_{1}

^{*}

In this paper, we use daily stock returns from the Stockholm Stock Exchange in order to examine their volatility. For this reason, we estimate not only GARCH (1,1) symmetric model but also asymmetric models EGARCH (1,1) and GJR-GARCH (1,1) with different residual distributions. The parameters of the volatility models are estimated with the Maximum Likelihood (ML) using the Marquardt algorithm (Marquardt [1]). The findings reveal that negative shocks have a large impact than positive shocks in this market. Also, indices for the return of forecasting have shown that the ARIMA (0,0,1)-EGARCH (1,1) model with t-student provide more precise forecasting on volatilities and expected returns of the Stockholm Stock Exchange.

The development of econometrics led to the invention of adjusted methodologies for the modeling of mean value and variance. Models of generalized conditional autoregressive heteroscedasticity (GARCH) are based on the assumption that random components in models present changes on volatility. These models were developed by Engle [

The models of autoregressive conditional heteroscedasticity (GARCH) have a long and noteworthy history but they are not free of limitations. For example, Black [

GARCH models were applied with great success on the modeling of changing variability or the variance volatility on time series for measuring financial investments. After the determination of an asymmetric relationship between conditional volatility and conditional mean value, econometricians focused their efforts on planning methodologies for modeling this phenomenon.

Nelson [

The purpose of this paper is to quantify two asymmetric models using prices from the Stockholm stock market for the period 30 September 1986 until 11 May 2016, representing 7,434 observations. The first 7,000 values in the model were used for quantification and statistical verification and the last 434 values for the forecast demonstration in retrospect.

The remainder of the paper is organized as follows: Section 2 provides a brief literature review. Section 3 discusses the symmetric and asymmetric GARCH models. Section 4 summarizes the data. The results are discussed in Section 5 and Section 6 proposes the forecasting methodology. Finally, the last section offers the concluding remarks.

The ability of GARCH models that study the relationship between risk and return has been validated in many studies. For example, Donaldson and Kamstra [

Nam, Pyun and Arize [

Tudor [

Panait and Slavescu [

Gao, Zhang and Zhang [

Dutta [

Given the different backgrounds for each market it is expected that risk and return differ from country to country. This paper attempts to examine the volatility and return of the Stockholm stock market using symmetric and asymmetric models.

One of the fundamental hypotheses for a stationary time series is the stable variance. But there are time series, mostly financial, that have intervals of large volatility. These series are characterized with periods of sharp increases and downturns during which their variance is varying. Thus, researchers are not interested in examining the variance of such series throughout the sample period but only interested in the varying or conditional variance. Based on this division between conditional and unconditional variance, we can characterize time series models with conditional variance as conditional heteroskedastic models. The notion of “conditional” heteroscedasticity was first introduced by Engle [

The varying GARCH models consist of two equations. The first one (the equation of mean) describes the data as a function of other variables adding an error term. The second equation (the equation of variance) determines the evolution of the conditional variance of the error from the mean equation as a function of past conditional variances and lagged errors. The first equation (equation of the mean) on GARCH varying models is not of great interest, contrary to the second equation (equation of variance) which is the one that we pay more attention to in order to compare different variances on the same equation of mean.

The traditional measuring methods of volatility (variance or standard deviation) are absolute and cannot conceive the characteristics of financial data (time series) like volatility clustering, asymmetries, leverage effect and long memory. The basic model suggested by Engel [

ε t = z t σ t

where z t is an independent identically distributed (i.i.d.) process with mean zero and variance 1. σ t is the volatility that evolves over time. The volatility σ t 2 in the basic ARCH (q) model is defined as:

σ t 2 = ω + ∑ i = 1 q α i ε t − i 2 (1)

where σ t 2 is the conditional variance, ω > 0 and α i ≥ 0 for σ t 2 to be positive.

This model shows that after a large (small) shock, it is likely that a large (small) shock will follow. In other words, a large (small) ε t − 1 2 implies a large (small) ε t 2 on the current period according to Equation (1) thus a large (small) variance (volatility).

