Augmenting the Realized-GARCH: the role of signed-jumps, attenuation-biases and long-memory effects
-
Ioannis Papantonis
,
Leonidas S. Rompolis
Abstract
This paper extends the Realized-GARCH framework, by allowing the conditional variance equation to incorporate exogenous variables related to intra-day realized measures. The choice of these measures is motivated by the so-called heterogeneous auto-regressive (HAR) class of models. Our augmented model is found to outperform both the Realized-GARCH and the various HAR models in terms of in-sample fitting and out-of-sample forecasting accuracy. The new model specification is examined under alternative parametric density assumptions for the return innovations. Non-normality seems to be very important for filtering the return innovations to which variance responds and helps significantly upon the prediction performance of the suggested model.
1 Introduction
Modelling and accurately predicting the volatility of asset returns is a key issue behind many financial applications, such as portfolio allocation, risk management and option pricing. Since Engle (1982) and Bollerslev (1986), a first “generation” of volatility models evolved and has been extensively used; the well-documented generalized autoregressive conditional heteroscedasticity (GARCH) class of models. The simple autoregressive structure of these models provided an easy way to capture clusters in the latent volatility dynamics of returns. Numerous different parametric GARCH-type specifications have been suggested to extend the GARCH models, leading to more sophisticated volatility models which incorporate leverage/feedback effects, long-memory, regime-switches, among other stylized properties of volatility.[1]
Normally, standard GARCH models (and their extensions) use daily returns to extract information about the current level of volatility, treating volatility itself as unobservable. However, more recent advances in the high-frequency financial econometrics literature utilized intra-day data to construct observable proxies of variance, the so-called as realized variance (RV) estimators. These observable variance measures gave rise to a new econometric modelling framework, aiming to either incorporate this information into GARCH models or directly model the realized variance process. Engle (2002) and Engle and Gallo (2006), first, used RV in a Multiplicative Error Model (MEM) context, while Shephard and Sheppard (2010) presented a so-called HEAVY (high-frequency volatility) model for jointly modelling latent and observable variance dynamics. The most popular and easily applicable models of this research-strand were the HAR-type realized variance regressions popularized by Corsi (2009) and the Realized-GARCH framework of Hansen, Huang, and Shek (2012). These models, which we can think of as a second generation of volatility models, have been shown to boost the forecasting performance of the variance process, especially when a realized variance proxy is included in the GARCH.[2]
One of the most important benefits of GARCH over HAR-type models is that they can jointly model the dynamics of returns and variance.[3] Hence, in this paper we employ and effectively build on the Realized-GARCH framework of Hansen, Huang, and Shek (2012). We first extend the Realized-GARCH model to incorporate additional intra-day realized measures capturing different volatility asymmetry sources, i.e. asymmetries originating from both responses of variance to lagged return innovations and/or short-lived effects from realized upside and downside intra-day variations. In addition to these measures, we also consider extensions including a volatility-of-volatility measure, which can correct for attenuation-biases in measuring realized variance, as well as heterogeneous terms of realized variance, which approximate long-memory properties of variance. These extensions are well-justified by the literature on HAR models. We show that the inclusion of these metrics in parametric GARCH specifications may improve the efficiency of the conditional variance estimates. We therefore refer to these extensions of the Realized-GARCH model as a third generation of volatility models.
We empirically assess the ability of the above new extensions to improve upon the performance of the standard Realized-GARCH models to fit the data and predict realized volatility. We use intra-day data of the S & P500 data and we allow return innovations to follow alternative parametric probability density functions (PDFs), apart from the Normal (N) that is often used in practice. We consider the cases of the skewed-GED (SGED) and normal-inverse-Gaussian (NIG) PDFs that have been shown to outperform other parametric densities in fitting financial return data.[4] These PDFs are more flexible and can account for more pronounced levels of skewness and excess kurtosis in asset returns. Therefore, they can help to better filter the return innovations, and hence to improve the model-inference for both returns and variance.
Two main conclusions can be drawn from the results of our empirical analysis. First, the model-extensions that we suggest seem to improve both the in- and out-of-sample performance of the model to predict realized volatility compared to the standard Realized-GARCH. This finding is justified by a number of goodness-of-fit and prediction-accuracy metrics reported, as well as by a series of equal-prediction performance tests evaluating the relative out-of-sample forecasting performance of the models. Our results show that the significant improvement in the prediction performance of the model comes primarily from the inclusion of the upside and downside intra-day movements of realized variance. Second, allowing for asymmetric/fat-tailed return distributions also plays a crucial role in the accurate filtering of the innovations, and consistently helps the identification of the parameters of the volatility process. This boosts the prediction performance of all the augmented GARCH-type specifications that we consider in our analysis.
The remainder of the paper is organized as follows. Section 2 presents a generic framework for the joint dynamics of returns and variance and specifies the conditional variance model suggested by the paper. Section 3 provides details on the joint estimation procedure that we follow. Section 4 provides a detailed discussion on both the in-sample fit and the out-of-sample forecasting performance of the models. Section 5 concludes the paper.
2 The model
Let us write the continuously-compounded daily returns as r
t
= p
t
− pt−1, where p
t
= log(P
t
) is the log-price of an observed price process P
t
of a financial asset. The conditional PDF of r
t
is described by
Under any (generic) parametric specification for the returns r
t
and log-variance v
t
dynamics, we can write the joint conditional PDF as
where
2.1 Conditional variance function specifications
In this section we introduce a new conditional variance framework which nests the standard GJR threshold GARCH threshold specification (Glosten, Jagannathan, and Runkle 1993) and the HAR family of RV models. We can write the variance function as follows:
where
is the news impact function (NIF) of variance (with b3 capturing the variance asymmetry induced by the well-documented leverage effect),[7] and
is the component of h t driven by exogenous variables related to intra-day realized measures. Note that this NIF specification τ(t) is more flexible than the one in Hansen’s Realized GARCH, as it can also accommodate for asymmetric volatility effects.
The rationale behind the choice of the realized measures included in x(t) (Eq. (4)) is motivated by recent empirical findings, and primarily from the literature of HAR models aiming to predict the dynamics of RV. We first include RV as in Hansen, Huang, and Shek (2012), which has been shown to enhance the forecasting performance of conditional/realized variance models. Next, we consider the realized upside and downside semi-variances, denoted as RV+ and RV−, respectively. Patton and Sheppard (2015) provide clear evidence supporting the decomposition of RV into its two semi-variance components, as it can be beneficial for volatility forecasting when there are prevalent asymmetries in high-frequency data. They use an HAR-type regression framework to show that the response of RV to the two semi-variances can differ, with the negative realized semi-variance being more informative for predicting future RV. At this point, we should highlight that semi-variances have never been used before into a GARCH-based framework, and therefore it would be interesting to examine (see Section 4) whether the parametric NIF component of the model can absorb similar information to the two semi-variances regarding the asymmetric responses of variance.
Moreover, we incorporate in x(t) the 5- and 20-day (weekly and monthly) moving-average terms of RV, i.e. RV Andersen et al. (2009) and RV Bollerslev et al. (2009), which have been used in the HAR literature to approximate the long-memory behaviour of RV.[8] Interestingly, the inclusion of these so-called “heterogeneous” terms in x(t) and, hence, in the conditional variance model (2) allows capturing long-memory patterns of h t exogenously, without affecting the parsimonious parametric structure of the model (see also Huang, Liu, and Wang (2016)). Baillie et al. (2019) provide a very detailed discussion on the long-memory property of RV and its connection with the heterogeneous terms.
