 Methodology
 Open Access
 Published:
A modelfree approach to do longterm volatility forecasting and its variants
Financial Innovation volume 9, Article number: 59 (2023)
Abstract
Volatility forecasting is important in financial econometrics and is mainly based on the application of various GARCHtype models. However, it is difficult to choose a specific GARCH model that works uniformly well across datasets, and the traditional methods are unstable when dealing with highly volatile or shortsized datasets. The newly proposed normalizing and variance stabilizing (NoVaS) method is a more robust and accurate prediction technique that can help with such datasets. This modelfree method was originally developed by taking advantage of an inverse transformation based on the frame of the ARCH model. In this study, we conduct extensive empirical and simulation analyses to investigate whether it provides higherquality longterm volatility forecasting than standard GARCH models. Specifically, we found this advantage to be more prominent with short and volatile data. Next, we propose a variant of the NoVaS method that possesses a more complete form and generally outperforms the current stateoftheart NoVaS method. The uniformly superior performance of NoVaStype methods encourages their wide application in volatility forecasting. Our analyses also highlight the flexibility of the NoVaS idea that allows the exploration of other model structures to improve existing models or solve specific prediction problems.
Introduction
In financial econometrics, forecasting volatility accurately and robustly is an important task (Engle and Patton 2001; Du and Budescu 2007). Highquality volatility forecasting is crucial for practitioners and traders to make decisions on risk management, asset allocation, price of the derivative instrument, and fiscal policies (Fang et al. 2018; Ashiya 2003; Bansal et al. 2016; Kitsul and Wright 2013; Morikawa 2019). However, volatility forecasting is challenging due to factors such as a small sample size, heteroscedasticity, and structural change (Chudý et al. 2020). Standard methods for volatility forecasting are typically built upon GARCHtype models; these models’ abilities to forecast the absolute magnitude and quantiles or the entire density of squared financial logreturns (i.e., equivalent to volatility forecasting to some extent)^{Footnote 1} were shown by Engle and Patton (2001) using the Dow Jones Industrial Index. Later, many studies compared the performances of different GARCHtype models in volatility prediction; see Chortareas et al. (2011), GonzálezRivera et al. (2004), Herrera et al. (2018), Lim and Sek (2013), Peters (2001), Wilhelmsson (2006) and Zheng (2012). Some researchers attempted to develop the GARCH model further, such as by adopting smoothing parameters or adding more related information to estimate models (Breitung and Hafner 2016; Chen et al. 2012; Fiszeder and Perczak 2016; Taylor 2004). To model the proper process of volatility during the fluctuating period, Kim et al. (2011) applied time series models with stable and temperedstable innovations to measure market risk during the highly volatile period, Ben Nasr et al. (2014) applied a fractionally integrated timevarying GARCH (FITVGARCH) model to fit volatility, and Karmakar and Roy (2021) developed a Bayesian method to estimate timevarying analogs of ARCHtype models to describe frequent volatility changes. Although there are several types of GARCH models, it is difficult to determine which one outperforms others uniformly because the performances of these models heavily depend on the error distribution, length of the prediction horizon, and property of the dataset.
To overcome this dilemma, we adhere to a recently developed modelfree method, NoVaS, which applies normalizing and variancestabilizing transformation (NoVaS transformation) to perform predictions. The NoVaS method is guided by the Modelfree Prediction Principle, first proposed by Politis (2003). Previous studies showed that the NoVaS method performs better than GARCHtype models in forecasting squared logreturns. Notably, Gulay and Emec (2018) showed that the NoVaS method could beat GARCHtype models (GARCH, EGARCH, and GJRGARCH) with generalized error distributions by comparing the pseudooutofsample^{Footnote 2}(POOS) forecasting performance. Furthermore, Chen and Politis (2019) found an approach to perform multistepahead predictions of squared logreturns based on the NoVaS method. Wu and Karmakar (2021) further substantiated the effective performance of NoVaS methods on aggregated longterm (30steps ahead) predictions. In a recent study, Wang and Politis (2022) applied a modelfree idea to provide estimation and prediction inferences for a general class of time series. Although they adopted a twostage transformation approach to achieve the modelfree goal, which is different from the NoVaS method, the validity of such a Modelfree Prediction Principle was shown. From a practical aspect of forecasting volatility, to obtain some inference about the future situation at an overall level, we choose the timeaggregated prediction metric taken by Wu and Karmakar (2021) to measure the short and longterm forecasting performance of different methods. This aggregated metric has been applied to depict the future situation of electricity prices and financial data (Chudý et al. 2020; Karmakar et al. 2022; Fryzlewicz et al. 2008).
One drawback of the existing NoVaStype methods is that the parameters of the transformation must obey a specific form, which decreases its flexibility. Inspired by the development of the ARCH model (Engle 1982) to the GARCH model (Bollerslev 1986), this study attempts to build a novel NoVaS method derived by iterating the GARCH(1,1) structure. Our new method provides more freedom in the region of the parameters. Moreover, to achieve a fair and comprehensive comparison between NoVaStype and standard GARCH methods, we simulated data from various models to examine their robustness. On the empirical side, we split volatility forecasting into three main classes, that is, considering the volatility prediction of stock, currency, and index data. Through extensive data analyses, we show that all NoVaStype methods bring significant improvements compared with the standard GARCH model when the available data are short or volatile. Moreover, our new methods generally perform better than current NoVaS methods.
The remainder of this paper is organized as follows. Details about the existing NoVaS method and the motivations for proposing our new method are explained in “NoVaS method and evaluation metric” section. We also explain the evaluation metrics used throughout this study. In “New variants of the NoVaS method” section, we propose a new NoVaS transformation approach and its parsimonious variants. To compare all NoVaStype methods with the standard GARCH model, POOS predictions on simulated and realworld datasets were performed using “Simulation” and “Realworld data analysis” Sections. In “Comparison of predictive accuracy: Statistical tests” section, we present statistical test results to substantiate our new methods. Finally, the discussion and conclusion are presented in “Results and discussion” and “Conclusion” sections, respectively.
NoVaS method and evaluation metric
In this section, we first introduce the Modelfree Prediction Principle. We then present how the NoVaS transformation can be built from an ARCH model. Subsequently, the motivation to build a new NoVaS transformation and timeaggregated metric is provided.
Modelfree prediction principle
Before presenting the NoVaS method in detail, we throw some light on the insight of the Modelfree Prediction Principle. The main idea behind this is to apply an invertible transformation function \(H_T\) that can map a noni.i.d. vector \(\{Y_t~;t = 1,\ldots ,T\}\) to a vector \(\{\epsilon _t;~t=1,\ldots ,T\}\) with i.i.d. components. Because the prediction of i.i.d. data is somewhat standard, the prediction of \(Y_{T+1}\) can easily be obtained by inversely transforming \({\hat{\epsilon }}_{T+1}\) which is a prediction of \(\epsilon _{T+1}\) using \(H_T^{1}\). In other words, we can express prediction \({\hat{Y}}_{T+1}\) as a function of \(\varvec{Y}_T\), \(\varvec{X}_{T+1}\) and \({\hat{\epsilon }}_{T+1}\):
where \(\varvec{Y}_{T}\) denotes all historical data \(\{Y_t;~t =1,\ldots ,T\}\), \(\varvec{X}_{T+1}\) is the collection of all predictors, and it also contains the value of a future predictor \(X_{T+1}\). In this article, we show how to build NoVaS transformations related to ARCH and GARCH models. After acquiring Eq. (1), we can even predict \(g(Y_{T+1})\), where \(g(\cdot )\) is a general continuous function. Politis (2015) defined two databased optimal predictors of \(g(Y_{T+1})\) under \(L_1\) (Mean Absolute Deviation) and \(L_2\) (Mean Squared Error) loss criteria respectively as below:
In Eq. (2), \(\{{\hat{\epsilon }}_{T+1,m}\}_{m=1}^{M}\) are generated by Bootstrap or Monte Carlo method; see more details in “GENoVaS method” section; M takes a large number of 5000 in this study.
