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In this paper, we assessed volatility of Ghana’s inflation rates for 2000 to 2018 using the auto-regressive conditionally heteroskedasticity (ARCH), generalized ARCH (GARCH), and the exponential GARCH (EGARCH) models. The inflation data were obtained from the Ghana Statistical Service (GSS). The proposed model should be able to provide projections of inflation volatility from 2019 and beyond. The results showed that higher order models are required to properly explain Ghana’s inflation volatility and the EGARCH(12, 1) is the best fitting model for the data. The EGARCH(12, 1) model is robust to model and forecast volatility of inflation rates. Also, the results suggest that we are forecasting increasing volatility and there is increasing trend in general prices of goods and services for 2018 and beyond. The forecasts figures revealed that Ghana’s economy is likely to be unstable in 2018 and 2019. This study therefore recommends that policy makers and industry players need to put in place stringent monetary and fiscal policies that would put the anticipated increase in inflation under control. The models were implemented using R software.

Inflation is an economic indicator that measures the relative changes in the prices of commodities and services and also measures the persistent increase in the level of consumer prices or persistent decline in the purchasing power of money [

In recent years, inflation has become one of the major economic challenges facing most countries in the world [

It is known that the traditional time series models assume that the conditional variance is constant. However, most economic and financial time series data are heteroskedastic (non-constant variance) in nature. This means that time series models that assume constant variance for the data are likely under-performed and produce bias statistical inferences when applied to non-constant variance time series data. There is therefore the need to build times series models to accommodate data with non-constant variance [

Various authors, for example, [

Inflation in various countries has been studied/modeled by various authors [

The rest of the article is organized as follows. In Section 2, we give an overview of the Ghana’s inflation and then an exploratory analysis of the inflation data used in Section 3. We discuss the ARCH, GARCH, and EGARCH models in Section 4. We then implement these models to the inflation data in Section 5 and give summary remarks in Section 6.

In this section we present a brief history of Ghana’s inflation rates from 1965 to 2017.

In this section, we provide exploratory assessment of the characteristics of the inflation data we obtained from the Ghana Statistical Service (GSS). Our focus is to study/investigate volatility of inflation rates for 2000-2018 and thereafter, suggest a best fitting model for the data. All the analyses in this paper are analyzed using R software [

inflation rates for 2000 to 2017. The inflation rates plot exhibits downward trend which fluctuate over the study period. These fluctuations in the inflation rates give an indication that the mean as well as the variance of the inflation rates are not constant over time.

We performed a normality test on the mean and variance using the Anderson-Darling Normality Test [

In order to achieve stationarity in the inflation rates time series data, we carried out logarithm and differencing transformation. Achieving stationarity is important because the models that are used in this paper assume that the data are stationarity. This means that there is the need to achieve stationarity, in the inflation time series data, in order to produce valid statistical inferences.

transformation to achieve stationarity.

In

We now investigate for the presence of auto-correlation in the original, logarithm and differencing transformation datasets. It can be observed in

Also, it can be observed in

We recall that

of heteroskedasticity (ARCH effects) although it has been reduced as compared to the case of the original monthly rate of inflation series.

From the fore going analysis, it can be concluded that the first difference monthly rates of inflation satisfy all the data assumption or characteristics for a volatility model such as the ARCH-family models. Hence in subsequent analyses, the first differenced monthly rates of inflation would be used or considered.

This section presents notations and concepts of the ARCH, GARCH, and EGARCH ARCH-family models for volatility in times series data.

The ARCH models [

y t = x ′ t δ + ϵ t , t = 1,2, ⋯ , T , (1)

where x is a p × 1 vector of exogenous variables, which may include lagged values of the dependent variable, and δ is a p × 1 vector of regression parameter estimates. The ARCH model characterizes the distribution of the stochastic error term ϵ t conditional on the realized values of the set of variables ψ t − 1 = { y t − 1 , x t − 1 , y t − 2 , x t − 1 , ⋯ } . The original Engle’s [

ϵ t | ψ t − 1 ∼ N ( 0, h t ) , where h t = α 0 + α 1 ϵ t − 1 2 + α 2 ϵ t − 2 2 + ⋯ + α q ϵ t − q 2 , (2)

with constrains α 0 > 0 and α i ≥ 0 , i = 1 , 2 , ⋯ , q in order to ensure that the conditional variance is positive. Since ϵ t − 1 = y t − 1 − x ′ t − 1 δ , i = 1,2, ⋯ , q , h t is obviously a function of the elements of ψ t − 1 .

