Autoregressive conditional dynamic semivariance models with value-at-risk estimates

Abstract

A variant of the autoregressive conditional heteroscedastic (ARCH) process called as autoregressive conditional dynamic semivariance process (ARCDS) that closely relates to semivariance in the residuals is introduced and in the volatility formulation. As in ARCH formulation, the conditional volatility varies over time. The conditions for stationarity and regularity for the ARCDS process and the information matrix for the process are derived. To test whether the disturbances follow the ARCDS process, the Lagrangian multiplier test is adopted, where the squared ordinary least square residuals are regressed on the squares of the minimum of the past residuals and zero. A second model employs the peaks over the threshold (POT) approach. The Hill estimators are used to estimate the parameters and the threshold is computed based on the mean excess function. The model is used to forecast mean, volatility and value-at-risk (VaR) in the returns of the equity growth funds in India during November 2012 to December 2021. With an exception, the model provides superior 90% in-sample and out-samples forecasts. Simulations are performed. We find that combination of the ARCDS process with POT approach provides superior VaR forecasts in comparison to normal distribution across various significance levels.

Appendix

1.1 ARCDS process

Theorem 1

The ARCDS(p) process with ${\alpha }_{0}>0$ and ${\alpha }_{j}\ge 0 for j=\mathrm{1,2}\dots .p$ is stationary iff all the roots of the characteristic equation formed by the ${\alpha }^{\prime}s$ lie outside the unit circle.

Proof

$$ y_{t} /\psi_{t - 1} \sim N\left( {x_{t} ^{\prime}\beta , h_{t} } \right) $$

$$E\left({{y}_{t}}^{2}/{\psi }_{t-1}\right)={h}_{t}$$

Again, the constant relating variance to ${h}_{t}$ is ignored.

Suppose ${w}_{t}^{\prime}=\left({({\text{min}}\left({\epsilon }_{t},0\right))}^{2},{({\text{min}}\left({\epsilon }_{t-1},0\right))}^{2}\dots ..{({\text{min}}\left({\epsilon }_{t-p},0\right))}^{2}\right)$

Expanding $E\left({{y}_{t}}^{2}/{\psi }_{t-1}\right)$, we can see that the moment is a linear combination of ${w}_{t-1}.$

Therefore, $E\left( {w_{t} /\psi_{t - 1} } \right) = b + Aw_{t - 1}$

where $b=\left[\begin{array}{c}{\alpha }_{0}\\ 0\\ 0\\ .\\ .\\ 0\end{array}\right]$ and $A=\left(\begin{array}{*{20}llll}{\alpha}_{1 }& {\alpha }_{2 } &\dots . & {\alpha }_{p } & 0\\ 1 & 0 &\dots. & 0 & 0 \\ 0 & 1 & \dots . & 0 & 0\\ & & \dots .\\ 0 & 0 & \dots . & 1 & 0\end{array}\right)$

Now, expressing ${w}_{t-1}$ as $b+A{w}_{t-2}$, ${w}_{t-2}$ as $b+A{w}_{t-3}$ and continuing successively, we get

$$ E\left( {w_{t} /\psi_{t - k} } \right) = \left( {1 + A + A^{2} + \ldots + A^{k - 1} } \right)b + A^{k} w_{t - k} $$

Now, $\mathop {{\text{lim}}}\limits_{k \to \infty } E\left( {w_{t} /\psi_{t - k} } \right) = \left( {1 - A} \right)^{ - 1} b$ exists iff all the roots $\alpha$ of the characteristic equation are greater than 1 in absolute value implying that the roots lie outside the unit circle.

Since the expression is independent of t, the expression reflects the stationary variance for the unconditional distribution of y.

