A GARCH model with modified grey prediction model for US stock return volatility

Abstract

This study proposes a Generalized Autoregressive Conditional Heteroscedasticity (GARCH) with modified Grey prediction model to investigate the transmission of volatility through analysis of the error terms. Generally, the higher the sample size, the better GARCH models describe variation. However, the GARCH( $p$ , $q$ ) model often causes the problem of time delay by assuming that the conditional variance and the squared error term have lags $p$ and $q$ periods, respectively. Consequently, this paper utilizes a Grey Model (GM), modified for general residual sequences and generalizing the squared error terms to incorporate influence by unexpected factors such as previous process states or delayed impact of information. Furthermore, this study illustrates the proposed model with daily NASDAQ closing prices for a total of 1265 observations. The modified Grey-GARCH model demonstrates improved accuracy over the Grey-GARCH model and the traditional GARCH model. The results of this study have practical implications for optimal investment strategies.

Keywords

Grey model GARCH Grey-GARCH residual sequences modified Grey-GARCH

1. Introduction

In the practice of time series analysis, it is often the case that the conditional variance does not meet the assumption of homogeneity in traditional econometric models, especially for financial data. Some researcher discovered that heterogeneous conditional variance often occurs in conjunction with the phenomenon of volatility clustering [20, 13, 17]. Roughly speaking, volatility clustering, originally proposed by [8], means that large changes tend to follow large changes, and small changes tend to follow small changes. For these situations, Engle proposed the Autoregressive Conditional Heteroscedasticity (ARCH) model to reduce bias in traditional econometric models.

Extending the ARCH model by Engle, Bollerslev used the ARMA model to generalize the volatility equation and proposed the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model. The generalized model includes conditional variance in previous states to estimate the transmission of volatility, and it is characterized by fat tails and excess kurtosis. Its analysis provides important insights into the transmission of volatility, and, in general, the GARCH model usually produces better descriptions for large samples. For these reasons, the GARCH model has been used frequently to research returns and transmissions of volatility in financial time series by many scholars; see for example in [6, 7, 1, 14, 18, 3, 21, 19, 9].

However, the GARCH(p,q) model often causes the problem of time delay by assuming that the conditional variance and the squared error terms have lags p and q periods, respectively. Typical research ignores that the squared error term should be subject to unexpected factors such as previous period events or delayed impact of information. Because there are many inestimable factors that influence the squared error term, this study assumes that it is a function of time. Numerous attempts and various applications have been made to show the validity of the prediction results obtained using GM(1,1), such as using in engineering field [22, 23, 15], medical system field [2] and management field [26]. Most of the existing grey models are essentially linear models, which limits the applicability of the grey models. Ma and Liu introduce a novel nonlinear multivariate grey model which is based on the kernel method [24]. The KGM(1, n) model contains an unknown function of the input series, which can be estimated using the kernel function, and then the KGM(1, n) model is available to describe the nonlinear relationship between the input and output seriesî“‰The case studies of predicting the oilfield production, the condensate gas well production and coal gas emission are carried out, and the results show that the KGM(1, n) model is much more efficient than the existing linear multivariate grey models.

More precisely, the GM(1,1) in grey system control is used because it performs well when system modeling is unclear or only incomplete information for prediction is available. Deng used the Grey prediction of GM(1,1) to deduce GARCH model or namely it as Grey-GARCH model [12]. Furthermore, this study will examine a practical Grey-GARCH prediction model to display an improvement for the better accuracy than the traditional GARCH model.

Tseng used the forecasting property of GM(1,1) model to modify the error terms of GARCH model and proposed GM-GARCH model [5]. Later, Tseng and Wang provided GM-EGARCH [4] and GM-GJR-GARCH models [25], utilizing GM(1,1) model to modify the error terms of EGARCH. The results indicated that the introduction of GM(1,1) model improved the short-term forecasting accuracy of the GARCH-type models to a certain degree. However, due to the theoretical shortcomings, GM(1,1) model may produce larger forecast error when forecasting the error sequences which are highly volatile. With the advantages of selforganizing and self-adaption, Modified Grey-GARCH can enhance the forecasting accuracy of GM(1,1) model.

