Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
63,100 characters · 15 sections · 65 citation commands
Deep Learning Enhanced Realized GARCH
Volatility modeling is an active area of research in financial econometrics with implications for risk management, portfolio allocation, and option pricing. The GARCH model (engle_autoregressive_1982; bollerslev_generalized_1986), and its variants including the exponential GARCH (nelson_conditional_1991) and the GJR model (glosten_relation_1993), provide an excellent modeling framework for understanding and forecasting volatility. While GARCH and its derivatives work well in many scenarios, when it comes to daily volatility modeling, they predominantly rely on the signal in daily squared returns. This can sometimes fall short in capturing swift intraday volatility fluctuations. This problem can be mitigated by incorporating more effective volatility proxies based on high-frequency intraday return data.
In the past two decades, many ex-post estimators of asset return volatility using high-frequency data have been introduced to the literature. Examples include the realized variance of andersen_answering_1998, realized kernel variance of barndorff-nielsen_designing_2008 and bipower variation of barndorff-nielsen_power_2004. These estimators, collectively referred to as realized volatility measures, are more informative than daily squared returns about representing the underlying volatility level, thus providing a better tool for modeling and forecasting volatility. engle_new_2002 explored the idea of including realized volatility measures in the GARCH model and found that it significantly improves its fit to return data. Engle's model only uses a realized volatility measure as a deterministic input to the GARCH equation and pays no attention to explaining the variation in realized volatility measures which should be viewed as noisy proxies of the underlying volatility. hansen_realized_2012 developed a complete model, called the realized GARCH (RealGARCH) model, that incorporates a realized measure into a GARCH framework and at the same time provides a measurement equation to explain the realized volatility measure dynamics. This model has been proven superior empirically jiang_modeling_2018,xie_realized_2020,li_efficient_2021. hansen_exponential_2016 further extended the RealGARCH model to realized EGARCH which incorporates multiple realized volatility measures of volatility.
The RealGARCH model expresses the underlying volatility as a {\it linear} combination of several lagged realized measures. This approach might fall short in grasping the nonlinear relationships and enduring impacts that prior realized measures have on the present intrinsic volatility. This issue is similar to the well-known problem of GARCH that has led to the development of long-memmory and non-linear GARCH models such as Fractionally Integrated GARCH of baillie1996fractionally and Recurrent Conditional Heteroskedasticity (RECH) of nguyen_recurrent_2022. This paper aims to extend RealGARCH and address its limitation by leveraging modern deep learning techniques together with recent advancements in Bayesian computation techniques.
With the advent of deep learning models and advancements in computational power, neural networks (NN) have recently been introduced in volatility modeling in the mainstream econometric literature. NNs are capable of learning complex non-linear functions and capturing long-range dependence in time series data. liu_novel_2019 and bucci_realized_2020 compared the predictive performance of feed-forward neural networks (FNN) and recurrent neural networks (RNN) with traditional econometric approaches, and found that deep learning models generally outperform econometric models on several stock markets. Some hybrid models, proposed by hyup_roh_forecasting_2007 and kim_forecasting_2018, add a neural network as another layer on top of an econometric model, using the volatility estimates produced by an econometric model as the input to a neural network, which then outputs the final estimate of the volatility. Although these models perform well on specific stock datasets, they are often engineering-oriented and lack interpretability in a financially or economically meaningful way, ignoring important stylized facts such as the leverage and clustering effects commonly observed in financial time series data. An exception is the recurrent conditional heteroscedastic model (RECH) of nguyen_recurrent_2022, that provides a flexible framework for combining deep learning with GARCH-type models. The RECH model represents the volatility as a sum of two components. The first component is governed by a GARCH-type model that retains the characteristics and interpretability of econometric models. The second component is governed by a RNN that can capture non-linear and long-term serial dependence structure in financial time series. As demonstrated in nguyen_recurrent_2022, RECH retains much of the interpretable characteristics from the GARCH model, while enjoying the modeling flexibility and prediction accuracy from RNN.
