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This study investigated the performance of eleven competing time series GARCH models for fitting the rate of returns data, monthly observations on the index returns series of the market over the period of January 1996 to December 2015 was used. From the results obtained from the Log Likelihood (Log L), Schwarzs Bayesian Criterion (SBC) and the Akaike Information Criterion (AIC) values it was found that the models identified was not the same for the two periods (Training and Testing period) that is for Training period were CGARCH (1,1) and EGARCH (1,1) while for Testing period were ARCH (1) and GARCH (2,1) . The two extreme classes of models are identified to represent the best and the worst groups respectively. The overall effect of this will tend to increase the volatility of the market returns. The paper therefore recommended that the Nigeria government should as a matter of urgency take appropriate positive measure s through the security and exchange commission to regulate the market volatility so that the provided market index could be safely used as predictive index for measuring the performance of the firms and as a guide for investment purpose.

The study employ different univariate specifications of GARCH type model for monthly observations on the index returns series. Furthermore, because of the sensitivity of global and regional economics models, there is more increasing attention in research in these areastime series GARCH models for fitting the rate of returns data. Studies involving stock market return, foreign exchange rates, inflation rates are wide. In addition, stock market exhibits changes in variance over time in such circumstances, that the assumption of constant variance (homoscedasticity) is inappropriate. The variability in the financial data could very well be due to the volatility of the financial market. More importantly, the extended financial market as well as globalization due to the markets is known to be sensitive to factors such as rum ours, political upheavals and changes in the government monetary and fiscal policies [

The objective of this study is to employ different univariate specifications of GARCH type model for monthly observations on the index returns series of the market over the period of January 1996 to December 2015 and to model stock returns volatility in Nigeria Stock Markets.

It has been a large amount of literature on modeling stock market return volatility in both developed and developing countries around the world. The volatility characteristics have been investigated using econometrics models. However, no single model is superior. The idea of using factor models with GARCH goes back to Engle, [

The univariate generalised autoregressive conditional heteroscedasticity (GARCH) models that were introduced [

Autoregressive Conditional Heteroscedasticity (ARCH) and its Generalization (GARCH) models represent the main methodologies that have been applied in modeling stock market volatility in finance time series. These models can be effective in removing the excess kurtosis. In this research different univariate GARCH specifications are employed to model stock returns volatility in Nigeria Stock Market Returns the models are to be used for testing symmetric volatility.

The data used in this research work, is the monthly rate of returns of the (Nigeria Stock Market) (NSM), registered from January 1996 to December 2015. In the fourth quarter of 2006, political crisis which hit the Asian region had badly hurt the performance of most of the Oil Market in the world including the NSM (Nigeria Stock Market).

The data were divided in to two periods: Training Period from January 1996 to December 2006 and Testing Period from January 2007 to December 2015.

The monthly rate of returns r_{i} of the NSM are calculated using the following formula:

r t = log ( I t I t − 1 ) , t = 1 , 2 , ⋯ , T (1)

where I_{T} denotes the reading on the composite index at the close of t^{th} trading day. As noted earlier, the rate of monthly returns of the NSM displays a changing variance over time. There are many ways to describe the changes in variance and one of them is by considering the Autoregressive Conditional Heteroscedasticity (ARCH) model.

Suppose for now that ε 1 , ε 2 , ⋯ is Gaussian white noise with unit variance. Later we will allow the noise to be independent white noise with a possibly non normal distribution, such as, a standardized t-distribution. Then

E ( ε t | ε t − 1 , ⋯ ) = 0 (2)

and

V a r ( ε t | ε t − 1 , ⋯ ) = 1 (3)

Property (2) is called conditional homoskedasticity.

The process a_{t} is an ARCH (q) process under the model

a t = ω + α 1 a t − 1 2 ε t (4)

Equation (4) is a special case of (3) with f equal to 0 and σ equal to ω + α 1 a t − 1 2 .

