an estimate for $\text{df}$, the number of degrees of freedom of the Student’s $t$-distribution.

If $mn=1$, the next element must contain the mean term $b_o$.

If $copts=False$, the remaining $\text{nreg}$ elements are taken as initial estimates of the linear regression coefficients $b_{\text{i}}$, for $\text{i}=1,2,\dots ,k$.

phtfloat

If $copts=False$ then $pht$ is the value to be used for the pre-observed conditional variance, otherwise $pht$ is not referenced.

coptsbool

If $copts=True$, the function provides initial parameter estimates of the regression terms, otherwise these are provided by you.

maxitint

The maximum number of iterations to be used by the optimization function when estimating the $\text{GARCH}(p,q)$ parameters.

tolfloat

The tolerance to be used by the optimization function when estimating the $\text{GARCH}(p,q)$ parameters.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

thetafloat, ndarray, shape $\left(\text{npar}\right)$

The estimated values $\hat{\theta }$ for the vector $\theta$.

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

The next $iq$ elements contain the coefficients ${\varphi }_{\text{i}}$, for $\text{i}=1,2,\dots ,q$.

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

If $dist=\text{'T'}$, the next element contains an estimate for $\text{df}$ then the number of degrees of freedom of the Student’s $t$-distribution.

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

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

sefloat, ndarray, shape $\left(\text{npar}\right)$

The standard errors for $\hat{\theta }$.

The first element contains the standard error for ${\alpha }_o$ and the next $iq$ elements contain the standard errors for ${\alpha }_{\text{i}}$, for $\text{i}=1,2,\dots ,q$.

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

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

If $dist=\text{'T'}$, the next element contains the standard error for $\text{df}$, the number of degrees of freedom of the Student’s $t$-distribution.

If $mn=1$, the next element contains the standard error for $b_o$.

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

scfloat, ndarray, shape $\left(\text{npar}\right)$

The scores for $\hat{\theta }$.

The first element contains the scores for ${\alpha }_o$, the next $iq$ elements contain the scores for ${\alpha }_{\text{i}}$, for $\text{i}=1,2,\dots ,q$, the next $iq$ elements contain the scores for ${\varphi }_{\text{i}}$, for $\text{i}=1,2,\dots ,q$, the next $ip$ elements are the scores for ${\beta }_{\text{j}}$, for $\text{j}=1,2,\dots ,p$.

If $dist=\text{'T'}$, the next element contains the scores for $\text{df}$, the number of degrees of freedom of the Student’s $t$-distribution.

If $mn=1$, the next element contains the score for $b_o$.

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

covrfloat, ndarray, shape $(\text{npar},\text{npar})$

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

phtfloat

If $copts=True$ then $pht$ is the estimated value of the pre-observed conditional variance.

etfloat, ndarray, shape $\left(\text{num}\right)$

The estimated residuals, ${ϵ}_{\text{t}}$, for $\text{t}=1,2,\dots ,T$.

htfloat, ndarray, shape $\left(\text{num}\right)$

The estimated conditional variances, $h_{\text{t}}$, for $\text{t}=1,2,\dots ,T$.

lgffloat

The value of the log-likelihood function at $\hat{\theta }$.

Raises

NagValueError

(errno $1$)

On entry, $\text{num}=⟨value⟩$.

Constraint: $\text{num}\ge \text{nreg}+mn$.

(errno $1$)

On entry, $\text{npar}=⟨value⟩$.

Constraint: $\text{npar}<20$.

(errno $1$)

On entry, $\text{npar}=⟨value⟩$.

Constraint: if $dist=\text{'N'}$ then $\text{npar}=1+2\times iq+ip+mn+\text{nreg}$, else $\text{npar}=2+2\times iq+ip+mn+\text{nreg}$.

(errno $1$)

On entry, $\text{num}=⟨value⟩$.

Constraint: $max(ip,iq)\le \text{num}$.

(errno $1$)

On entry, $maxit=⟨value⟩$.

Constraint: $maxit>0$.

(errno $1$)

On entry, $dist=⟨value⟩$.

Constraint: $dist=\text{'N'}$ or $\text{'T'}$.

(errno $1$)

On entry, $iq=⟨value⟩$.

Constraint: $iq\ge 1$.

(errno $1$)

On entry, $ip=⟨value⟩$.

Constraint: $ip\ge 0$.

(errno $1$)

On entry, $mn=⟨value⟩$.

Constraint: $mn=0$ or $1$.

(errno $1$)

On entry, $\text{nreg}=⟨value⟩$.

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

(errno $3$)

On entry, the matrix $X$ is not full rank.

(errno $4$)

The information matrix is not positive definite.

(errno $7$)

No feasible model parameters could be found.

Warns

NagAlgorithmicWarning

(errno $5$)

The maximum number of iterations has been reached.