Finally, we can point out that large values on time lags on ARCH models presuppose large periods of volatility contrary to small values of time lags that foresee smooth periods. This may not occur in reality. In order to overcome this problem, Bollerslev, Chou and Kroner [

The generalized form of GARCH(p, q) is given as:

R t = μ + ε t (mean equation)

σ t 2 = ω + ∑ i = 1 q α i ε t − i 2 + ∑ j = 1 p β j σ t − j 2 (variance equation) (2)

where R t are returns of time series at time t, μ is the mean value of the returns, ε t is the error term at time t, which is assumed to be normally distributed with zero mean and conditional variance σ t 2 , p is the order of GARCH and q is the order of ARCH process, μ , ω , α i and β j are parameters for estimation. All parameters in variance equation must be positive ( μ > 0 , ω > 0 , α i ≥ 0 , and β j ≥ 0 for σ t 2 to be positive). Also, we expect the value of parameter ω to be small. Parameter α i measures the response of volatility on market variances and parameter β j expresses the difference which was caused from outliers on conditional variance. Finally, we expect the sum α i + β j < 1

The model GARCH (1,1) has the following form:

R t = μ + ε t (mean equation)

σ t 2 = ω + α 1 ε t − 1 2 + β 1 σ t − 1 2 (variance equation) (3)

As we expect a positive variance, we can argue that regression coefficients are always positive ω ≥ 0 , α 1 ≥ 0 and β 1 ≥ 0 . Also, we should point out that in order to achieve stationarity on the variance, regression coefficients α 1 and β 1 should be less than one ( α 1 < 1 ) and ( β 1 < 1 ) . Thus, on the previous model, the following relations are valid: ω ≥ 0 α 1 ≥ 0 and β 1 ≥ 0 for a positive value of σ t 2 and α 1 < 1 and β 1 < 1 .

The conditional variance of the returns of Equation (3) is defined from three outcomes:

・ The constant given by ω coefficient.

・ The variance part expressed from the relationship α 1 ε t − 1 2 defined as ARCH component.

・ The part of predicted variance from past period expressed by β 1 σ t − 1 2 and is called GARCH.

The sum of regression coefficients α 1 + β 1 expresses the impact of variables’ variance of the previous period regarding the current value of volatility. This value is usually near to one and is regarded as a sign of increasing inactivity of shocks of the volatility of returns on the financial assets.

The main disadvantage of GARCH models is their inappropriateness in the cases where an asymmetric effect is usually observed and is registered from a different instability in the case of good and bad news. In the asymmetric models, upward and downward trends of returns are interpreted as bad and good news. If the decline of a return is accompanied with an increase of instability larger than the instability caused by the increase then it is said to have a leverage effect.

Given that all terms in a GARCH model are squared, there will always be an asymmetric response in positive and negative periods. However, due to natural leverage in most companies, a negative shock is more damaging than a positive shock because it produces larger volatility.

Among the most widely known asymmetric models are the Exponential GARCH model (EGARCH) and the asymmetric GJR model.

One of the most popular asymmetric ARCH models is the EGARCH model proposed by Nelson [

log σ t 2 = ω + ∑ i = 1 p α i | ε t − i σ t − i | + ∑ j = 1 q β j log σ t − j 2 + ∑ k = 1 r γ k ε t − k σ t − k (4)

where ω , α i , β j , and γ k are parameters which can be estimated using the maximum likelihood method. We should also point out that | β j | < 1 and γ k parameter is the one that gives the result of leverage effect. In other words, we consider that ε t − k term is the one that establishes the asymmetry of EGARCH (p, q) when parameter γ k ≠ 0 . Also, when parameter γ k < 0 , then positive shocks cause short volatility in relation to negative shocks. Furthermore, we expect that parameters γ k + α i > 0 , given that parameter γ k < 0 .

The conditional variance of the above model is expressed in logarithmic form which ensures the non-negativity without imposing more constraints of non-

negativity. The term ε t − k σ t − k on the above equation represents the asymmetric

effect of shocks. According to Poon and Granger [

The EGARCH (1,1) model is often used for the estimation of variance σ 2 and has the following form:

log σ t 2 = ω + α 1 | ε t − 1 σ t − 1 | + β 1 log σ t − 1 2 + γ 1 ε t − 1 σ t − 1 (5)

For a positive shock ε t − 1 σ t − 1 > 0 the above equation becomes

log σ t 2 = ω + β 1 log σ t − 1 2 + ( α 1 + γ 1 ) ε t − 1 σ t − 1 (6)

whereas for a negative shock ε t − 1 σ t − 1 < 0 the above equation becomes

log σ t 2 = ω + β 1 log σ t − 1 2 + ( α 1 − γ 1 ) ε t − 1 σ t − 1 (7)

The EGARCH model has many advantages when compared to the GARCH (p, q) model.