NoVaS transformation
The NoVaS transformation is a straightforward application of the Modelfree Prediction Principle,^{Footnote 3} which is based on the ARCH model introduced by Engle (1982), as follows:
In Eq. (3), these parameters satisfy \(a\ge 0\) and \(a_i\ge 0\) for all \(i = 1,\ldots ,p\) and \(W_t\sim i.i.d.~N(0,1)\). In other words, the structure of the ARCH model provides a readymade \(H_T\). We express \(W_t\) in Eq. (3) using the following terms:
Subsequently, Eq. (4) can be considered a potential form of \(H_T\). Additional adjustments were performed by Politis (2003) to obtain the modified Eq. (5):
In Eq. (5), \(\{Y_t;~t=1,\ldots ,T\}\) are the target data, such as financial logreturns in this study; \(\{W_{t};~t=p+1,\ldots ,T\}\) is the transformed vector; \(\alpha\) is a fixed scale invariant constant; \(s_{t1}^2\) is an estimator of the variance of \(\{Y_i;~i = 1,\ldots ,t1\}\) and can be calculated by \((t1)^{1}\sum _{i=1}^{t1}(Y_i\mu )^2\), where \(\mu\) is the mean of \(\{Y_i;~i = 1,\ldots ,t1\}\). \(\{W_t\}_{t = p+1}^{T}\) expressed in Eq. (5) are assumed to be i.i.d. N(0, 1); however, this is not the case. To make Eq. (5) a qualified function \(H_T\), that is, making \(\{W_t\}_{t=p+1}^{T}\) obey the standard normal distribution, we still need to impose some restrictions on \(\alpha\) and \(\beta , a_1,\ldots ,a_p\). Hence, first, we stabilize the variance by requiring
By imposing the above requirement, we can make \(\{W_t\}_{t=p+1}^{T}\) series possess approximate unit variance. Importantly, we must also make \(\{W_t\}_{t=p+1}^{T}\) independent. In practice, \(\{W_t\}_{t=p+1}^{T}\) transformed from financial logreturns by the NoVaS transformation are usually uncorrelated.^{Footnote 4} Therefore, if we make \(\{W_t\}_{t=p+1}^{T}\) close to a Gaussian series, that is, normalizing \(\{W_t\}_{t=p+1}^{T}\), we can obtain the desired i.i.d. transformed series. Note that the distribution of financial logreturns is usually symmetric; thus, kurtosis can serve as a simple distance to measure the departure of a nonskewed dataset from that of the standard normal distribution (Politis 2015). In addition, matching the marginal distribution seems sufficient to normalize the joint distribution of \(\{W_t\}_{t=p+1}^{T}\) for practical purposes, based on empirical results. We use \({\hat{F}}_w\) to denote the marginal distribution of \(\{W_t\}_{t=p+1}^{T}\) and use \(KURT(W_t)\) to denote the kurtosis of \({\hat{F}}_w\). Thus, to bring \({\hat{F}}_w\) close to the standard normal distribution, we attempt to minimize \(KURT(W_t)3\)^{Footnote 5} to obtain an optimal combination of \(\alpha ,\beta ,a_1,\ldots ,a_p\). Consequently, the NoVaS transformation was determined.
According to Chen (2018), based on the prediction accuracy and model structure, the Generalized Exponential NoVaS (GENoVaS) method is the most reasonable among the different NoVaStype methods with an exponentially decayed form of \(\{a_i\}_{i=1}^p\):
In this study, we verified the advantages of our new methods by comparing them with the GENoVaS method. Before further proposing the new NoVaS transformation, we discuss in more detail the GENoVaS method and our motivations for creating new methods.
GENoVaS method
For the GENoVaS method, the fixed \(\alpha\) is larger than 0 and selected from a grid of possible values based on prediction performance. In this study, we define this grid as \((0.1,0.2,\ldots ,0.8)\), containing eight discrete values.^{Footnote 6} From Eq. (2.2), using the Modelfree Prediction Principle, we can obtain the function \(H_T\) of the GENoVaS method by requiring the parameters to satisfy Eq. (7) and minimizing \(KURT(W_t)3\). To complete the modelfree prediction process, we must still determine the form of \(H_T^{1}\). From Eq. (5), \(H_T^{1}\) can be written as follows:
We can easily obtain the analytical form of \(Y_{T+1}\), which can be expressed as
In Eq. (9), \(s_T^2\) is an estimator of the variance of \(\{Y_t;~t=1,\ldots ,T\}\) and can be calculated using \(T^{1}\sum _{i=1}^T(Y_i\mu )^2\), \(\mu\) is the mean of the data. Based on Eq. (2), we can define \(L_1\) and \(L_2\) optimal predictors of \(Y_{T+1}^2\) after observing the historical information set \({\mathscr {F}}_{T} = \{Y_t,1\le t \le T\}\) as follow:
where \(\{W_{T+1,m}\}_{m=1}^{M}\) is bootstrapped M times from their empirical distribution or generated from a trimmed standard normal distribution^{Footnote 7} by using the Monte Carlo method. That is, \(Y_{T+1}\) can be represented as a function of \(W_{T+1}\) and \({\mathscr {F}}_{T}\) as follows:
To remind us of the relationship between \(Y_{T+1}\) and \(W_{T+1}, Y_1, \ldots , Y_T\) derived from the GENoVaS method, we use \(f_{GE}(\cdot )\) to denote this function. It is not difficult to determine that \(Y_{T+2}\) can be expressed as
We can generate \(\{W_{T+1,m},W_{T+2,m}\}_{m=1}^{M}\) M times to compute the \(L_1\) and \(L_2\) optimal predictors of \(Y_{T+2}^{2}\) as we did for the 1step ahead optimal prediction. Similarly, with \(\{W_{T+1,m},\ldots ,\) \(W_{T+h,m}\}_{m=1}^{M}\), we can accomplish the multistep ahead optimal prediction of \(Y_{T+h}^{2}\) for any \(h\ge 3\). In summary, we can express \(Y_{T+h}\) as
Note that the analytical form of \(Y_{T+h}\) from the GENoVaS transformation depends only on \(i.i.d.~\{W_{T+1},\ldots ,W_{T+h}\}\) and \({\mathscr {F}}_{T}\).
Motivations of building a new NoVaS transformation
Structured form of coefficients The current GENoVaS method simply sets \(\beta , a_1,\ldots ,a_p\) to be exponentially decayed. This allows us to propose the following idea. Can we build a more rigorous form of \(\beta , a_1,\ldots ,a_p\) based on the relevant model itself without assigning any prior form to the coefficients? In this study, a new approach to exploring the form of \(\beta , a_1,\ldots ,a_p\) based on the GARCH(1,1) model is proposed. Subsequently, the GARCHNoVaS (GANoVaS) transformation was built. This is discussed in “GANoVaS transformation” section.
Changing the NoVaS transformation Wu and Karmakar (2021) showed that the current stateoftheart GENoVaS method still renders extremely large timeaggregated multistep ahead predictions under \(L_2\) risk measure sometimes. The reason for this phenomenon is that the denominator of Eq. (9) is small when the generated \(W_t^*\) is very close to \(\sqrt{1/\beta }\). In this situation, the prediction error is amplified. Moreover, when a longstepahead prediction is desired, this amplification will accumulate, and the final prediction will be ruined. Thus, a \(\beta\)removing technique was applied to the GENoVaS method to obtain a GENoVaSwithout\(\beta\) method. This is a parsimonious version of the GENoVaS method. Henceforth, we call this method the PGENoVaS. Similarly, we can obtain a parsimonious variant of the GANoVaS method (PGANoVaS) by reusing this technique. A discussion of these parsimonious methods is presented in “Parsimonious variant of the GANoVaS method” and “Connection of two parsimonious methods” sections.