However, when values in the residual series are negative, it is not appropriate to use the GM(1,1) directly because simulation precision is often diminished. Therefore, this research proposes a modified GM(1,1), suitable for general residual series, to predict the squared errors in the GARCH model; the resulting model is termed a modified Grey-GARCH model. The modified Grey-GARCH model overcomes limitations in the GARCH model and produces reasonable results to assist understanding of various time series features. In particular, the parameters reflect volatility and provide important economic insights into its transmission.

The rest of this paper is organized as following: Section 2 presents the basic methodologies review. Section 3 explains a proposed modified Grey-GARCH model. Summary analysis of the financial data considered and provides empirical analysis of the series is shown in Section 4. The final section contains conclusions about the proposed method and words on future integration.

2. Methodologies review

This section provides a simple introduction to the GARCH model, discusses the grey and the Grey-GARCH models.

2.1 GARCH model

Many financial asset time series exhibit autocorrelation and volatility clustering, which violate assumptions of traditional econometric models. Extending research addressing these violations, Bollerslev modified the conditional variances in the ARCH( $q$ ) model by assuming that conditional variances are not only subject to influence by the squared errors but also by previous conditional variances; the generalized model is the GARCH( $p$ , $q$ ) model.

The GARCH( $p$ , $q$ ) model can be written as

$\displaystyle y_{t}=a_{t},\mbox{ }a_{t}=\sqrt{h_{t}}\varepsilon_{t},\mbox{ }h_% {t}=\alpha_{0}+\sum\limits_{i=1}^{q}{\alpha_{i}a_{t-i}^{2}}+\sum\limits_{j=1}^% {p}{\beta_{j}h_{t-j}}$ (1)

where:

$y_{t}$ is the original time series

ii.

$\varepsilon_{t}$ is a white noise random process ( $\varepsilon_{t}\sim N(0,1)$ )

iii.

$a_{t}\left|{\Omega_{t-1}\sim N(0,h_{t})}\right.$

iv.

$h_{t}$ is the conditional variance of $a_{t}$ at time $t$

$\Omega_{t-1}=\left\{{y_{1},y_{2},\ldots,y_{t-1}}\right\}$ represents the information acquired by time $t-1$

vi.

$p$ and $q$ are lag lengths.

In order to stabilize the model, the following parameter constraints are used:

$\alpha_{0}^{(1)},\alpha_{0}^{(2)}>0;\alpha_{i}^{(1)},\alpha_{i}^{(2)}\geqslant 0% ,i=1,2,3,\ldots,q;\beta_{i}^{(1)},\beta_{i}^{(2)}\geqslant 0,j=1,2,3,\ldots,p;% p,q>0;$

$\sum\limits_{i=1}^{q}{\alpha_{i}^{(1)}}+\sum\limits_{j=1}^{p}{\beta_{j}^{(1)}}% <1;\sum\limits_{i=1}^{q}{\alpha_{i}^{(2)}}+\sum\limits_{j=1}^{p}{\beta_{j}^{(2% )}}<1.$

Unfortunately, due to the time delay problem in the GARCH model, it does not always exhibit realistic behavior. In order to improve this shortcoming, this study uses the GM(1,1) to investigate the squared error terms. The next subsection introduces the theoretical model and calculation of the GM(1,1).

2.2 Grey model

Grey system theory was initially introduced by Deng and mainly focuses on situations where there is incomplete information for prediction or when model specification is unclear. It assumes that any random process can be treated as a grey quantity that varies within certain ranges, and the process is called a grey process. It cumulatively sums over the original discrete data series to generate an obvious index rule; this is followed by further summation operations to construct differential equation s. In the end, the least squares method is employed to obtain the coefficients of the first derivative for system evaluation and prediction.