This paper proposes a new approach to volatility modeling by combining deep learning and realized volatility measures. Our deep-learning-enhanced realized GARCH framework, under the acronym DeepRGARCH, incorporates and distills modeling advances from financial econometrics, high frequency trading data and deep learning. Inspired by RECH, we incorporate the long short-term memory model (LSTM) of hochreiter_long_1997 into RealGARCH. The LSTM architecture is one of the most advanced and sophisticated RNN techniques and has proven highly efficient for time series modeling. By incorporating LSTM into RealGARCH, we unlock its modeling power and allow it to be able to capture complex underlying dynamics in financial volatility. We compare the performance of the new model with several existing benchmark models on 31 stock market indices. We show that the new model substantially improves on previous approaches in both in-sample fit and out-of-sample forecasting. The code and examples reported in the paper can be found at \url{https://github.com/VBayesLab/DeepRGARCH}.
The rest of the article is organized as follows. Section (ref) briefly reviews the model components (the GARCH-type models, realized volatility measures and recurrent neural networks) before describing the proposed DeepRGARCH model. Section (ref) presents the Bayesian inference method for DeepRGARCH and Section (ref) presents the empirical analysis. Section (ref) concludes. The Appendix provides further implementation and empirical details.
Let $\boldsymbol y=\left\{y_t, t=1, \ldots, T\right\}$ be a time series of daily returns. The key term of interest in volatility modeling is the conditional variances, $\sigma_t^2=\operatorname{var}\left(y_t \mid \mathcal{F}_{t-1}\right),$ where $\mathcal{F}_{t-1}$ denotes the $\sigma$-field of information up to and including time $t-1$. We assume here that ${\mathbb E}(y_t|\mathcal{F}_{t-1})=0$, but the present method is easily extended to model the conditional mean ${\mathbb E}(y_t|\mathcal{F}_{t-1})$. The GARCH model expresses the conditional variance $\sigma_t^2$ as a linear combination of the previous squared returns and conditional variances as an ARMA($p$, $q$) model:
The restriction $\omega>0, \alpha_{i}, \beta_{j} \geq 0, i=1, \ldots, p, j=1, \ldots, q$ is used to ensure positivity of $\sigma_{t}^{2}$, and $\sum_{i=1}^{p} \alpha_{i}+\sum_{j=1}^{q} \beta_{j}<1$ is needed to ensure the stationarity of the time series $y_t$. The errors $\epsilon_t$ are independently and identically distributed as normal distributions with zero mean and unit variance; other distributions for $\epsilon_t$ such as Student's $t$ are also considered in the literature, e.g., gerlach_forecasting_2016. For other GARCH-type models, the reader is referred to nelson_conditional_1991, glosten_relation_1993 and bollerslev_glossary_2008. The GARCH model relies on daily squared returns, which only contain a weak signal of the daily volatility $\sigma_t^2$. It is widely known in the financial econometric literature that high-frequency return data, such as 5-minute data, can be used to estimate daily volatility with high accuracy. In the past twenty years, many estimators of daily volatility using high-frequency data were developed and are referred to as “realized volatility measures" andersen_answering_1998,barndorff-nielsen_power_2004,barndorff-nielsen_designing_2008. As realized volatility measures are ex-post, they cannot be directly used for volatility forecasting but they are effective volatility proxies for volatility modeling. engle_new_2002 is among the first to explore this idea by incorporating the realized volatility measure of andersen_answering_1998 into the GARCH model. Since then, many volatility models incorporating realized volatility measures have been developed, e.g., forsberg_bridging_2002, engle_multiple_2006, corsi_simple_2009, shephard_realising_2010. The realized GARCH model (RealGARCH) of hansen_realized_2012
is an important development in this direction. Here $\epsilon_{t}\stackrel{i.i.d}{\sim} N(0,1)$, $u_{t} \stackrel{i.i.d}{\sim} N\left(0, \sigma_{u}^{2}\right)$, $\text{\rm rv}_t$ is a realized volatility measure, and $\tau(\epsilon)$ is regarded as the leverage function and used to capture the leverage effect often observed in volatility. hansen_realized_2012 set $\tau(\epsilon)=\tau_{1} \epsilon+\tau_{2}\left(\epsilon^{2}-1\right)$. An attractive feature of the RealGARCH model is that it contains the measurement equation (ref) that accounts for the variation in the realized volatility measure $\text{\rm rv}_t$. It associates the observed realized volatility measure with the underlying latent volatility, in which the integrated high-frequency variance $\text{\rm rv}_t$ is explained as a linear combination of $\sigma_t^2$ plus a random innovation.