Longterm forecasting evaluation metric
We first describe how logreturns can be calculated from the following equation:
where \(\{X_t\}_{t = 1}^{250}\) and \(\{X_t\}_{t = 1}^{500}\) are 1year and 2year price series, respectively. Next, we define the timeaggregated predictions of squared logreturns with three different lengths of the prediction horizon as
In Eq. (15), \({\hat{Y}}_{k+1}^2,{\hat{Y}}_{i+m}^2,{\hat{Y}}_{j+m}^2\) are singlepoint predictions of realized squared logreturns by NoVaStype methods or the benchmark method; \({\bar{Y}}_{k,1}^2\), \({\bar{Y}}_{i,5}^2\) and \({\bar{Y}}_{j,30}^2\) represent 1step, 5steps and 30steps ahead aggregated predictions, respectively. More specifically, for exhausting the information contained in the dataset, we roll the 250 data points window through the whole dataset, that is, we use \(\{Y_1,\ldots ,Y_{250}\}\) to predict \(Y_{251}^2,\{Y_{251}^2,\ldots ,Y_{255}^2\}\) and \(\{Y_{251}^2,\ldots ,Y_{280}^2\}\); then we use \(\{Y_2,\ldots ,Y_{251}\}\) to predict \(Y_{252}^2,\{Y_{252}^2,\ldots ,Y_{256}^2\}\) and \(\{Y_{252}^2,\ldots ,Y_{281}^2\}\), for 1step, 5steps, and 30steps aggregated predictions respectively, and so on. To explore the performance of three different prediction lengths with small data size, we roll the 100 data point window through the entire dataset. For example, with a prediction horizon of 30, we perform timeaggregated predictions on a large dataset 220 times.
To measure the forecasting performance of the different methods, we propose a timeaggregated metric based on Eq. (16).
In Eq. (16), setting \(l = k,i,j\) means we consider 1step, 5steps, and 30steps ahead timeaggregated predictions, respectively; \({\bar{Y}}_{l,h}^2\) is the hstep (\(h\in \{1,5,30\}\)) ahead timeaggregated volatility prediction, defined in Eq. (15); \(\sum _{m=1}^h(Y_{l+m}^2/h)\) is the corresponding true aggregated value calculated from the realized squared logreturns. To compare various NoVaStype methods with the traditional method, we set a benchmark method to fit one GARCH(1,1) model directly (GARCHdirect). In “Simulation” and “Realworld data analysis” sections, we applied this metric to the simulation and real data analyses. In addition, in “Comparison of predictive accuracy: statistical tests” section, statistical tests are deployed to explore the predictive accuracy of NoVaS methods further.
New variants of the NoVaS method
In this section, we first propose the GANoVaS method which is directly developed from the GARCH(1,1) model without assigning any specific form of \(\beta , a_1,\ldots ,a_p\). Then, the PGANoVaS method is introduced by applying the \(\beta\)removing technique. We also provide algorithms for these two new methods at the end.
GANoVaS transformation
Recall that the GENoVaS method mentioned in “GENoVaS method” section, was built by exploiting the ARCH(p) model for a large p. Although the ARCH model is the basis of the GENoVaS method, the free parameters of the GENoVaS method are only c and \(\alpha\). To represent \(p+1\) number of coefficients using only two free parameters, some specific forms are assigned to \(\beta , a_1,\ldots ,a_p\). Here, we attempt to use a more convincing approach to find \(\beta , a_1,\ldots ,a_p\) directly, without assigning any prior form to these parameters. We call this NoVaS transformation method the GANoVaS.
The idea behind this new method was inspired by the fact that the GARCH(1,1) model is equivalent to the corresponding ARCH(\(\infty\)) model. If we want to build a NoVaS transformation based on the GARCH(1,1) model, the denominator on the righthand side of Eq. (4) should be replaced by the structure of the GARCH(1,1) model, which has the form Eq. (17):
In Eq. (17), \(a \ge 0\), \(a_1 > 0\), \(b_1 > 0\), and \(W_t\sim i.i.d.~N(0,1)\). In other words, a potentially qualified transformation related to the GARCH(1,1) or ARCH(\(\infty\)) model can be expressed as:
However, recall that the core insight of the NoVaS method connects the original data with the transformed data using a qualified transformation function. A primary problem here is that the righthand side of Eq. (18) contains terms other than \(\{Y_t\}\). Thus, additional manipulations are required to build the GANoVaS method. In fact, we can finally derive the transformation functions \(H_{T}\) and \(H_{T}^{1}\) corresponding to the GANoVaS method as follows:
where \(t=q+1,\ldots ,T\); see “Appendix 1” for details of this deduction process and the form of \(\{c_i\}_{i = 0}^{q}\).
Remark 1
(The difference between GANoVaS and GENoVaS methods) Compared with the existing GENoVaS method, the GANoVaS method possesses a completely different transformation structure. All coefficients except for \(\alpha\) implied by the GENoVaS method are expressed as \(\beta = c', a_i = c'e^{ci}~\) \(\text {for all}~1\le\) \(i\le p\), \(c' = \frac{1\alpha }{\sum _{j=0}^pe^{cj}}\). There are only two free parameters, c and \(\alpha\). However, there are four free parameters \(\beta , a_1, b_1\) and \(\alpha\) in Eq. (35). For example, the coefficient of \(Y_t^2\) in the GENoVaS method is \((1\alpha )/(\sum _{j=0}^pe^{cj})\). By contrast, the corresponding coefficient in the GANoVaS structure is \(\beta (1\alpha )/(\beta +(1b_1)\sum _{i=1}^{q}a_1b_1^{i1})\). We can assume that the freedom of coefficients within the GANoVaS is larger than the freedom in the GENoVaS. Simultaneously, the structure of the GANoVaS method is built from the GARCH(1,1) model directly without imposing any prior assumptions on the coefficients. We believe this is the reason why our GANoVaS method shows a better prediction performance in “Simulation” and “Realworld data analysis” sections.
Next, it is not difficult to express \(Y_{T+h}\) as a function of \(W_{T+1},\ldots , W_{T+h}\) and \({\mathscr {F}}_{T}\) using the GANoVaS method, as we did in “GENoVaS method” section:
Once the expression of \(Y_{T+h}\) is determined, we can apply the same procedure with the GENoVaS method to obtain the optimal predictor of \(Y_{T+h}\) under \(L_1\) or \(L_2\) risk criterion. To address \(\alpha\), we adopt the same strategy used in the GENoVaS method. Note that the value of \(\alpha\) is invariant during the optimization process once it is fixed as a specific value. More details regarding the algorithm of this new method can be found in “Algorithms of new methods” section.
Parsimonious variant of the GANoVaS method
According to the \(\beta\)removing concept, we can continue to propose the PGANoVaS method, which is a parsimonious variant of the GANoVaS method. First, we present the PGENoVaS method from Wu and Karmakar (2021).
Equation (21) still needs to satisfy the requirement of normalizing and variancestabilizing transformation. Therefore, we restrict \(\alpha + \sum _{i=1}^pa_i = 1\) and select the optimal combination of \(\alpha , a_1,\ldots ,a_p\) by minimizing \(KURT(W_t)3\). Then, \(Y_{T+1}\) can be expressed as Eq. (22):
Remark 2
Even though we do not include the effect of \(Y_T\) when we build \(H_T\), the expression of \(Y_{T+1}\) still contains the current value \(Y_T\). This means that the PGENoVaS method does not disobey the causal prediction rule.