These research require that ω > 0 and α_{1} ≥ 0 so that α 0 + α 1 a t − 1 2 > 0 . It is also required that α_{1} < 1 in order for a_{t} to be stationary with a finite variance. Equation (4) can be written as

a t 2 = ( ω + α 1 a t − 1 2 ) ε t 2 (5)

Which is very much like an AR (q) but in a t 2 , not a_{t}, and with multiplicative noise with a mean of 1 rather than additive noise with a mean of 0.

[

σ t δ = ω + β 1 σ t − 1 δ + α 1 ( | ε t − 1 | − γ 1 ε t − 1 ) δ (6)

where α 1 and β 1 are the standard ARCH and GARCH parameters, γ 1 is the leverage parameter and δ is the parameter for the power term. When δ = 2 Equation (6) becomes a classic GARCH model that allows for leverage effects, and when δ = 1 , the conditional standard deviation will be estimated.

The similarities seen in between GARCH and ARMA models are not a coincidence. If a_{t} is a GARCH process, then a t 2 is an ARMA process but with weak white noise, not i.i.d. white noise. To show this, we will start with the GARCH (1,1) model, where a t = σ t 2 ε t . Here ε_{t} is i.i.d. white noise and

E t − 1 ( a t 2 ) = σ t 2 = ω + α 1 a t − 1 2 + β 1 σ t − 1 2 (7)

where e_{t}_{−1} is the conditional expectation given the information set at time t − 1. Define η t = a t 2 − σ t 2 . Since E t − 1 ( η t ) = E t − 1 ( a t 2 ) − σ t 2 = 0 , by η_{t} is an uncorrelated process, that is, a weak white noise process. The conditional heteroskedasticity of a_{t} is inherited by η_{t}, so η_{t} is not i.i.d. white noise.

Simple algebra shows that

σ t 2 = ω + ( α 1 + β 1 ) a t − 1 2 − β 1 η t − 1 (8)

And therefore

a t 2 = σ t 2 + η t = ω + ( α 1 + β 1 ) a t − 1 2 − β 1 η t − 1 + η t (9)

Assume that α 1 + β 1 < 1 . If μ = ω / { 1 − ( α 1 + β 1 ) } , then

a t 2 − μ = ( α 1 + β 1 ) ( a t − 1 2 − μ ) + β 1 η t − 1 + η t (10)

From (9) one sees that a t 2 is an ARMA (1,1) process with mean µ. Using the notation of the AR (1) coefficient is φ 1 = α 1 + β 1 and the MA (1) coefficient is θ 1 = − β 1 .

For the general case, assume that σ_{t} follows so that

σ t 2 = ω + ∑ i = 1 p α i a t − i 2 + ∑ i = 1 q β i σ t − i 2 (11)

Assume also that p ≤ q―this assumption causes no loss of generality because, if q > p, then we can increase p to equal q by defining α i = 0 for i = p + 1 , ⋯ , q .

Define μ = ω / { 1 − ∑ i = 1 p ( α i + β i ) } . Straightforward algebra similar to the GARCH (1,1) case shows that

a t 2 − μ = ∑ i = 1 p ( α i + β i ) ( a t − 1 2 − μ ) − ∑ i = 1 q β i η t − i + η t (12)

So that a t 2 is an ARMA (p,q) process with mean µ. As a byproduct of these calculations, we obtain a necessary condition for a_{t} to be stationary:

∑ i = 1 p ( α i + β i ) < 1 (13)

[

I n ( h t ) = ω + ∑ i − 1 q α i g ( e t − 1 2 ) + ∑ j − 1 p β j I n ( h t − 1 )

where

g ( e t ) = θ e t + γ | e t | − γ E | e t |

The coefficient of the second term in g(e_{t}) is set to be 1( γ = 1 ) in this formulation. Unlike the linear GARCH model there are no restrictions on the parameters to ensure non-negativity of the conditional variances.

In the GARCH-in-Mean or GARCH-M model, the GARCH effects appear in the mean of the process, given by

ε t = h t e t

where e t ∼ N ( 0 , 1 ) and r t = μ + δ h t + ε t for the model with intercept and r t = δ h t + ε t for the non-intercept model. For the model GARCH (p,q) specification, [

The GARCH (p,q) is the most widely used GARCH process, so it is worthwhile to study it in some detail. If a_{t} is GARCH (p,q), then as we have just seen, a t 2 is ARMA (p,q).