Last, the term
The above general framework, to which we henceforth refer as GARCH-HAR-X class and is described by Eqs. (2)–(4), nests a number of specifications suggested in the literature for modelling conditional and/or realized variance dynamics. These nested specifications include several HAR-type models, as well as the Realized-GARCH and its extensions produced by the additional exogenous realized measures; i.e. from the factors included in x(t). More specifically, for b1 = b2 = b3 = c1 = c2 = c5 = 0 we obtain the standard HAR model of Corsi (2009). Similarly, for b1 = b2 = b3 = c0 = c5 = 0, we get the more flexible SHAR model of Patton and Sheppard (2015), where the capital letter “S” indicates the inclusion of semi-variances in (4). Finally, for c1 = c2 = c3 = c4 = c5 = 0, the model reduces to the Realized-GARCH, which for simplicity in this paper we denote as GARCH-R. Again, note that our GARCH-R is more generic than the standard Realized-GARCH as it allows for volatility asymmetries through the b3 term in the NIF (3).
We further utilize this framework to study several extensions to the aforementioned existing models. First, augmenting the standard HAR model to include information regarding the realized variance-of-variance (setting b1 = b2 = b3 = c1 = c2 = 0) yields a representation that we denote as HARQ, where the capital letter “Q” reflects the realized quarticity term in (3).[9] This can be combined with the two semi-variances (when b1 = b2 = b3 = c0 = 0) to generate the SHARQ model.
Next, if we allow for b2 ≠ 0 and (possibly) b3 ≠ 0, the above specifications of the HAR model (namely, the HAR, SHAR, HARQ and SHARQ) are extended to include information from the filtered return innovations. This extension allows capturing magnitude and leverage effects on RV, through out return innovations functions
Furthermore, we present extensions of the GARCH-R model nested in the above framework. For c0 = c3 = c4 = c5 = 0 we obtain the GARCH-S model which augments the GARCH-R by allowing for upside and downside semi-variances. When c1 = c2 = c5 = 0 and c0 = c5 = 0, we obtain the GARCH-HAR and GARCH-SHAR models, respectively. These two specifications include the heterogeneous (moving-average) terms of RV, as in the HAR-type models literature. Finally, for c1 = c2 = 0 and c0 = 0, we get the GARCH-HARQ and GARCH-SHARQ models, which incorporate realized quarticity information in the GARCH-HAR and GARCH-SHAR models. Note that the GARCH-SHARQ model constitutes what we call the “full” specification produced by Eqs. (2)–(4), and it encompasses all the specifications considered. We will use GARCH-SHARQ as benchmark for future model comparisons, especially vis-a-vis the GARCH-R and the augmented HAR-type representations using the same exogenous variables. The parameter restrictions for the nested specifications that we discussed are also summarized in Table 1.
Nested models within the GARCH-HAR-X class.
| h t |
|
|
RV t |
|
|
|
|
|
|
|---|---|---|---|---|---|---|---|---|---|
| HAR | – | – | – | X | – | – | X | X | – |
| SHAR | – | – | – | – | X | X | X | X | – |
| HARz | – | X | X | X | – | – | X | X | – |
| SHARz | – | X | X | – | X | X | X | X | – |
| HARQ | – | – | – | X | – | – | X | X | X |
| SHARQ | – | – | – | – | X | X | X | X | X |
| HARQz | – | X | X | X | – | – | X | X | X |
| SHARQz | – | X | X | – | X | X | X | X | X |
| GARCH-R | X | X | X | X | – | – | – | – | – |
| GARCH-S | X | X | X | – | X | X | – | – | – |
| GARCH-HAR | X | X | X | X | – | – | X | X | – |
| GARCH-SHAR | X | X | X | – | X | X | X | X | – |
| GARCH-HARQ | X | X | X | X | – | – | X | X | X |
| GARCH-SHARQ | X | X | X | – | X | X | X | X | X |
-
This table presents the variables included in the different models nested within the GARCH-HAR-X class. The first column corresponds to the model abbreviations used throughout the paper. Recall that
3 Data & estimation
3.1 Data
We implement our generalized Realized-GARCH-HAR framework on the S&P500 index.[10] We obtain ultra-high-frequency data (UHF) from the trade and quote (TAQ) Database. This allows us to monitor all the recorded tick-level data on the S&P500 index, at millisecond-level precision. Our dataset covers the period of 1995–2016. Given the UHF data on the index, we construct evenly-spaced observations of 5 min intervals for the trading hours between 9:30 and 16:00 (EST) by taking the last price that was recorded within the previous 5 min period. This gives us 78 intra-day observations per day, which we use to calculate the realized measures. In case there is no trading in a specific interval, then the corresponding return is set to zero.
To calculate the realized measure of variance RV t , included in relation (4), we rely on the heavily-used estimator (see Andersen et al. (2001), Barndorff-Nielsen and Shephard (2002a), Barndorff-Nielsen and Shephard (2002b), Barndorff-Nielsen and Shephard (2004a), Barndorff-Nielsen and Shephard (2004b), and Barndorff-Nielsen and Shephard (2006) among others):
where
respectively. The RV decomposition simply implies that
which, in the absence of jumps, it can be shown to provide a measure proportional to the asymptotic variance of RV; see Barndorff-Nielsen and Shephard (2002a) and Andersen, Bollerslev, and Diebold (2007).
3.2 A joint maximum likelihood estimation approach
We apply a joint maximum likelihood estimation (MLE) of returns r t and log-quadratic variation v t on the system of equations defined by (1a) and (1b), where v t is approximated by log(RV t ) and conditional variance h t is described by (2)–(4). Responding to heavily-documented evidence that return distributions exhibit pronounced skewness and/or kurtosis levels, we consider different parametric PDFs for the return innovations z t . Beyond the Normal (N) distribution, we consider the cases of the skewed-GED (SGED), as well as the very flexible normal-inverse-Gaussian (denoted as NIG).[11]
Both the SGED and the (standardized) NIG are two-parameter distributions that have been shown to outperform other parametric distributions in fitting financial return data. This is because they are able to span a significantly wider domain of theoretically feasible skewness-kurtosis combinations, compared to other conventional parametric densities; see discussion in Jondeau and Rockinger (2003).[12] They also nest a range of popular parametric densities (including the Normal) as limiting cases. Capturing asymmetry and fat-tails in returns helps to better filter the return innovations and hence to improve the model-inference for both returns and variance. Ignoring these features might induce inefficiency and bias in the parameter estimates which would impact both the in-sample and out-of-sample performance of the system of Eqs. (1a) and (1b); see discussion in Papantonis, Rompolis, and Tzavalis (2021).
On the other hand, evidence suggests that the distribution of log(RV) is approximately Normal; see Andersen et al. (2001) and Andersen et al. (2003). Therefore, it follows naturally to assume that the innovation term u in Eq. (1b) is Normally distributed, i.e. u ∼ N(0, 1). Also, note that working with log(RV), instead of RV, absorbs almost entirely the heteroscedasticity in the error term of Eq. (1b); see Bollerslev et al. (2009). Simulation results in Papantonis (2016) and Papantonis, Rompolis, and Tzavalis (2021) show significant improvements in terms of bias and efficiency for the parameter estimates of volatility models when a similar joint MLE framework is considered.
The joint MLE provides a very straightforward and robust method of obtaining parameter estimates and has been used in numerous similar applications (see Hansen, Huang, and Shek (2012), Hansen and Huang (2016), Papantonis (2016), Papantonis, Rompolis, and Tzavalis (2021) and references therein). The joint log-likelihood function that we maximize can be simply expressed as follows:
where T is the sample size. The total log-likelihood
The standard errors of the parameter estimator
where the Hessian matrix H is obtained numerically. The standard errors of
4 Estimation results
4.1 In-sample estimates
The in-sample MLE parameter estimates, along with their standard errors in parenthesis, are presented in Tables 2–4. Each table presents results under the three alternative density specifications for the return innovation term z
t
that we consider; i.e. for the Normal (Table 2), SGED (Table 3) and NIG (Table 4) distributions. Note that the tables include not only the results of the full model, but also those of the several nested specifications discussed in Section 2. For all the specifications considered, the tables report the values of both components of
Full sample maximum likelihood parameter estimates for linear GARCH-HAR-X(N) models.