It is important to note that distinguishing feature of the model (2) is not only that the conditional variance h t is a function of the conditioning set ψ t − 1 , but rather it is the particular functional form that is specified [

The ARCH model was first empirically applied to the relationship between the level and the volatility of inflation [

h t = α 0 + α 1 ∑ i = 1 q w i ϵ t − i 2 , where the weights w i = ( q + 1 ) − i 1 2 q ( q + 1 )

tend to decline linearly over time and constructed in such a way that ∑ i = 1 q w i = 1 . Given this parameterization, a large lag q can be specified and yet only two parameters are required to be estimated in the conditional variance function h t ( [

Bollerslev [

h t = α 0 + α 1 ϵ t − 1 2 + α 2 ϵ t − 2 2 + ⋯ + α q ϵ t − q 2 + β 1 h t − 1 + β 2 h t − 2 + ⋯ + β p h t − p , (3)

where the following inequality constrains are required in order to ensure that the conditional variance h t is strictly positive:

α 0 > 0 , α i ≥ 0 for i = 1 , ⋯ , q and β j ≥ 0 for j = 1 , ⋯ , p . (4)

So we say that a GARCH process with order p and q is denoted as GARCH(p, q). The GARCH(p, q) model consists of three components:

1) α 0 is the weighted long run variance.

2) ∑ i = 1 q α i ϵ t − i 2 is the moving average term, which is the sum of the m previous lags of squared-innovations multiplied by the assigned weight α i for each lagged square innovation.

3) ∑ j = 1 p β j h t − j is the auto-regressive term, which is the sum of the s previous lagged variances multiplied by the assigned β j for each lagged variance.

It is important to point out that the motivation of the GARCH process can be observed in the expression (3) as the conditional variance h t can be expressed as

h t = α 0 + α ( B ) ϵ t 2 + β ( B ) h t ,

where

α ( B ) = α 1 B 1 + α 2 B 2 + ⋯ + α q B q and β ( B ) = β 1 B 1 + β 2 B 2 + ⋯ + β p B p

are polynomials in the back-shift operator B ( [

h t = α 0 1 − β ( 1 ) + α ( B ) 1 − β ( B ) ϵ t 2 = α 0 * + ∑ i = 1 ∞ ξ i ϵ t − 1 2 (5)

where α 0 * = α 0 1 − β ( 1 ) and the coefficient ξ i is the coefficient of B i in the expansion of [ α ( B ) 1 − β ( B ) ] − 1 . The conditional variance function (5) reveals that a

GARCH(p, q) is an infinite order ARCH process with a rational lag structure imposed on the coefficients [

Bera and Higgins [

α 0 * > 0 and ξ i ≥ 0, i = 1, ⋯ , ∞ (6)

are sufficient to ensure that the conditional variance is strictly positive. Nelson and Cao expressed α 0 * and ξ i ’s in terms of the original parameters of the GARCH model and showed that the expression (6) does not require all the inequalities in (4) to hold. That is in GARCH(1, 2) process, α 0 > 0 , α 1 ≥ 0 , β 1 ≥ 0 , and β 1 α 1 + α 2 ≥ 0 are sufficient to ensure that h t > 0 . This means that in GARCH(1, 2), α 2 may be negative. Nelson and Cao present general results on GARCH(1, q) and GARCH(2, q), but stated that a derivation for GARCH process with p ≥ 3 is difficult. Several empirical studies [

