Modeling inflation rates and exchange rates in Ghana: application of multivariate GARCH models
- Ezekiel NN Nortey^{1}Email author,
- Delali D Ngoh^{1},
- Kwabena Doku-Amponsah^{1} and
- Kenneth Ofori-Boateng^{2}
https://doi.org/10.1186/s40064-015-0837-6
© Nortey et al.; licensee Springer. 2015
Received: 25 September 2014
Accepted: 19 January 2015
Published: 24 February 2015
Abstract
This paper was aimed at investigating the volatility and conditional relationship among inflation rates, exchange rates and interest rates as well as to construct a model using multivariate GARCH DCC and BEKK models using Ghana data from January 1990 to December 2013. The study revealed that the cumulative depreciation of the cedi to the US dollar from 1990 to 2013 is 7,010.2% and the yearly weighted depreciation of the cedi to the US dollar for the period is 20.4%. There was evidence that, the fact that inflation rate was stable, does not mean that exchange rates and interest rates are expected to be stable. Rather, when the cedi performs well on the forex, inflation rates and interest rates react positively and become stable in the long run. The BEKK model is robust to modelling and forecasting volatility of inflation rates, exchange rates and interest rates. The DCC model is robust to model the conditional and unconditional correlation among inflation rates, exchange rates and interest rates. The BEKK model, which forecasted high exchange rate volatility for the year 2014, is very robust for modelling the exchange rates in Ghana. The mean equation of the DCC model is also robust to forecast inflation rates in Ghana.
Keywords
Introduction
When the general level of prices is relatively stable, the uncertainties of time-related activities such as investment diminish. This helps to promote full employment and strong economic growth. When price stability is achieved and maintained, monetary policy makers have done their job well (Sobel et al. 2006). Conceivably, one of the most important responsibilities of every government is fostering a healthy economy, which benefits all her citizens. The government through its ability to tax, spend and control money supply, attempts to promote full employment, price stability and economic growth.
The importance of price stability is also emphasized in the Maastricht agreement, which defined the framework for a single European Currency, Euro, and identified price stability as the main objective of the new European Central Bank (McEachern 2006). Deflation could result to doom for an economy; that is, it weakens consumer demand for goods and services as households are likely not to spend, believing that prices will continue to fall. This means that businesses as well as government may be unable to pay debts and could result in retrenchment. Emphasizing this point is Lagarde; the Managing Director of the IMF, in April 2014 who cautioned the euro area that, a prolonged period of “low-inflation” or deflation can suppress demand and output, and overturn growth and jobs.
According to Goldberg and Knetter (1997) exchange rate pass-through is the percentage change in local currency import prices resulting from a one percent change in the exchange rate between the exporting and importing countries. Exchange rate pass-through therefore is the effect (positive or negative) of exchange rates on import and export prices, consumer prices or inflation, investments as well as trade volumes. Engel and Rogers (1996) established that crossing the US-Canada border can considerably raise relative price volatility and that exchange rate fluctuations explain about one-third of the volatility increase. That is US-Canada border is an important determinant of relative price volatility even after making due allowance for the role of distance. Parsley and Wei ( 2001) confirmed previous findings that crossing national borders adds significantly to price dispersion.
The demand for and supply of money are the key determinants of exchange rates. Interest Rate Parity is an important concept that explains the equilibrium state of the relationship between interest rate and exchange rate of two countries. The foreign exchange market is in equilibrium when deposits of all currencies offer the same expected rate of return. The condition that the expected returns on deposits of any two currencies are equal when measured in the same currency is called the interest parity condition. It implies that potential holders of foreign currency deposits view them all as equally desirable assets, provided their expected rates of return are the same. Given that the expected return on say US dollar deposits is 4 percent greater than that on Ghana cedi deposits, all things being equal, no one will be willing to continue holding Ghana cedi deposits, and holders of Ghana cedi deposits will be trying to sell them for US dollar deposits. There will therefore be an excess supply of Ghana cedi deposits and an excess demand for US dollar deposits in the foreign exchange market (Krugman et al. 2012).
An important theory of the relationship between inflation rate and interest rate is the Fisher effect; sometimes referred to as the Fisher hypothesis by Irvin Fisher. Fisher proved mathematically that the nominal interest rate is equal to the real interest rate minus the expected (predicted) inflation rate. The Fisher effect simply explains for example that; if the nominal interest rate is say 50 per cent for a given period, and the predicted inflation rate during that same period is 20 per cent, then the real interest rate is 30 per cent. The movement in short term interest rates primarily reflects fluctuation in expected inflation, which in effect has a predictive ability for future inflation (Mishkin and Simon 1995).
