Table 9: Diagnostics Estimates of the Selected Banks BL-GARCH (1, 1) Model

Bank Jarque-Bera P-Value (Chi^2)

These tests show significant evidence in support of GARCH effects (heteroskedasticity). Each of these extracts the sample mean from the actual returns. This is consistent with the definition of the conditional mean equation in which the innovations process is _t  y_t _t, and



_t

is the mean of

y

t. Evidence of time dependence is found using Ljung-Box statistics, which is robust to heteroskedasticity and reported for autocorrelations up to 20 lags. The statistics show strong serial correlations in both levels of the return series. This is consistent with the results of Pitt and Shephard (1999), Storvik (2002), Johannes, Polson and Stroud (2006), Raggi and Bordignon

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(2006), who found that serial correlations in DJIA returns are significant but unstable and depend on the sample period.

A non- constant variance of asset returns should lead to a non-normal distribution. Figure 17 represents the histogram and the student-t distribution of the stock market prices plotted in Figures 6, 7, 8 and 9. Assets returns are highly leptokurtic and slightly asymmetric, a phenomenon correctly observed by Mandelbrot (1963): “The empirical distributions of price changes are usually too

“peaked” in relation to samples from Gaussian populations … the histograms of price changes are indeed unimodal and their central bells remain the Gaussian ogive. But there are typically so many outliers that ogives fitted to the mean square of price changes are much lower and flatter than the distribution of the data themselves.” Non-stationarity in the conditional variances is not the only possible source of non-stationarity for stock return series. The level of stock prices and daily trading volume may also be non-stationary. The volatile behaviour of the disturbances contradicts the assumption of normally distributed disturbances. The disturbances are therefore modelled with the BL-GARCH (1, 1) model using independent generalized error and generalized student-t distributed random variables with unknown common degrees of freedom. The increase in the variance does not occur when the disturbances are generalized error and generalized student-t distributed because with a lower value of the parameter  one can also explain occurrence of several rather large values of the disturbances, that is, heteroskedasticity.

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-0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06

Returns Fig. 17: Histogram and Student-t Distribution of the Selected Banks

The series show evidence of fat tails since the kurtosis is positive and evidence of skewness, which means that the tails are either heavier or lighter than the usual student-t distribution. Looking at the plot of GTB stock prices, the student-t distribution tends to infinity (Fig. 17). This prompts the study to use the generalized student-t distribution to capture the extreme values in Figure 18.

Kurtosis and the narrow bands in plot are hints of conditional heteroskedasticity. These models, although able to capture the leptokurtosis, could not account for the existence of non-linear temporal dependence as the volatility clustering observed from the data. For example, applying an autoregressive model to remove the linear dependence from an asset return series and testing the residuals for a higher-order dependence using the Brock, Dechert and Scheinkman (BDS test), the null hypothesis, that the residuals are independent identically distributed, is rejected. . The Jarque-Bera (JB) test decisively rejects the normal distribution (see Table 9). The JB test is a test statistic for testing whether the series is normally distributed. The test statistic measures the difference of the skewness and kurtosis of the series with those from the normal distribution.

Jarque-Bera = 

109

where S is the skewness, K is the kurtosis, and k represents the number of estimated coefficients used to create the series. Under the null hypothesis of a normal distribution, the JB statistic is distributed as ² with 2 degrees of freedom.

Figure 18 shows that the generalized student-t distribution seems to be a more appropriate distribution for the selected banks data. The generalized student-t distribution allows for situations where the tails are heavier or lighter than the usual student-t distribution. The extreme values (left

)

(₁ and right tails(₂)) for different banks are estimated automatically together with the plot of the P-P plots of the selected banks in Figure 19.

GTB

ZEB

UBA

-3 100 200 300 400 500 600 700 800 ⁹⁰⁰

0

0.05 0.1 0.15

0.2 0.25 0.3 0.35 0.4 0.45

Days

Fig. 18: Plots of Generalized Student-t Distribution for Banks that exhibit Heavier and Lighter Tails.