・ The first is the logarithmic form which does not allow the positive constraint among parameters.

・ Another advantage of EGARCH model is that it incorporates asymmetries in the change of volatility of returns.

・ Parameters α and γ define two important asymmetries in the conditional variance. If γ 1 < 0 then negative changes increase volatility (instability) more than positive changes of the same size.

・ EGARCH model can successfully define the change of volatility.

The form of the EGARCH model denotes that the conditional variance is an exponential function of the examined variables which ensures a positive character. In other words, conditional variance ensures the exponential nature of the EGARCH model where external changes will have a stronger influence on the predicted volatility than TGARCH. An asymmetric influence is indicated by the no null value of γ_{1} coefficient whereas the presence of leverage is indicated by the negative value of the same coefficient.

The GJR-GARCH(p, q) model is another asymmetric GARCH model proposed by Glosten, Jagannatahan and Runkle [

σ t δ = ω + ∑ i = 1 p α i ε t − i 2 + ∑ j = 1 q β j σ t − j 2 + γ i I t − i ε t − i 2 (8)

I t − i = { 1 when ε t − i < 0 0 when ε t − i ≥ 0

when ω , α i , β j and γ i are parameters under estimation

I t − i is a dummy variable, meaning that I t − i is a functional index which takes zero value when ε t − i is positive and value one when ε t − i is negative. If parameter γ i > 0 then negative errors are leveraged meaning that negative innovations or bad news have larger impact than good news. Finally, we assume that on the GJR-GARCH model parameters are positive and the relationship

α i + β j + γ i 2 < 1 is valid.

The GJR-GARCH (1,1) model is the one that is more often used for the estimation of σ 2 variance and has the following form:

σ t 2 = ω + α 1 ε t − 1 2 + β 1 σ t 2 + γ 1 I t − 1 ε t − 1 2 (9)

I t − 1 = { 1 when ε t − 1 < 0 0 when ε t − 1 ≥ 0

Engle and Ng [

The estimation of GARCH models can be done with the Ordinary Least Squares method. Due to the fact that error terms are not independently and identically distributed iid(0,1), it is better to avoid using the OLS method mainly on small samples. In this case, it is better to use the maximum likelihood method(see Greene, [

ln L [ ( y t ) , θ ] = ∑ t = 1 T [ ln [ D ( z t ( θ ) , υ ) ] − 1 2 ln [ σ t 2 ( θ ) ] ] (10)

where θ is the vector of the parameters that have to be estimated for the conditional mean, conditional variance and density function, z t denoting their density function, D ( z t ( θ ) , υ ) , is the log-likelihood function of [ y t ( θ ) ] , for a sample of T observation. The maximum likelihood estimator θ ^ for the true parameter vector is found by maximizing (10).

The models GARCH assumed Gaussian innovations, but nonetheless imply non-Gaussian unconditional distributions. However, time-varying volatility models with Gaussian innovations generally do not generate sufficient unconditional non-Gaussianity to match certain financial asset return data (see, Poon and Granger [

In this section we describe the log-likelihood functions used for the estimation of parameters on volatility models for all theoretical distributions.

1) Normal Distribution

In the case of a standard normal distribution for the i.i.d. random variables { z t } , the following log-likelihood function needs to be maximized.

ln L [ ( y t ) , θ ] = − 1 2 [ T ln ( 2 π ) + ∑ t = 1 T z t 2 + ∑ t = 1 T ln ( σ t 2 ) ] (11)

where θ is the vector of the parameters that have to be estimated for the conditional mean, conditional variance and density function, T is observations.

2) Student-t Distribution

The Student-t distribution can handle more severe leptokurtosis. The log- likelihood function is defined as

ln L [ ( y t ) , θ ] = T [ ln Γ ( υ + 1 2 ) − ln Γ ( υ 2 ) − 1 2 ln [ π ( υ − 2 ) ] ] − 1 2 ∑ t = 1 T [ ln ( σ t 2 ) + ( 1 + υ ) ln ( 1 + z t 2 υ − 2 ) ] (12)

where Γ ( υ ) = ∫ 0 ∞ e − x x υ − 1 d x is the gamma function and υ is the degree of freedom.

The t-Student is symmetric around zero. The Student-t distribution incorporates the standard normal distribution as a special case when υ = ∞ and the Cauchy distribution when υ = 1 . Hence, a lower value, υ yields a distribution with “fatter tails”.