| HAR | SHAR | HARz | SHARz | HARQ | SHARQ | HARQz | SHARQz | |
|---|---|---|---|---|---|---|---|---|
| μ |
|
|
|
|
|
|
|
|
| b 0 |
|
|
|
|
|
|
|
|
| b 1 | – | – | – | – | – | – | – | – |
| b 2 | – | – |
|
|
– | – |
|
|
| b 3 | – | – |
|
|
– | – |
|
|
| c 0 |
|
– |
|
– |
|
– |
|
– |
| c 1 | – |
|
– |
|
– |
|
– |
|
| c 2 | – |
|
– |
|
– |
|
– |
|
| c 3 |
|
|
|
|
|
|
|
|
| c 4 |
|
|
|
|
|
|
|
|
| c 5 | – | – | – | – |
|
|
|
|
| η | – | – | – | – | – | – | – | – |
| λ | – | – | – | – | – | – | – | – |
|
|
17,696.22 | 17,718.79 | 17,704.42 | 17,709.31 | 17,732.42 | 17,740.41 | 17,732.05 | 17,732.45 |
|
|
−5019.80 | −4871.16 | −4863.74 | −4841.68 | −4768.40 | −4725.40 | −4663.16 | −4662.86 |
|
|
12,676.42 | 12,847.63 | 12,840.68 | 12,867.64 | 12,964.02 | 13,015.02 | 13,068.89 | 13,069.59 |
| AIC | −25,342.84 | −25,683.27 | −25,667.36 | −25,719.27 | −25,916.03 | −26,016.03 | −26,121.78 | −26,121.18 |
| BIC | −25,309.75 | −25,643.55 | −25,621.03 | −25,666.32 | −25,876.31 | −25,969.69 | −26,068.82 | −26,061.60 |
| K | 5 | 6 | 7 | 8 | 6 | 7 | 8 | 9 |
| GARCH-R | GARCH-S | GARCH-HAR | GARCH-SHAR | GARCH-HARQ | GARCH-SHARQ | |||
| μ |
|
|
|
|
|
|
||
| b 0 |
|
|
|
|
|
|
||
| b 1 |
|
|
|
|
|
|
||
| b 2 |
|
|
|
|
|
|
||
| b 3 |
|
|
|
|
|
|
||
| c 0 |
|
– |
|
– |
|
– | ||
| c 1 | – |
|
– |
|
– |
|
||
| c 2 | – |
|
– |
|
– |
|
||
| c 3 | – | – |
|
|
|
|
||
| c 4 | – | – |
|
|
|
|
||
| c 5 | – | – | – | – |
|
|
||
| η | – | – | – | – | – | – | ||
| λ | – | – | – | – | – | – | ||
|
|
17,758.16 | 17,775.49 | 17,748.91 | 17,765.45 | 17,774.01 | 17,777.40 | ||
|
|
−4803.51 | −4755.93 | −4739.31 | −4691.87 | −4576.14 | −4568.83 | ||
|
|
12,954.64 | 13,019.56 | 13,009.61 | 13,073.58 | 13,197.88 | 13,208.57 | ||
| AIC | −25,897.29 | −26,025.11 | −26,003.21 | −26,129.15 | −26,377.75 | −26,397.13 | ||
| BIC | −25,857.57 | −25,978.78 | −25,950.26 | −26,069.58 | −26,318.18 | −26,330.94 | ||
| K | 6 | 7 | 8 | 9 | 9 | 10 |
-
This table presents maximum likelihood estimation (MLE) results for the GARCH-HAR-X class of models defined in Table 1 assuming that the return innovations follow the Normal distribution.
Full sample maximum likelihood parameter estimates for linear GARCH-HAR-X(SGED) models.
| HAR | SHAR | HARz | SHARz | HARQ | SHARQ | HARQz | SHARQz | |
|---|---|---|---|---|---|---|---|---|
| μ |
|
|
|
|
|
|
|
|
| b 0 |
|
|
|
|
|
|
|
|
| b 1 | – | – | – | – | – | – | – | – |
| b 2 | – | – |
|
|
– | – |
|
|
| b 3 | – | – |
|
|
– | – |
|
|
| c 0 |
|
– |
|
– |
|
– |
|
– |
| c 1 | – |
|
– |
|
– |
|
– |
|
| c 2 | – |
|
– |
|
– |
|
– |
|
| c 3 |
|
|
|
|
|
|
|
|
| c 4 |
|
|
|
|
|
|
|
|
| c 5 | – | – | – | – |
|
|
|
|
| η |
|
|
|
|
|
|
|
|
| λ |
|
|
|
|
|
|
|
|
| Skew | −0.1817 | −0.1810 | −0.1816 | −0.1809 | −0.1854 | −0.1831 | −0.1825 | −0.1823 |
| Kurt | 3.9491 | 3.9375 | 3.9595 | 3.9536 | 3.9242 | 3.9183 | 3.9362 | 3.9356 |
|
|
17,828.06 | 17,849.71 | 17,836.58 | 17,840.82 | 17,861.87 | 17,868.51 | 17,861.68 | 17,861.88 |
|
|
−4988.08 | −4841.32 | −4830.21 | −4808.78 | −4739.42 | −4696.98 | −4632.22 | −4631.92 |
|
|
12,839.97 | 13,008.39 | 13,006.37 | 13,032.04 | 13,122.45 | 13,171.54 | 13,229.46 | 13,229.97 |
| AIC | −25,665.95 | −26,000.77 | −25,994.74 | −26,044.08 | −26,228.91 | −26,325.07 | −26,438.92 | −26,437.93 |
| BIC | −25,619.61 | −25,947.82 | −25,935.16 | −25,977.88 | −26,175.95 | −26,265.50 | −26,372.73 | −26,365.12 |
| K | 7 | 8 | 9 | 10 | 8 | 9 | 10 | 11 |
| GARCH-R | GARCH-S | GARCH-HAR | GARCH-SHAR | GARCH-HARQ | GARCH-SHARQ | |||
| μ |
|
|
|
|
|
|
||
| b 0 |
|
|
|
|
|
|
||
| b 1 |
|
|
|
|
|
|
||
| b 2 |
|
|
|
|
|
|
||
| b 3 |
|
|
|
|
|
|
||
| c 0 |
|
– |
|
– |
|
– | ||
| c 1 | – |
|
– |
|
– |
|
||
| c 2 | – |
|
– |
|
– |
|
||
| c 3 | – | – |
|
|
|
|
||
| c 4 | – | – |
|
|
|
|
||
| c 5 | – | – | – | – |
|
|
||
| η |
|
|
|
|
|
|
||
| λ |
|
|
|
|
|
|
||
| Skew | −0.1800 | −0.1790 | −0.1795 | −0.1777 | −0.1815 | −0.1799 | ||
| Kurt | 3.9047 | 3.8863 | 3.9141 | 3.8962 | 3.8921 | 3.8885 | ||
|
|
17,875.72 | 17,889.54 | 17,870.04 | 17,882.95 | 17,893.30 | 17,895.70 | ||
|
|
−4771.74 | −4725.34 | −4707.22 | −4660.95 | −4546.11 | −4538.94 | ||
|
|
13,103.98 | 13,164.20 | 13,162.83 | 13,222.00 | 13,347.19 | 13,356.76 | ||
| GARCH-R | GARCH-S | GARCH-HAR | GARCH-SHAR | GARCH-HARQ | GARCH-SHARQ | |||
| AIC | −26,191.96 | −26,310.40 | −26,305.65 | −26,422.00 | −26,672.39 | −26,689.52 | ||
| BIC | −26,139.01 | −26,250.83 | −26,239.46 | −26,349.19 | −26,599.57 | −26,610.08 | ||
| K | 8 | 9 | 10 | 11 | 11 | 12 |
-
This table presents maximum likelihood estimation (MLE) results for the GARCH-HAR-X class of models defined in Table 1 assuming that the return innovations follow the SGED distribution.