Density

(x10

^-3

)

110

Probability-Probability Plot of FBN Distribution: Extreme(23.382, 10.7655)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

T heoretical cumulative distribution -0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Empirical cumulative distribution

Probability-Probability Plot of GTB Distribution: Extreme(18.443, 7.82681)

-0.2 0.0 0.2 0.4 0.6 0.8 1.0

T heoretical cumulative distribution -0.4

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Empirical cumulative distribution

Probability-Probability Plot of UBA Distribution: Extreme(20.896, 13.8663)

0.0 0.2 0.4 0.6 0.8 1.0

T heoretical cumulative distribution -0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Empirical cumulative distribution

Probability-Probability Plot of ZEB Distribution: Extreme(24.0911, 13.4872)

0.0 0.2 0.4 0.6 0.8 1.0 1.2

T heoretical cumulative distribution -0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Empirical cumulative distribution

Fig. 19: Probability-Probability (P-P) Plots with Extreme Values for the Four Banks

The probability-probability plots of the four banks show that they are primarily a few large outliers that cause the departures of the system from normality as obtainable from Figure 19. These departures are pointing out that there are other factors that interrupt the expected volatility of stock market prices of these banks on the floor of the Nigeria Stock Exchange. The factors may include among other things, giving loans to private and corporate firms to buy shares without due process, lack of strategic management and regular supervision. If the residuals are normally distributed, the P-P plots should lie on a straight line.

111

The data was estimated by methods of the Maximum Likelihood Estimator (MLE) using MATLAB (R2008b) soft ware. The parameter estimates are presented in Tables 10, 11 and 12 below:

Table 10: Conditional Variance GARCH (1, 1) Model Parameter Estimation Results

ˆ

0

(0.26107, 15.9543) Gaussian

The values in parenthesis, say (a, b), are the standard errors and t-statistics respectively.

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Table 11: Conditional Variance BL-GARCH (1, 1) Model Parameter Estimation Results

ˆ

0

113

Table 12: Conditional Variance BL-GARCH (1, 1)-Volume Model Parameter Estimation Results

ˆ

0

(0.00852, 29.7183)

0.7312 0.01060* BL-GARCH (1, 1)-Volume model parameter estimation results respectively. Results reveal that parameter estimates are satisfactory (asymptotically unbiased, efficient and consistent) in that the standard errors are small and the t-statistic for GARCH parameters () is high. It is clear from the

114

analysis that estimate ˆ₁ and ^ˆ₁ in the BL-GARCH (1, 1) and BL-GARCH (1, 1)-Volume model are significant at the 5% level with the volatility coefficient greater in magnitude. Hence, the hypothesis of constant variance is rejected, at least within sample. Furthermore, the stationarity condition is satisfied for the four distributions, as ˆ₁ˆ₁1 at the maximum of the respective log-likelihood functions. Even whenˆ₁ˆ₁ 1, so long as E[ln(₁z_t² ₁)]0, covariance stationarity

is established. The estimated asymmetric volatility response

ˆc

₁ is negative and significant for all models confirming the usual expectation in stock markets where downward movements (falling returns) are followed by higher volatility than upward movements (increasing returns). The results also follow the empirical findings of Storti and Vitale (2003), in that the kurtosis strongly depends on the leverage-effect response parameter. The results indicate that the BL-GARCH (1, 1) as well as the GARCH (1, 1) processes is appropriate for modelling the conditional variance of the selected banks return. However, from Table 9, the goodness-of-fit statistics indicate that the BL-GARCH (1, 1) performs better in describing the conditional variance of the selected banks return. Moreover, the possible usefulness of using fat-tailed innovations for the GARCH (1, 1) and BL-GARCH (1, 1) models seem to be confirmed by the log-likelihood values and the AIC in Table 9. Using Akaike (1974), the BL-GARCH (1, 1) model with minimum AIC was selected as the best.

The BL-GARCH (1, 1) conditional variance model that best fits the observed data is _t² 0.31060.3916_t²_1 0.9162_t²_1 0.5314_t1_t1