Nelson [

ln L [ ( y t ) , θ ] = ∑ t = 1 T [ ln ( υ λ ) − 1 2 | z t λ | υ − ( 1 + υ − 1 ) ln ( 2 ) − ln Γ ( 1 υ ) − 1 2 ln ( σ t 2 ) ] (13)

where λ = [ 2 − 2 / υ Γ ( 1 υ ) Γ ( 3 υ ) ] 1 / 2

The distribution of generalized error (GED) incorporates both normal distribution when ( υ = 2 ), Laplace distribution when ( υ = 1 ), and the unique distribution for υ = ∞ . Specifically, we would say when υ = 2 the distribution of the random variable z_{t} would be the standard normal distribution. When υ < 2 , the distribution of the random variable z_{t}, will have thicker tails than that of normal distribution. For υ = 1 the distribution of the random variable z_{t} will have a double exponential distribution. For υ > 2 the distribution of the random variable z_{t} will have thinner tails than normal distribution, and for υ = ∞ the distribution of the random variable z_{t} will be a uniform distribution.

In our paper we estimate the conditional volatility using normal distribution, t-student and generalized error distribution. Engle [

The data in our study are collected from the official website www.nasdaqomxnordic.com. The data span is the period from 30 of September 1986 to 11 May 2016 and comprise 7434 observations. The daily stock return is calculated as:

R t = ln ( I t I t − 1 ) × 100 = ( ln I t − ln I t − 1 ) × 100 (14)

where I t is the daily closing value of the stock market on day t, and R t is the daily stock return.

The daily closing values of OMX Stockholm 30 Index and its returns are displayed in

As it can be seen in

As it can be seen in

The Ljung and Box Q-statistics [

In addition, in

ARCH effect. Since we have detected that there is an ARCH effect (

The summary of the descriptive statistics for the daily logarithmic stock index returns of the OMX Stockholm 30 Index is presented in

The results in

The Q-Q plot, which displays the quantiles of return data series against the quantiles of the normal distribution, shows that there is a low degree of fit of the empirical distribution to the normal distribution.

The leptokurtic behavior of the data is confirmed by the normal quintile and empirical density graph presented in

The summary of the descriptive statistics, normality tests, ARCH tests and unit root tests for the daily stock index returns of the OMX Stockholm 30 Index is presented in

After the detection of series stationarity, we define the form of the ARMA (p, q) model from the correlogram of

Descriptive statistics | Normality tests | ARCH tests | Unit root tests | ||||
---|---|---|---|---|---|---|---|

Mean | 0.031 | J-B | 5601.1 | Q^{2}(10) | 2857.0 | ADF | −84.319 |

Median | 0.070 | p-value | 0.000 | p-value | 0.000 | p-value | 0.000 |

Maximum | 11.02 | Lilliefors | 0.057 | Q^{2}(20) | 4467.3 | P-P | −84.332 |

Minimum | −8.52 | p-value | 0.000 | p-value | 0.000 | p-value | 0.000 |

Std. Dev. | 1.462 | Q^{2}(30) | 5521.5 | ||||

Skewness | 0.025 | p-value | 0.000 | ||||

Kurtosis | 7.252 |

respectively, compared to the critical value ± 2 n = ± 2 7433 = ± 0.023 . There-,

fore the value of p will be between 0 ≤ p ≤ 3 , and respectively, the value of q will be between 0 ≤ q ≤ 3 . Thereafter, we create

The results from

After the estimation of the above model in

ARIMA model | AIC | SC | HQ |
---|---|---|---|

(1,0,0) | 3.5977 | 3.6005 | 3.5986 |

(2,0,0) | 3.5977 | 3.6005 | 3.5986 |

(3,0,0) | 3.5977 | 3.6005 | 3.5986 |

(1,0,1) | 3.5977 | 3.6005 | 3.5986 |

(2,0,1) | 3.5977 | 3.6005 | 3.5986 |

(3,0,1) | 3.5977 | 3.6006 | 3.5987 |

(1,0,2) | 3.5977 | 3.6006 | 3.5987 |

(2,0,2) | 3.5977 | 3.6006 | 3.5987 |

(3,0,2) | 3.5968 | 3.6033 | 3.5990 |

(1,0,3) | 3.5968 | 3.6021 | 3.5986 |

(2,0,3) | 3.5967 | 3.6032 | 3.5990 |

(3,0,3) | 3.5977 | 3.6007 | 3.5987 |

(0,0,1) | 3.5966 | 3.6004 | 3.5984 |

(0,0,2) | 3.5977 | 3.6006 | 3.5987 |

(0,0,3) | 3.5977 | 3.6007 | 3.5989 |

of conditional heteroscedasticity (ARCH(q) test) from the squared residuals of the above model.