Similarly, the PGANoVaS can be represented by the following equation:
Note that \(\{\tilde{c}_1,\ldots ,\tilde{c}_q\}\) represents \(\{a_1,a_1b_1\) \(,a_1b_1^{2},\) \(\ldots ,a_1b_1^{q1} \}\) scaled by multiplying a scalar \(\frac{1\alpha }{\sum _{j=1}^{q}a_1b_1^{j1}}\) and the optimal combination of \(\alpha , a_1,b_1\) is selected by minimizing \(KURT(W_t)3\) to satisfy the normalizing requirement. For the PGENoVaS and PGANoVaS methods, we can express \(Y_{T+h}\) as a function of \(\{W_{T+1},\ldots ,W_{T+h}\}\) and repeat the aforementioned procedure to obtain the optimal \(L_1\) and \(L_2\) predictors. For example, we can derive the expression for \(Y_{T+h}\) using the PGANoVaS method:
Remark 3
(Slight computational efficiency from removing \(\beta\)) Note that the computation cost of NoVaStype methods without \(\beta\) term is less than that of the current ones because: recall \(1/\sqrt{\beta }\) is required to be larger than or equal to three to ensure that \(\{W_t\}\) has a sufficiently large range, that is, \(\beta\) is required to be less than or equal to 0.111. However, the optimal combination of NoVaS coefficients may not render a suitable \(\beta\). Therefore, we need to increase the time series order (p or q) and repeat the normalizing and variancestabilizing processes until \(\beta\) in the optimal combination of coefficients is appropriate. This replication process increased the computational workload.
Connection of two parsimonious methods
In this subsection, we reveal that the PGENoVaS and PGANoVaS methods have the same structure. The difference between these two methods lies in the region of free parameters. To observe this phenomenon, let us consider the scaled coefficients of the PGANoVaS method, except for \(\alpha\).
Recall that the parameters of the PGENoVaS method, except for \(\alpha\) implied by Eq. (7), are:
Observing the above two equations, although we can discover that Eqs. (25) and (26) are equivalent if we set \(b_1\) to be equal to \(e^{c}\), these two methods are still different because the regions \(b_1\) and c play a role in the process of optimization. The complete region for c is \((0,\infty )\). However, Politis (2015) indicated that c cannot take a large value^{Footnote 8} and the region c should be an interval of type (0, m) for some m. In other words, a formidable search problem for determining the optimal c is avoided by choosing a trimmed interval. However, \(b_1\) is explicitly searched from (0, 1) which corresponds to c taking values from \((0,\infty )\). Similarly, by applying the PGANoVaS method, the aforementioned burdensome search problem is eliminated. Moreover, we can construct a transformation based on the entire available region of the unknown parameter. Therefore, we argue that the PGANoVaS method is more stable and reasonable than the PGENoVaS method. Based on empirical comparisons, the PGANoVaS method can achieve significantly superior prediction performance in some cases; see “Appendix 2” for more details.
Algorithms of new methods
In this section, we provide algorithms of the two methods. For the GANoVaS method, the unknown parameters \(\beta , a_1, b_1\) are selected from three grids of possible values to normalize \(\{W_t;~t = q+1,\ldots ,T\}\). If our goal is the hstepahead prediction of \(g(Y_{T+h})\) using past \(\{Y_t;~t=1,\ldots ,T\}\), the algorithm of the GANoVaS method can be summarized in Algorithm 1.
To apply the PGANoVaS method, we only need to change Algorithm 1 slightly to obtain Algorithm 2.
In our experimental setting, we choose regions of \(\beta ,a_1,b_1\) being (0, 1) and set a 0.02 grid interval to find all parameters. In addition, for the GANoVaS method, we ensure that the sum of \(\beta ,a_1,b_1\) is less than 1, and the coefficient of \(Y_t^{2}\) is the largest.
Simulation
In simulation studies, for controlling the dependence of prediction performance on the length of the dataset, 16 datasets (2 from each setting) are generated from 8 different GARCH(1,1)type models separately and the size of each dataset is 250 (short data mimics 1year of econometric data) or 500 (large data mimics 2year of econometric data).
Model 1 Timevarying GARCH(1,1) with Gaussian errors
\(X_t = \sigma _t\epsilon _t,~\sigma _t^2 = \omega _{0,t} + \beta _{1,t}\sigma _{t1}^2+\alpha _{1,t}X_{t1}^2,~\{\epsilon _t\}\sim i.i.d.~N(0,1)\)
\(g_t = t/n; \omega _{0,t}= 4sin(0.5\pi g_t)+5; \alpha _{1,t} = 1(g_t0.3)^2 + 0.5; \beta _{1,t} = 0.2sin(0.5\pi g_t)+0.2,~n = 250~\text {or}~500\).
Model 2 Another timevarying GARCH(1,1) with Gaussian errors
\(X_t = \sigma _t\epsilon _t,~\sigma _t^2 = 0.00001 + \beta _{1,t}\sigma _{t1}^2+\alpha _{1,t}X_{t1}^2,~\{\epsilon _t\}\sim i.i.d.~N(0,1)\)
\(g_t = t/n\); \(\alpha _{1,t} = 0.1  0.05g_t\); \(\beta _{1,t} = 0.73 + 0.2g_t,~n = 250~\text {or}~500\).
Model 3 Standard GARCH(1,1) with Gaussian errors
\(X_t = \sigma _t\epsilon _t,~\sigma _t^2 = 0.00001 + 0.73\sigma _{t1}^2+0.1X_{t1}^2,~\{\epsilon _t\}\sim i.i.d.~N(0,1)\).
Model 4 Standard GARCH(1,1) with Gaussian errors
\(X_t = \sigma _t\epsilon _t,~\sigma _t^2 = 0.00001 + 0.8895\sigma _{t1}^2+0.1X_{t1}^2,~\{\epsilon _t\}\sim i.i.d.~N(0,1)\).
Model 5 Standard GARCH(1,1) with Studentt errors
\(X_t = \sigma _t\epsilon _t,\) \(~\sigma _t^2 = 0.00001 + 0.73\sigma _{t1}^2+0.1X_{t1}^2,\)
\(~\{\epsilon _t\}\sim i.i.d.~t\) \(\text {distribution with five degrees of freedom}\).
Model 6 Exponential GARCH(1,1) with Gaussian errors
\(X_t = \sigma _t\epsilon _t,~\log (\sigma _t^2) = 0.00001 + 0.8895\log (\sigma ^2_{t1})+0.1\epsilon _{t1}+0.3(\epsilon _{t1}E\epsilon _{t1}),\)
\(~\{\epsilon _t\}\sim i.i.d.~N(0,1)\).
Model 7 GJRGARCH(1,1) with Gaussian errors
\(X_t = \sigma _t\epsilon _t,~\sigma _t^2 = 0.00001 + 0.5\sigma ^2_{t1}+0.5X_{t1}^20.5I_{t1}X_{t1}^2,~\{\epsilon _t\}\sim i.i.d.~N(0,1)\)
\(I_{t} = 1~\text {if}~ X_t \le 0, and I_{t} = 0~ \text {otherwise}\).
Model 8 Another GJRGARCH(1,1) with Gaussian errors
\(X_t = \sigma _t\epsilon _t,~\sigma _t^2 = 0.00001 + 0.73\sigma ^2_{t1}+0.1X_{t1}^2+0.3I_{t1}X_{t1}^2,~\{\epsilon _t\}\sim i.i.d.~N(0,1)\)
\(I_{t} = 1~\text {if}~ X_t \le 0, and I_{t} = 0~ \text {otherwise}\).