The primary objective of the Central Bank of Ghana is to maintain stability in the general level of prices (Bank of Ghana Act 2002). Price Stability is, therefore, one of the most important indicators of the health of a nation’s economy. It must be noted that price stability alone might not be enough for a healthy economy.
Several studies have been conducted on modelling inflation rates in Ghana, and majority of these used the constant variance assumption model. Although Mbeah-Baiden (2013) used non-constant variance models to model inflation rates in Ghana, his work only considered a univariate analysis of inflation rates. In the developed countries where a number of the researchers have modelled financial data series using Multivariate Generalized Autoregressive Conditional Heteroscedastic (MGARCH) models, none has modelled the co-movements of inflation rates, exchange rates and interest rates. The MGARCH models have not been explored enough on Ghanaian data and to a very large extent, Africa. It must be noted that Atta-Mensah and Bawumia (2003) used Vector Error Correction forecasting model for Ghana and concluded that growth rate, broad money supply (M2+) and depreciation of exchange rate are the main drivers of higher inflation.
The main objective of the study is to investigate the volatility and conditional relationship of inflation, exchange and interest rates and to construct a model using the multivariate GARCH BEKK (Baba, Engle, Kraft and Kroner) and DCC (Dynamic Conditional Correlation) models.
A researcher can apply all these models on data series and the best model is chosen based on the performance of the model using a criterion. According to (Doan: RATS Handbook for ARCH/GARCH and Volatility Models. pp: 38. Evanston, United States: Estima, Unpublished Draft Book), the application of BEKK and DCC in modelling the conditional variance generally achieved similar results and the difference is negligible.
Data and methodology
The monthly inflation rates, average monthly exchange rates (cedi to US dollar) and interest rates (lending rate to the public) in Ghana spanning the period January 1990 to December 2013 were used for the study. This means that a total of 288 data points were considered for each variable. The sources of data were the Ghana Statistical Service (GSS) and Ghana Commercial Bank (GCB).
The data were analyzed using multivariate GARCH, DCC and BEKK models. The procedure most often used in the model estimation involves the maximization of a likelihood function constructed on the assumption of independently and identically distributed standardized residuals.
According to Engle and Sheppard (2001), analyzing and understanding how the univariate GARCH works is fundamental for the study of the Dynamic Conditional Correlation multivariate GARCH model. The DCC model is a nonlinear combination of univariate GARCH and its matrix is based on how the univariate GARCH (1, 1) process works.
Suppose that the stochastic process \( {\left\{{x}_t\right\}}_t^T \) denotes the return during a specific time period, where x _{ t } is the return observed at time t. Assuming for instance that the model for a return is given as: x _{ t } = μ _{ t } + ε _{ t }, where μ _{ t } = Ε(x _{ t }/λ _{ t − 1}) denotes the conditional expectation of the return series, ε _{ t } is the condition error and λ _{ t − 1} = σ(x : s ≤ t − 1) represent the sigma field (information set) generated by the values of the return until time t - 1. Suppose that the conditional errors are conditional standard deviations of the returns \( {h}_t^{1/2}=Var{\left({x}_t/{\lambda}_{t-1}\right)}^{1/2} \) times is independent and identically normally distributed with zero mean and a unit variance stochastic variable y _{ t }. Note that h _{ t } and y _{ t } are independent for all time t, \( {\varepsilon}_t=\sqrt{h_t{y}_t}\sim N\left(0,{h}_t\right) \). Lastly, assume that the conditional expectation μ _{ t } = 0, which implies that \( {x}_t=\sqrt{h_t{y}_t} \) and x _{ t }/λ _{ t − 1} ∼ N(0, h _{ t }).
- (i)
ϕ - the weighted long run variance
- (ii)
\( \sum_{i=1}^p{\alpha}_i{\varepsilon}_{t-1}^2 \) - the moving average term, which is the sum of the p previous lags of squared-innovations multiplied by the assigned weight α _{ i } for each lagged square innovation
- (iii)
\( \sum_{i=1}^q{\beta}_i{h}_{t-1} \) - the autoregressive term, which is the sum of the q previous lagged variances multiplied by the assigned β _{ i } for each lagged variance.
Since the variance is non-negative by definition, the process \( {\left\{{h}_t\right\}}_{t=0}^{\infty } \) must also be non-negative valued.