Calculation of the GM(1,1) is as follows.

1.
Define the sequence $x^{(0)}$ , where $x^{(0)}(i)\in R^{+}$ , $i=1,2,3,\ldots,n$ , for all $x^{(0)}(i)\in x^{(0)}$ , as

$\displaystyle x^{(0)}=\{x^{(0)}(1),x^{(0)}(2),x^{(0)}(3),\ldots,x^{(0)}(n)\}$ (2)
2.
Obtain the sequence $x^{(1)}$ from $x^{(0)}$ by one application of the Accumulated Generating Operation.

$\displaystyle x^{(1)}=\{x^{(1)}(1),x^{(1)}(2),x^{(1)}(3),\ldots,x^{(1)}(n)\}$ (3)

for all $x^{(1)}(k)\in x^{(1)}$

$\displaystyle x(1)(k)=\sum\limits_{i=1,x^{(0)}(i)\in x^{(0)}}^{k}{x^{(0)}(i)}$ (4)

The differential equation that results after one application of the AGO to the sequence $x^{(1)}(k)$ can be written as

$\displaystyle\frac{dx^{(1)}(k)}{dk}+ax^{(1)}(k)=b$ (5)

in Eq. (5), $a$ is the development coefficient and $b$ is the grey control parameter. This equation is the grey model GM(1,1). From the time response function of the first derivative, one obtains the general solution

$\displaystyle\hat{x}^{(1)}(k+1)=\left(x^{(0)}(1)+\frac{b}{a}\right)e^{-ak}+% \frac{b}{a}$ (6)

By definition of the differential equation

$\displaystyle\frac{dx^{(1)}(k)}{dk}=\mathop{\lim}\limits_{\Delta k\to 0}\frac{% x^{(1)}(k+1)-x^{(1)}(k)}{\Delta k}$ (7)

if $\Delta(k)=1$ , then Eq. (7) can be written as

$\displaystyle\frac{x^{(1)}(k+1)-x^{(1)}(k)}{1}=x^{(0)}(k)$ (8)

the original differential equation now becomes a quasi differential equation described by

$\displaystyle x^{(0)}(k)+az^{(1)}(k)=b$ (9)

where the value of $z^{(1)}(k)$ is the mean defined by

$\displaystyle z^{(1)}(k)=\frac{x^{(1)}(k+1)+x^{(1)}(k)}{2}$ (10)

and where $z^{(1)}(k)$ is the accumulated value of $\frac{dx^{(1)}(k)}{dk}$ . According to Eq. (8), the formula can be rewritten as follows

$\displaystyle x^{(0)}(k)+az^{(1)}(k)=b,k=2,3,4,\ldots,n$ (11)

that is

$\displaystyle-az^{(1)}(k)+b=x^{(0)}(k),k=2,3,4,\ldots,n$ (12)

where the parameters $a$ and $b$ are obtained from

$\displaystyle\left[{{\begin{array}[]{c}a\\ b\\ \end{array}}}\right]=(B^{T}B)^{-1}B^{T}yN$ (13)

with

$\displaystyle B=\left[{\begin{array}[]{cc}-\frac{1}{2}(x^{(1)}(1)+x^{(1)}(2))&% 1\\ -\frac{1}{2}(x^{(1)}(2)+x^{(1)}(3))&1\\ \vdots\\ -\frac{1}{2}(x^{(1)}(n-1)+x^{(1)}(n))&1\\ \end{array}}\right]$ (14)

$\displaystyle y_{N}=(x^{(0)}(2),x^{(0)}(3),\ldots,x^{(0)}(n))^{T}$ (15)

