# Stable-GARCH Models for Financial Returns: Fast Estimation and Tests for Stability

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## Abstract

**:**

## 1. Introduction

## 2. Critique of the Stable Paretian Assumption

## 3. Stable Paretian Distribution: Evaluation and Estimation

#### 3.1. Density, VaR and ES Calculation

#### 3.2. Parameter Estimation

**Remarks**:

- There exist other methods for estimation of the four-parameter location-scale asymmetric stable distribution in the i.i.d. case which obviate the need to evaluate the pdf. For example, the estimator of Kogon and Williams [83] is based on the sample characteristic function, is very fast to calculate, and results in estimates which are close in performance to the MLE (though less so for the asymmetry parameter); see Section 5.4 below. Another method is via indirect inference, which requires simulating stable realizations, but not evaluating its likelihood; see Lombardi and Veredas [84] and Garcia et al. [85].
- Most, but certainly not all, applications with stable distributions, including our testing procedures herein, assume existence of the mean, i.e., $1<\alpha \le 2$. This is convenient because, as α decreases towards zero, evaluation of the pdf, and inference about the parameters, becomes more difficult. This problem is addressed in Koblents et al. [86], in which an estimation procedure is proposed for i.i.d. asymmetric stable Paretian data that is applicable for all, but particularly small, α. Use of their method confirms that it performs well, and can outperform the MLE in terms of mean squared error for some parameter constellations, including values of $\alpha >1$. However, it is on the order of 1000 times slower than use of the MLE when using spline approximations to the pdf.

#### 3.3. Nonparametric Estimation of the Tail Index

## 4. Testing Procedures

#### 4.1. Summability Tests: ${\tau}_{0}$ and ${\tau}_{20}$

#### 4.2. ALHADI: The α-Hat Discrepancy Test

#### 4.3. Combined Test: $A+{\tau}_{20}$

#### 4.4. Extension to Testing the Asymmetric Stable Paretian Case

#### 4.5. Likelihood Ratio Test in the Asymmetric Stable Paretian Case

- Estimate the parameters of the ${S}_{\alpha ,\beta}$ distribution, say ${\widehat{\mathit{\theta}}}_{0}=(\widehat{\alpha},\widehat{\beta},\widehat{\sigma},\widehat{\mu})$, using the MLE, with associated log-likelihood denoted by ${\ell}_{{S}_{\alpha ,\beta}}({\widehat{\mathit{\theta}}}_{0};\mathbf{X})$.
- Estimate the parameters of the location-scale NCT distribution and compute the associated log-likelihood ${\ell}_{\mathrm{NCT}}(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}};\mathbf{X})$.
- Compute ratio$${\text{LR}}_{0}\left(\mathbf{X}\right)=2\times \left({\ell}_{\mathrm{NCT}}(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}};\mathbf{X})-{\ell}_{{S}_{\alpha ,\beta}}({\widehat{\mathit{\theta}}}_{0};\mathbf{X})\right).$$
- For $i=1,\dots ,{s}_{1}$,
- (a)
- Simulate ${\mathbf{X}}_{\left(i\right)}$, consisting of T copies of ${S}_{\alpha ,\beta}$ realizations with parameter vector ${\widehat{\mathit{\theta}}}_{0}$.
- (b)
- Similar to steps 1 to 3, compute ratio ${\text{LR}}_{i}\left({\mathbf{X}}_{\left(i\right)}\right)=2\times \left({\ell}_{\mathrm{NCT}}(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}};{\mathbf{X}}_{\left(i\right)})-{\ell}_{{S}_{\alpha ,\beta}}({\widehat{\mathit{\theta}}}_{i};{\mathbf{X}}_{\left(i\right)})\right)$.

- Reject the ${S}_{\alpha ,\beta}$ null hypothesis (formally in favor of the NCT alternative) if ${\text{LR}}_{0}$ is equal to or exceeds the 95% empirical quantile of $({\text{LR}}_{1},\dots ,{\text{LR}}_{{s}_{1}})$.

## 5. Estimation of the Stable-APARCH Process

#### 5.1. Model

#### 5.2. Model Estimation: Likelihood-Based

**θ**(as chosen by a generic multivariate optimization algorithm), the STE method quickly delivers estimates of ${a}_{0}$, α and β. However, the likelihood of the model still needs to be computed, and this requires the evaluation of the stable pdf, for which we use the fast spline approximation available in the Nolan Stable toolbox. As such, the proposed method is related to the use of profile likelihood, enabling a partition of the parameter set for estimation. As the STE method is not likelihood-based, the resulting estimator is not the MLE, but enjoys similar properties of the MLE such as consistency, and can actually outperform the MLE, first in terms of numeric reliability and speed, as already mentioned, but also in terms of mean squared error (MSE), as discussed next.