ρ a 2 ( 1 ) = α 1 ( 1 − α 1 β 1 − β 1 2 ) 1 − 2 α 1 β 1 − β 1 2 (14)

and

ρ a 2 ( k ) = ( α 1 + β 1 ) k − 1 ρ a 2 ( 1 ) , k ≥ 2 (15)

By (14), there are infinitely many values of (α_{1}, β_{1}) with the same value of ρ a 2 ( 1 ) . By (15), a higher value of α_{1} + β_{1} means a slower decay of ρ a 2 after the first lag.

It allows mean reversion to a varying level μ t

σ t 2 − m t = ϖ + α ( u t − 1 2 − ϖ ) + β ( σ t − 1 2 − ϖ ) (16)

m t = ω + ρ ( m t − 1 − ω ) + ϕ ( u t − 1 2 − σ t − 1 2 ) (17)

σ t − 1 2 is still the volatility, while m_{t} takes the place of ω and is the time varying long-run volatility. The first equation describes the transitory component, σ t 2 − m t which converges to zero with powers of ( α + β ) . The second equation describes the long run component m_{t}, which converges to ω with powers of ρ .

[

σ t 2 = ω + α ε t − 1 2 + γ ε t − 1 2 I ( ε t − 1 < 0 ) + β σ t − 1 2 (18)

where I denotes the indicator function. The model is also sometimes referred to as a Sign-GARCH model.

Some descriptive statistics for the monthly return of the Nigeria Stock Market are presented in

From the results presented in

Period | N | Mean | Sd | Variance | Skewness | Kurtosis |
---|---|---|---|---|---|---|

Training period | 132 | −0.063207 | 0.992520 | 0.985096 | 0.274531 | 3.462173 |

Testing period | 108 | −0.024417 | 1.006990 | 1.014029 | 0.312870 | 3.485325 |

Training Period was from January 1996 to December 2006.

From

From

The DF-GLS statistic test the null hypothesis of unit root test against the alternative of no unit root test and the decision rule is to reject the null hypothesis is when the value of the test statistic is less than the critical value. The Ng-Perron statistic test the null hypothesis of stationary against the alternative of no stationary and the decision rule is to accept the null hypothesis when the value of the test statistic is less than the critical value. The results of the DF-GLS and Ng-Perron tests are in

Critical Value | DF-GLS Test Statistics: −8.831925 | Ng-Perron Test Statistics: 0.40208 |
---|---|---|

1% | −2.582872 | 1.78000 |

5% | −1.943304 | 3.17000 |

10% | −1.615087 | 4.45000 |

To achieve the overall objective of the research, we examine the characteristics of the unconditional distribution of the training period of Nigeria stock market returns. This will enable us to explore and explain some stylized facts embedded in the financial time series. Jarque Bera normality test is used to demonstrate this and the results are given in

From

From the results obtained in

The results obtained in _{1} for GJR-GARCH (1,1), β_{1} for GARCH (1,1) and EGARCH(1,1), α_{2} for ARCH (2), β_{2} for CGARCH (1,1) and GARCH (2,2) and θ for CGARCH (1,1) are significant at 5% level.

Testing Period was from January 2007 to December 2015.

Having discovered that the Nigeria Stock Market returns series could be modeled as ARCH and GARCH, the next is to examine the ACF and PACF to see the degree of correlation in the data point of the series.