Using the first-order differential equation, one obtains

$\displaystyle\hat{x}^{(1)}(k+1)=(x^{(0)}(1)+\frac{b}{a})e^{at}+\frac{b}{a}$ (16)

where $\mathord{\buildrel\lower 3.0pt\hbox{$\scriptscriptstyle\frown$}\over{x}}^{(1)}% =x^{(0)}(1)=x^{(1)}(1)$ . One may now substitute $a$ and $b$ obtained from the grey differential equation in the general equation with $\hat{x}^{(1)}=x^{(0)}(1)=x^{(1)}(1)$ . Since the prediction model is not constructed with original data but is modified by one application of cumulative addition, reverse addition is required to recover the predicted data. From $\hat{x}^{(0)}(k+1)=\hat{x}^{(1)}(k+1)-\hat{x}^{(1)}(k)$ one obtains equation Eq. (17), which is the dynamic generator of future values in the GM(1,1):

$\displaystyle x^{(0)}(k)=x^{(1)}(k)-\hat{x}^{(1)}(k-1)$ (17)
2.3 Grey-GARCH model

Because there are still many complex factors which might affect the descriptive ability of the squared error term in the GARCH model, the square errors have some nice mathematical advantages, especially simple analytic formulas for time series. This is very handy when you want to fit a set of data with a function and you have to determine the coefficients in the function that give the best fit. the data of square error is relative simple and seldom. The GM(1,1) can solve these problems for a good technique it can find the law in these data and predict the unknown data. It is one of the effective methods to investigate the square error term with the poor information system. this study adopts the characteristics of grey system theory to modify the error term in equation Eq. (1) and propose the Grey-GARCH model [10].

The Grey-GARCH( $p$ , $q$ ) model is defined as

$\displaystyle y_{t}=a_{t},a_{t}=\sqrt{h_{t}}\varepsilon_{t},h_{t}=\alpha_{0}+% \sum\limits_{i=1}^{q}{\alpha_{i}(a_{t-i}^{2}+}\hat{a}_{t}^{2})+\sum\limits_{j=% 1}^{p}{\beta_{j}h_{t-j}}$ (18)

where $\hat{a}_{i}^{2}$ in equation Eq. (18) is obtained via the GM(1,1).

Because simulation precision with the GM(1,1) is often unsatisfactory when some residual values are negative, this paper introduces a modified GM(1,1) to address this shortcoming.

3. A modified Grey-GARCH model

The goal of developing the modified GM(1,1) is to improve the method of residual recognition by past research [11]. In the unmodified model, a given residual sequence $\varepsilon^{(0)}(k)$ may not be appropriate when some terms $\varepsilon^{(0)}(l)$ are negative. Deng employed the method of marginal residuals to deal with this problem, and this approach is feasible in many situations. For instance, given a residual sequence $\varepsilon^{(0)}(i)=$ {0.5, 0.6, 0.3, $-$ 0.1, 0.4}, cumulative summation yields the marginal residual sequence $\varepsilon^{(0)}(i)=$ {0.5, 1.1, 1.4, 1.3, 1.7}, with all terms non-negative and therefore meets the conditions for setting up the GM(1,1). But for residual sequences like $\varepsilon^{(0)}(i)=$ { $-$ 0.3, $-$ 0.2, 0.1, 0.2}, summation yields $\varepsilon^{(1)}(i)=$ { $-$ 0.3, $-$ 0.5, 0.4, $-$ 0.2}, and no solution sequence to the differential model is possible regardless of the number of subsequent summations. In addition, positive and negative terms cancel during summation, leading to distortion of information and complicating forecasting. In light of the above considerations, this paper proposes a new method of residual recognition, the modified GM(1,1), applicable to general residual sequences and exhibiting improved forecast precision.