where

5314 . 0

9162 . 0 ) 1 ˆ (

, 3916 . 0 ) 1 ˆ (

, 3106 . ˆ 0

1

1 1

0



















effect leverage c

and

GARCH

ARCH 





115 The model for individual bank estimates are given as

_t² 2e^007 0.3634_t²_10.6765_t²_1 0.0856_t1_t1

--- FBN

1 1 2

1 2

1

2 3.18930.5190 _t 0.4398 _t 0.0980 _{t t}

t    

 --- GTB

1 1 2

1 2

1

2 0.61340.4257 _t 0.8419 _t 0.1760 _{t t}

t    



--- UBA

1 1 2

1 2

1 005

2 2 ^ 0.3176 _t 0.6543 _t 0.0297 _{t t}

t e    



--- ZEB The model for tuberculosis (TB) patients are given as

2 1 2

1

2 0.01870.2546 _t 0.6553 _t

t  



From the results obtained, the BL-GARCH (1, 1) model with generalized student-t distribution fits GTB, UBA and ZEB data better while the First Bank of Nigeria data follows the student-t BL-GARCH (1, 1) models. This is because adding more parameters in modelling the FBN data does not improve the parameter estimates of the FBN. The parameteris therefore a good approximation of the degree up to which one is able to explain the variance/kurtosis of the disturbances. The GTB, UBA and ZEB series confirm these statements as seen in Figure 18. Table 11 shows that imposition of a constant variance leads to student-t distributed disturbances whose degrees of freedom parameter is almost approximately 2.0. The variance of the disturbances is infinite as a consequence in GTB, for example. The generalized student-t distribution of  for BL-GARCH conditional variances lies almost completely above 2.0 such that the conditional variances of the disturbances are finite. In Table 12, the coefficients for daily trading volume variable,  is positive, greater than 0 and significant at 5% level for GTB, UBA and ZEB and negative for FBN. This analysis is suggesting that the daily trading volume increases for GTB, UBA and ZEB due to good news in the market and decreases for FBN trading volume establishing that its stocks are more volatile.

Coefficients  shows that a positive impact of volume on stock returns also generate less impact on volatility of the market. Thus, the inclusion of daily trading volume variable in the variance process

116

accounts for some of the observed GARCH persistence and asymmetric effect embedded in the volatility of returns for the sampled period. This analysis also shows that the recent news of daily trading volume can be used to improve the prediction of stock price volatility.

The diagnostics estimates are provided in Table 9. In order to compare objectively, various goodness-of-fit statistics are used. The diagnostics, summarized are the log-likelihood function at its maximum, the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC).

The study adapts a number of criteria that have been proposed in the literature to evaluate the fit of the model versus the number of parameters, see Marquardt (1963), Bernt et al (1974), Lee and Hansen (1994), Lumsdaine (1996). These criteria were developed for pure autoregressive models but have been extended for autoregressive moving average models. The more applied model selection criteria are the Akaike Information Criterion, AIC, (Akaike (1994), the Bayesian Information Criterion, BIC, (Schwarz (1978) and Ljung-Box, Q². The criteria were given as:

N

AIC 2k

ˆ ) ln( ² 

 _

) ln(

ˆ )

ln( ² N

N BIC _  k



 



 ^q

i N i

N i N q Q

1

2 ( )

) 2 ( )

(



where k is the number of the estimated ARMA parameters (p + q) and N is the number of

observations used for estimation. The AIC and BIC criteria are based on the estimated variance

 ˆ

_²

plus a penalty adjustment depending on the number of estimated parameters and it is in the extent of this penalty that these criteria differ. The AIC uses the observed data to simultaneously estimate the parameters of the model by the method of maximum likelihood and calculate the model selection criteria. The term 2k in the definition of the AIC is used to correct the bias in the estimated

117

maximum log likelihood. The penalty proposed by BIC is larger than AIC's since ln(N)2 for

8

N . Therefore, the difference between both criteria can be very large if N is large; BIC tends to select simpler models than those chosen by AIC. In practical work, both criteria are usually examined. If they do not select the same model, many authors tend to recommend to use the more parsimonious model selected by AIC. In MATLAB, most of the residual diagnostics for BL-GARCH models are in terms of the standardized residuals [which should be N (0, 1)]. Note that kurtosis is greater than 3.

As shown under parameter estimates, all two models (GARCH (1, 1) and BL-GARCH (1, 1)) appear to have statistical strength in terms of large t-statistics. The models pass the residual diagnostics with very similar results. This is evident from the plot of the relationship between the innovations (residuals) derived from the fitted model, the corresponding conditional standard deviations and the observed returns of the selected banks (FBN, GTB, UBA and ZEB) in the two models shown in Figure 20 below.

118

119

120

121

Fig. 20: Graph of Innovations, Conditional Deviations and Observed Returns Compared for the Selected Banks

122

Fig. 21: Graph of Innovations, Conditional Deviations and Observed Returns of Tuberculosis Patients.

In all the models, the innovations exhibit volatility clustering just like the returns. The spikes in the conditional standard deviation in FBN, UBA and ZEB BL-GARCH models indicate high volatility, asymmetric effect and confirmation of heavy tails in high frequency financial data while GARCH models capture GTB bank as well as TB volatility better.

0 20 40 60 80 100 120

-2 0 2

Innovations

Innovation

0 20 40 60 80 100 120

0 0.5 1

Conditional Standard Deviations

Standard Deviation

0 20 40 60 80 100 120

-2 0 2

Returns

Return

123

In document CHAPTER ONE INTRODUCTION (Page 106-123)