Garch modeling of Nigerian stock exchange market capitalization

Introduction

The stock exchange is a well-organized marketplace where investors can purchase and sell securities such as shares, stocks, and debentures. It is a market where those interested in purchasing stocks are brought into direct touch with the seller. It is a crucial component of the capital market, serving as a source of cash for publicly traded firms and government institutions, as well as a platform for individual financial participation. The market exclusively deals with existing shares; no new ones are exchanged there. The stock exchange operates a daily auction market in which specialist traders known as stock brokers purchase and sell shares of quoted firms both for their own account and on behalf of their clients. Nigeria only has one stock exchange market. The Nigerian stock exchange market, which has seven branches in seven states of the federation plus Abuja, is linked to its headquarters in Lagos via computer networks. The exchange runs an auction market, which is a system in which securities are bought and sold through a large number of brokers at prices determined by competitive bidding. Stock brokers dealing on their own account and on behalf of their clients enter their bids to buy and sell securities in their computer system on the trading floor. Some stock broking firms now have remote trading tools and trade from their offices during trading hours. The Nigeria Stock Exchange was established on June 5, 1961, when the Lagos Stock Exchange Act was enacted by the Federal Government of Nigeria. The business began officially on August 25th, 1961, with 19 stocks registered for trading, while informal operations began earlier in June 1961. The operation was initially handled inside the central bank (CBN) premises, with four firms serving as market dealers: Inlaks, John Holt, Bowring, and Icon (Investment Company of Nigeria). According to Marc Reinganum (1999) [13] , market capitalization is a major driver of portfolio performance and policy decisions. Gopinath (2005) [11] investigates market capitalization based on the patterns of share price fluctuation and the behavior of ten (10) companies, revealing the adequate performance of share price movements and the performance of the same company were cyclical.

The study also identifies a significant relationship between share price and price earnings ratio, market capitalizations, turnover ratio, book value, and market returns. In this study, we continue our investigation into stock exchange market capitalization movement by identifying different aspects in model and time frames, as well as the models that can be used to capture stock exchange market capitalization volatility using the ARIMA and GARCH models, as described by Bollerslev (1986) [4] and Eagle (1982) [9] , respectively. As a result, we strive to discover the model that best captures the features of Nigeria's stock exchange market capitalization. GARCH and ARIMA models were chosen to compare the volatility clustering and capitalization of the Nigeria stock exchange market capitalization. Model selection approaches such as Augmented Dickey Fuller (ADF) and Akiake Information Criterion (AIC) were used to assess the volatility performance of the Nigeria stock exchange market capitalization in order to determine the basis for claiming its model performances.

GARCH is a statistical model that may be used to examine a variety of financial data, including macroeconomic data.

Related Literatures

GARCH is a statistical model that may be used to analyze various types of financial data, including macroeconomic data. However, Goudazri and Ramanarayanan (2010) studied the volatility of the Indian stock market over ten years using the BSE 500 stock index as a proxy. ARCH and GARCH models were estimated, and the best model was chosen based on the model selection criteria Viz, Akaike Information Criterion (AIC), and Schwarz Information Criterion (SIC). The analysis discovered that GARCH (1,1) was the best model for explaining volatility clustering and mean reversion in the series over the study period. Usman et al. (2017) [14] studied and captured the fitting of the Nigeria stock market returns series using GARCH model, the performance of eleven competing time series GARCH model for fitting the rate of returns data, and monthly observations on the market index returns series from January 1996 to December 2015. According to their findings, the log likelihood (Log L), Schwartz's Bayesian Criterion (SBC), and Akaike information criterion (AIC) values indicated that the models discovered were not the same. CGARCH (1,1) and EGARCH (1,1) were used for training, and ARCH (1) and GARCH (2,1) for testing. Dallah and Ade (2010) [7] investigated the volatility of daily stock returns in Nigerian insurance stocks using daily data from 26 insurance companies from December 15, 2000 to June 9, 2008 as training data and from June 10, 2008 to September 9, 2008 as out-of-sample data. The results of ARCH (1), GARCH (1,1), TARCH (1,1), and EGARCH (1,1) reveal that EGARCH is better suited to modeling stock price returns because it beats the other models in model evaluation and out-of-sample forecasting.

Model Equations

According to Eagle R. F. (1982) [9] , the general GARCH model in terms of (p,q) is written as

Where δ 2 t -1 and ∝ 2 t -1 = explanatory variable and independent ∝ 0 , ∝ 1 and β1=parameters to be estimated GARCII (1,1) Model is given by

Where δ 2 t =The variance and the dependent variable with time (t) 𝜀 t =Measure the effect of error term (white noise) when 0 ≤∝ 1 ≤ 1 ∝ 0 , ∝ 1 and β1= Non-negative parameters to be estimated, it measures the variance of past volatility and past residual of the current volatility. δ 2 t =The explanatory variable and independent variable at time(t). We must note that GRACH is a model for variance. Another proposed model called ARIMA model was adopted based on if an autoregressive moving average (ARIMA) is assumed for the error variance, the model is generalized autoregressive conditional heteroscedasticity (GARCH) model. Hence, the general ARIMA model is written as

Where X t =Measures the volatility in the stock exchange market capitalization 𝜀 t = Control the error term (white noise) Ø=The AR Parameters to be estimated which measure the Coefficient of the effect of variation in the past volatility θ =The MA Parameters to be estimated which measures the coefficient of the past residual of the current volatility × 𝑡-2 × 𝑡-3 &ε 𝑡-1 , ε 𝑡-2 , ε 𝑡-3 are lag order often referred to AR and MA respectively

Analysis and Estimation

Statistical estimations were analyzed accordingly.