Baba, Engle, Kraft and Kroner (BEKK) model
The BEKK model specified in equation (3) also has its diagonal form by assuming that the matrices A _{ kj } and B _{ kj } are diagonal. It is a restrictive version of the DVEC model. The most restricted version of the diagonal BEKK is the scalar BEKK one with A = aI and B = bI where α and b are scalars.
Estimation of the BEKK model still bears large computations due to several matrix transpositions. The number of parameters of a complete BEKK model is (p + q)KN ^{2} + N(N + 1)/2 whereas in the diagonal BEKK, the number of parameters reduces to (p + q)KN + N(N + 1)/2. The BEKK form is not linear in the parameters, which makes the convergence of the model difficult. However, the model structure automatically guarantees the positive definiteness of H _{ t }. Under the overall consideration, it is assumed that p = q = k = 1 in BEKK forms of application. The difference between the results of BEKK model and the DCC model is highly negligible.
The Dynamic Conditional Correlation (DCC) Model
To extend the assumptions in the univariate GARCH to multivariate case, suppose that we have n assets in a portfolio and the return vector is x _{ t } = (x _{1t }, x _{2t }, x _{3t }, …, x _{ nt }) '. Furthermore, assume that the conditional returns are normally distributed with zero mean and conditional covariance matrix \( {H}_t=E\left\{{x}_t{x}_t^{,}/{\lambda}_{t-1}\right\} \). This implies that \( {x}_t={H}_t^{1/2}{y}_t \) and x _{ t }/λ _{ t − 1} ∼ N(0, H _{ t }), z _{ t } = (z _{1t }, z _{2t }, z _{3t }, …, z _{ nt }) ' ∼ N(0, I _{ n }) and I _{ n } is the identity matrix of order n. \( {H}_t^{1/2} \) may be obtained by Cholesky decomposition of H _{ t }.
The dynamic structure defined above is the simplest multivariate GARCH called Scalar GARCH. A major caveat of this structure is the all correlations obey the same structure.
In this work, only the DCC(1,1) will be utilized.
Constraints of the DCC(1,1) Model
If the covariance matrix is not positive definite then it is impossible to invert the covariance matrix which is essential in portfolio optimization. To guarantee a positive definite H _{ t } for all t, simple conditions on the parameters are imposed. Firstly, the conditions for the univariate GARCH model has to be satisfied. Similar conditions on the dynamic correlations are also required, namely: β ≥ 0 and α ≥ 0, α + β < 1, Q _{0} has to be positive definite.
Estimation of the DCC(1,1) Model
The estimation follows the following two steps.
Step one
It is obvious that this quasi-likelihood function is the sum of the univariate GARCH log-likelihood functions. Therefore, one can use the algorithm to estimate parameter θ _{1} = (ϕ _{1}, α _{1}, β _{1}, ϕ _{2}, α _{2}, β _{2} …, ϕ _{ n }, α _{ n }, β _{ n }) for each univariate GARCH process. Since the variance h _{ it } for asset i = 1, 2, 3, … n is estimated for t ∈ {1, T}, then also the element of the D_{t} matrix under the same time period is estimated.
Step two
Variance targeting is used in the dynamic structure and therefore \( {\widehat{Q}}_0={\varepsilon}_0^{\hbox{'}}{\varepsilon}_0 \) and since the conditional correlation matrix also is the covariance matrix of the standardized residuals, \( {\widehat{X}}_0={\varepsilon}_0^{\hbox{'}}{\varepsilon}_0 \).
Results and discussion
Multivariate-GARCH modeling
Multivariate GARCH models are estimated by the quasi maximum likelihood technique. Regression Analysis of Time Series (RATS) 8.3 is a widely used software in estimating MGARCH models as a result of its flexible maximum likelihood estimation capabilities and has advantages over many other software packages on estimating MGARCH models. The optimization algorithm used for the maximum likelihood estimation in RATS is BFGS; proposed independently by Broyden (1970), Fletcher (1970), Goldfarb (1970) and Shanno (1970), (Estima 2013). This optimization algorithm uses iteration routines to obtain the coefficient estimation. As such, convergence is assumed to occur if the change in the coefficient to be estimated is less than the criterion option 0.00001 specified. RATS was used in estimating the MGARCH models for this study.