Our article employs a simple yet effective realized volatility measure, the 5-minute realized variance ($\text{RV}_5$) of andersen_answering_1998. Suppose that we observe the asset price at $n$ trading times within a trading day $t$, $t_{j}=t-1+j/n, j=1, \ldots, n$. Let $\{P(t_j),j=1,...,n\}$ be the observed prices and $r_{t_{j}}=\log P\left(t_{j}\right)-\log P\left(t_{j-1}\right)$ be the log-returns. The RV for the trading day $t$ is defined as
It can be shown that andersen_answering_1998, as $n\to\infty$, $\text{\rm rv}_t$ converges in probability to the true latent variance $\sigma_t^2$. For $\text{RV}_5$, the return $r_{t_j}$ in the above equation is recorded at 5 minutes frequency. There are various definitions of realized volatility measures but there is little evidence that any outperform $\text{RV}_5$ as a volatility proxy; see liu_does_2015 for a detailed comparison of more than 400 realized volatility measures.
RNN is a special class of neural network designed for modeling sequential data. Let $\{D_t=(x_t,y_t),t=1,2,...\}$ be the data with $x_t$ the input and $y_t$ the output. We use $(x_t,y_t)$ in this section as generic notation, not necessarily applicable to the return data in other sections. The task is to model the conditional mean $\widehat{y_t}={\mathbb E}(y_t|x_t,D_{1:t-1})$. The basic RNN framework is
The main feature of the RNN structure is its vector of hidden states $h_t$ which is defined recurrently. At each time $t$, two information sources are fed into $h_t$: the historical information stored in $h_{t-1}$ and the current information from the input $x_t$. The functions $g_h$ and $g_y$ are activation functions such as $\text{sig} (z):=1/(1+e^{-z})$, or $\text{tanh} (z):=(e^{z}-e^{-z})/(e^{z}+e^{-z})$. Finally, the $W$ and $b$ are trainable model parameters.
The basic RNN model in (ref)-(ref) has some limitations in terms of both modeling flexibility and training difficulty. Many sophisticated RNN structures are proposed to overcome these limitations, and the LSTM model of hochreiter_long_1997 stands out as one of the most successful methods. LSTM uses a gate structure to control the memory in the data. It is written as follows:
Unlike the basic RNN that fully overwrites the memory stored in the hidden states at each step, LSTM can decide to keep, forget or update the memory via the memory cell $c_t$ in (ref). This memory cell $c_t$ is updated by partially forgetting the previous memory from $c_{t-1}$ and adding new memory from $\tilde{c}_t$. The extent of forgetting the history and adding new information is controlled by the forget gate $g^f_t$ and input gate $g^i_t$, respectively. Finally, the degree of current memory usage for final output is controlled by the output gate $g^o_t$. See goodfellow_deep_2016, for a book length discussion of RNN and deep learning methods in general.
The ability of LSTM to control its memory and quickly adapt to new data patterns makes it very suitable to model volatility dynamics. It is well-known that volatility dynamics has long memory as well as a clustering effect in which the volatility can be highly volatile from period to period Benoit1967,Ding:1993,Christensen_Nielsen_long_memory. Section (ref) describes how to incorporate LSTM into volatility modeling.
This section presents our DeepRGARCH model, that incorporates LSTM into RealGARCH for flexible and accurate volatility modeling. Its key motivation is using an additive component governed by LSTM, that takes the realized measure as its input, to capture complex serial dependence structure in the volatility dynamics. As intraday realized volatility measures are accurate volatility proxies andersen_answering_1998,barndorff-nielsen_designing_2008, feeding them to the LSTM unit further empowers its modeling capacity. The DeepRGARCH model of order $p$ and $q$, DeepRGARCH($p,q$), is written as:
Compared to the RealGARCH model, equation (ref), that only allows a linear and short-term dependence of the true latent conditional variance $\sigma_t^2$ on the realized volatility measure, the DeepRGARCH model is much more flexible. The latter, via its LSTM component $g(\omega_t)$, allows for both non-linear and long-term dependence that the previous realized volatility measures might have on $\sigma_t^2$. As $\sigma_{t-1}^2$ is also an input to the LSTM component, DeepRGARCH also allows for complex serial dependence that the previous volatility might have on $\sigma_t^2$. We found empirical evidence for the inclusion of $\omega_{t-1}$ and $y_{t-1}$ in the input vector $x_{t}$. The function $g$ in (ref) is a non-negative activation function, applied to the $\omega_t$ to ensure the positive conditional variance. We adopt the RELU function, $g\left(\omega_t\right):=\max \left\{\omega_t, 0\right\}$, for this purpose.