Full sample maximum likelihood parameter estimates for linear GARCH-HAR-X(NIG) models.
| HAR | SHAR | HARz | SHARz | HARQ | SHARQ | HARQz | SHARQz | |
|---|---|---|---|---|---|---|---|---|
| μ |
|
|
|
|
|
|
|
|
| b 0 |
|
|
|
|
|
|
|
|
| b 1 | – | – | – | – | – | – | – | – |
| b 2 | – | – |
|
|
– | – |
|
|
| b 3 | – | – |
|
|
– | – |
|
|
| c 0 |
|
– |
|
– |
|
– |
|
– |
| c 1 | – |
|
– |
|
– |
|
– |
|
| c 2 | – |
|
– |
|
– |
|
– |
|
| c 3 |
|
|
|
|
|
|
|
|
| c 4 |
|
|
|
|
|
|
|
|
| c 5 | – | – | – | – |
|
|
|
|
| η |
|
|
|
|
|
|
|
|
| λ |
|
|
|
|
|
|
|
|
| Skew | −0.2819 | −0.2768 | −0.2867 | −0.2818 | −0.2893 | −0.2830 | −0.2902 | −0.2895 |
| Kurt | 3.1602 | 3.1549 | 3.1661 | 3.1605 | 3.1697 | 3.1626 | 3.1709 | 3.1701 |
|
|
18,026.91 | 18,045.10 | 18,035.71 | 18,038.15 | 18,060.83 | 18,065.43 | 18,061.25 | 18,061.25 |
|
|
−4953.26 | −4808.52 | −4796.64 | −4775.34 | −4708.11 | −4666.18 | −4601.30 | −4600.98 |
|
|
13,073.66 | 13,236.58 | 13,239.07 | 13,262.81 | 13,352.72 | 13,399.25 | 13,459.96 | 13,460.27 |
| AIC | −26,133.31 | −26,457.16 | −26,460.13 | −26,505.62 | −26,689.45 | −26,780.50 | −26,899.92 | −26,898.55 |
| BIC | −26,086.97 | −26,404.21 | −26,400.56 | −26,439.43 | −26,636.49 | −26,720.92 | −26,833.72 | −26,825.74 |
| K | 7 | 8 | 9 | 10 | 8 | 9 | 10 | 11 |
| GARCH-R | GARCH-S | GARCH-HAR | GARCH-SHAR | GARCH-HARQ | GARCH-SHARQ | |||
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| c 3 | – | – |
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| GARCH-R | GARCH-S | GARCH-HAR | GARCH-SHAR | GARCH-HARQ | GARCH-SHARQ | |||
| c 4 | – | – |
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| Skew | −0.2918 | −0.2846 | −0.2946 | −0.2854 | −0.2916 | −0.2877 | ||
| Kurt | 3.1733 | 3.1651 | 3.1765 | 3.1659 | 3.1735 | 3.1690 | ||
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18,073.95 | 18,084.96 | 18,070.49 | 18,080.10 | 18,091.53 | 18,092.74 | ||
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−4739.17 | −4693.48 | −4675.14 | −4629.49 | −4516.30 | −4509.27 | ||
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13,334.78 | 13,391.48 | 13,395.35 | 13,450.61 | 13,575.24 | 13,583.46 | ||
| AIC | −26,653.57 | −26,764.96 | −26,770.70 | −26,879.23 | −27,128.48 | −27,142.93 | ||
| BIC | −26,600.61 | −26,705.38 | −26,704.51 | −26,806.41 | −27,055.66 | −27,063.49 | ||
| K | 8 | 9 | 10 | 11 | 11 | 12 |
-
This table presents maximum likelihood estimation (MLE) results for the GARCH-HAR-X class of models defined in Table 1 assuming that the return innovations follow the NIG distribution.
Full sample variance loss-function results for GARCH-HAR-X models.
| MSE | MAE | HMSE | MAPE | |
|---|---|---|---|---|
| Panel A: GARCH-HAR-X(N) | ||||
| HAR | 0.3587 | 0.4649 | 0.8972 | 0.5752 |
| SHAR | 0.3400 | 0.4520 | 0.8343 | 0.5526 |
| HARz | 0.3390 | 0.4514 | 0.8211 | 0.5518 |
| SHARz | 0.3364 | 0.4495 | 0.8124 | 0.5483 |
| HARQ | 0.3276 | 0.4459 | 0.6897 | 0.5313 |
| SHARQ | 0.3225 | 0.4423 | 0.6880 | 0.5267 |
| HARQz | 0.3154 | 0.4364 | 0.6659 | 0.5177 |
| SHARQz | 0.3153 | 0.4364 | 0.6664 | 0.5178 |
| GARCH-R | 0.3318 | 0.4458 | 0.7897 | 0.5428 |
| GARCH-S | 0.3261 | 0.4427 | 0.7478 | 0.5356 |
| GARCH-HAR | 0.3242 | 0.4417 | 0.7455 | 0.5337 |
| GARCH-SHAR | 0.3187 | 0.4382 | 0.7146 | 0.5268 |
| GARCH-HARQ | 0.3056 | 0.4292 | 0.6258 | 0.5054 |
| GARCH-SHARQ | 0.3048 | 0.4288 | 0.6252 | 0.5050 |
| Panel B: GARCH-HAR-X(SGED) | ||||
| HAR | 0.3547 | 0.4618 | 0.8389 | 0.5578 |
| SHAR | 0.3364 | 0.4489 | 0.7824 | 0.5366 |
| HARz | 0.3350 | 0.4481 | 0.7665 | 0.5351 |
| SHARz | 0.3324 | 0.4463 | 0.7591 | 0.5319 |
| HARQ | 0.3242 | 0.4430 | 0.6449 | 0.5162 |
| SHARQ | 0.3193 | 0.4394 | 0.6443 | 0.5119 |
| HARQz | 0.3119 | 0.4331 | 0.6217 | 0.5026 |
| SHARQz | 0.3119 | 0.4332 | 0.6222 | 0.5027 |
| GARCH-R | 0.3280 | 0.4427 | 0.7402 | 0.5276 |
| GARCH-S | 0.3226 | 0.4397 | 0.7024 | 0.5210 |
| Panel B: GARCH-HAR-X(SGED) | ||||
| GARCH-HAR | 0.3205 | 0.4387 | 0.6975 | 0.5188 |
| GARCH-SHAR | 0.3152 | 0.4352 | 0.6701 | 0.5123 |
| GARCH-HARQ | 0.3024 | 0.4262 | 0.5856 | 0.4916 |
| GARCH-SHARQ | 0.3016 | 0.4259 | 0.5854 | 0.4914 |
| Panel C: GARCH-HAR-X(NIG) | ||||
| HAR | 0.3503 | 0.4577 | 0.7069 | 0.5183 |
| SHAR | 0.3324 | 0.4452 | 0.6647 | 0.5007 |
| HARz | 0.3310 | 0.4441 | 0.6522 | 0.4991 |
| SHARz | 0.3285 | 0.4425 | 0.6457 | 0.4964 |
| HARQ | 0.3206 | 0.4384 | 0.5505 | 0.4819 |
| SHARQ | 0.3158 | 0.4345 | 0.5514 | 0.4781 |
| HARQz | 0.3085 | 0.4281 | 0.5338 | 0.4698 |
| SHARQz | 0.3084 | 0.4282 | 0.5342 | 0.4699 |
| GARCH-R | 0.3242 | 0.4388 | 0.6336 | 0.4932 |
| GARCH-S | 0.3189 | 0.4360 | 0.6022 | 0.4878 |
| GARCH-HAR | 0.3168 | 0.4348 | 0.5978 | 0.4854 |
| GARCH-SHAR | 0.3116 | 0.4313 | 0.5750 | 0.4798 |
| GARCH-HARQ | 0.2991 | 0.4216 | 0.5057 | 0.4608 |
| GARCH-SHARQ | 0.2984 | 0.4214 | 0.5056 | 0.4608 |
-
This table presents in-sample results for one-step ahead predictions of log RV generated by the models shown in the first column of the table. It reports the mean squared error (MAE), mean absolute error (MAE), heteroskedasticity-adjusted mean squared error (HMSE), and mean absolute percentage error (MAPE). Panel A reports the results assuming that return innovations follow the standard Normal distribution. In Panel B and C return innovations follow the SGED and NIG distributions, respectively.