From the results of

Since there are ARCH effects in the Stockholm stock return data, we can proceed with the estimation of the ARIMA(0,0,1)-GARCH models.

First of all, we estimate the symmetric ARIMA(0,0,1)-GARCH(1,1) model with normal distribution, t-student distribution as well as the Generalized error distribution (GED). The estimation of parameters is done with the maximum likelihood method using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) (see Press et al. [

We proceed with the following asymmetric (non linear) GARCH models such as the ARIMA(0,0,1)-EGARCH (1,1) model as well as the ARIMA(0,0,1)-GJR- GARCH(1,1) model with the normal distribution, t-student distribution and Generalized error distribution (GED). The estimation of parameters is done with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm Marquardt [

From the results of

ARIMA(0,0,1)-GARCH(1,1) | |||
---|---|---|---|

Parameter | Normal | Student’s-t | GED |

ω | 0.037(0.000) | 0.025(0.000) | 0.030(0.000) |

α_{1} | 0.096(0.000) | 0.095(0.000) | 0.096(0.000) |

β_{1} | 0.885(0.000) | 0.894(0.000) | 0.890(0.000) |

D.O.F = 9.056 (0.000) | PAR = 1.499(0.000) | ||

Persistence | 0.981 | 0.989 | 0.986 |

LL | −12245.51 | −12112.3 | −12148.7 |

Jarque-Bera | 2610.0(0.000) | 3408.1(0.000) | 2982.4(0.000) |

ARCH(10) | 2.783(0.986) | 3.034(0.980) | 2.911(0.983) |

Q^{2}( 30) | 12.142(0.998) | 16.335(0.980) | 14.439(0.993) |

Notes: 1. The persistence is calculated as α_{1} + β_{1} for the ARMA(0,0,1)-GARCH(1,1) model. 2. Values in parentheses denote the p-values. 3. LL is the value of the log-likelihood.

ARIMA(0,0,1)-EGARCH(1,1) | |||
---|---|---|---|

Parameter | Normal | t-Student | GED |

Ω | −0.111(0.000) | −0.122(0.000) | −0.118(0.000) |

α_{1} | 0.158(0.000) | 0.167(0.000) | 0.163(0.000) |

β_{1} | 0.975(0.000) | 0.978(0.000) | 0.977(0.000) |

γ_{1 } | −0.088(0.000) | −0.089(0.000) | −0.087(0.000) |

T-Dist.Dof | 10.128(0.000) | 1.546(0.000) | |

Persistence | 0.975 | 0.978 | 0.977 |

LL | −12159.04 | −12044.66 | −12082.75 |

Jarque-Bera | 2199.27(0.000) | 2601.59(0.000) | 2397.07(0.000) |

ARCH(10) | 4.521(0.920) | 3.162(0.977) | 3.647(0.961) |

Q^{2}(30) | 12.667(0.998) | 13.641(0.995) | 12.869(0.997) |

ARIMA(0,0,1)-GJR-GARCH(1,1) | |||

Parameter | Normal | t-Student | GED |

Ω | 0.040(0.000) | 0.033(0.000) | 0.036(0.000) |

α_{1} | 0.023(0.000) | 0.026(0.000) | 0.025(0.000) |

β_{1} | 0.891(0.000) | 0.890(0.000) | 0.890(0.000) |

γ_{1} | 0.127(0.000) | 0.131(0.000) | 0.128(0.000) |

T-Dist.Dof | 10.198(0.000) | 1.555(0.000) | |

Persistence | 0.9775 | 0.9815 | 0.979 |

LL | −12155.06 | −12042.88 | −12080.43 |

Jarque-Bera | 2105.14(0.000) | 2534.87(0.000) | 2308.25(0.000) |

ARCH(10) | 3.521(0.966) | 4.555(0.918) | 3.989(0.947) |

Q^{2}(30) | 12.492(0.998) | 15.569(0.986) | 13.923(0.995) |

Notes: 1. The persistence is calculated as β 1 for ARIMA(0,0,1)-EGARCH(1,1) model, and α 1 + γ 1 / 2 + β 1 for ARIMA(0,0,1)-GJR-GARCH(1,1) model. 2. Values in parentheses denote the p-values. 3. LL is the value of the log-likelihood.

satisfactory. Also, the results from the models show that Q-statistics for the standardized squared residuals and the ARCH-LM test are insignificant with high p values. From the above table we can see that ARIMA(0,0,1)-GJR- GARCH(1,1) model has the largest logarithmic likelihood (LL) value with t-student distribution. Thus, this model can be used for forecasting.

