Measure of Volatility and Its Forecasting: Evidence from Naira / Dollar Exchange Rate

Time series is a wide area in statistics focusing on forecasting, and understanding the structural behaviors of series (stochastic random variable) [1]. Box Jenkins has dominantly analyzed time series data for forecasting in the last few decades and it is regarded with great usage by the scientific community. Also, structural models have been developed and applied in the last decades as well but were not able to outperform the originality of the random walk model proposed by Box and Jenkins in 1976 in aid of forecasting of the stochastic random variable [1, 2].

Forecasting techniques used in practice can be categorized into univariate time series models (GARCH) which capture volatility, and it has an edge above Box and Jenkins methodology of univariate time series models [3]. One of the limitations of the Box Jenkins method is that when there is the presence of threshold value, an exhibition of volatility clustering, the underlying structure of the series under study cannot be modeled but can be modeled with the idea of non-linear time series models.

Other types of forecasting techniques that have been in existence include but are not limited to market-based forecasting techniques. A recent development on forecasting time series or economic time series includes chaos theory and Artificial Neural Networks according to [4-6].

Hsieh [7] shows that different specifications of ARCH/GARCH models usually describe different currencies better than a unique model (STM or ARIMA models) because some currencies show a higher degree of seasonality than others due to higher amounts of export of goods around Christmas.

The multivariate regression approach has been used in many research to study and predict the exchange rate based on some listed variables but these have a limitation in the sense that macro-economic variables are available at most monthly periods precisely modeling of such explanatory variables will lead to a proportion change in the exchange rate [4, 8].

Majorly, applied econometricians use the giant statistical tool (least-squares model) as a natural choice to describe the nature of the exchange rate based on another macroeconomic variable listed above. The ARCH and GARCH models, which stand for autoregressive conditional heteroskedasticity and generalized autoregressive conditional heteroscedasticity respectively are designed to deal with just this issue [9].

In most recent times, they have been declared as a time series tool for capturing volatility or measuring volatility through the aid of measures of dispersion (variance or standard deviation) and these can be used in financial decisions concerning risk analysis, portfolio selecting, asset pricing and derivative pricing [10].

The volatility of financial assets has been a growing area of research. The traditional measure of volatility is represented by variance or standard deviation as stated above is unconditional and does not recognize the interesting patterns in asset volatility such as time-varying and cluster properties [11].

In Nigeria's context, two series were examined; structural adjustment program (SAP) where the exchange rate of Nigeria depreciated against the major intervention currency (US dollar) from 1970 to 1994). During this regime, investors and the capital market were not functioning very well. Some few business tycoons only invested in the capital market as a result of poor awareness. Also, decision-makers, business forecasters were not able to generate forecasts since the introduction of the regime. In 1994, the Nigerian government introduces foreign exchange markets (FEM) where little depreciation of naira was examined in 1994 and steady depreciation was examined in 1998 and 1999 and appreciated in 2005, 2006. A continuation of depreciation still exists to date. With this volatility in the exchange rate, little work has been done on modeling naira exchange rate volatility in Nigeria or how to capture the volatility using the GARCH models.

Recently, Musa et al. [12] makes use of GARCH models and the extension of GARCH models to model naira/dollar exchange using a short spine of periodicity. He found out that GJR – GARCH, and T- GARCH model shows the existence of a statistically significant asymmetric effect. But in this research work, the aim is to check for volatility measurement and its accuracy of predicting the naira exchange rate by applying a wide periodicity to check for the existence of leverage effect. The aim is attained by implementing extended GARCH families. The remainder of this research is section as section 2 focuses on data sources and empirical framework of ARCH models, section 3 describes the methodology, section 4 focuses on results and discussions, and section 5 describes the conclusion and possible recommendation for the research.

This research work aims to identify volatility, measure volatility, and forecast using GARCH models and the selected extension of GARCH models via E- GARCH, and T- GARCH models using annual financial time series data which covers from 1981 to 2021 i.e., a widespread of data (exchange rate).

The E-GARCH model will capture the asymmetric effect of the response of disturbance on volatility while T – GARCH model will capture the leverage effect (the reaction of volatility to changes in exchange rate prices. This aim requires several steps to be taken, firstly a descriptive statistic is encouraged to be found to understand the nature of the exchange rate returns. It is expected that the mean return would be zero identifying stationarity else non-stationary.