Calculation of the modified GM(1,1) is as follows

1.
Given a time series $\hat{x}^{(0)}(i),i=1,2,3,\ldots N$ , use the equation

$\displaystyle\frac{dx^{(1)}}{dt}-ax^{(1)}=b$ (19)

to obtain the forecast sequence $\hat{x}^{(0)}(i),i=1,2,3,\ldots,N$ , where the difference between the original value $x^{(0)}(l)$ and the forecast value $\hat{x}^{(0)}(l)$ at time $l$ is called the residual at time $l$ and is recorded as

$\displaystyle\varepsilon^{(0)}(l)=x^{(0)}(l)-\mathord{\buildrel\lower 3.0pt% \hbox{$\scriptscriptstyle\frown$}\over{x}}^{(0)}(l)$ (20)
2.
In order to improve forecast precision, one may subtract from the residual process $\{\varepsilon^{(0)}(l)\}$ the minimum observed value. Let

$\displaystyle m=\mathop{\min}\limits_{l}\{\varepsilon^{(0)}(l)\}$ (21)

and form the new sequence $\{\eta^{(0)}(l)\}$

$\displaystyle\eta^{(0)}(l)=\varepsilon^{(0)}(l)+m,l=1,2,3,\ldots,N$ (22)

Regardless of the original $\varepsilon^{(0)}(l)$ , it follows $\eta^{(0)}(l)\geqslant 0$ and $\eta^{(1)}(l)$ is increasing.
3.
Set up the GM(1,1) for the sequence $\eta^{(1)}(l)$

$\displaystyle\frac{d\eta}{dt}-a\eta^{(1)}=b$ (23)

where $a$ and $b$ are parameters to be determined for the GM(1,1). Note that $\eta^{(1)}$ is the sum over the sequence $\eta^{(0)}$

$\displaystyle\eta^{(1)}(k)=\sum\limits_{i=1}^{k}{\eta^{(0)}(i)}$ (24)
4.
Determine parameter column $\mathord{\buildrel\lower 3.0pt\hbox{$\scriptscriptstyle\frown$}\over{a}}=[a,b]% ^{T}$

$\displaystyle\mathord{\buildrel\lower 3.0pt\hbox{$\scriptscriptstyle\frown$}% \over{a}}=(B^{T}B)^{-1}B^{T}Y_{N}(\eta)$ (25)

where

$\displaystyle B=\left[{{\begin{array}[]{cc}{\frac{1}{2}(\eta^{(1)}(2)+\eta^{(1% )}(1))}\hfill&1\\ {\frac{1}{2}(\eta^{(1)}(3)+\eta^{(1)}(2)}\hfill&1\\ \cdots\hfill&\ldots\\ {\frac{1}{2}(\eta^{(1)}(N)+\eta^{(1)}(N-1))}&1\\ \end{array}}}\right]$ (26)

$\displaystyle Y_{N}(\eta)=[\eta^{(0)}(2),\eta^{(0)}(3),\ldots,\eta^{(0)}(N)]^{T}$ (27)
5.
Determine the new forecast values of $\{x^{(0)}(l)\}$ .

The differential equation can be determined with the parameter $\hat{a}$ given above:

$\displaystyle\frac{d\eta^{(1)}}{dt}-a\eta^{(1)}=b$ (28)

let the initial value be $\eta^{(0)}(1)$ and response function be given by

$\displaystyle\eta^{(1)}(t)=(\eta^{(0)}(1)+\frac{b}{a})e^{at}-\frac{b}{a}$ (29)

the discrete response function is

$\displaystyle\hat{\eta}^{(1)}(t)=(\eta^{(0)}(1)+\frac{b}{a})e^{at}-\frac{b}{a}% ,k=1,2,3,\ldots,N$ (30)

where the forecast sequence $\{\hat{\eta}^{(1)}(l)\}$ of $\{\eta^{(1)}(l)\}$ is obtained as above. Letting

$\displaystyle\hat{\eta}^{(0)}(k)=\hat{\eta}^{(1)}(k)-\hat{\eta}^{(1)}(k-1),k=2% ,3,4,\ldots,N$ (31)

the forecast sequence $\{\hat{\eta}^{(0)}(l)\}$ of $\{\eta^{(0)}(l)\}$ is also obtained. A new forecast sequence $\{\hat{x}^{(0)}(i,1)\}$ can now be formulated as follows