(errno $6$)

The log-likelihood cannot be optimized any further.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

A univariate regression-exponential $\text{GARCH}(p,q)$ process, with $q$ coefficients ${\alpha }_{\text{i}}$, for $\text{i}=1,2,\dots ,q$, $q$ coefficients ${\varphi }_{\text{i}}$, for $\text{i}=1,2,\dots ,q$, $p$ coefficients, ${\beta }_{\text{i}}$, for $\text{i}=1,2,\dots ,p$, and $k$ linear regression coefficients $b_{\text{i}}$, for $\text{i}=1,2,\dots ,k$, can be represented by:

$$\begin{array}{c}\begin{array}{c}{y}_t=b_o+x_t^{T}b+{ϵ}_t\\ ln\left(h_t\right)={\alpha }_0+{\sum }_{i=1}^q{\alpha }_i{z}_{t-i}+{\sum }_{i=1}^q{\varphi }_i(\left|z_{t-i}\right|-E\left[\left|z_{t-i}\right|\right])+{\sum }_{i=1}^p{\beta }_{i}ln\left(h_{t-i}\right)\text{,}t=1,2,\dots ,T\end{array}\end{array}$$

where $z_t=\frac{{ϵ}_t}{\sqrt{h_t}}$, $E\left[\left|z_{t-i}\right|\right]$ denotes the expected value of $\left|z_{t-i}\right|$ and ${ϵ}_t|{\psi }_{t-1}=N(0,h_t)$ or ${ϵ}_t|{\psi }_{t-1}=S_t(\text{df},h_t)$. Here $S_t$ is a standardized Student’s $t$-distribution with $\text{df}$ degrees of freedom and variance $h_t$, $T$ is the number of terms in the sequence, $y_t$ denotes the endogenous variables, $x_t$ the exogenous variables, $b_o$ the regression mean, $b$ the regression coefficients, ${ϵ}_t$ the residuals, $h_t$ the conditional variance, $\text{df}$ the number of degrees of freedom of the Student’s $t$-distribution, and ${\psi }_t$ the set of all information up to time $t$.

uni_garch_exp_estim provides an estimate $\hat{\theta }$, for the vector $\theta =(b_o,b^T,{\omega }^T)$ where $b^T=(b_1,\dots ,b_k)$, ${\omega }^T=({\alpha }_0,{\alpha }_1,\dots ,{\alpha }_q,{\varphi }_1,\dots ,{\varphi }_q,{\beta }_1,\dots ,{\beta }_p,\gamma )$ when $dist=\text{'N'}$, and ${\omega }^T=({\alpha }_0,{\alpha }_1,\dots ,{\alpha }_q,{\varphi }_1,\dots ,{\varphi }_q,{\beta }_1,\dots ,{\beta }_p,\gamma ,\text{df})$ when $dist=\text{'T'}$.

$mn$, $\text{nreg}$ can be used to simplify the $\text{GARCH}(p,q)$ expression in (1) as follows:

No Regression and No Mean

$y_t={ϵ}_t$,

$mn=0$,

$\text{nreg}=0$ and

$\theta$ is a $(2\times q+p+1)$ vector when $dist=\text{'N'}$, and a $(2\times q+p+2)$ vector, when $dist=\text{'T'}$.

No Regression

$y_t=b_o+{ϵ}_t$,

$mn=1$,

$\text{nreg}=0$ and

$\theta$ is a $(2\times q+p+2)$ vector when $dist=\text{'N'}$ and a $(2\times q+p+3)$ vector, when $dist=\text{'T'}$.

Note: if the $y_t=\mu +{ϵ}_t$, where $\mu$ is known (not to be estimated by uni_garch_exp_estim) then (1) can be written as $y_t^{\mu }={ϵ}_t$, where $y_t^{\mu }=y_t-\mu$. This corresponds to the case No Regression and No Mean, with $y_t$ replaced by $y_t-\mu$.

No Mean

$y_t=x_t^{T}b+{ϵ}_t$,

$mn=0$,

$\text{nreg}=k$ and

$\theta$ is a $(2\times q+p+1+k)$ vector when $dist=\text{'N'}$ and a $(2\times q+p+2+k)$ vector, when $dist=\text{'T'}$.

References

Bollerslev, T, 1986, Generalised autoregressive conditional heteroskedasticity, Journal of Econometrics (31), 307–327

Engle, R, 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica (50), 987–1008

Engle, R and Ng, V, 1993, Measuring and testing the impact of news on volatility, Journal of Finance (48), 1749–1777

Glosten, L, Jagannathan, R and Runkle, D, 1993, Relationship between the expected value and the volatility of nominal excess return on stocks, Journal of Finance (48), 1779–1801

Hamilton, J, 1994, Time Series Analysis, Princeton University Press