Several interesting conclusions can be drawn from the results shown in Tables 2–5. First, they clearly demonstrate that the full GARCH-SHARQ specification considerably improves upon the in-sample fit and predictive performance of realized variance compared to the HAR model of Corsi (2009) or the GARCH-R model of Hansen, Huang, and Shek (2012). This result can be clearly justified by all the prediction and fit performance metrics reported in the tables. Recall that the GARCH-R and HAR-type models are the key reference models for our comparisons, since the GARCH-SHARQ follows as direct extension of these.
The enhanced performance of the GARCH-SHARQ model with respect to GARCH-R comes primarily from the decomposition of RV into upside and downside realized semi-variances (RV+ and RV−) therefore introducing a source of asymmetry (this also holds for the GARCH-S/SHAR/SHARQ models), as well as from the inclusion of realized quarticity RQ in x(t), which helps controlling for attenuation biases in realized variance measurement (refer to the GARCH-HARQ/SHARQ). It becomes obvious that this particular extension of the GARCH-R model increases the variance prediction performance significantly (e.g. the HMSE metric drops from 0.6336 to 0.5056 under the NIG distribution). A similar conclusion can be drawn from the values of the log-likelihood function
Second, our results clearly indicate that this generalized/augmented GARCH-HAR-X framework also provides the means for useful extensions to the standard HAR-type model-variants suggested in the literature to predict RV. As can be seen from Tables 2–4, comparing the standard HAR-type models to their corresponding GARCH-HAR-X counterparts shows that all the GARCH-model features have significant impact on RV
t
, even in the presence of the other HAR-model terms. Their sign and magnitude is in line with other studies in the standard GARCH model literature, and they seem to improve both the joint fit to returns and variance and the performance of the HAR model to predict the realized variance measure v
t
. The values of the loss-metrics reported in Table 5 indicate a sizeable performance improvement (for instance the HMSE drops from 0.6647 to 0.5750 when comparing the SHAR of Patton and Sheppard (2015) to the corresponding GARCH-SHAR), while those of
The improved performance of the augmented HAR-type models that include the GARCH terms stems firstly from the autoregressive conditional variance component ht−1, differentiating the impact from that of lagged RV. As seen by the estimates of the models, the inclusion of ht−1 in the HAR model does not interact with the lagged realized measures, which means that it reflects different dynamic effects of conditional variance. A second source of performance improvement over the standard HAR models comes from adding the return innovation functions, i.e. through the NIF τ(t); refer to the HARz, SHARz, HARQz, SHARQz models. Doing so allows us to capture asymmetries in realized variance in a similar way as in GARCH models. Other studies have tried to capture leverage effects in HAR models by simply using lagged returns (Audrino and Hu 2016; Corsi and Renò 2012). Even though this approach might look very similar conceptually, we have found that using lagged filtered innovations z t instead (as in a GARCH model) performs much better in fitting the data. This is an interesting finding and can be justified by the fact that filtered innovations provide a “cleaner” source of news information and have time-invariant distributional properties (at least for the first two moments of the distribution) since they are not contaminated by conditional variance or variance-in-mean effects, unlike raw returns.
A third conclusion that can be drawn from our results is related to the importance of the NIF component τ(t). It appears that including in h t the exogenous realized variables through x(t) cannot fully absorb the parameters b2 and b3 of τ(t), which control for asymmetric effects of lagged innovations on h t . This can be seen by directly comparing the estimates of the HAR-type models (HAR, SHAR, HARQ and SHARQ) to those of their corresponding NIF-augmented representations (HARz, SHARz, HARQz and SHARQz), which have been intentionally designed to include the τ(t) terms. The parameters of τ(t) are statistically significant even in the presence of the two semi-variances RV+ and RV−.[13] This result implies that the realized semi-variances bring different information to the conditional variance equation. Almost certainly, this source of information relates mostly to short-term effects from signed-jump variation. Moreover, the different values of the coefficient estimates for RV+ and RV− indicate that there are prevalent asymmetries in the responses of the conditional variance to positive and negative jumps. These asymmetric responses are directly related to realized skewness in intra-day data. In the absence of realized skewness the two semi-variances would be almost identical, hence the signed jump variation would collapse to zero, therefore having no contribution in forecasting conditional variance. This finding is in line with the HAR-type regression results discussed in Patton and Sheppard (2015). Summing up, the above results highlight that there are two channels of volatility asymmetry; the one originates from the asymmetric responses to intra-day realized upside/downside variation, while the other is driven from non-linearities with respect to lagged return innovations. Both of these channels are found to carry statistically significant information for volatility forecasting.
Fourth, the impact of realized quarticity RQ on conditional variance is found to be negative and significant, with a sizeable contribution to the likelihood. This indicates that beyond the skewness effect on conditional variance that comes from signed-jump variations, there is also a material volatility-of-volatility effect related to intra-day realized kurtosis. Bollerslev, Patton, and Quaedvlieg (2016) and Cipollini, Gallo, and Otranto (2020), in a slightly different context, use the RQ in order to indirectly account for time-variation in the HAR parameters and they also report results similar in direction and magnitude; they attribute this effect to attenuation-biases in measuring realized variance.
Finally, regarding the alternative densities that we used to model the returns innovations z t , our results show that the NIG distribution provides much better fit to the actual data, followed by the SGED. This improvement is significant not only in terms of joint log-likelihood and information criteria, but also in terms of in-sample variance loss-functions. This result is robust across all the restricted specifications nested within our generalized GARCH-HAR-X framework. In particular, our results indicate that the NIG distribution is able to capture a much higher level of negative skewness in the data (varying around −0.29 across models), while producing less pronounced kurtosis (approximately 3.17). The SGED distribution on the other hand generates lower values of negative skewness (approximately −0.19), while generating higher values for kurtosis (approximately 3.86). It is evident that the fit improves significantly when accounting for higher negative skewness levels. The semi-heavy behaviour of the NIG tails seems to be more flexible in capturing the skewness of the actual data, hence improving the filtering of z t and the identification of the parameters of h t , especially for the NIF component.
4.2 Out-of-sample results
In order to assess the out-of-sample (OOS) performance of our models, we split the sample in two parts; we use the first 3000 observations (4/01/1995 to 30/11/2006) to initialize fitting the models (“training” sample) and the remaining approximately 2500 observations (that is from 1/12/2006 onwards) to test the OOS performance. In this way, we maintain a good balance between having a significantly large sample that allows us to accurately estimate model-parameters and also keeping a considerably large period for our OOS testing (Diebold 2015). Note that our OOS period covers the major financial crisis of 2007–08 and other short-lived shocks such as the flash-crash of 2010; this will enable us to examine the robustness of our models against those severe events as well.
We carry out a daily rolling re-estimation of the models to obtain one-step ahead predictions, always keeping the window-size fixed at 3000 observations.[14] Despite the fact that the GARCH-based framework is by construction optimized to provide one-step ahead conditional variance forecasts at a daily frequency, another reason we focus on evaluating the one-step ahead variance predictions is because it has been shown in numerous theoretical econometric studies that the test statistics used in the assessment of equivalent prediction accuracy in pair-wise model-comparisons have good size and power properties primarily for one-step ahead forecasts.[15]
We carry out a series of comparative tests; these will help us shed more light to the model-performance and ensure that our findings from the full-sample analysis also hold OOS. In order to facilitate the comparisons we use the same variance loss-functions as in the in-sample analysis and we also employ the reality check (RC) introduced in White (2000). This test is asymptotically valid for both non-nested and nested model-comparisons when the OOS estimates are obtained using a rolling-window. RC is a very common choice among similar studies (see for instance Hansen and Lunde (2005) as well as Bollerslev, Patton, and Quaedvlieg (2016)), since it provides a very straightforward framework for comparing whether the loss of a model A is (statistically) significantly lower than that of a benchmark model B.[16] The statistical hypotheses for the RC test can be simply formulated as
where
We compare the prediction performance of the full GARCH-SHARQ model relative to all the restricted cases. As before, we are primarily interested in the comparisons vis-a-vis the standard GARCH-R model and the nested HAR-type variants. In Tables 6 and 7 we show the OOS variance loss-functions and RC p-values, respectively, based on the MSE, MAE, HMSE and MAPE metrics. To save some space we report the results for the NIG distribution, since it was found to outperform both the Normal and SGED.