It is also expectant that most financial data series have heavy tails i.e., their kurtosis exceeds 3 signifying that the distribution is leptokurtic and has a high value or peak; more also for normality the Jarque test will signify for the randomness of the return series i.e., are the data followed a normal distribution. It is also not to be forgotten the unconditional standard deviation which measures the spread of data values and the volatile measure, a high value of standard deviation signifies high volatility which reverse is the low volatility. Less, the asymmetric distribution (Skewness) which equals zero indicates normality, a negative value of Skewness indicates negative Skewness or left tail is particularly extreme while a positive value indicates positive Skewness or right tail is particularly extreme.

Once descriptive statistics are defined, we performed ADF test for test of stationarity of the return series. Once the unit root is tested, we developed a series of ARMA models of Eq. (1) below using least square estimation methods or a well-defined maximum likelihood estimation.

$\mu_{t}(\theta)=\varphi_{0}+\varphi_{1} y_{t-1}+\cdots+\varphi_{p} y_{t-p}+\theta_{1} \varepsilon_{t-1}+\cdots$

$+\theta_{q} \varepsilon_{t-q}+\epsilon_{t}$    (1)

where $\varphi_{0}=$ constant term, $\varphi_{1}, \ldots . \varphi_{p}=$ autoregressive coefficients from order 1 to $p^{t h}$ order and $\varepsilon_{1}, \ldots . \varepsilon_{q}=$ moving average coefficients from order 1 to $q^{t h}$ order and $\epsilon_{t}=$ ARMA error term.

According to Ref. [13], both methods of estimations give the same results. Akaike information and Bayesian information criteria are the two criteria used to select the best ARMA model, once the best ARMA model is developed we generate the error ($\varepsilon_{t}$)  and check for the optimality of the best ARMA model, and plot the ACF (autocorrelation function) of the squared disturbance terms and also disturbance terms; if the disturbances hover the zero line, hence the best ARMA model is good. Once determine the optimality of the best ARMA model, we test for the ARCH effect (serial correlation in conditional variance).

Let

$\epsilon_{t}=y_{t}-\varphi_{0}-\sum_{i=1}^{p} \varphi_{i} y_{t-i}+\sum_{j=1}^{q} \theta_{j} \varepsilon_{t-j}$     (2)

Be the residuals of ARMA model of Eq. (1). The squared error series $\varepsilon_{i}^{2}$ is then check for conditional heteroscedasticity, which is known as the ARCH effect. The test for conditional heteroscedasticity is the Lagrange multiplier test of Engle [14]. The test is equivalent to usual F – statistics for testing $\alpha_{i}=0(i=1,2, \ldots \ldots q)$ in Eq. (3) below.

$\sigma_{t}^{2}=\alpha_{0}+\sum_{i=1}^{q} \alpha_{i} \varepsilon_{t-i}^{2}$     （3）

where $t=q+1, \ldots T$, where $\varepsilon_{t}$ denotes the error term, $q$ is the specified positive integer, and $T$ is the sample size. Specifically, the null hypothesis is:

$H_{0}: \alpha_{1}=\alpha_{2}=\ldots \ldots \alpha_{q}($ No Arch effect $)$

At least one parameter is not equal to zero (Arch effect). Let $S S R_{0}=\sum_{t=q+1}^{T}\left(\varepsilon_{t}^{2}-K\right)^{2}$, where $K=\frac{\sum_{t=q+1}^{T} \varepsilon^{2} t}{T}$, the sample is mean of $\varepsilon_{t}^{2}$ and let $S S R_{1}=\sum_{t=q+1}^{T} \hat{\varepsilon}_{t}^{2}$, where $\hat{\varepsilon}_{t}$, the least square residual of Eq (1) is, then we have:

$F_{s t a t}=\frac{\left(S S R_{1}-S S R_{2}\right) / q}{S S R_{1}(T-2 Q-1)} \sim X^{2}(q)$    (4)

where $q=$ the degree of freedom. The decision rule is to reject the null hypothesis if $F_{\text {stat }}>X^{2}{ }_{q}(\alpha)$ is the upper $100(1-$ $\alpha)^{t h}$ percentile of $X_{q}^{2}$ or the $\mathrm{p}$-value of $\mathrm{F}_{\text {stat }}$ is less than $\alpha$. Otherwise, no rejection of $\mathrm{H}_{0}$. if found ARCH effect is present i.e. a serial correlation in conditional variance of the error derived in Eq. (1), we estimate GARCH models to measure volatility. The essence of this ARCH effect will aid in the effect of exchange rate volatility which is been caused by some factors such as government policies or trade in balance etc. in fact this point out how effective are these factors have on naira dollar exchange rate.

