Modeling variations in the cedi/dollar exchange rate in Ghana: an autoregressive conditional heteroscedastic (ARCH) models
- Michael Techie Quaicoe^{1},
- Frank B K Twenefour^{2}Email authorView ORCID ID profile,
- Emmanuel M Baah^{2} and
- Ezekiel N N Nortey^{3}
Received: 25 May 2015
Accepted: 26 June 2015
Published: 8 July 2015
Abstract
This research article aimed at modeling the variations in the dollar/cedi exchange rate. It examines the applicability of a range of ARCH/GARCH specifications for modeling volatility of the series. The variants considered include the ARMA, GARCH, IGARCH, EGARCH and M-GARCH specifications. The results show that the series was non stationary which resulted from the presence of a unit root in it. The ARMA (1, 1) was found to be the most suitable model for the conditional mean. From the Box–Ljung test statistics x-squared of 1476.338 with p value 0.00217 for squared returns and 16.918 with 0.0153 p values for squared residuals, the null hypothesis of no ARCH effect was rejected at 5% significance level indicating the presence of an ARCH effect in the series. ARMA (1, 1) + GARCH (1, 1) which has all parameters significant was found to be the most suitable model for the conditional mean with conditional variance, thus showing adequacy in describing the conditional mean with variance of the return series at 5% significant level. A 24 months forecast for the mean actual exchange rates and mean returns from January, 2013 to December, 2014 made also showed that the fitted model is appropriate for the data and a depreciating trend of the cedi against the dollar for forecasted period respectively.
Keywords
ARMA GARCH ACF PACF and ARCH effect Conditional mean with variance Dollar/cedi exchange rate Forecasting Time series modelsBackground
Time series models play an important role in the financial market by describing the underlying structure of an economics variable. With available data for financial market analyses in recent times, there has been an increase in the studies concerning persistent shocks both in the mean as well as the variance of the returns of financial instruments in the market. Many time series especially those occurring in natural sciences and engineering cannot be modeled by linear processes. These kinds of time series can have trends which can be modeled by nonlinear processes.
The particular type of non-linear model that is used in finance is known as the autoregressive conditional heteroscedastic—ARCH model (Engle 1982). In the application of financial time series where the variance of the error term is very unlikely to be constant over time, ARCH models are used to describe the behavior of the volatility structure of the error term. They are employed commonly in modeling the volatility structure of financial data and financial indices in order to identify similarities and differences in the structure of the variance of the error term of the observed series.
The basic underlying assumption of the least squares model is that the expected value of all error terms, when squared, is the same at any given point. This assumption is called homoscedasticity (Engle 2001). In volatility analysis however, the variance of the residuals depends on past history and we face heteroscedasticity because the variance is changing over time. A basic means of dealing with heteroscedasticity is to have the variance depending on the lagged period of the squared error terms. ARCH models allows the conditional variance to be dependent upon its own previous lags. In the GARCH (p, q) model (Bollerslev 1986), the conditional variance is dependent upon q lags of the squared error term and p lags of the conditional variance which is very effective to capture the volatility nature of data in financial time series. ARCH and GARCH models treat heteroscedasticity as a variance to be modeled and are most often used in financial theory and practice.
In this article, the authors’ main idea is to use the ARCH/GARCH Specification for modeling the volatility structure of the monthly exchange rate of the Cedi and the US dollar by explaining the volatility structure of the residuals obtained under the best suited mean model for the observed series. This study is significant since the exchange rate of a currency is essential in determining: the level of imports and exports as companies/institutions that rely on import/export can estimate the cost of these import/export with respect to variations in the exchange rate; the country’s level of business activities, Gross Domestic Product (GDP) and employment level and the purchasing power of a local currency i.e. to see whether the currency is appreciating or depreciating against other foreign currencies.
Analysis of results and discussion
Exchange rate distribution
Determining order of dependency of returns series
ARMA (1, 1) model parameter estimates
Variable | Coefficient | Standard error | T-statistics | Probability |
---|---|---|---|---|
AR (1) | 0.999851 | 0.004673 | 213.985 | 2.00E−16 |
MA (1) | 0.518176 | 0.077013 | 6.728 | 1.72E−11 |
Test for ARCH effect
To eliminate the ARCH effect, GARCH (1, 1) was found to be the most suitable model for the conditional variance, with standardized residual test showing no serial correlation in standardized squared residual at different lags indicating that it is adequate in describing the dynamic volatility of the return series.