Out-of-sample variance loss-function results for GARCH-HAR-X(NIG) models.
| MSE | MAE | HMSE | MAPE | |
|---|---|---|---|---|
| GARCH-SHARQ | 0.3613 | 0.4627 | 0.6325 | 0.5062 |
| GARCH-HARQ | 0.3603 | 0.4616 | 0.6274 | 0.5027 |
| GARCH-SHAR | 0.3807 | 0.4808 | 0.8222 | 0.5715 |
| GARCH-HAR | 0.3893 | 0.4863 | 0.8735 | 0.5815 |
| GARCH-S | 0.3869 | 0.4841 | 0.8461 | 0.5762 |
| GARCH-R | 0.3957 | 0.4884 | 0.9150 | 0.5858 |
| SHARQz | 0.3725 | 0.4697 | 0.7074 | 0.5253 |
| HARQz | 0.3717 | 0.4690 | 0.7049 | 0.5242 |
| SHARQ | 0.3791 | 0.4744 | 0.7240 | 0.5333 |
| HARQ | 0.3797 | 0.4743 | 0.7036 | 0.5269 |
| SHARz | 0.4081 | 0.4948 | 0.9918 | 0.5977 |
| HARz | 0.4130 | 0.4979 | 1.0059 | 0.6032 |
| SHAR | 0.4082 | 0.4946 | 1.0076 | 0.5985 |
| HAR | 0.4314 | 0.5100 | 1.0892 | 0.6241 |
-
This table presents out-of-sample results for one-step ahead predictions of log RV generated by the models shown in the first column of the table assuming that return innovations follow the NIG distribution. It reports the mean squared error (MAE), mean absolute error (MAE), heteroskedasticity-adjusted mean squared error (HMSE), and mean absolute percentage error (MAPE). We apply a daily rolling re-estimation of the models with a fixed window size of 3000 observations.
Reality check tests results comparing restricted models within the GARCH-HAR-X(NIG) to the full GARCH-SHARQ specification.
| MSE | MAE | HMSE | MAPE | |
|---|---|---|---|---|
| GARCH-HARQ | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
| GARCH-SHAR | 0.0001 | 0.0000 | 0.0000 | 0.0000 |
| GARCH-HAR | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| GARCH-S | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| GARCH-R | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| SHARQz | 0.0136 | 0.0074 | 0.0014 | 0.0001 |
| HARQz | 0.0175 | 0.0168 | 0.0023 | 0.0001 |
| SHARQ | 0.0021 | 0.0005 | 0.0002 | 0.0000 |
| HARQ | 0.0006 | 0.0003 | 0.0010 | 0.0000 |
| SHARz | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| HARz | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| SHAR | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| HAR | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
-
This table reports the p-values of Reality Check tests for the GARCH-HAR-X class of models with NIG-distributed return innovations. All (nested) models, shown in the first column of the table, are compared to the full GARCH-SHARQ specification.
Table 6 results indicate that our extended specifications outperform the benchmark GARCH-R and HAR-type models, corroborating the findings of the in-sample analysis. Overall, the GARCH-HARQ model was found to yield the best OOS performance, followed closely by the GARCH-SHARQ. For example, the GARCH-HARQ representation reduces the MSE from 0.3957 under the GARCH-R or 0.3797 under the HARQ model to 0.3603. Results for the variance percentage errors (i.e. for the HMSE metric) are also very interesting. Again, if we compare the GARCH-R and HARQ models to our augmented GARCH-HARQ specification we see that HMSE drops significantly from 0.9150 and 0.7036 to 0.6274. These findings reveal that the GARCH-HARQ (and the full GARCH-SHARQ) model can not only shrink the magnitude of the average errors, but also better capture extreme errors relative to the actual realized variance levels. The above arguments can be further justified by the p-values of the RC tests that we have performed. Results in Table 7 suggest that all of the aforementioned loss-function improvements are statistically significant at conventional levels of statistical significance (bootstrapped p-values are below 5%, in most cases close to zero).
Turning again the discussion to the relative contribution of the additional variables that extend the GARCH-R and HAR-type models to the GARCH-SHARQ framework, we observe that the results of the OOS exercise are consistent with those of the in-sample (full-sample) analysis. In particular, the performance metrics indicate that the GARCH-S improves upon the standard GARCH-R, highlighting the importance of the RV+ and RV− semi-variances. Similar conclusions can be drawn for the heterogeneous components RV (Andersen et al. 2009) and RV (Bollerslev et al. 2009) (e.g. by comparing the GARCH-HAR to the GARCH-R) and the realized quarticity measure RQ (by comparing the GARCH-SHARQ to the GARCH-SHAR). The above results become more clear when looking at the HMSE or MAPE metrics. As can be seen by the estimates of these metrics the majority of the prediction gains come firstly from correcting for attenuation-biases through RQ and secondly through decomposing RV into its two upside/downside semi-variance components RV+ and RV−. The relative contribution of the heterogeneous terms is less pronounced but still very significant, accommodating for long-memory patterns beyond the persistency inherent in the parametric structure of the GARCH-based framework.
Looking into the HAR-type model extensions, we conclude again that the OOS results confirm the in-sample ones. The reported prediction performance metrics indicate that the OOS gains come primarily from the extension of the HAR to include information from filtered return innovations (e.g. compare SHARQ to SHARQz), followed by the extensions also incorporating filtered conditional variance ht−1 (compare for instance the SHARQz to the GARCH-SHARQ).
Finally, results for the OOS exercise evaluating the impact of the distributional assumption for the return innovations z t on the prediction performance of the models are presented in Tables 8 and 9. In particular, Table 8 presents the variance loss-function results under all three parametric densities that we employ for the returns distribution and, similarly, Table 9 presents the RC test results for the models using NIG innovations for z t versus those assuming SGED or Normal. Note that the models with SGED and/or Normal are the benchmark models when compared to the NIG, since the NIG has been generally found to perform more flexibly. The table results clearly indicate that, for all loss/prediction metrics, the NIG distribution enhances the performance of variance forecasts compared to both SGED and Normal distribution, and this holds across all model specifications. The superiority of the NIG distribution can be statistically confirmed by the p-values of the RC tests. These are almost zero, thus clearly rejecting the null hypothesis suggesting that the SGED (or Normal) distribution is equivalent to the NIG in terms of prediction performance.
Out-of-sample variance loss-function results for GARCH-HAR-X models under different distributional assumptions for the return innovations.