Test of multivariate ARCH effect and serial correlation of DCC and BEKK
DCC | BEKK | |
---|---|---|
Lag | 10 | 10 |
D.F | 360 | 360 |
Stats | 291.6 | 303.07 |
P-value | 0.99659 | 0.98679 |
DCC model
Seven months of 2014 out-sample forecast of inflation rate from the mean equation of the DCC model
Month/Year | Observed (%) | Forecast (%) | Error (%) |
---|---|---|---|
Jan-2014 | 13.80 | 13.82 | 0.02 |
Feb-2014 | 14.00 | 13.70 | −0.30 |
Mar-2014 | 14.50 | 13.58 | −0.92 |
Apr-2014 | 14.70 | 13.46 | −1.24 |
May-2014 | 14.80 | 13.22 | −1.58 |
Jun-2014 | 15.00 | 13.10 | −1.90 |
Jul-2014 | 15.30 | 12.98 | −2.32 |
Aug-2014 | 15.90 | 12.86 | −3.04 |
BEKK model
Certainly, it is evident that the BEKK model is robust in modeling volatility in the depreciation of the cedi to other foreign currencies.
Vector error correction model and granger causality
Johansen test of cointegration among the variables using STATA 12
Maximum rank | Eigen values | Trace statistics | 5% critical value |
---|---|---|---|
0 | 29.73 | 29.68 | |
1 | 0.06181 | 11.6746 | 15.41 |
2 | 0.03089 | 2.7963 | 3.76 |
3 | 0.00983 |
The Vector Correction Model evidence long run and short run causality among the variables after the null hypothesis of both “no long run causality and no short run causality” were rejected. After a pair-wise Granger-causality tests at 5% significant level, the result show that, exchange rate Granger-cause inflation rate but the converse does not. Similarly, inflation rate Granger cause interest rate but the reverse does not.
Conclusions
Multivariate GARCH, DCC and BEKK models were fitted to the variances of the data. Both models passed the diagnostic test. The mean equation of the DCC model was used to predict the expected inflation rate and proved to be robust in the short to medium term, similarly, the BEKK model was used to predict the expected exchange rate volatility.
These predictions suggest that, inflation rates are expected to increase at a very slow rate in 2014. Also, the forecast of exchange rate volatility suggested that, there is a very high risk of abrupt depreciation of the cedi to the US dollar. This implies that the rates of inflation as well as interest rates are likely to react in the long run to the expected volatility in exchange rate for the year 2014.
There was generally positive conditional and unconditional correlation between inflation rates and exchange rates, inflation rates and interest rates as well as exchange rates and interest rates. This implies that there is some evidence that when the general prices of goods and services are stable, interest rates are expected to be stable and possibly low. That of inflation and exchange rates implies that the stability of inflation means that the cedi depreciated to the dollar at low rate. There was evidence that the cedi has depreciated cumulatively to the US dollar of 7010.02% from 1990 to 2013 with a weighted annual average depreciation of 20.4%.
The volatility experienced in inflation, exchange and interest rates in the study, to a large extent were not in elections year. It is therefore factually inaccurate to assert that during election years, the cedi depreciates faster to the US dollar. The evidence rather suggests, there seem to be volatility in these economic variables, periods after elections were held rather than during elections year and also during the first quarter of every year.
It was also evident that, the fact that inflation rates were stable, does not mean that exchange rates and interest rates are expected to be stable. Rather, when the cedi performs well on the forex, inflation and interest rates react positively in the long run. All things being equal, this reaction tickles down to all aspects of the economy thus, occasioning improved standards of living.
The economy of Ghana reacts positively in most instances when the cedi performs strongly on the forex market. Such performance was evidenced in 2003 when the cedi depreciated to the US dollar at an average of 3.81%, during that same year the Ghana Stock Exchange recorded returns on investments of about 155%, the highest since its inception. The success of the cedi during this year could be traced to foreign inflows of HIPC benefits into the country. This implies that the health of the economy of Ghana is highly dependent on the strength of the cedi against foreign currencies such as the US dollar, Euro and the British pound sterling.
Recommendations
Recommendations are made for both policy formulation and areas of further research based on the findings of the study.
To begin with, it is recommended that policy makers use multivariate GARCH models to study the dynamics of economic and financial data. The DCC model proved to be robust in modeling the correlation among inflation, exchange and interest rates, and the mean equation of the model was robust for modelling inflation rates in the short to medium term. Similarly, the BEKK model was found to be robust in modeling volatility as well as forecasting.