Model description Models 1 and 2 present a timevarying GARCH model where coefficients \(a_0, a_1, b_1\) change over time slowly. They differ significantly in the intercept term of \(\sigma _t^2\) as we intentionally kept it low in the second setting. Models 3 and 4 are from a standard GARCH, where, in Model 4, we wanted to explore a scenario in which \(\alpha _1+\beta _1\) is very close to 1 and thus mimics what would happen for the iGARCH situation. Model 5 allows the error distribution to originate from a studentt distribution instead of a Gaussian distribution. For fair competition with the existing GENoVaS method, we chose Models 2 to 5, similar to the simulation settings of Chen and Politis (2019). Models 6, 7, and 8 present the different types of GARCH models. These settings allow us to check the robustness of our method against model misspecification. In the real world, it is difficult to convincingly tell if the data obey one particular type of GARCH model; hence, we shall pursue this exercise to see if our methods are satisfactory, regardless of the underlying distribution and the GARCHtype model. This approach to test the performance of a method under model misspecification is standard; see Olubusoye et al. (2016) and Bellini and Bottolo (2008) for more examples.
Window size Using these datasets, we perform 1step, 5steps, and 30steps ahead timeaggregated POOS predictions. To measure the prediction performance of different methods on larger datasets (i.e., data size of 500), we use 250 data as a window to perform predictions and roll this window through the entire dataset. To evaluate the performance of different methods on smaller datasets (i.e., data size of 250), we use 100 data as a window.
Different variants of methods Note that we can perform GENoVaStype and GANoVaStype methods to predict \(Y_{T+h}\) by generating \(\{W_{T+1,m},\ldots , W_{T+h,m}\}_{m=1}^{M}\) from a standard normal distribution or the empirical distribution of \(\{W_t\}\) series, then we can calculate the optimal predictor based on \(L_1\) or \(L_2\) risk criterion. This means that each NoVaStype method has four variants.
When performing POOS forecasting, we do not know which \(\alpha\) is optimal. Therefore, we perform every NoVaS variant using \(\alpha\) from eight potential values \(\{0.1, 0.2, \ldots ,0.8\}\) and then select the optimal result. To simplify the presentation, we further select the final prediction from the optimal results of the four variants of the NoVaS method and use this result to be the best prediction to which each NoVaS method can reach. This procedure allows us to take a computationally intensive approach to compare the potentially best performances of different methods.
Simulation results
In this subsection, we compare the performances of our new methods (GANoVaS and PGANoVaS) with GARCHdirect and existing GENoVaS methods on forecasting 250 and 500 simulated data. Based on the timeaggregated prediction metric Eq. (16), the results are tabulated in Table 1.^{Footnote 9}
Simulation results of Models 1 to 5
From Table 1, we conclusively find that NoVaStype methods outperform the GARCHdirect method. Especially when using the 500 Model 1 data to perform 30steps ahead of the aggregated prediction, the performance of the GARCHdirect method is poor. NoVaStype methods are almost 30 times better than the GARCHdirect method indicating that the standard prediction method may be affected by the error accumulation problem when longterm predictions are required. However, modelfree methods can overcome this problem.
In addition to the overall advantage of NoVaStype methods over the GARCHdirect method, we find that the GANoVaS method is generally better than the GENoVaS method in predicting both short and large data. This conclusion is twofold: (1) GANoVaS consumes less time than the GENoVaS method; (2) because we want to compare the forecasting ability of the GENoVaS and GANoVaS methods, we use \(*\) symbol to represent cases where the GANoVaS method works at least 10\(\%\) better than the GENoVaS method, or inversely, the GENoVaS method is 10\(\%\) better. We find no case to support that the GENoVaS works better than the GANoVaS with at least 10\(\%\) improvement. On the other hand, the GANoVaS method exhibits a significant improvement when longterm predictions are required. Moreover, the PGANoVaS dominates the other two NoVaStype methods.
Models 6 to 8: different GARCH specifications
Since the main crux of Modelfree methods is how such nonparametric methods are robust to different underlying datageneration processes. The GANoVaS method is based on the GARCH model, so it is interesting to explore whether these methods can sustain a different type of true underlying data generation process. The simulation results for Models 6–8 are tabulated in Table 1.
In general, NoVaStype methods still outperform the GARCHdirect method for these cases. The GANoVaS method is better than the GENoVaS method in terms of longterm forecasting. In addition, the GANoVaS method can also bring about significant improvement for shortsize data, such as the 30steps ahead aggregated prediction of 250 Model 6 simulated data. Improving prediction with short data is always a significant challenge; thus, it is valuable to discover whether the GANoVaS method gives superior performance in this scenario. Unsurprisingly, the PGANoVaS method performed well.
Simulation summary
Simulation data analysis shows that NoVaStype methods can sustain great performance against short data and model misspecification. Overall, our new method outperforms the GENoVaS method with notable improvements in some cases where longterm predictions are desired, such as the 500size simulation of Model 8. Table 1, clearly shows that the GARCHdirect method is unsuitable for this case. To further compare the different methods in an absolute sense for this case, we plot the predictions of different methods and actual values in the same figure. Based on these plots, it is clear that the GARCH method is unstable and far from the true curve for longterm aggregated predictions. However, NoVaStype methods work well and fit the trend of the true curve in an absolute sense. The corresponding plots are shown in “Appendix 3”. Furthermore, the NoVaStype methods outperform the GARCH method for Models 3 and 4, even if the underlying model is also GARCH(1,1). Moreover, we find that NoVaStype methods are competitive when applying the estimated Exponential GARCH(1,1) and GJRGARCH(1,1) models to predict Models 6 and 8, respectively. These results further support the claim that NoVaStype methods are robust against model misspecification. The efficiency of the modelfree prediction concept is demonstrated. The corresponding analyses are provided in “Appendix 3”.
Realworld data analysis
This section is devoted to exploring, in the context of real datasets forecasting, whether NoVaStype methods can provide good longterm timeaggregated forecasting and how our new methods compare to the existing Modelfree method.
To conduct extensive analyses and subsequently obtain a convincing conclusion, we use three types of data—stock, index, and currency—to perform predictions. Moreover, as in the simulation studies, we apply this exercise to two different data lengths. To build large datasets (2year period data), we take more recent datasets from January 2018 to December 2019 and previous data from approximately 20 years ago, separately. The dynamics of these econometric datasets have changed significantly over the past 20 years; therefore, we wanted to explore whether our methods are suitable for both old and new data. Subsequently, we challenge our methods using short (1year) reallife data. We also perform forecasting using volatile data, that is, data from November 2019 to October 2020. Note that economies across the world went through a recession due to the COVID19 pandemic and then slowly recovered during this period; typically, these types of situations introduce systematic perturbation in the dynamics of econometric datasets. We aimed to determine whether our methods could sustain such perturbations or abrupt changes.
2year data
For mimicking the 2year period data, we adopt several stock datasets—AAPL, BAC, MSFT and MCD—with 500 data size to perform forecasting. In summary, we compare the performances of different methods on 1step, 5steps, and 30steps ahead POOS timeaggregated predictions. All results obtained through a procedure similar to that in “Simulation” section are shown in Table 2. The NoVaStype methods still outperform the GARCHdirect method. Additionally, our new method is more robust than the GENoVaS method; see the 30steps ahead prediction of the previous 2year BAC and MSFT cases. We can also see that the PGANoVaS method is more robust than the other two NoVaS methods. The \(\beta\)removing idea proposed by Wu and Karmakar (2021) was substantiated again.
Because the main objective of this study is to offer a new type of NoVaS method that performs better than the GENoVaS method in dealing with short and volatile data, we provide more extensive data analyses to support our new methods in the sections ahead.