3.1 Estimation techniques (arch models)

Three likelihood functions proposed by Fan and Yao [15] were used in ARCH estimation. If the series assumes normality, then, the likelihood function for the ARCH(q) model is:

$f\left(y_{1}, y_{2}, \ldots . y_{T} \mid \alpha\right)=f\left(y_{T} \mid \varphi_{t-1}\right) f\left(y_{T-1} \mid \varphi_{t-2}\right)$

$f\left(y_{q+1} \mid \varphi_{q}\right) f\left(y_{1}, y_{2}, \ldots \ldots y_{q} \mid \alpha\right)$    （5）

where $\alpha=\left(\alpha_{0}, \alpha_{1}, \ldots \ldots \alpha_{q}\right)^{\prime}$ and $f\left(y_{1}, y_{2} \ldots \ldots y_{q} \mid \alpha\right)$ is the joint probability density function of $y_{1}, y_{2}, \ldots y_{q}$,. Since the exact form of $f\left(y_{1}, y_{2} \ldots y_{q} \mid \alpha\right)$ is complicated, it is commonly dropped from the prior likelihood function, especially when the sample size is sufficiently large. This results in using the conditional likelihood function:

$f\left(y_{q+1}, y_{q+2} \ldots y_{T} \mid \alpha, y_{1}, y_{2}, \ldots . y_{q}\right)$    (6)

$=\prod_{t=q+1}^{T} \frac{1}{\sqrt{2 \pi} \sigma_{t}} e^{\left(\frac{-y^{2} t}{2 \sigma^{2} t}\right)}$

where, $\sigma_{t}^{2}$ can be evacuated recursively. We prefer the estimates obtained by maximizing likelihood estimates (MLE's) under which is easier to handle. In addition, it ensures that the maximum value of the log of the probability of Eq. (7) occurs at the same point as the original probability function in Eq. (5). Moreover, one of the usefulness of maximum likelihood estimate is that the population parameters in Eq. (5) are selected, by selecting the parameters such that the population distribution has the moments that are equivalent to the observed moments in the sample. Therefore, we can work with the simpler log-likelihood instead of the original likelihood. The conditional log likelihood function is:

$f\left(y_{q+1}, y_{q+2} \ldots y_{T} \mid \alpha, y_{1}, y_{2}, \ldots y_{q}\right)$

$=\sum_{T=q+1}^{T}\left(-\frac{1}{2} \ln (2 \Pi)-\frac{1}{2} \ln \left(\sigma_{t}\right)-\frac{1}{2} \frac{y_{t}^{2}}{\sigma_{t}^{2}}\right)$    (7)

Since the first two terms $\ln (2 \Pi)$ do not involve any parameter, the log likelihood function becomes:

$f\left(y_{q+1}, y_{q+2} \ldots y_{T} \mid \alpha, y_{1}, y_{2}, \ldots y_{q}\right)$

$=-\sum_{t=q+1}^{T}\left(\frac{1}{2} \ln \left(\sigma_{t}\right)+\frac{1}{2}\left(\frac{y_{t}^{2}}{\sigma^{2}}\right)\right)$    (8)

where, $\sigma_{t}^{2}=\alpha_{0}+\alpha_{1} \varepsilon_{t-1}^{2}+\alpha_{2} \varepsilon_{t-2}^{2}+\ldots \ldots \ldots \ldots \alpha_{q} \varepsilon_{t-q}$ can be evaluated recursively. Now in some applications, it is more appropriate to ensure that $\varepsilon_{t}$ follows a heavy tailed distribution such as standard t – distribution. There are two approaches to deal with the problem via robust inference and fat tail conditional distribution. For robust inference, see Ref. [16], for fat tailed conditional distribution; two approaches were involved parametric [17], non-parametric (semi parametric, [18], semi – non parametric density estimation; [19]. Following Ref. [17],

Let $X_{v}$ be a student $\mathrm{t}-$ distribution with $\mathrm{v}$ degrees of freedom. Then $v \operatorname{ar}\left(x_{v}\right)=\frac{v}{v-2}$ for $v>2$ and use $\varepsilon_{t}=\frac{x_{v}}{\sqrt{v(v-2)}}$, then probability density function of $\varepsilon_{t}$ is:

