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On Evaluating the Volatility of Nigerian Gross Domestic Product Using Smooth Transition Autoregressive-GARCH (STAR - GARCH) Models

Akintunde Mutairu Oyewale

Department of Statistics, Federal Polytechnic, Ede, Osun State, Nigeria

Abstract: STAR-GARCH models are hybrid models that combine the functional form of smooth transition autoregressive models and Generalized autoregressive conditional heteroscedasticity models. The two classes of STAR models considered in this paper are Exponential and Logistic Smooth transition autoregressive models (ESTAR and LSTAR). The functional form of each of this was combined with that of GARCH model and the resulting models becomes ESTAR-GARCH and LSTAR-GARCH models. The derived equations were applied to Nigerian gross domestic product (Real estate) for empirical illustration. Statonarity tests (Unit root test Graphical and correlogrom methods) conducted revealed that the series was stationary at Second difference. The hybrid models equations so derived were used to determine the model that performed better using the information criteria (AIC, SIC and HQIC), variances obtained from the data, performance measure indices (RMSE, MAE, MAPE THEIL U, Bias proportion, variance Bias proportion and covariance Bias proportion) analysis and in - sample forecast accuracy for the models. From all the criteria used it was observed that the duo of LSTAR-GARCH and ESTAR-GARCH models performed far better than classical GARCH model. However, LSTAR-GARCH performs slightly better than ESTAR-GARCH. From these results it is evident that volatility in Nigerian gross domestic product (Real estate) is best captured using Logistic smooth transition GARCH (LSTAR-GARCH) models, it is therefore, recommended for would be forecasters, investors and other end users to make use of LSTAR-GARCH models.

Keywords: GARCH model, STAR model, ESTAR-GARCH, LSTAR-GARCH, performance measure indices, volatility

Received August 28, 2021; Revised October 02, 2021; Accepted October 11, 2021

Citation: Akintunde Mutairu Oyewale, On Evaluating the Volatility of Nigerian Gross Domestic Product Using Smooth Transition Autoregressive-GARCH (STAR - GARCH) Models. American Journal of Applied Mathematics and Statistics, 9 (3). 102-106. January 2021. doi:10.12691/ajams-9-3-4

1. Introduction

Nigeria is regarded as the largest economy in West Africa and is currently at per or has surpassed South Africa as the largest economy in Africa (previously regarded as the second largest economy in Africa), African Development Bank, [1] asserted that Nigerian economy is believed to be about 55 percent of West Africa’s Gross domestic product so also Ignite, [2] said Nigeria economy accounted for 64 percent of GDP using purchasing power parity (PPP) indicators of the fifteen member countries in the ECOWAS sub-region. The country projected population is about 200 million people of highly industrious inhabitants. Osagie, [3] said Nigeria natural resources made up of over 80 million hectares of arable land, several solid minerals resources and abundant crude oil and gas reserves.

Nwachukwu V.O. [4] refers to the gross domestic product (GDP) asa measure of national income and output of a particular country's economy. The gross domestic product (GDP) is equivalent to the sum total of all expenditures for all final goods and services produced within the country’s under study at a given period of time. In 2019, according to the World Bank official statistics, the Gross Domestic Product (GDP) of Nigeria was valued to be 448.10 billion US dollar, accidentally the above quoted value of GDP is equivalent to 0.37 percent of the total world economy.

Many financial and economic time series data in most cases are noisy, chaotic and volatile because of these behaviors the markets show marked breaks in their characteristics, in a way that the series changes markedly compared to what they exhibited initially. The changes that come along the features could be temporary (If the changes are for a while before returning to its original behaviour or switching to yet another style of behaviour, this is termed a regime shift or regime switch) but if such characteristics become permanent then the problem of structural break is then established. Such features are associated with events such as financial crises [5,6,7] or sudden changes in government policy [8,9,10]. Of interest to financial econometrician or time series practitioners is the seeming behaviour of many economic variables to change differently during economic recession and posterity [11,12]. These unexpected changes are common features of financial and economic data.