GARCH (1, 1) model’s parameter estimates
Variable | Coefficients | Standard error | T-statistic | Probability |
---|---|---|---|---|
Mean | 0.001923 | 0.0004442 | 4.329 | 0.000015 |
Omega | 0.00001427 | 0.000004939 | 2.89 | 0.003843 |
Alpha | 1 | 0.2271 | 4.403 | 0.0000107 |
Beta | 0.2711 | 0.02162 | 3.785 | 0.000154 |
Standardized residual test for GARCH (1, 1)
Standardized residual test of GARCH (1, 1)
Residual test | Variable | Test statistic | Test value | Probability |
---|---|---|---|---|
Jarque–Bera | R | \(\chi^{2}\) | 152.9664 | 0 |
Shapiro–Wilk | R | W | 0.8465621 | 5.965E−12 |
Ljung–Box | R | \(Q(10)\) | 23.50554 | 0.009026764 |
Ljung–Box | R | \(Q(15)\) | 27.92452 | 0.022046764 |
Ljung–Box | R | \(Q(20)\) | 31.84835 | 0.0449413 |
Ljung–Box | R | \(Q(10)\) | 5.126632 | 0.8825598 |
Ljung–Box | R | \(Q(15)\) | 9.760762 | 0.8345127 |
Ljung–Box | R | \(Q(20)\) | 11.83056 | 0.9217843 |
LM Arch | R | \(TR^{2}\) | 4.663342 | 0.9682894 |
Model diagnostics of GARCH (1, 1)
Returns and variance equation
ARMA (1, 1) + GARCH (1, 1) model’s parameter estimates
Variable | Coefficients | Standard error | T-statistic | Probability |
---|---|---|---|---|
AR (1) | 0.8097 | 0.03542 | 22.862 | 2.00E−16 |
MA (1) | −0.5749 | 0.107 | −5.194 | 2.06E−07 |
Omega | 0.00002507 | 0.000006437 | 3.894 | 9.85E−05 |
Alpha | 1.00000 | 0.12798 | 3.574 | 0.000352 |
Beta 1 | 0.1984 | 0.09501 | 2.088 | 0.03688 |
Standardized residual test of ARMA (1, 1) + GARCH (1, 1)
Standardized residual test of ARMA (1, 1) + GARCH (1, 1)
Residual test | Variable | Test statistic | Test value | Probability |
---|---|---|---|---|
Jarque–Bera | R | \(\chi^{2}\) | 170.0431 | 0 |
Shapiro–Wilk | R | W | 0.8750428 | 1.368657E−10 |
Ljung–Box | R | \(Q(10)\) | 8.937746 | 0.5380217 |
Ljung–Box | R | \(Q(15)\) | 16.65662 | 0.3398028 |
Ljung–Box | R | \(Q(20)\) | 20.21719 | 0.4444191 |
Ljung–Box | R ^{2} | \(Q(10)\) | 6.394848 | 0.7810711 |
Ljung–Box | R ^{2} | \(Q(15)\) | 8.093766 | 0.9199685 |
Ljung–Box | R ^{2} | \(Q(20)\) | 10.61964 | 0.9554965 |
LM ARCH | R | \(TR^{2}\) | 9.613653 | 0.6498134 |
Information criteria statistics of ARMA (1,1) +GARCH (1,1)
AIC | BIC | SIC | HQIC |
---|---|---|---|
−6.319293 | −6.225940 | −6.321017 | −6.281403 |
Model validation
Mean forecast of actual exchange rates for 2013/2014
Year (2013) | Actual rates | Forecasted rates | Year (2014) | Actual rates | Forecasted rates |
---|---|---|---|---|---|
January | 1.88 | 1.89 | January | 2.40 | 2.00 |
February | 1.88 | 1.90 | February | 2.52 | 2.00 |
March | 1.89 | 1.90 | March | 2.68 | 2.00 |
April | 1.90 | 1.90 | April | 2.80 | 2.00 |
May | 1.92 | 1.90 | May | 2.90 | 2.00 |
June | 1.95 | 1.95 | June | 3.00 | 2.00 |
July | 1.95 | 1.95 | July | 3.03 | 2.00 |
August | 1.95 | 1.98 | August | – | 2.10 |
September | 1.96 | 1.98 | September | – | 2.35 |
October | 1.99 | 1.99 | October | – | 3.00 |
November | 2.06 | 2.01 | November | – | 3.08 |