| MSE | MAE | HMSE | MAPE | |
|---|---|---|---|---|
| GARCH-SHARQ(NIG) | 0.3613 | 0.4627 | 0.6325 | 0.5062 |
| GARCH-SHARQ(SGED) | 0.3689 | 0.4660 | 0.7088 | 0.5314 |
| GARCH-SHARQ(N) | 0.3660 | 0.4675 | 0.7452 | 0.5425 |
| GARCH-HAR(NIG) | 0.3893 | 0.4863 | 0.8735 | 0.5815 |
| GARCH-HAR(SGED) | 0.3974 | 0.4926 | 0.9946 | 0.6165 |
| GARCH-HAR(N) | 0.4023 | 0.4959 | 1.0471 | 0.6308 |
| GARCH-SHAR(NIG) | 0.3807 | 0.4808 | 0.8222 | 0.5715 |
| GARCH-SHAR(SGED) | 0.3885 | 0.4874 | 0.9346 | 0.6057 |
| GARCH-SHAR(N) | 0.3932 | 0.4906 | 0.9814 | 0.6192 |
| GARCH-S(NIG) | 0.3869 | 0.4841 | 0.8461 | 0.5762 |
| GARCH-S(SGED) | 0.3942 | 0.4900 | 0.9588 | 0.6101 |
| GARCH-S(N) | 0.3991 | 0.4935 | 1.0093 | 0.6242 |
| GARCH-R(NIG) | 0.3957 | 0.4884 | 0.9150 | 0.5858 |
| GARCH-R(SGED) | 0.4033 | 0.4945 | 1.0397 | 0.6212 |
| GARCH-R(N) | 0.4084 | 0.4980 | 1.0962 | 0.6361 |
| SHAR(NIG) | 0.4082 | 0.4946 | 1.0076 | 0.5985 |
| SHAR(SGED) | 0.4163 | 0.5009 | 1.1500 | 0.6359 |
| SHAR(N) | 0.4211 | 0.5043 | 1.2106 | 0.6515 |
| HAR(NIG) | 0.4314 | 0.5100 | 1.0892 | 0.6241 |
| HAR(SGED) | 0.4403 | 0.5160 | 1.2526 | 0.6648 |
| HAR(N) | 0.4456 | 0.5196 | 1.3217 | 0.6820 |
-
This table presents out-of-sample results for one-step ahead predictions of log RV generated by the models shown in the first column of the table assuming that return innovations follow the standard Normal, SGED, and NIG distribution. It reports the mean squared error (MAE), mean absolute error (MAE), heteroskedasticity-adjusted mean squared error (HMSE), and mean absolute percentage error (MAPE). The definitions of these metrics are given in (6). We apply a daily rolling re-estimation of the models with a fixed window size of 3000 observations.
Reality Check tests results comparing different distributions for the return innovations.
| MSE | MAE | HMSE | MAPE | |
|---|---|---|---|---|
| Panel A: NIG vs SGED | ||||
| GARCH-SHARQ | 0.1229 | 0.0092 | 0.0000 | 0.0000 |
| GARCH-SHAR | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| GARCH-HAR | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| GARCH-S | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| GARCH-R | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| SHAR | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| HAR | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| Panel B: NIG vs N | ||||
| GARCH-SHARQ | 0.0187 | 0.0000 | 0.0000 | 0.0000 |
| GARCH-SHAR | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| GARCH-HAR | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| GARCH-S | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| GARCH-R | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| SHAR | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| HAR | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
-
This table reports the p-values of reality check tests comparing similar models (reported in the first column) under different distributions for the return innovations. Panel A compares the NIG versus SGED, while panel B compares the NIG versus the standard Normal distribution.
4.3 Robustness of OOS results to longer horizons
The OOS results of the previous section are focused on one-period ahead, where inference procedures on the prediction performance have better size and power properties as noted before. In this section, we examine the OOS forecasting performance of the models for longer horizons ahead to see if our results remain robust. To obtain the multi-period ahead predictions, we rely on the “direct” forecasting approach (see for instance Clark and McCracken (2005), Marcellino, Stock, and Watson (2006), Clark and McCracken (2013), and Ghysels et al. (2019)) and we employ the following horizon-specific regression model as implied by the full GARCH-SHARQ specification:
where we forecast the log-realized variance
There are several reasons behind our decision to run the OOS predictive regressions using log-transformations. Firstly, we want to maintain consistency with log-variance Eq. (1b), as this is the one we actually estimate in the rolling-sample MLE. Secondly, predicting the log-transformation of variance (at any horizon) is the best way to ensure that we will obtain non-negative variance estimates OOS, without having to impose any parameter restrictions. Additionally, as we mentioned in our empirical estimation section, the log-realized variance approximately follows a Normal distribution, which is a desired property for our multi-step OOS regression framework, as the OLS estimator will demonstrate lower bias and improved efficiency (see also Papantonis, Rompolis, and Tzavalis (2021)). Effectively, what we do here is very similar to a log-HAR regression specification for predicting variance at multiple-horizons ahead; the only difference being the addition of the log-conditional variance in the regression, which encapsulates in
In Table 10, we present estimates of the OOS prediction loss-metrics for all the alternative specifications nested with the GARCH-SHARQ. This is done for s = {5, 10, 20} days ahead. For reasons of space, we present results for the full set of models only for the NIG distribution, since it has been found to perform better under all circumstances.[18] The results of the table are consistent with those for the case of one-period ahead for all models, and indicate that the full GARCH-SHARQ model provides the best OOS performance compared to all the other nested specifications. This result is true for all the different horizons ahead considered and under all the forecasting performance metrics reported in the table. Especially when comparing with the GARCH-R model, the improvements in variance forecasts appear to be very substantial. As can be seen from the table, these improvements are mainly due to the inclusion of the
Out-of-sample variance loss-function results for multi-step ahead predictions.
| Horizon s = 5 | Horizon s = 10 | Horizon s = 20 | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| MSE | MAE | HMSE | MAPE | MSE | MAE | HMSE | MAPE | MSE | MAE | HMSE | MAPE | |
| GARCH-SHARQ | 0.6088 | 0.6013 | 1.6880 | 0.7307 | 0.7386 | 0.6657 | 2.1816 | 0.8239 | 0.8699 | 0.7265 | 2.4475 | 0.9066 |
| GARCH-HARQ | 0.6134 | 0.6027 | 1.7160 | 0.7323 | 0.7399 | 0.6666 | 2.2092 | 0.8254 | 0.8709 | 0.7273 | 2.4589 | 0.9083 |
| GARCH-SHAR | 0.6119 | 0.6037 | 1.6152 | 0.7314 | 0.7421 | 0.6676 | 2.0901 | 0.8262 | 0.8738 | 0.7284 | 2.4402 | 0.9097 |
| GARCH-HAR | 0.6240 | 0.6099 | 1.6885 | 0.7422 | 0.7455 | 0.6702 | 2.0859 | 0.8303 | 0.8768 | 0.7300 | 2.4529 | 0.9142 |
| GARCH-S | 0.6324 | 0.6173 | 1.9495 | 0.7697 | 0.7697 | 0.6791 | 2.5397 | 0.8683 | 0.8982 | 0.7411 | 2.6443 | 0.9391 |
| GARCH-R | 0.6441 | 0.6235 | 2.1038 | 0.7823 | 0.7693 | 0.6792 | 2.4811 | 0.8673 | 0.8987 | 0.7429 | 2.6430 | 0.9425 |
| SHARQz | 0.6159 | 0.6029 | 1.7055 | 0.7331 | 0.7431 | 0.6676 | 2.1937 | 0.8258 | 0.8729 | 0.7271 | 2.4710 | 0.9074 |
| SHARQ | 0.6187 | 0.6045 | 1.6795 | 0.7336 | 0.7452 | 0.6681 | 2.2252 | 0.8266 | 0.8731 | 0.7271 | 2.4718 | 0.9073 |
| HARQ | 0.6198 | 0.6037 | 1.7009 | 0.7327 | 0.7449 | 0.6684 | 2.2332 | 0.8271 | 0.8737 | 0.7276 | 2.4798 | 0.9086 |
| SHAR | 0.6227 | 0.6069 | 1.6693 | 0.7369 | 0.7485 | 0.6702 | 2.1848 | 0.8295 | 0.8761 | 0.7287 | 2.4682 | 0.9103 |
| HAR | 0.6312 | 0.6113 | 1.7184 | 0.7449 | 0.7525 | 0.6728 | 2.1832 | 0.8336 | 0.8798 | 0.7304 | 2.4820 | 0.9144 |
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This table presents out-of-sample results for 5, 10, and 20 days ahead predictions of log RV generated by the models shown in the first column of the table assuming that return innovations follow the NIG distribution. The multi-period ahead predictions of log RV is estimated using the OOS regression model (6). It reports the mean squared error (MAE), mean absolute error (MAE), heteroskedasticity-adjusted mean squared error (HMSE), and mean absolute percentage error (MAPE).