Discussion of Results

A Time plot of capitalization

Stationarity Test and Volatility Presence

Figure 2 depicts the time plot of the first difference in capitalization data (DCPT). There is evidence of heteroscedasticity. Essentially, when there is heteroscedasticity, observations tend to cluster rather than corroborate a linear trend. Heteroscedasticity refers to an irregular pattern of variation in an error term or variable in a statistical model. We required stationarity in the series, but if the series is not stationary, the essential test statistics used to evaluate the results become incorrect. The research above revealed that the mean and variance are stable and near, and that the mean, variance, and autocorrelation structure did not vary over time. We also detected upward and downward volatility as the series increased. Without a trend, stable variance over time, a consistent autocorrelation structure over time, and non-periodic fluctuations, the test statistics used to evaluate this estimation become valid. The stationarity test was carried out using the Augmented Dickey-Fuller Test (ADF) method, and the results reveal the Nigeria stock series. Only after the first Differencing of determining if the variable is stationary can the exchange market capitalization series be considered stationary. The series was stable for a few years from 1986 until it increased slightly, and it continued at this level until about 2006, when the variance suddenly rose, and then the variance decreased from about 2013 until 2015, before increasing slightly until 2016, indicating that there is volatility clustering in the series. The main reason could be traced back to market events such as economic liberalization, enterprise promotion, policies, financial crisis, political instability, and national economic reforms such as Nigeria's Economic Empowerment Development Strategy (NEEDS), Structural Adjustment Programme (SAP), and an alarming decline in Nigerian economic activity in the Nigerian capital market. It provides a foundation for estimating the time series model's performance. Table 2 shows the test critical values at the 1%, 5%, and 10% levels of significance. The Augmented Dickey-Fuller (ADF) test result of -8.6622 is smaller than all critical values. Thus, the test is significant (p=0.0000 < 0.05). As a result, the DCPT is considered stationary. Fitting the model according to their respective coefficient, we have as follows;

Summary Statistics for the

The above is ARIMA ( The GARCH model is of the form

From the analysis below the values of those parameters are a 0 = 𝑐 = 0.006446 a 0 = Residual(-1) 2 = 0.170195 a 0 = Residual (-1) = 0.924814

The proposed GARCH (1,1) model is represented as

Where, a 0 = C = 0.0006446, Non-negative constant a 1 = Residual (-1) 2 = 0.170195, measures the effect of previous residual of Nigeria stock exchange market capitalization. The first coefficients (a 0 ) the usual constant in the model, the second coefficient measures the effect of previous year residual of variance on current variance. It shows positive correlation between past residual and current variance in the stock exchange market capitalization series, the third coefficient measures the correlation between the past value and current value which found to be positive. The implication of the statistic findings shows average risk in Nigeria stock exchange market capitalization of the current period regarding investment risk and previous investment risk in the stock exchange market in capitalization in Nigeria till date.

Conclusion and Policy Implication

Assessment of volatility and estimation in the Nigeria stock exchange market capitalization series was conducted and the presence of volatility was evident, therefore, this presented justification to proceed to modelling of the volatility in the Nigeria stock exchange market capitalization series. Two major models, the autoregressive integrated moving average (ARIMA) model and the generalized autoregressive conditional heteroscedasticity (GARCH) model, were objectively estimated for comparison in order to develop an efficient and better model to effectively model the volatility of Nigeria's stock exchange market capitalization. Therefore, the result obtained from the statistical estimation revealed clearly that the ARIMA (3, 1, 2) model is efficient but not better in constructing a volatility to find efficient and better for the series to model the volatility of the Nigerian stock exchange market. The Akaike Information Criterion is 14.78665, while the Schwarz Criterion is 14.85881. Similarly, the Criterion is 9.900357 and the Schwarz Criterion is 9.931282. According to the statistical estimation above, (1,1) is an efficient and better model for the series since it has lower Akaike Information, Schwarz capitalization is variable over time, and trading is not continuous (increasing and decreasing).

The policy implications of the findings in the Nigerian economy show that the stock exchange market capitalization is not stable, that the volatility in the Nigerian stock exchange market capitalization is gradually increasing, and that trading in the Nigerian stock exchange market capitalization tends to become more risky over time.

We propose that both indigenous and foreign investors develop their own investment policies based on this to direct their trading and, ultimately, improve the Nigerian economy through stock exchange market capitalization. This is a good tool for anticipating trade and returns on investments in Nigeria's stock exchange market capitalization.

Acknowledgement

, which can compensate for dynamism in variance, was discovered to be the better and more efficient model for the Nigeria stock exchange market capitalization series. Because GARCH (1,1) has a lower Akaike Information and Schwarz Criteria value, it indicates that the stock exchange market capitalization is volatile (up and down), implying that the stock market rises and falls according to economic conditions.As a result, GARCH (1,1), which can compensate for dynamism in variance, was discovered to be the better and more efficient model for the Nigeria stock exchange market capitalization series. Because GARCH (1,1) has a lower Akaike Information and Schwarz Criteria value, it indicates that the stock exchange market capitalization is volatile (up and down), implying that the stock market rises and falls according to economic conditions.

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