2018 and 2019 1year data
For challenging our new methods in contrast to other methods for small reallife datasets, we separate every new 2year period data in “2year data” section into two 1year period datasets, that is, separate four new stock datasets to eight samples. We believe that evaluating the prediction performance using shorter data is a more important problem, and thus, we wanted to make our analysis very comprehensive. Therefore, for this exercise, we add seven index datasets: NASDAQ, NYSE, Small Cap, Dow Jones, S &P 500, BSE and BIST; and two stock datasets: Tesla and Bitcoin into our analysis.
From Table 3, which presents the prediction results of different methods on the 2018 and 2019 stock data, we still observe that NoVaStype methods outperform the GARCHdirect method for almost all cases. Among the different NoVaS methods, it is obvious that our new methods are superior to the existing GENoVaS method. After applying the \(\beta\)removing concept, the PGANoVaS method significantly outperforms the other methods in almost all cases.
From Table 4, which presents the prediction results of different methods on the 2018 and 2019 index data, we obtain the same conclusion as before. NoVaStype methods are far superior to the GARCHdirect and our new NoVaS methods outperform the existing GENoVaS method. Interestingly, the GENoVaS method is beaten by the GARCHdirect method in some cases, such as the 2019NASDAQ, Smallcap, and BIST. However, the new methods still exhibit stable performance.
Volatile 1year data
In this subsection, we perform POOS forecasting using volatile 1year data (i.e., data from November 2019 to October 2020). We tactically choose this period data to challenge our new methods for checking whether it can selfadapt to the structural incoherence between pre and postpandemic, and compare our new methods with the existing GENoVaS method. To observe the effects of the pandemic, we take the price of the S &P500 index as an example. From Fig. 1, it is apparent that the price grew slowly during the normal period from January 2017 to December 2017. However, during the period from November 2019 to October 2020, prices fluctuated severely due to the pandemic.
Stock data
The POOS forecasting results of volatile 1year stock datasets are presented in Table 5. NoVaStype methods dominate the GARCHdirect method. The performance of the GARCHdirect method is poor, especially for the Bitcoin case. Apart from this overall advantage of NoVaStype methods, there is no doubt that the GANoVaS method exhibits greater prediction results than the GENoVaS method because it occupies 13 out of 27 optimal choices and represents at least a 10\(\%\) improvement for five cases. The PGANoVaS method also shows better results than those of the GENoVaS method.
Currency data
The POOS forecasting results of selected most recent 1year currency datasets are presented in Table 6. The meaning of bold values and values with asterisk marks is the same as the definition in Table 5. Note that Fryzlewicz et al. (2008) showed that the ARCH framework appears to be a superior methodology for dealing with currency exchange data. Therefore, we should not anticipate that GANoVaStype methods can attain significant improvements for this data case. However, the GANoVaS method still results in approximately 26\(\%\) and 37\(\%\) improvement for 30steps ahead aggregated predictions of CADJPY and CNYJPY, respectively. Besides, the PGANoVaS method also remains a great performance.
Index data
The POOS forecasting results of the most recent 1year index datasets are presented in Table 7. The meaning of bold values and values with asterisk marks is the same as the definition in Table 5. Consistent with the conclusions corresponding to the previous two classes of data, NoVaStype methods still exhibit obviously better performance than the GARCHdirect method. In addition to this advantage of NoVaS methods, new methods still govern the existing GENoVaS method. In addition to these expected results, we find that the GENoVaS method is 14\(\%\) worse than the GARCHdirect method for 1step USDX future case. However, GANoVaStype methods still perform well. This phenomenon also appears in “Simulation results of Models 1 to 5”, “Models 6 to 8: Different GARCH specifications”, “Simulation summary”, “2year data” and “2018 and 2019 1year data” sections. Beyond this, there are 12 cases in which the GANoVaS method renders more than a 10\(\%\) improvement compared to the GENoVaS method.
Summary of realworld data analysis
After extensive realworld data analysis, we can conclude that NoVaStype methods generally perform better than the GARCHdirect method. Sometimes, the longterm prediction of the GARCHdirect method is impaired because of accumulated errors. Applying NoVaStype methods helps avoid this issue. In addition to this encouraging result, the two new NoVaS methods proposed in this study perform better than the existing GENoVaS method, especially for analyzing short and volatile data. We present some plots to compare various methods in “Appendix 4”, as we did in “Simulation summary” section. In addition, the satisfactory performance of NoVaStype methods in predicting Bitcoin data may also open up the application of NoVaStype methods to forecasting cryptocurrency data.
Comparison of predictive accuracy: statistical tests
In this section, we determine whether the victory of our new methods is statistically significant. We note that Wu and Karmakar (2021) applied CW tests to show that removing the\(\beta\) idea is appropriate for refining the GENoVaS method. Likewise, we are curious whether this refinement is reasonable for deriving the PGANoVaS method from the GANoVaS method. In this study, we focus on the CW test built by Clark and West (2007)^{Footnote 10} which applied an adjusted Mean Squared Prediction Error (MSPE) statistic to test if the parsimonious null model and larger model have equal predictive accuracy; see Dangl and Halling (2012), Kong et al. (2011) and Dai and Chang (2021) for examples of applying this CW test. In addition, we also take the Model Confidence Set (MCS) proposed by Hansen et al. (2011) to eliminate inferior models.
CWtest
Note that the PGANoVaS method is parsimonious compared to the GANoVaS method. The reason for removing the \(\beta\) term is described in “Motivations of building a new NoVaS transformation” section. Here, we want to deploy the CW test to ensure that the \(\beta\)removing idea is not only empirically adoptable, but also statistically reasonable. We use several results from “Realworld data analysis” section to run the CW tests. However, it is difficult to apply the CW test to compare 5steps and 30steps aggregated predictions. In other words, the CW test results for the aggregated predictions are ambiguous. It is difficult to explain the significance of a significantly small p value. Does this mean that the method outperforms the opposite for all singlestep horizons? Alternatively, does this mean that the method achieves better performance in some specific future steps? Therefore, we consider the 1step ahead prediction horizon, and the CW test results are tabulated in Table 8.
From Table 8, under a onesided 5\(\%\) significance level, there is only one case out of the 28 cases that reject the null hypothesis. This result implies that GANoVaS and PGANoVaS methods are statistically equivalent. However, the PGANoVaS method is more computationally efficient than the GANoVaS method because it uses a more concise format. More importantly, it provides a better empirical prediction performance. Thus, the reasonability of removing \(\beta\) term is demonstrated again, and we favor the PGANoVaS method in practice.
MCStest
Accompanied by the CW test, we utilize the Model Confidence Set (MCS)^{Footnote 11} to determine a set of preferred models. The procedure for building the MCS is made up of a sequence of tests (henceforth MCStest), where the null hypothesis of equal predictive ability (EPA) is not rejected at a specific confidence level. The advantage of the MCS test is that we can apply different loss functions, such as MSE and QLIKE, to compute the test statistics corresponding to different models and then select the best models. Moreover, we can rank the models in the MCS based on their prediction performance. Here, according to Eq. 8 of Bernardi and Catania (2018), we use the second test statistic \(T_{max,M}\) to rank all models. To compare the different models, we propose three criteria: (1) Average Rank Order (ARO), which is the average of rank orders with respect to each model; (2) Confidence Set Rate (CSR), which is the relative frequency of each model belonging to the MCS; and (3) Best Model Rate (BMR), which is the relative frequency of each model ranking first. Similar criteria are defined in Amendola and Candila (2016). The three criteria are as follows:
where \(ARO_i\), \(CSR_i\), and \(BMR_i\) represent the ARO, CSR, and BMR for the ith interested model, respectively; D represents the number of datasets we apply for comparisons; \(r_{d,i}\) stands for the rank order of the ith model on the dth dataset; \(I(M_{d,i}\subset {\hat{M}}^{*}_{d,1\alpha })\) is the indicator function, which is equal to 1 if the ith model belongs to the MCS \({\hat{M}}^{*}_{d,1\alpha }\) for the dth dataset, otherwise it equals 0; \(FM_{d,i}\) equals 1 if the ith ranks first for the dth dataset and equals 0 otherwise.