5 Summary & conclusions
In this paper, we extend the Realized-GARCH model (or GARCH-R under our notation) with exogenous variates related to measurable features of realized variance, the selection of which is motivated by the ongoing literature on HAR models for realized variance. The suggested GARCH-HAR-X framework can be also seen as an a dual extension of the HAR model to include GARCH terms. We analyse both the in-sample and out-of-sample prediction performance of the generalized model, as well as the several specifications nested within, and we do that under different assumptions for the probability density function of the return innovations. We consider not only the Normal, often used in practice, but also the SGED and the NIG distribution which have been shown to perform very well in capturing distributional asymmetries of financial data.
We find our “full” GARCH-SHARQ specification to perform significantly better than the standard GARCH-R in fitting the data and forecasting realized variance. We show that its enhanced performance originates from the following three sources/extensions to the conditional variance function: (i) the decomposition of realized variance into upside downside semi-variances, which allows to better capture short-term asymmetric behaviours in variance due to the signed-jump variation; (ii) the addition of the heterogeneous components, which effectively approximate a long-memory effect in conditional variance dynamics and, most importantly, (iii) the inclusion of a variance-of-variance (quarticity) proxy responsible for capturing attenuation-biases in variance forecasts. The in-sample and out-of-sample performance of the models is assessed based on a number of different fit and prediction-accuracy metrics; these clearly indicate that the inclusion of upside/downside realized semi-variances, as well as realized quarticity, are the main drivers for the significant gains in the prediction performance of the augmented GARCH-SHARQ model.
Furthermore, our GARCH-SHARQ model has also been found to perform better than the several nested HAR-type representations. We show that this is primarily due to the filtered return innovations when included in conditional variance through the news-impact function. This allows for two separate channels of leverage-effect; through asymmetric responses to both return innovations and semi-variances. Finally, regarding the performance of the alternative model specifications under different parametric density assumptions, our analysis shows that the NIG distribution enhances the in-sample fit and out-of-sample forecasting performance compared to both the SGED and the Normal distribution, and this holds across all model specifications. We show that this may be attributed to the enhanced flexibility of the NIG in capturing asymmetries in the data.
Funding source: European Social Fund
Award Identifier / Grant number: MIS-5049538
Acknowledgement
We would like to thank the editor Bruce Mizrach, as well as the two anonymous reviewers for providing useful comments and suggestions on an earlier version of the paper. We also acknowledge financial support from the Operational Programme “Human Resources Development, Education and Lifelong Learning 2014–2020” in the context of the project MIS-5049538, co-financed by Greece and the European Union (European Social Fund, ESF).
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Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
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Research funding: This research is co-financed by Greece and the European Union (European Social Fund, ESF) through the Operational Programme “Human Resources Development, Education and Lifelong Learning 2014–2020” in the context of the project MIS-5,049,538.
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Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
Appendix A: Distributions
A.1 The standardized normal-inverse-Gaussian distribution
A.1.1 The normal-inverse-Gaussian (NIG) distribution
The normal-inverse-Gaussian, henceforth NIG, results as a special case of the generalized hyperbolic (GH) family of (mean-variance) mixture distributions popularized by Barndorff-Nielsen (1978), with the mixing distribution in this case being the inverse-Gaussian (IG). A very detailed description of this family of distributions can be found in McNeil, Frey, and Embrechts (2005). The location-scale invariant parametrization of the PDF of a random variable X that follows an NIG distribution, i.e.
where
while the moment-generating function (MGF) can be expressed as:
To get the first four central moments we solve the above MGF:
where
A.1.2 The zero-mean & unit-variance NIG specification
We want to ensure that the innovation z follows a zero-mean and unit-variance NIG distribution. Using the MGF of the NIG to solve for the first central moments and equating the first two moments of rt+1, we can show that the returns follow a conditional NIG distribution as:
with
i.e. it is entirely driven by the constant values
which we can think of as two quantities driving the “asymmetry” and “steepness” of the density. Finally, we solve the system of equations implied in (A.1) for
which we substitute in the density function above to get the log-likelihood of returns. Using the parametrization above, we can think of the standardized innovation z as having a standardized NIG density purely defined by two parameters, i.e. z ∼ NIG(η, λ).
A.2 The SGED (skewed generalized error) distribution
A zero-mean and unit-variance random variable Z is SGED-distributed as z ∼ SGED(η, λ) (Theodossiou 1998) when it adheres to the PDF:
where the following parametrizations hold:
with Γ being the gamma-function
This density nests several conventional densities as well. For instance, for λ = 0 the density reduces to the GED distribution of Nelson (1991), which is symmetric but fat-tailed. When λ = 0 and η = 2, the density collapses to the standard Normal. For λ = 0 and η = 1 we get the double-exponential distribution and finally for λ = 0 and η = +∞ we get a uniform in the
Regarding the higher moments of an SGED-distributed random variable, the following mapping holds:
with
A.3 Robustness of the GARCH-SHARQ model to alternative specifications of the NIF
The NIF given by Eq. (3) assumes abrupt shifts in variance ht+1 driven by the sign of ϵ t . Alternative specifications of this function considered smooth transition shifts have been also suggested in the literature (see Franses and Van Dijk (2000), for a survey). In this direction, the logistic smooth transition model of González-Rivera (1998) is one popular alternative. This model assumes that τ(t) is given as
where
Another direction towards which Eq. (3) can be extended is to allow for an unknown threshold parameter, instead of assuming δ = 0 as in (3), i.e.,
The threshold parameter δ can be estimated endogenously from the data, using a grid-search approach.[20]
In Table 11, we present estimates of the fully specified GARCH-SHARQ model under the three distributions considered. The results show that the estimates of γ are quite large (implying almost abrupt shifts between the two regimes of the NIF) and the threshold parameter δ is very close to zero; the bootstrapped standard errors reported in square brackets indicate that the null hypothesis of δ = 0 cannot be rejected. These results hold for all distributions considered. This allows us to conclude that the NIF function given by (3) provides a correct specification of the data. Further support of (3) can be obtained by the values of the likelihood components
Full sample maximum likelihood parameter estimates for linear GARCH-HAR-X models under different NIF specifications.
| NIG | SGED | Normal | |||||||
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| γ | – |
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– | – |
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– | – |
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– |
| δ | – | – |
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– | – |
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– | – |
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| η |
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| λ |
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18,092.74 | 18,092.74 | 18,092.76 | 17,895.70 | 17,895.70 | 17,895.73 | 17,777.40 | 17,777.39 | 17,777.42 |
|
|
−4509.27 | −4509.27 | −4509.23 | −4538.94 | −4538.94 | −4538.90 | −4568.83 | −4568.82 | −4568.77 |
|
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13,583.46 | 13,583.47 | 13,583.53 | 13,356.76 | 13,356.76 | 13,356.83 | 13,208.57 | 13,208.57 | 13,208.65 |
| AIC | −27,142.93 | −27,140.93 | −27,141.05 | −26,689.52 | −26,687.53 | −26,687.65 | −26,397.13 | −26,395.14 | −26,395.29 |
| BIC | −27,063.49 | −27,054.88 | −27,055.00 | −26,610.08 | −26,601.47 | −26,601.60 | −26,330.94 | −26,322.33 | −26,322.48 |
| MSE | 0.2984 | 0.2984 | 0.2984 | 0.3016 | 0.3016 | 0.3016 | 0.3048 | 0.3048 | 0.3048 |
| MAE | 0.4214 | 0.4214 | 0.4214 | 0.4259 | 0.4259 | 0.4259 | 0.4288 | 0.4288 | 0.4288 |
| HMSE | 0.5056 | 0.5056 | 0.5056 | 0.5854 | 0.5854 | 0.5854 | 0.6252 | 0.6252 | 0.6252 |
| MAPE | 0.4608 | 0.4608 | 0.4608 | 0.4914 | 0.4914 | 0.4914 | 0.5050 | 0.5050 | 0.5050 |
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This table presents maximum likelihood estimation (MLE) results for the GARCH-SHARQ model under different NIF specifications assuming that the return innovations follow the standard Normal, SGED, and NIG distribution. The NIF specifications considered are the GJR, LST-GARCH, and T-GARCH.
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