Model | LOGL | SBC | AIC |
---|---|---|---|

ARCH (1) | −386.6639 | 6.312633 | 6.071205 |

ARCH (2) | −386.1990 | 6.342750 | 6.079373 |

PARCH (1,1) | −384.5735 | 6.466795 | 6.115626 |

GARCH (1,1) | −386.2125 | 6.342958 | 6.079581 |

GARCH (1,2) | −382.5851 | 6.324792 | 6.039467 |

GARCH (2,1) | −380.4210 | 6.291753 | 6.006428 |

EGARCH (1,1) | −380.1429 | 6.287507 | 6.002182 |

GJR-GARCH (1,1) | −389.1248 | 6.424635 | 6.139310 |

CGARCH (1,1) | −389.5262 | 6.579624 | 6.206507 |

GARCH-M (1,1) | −382.2043 | 6.356194 | 6.048921 |

GARCH (2,2) | −382.1420 | 6.355243 | 6.047970 |

MODEL | ω × 10^{−5 } | t-ratio | α_{1} | t-ratio | β_{1} | t-ratio | α_{2} | t-ratio | β_{2} | t-ratio | δ | t-ratio | θ | t-ratio |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ARCH (1) | 10.023 | 3.29 | 0.82 | 3.12 | ||||||||||

ARCH (2) | 10.605 | 3.14 | 0.78 | 3.07 | −0.02 | −0.42 | ||||||||

PARCH (1,1) | 2.3729 | 1.21 | 0.32 | 2.13 | 0.36 | 1.81 | 0.08 | 0.92 | 0.81 | 0.56 | ||||

GARCH (1,1) | 10.891 | 2.89 | 0.79 | 3.07 | −0.03 | −0.41 | ||||||||

GARCH (1,2) | 10.186 | 2.39 | 0.61 | 2.47 | −0.14 | −1.51 | 0.23 | 2.19 | ||||||

GARCH (2,1) | 1.8803 | 2.85 | 0.77 | 3.44 | −0.74 | −3.68 | 0.91 | 22.8 | ||||||

EGARCH (1,1) | 3.0994 | 4.80 | 1.17 | 4.45 | 0.01 | 0.05 | −0.37 | −2.29 | ||||||

GJRGARCH (1,1) | 11.098 | 0.86 | −0.01 | −0.79 | −0.08 | −2.50 | 0.59 | 1.19 | ||||||

CGARCH (1,1) | 23.121 | 5.11 | 0.02 | 0.02 | 0.07 | 0.02 | 0.05 | 0.02 | −0.03 | −0.36 | 0.00 | 0.00 | ||

GARCH−M (1,1) | 15.346 | 3.96 | 0.35 | 2.06 | 0.39 | 1.07 | −0.12 | −1.12 | ||||||

GARCH (2,2) | 3.8942 | 2.86 | 0.88 | 3.37 | −0.67 | −4.04 | 0.74 | 4.09 | −0.03 | −1.01 |

The DF-GLS statistic test the null hypothesis of unit root against the alternative of no unit root and the decision rule is to reject the null hypothesis is when the value of the test statistic is less than the critical value. The Ng-Perron statistic test the null hypothesis of stationary against the alternative of no stationary and the decision rule is to accept the null hypothesis when the value of the test statistic is less than the critical value. The results of the DF-GLS and Ng-Perron tests are in

From

To achieve the overall objective of the research, we examine the characteristics of the unconditional distribution of the training period of Nigeria stock market returns. This will enable us to explore and explain some stylized facts embedded in the financial time series. Tarque Bera normality test is used to demonstrate this and the results are given in

Critical Value | DF-GLS Test Statistics: −2.245094 | Ng-Perron Test Statistics: 3.11222 |
---|---|---|

1% | −2.587387 | 1.78000 |

5% | −1.943943 | 3.17000 |

10% | −1.614694 | 4.45000 |

Model | LOGL | SBC | AIC |
---|---|---|---|

ARCH (1) | −354.5865 | 7.108170 | 6.833393 |

ARCH (2) | −339.1416 | 6.863150 | 6.563394 |

PARCH (1,1) | −329.9103 | 6.821617 | 6.446921 |

GARCH (1,1) | −335.0621 | 6.786899 | 6.487142 |

GARCH (1,2) | −329.8165 | 6.732520 | 6.407784 |

GARCH (2,1) | −331.6776 | 6.767307 | 6.442571 |

EGARCH (1,1) | −351.1336 | 7.130971 | 6.806235 |

GJR−GARCH (1,1) | −333.9890 | 6.810512 | 6.485776 |

CGARCH (1,1) | −335.1591 | 7.007068 | 6.582413 |

GARCH−M (1,1) | −332.9424 | 6.834621 | 6.484905 |

GARCH (2,2) | −330.8776 | 6.796026 | 6.446310 |

tion Criterion (AIC) values of the ARCH and GARCH models that is used in choosing the best fit model from Testing period of Nigeria stock market returns.