Secondly, the research work has revealed that, the health of Ghana’s economy is highly dependent on the strength of the Ghanaian currency: cedi against the foreign currencies since the country is import dependent, as such there must be a national agenda to increase foreign inflows and introduce a policy aimed at Exchange Rate Targeting (ERT). The forecast is also an indication that policy makers and industry players can effectively plan to curb uncertainties in the Ghanaian economy given these models are used.
Thirdly, there must be a national consensus to reduce imports into the country by improving production and in the long run increase non-traditional exports. The government could adopt a policy through consensus with private sectors (services) to list on the Ghana Stock Exchange to attract Ghanaians to own shares, tax incentives could be used as a stimulus package. This is to ensure that 100% of the profit is not repatriated. Government could also dialogue with the private sector and propose a policy that mandates foreign owned companies to delay about 50% repatriation of their profit in the economy of Ghana for about two years. Government must also adopt a policy to reduce the number of State delegations to international events abroad to about 20%, this could also reduce the pressure on the Ghanaian cedi.
Lastly, a study into the dynamics of interest rates, stock returns and exchange rates is recommended. Other economic indicators such as money supply, balance of payment and budget deficit could be added to inflation rate, exchange rate and interest for modelling using multivariate GARCH models. Modelling the volatility in the five most traded currencies in Ghana is also recommended. Impulse analysis of inflation rates, exchange rates and interest rates is suggested as well.
Declarations
Authors’ Affiliations
References
- Atta-Mensah J, Bawumia M (2003) A Simple vector error correction forecasting Model for Ghana. Bank of Ghana Working Paper SWP/BOG-2003/01, Accra, GhanaGoogle Scholar
- Bank of Ghana Act (2002). Thursday, 05 June 2014. Retrieved from www.bog.gov.gh/privatecontent/Banking/Banking_Acts/bank%20of%20ghana%20act%202002%20act%20612.pdf
- Bank of Ghana (2014). Thursday, 05 June 2014. Retrieved from www.bog.gov.gh/index.php?option=com_wrapper&view=wrapper&Itemid=298
- Bolleslev T (1986) Generalized autoregressive conditional heteroscedasticity. J Econ 31(3):307–327View ArticleGoogle Scholar
- Broyden CG (1970) The convergence of a class of double rank minimization algorithms to the new algorithm. J Inst Math Appl 6:222–231View ArticleGoogle Scholar
- Engel C, Rogers JH (1996) How wide is the border? Am Econ Rev 86(5):1112–1125Google Scholar
- Engle R, Sheppard K (2001) Theoretical and Empirical Properties of Dynamic Conditional Correlation Multivariate GARCH. UCSD, MimeoView ArticleGoogle Scholar
- Estima (2013) Regression Analysis of Time Series Version 8.2 User Guide, p: UG-115. Estima, Evanston, United StatesGoogle Scholar
- Fletcher R (1970) A new approach to variable metric algorithms. Computer J 13:317–322View ArticleGoogle Scholar
- Goldberg PK, Knetter MM (1997) Goods prices and exchange rates. J Econ Lit 35(3):1243–1272Google Scholar
- Goldfarb D (1970) A family of variable metric methods derived by variational means. Math Comp 24:23–26View ArticleGoogle Scholar
- Krugman PK, Obstfeld M, Melitz MJ (2012) Harvard University International Economics Theory & Policy. Addison-Wesley, Boston, USA, p 337Google Scholar
- Mbeah-Baiden B (2013) Modeling Inflation in Ghana; An Application of Autoregressive Conditional Heteroscedastic (ARCH) type model. (Unpublished M.Phil thesis). Department of Statistics, University of Ghana, AccraGoogle Scholar
- McEachern WA (2006) Microeconomics – A Contemporary Introduction. Thompson South-Western, Mason, USA, pp 55–336Google Scholar
- Mishkin FS, Simon J (1995) An Empirical Examination of the Fisher Effect in Australia. National Bureau of Economic Research, Massachusetts, Working Paper number 5080View ArticleGoogle Scholar
- Parsley DC, Wei S-J (2001) Explaining the border effect: the role of exchange rate variability, shipping costs, and geography. J Int Econ 55:87–105View ArticleGoogle Scholar
- Shanno DF (1970) Conditioning of quasi-Newton methods for function minimisation. Math Comp 24:647–650View ArticleGoogle Scholar
- Sobel RS, Stroup RL, Macpherson DA, Gwartney JD (2006) Understanding Economics. Thompson South-Western, Mason, USA, p 343Google Scholar
Copyright
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.