$f\left(\varepsilon_{t} \mid v\right)=\frac{\gamma\left(\frac{v+1}{2}\right)}{\gamma\left(\frac{v}{2}\right) \sqrt{(v-2) \Pi}}\left(1+\frac{\varepsilon_{t}^{2}}{v-2}\right)^{-\frac{v+1}{2}}$   $v>2$     (9)

where, $\gamma(x)$ is the use of gamma function i.e. $\gamma(x)=$ $\int_{0}^{\alpha} t^{x-1} e^{-t} \partial t$ now using $y_{t}=\varepsilon_{t} \sigma_{t}$ we obtain the conditional likelihood function of $y_{t}$ as:

$f\left(y_{q+1}, y_{q+2} \ldots y_{T} \mid \alpha, y_{1}, y_{2}, \ldots y_{q}\right)$$=\prod_{t=q+1}^{T} \frac{\gamma\left(\frac{v+1}{2}\right)}{\gamma\left(\frac{v}{2}\right) \sqrt{v-2} \pi} \frac{1}{\sigma_{t}}\left(1+\frac{y_{t}^{2}}{(v-2) \sigma_{t}^{2}}\right)^{-\frac{v+1}{2}}$$v>2$     (10)

Now we now refer to the estimates that maximize the prior likelihood function as the conditional MLE’s under the t – distribution. The degrees of freedom of the t – distribution can be specified a prior or estimated jointly with other parameters. Now a value between 3 or 6 is often used if it is per specified, then the conditional log likelihood function is:

$\ln \left(y_{q+1}, y_{q+2} \ldots y_{T} \mid \alpha, y_{1}, y_{2}, \ldots y_{q}\right)$

$=-\sum_{t=q+1}^{T}\left[\frac{v+1}{2} \ln \left(1+\frac{y_{t}^{2}}{(v-2) \sigma_{t}^{2}}+\frac{1}{2} \ln \left(\sigma_{t}^{2}\right)\right)\right]$     (11)

Now finally, it may assume a generalized error distribution GED), a situation where errors (the difference between the expected value of Eq. (3) and the observed values) are not normally distributed, i.e. if it does not obey the rule of symmetry or asymptotical rule. The probability function: is defined as:

$f(v s)=\frac{\operatorname{vexp}\left(-\frac{1}{2}\left|\frac{S}{v}\right|^{v}\right)}{\lambda 2\left(1+\frac{1}{v}\right) \gamma\left(\frac{1}{v}\right)}$$-\alpha<x \leq \alpha ; 0<v<\alpha$    (12)

And $\lambda=\left[2\left(\frac{-2}{v}\right) \frac{\gamma\left(\frac{1}{v}\right)}{\gamma\left(\frac{3}{v}\right)}\right]^{\frac{1}{2}}$, this distribution reduces to Gaussian distribution if $v=2$, and it has heavy tail i.e. the distribution will have a high kurtosis value indicating high peak of outliers when $v<2$. But if $v>2$, the distribution diverges as a result of convolutions problems or gradient problems.

3.2 Estimation technique (GARCH models)

From Fan and Yao [15], the conditional normal GARCH process is:

$\left(y_{t} \mid \varphi_{t-1}\right) \sim N\left(u_{t}(\theta), \sigma_{t}^{2}\right)$    (13)

And their conditional density is:

$f\left(y_{t} \mid \varphi_{t-1}\right) ; \theta=\frac{1}{\sqrt{2 \pi}} \sigma^{2} e^{\left(-\frac{1}{2}\left(\frac{y_{t}-\mu_{t}(\theta)}{\sigma^{2} t}\right)\right)^{2}}$     (14)

Its prediction error decomposition is given by:

$f\left(y_{T}, y_{T-1}, \ldots y_{1} ; \theta\right)$

$=f\left(y_{T} \mid \varphi_{t-1}\right) \times f\left(y_{T-1} \mid \varphi_{t-2}\right) \times f\left(y_{1} \mid \varphi_{0}\right)$     (15)

The log likelihood function is given by:

$\log \left(y_{T}, y_{T-1}, \ldots y_{1} ; \theta\right)=\log f\left(y_{T}, y_{T-1}, \ldots y_{1} ; \theta\right)$

$=-\frac{T}{2} \log 2 \pi-\frac{1}{2} \sum_{i=1}^{T} \log \sigma_{t}^{2}-\frac{1}{2} \sum_{t=1}^{T}\left(y_{t}-\mu_{t}(\theta)\right)^{2}$   (16)