Since one of the first challenges encountered using the linear ARCH model was that the estimated α i were most often negative, Geweke [

log ( h t ) = α 0 + α 1 log ( ϵ t − i 2 ) + ⋯ + α q log ( ϵ t − q 2 ) (7)

in order to avoid negative estimates of α i . It can be observed that taking exponential on both sides of the expression (7), the conditional variance becomes

h t = exp ( α 0 + α 1 log ( ϵ t − i 2 ) + ⋯ + α q log ( ϵ t − q 2 ) )

which is strictly positive. This means that no restrictions are required for estimates of α i to ensure that the conditional variance is strictly positive. In order to assess whether the expression (2) or (7) provides best fit of the actual data, Higgins and Bera (1992) suggested the non-linear ARCH (NARCH) model with non-negativity restrictions, but includes linear ARCH as a special case and log ARCH as a limiting case. Higgins and Bera (1992) concluded that data favored logarithmic ARCH model than the linear ARCH model (also see the h t formulations in Bera and Higgins ( [

One potential limitation of the functional form of the h t described above is that this conditional variance is symmetric in the lagged ϵ t ’s. Nelson [

h t = h ( η t − 1 , η t − 2 , ⋯ , η t − q , h t − 1 , h t − 2 , ⋯ , h t − p ) , (8)

h t can be viewed as a stochastic process in which η t serves as a “forcing variable” for the both the conditional variance and the error term. Nelson nelson1991conditional used the expression (8) to produce the desire dependencies and to avoid non-negativity restriction on the parameter estimates, he maintained the expression (7) and proposed

log ( h t ) = α 0 + ∑ i = 1 q α i g ( η t − i ) + ∑ i = 1 p β i log ( h t − i ) ; g ( η t ) = θ η t + γ [ | η t | − E | η t | ] . (9)

The conditional variance (9) is known as the exponential GARCH (EGARCH). For detailed information about the properties of the EGARCH model, see Bera and Higgins ( [

1) The conditional variance is pairwise linear in η t with slopes α i ( θ + γ ) when η t is positive and α i ( θ + γ ) when η t is negative.

2) The first term in log ( h t ) in the expression (9) allows for the correlation between the error term and future conditional variances. For instance, if γ = 0 and θ < 0 , then a negative η t will result in negative error and the variance process will be positive.

3) The second term in log ( h t ) in the expression (9) produces ARCH effect. Assume that γ > 0 and θ = 0 . Whenever the absolute magnitude of η t exceeds its expected value, the innovation g ( η t ) is positive. This means that large shocks increase the conditional variance.

In this section, we fitted ACRH, GARCH, and EGARCH models to the inflation data for 2000 to 2017. Candidates models will be fitted to the data in order to determine the besting fitting model for the inflation data.

For the ARCH model, we used the function garch() in the tseries package. We note that the function garch() is used for GARCH model fitting. However, function garch() becomes an ARCH model when used with the order argument equal to c(q > 0, 0). The PACF chart in

On the other hand, various GARCH(p, q) and EGARCH(p, q) candidate models were fitted to the differenced inflation data for 2000-2017. The results from these candidate models are respectively shown in columns two and three of

ARCH | GARCH | EGARCH | ||||||
---|---|---|---|---|---|---|---|---|

ARCH(q) | AIC | −2ℓ | GARCH(p, q) | AIC | −2ℓ | EGARCH(p, q) | AIC | −2ℓ |

ARCH(1) | −2.28 | 248.29 | GARCH(1, 1) | −2.33 | 254.12 | EGARCH(1, 1) | −3.12 | 343.22 |

ARCH(2) | −2.27 | 247.95 | GARCH(1, 2) | −2.34 | 256.21 | EGARCH(1, 2) | −3.15 | 347.36 |

ARCH(3) | −2.26 | 248.33 | GARCH(2, 1) | −2.32 | 254.25 | EGARCH(2, 1) | −3.13 | 346.39 |

ARCH(4) | −2.25 | 247.70 | GARCH(2, 2) | −2.33 | 256.21 | EGARCH(2, 2) | −3.14 | 348.02 |

ARCH(12)^{*} | −2.75 | 309.68 | GARCH(12, 1)* | −2.74 | 309.68 | EGARCH(12, 1)* | −3.21 | 374.76 |

*denotes the best candidate model and −2ℓ denotes the log-likelihood.