Asymmetry and leverage effects results are examined for non-linear variances of ARIMA(0,0,1)-EGARCH(1,1) and ARIMA(0,0,1)-GJR-GARCH(1,1) models from three different distributions. Since the coefficients are statistically significant in all cases, asymmetry exists. Positive signs of the coefficients on the ARIMA(0,0,1)-GJR-GARCH(1,1) models as well as negative signs on the ARIMA(0,0,1)-EGARCH(1,1) models indicate that there are leverage effects. In addition, bad news has more impact on volatility than good news in all distributions that we used. In the following

In order to examine if an asymmetric model is suitable for forecasting, Engle and Ng [_{1} coefficient of the following regression:

ε ^ t 2 = b 0 + b 1 D i , t − 1 − + v t (15)

where

ε ^ t 2 are the squared residuals from the symmetric GARCH model.

D i , t − 1 − is a dummy variable which takes the value 1 if ε t − i is negative and 0 otherwise, and gives the slope dummy value.

v t is an i.i.d. error term. (see Dutta [

If b_{1} coefficient is statistically significant in positive and negative changes relatively to conditional variance then there is asymmetry on the GARCH model.

A test for sign bias can also be conducted using the following regression:

ε ^ t 2 = b 0 + b 1 D t − 1 − ε t − 1 + v t (16)

Like regression (15), the statistical significance of b_{1} coefficienton regression (16) indicates that the size of a shock will have an asymmetric impact on volatility. Regression (16) tests the negative bias size. For the positive bias size we use the following regression: (see Dutta [

ε ^ t 2 = b 0 + b 1 ( 1 − D t − 1 − ) ε t − 1 + v t (17)

A joint test can be conducted through defining D t − 1 + as 1 − D t − 1 − which indicates a positive size bias. The joint test for positive sign bias and positive or negative size bias is presented on the following regression:

ARIMA(0,0,1)-EGARCH(1,1) | ARIMA(0,0,1)-GJR-GARCH(1,1) | |||||
---|---|---|---|---|---|---|

Normal | t-Student | GED | Normal | t-Student | GED | |

Bad News | 1.088 | 1.089 | 1.089 | 0.150 | 0.157 | 0.153 |

Good News | 1.012 | 0.911 | 0.913 | 0.023 | 0.026 | 0.025 |

Notes: The asymmetry is calculated as 1 − γ 1 , and 1 + γ 1 for the ARIMA(0,0,1)-EGARCH(1,1) model, α 1 + γ 1 _{ , }and α 1 for the ARIMA(0,0,1)-GJR-GARCH(1,1) model.

ε ^ t 2 = b 0 + b 1 D t − 1 − + b 2 D t − 1 − ε t − 1 + b 3 D t − 1 + ε t − 1 + v t (18)

The significance of b_{1} coefficient of regression (17) shows the existence of sign bias where positive and negative changes have different consequences in volatility compared to the symmetric GARCH model. On the other hand, the significance of b_{2} and b_{3} coefficients of regression (18) indicates not only size bias but also if the size of change is significant. The test follows a χ^{2} distribution with degrees of freedom equal to 3. The joint test statistic is given from the formula TR^{2}. The null hypothesis for the joint test is that there is no asymmetric result. (see Brooks, [

The results of

The Likelihood ratio (LR) tests consist of estimations on two models, (an unrestricted model and a restricted one). The null hypothesis that is examined is H 0 : γ 1 = 0 . The maximized values of log-likelihood function (LLF) are used on this test according to the following:

L R = − 2 ( L L F r − L L F u ) → χ 2 ( m ) (19)

where

L L F r is the value of maximum likelihood function from the constrained model

L L F u is the value of maximum likelihood function from the unconstrained model

m the number of constraints.

The Likelihood ratio (LR) tests follow asymptotically χ^{2} distribution with m degrees of freedom.