Inspired by RealGARCH, DeepRGARCH includes the measurement equation (ref) to account for the variation in the realized volatility measures. Rather than using the linear structure as in (ref), one could easily use an RNN model to explain $\text{\rm rv}_t$ based on $\sigma_t^2$; we have tried this, however, did not observe any meaningful improvement. We only include one realized volatility measure in the DeepRGARCH model; however, it is easy to incorporate as many realized volatility measures as possible by using them as additional inputs into $x_t$. We only consider DeepRGARCH(1,1) in this paper, which, for ease of reading and later cross-reference, is expressed as
We adopt the Bayesian inference approach for the estimation of DeepRGARCH. Following hansen_realized_2012, we assume Gaussian errors $\epsilon_t \stackrel{i.i.d}{\sim} N(0,1)$ and $u_t \stackrel{i.i.d}{\sim} N(0, \sigma_u)$. Hence, given the observed data $(\boldsymbol y=\{y_1,...,y_T\},\boldsymbol{rv}=(\text{\rm rv}_1,...,\text{\rm rv}_T))$ the log-likelihood function of the DeepRGARCH model is:
where $u_t=\text{\rm rv}_t-\xi-\varphi \sigma_t^2-\tau_1 y_t/\sigma_t-\tau_2\left(y_t^2/\sigma_t^2-1\right)$. Recall that the vector of model parameters $\theta$ consists of the GARCH and LSTM parameters. For example, the DeepRGARCH(1,1) model with a single hidden state LSTM has 7 GARCH parameters, $(\beta, \gamma, \xi, \varphi, \tau_1, \tau_2,\sigma_{u})$, and 26 LSTM parameters including $\beta_0$, $\beta_1$ and 24 parameters within the LSTM structure.
For the prior distributions on the GARCH parameters, we use the commonly used priors in the literature; see, e.g., gerlach_forecasting_2016. The prior $N(0,0.1)$ is used for the LSTM parameters.
For Bayesian inference and prediction in volatility modeling, the Sequential Monte Carlo (SMC) method neal_annealed_2001,chopin_sequential_2002,del_moral_adaptive_2012 is an effective approach for computing rolling-window volatility forecasts which can effectively sample from non-standard posteriors, while also providing the marginal likelihood estimate as a by-product. The SMC technique uses a set of $M$ weighted particles initially sampled from an easy-to-sample distribution, such as the prior $p(\theta)$, with the particles then traversed through intermediate distributions which eventually become the target distribution. See gunawan_flexible_2022 for a review of the SMC method.
For in-sample model estimation and inference, we use the likelihood annealing version of SMC that samples from the sequence of distributions
here, $0=\gamma_0<\gamma_1<\gamma_2<\ldots<\gamma_K=1$ are called the temperature levels. Reweighting, resampling, and a Markov transition are the three primary components of the SMC approach. Several methods exist to implement SMC in practice, and we briefly describe one of them now. The collection of weighted particles $\left\{W_{k-1}^j, \theta_{k-1}^j\right\}_{j=1}^M$ that approximate the intermediate distribution $\pi_{k-1}(\theta)$ is reweighted at the start of iteration $k$ to approximate the target $\pi_k(\theta)$. The efficiency of these weighted particles is measured by the effective sample size (ESS) kass_markov_1998
where $W_t:=(W_t^1, \dots, W_t^M)$ is the weight vector for the $M$ particles at time $t$. If the ESS is below a threshold, the particles are resampled to obtain equally weighted particles, and a Markov kernel with the invariant distribution $\pi_k(\theta)$ is then applied to refresh these equally weighted particles. Following del_moral_adaptive_2012, we adaptively choose the tempering sequence $\gamma_k$ in order to maintain an adequate particle diversity.