Smooth Transition Autoregressive (STAR) models are applied to time series data as an extension of autoregressive models in order to allow for higher degree of easiness in model parameters through a smooth transition. Also, STAR models are introduced, according to Terasvirta and Anderson [13] and Terasvirta [14], because of the existence of two distinct regimes with potentially different dynamic properties and because the transition between the regimes is smooth. STAR models allow economic variables to follow a given number of regimes with switches between regimes achieved in a smooth and continuous fashion and governed by the value of a particular variable or group of variables. The transition parameter denoted by is a slope of parameter that determines the speed of transition between the two extreme regimes with low absolute values resulting in slower transition. It should be noted that are generated by data series.

The model presented in this study is inspired by the paper published by Akintunde et.al. [15] in which STAR-GARCH models was used to forecast exchange rate data of Nigeria, Botswana, Britain and Japan using American dollar as a benchmark. The results were related to GARCH models.

2. Mathematical Specification

The GARCH model used for the study is represented by

[picture] | (1)

The general STAR model can be represented as:

[picture] | (2)

where is the error term distributed independently and identically with mean zero and variance one. is the transition function bounded between zero and unity (0,1).

The Logistic Smooth Transition (LSTAR) model is as follows:

[picture] | (3)

The Exponential Smooth Transition (ESTAR) model is as follows:

[picture] | (4)

Based on the conditions stated above, the STAR model offers the possibility to investigate the presence of non-linearity in time series data which may account for the weakness of GARCH model mentioned in Chapter Four. Without loss of generality, we can strengthen the GARCH model with STAR models by adjusting the error terms.

The LSTAR-GARCH model is

[picture] | (5)

The ESTAR-GARCH model is

[picture] | (6)

Generally, the STAR-GARCH model used for the study is of the form

[picture] | (7)

3. Empirical Analysis with Nigerian Gross Domestic Product Data

Data was analysed using Econometrics view software and set of programme was written to accomplish the goal using monthly Nigerian Gross domestic product data from 1997 to 2019. The data was obtained from National Bureau of Statistics website. The results obtained from the analysis are shown below.

Figure 1. DESCRIPTIVE STATISTICS

Figure 1 above shows the distributional properties of the data used in the study. There is evidence of fluctuation in data as shown by the high standard deviation obtained. The distribution of the data is almost symmetric. The data is platykurtic in nature (1.7 < 3). The hypothesis of normality is rejected as evidence by Jarque-Bera test shown above. So before proceeding to the analysis of the data there is the need to make the data stationary.

Table 1. The GARCH model fitted for the series

The general form of our STAR-GARCH is

ESTAR-GARCH

with var. of 3.2435

LSTAR-GARCH

with var. 2.1311

FITTED VALUES FOR STAR-GARCH MODELS

Table 2. Fitted model for STAR-GARCH

3.2. Empirical Comparison of the Models

The models under consideration are GARCH and ESTAR-GARCH and LSTAR-GARCH models. Table 3 below shows the variances of models as obtained from the analysis. The performance of ESTAR-GARCH and LSTAR-GARCH is comparable (3.2435 for ESTAR- GARCH and 2.1311 for LSTAR- GARCH), however the two models performed far better than classical GARCH model with variance 9.7312. This implies that to would be analyist, investors and other would-be users LSTAR model is recommended, this is closely followed by ESTAR-GARCH. The policy implication of this is that the users can make use of ESTAR-GARCH in the absence of LSTAR-GARCH, while the performance of classical is nothing to write home about.

Table 3. Variances of all models

From Table 3 above, the variances obtained for the three models were considered. LSTAR-GARCH produced the least variance, making it the best variance wise closely followed by ESTAR-GARCH and GARCH models in that order. Variance is a measure of error, and the model that has the least is preferred over and above the model(s) with higher error.

Table 4. Information criteria of all models

Table 4 above revealed the performance of the three models on the basis of information criteria used. Considering the first column which is Akaike information criteria (AIC), LSTAR-GARCH produced the best information closely followed by ESTAR-GARCH. Column 2 is the column of Schwarz information criteria, here also LSTAR-GARCH produced the best results followed by ESTAR-GARCH and GARCH models in that order. The last column is the column of Hannan-Quinn information criteria, here ESTAR-GARCH had the best results followed by LSTAR-GARCH and GARCH models in that order. In all the three criteria used LSTAR-GARCH performed best in two (AIC and SIC) while ESTAR-GARCH model performed best in one (HQIC). It should be noted that the asterisk sign used here shows the column by column of the model that performed best using information criteria.