$\displaystyle\hat{x}^{(0)}(i,1)=\hat{\varepsilon}^{(0)}(i)+\hat{x}^{(0)}(i),i=% 1,2,3,\ldots,N,\hat{\varepsilon}^{(0)}(i)=\hat{\eta}^{(0)}(1)+m,i=1,2,3,\ldots,N$ (32)

where: $\{\hat{x}^{(0)}(i,1)\}$ is the forecast sequence of $\{x^{(0)}(i,1)\}$ after correction.
3.1 Modified Grey-GARCH model

Much financial data exhibits volatility clustering and price limits, indicating that such series may present problems in the unmodified GM(1,1). This paper adopts the characteristics of the modified GM(1,1) coupled with GARCH methodology to gauge information transmission and increased forecasting precision to investigate underlying market dynamics.

The modified Grey-GARCH( $p$ , $q$ ) model is defined as

$\displaystyle y_{t}=a_{t},a_{t}=\sqrt{h_{t}}\varepsilon_{t},h_{t}=\alpha_{0}+% \sum\limits_{i=1}^{q}{\alpha_{i}(a_{t-i}^{2}+}\hat{a}_{m,t}^{2}+\sum\limits_{j% =1}^{p}{\beta_{j}h_{t-j}}$ (33)

where $\hat{a}_{t}^{2}$ in Eq. (33) is obtained via the modified GM(1,1).

4. Empirical study

In this analysis, interaction in the transmissions of volatility in the U.S. NASDAQ index is explored. The data used in this paper are the daily closing prices of the NASDAQ index from January 6, 1992, to December 31, 1996; the number of observations is 1265. All data were taken from the Taiwan Economics Journal (TEJ) database. The daily closing price series is shown Fig. 1.

Figure 1.

Daily closing stock-price index of NASDAQ index.

4.1 Data characteristics

As shown in Fig. 1, the NASDAQ index is non-stationary, and it is necessary to convert it into a stationary sequence. Daily stock returns are generated by taking the logarithmic difference of the daily stock index times 100 to transform the I(0) series; more precisely, daily returns $R_{t}$ are given by $R_{t}=100\times(\log P_{t}-\log P_{t-1})$ , where $P_{t}$ is the closing price on day $t$ . The daily return series of the NASDAQ index is shown in Fig. 2. It is visually clear that the behavior of the NASDAQ index is also consistent with the phenomenon of volatility clustering.

To provide a general understanding of the market return structure, summary statistics of daily returns are presented in Table 1. Table 1 indicates the mean and variance of the NASDAQ index are 0.000605 and 0.008227, respectively. Obviously, the mean return is higher and the standard deviation smaller than in a standard normal distribution; in fact, the series exhibits excess kurtosis and fat tails. At the 5% significance level, the Jarque-Bera statistic is 272.8962, indicating rejection of normality for the series.

Table 1
Daily return of NASDAQ index

Sample period	2011/1/2 $\sim$ 2015/12/31
Observations	1265
Mean	0.000605	Standard deviation	0.008227
Skewness	$-$ 0.535742	Kurtosis	5.011422
Maximum	0.031849	Minimum	$-$ 0.040030
J-B N	272.8962*	ADF value	$-$ 16.05436*
L-B Q(6)	45.4337*	L-B Q(12)	63.2259*
L-B ${}^{2}$ Q(6)	145.6718*	L-B ${}^{2}$ Q(12)	184.3219*

Figure 2.

Daily return of NASDAQ index.

Notes: J-B N is Jacque-Bera statistics testing for the normal distribution. L-B Q(k)and L-B ${}^{2}$ Q(k) are Ljung-Box statistics testing for autocorrelation in the level of returns and the squared returns up to the kth lag. * is denoted significance at the 5% level. ADF is unit root testing for diagnosing the data is stationary or not.