For out-of-sample rolling-window volatility forecasting which updates the posterior each time a new data observation arrives, we employ the data annealing SMC method that samples from the sequence
This section studies the performance of the DeepRGARCH(1,1) model and compares it to GARCH, RealGARCH and RECH using 31 stock indices including the Amsterdam Exchange Index (AEX), the Dow Jones Index (DJI), the Frankfurt Stock Exchange (GDAXI) and the Standard and Poor’s 500 Index (SP500). The datasets were downloaded from the Realized Library of The Oxford-Man Institute. In the main text, we present results for the above mentioned four representative indices and the average over 31 indices to conserve space; the detailed results for all 31 indices are shown in the appendix. Given the closing prices $\{P_{t},t=0,...,T\}$, we compute the demeaned close-to-close return process as
We adopt the 5 mins realized variance andersen_modeling_2003 as the realized volatility measure, $\text{\rm rv}_t$, in RealGARCH and DeepRGARCH. As the realized volatility measures ignore the overnight variation of the prices and sometimes the variation in the first few minutes of the trading day when recorded prices may contain large errors shephard_realising_2010, we follow hansen_forecast_2005 and scale the realized volatility measure as
and use $\tilde{\sigma}_t^2$ as the estimate of the latent conditional variance $ \sigma_t^2$. Our sample is from 1 January 2004 to 1 January 2022, including the COVID-19 pandemic period, which we divide it into a first half for training and the second half for out of sample analysis. There are, on average, 2139 trading days for both. See appendix for a detailed data description.
As mentioned in Section (ref), the marginal likelihood estimate is a by-product of SMC, which is useful for in-sample model comparison using a Bayes factor. Given the training data $(\boldsymbol y = y_{1:T},\boldsymbol{rv} = \text{\rm rv}_{1:T})$, let $\widehat{p}\left(\boldsymbol y,\boldsymbol{rv}|M_i\right)$ be the marginal likelihood estimate of model $M_i$, $i=1,2$. The Bayes factor of model $M_1$ relative to $M_2$ is
The higher the Bayes factor, the more decisive is the evidence that model $M_1$ is superior to $M_2$ kass_bayes_1995. Table (ref) summarizes the parameter estimation (for the five main parameters) and the Bayes factor (with GARCH used as the baseline model) for the four models, GARCH, RealGARCH, RECH and DeepRGARCH. The last panel reports the results for the average over the 31 indices.
We draw the following conclusions from the estimation results. First, the marginal likelihood estimates show that the DeepRGARCH model fits the index datasets better than the competing models for 28 of 31 indices. On average, the Bayes factors of the DeepRGARCH model compared with the GARCH, RECH, and RealGARCH models are roughly $e^{58}$, $e^{30}$ and $e^{14}$, respectively, which according to Jeffery's scale for interpreting the Bayes factor jeffreys_theory_1998, decisively support the DeepRGARCH model. Second, the estimated posterior mean of the parameter $\beta_1$ in the DeepRGARCH model is significantly different from zero in all cases, providing evidence of the volatility effects rather than linearity that the previous conditional variance $\sigma_{t-1}^2$ and realized volatility measure $\text{\rm rv}_{t-1}$ have on $\sigma_t^2$. This also suggests that the LSTM component $g(\omega_t)$ in DeepRGARCH can detect these effects effectively. Additionally, the estimated value of the parameter $\gamma$ (the coefficient concerning the realized volatility measure) in DeepRGARCH is always smaller than that in RealGARCH. This is perhaps because the effect of the realized volatility measure $\text{\rm rv}_{t-1}$ on the underlying volatility $\sigma_t^2$ is well captured by the LSTM component, which, unlike RealGARCH, allows non-linear dependence of $\sigma_t^2$ on $\text{\rm rv}_{t-1}$.