Looking at Eq. (16), there is a non – linear function θ, so according to [15], numerical optimization techniques would be applied to estimate the persistent parameters in Eq. (17) below:

$\begin{aligned} \sigma_{t}^{2} &=\alpha_{0}+\alpha_{1} \varepsilon_{t-1}^{2}+\alpha_{2} \varepsilon_{t-2}^{2}+\cdots \alpha_{p} \varepsilon_{t-p}^{2} \\ &+\beta_{1} \sigma_{t-1}^{2}+\beta_{2} \sigma_{t-2}^{2}+\cdots \beta_{q} \sigma_{t-q}^{2} \end{aligned}$     (17)

We fit different types of GARCH models using the numerical optimization techniques as described above to estimate significant parameters and the best GARCH model is selected base on the smallest AIC and BIC criteria. The optimality of the GARCH model selected passed through diagnosis testing (plot of ACF of the standardized squared errors terms generated to test for continuous arch effect and the ACF of the standardized residuals for normality. If the ACF hovers around the line then the GARCH model estimated is satisfied. Hence the sum of coefficients of ARCH lags with GARCH lags would be less than unity and the coefficients of GARCH LAGS must be greater than zero. All the parameters must be (statistically significant). We proceed to perform the presence of ARCH effect from the ACF of the standardized squared residual if found out no ARCH effect we base our interpretation on the persistent parameter $\left(\beta_{j}\right)$ in Eq. (17). Estimation of E – GARCH and T- GARCH are estimated and the optimality of the models are followed in the same way as for GARCH models. For E – GARCH models the parameters that measure the leverage effect must be less than zero for the positivity of the variance while for the T – GARCH model; the parameter that measures the leverage effect must be greater than zero for the positivity of conditional variance.

3.3 H-step ahead forecast of the conditional variance of the error

Detail on the forecast conditional variance of GARCH model to be estimated in this research is given by:

Recall from GARCH $(1,1)$

$\sigma_{t}^{2}=\alpha_{0}+\alpha_{1} \varepsilon_{t}^{2}+\beta \sigma_{t}^{2}$   (18)

1 - step Ahead prediction is given by:

 $\left(\sigma^{2}{ }_{h+k} \mid t\right)=\alpha_{0}+\alpha_{1} \varepsilon^{2}+\beta \sigma_{t}^{2}$    (19)

$k=$ step Ahead prediction is given by:

$\left(\sigma_{h+k}^{2} \mid t\right)=\alpha_{0}+\alpha_{1} \varepsilon_{t+k-1}^{2}+\beta \sigma_{t+k-1}^{2}$    （20）

For the long run – predictions:

$\lim _{k \rightarrow \infty}\left(h_{t+k} \mid t\right)=\frac{\alpha_{0}}{1-\alpha-\beta}=\bar{\alpha}_{0}$     (21)

For higher order $G A R C H(p, q)$

$\sigma_{t}^{2}=\alpha_{0}+\alpha_{1} \sum_{i=1}^{p} \varepsilon_{t-i}^{2}+\beta \sum_{j=1}^{q} \sigma_{t-j}^{2}$    (22)

Taking conditional expectations and assuming that a sample size of T, and for the conveniences that parameters in the forecast function are known, a forecast function for the optimal h – step ahead forecast of the conditional variance is written as:

$E\left(\sigma_{t+k}^{2} \mid \varphi_{t}\right)=\alpha_{0}$

$+\sum_{i=1}^{p} \alpha_{i} E\left(\varepsilon^{2} t \pm i \mid \varphi_{T}\right)+\sum_{j=1}^{q} \beta_{j} E\left(h_{t+k-j} \mid \varphi_{t}\right)$    (23)

When $\psi_{t}$ is the other relevant information pertaining to the forecast of $\sigma^{2}{ }_{t} \cdot E\left(\varepsilon^{2}{ }_{t+i} \mid \psi_{t}\right)=E\left(\sigma^{2}{ }_{t+i} \mid \psi_{t}\right)$ For $i>0$, $E\left(\varepsilon^{2}{ }_{t+i} \mid \psi_{T}\right)=\sigma^{2}{ }_{t+i}$ for $i>1 . E\left(\sigma^{2}{ }_{t+i} \mid \psi_{t}\right)$ is computed recursively. Thus, obtaining one- head forecast of $\sigma^{2}{ }_{t}$ is giving by $E\left(\sigma^{2}{ }_{t+1} \mid \psi_{t}\right)=\alpha_{0}+\alpha_{1} \varepsilon^{2}{ }_{t}+\beta \sigma^{2}{ }_{t}$ which is equivalent to Eq. (20). The procedure is continued and the long run predictor is equivalent to Eq. (21). According to Ref. [20], He stated that for GARCH $(1,1)$, the two step forecast is a little closer to the long run average variance than one - step forecast. Ultimately, the long run prediction is the same for all time periods as long as $\alpha_{i}+\beta_{i}<1$. Engle [20] regards that it is called the unconditional variance, hence GARCH models are mean reverting and conditionally heteroscedasticity but have a constant unconditional variance.