Theorem 2

The ARCDS process defined by the specification.

$$ y_{t} /\psi_{t - 1} \sim N\left( {0, h_{t} } \right), $$

$$ h_{t} = \alpha_{0} + \alpha_{1} \left( {\min \left( {y_{t - 1} ,0} \right)} \right)^{2} + \ldots + \alpha_{p} \left( {\min \left( {y_{t - p} ,0} \right)} \right)^{2} $$

is regular if ${\alpha }_{0}>0$ and ${\alpha }_{j}\ge 0 for j=\mathrm{1,2}\dots .p$

Proof

The model $y_{t} /\psi_{t - 1} \sim N\left( {0, h_{t} } \right)$,

$$ h_{t} = \alpha_{0} + \alpha_{1} \left( {\min \left( {y_{t - 1} ,0} \right)} \right)^{2} + \ldots + \alpha_{p} \left( {\min \left( {y_{t - p} ,0} \right)} \right)^{2} $$

is regular if:

• (a) for any random vector

  $${\xi }_{t}=\left[\begin{array}{c}{\xi }_{t-1}\\ {\xi }_{t-2}\\ {\xi }_{t-3}\\ .\\ .\\ {\xi }_{t-p}\end{array}\right]$$

  $min h\left({\xi }_{t}\right)\ge \delta$ for positive $\delta$

• (b) $E\left(\left|\frac{\partial h\left({\xi }_{t}\right)}{\partial {\alpha }_{i}}\right|\left|\frac{\partial h\left({\xi }_{t}\right)}{\partial {\xi }_{t-m}}\right||{\psi }_{t-m-1}\right)$ exists for all i, m and t.

Now, a) is satisfied as volatility is positive. $h\left({\xi }_{t}\right)\ge {\alpha }_{0}>0$

Consider b)

$$E\left(\left|\frac{\partial h\left({\xi }_{t}\right)}{\partial {\alpha }_{i}}\right|\left|\frac{\partial h\left({\xi }_{t}\right)}{\partial {\xi }_{t-m}}\right||{\psi }_{t-m-1}\right)=2{\alpha }_{m}E[{\left|{\text{min}}\left({\xi }_{t-i},0\right)\right|}^{2}|{\text{min}}({\xi }_{t-m},0)|\left|{\psi }_{t-m-1}\right]$$

as ${\alpha }_{m}>0$.

If i > m, $E[|{\text{min}}({\xi }_{t-m},0)|\left|{\psi }_{t-m-1}\right]$ is finite.

If i = m, $E[{\left|{\text{min}}\left({\xi }_{t-i},0\right)\right|}^{3}\left|{\psi }_{t-m-1}\right]$ is finite as conditional density is normal.

If i < m, we have,

$$ \begin{aligned} & 2\alpha_{m} E[\left| {{\text{min}}\left( {\xi_{t - m} ,0} \right)} \right|E(\left| {\min \left( {\xi_{t - i} ,0} \right)} \right|^{2} {|}\psi_{t - i - 1} )|\psi_{t - m - 1} {]} = \phi_{i.m,t} \\ & \quad = 2\alpha_{m} E[\left| {{\text{min}}\left( {\xi_{t - m} ,0} \right)} \right|\left( {\alpha_{0} + \mathop \sum \limits_{j = 1}^{p} \alpha_{j} (\min \left( {\xi_{t - i - j} ,0} \right))^{2} } \right){|}\psi_{t - m - 1} {]} \\ & \quad = 2\alpha_{m} \alpha_{0} E[\left| {{\text{min}}\left( {\xi_{t - m} ,0} \right)} \right||\psi_{t - m - 1} ] + \mathop \sum \limits_{j = 1}^{p} \alpha_{j} \phi_{i + j.m,t} \\ & \quad \quad 2\alpha_{m} E[\left| {{\text{min}}\left( {\xi_{t - m} ,0} \right)} \right| + \mathop \sum \limits_{j = 1}^{p} \alpha_{j} (\min \left( {\xi_{t - i - j} ,0} \right))^{2} {|}\psi_{t - m - 1} {]} \\ \end{aligned} $$

If the index on $\phi$ is large such that i + j is greater that m or equal to m, we refer to the earlier cases of i > m and i = m. If i + j < m, the recursive process is administered.

Finally, we get an expression involving a constant times $E{[(\left|{\text{min}}\left({\xi }_{t-i},0\right)\right|}^{3}\left|{\psi }_{t-m-1})\right]$ and another constant times $E[|{\text{min}}({\xi }_{t-m},0)|]|{\psi }_{t-m-1}$. With finite no. of lags and finite constants as they are associated with finite no. of terms, requirement b) of the regularity condition is satisfied.