We now test the predictive performance of the DeepRGARCH model for volatility forecasting. The forecast performance is measured by mean squared error (MSE) and mean absolute deviation (MAD) computed on test data $D_{test}$ of size $T_{test}$
where $\widehat{\sigma}_t$ is the one-step-ahead rolling-window forecast of the latent $\sigma_t$ and $\widetilde{\sigma_t}$ is the squared root of an ex-post realized volatility measure after rescaling as in (ref). We use five ex-post volatility proxies, the Realized Variance (RV), Bipower Variation (BV), Median Realized Volatility (MedRV), Realized Kernel Variance with the Non-flat Parzen kernel (RK-Parzen) and the Tukey-Hanning kernel (RSV). See shephard_realising_2010 for details about the Realized Library.
Tables (ref) and (ref) summarize the forecast performance of the four models. The last column in each table shows the number of times a model has the best predictive score across the five volatility proxies. The last panel in each table shows the average scores over the 31 indices. Using the five ex-post realized volatility measures as the ground truth, the results show that the DeepRGARCH model performs the best in terms of volatility forecasting.
An attractive feature of GARCH-type models including DeepRGARCH is that they model both the volatility and return processes. This feature is crucial to risk management which is one of the most important applications of volatility modeling. For risk management, the key task is to forecast the Value at Risk (VaR) and Expected Shortfall (ES). An $\alpha$-level VaR is the $\alpha$-level quantile of the distribution of the return, and the $\alpha$-level ES is the conditional expectation of return values that exceed the corresponding $\alpha$-level VaR. Both VaR and ES are used as the two key risk measures in financial regulation and recommended by the Basel Accord.
To evaluate the quality of the VaR forecast, we adopt the standard quantile loss function
where $Q_t^\alpha$ is the forecast of $\alpha$-level VaR of $y_t$ koenker_regression_1978. We note that, given the volatility $\sigma_t$ and the normality assumption of the random shock $\epsilon_t$ in (ref), it is straightforward to compute $Q_t^\alpha$. The quantile loss function is strictly consistent fissler_higher_2016, i.e., the expected loss is lowest at the true quantile series. The most accurate VaR forecasting model should therefore minimize the quantile loss function.
There is no strictly consistent loss function for ES; however, fissler_higher_2016 found that ES and VaR are jointly elicitable, i.e., there is a class of strictly consistent loss functions for evaluating VaR and ES forecasts jointly. taylor_forecasting_2019 showed that the negative logarithm of the likelihood function built from the Asymmetric Laplace (AL) distribution is strictly consistent for VaR and ES considered jointly and fits into the class developed by fissler_higher_2016. This AL based joint loss function is given as
with $\mathrm{ES}_t^\alpha$ the forecast of $\alpha$-level ES of $y_t$.
To measure the fit of a volatility model to the return series, we also consider the Partial Predictive Score (PPS), which is one of the most commonly used metrics for evaluating predictive performance in statistical modeling. It measures the negative log-likelihood of observing the return series based on our volatility forecast. The model with the smallest PPS is preferred. The PPS score is defined as
Table (ref) reports the quantile loss, joint loss for 1% and 5% VaR and ES forecast, and PPS score. The Count panel reports the number of indices where a model achieves the best predictive score. The results show that for most of the indices, the DeepRGARCH model produces the best VaR and ES forecasts. DeepRGARCH is also superior to its competitors in terms of overall fit to the return series. It reports the lowest PPS on most returns series and on average. Additionally, we find that for most of the metrics, the RECH model performs the best on more indices than RealGARCH. In contrast, in the previous section, RealGARCH dominates RECH regarding forecast error compared ex-post proxies. This suggests that ranking conditional volatility forecasts by ex-post proxies can sometimes lead to undesirable outcomes, and we should also evaluate predictive performance by economic loss.