#### 5.3. Model Estimation: Nearly Instantaneous Method

**θ**that is determined by finding the compromise values which optimize the sum of log-likelihoods of (16) for numerous typical daily financial return series. The reason this works is statistically motivated in Krause and Paolella [55]. This is nothing but an (extreme) form of shrinkage estimation, and, as demonstrated in Krause and Paolella [55], often leads to better VaR forecasts than when estimating

**θ**separately for each return series. A similar analysis as done in Krause and Paolella [55] leads to the choice of

**θ**as

#### 5.4. Model Estimation: Characteristic Function Method

**Remark**:

## 6. Simulation Study Under True Model and Variations

## 7. Empirical Illustration

#### 7.1. Detailed Analysis for Four Stocks from the DJIA Index

- moving windows of length $T=500$ and moving the window ahead by 100 days (resulting in 33 windows of data for each return series);
- moving windows of length $T=1000$ and moving the window ahead by 250 days (resulting in 12 windows of data for each return series);
- the full data set.

#### 7.2. Summary of p-Values from the 29 DJIA Index Stocks

**Remark**:

#### 7.3. Summary of p-Values from 100 S&P500 Stocks

## 8. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Hill Estimates for simulated Pareto, symmetric stable Paretian, and Student’s t, with tail index $\alpha =1.3$ (left) and $\alpha =1.7$ (right), and sample size $T=10,000$.

**Figure 2.**Performance comparison via boxplots of the Hint, McCulloch, and ML estimators of tail index α for i.i.d. symmetric stable Paretian data based on sample size T.

**Figure 3.**Plots associated with the ${\tau}_{0}$ summability test, based on $T=2000$. Left uses symmetric stable data with $\alpha =1.5$; right uses Student’s t with three degrees of freedom.

**Figure 4.**Simulation results of the small sample properties of $\widehat{\alpha}$ and $\widehat{\beta}$ for the Kogon and Williams [83] (K-W) (top) and stable-table estimator (STE) (bottom) methods, using ${X}_{t}\stackrel{\mathrm{iid}}{\sim}{S}_{\alpha ,\beta}(0,1)$, $t=1,\dots ,T$, for $T=250$, and based on 100,000 replications. The vertical dashed line indicates the true values of $\alpha =1.85$ and $\beta =0$.

**Figure 5.**Left: histogram of the estimates of α of the fitted ${S}_{\alpha ,\beta}$-APARCH process, for the 348 time series of length $T=1000$ from the 29 DJIA stock returns; Right: Same but for the 1500 time series of length $T=1000$ from the 100 largest cap stocks from the S&P500 index.

**Figure 6.**Simulation results, based on 10,000 replications with sample size $T=500$, using the fixed APARCH parameters (18) and the trimmed mean technique for ${\widehat{a}}_{0}$ and the fast table lookup method for the two stable shape parameters.

**Figure 7.**Similar to Figure 6, but having used the jointly computed MLE of all seven model parameters.

**Figure 8.**Simulation results under the fixed APARCH (left) and MLE (right) settings, when taking ${c}_{1}=0.01$ and ${d}_{1}=0.95$ for the true data generating process, and the remaining parameters as before, i.e., ${a}_{0}=0$, $\alpha =1.85$, $\beta =0$, ${c}_{0}=0.04$ and ${g}_{1}=0.4$.

**Figure 9.**Application of the testing procedures to moving windows of data of length $T=500$, in increments of 100, for Procter & Gamble (

**left**) and JPMorgan Chase (

**right**). Estimation is based on the method in (16).

**Figure 10.**Similar to Figure 9, but for Cisco Systems and McDonalds Corporation. Estimation is based on the method in (16).

**Figure 12.**Histograms of the p-values over all 33 windows using $T=500$, and the 29 available stocks of the DJIA. Estimation is based on the method in (16).

**Figure 13.**Histograms of the p-values for all 29 stocks in the DJIA using the entire data set of $T=3773$. Estimation is based on the method in (16).

**Figure 14.**Same as Figure 12, i.e., p-values over windows using $T=500$, but having used the 100 largest market-cap stocks from the S&P500 index, resulting in 4100 p-values.

^{1.}Notice that, unless all theoretically possible permutations are used (or, taking $B=\infty $ if they are randomly drawn), the outlined procedure will still return different test statistics for the same data set (unless the set of seed values for the random permutations is held constant to some arbitrary choice). While this feature is still undesirable, it cannot be avoided with finite B and random permutations.^{2.}The author is grateful to anonymous referee for suggesting to consider this method.

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license ( http://creativecommons.org/licenses/by/4.0/).

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Paolella, M.S.
Stable-GARCH Models for Financial Returns: Fast Estimation and Tests for Stability. *Econometrics* **2016**, *4*, 25.
https://doi.org/10.3390/econometrics4020025

**AMA Style**

Paolella MS.
Stable-GARCH Models for Financial Returns: Fast Estimation and Tests for Stability. *Econometrics*. 2016; 4(2):25.
https://doi.org/10.3390/econometrics4020025

**Chicago/Turabian Style**

Paolella, Marc S.
2016. "Stable-GARCH Models for Financial Returns: Fast Estimation and Tests for Stability" *Econometrics* 4, no. 2: 25.
https://doi.org/10.3390/econometrics4020025