data. It follows that the EGARCH(12, 1) model, with the lowest AIC = −3.21 and largest −2ℓ = 374.76, is best fitting model. The auto-correlation function charts for the GARCH(12, 1) and EGARCH(12, 1) are shown

Now we present, compare, and contrast results from these best fitting models (ARCH(12), GARCH(12, 1), and EGARCH(12, 1). The parameter estimates from these models are presented in

It can be observed that some of the parameter estimates α 2 , α 3 , α 7 and α 9 and their corresponding standard errors are approximately zeros. The test for ARCH effect results indicates that there is no ARCH effect in the ARCH, GARCH, and EGARCH models considered. Because higher order q = 12 is required to eliminate auto-correlation in the residuals and restrictions imposed on the parameters in the ARCH and GARCH models, there was issue of convergence as some of the diagonals of the matrix of these parameters are negatives. These results confirmed the results from various authors [

ARCH(12) | GARCH(12, 1) | EGARCH(12, 1) | |||||||
---|---|---|---|---|---|---|---|---|---|

Est | s.e. | p-value | Est | s.e. | p-value | Est. | s.e. | p-value | |

μ | 0.0012 | 0.0020 | 0.7053 | 0.0012 | 0.0031 | 0.6933 | −0.0068 | 0.0002 | 0.0000 |

ω | 0.0007 | 0.0002 | 0.0001 | 0.0007 | 0.0002 | 0.0001 | −1.1825 | 0.0172 | 0.0000 |

α 1 | 0.8719 | 0.2916 | 0.0029 | 0.8719 | 0.0119 | 0.0001 | −0.2028 | 1.0702 | 0.8497 |

α 2 | 0.0001 | 0.0000 | 0.0000 | 0.0001 | 0.0000 | 0.0000 | −0.1862 | 0.9804 | 0.8494 |

α 3 | 0.0001 | 0.0000 | 0.0000 | 0.0001 | 0.0000 | 0.0000 | −0.0366 | 0.3328 | 0.9125 |

α 4 | 0.0029 | 0.0038 | 0.4469 | 0.0029 | 0.0038 | 0.4468 | 0.1366 | 0.9010 | 0.8795 |

α 5 | 0.0001 | 0.0028 | 0.9999 | 0.0001 | 0.0028 | 0.9999 | −0.6129 | 1.4908 | 0.6810 |

α 6 | 0.0001 | 0.0040 | 0.9999 | 0.0001 | 0.0040 | 0.9999 | 0.0628 | 1.1507 | 0.9565 |

α 7 | 0.0001 | 0.0000 | 0.0000 | 0.0001 | 0.0000 | 0.0000 | 0.4518 | 0.7132 | 0.5264 |

α 8 | 0.0029 | 0.0061 | 0.6376 | 0.0029 | 0.0060 | 0.6350 | 0.1197 | 0.4016 | 0.7657 |
---|---|---|---|---|---|---|---|---|---|

α 9 | 0.0001 | 0.0000 | 0.0000 | 0.0001 | 0.0000 | 0.0000 | −0.1064 | 0.3412 | 0.7551 |

α 10 | 0.0012 | 0.0023 | 0.5872 | 0.0012 | 0.0023 | 0.5871 | −0.2842 | 0.9506 | 0.7649 |

α 11 | 0.0001 | 0.0306 | 1.0000 | 0.0001 | 0.0303 | 1.0011 | 0.0830 | 0.7530 | 0.9123 |

α 12 | 0.4362 | 0.1873 | 0.0198 | 0.4362 | 0.1626 | 0.0073 | 0.3748 | 1.8671 | 0.8409 |