Apart from risk management, options trading is also one of the most attractive applications of volatility forecasting. In a theoretical pricing model, volatility is the most difficult input for traders to predict among all the inputs required for option evaluation. At the same time, volatility often plays the most crucial role in actual trading decisions. Consider the inputs of a Black-Scholes model for European options:
Volatility is the only unknown input here; hence the profitability of an options trader is greatly affected by their ability to forecast volatility. This section examines model performance based on its ability to price options correctly. We follow engle_valuation_1990 to design a hypothetical option market where each agent uses their volatility forecast and the Black-Scholes model to price options and trade with competing agents. The experiment is organized as follows:
Table (ref) reports the daily, annual profit (in cents) and Sharpe ratio of the options agents that use GARCH, RealGARCH, RECH and DeepRGARCH as their forecast models. The simulations are repeated on 31 stocks indices and the following four scenarios:
In scenario (1), where all agents are trading against each other, the DeepRGARCH agent generates the highest profits and Sharpe ratio in most stock indices on average. Scenarios (2), (3) to (4) further illustrate the consistent profitability of the DeepRGARCH agent against its two direct ancestors, RealGARCH and RECH, in our hypothetical market. A surprising finding is that the RealGARCH performs almost as badly as the GARCH model.
The previous sections show that DeepRGARCH outperforms the competing models in terms of in-sample fit, forecasting error, tail risk forecast and option pricing. This section tests whether these improvements are statistically significant using the Model Confidence Set (MCS) introduced by hansen_model_2011. Let $\mathcal M$ be a set of competing models. A set of superior models (SSM) is established under the MCS procedure, which consists of a series of equal predictive accuracy tests given a specific confidence level. Let $L_{i,t}$ be a performance loss, such as the MSE or quantile loss, incurred by model $i\in \mathcal M$ at time $t$. Define $d_{i,j,t}=L_{i,t}-L_{j,t}$ to be the relative loss of model $i$ compared to model $j$ at time $t$. The MCS test assumes that $d_{i,j,t}$ is a stationary time series for all $i,j$ in $\mathcal M$, i.e., $\mu_{i,j}={\mathbb E}(d_{i,j,t})$ for all $t$. By testing the equality of the expected loss difference $\mu_{i,j}$, MCS determines if all models have the same level of predictive accuracy. The null hypothesis is
A model is eliminated when the null hypothesis $H_{0}$ of equal forecasting ability is rejected. The collection of models that do not reject the null hypothesis $H_{0}$ is then defined as the SSM. For each model $i\in\mathcal M$, the MCS produces a $p$-value $p_i$. The lower the $p$-value of a model, the less likely that it will be included in the SSM. See hansen_model_2011 for more details.
Table (ref) reports the model confidence sets computed by all the predictive scores. For each model, we report the total number of times, across the 31 indices, that the model is included in the MCS and its average $p$-value (in parentheses). We note that a small $p$-value indicates that the model is unlikely to be the best model hansen_model_2011. At a 75% confidence level and across the predictive scores, the DeepRGARCH models are included in SSM for 24 indices, and have the highest $p$-value of 1 for 20 indices. The result shows that DeepRGARCH is the most likely to be included in the SSM and have statistically significant superiority over the other models in all the considered predictive metrics: forecasting error (MSE and MAD), fit to return series (PPS), risk forecast (quantile loss and joint loss) and option pricing.
The paper presents a novel methodology for volatility modeling, effectively harnessing the strengths of deep learning and the invaluable insights from realized volatility data. Bayesian inference and forecasting are performed using Sequential Monte Carlo with likelihood and data annealing. The DeepRGARCH model is evaluated on 31 major world stock markets, demonstrating improved performance in statistical criteria (in-sample fit, forecast error to realized measures) and economic criteria (tail risk forecast and option pricing) compared to the standard GARCH, RECH, and RealGARCH models.
Our proposed framework could be extended in several ways. First, it is possible to include multiple realized volatility measures since hansen_exponential_2016 discovered that including multiple realized volatility measures evidently improves both the in-sample and out-of-sample fit. The multiple measurement equations then could be replaced by a single LSTM with multiple outputs which would allow us to capture the non-linear relationship between conditional volatility and realized volatility measures as well as the interactions between the realized volatility measures. Second, incorporating financial news into the input of DeepRGARCH is an interesting topic to study, as news has been shown to have a major influence on volatility movement (atkins_financial_2018, xing_sentiment-aware_2019, rahimikia_realised_2021). Lastly, incorporating transfer learning would be a natural extension. Transfer learning can help mitigate the problem of insufficient training data and enable us to use a large number of pre-trained deep learning models and traditional financial econometric models for volatility modeling.
\printbibliography \singlespacing