For $K$ - period's returns (exchange return):

$\varphi_{t+k}(K)=e_{t+k}+e_{t+k-1}+\cdots e_{t+1}$    (24)

For $K$-period return variance:

$\begin{aligned} \operatorname{var}\left(e_{t+k}(K) \mid \varphi_{t}\right) & \sim\left(\sigma_{t+k}^{2} \mid t\right)+\left(\sigma_{t+k-1}^{2} \mid t\right) \\ &+\cdots\left(\sigma_{t+1}^{2} \mid t\right) \end{aligned}$    (25)

3.4 Volatility forecast comparison

Four different measurement forecast accuracy tools would be used to compare the very best conditional heteroscedasticity models that provide an accurate forecast of the conditional variance via root mean squares, mean absolute error, mean absolute percentage error, and the Theil inequality. The mathematical representations of the above forecast measurement tool are as follows:

Root mean square $=\sqrt{\frac{\sum_{i=1}^{N}\left(\widehat{\sigma}^{2} t^{\left.-\sigma_{t}\right)^{2}}\right.}{N}}$    (26)

Mean absolute error $=\frac{\sum_{i=1}^{N}\left|\left(\hat{\sigma}^{2} t^{-\sigma_{t}}\right)^{2}\right|}{N}$    (27)

Mean absolute percentage error $=\sum_{i=1}^{N} \frac{\left|\left(\hat{\sigma}^{2} t^{-\sigma_{t}}\right)^{2}\right|}{\sigma_{t}}$   (28)

Theil inequality $=\frac{R M S E}{\sqrt{\frac{\sum_{i=1}^{N} \hat{\sigma}^{2} t}{N}}+\sqrt{\frac{\sum_{i=1}^{N} \sigma^{2} t}{N}}}$    (29)

where N is the number of out sample observations, $\sigma_{t}$ is the actual volatility at forecasting period “t” measured as the squared daily return, and $\hat{\sigma}^{2} t$ is the forecast at volatility at “t”. The first measurement forecasting tool would be used to forecast for the same series across all different conditional heteroscedasticity models. The smallest error indicates the better forecasting ability of that model according to the criteria. Under the Theil inequality, it is expected that the coefficient or the value of Theil would be zero which would indicate a perfect fit. Theil lies between 0 and 1. It is of necessity to understand the variability between volatility forecast from the actual series and variability between the mean of volatility forecast from the actual mean series. The following proportions define both clearly.

Bias proportion $=\frac{\sum_{i=1}^{N}\left(\hat{\sigma}^{2} t-\bar{y}\right)^{2} /{ }_{N}}{\sum_{i=1}^{N}\left(\hat{\sigma}^{2} t^{\left.-\sigma_{t}\right)^{2}} /_{N}\right.}$    (30)

This measures how far the mean difference of the volatility is from the mean of the actual series. The absolute range of tolerance error between the mean difference of the volatility and that of the mean of the observed series should not exceed $\pm 1.5 \%$ and there is no need for normalization of this difference.

Variance proportion $=\frac{\left(\delta_{\hat{y}}-\delta_{y}\right)^{2}}{\sum_{i=1}^{N}\left(\hat{\sigma}^{2} t-\sigma_{t}\right)^{2} /_{N}}$    (31)

This measures the variance of the volatility forecast from the variation of the actual series. Where $\delta_{\hat{y}}, \delta_{y}$ are the biased standard deviation of $\hat{\sigma}_{t}$ and $\sigma_{t}$ respectively.

Covariance proportion $=\frac{2(1-r) \delta_{\hat{y}} \delta_{y}}{\frac{1}{N} \sum_{i=1}^{N}\left(\widehat{\sigma}_{t}-\sigma_{t}\right)}$  (32)

This measures the remaining unsymmetrical forecasting errors. It is expected to be close to unity so that all the bias would be concentrated on it and allow variance proportion and bias proportion to be small.