β 1 | - | - | - | 0.0001 | 0.0000 | 0.0000 | 0.8059 | 0.0002 | 0.0001 |

γ 1 | - | - | - | - | - | - | 0.4213 | 1.2387 | 0.7338 |

γ 2 | - | - | - | - | - | - | −0.378 | 0.4600 | 0.4109 |

γ 3 | - | - | - | - | - | - | 0.2411 | 0.8475 | 0.7760 |

γ 4 | - | - | - | - | - | - | −1.0571 | 2.7773 | 0.7035 |

γ 5 | - | - | - | - | - | - | 1.4244 | 0.1195 | 0.0000 |

γ 6 | - | - | - | - | - | - | −0.1117 | 0.5592 | 0.8417 |

γ 7 | - | - | - | - | - | - | −0.5213 | 0.5849 | 0.3728 |

γ 8 | - | - | - | - | - | - | 0.0240 | 0.2847 | 0.9327 |

γ 9 | - | - | - | - | - | - | −0.1529 | 0.1613 | 0.3432 |

γ 10 | - | - | - | - | - | - | 0.3374 | 0.7749 | 0.6633 |

γ 11 | - | - | - | - | - | - | 0.0107 | 1.3196 | 0.9935 |

γ 12 | - | - | - | - | - | - | −0.2610 | 3.1678 | 0.9343 |

θ | - | - | - | - | - | - | 3.3948 | 1.3887 | 0.0145 |

Test | 0.9997 | 0.9997 | 0.9997 |

were not encountered in the EGARCH since Geweke [

It is important to note that volatility for 2018 is relatively lower with a general decrease in inflation rates. This predicted volatility in the 2018 inflation rates is confirmed in

We also used the 2000 to 2018 inflation rates to forecast volatility for 2019.

In this paper, we modeled Ghana’s inflation volatility and provided projections of inflation volatility for 2019. That inflation rates data were obtained from the Ghana Statistical Service. These data consist of monthly inflation data from 2000 to 2018. The inflation volatility in these datasets was modeled and explained using the auto-regressive conditionally heteroskedasticity (ARCH), generalized ARCH (GARCH), and the exponential GARCH (EGARCH) models. These models were implemented in R software using fGARCH package. Results from these

models were then compared and best model selected using their respective AICs and log-likelihoods (−2ℓ).

We observed from the results of the analyses that some of the parameter estimates and their corresponding standard errors are approximately zeros, especially in the ARCH and GARCH models. This is because higher order q = 12 is required to eliminate auto-correlation in the residuals and restrictions imposed on the parameters in the ARCH and GARCH models results in issue of convergence as some of the diagonals of the matrix of these parameters are negatives. These results are in line with various authors [

The proposed best fitting model, in this paper, is then used to forecasts volatility in inflation rates from January 2018 to December 2018. The inflation rates forecast gives an indication that, in 2018, general prices of goods and services are likely to increase. These forecasts suggest that Ghana’s economy is likely to be slightly unstable in 2018 and beyond. Our forecast results suggest that we are forecasting increasing volatility in 2018 with the highest inflation rate of 10.4% in January 2018 and the lowest inflation rate of 9.3% in November 2018. At the time of writing this paper, inflation figure for December has not been reported by the GSS. It is important to note that volatility for 2018 is relatively lower with a general decrease in inflation rates. Our analyses also revealed the possibility of increasing volatility in 2019.

Per the results presented in this paper, we therefore advise the government of Ghana to put in measures such as, monetary, fiscal, and price-control policies, to combat this anticipated rise in the inflation rates.

The author would like to thank the Ghana Statistical Service for making data available for this study.

FundingThis study receives no funding.

Disclosure StatementThe author declares that they have no competing interests.

Notes on Contributor(s)AI carried out the literature review and statistical analyses. AI, DO, IWA, and SA interpreted results and proof-read the manuscript. All authors have read and approved the final version of the manuscript.

Consent to PublishNot applicable.

Ethics Approval (and Consent to Participate)Not applicable.

Availability of Data and MaterialsWe do not have permission to distribute the data.

Conflicts of InterestThe authors declare no conflicts of interest regarding the publication of this paper.

Iddrisu, A.-K., Otoo, D., Abdul, I.W. and Ankamah, S. (2019) Modeling and Forecasting of Ghana’s Inflation Volatility. American Journal of Industrial and Business Management, 9, 930-949. https://doi.org/10.4236/ajibm.2019.94064