December | 2.00 | 1.99 | December | – | 3.42 |
Model diagnostic of conditional returns with conditional variance
Prediction of next 24 observations of mean returns
Mean forecast of returns for 2014/2015
Mean forecast | Mean error | Stand. dev. | 95% confidence interval | |
---|---|---|---|---|
Lower limit | Lower limit | |||
−0.00036229 | 0.00545061 | 0.00545061 | −0.01032071 | 0.01104529 |
−0.00030768 | 0.0066149 | 0.00646497 | −0.0126573 | 0.01327265 |
−0.0002613 | 0.00753796 | 0.00725586 | −0.01451283 | 0.01503543 |
−0.00022191 | 0.00829912 | 0.00789875 | −0.01604408 | 0.01648789 |
−0.00018845 | 0.00894054 | 0.0084343 | −0.01733467 | 0.01771158 |
−0.00016005 | 0.00948834 | 0.00888767 | −0.01843694 | 0.01875703 |
−0.00013592 | 0.00996082 | 0.00927586 | −0.01938693 | 0.01965877 |
−0.00011343 | 0.01037087 | 0.00961104 | −0.0202111 | 0.02044196 |
−0.00009803 | 0.01072863 | 0.0099023 | −0.02092969 | 0.02112575 |
−0.00008325 | 0.01104203 | 0.01015669 | −0.02155874 | 0.02172524 |
−0.0000707 | 0.01131748 | 0.01037977 | −0.02211116 | 0.02225256 |
−0.00006004 | 0.01156023 | 0.01057605 | −0.02259758 | 0.02271767 |
−0.00005099 | 0.01177463 | 0.01074922 | −0.02302686 | 0.02312885 |
−0.00004331 | 0.01196438 | 0.01090236 | −0.02340644 | 0.02349305 |
−0.00003678 | 0.01213257 | 0.01103803 | −0.02374262 | 0.02381618 |
−0.00003423 | 0.01228188 | 0.01115844 | −0.02404081 | 0.02410327 |
−0.00002653 | 0.01241459 | 0.01126544 | −0.02430562 | 0.02435867 |
−0.00002253 | 0.01253267 | 0.01136065 | −0.02454106 | 0.02458611 |
−0.00001913 | 0.01263785 | 0.01144546 | −0.02475059 | 0.02478885 |
−0.00001625 | 0.0127316 | 0.01152107 | −0.02493724 | 0.02496973 |
−0.0000138 | 0.01281524 | 0.01158854 | −0.02510362 | 0.02513121 |
−0.00001172 | 0.01288991 | 0.01164878 | −0.02525204 | 0.02527547 |
−0.00000995 | 0.0129566 | 0.0117026 | −0.02538453 | 0.02540443 |
−0.00000845 | 0.01301621 | 0.0117507 | −0.02550285 | 0.02551976 |
Conclusions
The observed series does not change over time, thus showing that the series is stationary; hence the probability law that governs the behavior of the process does not change over time. The distribution of the return series is not normal with non-constant variance skewed to the right. The model that explains the stochastic mechanism of the observed series is ARMA (1, 1) + GARCH (1, 1). That is the optimum model for the dollar/cedi exchange rate returns (conditional mean with non-constant variance). The time series components found in the model were trend and random variation. It was also observed that, the ARMA (1, 1) + GARCH (1, 1) fitted is adequate for treating the series’ heteroscedasticity by modeling the variations in the series which is the main objective of this study. The forecasts were found to have upward trend for the 2 years (2014/2015) period, indicating that the cedi will continue to depreciate against the dollar for the forecasted period.