ARIMA(0,0,1)-EGARCH(1,1) | ARIMA(0,0,1)-GJR-GARCH(1,1) | |||||
---|---|---|---|---|---|---|

Normal | t-Stud. | GED | Normal | t-Stud. | GED | |

Sign bias | 0.030* | 0.056* | 0.039* | 0.203* | 0.054* | −0.053* |

Negative size bias | −0.265* | −0.055* | −0.015* | −0.684* | −0.048* | −0.030* |

Positive size bias | 0.331* | 0.043* | 0.111* | 0.039 | 0.036* | 0.019* |

Joint test (F-test) | 2614.8 | 853.4 | 9212.3 | 1115.06 | 432.36 | 13409.3 |

p-value | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) |

Notes: 1. (*) denotes significance at the (1%); 2. The t-statistics for the sign bias, negative size bias and positive size bias tests are those of coefficient b_{1} in regression (15), (16) and (17), respectively. 3. The F-statistic is basedon regression (18). 4. Values in parentheses denote the p-values.

LLF_{r} | LLF_{u} | LR | ||
---|---|---|---|---|

ARIMA(0,0,1)-GARCH(1,1) | Normal | −12245.51 | ||

t-Student | −12112.3 | |||

GED | −12148.7 | |||

ARIMA(0,0,1)-EGARCH(1,1) | Normal | −12159.04 | 172.94* | |

t-Student | −12044.66 | 135.28* | ||

GED | −12082.75 | 131.9* | ||

ARIMA(0,0,1)-GJR-GARCH(1,1) | Normal | −12155.06 | 180.9* | |

t-Student | −12042.88 | 138.84* | ||

GED | −12080.43 | 136.5* |

Note: (*) denotes significance at the (1%) level, LLF_{r} is the value from maximum likelihood function from the constrained model, LLF_{u} is the value from maximum likelihood function from the unconstrained model and LR is Likelihood ratiotest.

The results from

In this section we present the forecasting results from the two asymmetric models. In our paper we forecast for future values on rate of return and volatility on the Stockholm stock market using the static1-step ahead method based on estimated parameters of the two asymmetric models. The last 434 series observations were used for an ex-post forecast, with the main focus on the forecast of volatility. In the literature, a variety of statistics has been used which evaluates and compares the forecasts of returns. The optimal forecasting value is evaluated through Mean Squared Error. Other statistical indices usually used for the return of forecasting are the Mean Absolute Error (MAE), the Root Mean Square Error (RMSE), the Mean Absolute Percentage Error (MAPE) and the (Theil U-Theil index [

The above figures show that estimation intervals are stable on both models. However, there are some indices that help us to test the forecast of a model. The first one is referred to as the Mean Square Error. According to Patton [

mean absolute error index which is more robust to outliers. According to the Mean Absolute Error (MAE), the Root Mean Square Error(RMSE) and Mean Absolute Percentage Error (MAPE), the ARIMA(0,0,1)-EGARCH(1,1) model provides more exact forecasts on the returns of the Stockholm stock market.

The modeling and forecasting of volatility in financial markets used to be a fundamental issue for many researchers. The importance of this problem increased over the last years as there is upheaval in the financial world. The aim of this paper is to compare various volatility models and forecast for future values regarding the rate of return and volatility of the Stockholm stock market using 1-step ahead. Measuring the period from 30 September 1986 until 11 May 2016 and using as a sample 7,434 daily observations for different models we concluded that the asymmetric models give better results on the returns and volatility of the Stockholm stock market. More specifically, we estimated the symmetric ARIMA(0,0,1)-GARCH(1,1) model, as well as the asymmetric models ARIMA(0,0,1)-EGARCH(1,1) and ARIMA(0,0,1)-GJR-GARCH(1,1) models with different residual distributions. The analysis of estimations indicated that t-student distribution is considered the most suitable on the estimation of parameters for all models. These results are in accordance with the empirical works by (Hamilton and Susmel [

To sum up, the results of our paper confirm previous findings that GARCH models with normal errors do not seem to fully capture the leptokurtosis in empirical time series (see, e.g. Kim and White [

Dritsaki, C. (2017) An Empirical Evaluation in GARCH Volatility Modeling: Evidence from the Stockholm Stock Exchange. Journal of Mathematical Finance, 7, 366-390. https://doi.org/10.4236/jmf.2017.72020