The $Q$ -statistic proposed by Ljung-Box is employed to test for autocorrelation in the squared series. At the 5% significance level, the test values $Q$ (6) and $Q$ (12) are both greater than the critical values of the chi-square distribution on 6 and 12 degrees of freedom, indicating that the NASDAQ index exhibits autocorrelation and the squared series reveals high-order autocorrelation. This result implies heterogeneous return variance in the stock market over time, which is consistent with the phenomena of autocorrelation and volatility clustering. Consequently, before using the ARCH or GARCH model to analyze the series, it is necessary to test for volatility clustering in the error term variance using the ARCH test. To implement the ARCH test, the Lagrange Multiplier (LM) is employed, as proposed by Engle. If the null hypothesis is rejected, the series is suitable for ARCH or GARCH modeling. The test results are presented in Table 2.

Table 2

Estimates of ARCH test

Lags	NASDAQ index	Lags	NASDAQ index
1	15.2197*	7	92.1698*
2	49.3643*	8	96.1749*
3	58.4437*	9	96.3078*
4	73.3727*	10	96.5183*
5	75.4090*	11	96.7211*
6	91.8839*	12	97.4825*

*is denoted significance at the 5% level.

According to research by Bollerslev, Chou, and Droner, the GARCH(1,1) model provides a satisfactory description of volatility in financial and economic time series. Therefore, this paper uses ( $p$ , $q$ ) $=$ (1,1) to estimate parameters.

4.2 Empirical analysis

The Financial Toolbox in Matlab was developed to conduct practical studies with GARCH models. The GARCH(1,1) model equation s are as follows

$\displaystyle y_{t}=C+\varepsilon_{t},\mbox{ }h_{t}=K+G_{1}h_{t-1}+A_{1}% \varepsilon_{t-1}^{2}$ (34)

where:

$\varepsilon_{t}$ is a white noise random process ( $\varepsilon_{t}\in N(0,1))$ .

ii.

$h_{t}$ is the conditional variance of $a_{t}$ at time $t$ .

iii.

$G_{1}$ is the coefficient of the conditional variance.

iv.

$A_{1}$ is the coefficient of the squared error.

$C$ and $K$ are constants in the mean and variance equation s.

Convergence constraints are $G_{1}+A_{1}<1$ ; $G_{1},A_{1}\geqslant 0$ ; $K>0$ .

Table 3

Parameter estimates for NASDAQ index in GARCH model

Parameter	Value	Standard error	$t$ -Statistic
C	0.088692*	2.2859e-4*	3.8800*
K	6.534e-004*	1.705e-6*	3.8322*
$G_{1}$	0.78935*	0.042441*	18.5985*
A ${}_{1}$	0.11341*	0.021924*	5.1728*

*is denoted significance at the 5% level.

Table 4

Parameter estimates for NASDAQ index in Grey-GARCH model

Parameter	Value	Standard error	$t$ -Statistic
C	4.3311e-3*	5.2155e-4*	8.3043*
K	5.538e-5*	1.3483e-5*	4.1075*
$G_{1}$	0.69216*	0.054332*	12.7394*
A ${}_{1}$	0.18494*	0.03132*	5.9047*

*is denoted significance at the 5% level.

Table 5

Parameter estimates for NASDAQ index in modified grey-GARCH model

Parameter	Value	Standard error	$t$ -Statistic
C	2.9469e-3*	5.1729e-4*	5.6967*
K	4.6759e-005*	1.2638e-005*	3.6999*
$G_{1}$	0.72155*	0.057382*	12.5747*
A ${}_{1}$	0.15442*	0.030926*	4.9931*

Tables 3 to 5 list the parameter estimates for the NASDAQ index using the GARCH, Grey-GARCH, and modified Grey-GARCH models, respectively. Parameter estimates in all models conform to convergence criteria, and $G_{1}+A_{1}$ is close to unity, indicating conditional variances are positive and the models are stationary.