Recommendations
Based on the findings, it is recommended that; there should be an increase in the supply of foreign currencies into the local market and absorbing some of the excess liquidity from the economy. The government should consider the volatility of other macroeconomic variables in making both policy and investment decision. The fiscal and monetary policies adopted should be revised to help address the depreciation of the cedi.
Methods
This paper employs a model based on information and real data obtained from the Bank of Ghana and the Ghana Stock Exchange. The samples include monthly data excluding weekends and holidays, of the average nominal exchange rate of the Ghana cedi/the US dollar, for the period January, 2000 to December 2013, comprising 168 data points. ARCH type models of Time Series Analysis and the statistical computing package R were used for the modeling. The stationarity of data is usually described by the time plots and the correlogram. The unit root test determines whether a time series is stable around its level or stable around the difference in its level (Dickey–Fuller or the Augmented Dickey–Fuller root test).
ARMA (p, q) model
The polynomial \(1 - \phi_{1} \beta - \cdots - \phi_{p} \beta^{p}\) is the AR polynomial of the model. Similarly, \(1 - \theta_{1} \beta \cdots - \theta \beta^{q}\) is the MA polynomial. It is required that there are no common factors between the AR and MA polynomials; otherwise the order of the model can be reduced (Noh, Engle and Kane 1994; DeLurgio 1998; Keller and Warrack 2003).
ARCH (1) model
The first and simplest heteroscedastic model we will look at is the ARCH model, which stands for Autoregressive Conditional Heteroscedasticity. The AR comes from the fact that these models are autoregressive models in squared returns. The conditional comes from the fact that in these models, next period’s volatility is conditional on information this period. Heteroscedasticity means non constant volatility. In a standard linear regression where \(y_{i} = \alpha + \beta x_{i} + \varepsilon_{i}\), when the variance of the residuals, \(\varepsilon_{i}\) is constant, we call that homoscedastic and use ordinary least squares to estimate \(\alpha\) and \(\beta\). If, on the other hand, the variance of the residuals is not constant, we call that heteroscedastic and we can use weighted least squares to estimate the regression coefficients (Noh et al. 1994; DeLurgio 1998; Keller and Warrack 2003).
Let us assume that the return on an asset is \(r_{t} = \mu + \sigma_{t} \varepsilon_{t}\) where \(\varepsilon_{t}\) is a sequence of N (0, 1) i.i.d. random variables. We will define the residual return at time t, \(r_{t} - \mu\), as \(a_{t} = \sigma_{t} \varepsilon_{t} .\) In an ARCH (1) model, first developed by Engle (1982), \(\alpha_{t} = \alpha_{0} + \alpha_{1} a_{t - 1}^{2}\), where \(\alpha_{0}\) >0 and \(\alpha_{1} \ge 0\) to ensure positive variance and 1 < 1 for stationarity.
Under an ARCH (1) model, if the residual return, \(a_{t}\) is large in magnitude, our forecast for next period’s conditional volatility, \(\sigma_{t + 1}\) will be large. We say that in this model, the returns are conditionally normal (conditional on all information up to time \(t - 1\), the one period returns are normally distributed). Also, note that the returns; \(r_{t}\) are uncorrelated but are not identically independent distributed (Engle 1982, 2001).
This is an AR (1) process on squared residuals.
GARCH (1, 1) model
In this way, it is easy to see that next period’s conditional variance is a weighted combination of the unconditional variance of returns, \(E\left[ {\sigma^{2} } \right]\) last period’s squared residuals \(a_{t - 1}^{2}\), and last period’s conditional variance, \(\sigma_{t - 1}^{2}\), with weights \(\left( {1 - \alpha_{1} - \beta_{1} } \right)\), \(\alpha_{1}\), \(\beta_{1}\) which sum to one. It is often useful not only to forecast next period’s variance of returns, but also to make an \(l\)-step ahead forecast, especially if our goal is to price an option with \(l\) steps to expiration using our volatility model (Noh et al. 1994; DeLurgio 1998; Keller and Warrack 2003).