We run the MCS test on datasets that are applied to create Table 8. We take the confidence level of the MCS test to be 95\(\%\) and adopt the MSE loss function to compute the test statistics. If one model is eliminated from the MCS, we set the corresponding rank order to 6. All the results are shown in Table 9. The PGANoVaS method has the lowest ARO and the highest BMR. However, the GARCHdirect method does not win any firstrank title. In addition, we can see the “naive” GENoVaS method is dominated by other NoVaS methods. As indicated in Wu and Karmakar (2021), the PGENoVaS method is superior to the GENoVaS method. Here, the claim is verified by the MCStest again based on these three criteria. Interestingly, with the new transformation structure developed from the GARCH model, GANoVaS is competitive with PGENoVaS even without applying the \(\beta\)removing technique.
Results and discussion
We conducted substantial simulation analyses to demonstrate the advantages of NoVaS methods for longterm forecasting and contrast our new methods with the existing NoVaS method. We compared as many as eight different simulation setups, and the highlighted benefits of the new methods are fairly uniform. In addition, we have provided a comprehensive realdata study to show that the advantages we are discussing in this paper not only stem from analyzing a particular dataset or even a particular type of data. We covered two different sizes of three different data types: (1) traditional stocks, (2) currency data, and (3) index data. Moreover, we covered three different lengths of the prediction horizon. After such a comprehensive range of experiments, we are confident that these methods will perform adequately well with any financial economic data from this wide range of forecasting exercises. Overall, the current stateoftheart GENoVaS and our proposed new methods can avoid error accumulation problems, even when longstep ahead predictions are required. These methods outperform the GARCH(1,1) model in predicting either simulated or realworld data under different forecasting horizons.
In the future, we plan to explore the NoVaS method in various directions. Our new methods corroborate this and also open up avenues for exploring other specific transformation structures. In the financial market, stock data move together. Therefore, it would be interesting to see if one can make Modelfree predictions for multiple time series directly. In certain areas, integervalued time series have important applications. Thus, adjusting such Modelfree predictions to handle count data is desirable. Moreover, with the advent of widely accessible highfrequency financial data, researchers have begun to investigate methods for digesting this abundant information within data, such as the heterogeneous autoregressive (HAR) model of Corsi (2009) and the GARCH model of Hansen et al. (2012). Application of the NoVaS prediction framework to highfrequency data could be a meaningful extension. In addition, the volatility forecasting returned by NoVaS methods could be considered as a meaningful feature that serves financial purposes, for example, bankruptcy prediction of small and mediumsized enterprises (SMEs) or complicated financial risk analysis; see related discussions from the work of Kou et al. (2014, 2021). There is also much scope in proving the statistical validity of such predictions. First, we hope to provide a rigorous and systematic way to compare the predictive accuracy of NoVaStype and standard GARCH methods for timeaggregated forecasting. From a statistical inference point of view, one can also construct prediction intervals for these predictions using bootstrapping. Such prediction intervals are well sought in the econometrics literature, and some results on their asymptotic validity can be proved. We can also explore dividing the dataset into testing and training in an optimal manner and determine whether it can improve the performances of these methods. Beyond relying on Eq. (16) to measure different models, we can also consider proposing other measurements, such as the QLIKElossbased criterion. Simultaneously, the investigation of NoVaS methods for optimizing nonsymmetric loss could be a future work. Additionally, because determining the transformation function involves the optimization of unknown coefficients, designing a more efficient and precise algorithm may be a further direction for improving NoVaStype methods.
Conclusion
The NoVaS method is a nonparametric approach that can be used for many recursive timeseries models. This study sheds new light on an attractive feature of the NoVaS method in the regime of conditional heteroscedastic models and then builds on new variants that can improve the stateoftheart NoVaS methods designed for ARCH processes. Moreover, the newly proposed GANoVaS method has a more stable structure for handling volatile and short data than the already competent GENoVaS method. It can also bring about significant improvements when longterm prediction is desired. Additionally, although we reveal that parsimonious variants of GANoVaS and GENoVaS possess the same structure, the PGANoVaS method is still more favorable because the corresponding region of the model parameter is more complete by design. In addition, the result from the CW test also indicates the possibility of achieving a good forecasting performance with the parsimonious version of the GANoVaS. In summary, the approach to building the NoVaS transformation using the GARCH(1,1) model is sensible and results in superior GANoVaStype methods.
Methods/experimental
In this study, we consider five methods: (1) the current NoVaStype method, (2) GENoVaS and its parsimonious variant, (3) PGENoVaS, (4) the newly proposed GANoVaS and PGANoVaS, and (5) standard GARCH(1,1). To compare the performances of these methods with longterm timeaggregated predictions of volatility, we deployed simulations using eight GARCHtype models. We also selected comprehensive realworld datasets that cover traditional stock, currency, and index data. The prediction procedure and evaluation metrics are explained in “Longterm forecasting evaluation metric” section. Moreover, we substantiated the superiority of our methods using the CW and MCS tests. These tests are described in “Comparison of predictive accuracy: statistical tests” section. All data analyses are parallelly computed in the Rstudio.
Availability of data and materials
We have collected all data presented here from www.investing.com manually. Then, we transform the closing price data to financial logreturns based on Eq 4.1 in the manuscript.
Notes
Squared logreturns are unbiased, but a very noisy measure of volatility pointed out by Andersen and Bollerslev (1998). Additionally, Awartani and Corradi (2005) showed that using squared logreturns as a proxy for volatility can render a correct ranking of different GARCH models in terms of a quadratic loss function.
The POOS forecasting analysis means using data up to and including current time to predict future values.
The “Modelfree” in this context means we do not rely on a statistical model to do predictions. Although a transformation function needs to be estimated, it is just a “bridge” which connects original and transformed distributions.
If \(\{W_t\}_{t=p+1}^{T}\) is correlated, some additional manipulations need to be done, more details can be found in Politis (2015).
More details about this minimizing process can be found in Politis (2015).
It is possible to refine this grid to get a better transformation. However, the computation burden will also increase.
The reason for utilizing a trimmed standard normal distribution is transformed \(\{W_t\}_{t=p+1}^{T}\) are between \(1/\sqrt{\beta }\) and \(1/\sqrt{\beta }\) from Eq. (5).
When c is large, \(a_i \approx 0\) for all \(i > 0\). It is hard to make the kurtosis of transformed series be 3.
Due to the slidingwindow prediction property, we only repeat each simulation 5 times and present average results.
See Clark and West (2007) for the theoretical details of this test, explaining that these details are not within the scope of this study.
Abbreviations
 NoVaS:

Normalizing and variance stabilizing
 FITVGARCH:

Fractionally integrated timevarying GARCH
 POOS:

Seudoout of sample
 GENoVaS:

Generalized Exponential NoVaS
 GANoVaS:

GARCHNoVaS
 PGENoVaS:

Parsimonious GENoVaS
 PGANoVaS:

Parsimonious GANoVaS
 MCS:

Model confidence set
 EPA:

Equal predictive ability
 ARO:

Average rank order
 CSR:

Confidence set rate
 BMR:

Best model rate
 HAR:

Heterogeneous autoregressive model
 EXPGARCH(1,1):

Exponential GARCH(1,1)
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Acknowledgements
The first author is thankful to Professor Dimitris N. Politis for the introduction to the topic and useful discussions. The second author’s research is partially supported by NSFDMS 2124222
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The second author’s research is partially supported by NSFDMS 2124222.