Table 5. Performance measure indices

Table 5 is a table of performance measure indices, in all seven criteria were used out of which LSTAR-GARCH models is best in five, ESTAR-GARCH is best in two. The asterisk is used to denote the best performance measure indices per column as shown above. it could be remarked clearly that if the performance measure indices as criteria were used there is no gainsaying in the fact that LSTAR-GARCH is best and it is followed by ESTAR-GARCH.

3.3. Forecast Efficacy

The in-samples forecast performance for STAR-GARCH models (lower and upper forecast limits) for all models under show that ESTAR-GARCH and LSTAR-GARCH models forecast excellently well as shown in the Table 7 and Table 8 below. It is observed that the actual values of data obtained from record are at par with the upper forecast values. The implication of this is that, its performance asserts its superiority LSTAR-GARCH over and above GARCH, and ESTAR-GARCH models, as could be seen from 6 through 8. However, the performance of ESTAR-GARCH is very close to that of LSTAR-GARCH.

Table 6. In -samples forecast performance for GARCH

Table 7. In -samples forecast performance for ESTAR-GARCH

Table 8. In-samples forecast performance for LSTAR-GARCH

4. Conclusion

From Table 5 through Table 7, it is clear that the STAR-GARCH models are better than the classical GARCH model for all models in terms of their variances compared to classical GARCH models. However, within the group of STAR-GARCH, LSTAR-GARCH performed excellently better than others. This is closely followed by ESTAR-GARCH and LSTAR-GARCH in that order. Also, the forecast efficacy of STAR-GARCH model could not be compared with classical GARCH, as its forecast is excellent and could be compared to the original data.

References

[1]   | African Development Bank (2013). Country Strategy Paper 2013-2017.
[2]   | Ignite (2013). Fiscal Management Reform Gaining Traction in Nigeria Again.
[3]   | Osagie, C. (2011). FG Blames Sluggish GDP Growth on Poor Management, This Day, 25 October.
[4]   | Nwachukwu V.O. (2008). Principles of Statistical Infrence. Zelon Enterprises, Port-Harcourt.
[5]   | Jeanne, O. and Masson, P. (2000). “Currency Crises, Sunspots, and Markov-Switching Regimes,” Journal of International Economics 50, 327-350.
[6]   | Cerra, Valerie, and Sweta Chaman Saxena (2005). “Did Output Recover from the Asian Crisis?” IMF Staff Papers 52, 1-23.
[7]   | Hamilton, J.D. (2005). What’s Real About the Business Cycle? NBER Working Paper, w11161.
[8]   | Hamilton, James D. (1988). “Rational-Expectations Econometric Analysis of Changes in Regime: An Investigation of the Term Structure of Interest Rates,” Journal of Economic Dynamics and Control 12, 385-423.
[9]   | Sims, C., Zha, T., (2006). Were there regime switches in US monetary policy? American Economic Review 96, 54-81.
[10]   | Davig, Troy (2004). “Regime-Switching Debt and Taxation,” Journal of Monetary Economics. 51, 837-859.
[11]   | Hamilton J.D., (1989). A new approach to the economic analysis of non-stationary time series and the business cycle. Econometrica, 57(2): 357-384.
[12]   | Chauvet, Marcelle, and James D. Hamilton (2005). “Dating Business Cycle Turning Points,” in Nonlinear Analysis of Business Cycles, edited by Costas Milas, Philip Rothman, and Dick van Dijk.
[13]   | Teräsvirta, T.and Anderson, H., (1992). Characterizing nonlinearities in business cycles using smooth transition autoregressive models. Journal of Applied Econometrics, 7: S119-S136.
[14]   | Teräsvirta, T., (2006). Univariate nonlinear time series models. In K. Patterson and T. Mills, editors, Palgrave Handbook of Econometrics, volume I. Palgrave Macmillan, New York.
[15]   | Akintunde, M. O., Shangodoyin, D. K. and Kgosi, P.M. (2013). Smooth Transition Autoregressive-GARCH Model in Forecasting Non-linear Economic Time Series Data. Journal of Statistical and Econometric Methods, vol. 2, no.2, 11-19.