The integrated GARCH model
This model, first developed by Engle and Bollerslev (1986), is referred to an Integrated GARCH model, or an IGARCH model. Squared shocks are persistent, so the variance follows a random walk with a drift. Since we generally do not observe a drift in variance, we will assume \(\alpha_{0} = 0\). Just as a GARCH model is analogous to an ARMA model, the IGARCH model where the variance process has a unit root is analogous to an ARIMA model (Noh et al. 1994; DeLurgio 1998; Keller and Warrack 2003).
The GARCH-M model
Another variation of a GARCH model tests whether variance can impact the mean of future returns. These models are referred to as GARCH in the mean or GARCH-M models (Noh et al. 1994; DeLurgio 1998; Keller and Warrack 2003).
In some specifications, the volatility, rather than the variance, affects returns a, \(r_{t} = \mu + \sigma_{t} \varepsilon_{t} + \lambda \sigma_{t}\).
If \(\lambda = 0\), these models imply a serial correlation of returns since variance is serially correlated and the returns depend on the variance. Many studies have tried to determine whether \(\lambda\) is significantly different from zero, usually with mixed conclusions.
The EGARCH model
Here there is an asymmetric effect between positive and negative return. Also to avoid the possibility of a negative variance, the model is an AR (1) on \(In\left( {\sigma_{t}^{2} } \right)\) rather than \(\sigma_{t}^{2}\).
There are many other models that try to capture the asymmetry between up and down on future volatility. For example, a slight variation on a GARCH (1, 1) is \(\sigma_{t}^{2} = \alpha_{0} + \alpha_{1} \left( {a_{t - 1} - x} \right)^{2} + \beta_{1} \sigma_{t - 1}^{2}\) so that the next period’s variance is not necessarily minimized when the squared residual are zero.
ARCH effects
ARCH effects can be tested in pre-estimation or post-estimation analysis. In post-estimation, it test the remaining ARCH effect, thus whether or not the conditional heteroscedasticity has been removed. For the purpose of this research, it is applied on standardized residual of the fitted model which is an L M test for the ARCH effect in the residual. An indication of ARCH effects is that the residuals are uncorrelated, but the squared residuals are correlated. Normality test are used to test the behavior of ARCH effect if the normality can be described by the conditional error distribution. Another way is to inspect the autocorrelation structure of the residual and squared residuals using portmanteau tests. These tests are used for diagnostic checking of fitted time series model (Bollerslev 1986; Engle 1982; DeLurgio 1998).
Estimation of GARCH models is done with the normal distribution. Pre-estimation analysis is performed on the return and squared return, which includes important test applied to the two time series to ensure that conditional volatility modeling is appropriate. The main tests before actually estimating the conditional volatility are Engle’s ARCH test and Portmanteau (Bollerslev 1986; Engle 1982).
Forecast evaluation methods
The selected model for the time series is used to make prediction into the future. After making forecast and choosing a proxy for actual volatility, next step is to choose statistical loss function to see how close the forecast to their target and compare forecast performance of model. Evaluation of performance of different volatility model is built on statistical loss function present in literature. These are based on moment of forecast error such as root mean squared error (RMSE), mean absolute error (MAE), adjusted mean absolute percentage error (AMAPE). The model would be the one that minimize such a function of the forecast errors. Autoregressive forecasting will be used. Here the observed series depends linearly on its own previous values plus a combination of currents and previous values of a white noise error term. Predictions are made by constructing a model each with one-step-ahead prediction of the next observations (Noh et al. 1994; DeLurgio 1998; Keller and Warrack 2003).
Declarations
Authors’ contribution
MTQ and FBKT, formulated the research idea of the manuscript (Modeling Variations in the Cedi/Dollar Exchange Rate in Ghana: an Autoregressive Conditional Heteroscedastic (ARCH) Models) after our session in Advanced Time Series. We deliberated on the topic and disclosed our intention to Dr. EMB and Dr. ENNN who guided us through the intricacies in Autoregressive Conditional Heteroscedastic (ARCH) Models and gave us their support throughout the process. We were in charge of the acquisition of the data, the analysis and results of the manuscript. Review of the Statistical Methods and interpretation of the results, revision of the whole article from beginning to the final approval of the manuscript were done by Dr. EMB and Dr. ENNN respectively. All authors read and approved the final manuscript.
Compliance with ethical guidelines
Competing interest The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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