The coefficients $G_{1}$ exceed the coefficients $A_{1}$ in all models, which implies that the predicted volatility is highly sensitive to the AR model. This observation also supports the statement that the variance changes over time in the stock market and external impacts have long-term effects.

In the GM(1,1), it is necessary to have at least four observations to predict the next one. Hence, the analysis is carried out beginning with the fourth observation.

Table 6

RMSE, MAE, LL, and LINEX of four types of volatility models for NASDAQ index

Index	Models	RMSE	MAE	LL	LINEX
NASDAQ	Modified GM-GARCH	1.0326	0.7853	0.1586	0.0135
	GM-GARCH	1.2193	0.8284	0.1847	0.1229
	GARCH	1.6147	1.0882	0.4632	0.9584

Figure 3.

The returns of GARCH, Grey-GARCH, and modified Grey-GARCH models.

The out-of-sample forecasting performances of each type of models were evaluated by four statistical indices: the root mean squared error (RMSE), the mean absolute error (MAE), the logarithmic error statistic (LL), and the Linear.

Exponential index (LINEX). Table 6 lists the comparison of the results of the four types of models in forecasting volatility of NASDAQ. It can be seen from Table 6: Modified GM-GARCH generates smaller RMSE, MAE, LL, and LINEX compared to the other types of models for NASDAQ index,, which show a better performance than the other types of models in forecasting volatility.

Figure 3 illustrates predicted and actual returns of the NASDAQ index using the GARCH, Grey-GARCH and modified Grey-GARCH models. As is evident in the figure, the returns in the modified Grey-GARCH model are much more accurate than in the GARCH and Grey-GARCH models.

Based on this empirical study, it appears that the modified Grey-GARCH model is able to produce a reasonable, accurate description. Because predictions employ updated data for real time processing of changes in return, investment risk is reduced.

5. Conclusions

Most financial time series are characterized by autocorrelation and volatility clustering, violating assumptions in traditional econometric models. Among the research addressing this issue, Bollerslev modified the conditional variance structure in the ARCH model, pointing out that the conditional variance is not only affected by previous squared errors but by previous conditional variance as well. The more general GARCH model usually produces better descriptions for large samples. However, this model often causes the problem of time delay by assuming that the conditional variance and the squared error term have lags $p$ and $q$ periods, respectively.

To better understand information transmission, it is revealing to analyze the long-term effects on model error terms. This paper considers the influence of previous process states and the delayed impact of information on error volatility. Circumventing the imperfect residual recognition in the unmodified GM(1,1), this paper adopts the modified GM(1,1) to investigate the squared error term and proposes the modified Grey-GARCH model.

Empirical results indicate that the modified Grey-GARCH model is capable of attaining higher returns than the GARCH and the Grey-GARCH models due to increased forecasting precision. However, it is worthwhile to keep in mind that this model is only one of many proposed in financial market research, and it remains necessary to validate its practical merits relative to other approaches through extensive empirical studies. In addition, the Modified GM-GARCH models, as a whole, produce superior volatility forecasts compared to the GM-GARCH and GARCH models. While, the Modified GM-GARCH, GM-GARCH and GARCH models show different volatility forecasting abilities according to different evaluation criteria, which need to be further studied.

Footnotes

Acknowledgments

This work is supported in part by the Science Foundation of Ningde (20160036), School Innovation Team of the Ningde Normal University (2015T011), Project of the Fujian Provincial Natural Science Fund (Grant No. 2017I0016) and Recruiting High Level Talent program of Ningde Normal University (2017Y001).

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A GARCH model with modified grey prediction model for US stock return volatility

Abstract

Keywords

1. Introduction

2. Methodologies review

2.1 GARCH model

Table 1 Daily return of NASDAQ index

Footnotes

Acknowledgments

References

Table 1
Daily return of NASDAQ index