The results obtained in _{1} for GARCH-M (1,1), β_{1} for GARCH (2,2), α_{2} for CGARCH (1,1), and ω for GARCH (1,1) and GJR-GARCH (1,1) are significant at 5% level.

MODEL | ω × 10^{−5} | t-ratio | α_{1} | t-ratio | β_{1} | t-ratio | α_{2} | t-ratio | β_{2} | t-ratio | δ | t-ratio | θ | t-ratio |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ARCH (1) | 20.5 | 3.40 | 0.99 | 2.84 | ||||||||||

ARCH (2) | 12.6 | 2.97 | 0.07 | 0.46 | 0.85 | 3.58 | ||||||||

PARCH (1,1) | 0.41 | 0.14 | 0.29 | 0.95 | 0.31 | 2.98 | 0.45 | 1.75 | 5.63 | 1.39 | ||||

GARCH (1,1) | −0.03 | −0.06 | 0.39 | 2.29 | 0.68 | 5.88 | ||||||||

GARCH (1,2) | −0.59 | −4.09 | 0.41 | 2.36 | 0.33 | 1.01 | 0.34 | 1.22 | ||||||

GARCH (2,1) | −0.39 | −1.71 | 0.45 | 2.03 | −0.19 | −1.04 | 0.79 | 9.63 | ||||||

EGARCH (1,1) | 4.56 | 5.66 | 0.13 | 0.48 | 0.54 | 3.29 | −0.25 | −1.20 | ||||||

GJRGARCH (1,1) | 0.01 | 0.02 | 0.24 | 1.90 | 0.26 | 1.43 | 0.69 | 6.89 | ||||||

CGARCH (1,1) | 42.2 | 0.82 | 0.89 | 7.64 | 0.16 | 0.32 | −0.02 | −0.04 | 0.17 | 0.78 | 0.78 | 1.46 | ||

GARCH−M (1,1) | 1.89 | 0.75 | −0.02 | −0.18 | 0.67 | 2.16 | 0.73 | 5.33 | ||||||

GARCH (2,2) | −0.57 | −1.34 | 0.42 | 1.87 | −0.04 | −0.13 | 0.42 | 0.62 | 0.28 | 0.54 |

In this study the method for selecting the best model from a set of competing GARCH models for fitting the Nigeria Stock Market Return series was used. The method identified exactly the best and worst fit models as for the two periods. However, as a whole, the models occupying the intermediate positions differ in the method. The results obtained from the Log Likelihood (Log L), Schwarzs Bayesian Criterion (SBC) and the Akaike Information Criterion (AIC) values found out that the models identified by the method were not the same for the two periods i.e. for Training period were CGARCH (1,1) and EGARCH (1,1) while for Testing period were ARCH (1) and GARCH (2,1). The two extreme classes of models are identified to represent the best and the worst groups respectively. The overall effect of this will tend to increase the volatility of the market returns. Another advantage is that the method can help models to be classified in to several distinct groups ordered in such a way that each group is made up of models with about the same level of fitting ability. The two extreme classes of models are identified to represent the best and the worst groups respectively.

Based on our findings, this research has contributed to the knowledge in the following directions:

1) We find out that the result of the criteria (Log Likelihood (Log L), Schwarzs Bayesian Criterion (SBC) and the Akaike Information Criterion (AIC)) are use to identify the best fit model.

2) And the parameter estimate are being classified in to different groups and with those that have exceptional.

It is our suggestion that for future researchers can applied principal component analysis in testing the best fit model among the GARCH model.

Usman, U., Auwal, H.M. and Abdulmuhyi, M.A. (2017) Fitting the Nigeria Stock Market Return Series Using GARCH Models. Theoretical Economics Letters, 7, 2159-2176. https://doi.org/10.4236/tel.2017.77147