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Data curation, KW and SK; Formal analysis, SK; Investigation, KW and SK; Methodology, SK; Software, KW and SK; Visualization, KW; Writing original draft, KW; Writing and editing manuscript, SK.
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Appendix
Appendix
Appendix 1: The deduction of \(H_{T}\) and \(H_{T}^{1}\) corresponding with the GANoVaS method
Based on Eq. (17), we first find out expressions of \(\sigma _{t1}^2,\sigma _{t2}^2,\ldots\) as follow:
Plug all components in Eq. (28) into Eq. (17), one equation sequence can be gotten:
Iterating the process in Eq. (29), with the requirement of \(a_1+b_1<1\) for the stationarity, the limiting form of \(Y_t\) can be written as Eq. (30):
We can rewrite Eq. (30) to get a potential function \(H_T\) which is corresponding to the GANoVaS method:
Recall the adjustment taken in the existing GENoVaS method, the total difference between Eq. (4,5) can be seen as the term a being replaced by \(\alpha s_{t1}^2 + \beta Y_t^2\). Apply this same adjustment on Eq. (31), then this equation will be changed to the form as follows:
In Eq. (32), since \(\alpha /(1b_1)\) is also required to take a small positive value, this term can be seen as a \(\tilde{\alpha }\) (\(\tilde{\alpha } \ge 0\)) which is equivalent with \(\alpha\) in the existing GENoVaS method. Thus, we can simplify \(\alpha s_{t1}^2/(1b_1)\) to \(\tilde{\alpha } s_{t1}^2\). For keeping the same notation style with the GENoVaS method, we use \(\alpha s_{t1}^2\) to represent \(\alpha s_{t1}^2/(1b_1)\). Then Eq. (32) can be represented as:
For getting a qualified GANoVaS transformation, we still need to make the transformation function Eq. (33) satisfy the requirement of the Modelfree Prediction Principle. Recall that in the existing GENoVaS method, \(\alpha + \beta + \sum _{i=1}^pa_i\) in Eq. (5) is restricted to be 1 for meeting the requirement of variancestabilizing and the optimal combination of \(\alpha ,\beta , a_1,\ldots ,a_p\) is selected to make the empirical distribution of \(\{W_t\}\) as close to the standard normal distribution as possible (i.e., minimizing \(KURT(W_t)3\)). Similarly, for getting a qualified \(H_T\) from Eq. (33), we require:
Under this requirement, since \(a_1\) and \(b_1\) are both less than 1, \(a_1b_1^{i1}\) will converge to 0 as i converges to \(\infty\), i.e., \(a_1b_1^{i1}\) is neglectable when i takes large values. So it is reasonable to replace \(\sum _{i=1}^{\infty }a_1b_1^{i1}\) in Eq. (34) by \(\sum _{i=1}^{q}a_1b_1^{i1}\), where q takes a large value. Then, Eq. (35) is obtained:
Now, we take Eq. (35) as a potential function \(H_T\). Then, the requirement of variancestabilizing is changed to:
Akin to Eq. (7), we scale \(\{\frac{\beta }{1b_1},a_1,a_1b_1\) \(,a_1b_1^{2},\) \(\ldots ,a_1b_1^{q1} \}\) of Eq. (36) by timing a scalar \(\frac{1\alpha }{\frac{\beta }{1b_1} + \sum _{i=1}^{q}a_1b_1^{i1}}\), and then search optimal coefficients. For presenting Eq. (35) with scaling coefficients in a concise form, we use \(\{c_0,c_1,\ldots ,c_q\}\) to represent \(\{\frac{\beta }{1b_1},a_1,a_1b_1\) \(,a_1b_1^{2},\) \(\ldots ,a_1b_1^{q1} \}\) after scaling, which implies that we can rewrite Eq. (35) as:
Furthermore, for achieving the aim of normalizing, based on Eq. (37), we still fix \(\alpha\) to be one specific value from \(\{0.1,0.2,\ldots ,0.8\}\), and then search the optimal combination of \(\beta ,a_1,b_1\) from three grids of possible values of \(\beta ,a_1,b_1\) to minimize \(KURT(W_t)3\). After getting a qualified \(H_T\), \(H_T^{1}\) can be outlined immediately:
Based on Eq. (38), for example, \(Y_{T+1}\) can be expressed as the equation follows:
Appendix 2: The comparison of parsimonious GENoVaS and GANoVaS methods
We asserted that the PGANoVaS method works better than the PGENoVaS in Eq. (3.3). Although these two parsimonious variants of GENoVaS and GANoVaS have the same structure, we showed that the regions of their parameters are different. The PGANoVaS method has a wider parameter space, and this property implies that it is a more complete technique. For substantiating this idea, we compare the forecasting performance of these two parsimonious methods and present results in Table 10. We use the bold values mark cases where one of these two methods is at least 10% better than the other one based on the relative prediction performance. We can find most cases are accompanied by very small relative values, which indicates that these two methods stand almost the same performance and is in harmony with the fact that they share the same structure. However, we can find there are 21 cases where the PGANoVaS method works at least 10\(\%\) better than the PGENoVaS method. On the other hand, there are only 8 cases where the PGENoVaS method shows significantly better results. We shall notice that the PGANoVaS method is optimized by determining parameters from several grids of values. Therefore, we can imagine the performance of this method will further increase if more refined grids are used.
Appendix 3: The detailed analysis of simulation results with Models 6 and 8
From Table 1, a remarkable case is the 500size simulation of Model 8, where NoVaStype methods achieve incredible victory compared to the GARCHdirect method. Here, we want to present plots of different methods’ predictions and true values in the same figure to compare them directly in an absolute sense. Different methods’ performance on 1step, 5steps and 30steps ahead aggregated predictions of 500 simulated Model 8 data are figured in Figs. 2, 3 and 4, respectively. It is clear that the GARCHdirect method is quite unstable for longterm (5steps and 30steps ahead) aggregated predictions. On the other hand, NoVaStype methods can capture the basic trend of simulated data for both short and longterm aggregated predictions. In addition, we further investigate whether the NoVaS method is robust against the model misspecification. Thus, we estimate a GJRGARCH(1,1) model and then do predictions. Setting the benchmark method as GJRGARCH(1,1) model, we tabulated the comparison result among various NoVaS, GARCH(1,1) and GJRGARCH(1,1) methods in Table 11. For both short and longterm forecasting, NoVaStype methods stand for generally acceptable performance. Particularly, the GANoVaS method works even better than the GJRGARCH(1,1) model on 30steps ahead aggregated predictions. Similarly, we apply the above procedure to Model 6 which is an Exponential GARCH(1,1) (EXPGARCH(1,1)) model. We also estimate an Exponential GARCH(1,1) model to do predictions and set it as the benchmark. The comparison of different methods is tabulated in Table 12. The results show the PGANoVaS method is the best one for all three prediction horizons, which highlights the superior ability of NoVaS methods on forecasting short data.
Appendix 4: Comparison plots of the forecasting on the volatile CADJPY data
Based on the empirical analyses, a remarkable result is the forecasting of the volatile CADJPY case from Table 6, where the GANoVaS method dominates other models. As we did in “Appendix 3”, we present comparison plots of the 1step, 5steps and 30steps ahead aggregated forecast on volatile CADJPY data in Figs. 5, 6 and 7, respectively. Still, the GARCHdirect method is quite off the actual curve for longterm predictions. On the other hand, NoVaStype methods are obviously more stable in an absolute sense.
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Wu, K., Karmakar, S. A modelfree approach to do longterm volatility forecasting and its variants. Financ Innov 9, 59 (2023). https://doi.org/10.1186/s40854023004666
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DOI: https://doi.org/10.1186/s40854023004666
Keywords
 ARCHGARCH
 Model free
 Aggregated forecasting