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American Journal of  Applied 
Statistics and Economics (AJASE)

Predictive Modeling of  Ghana’s Private Sector Pensions Asset under Management 
Contribution Using ARIMA Model

Chinton Emmanuel1*, Donkoh Kojo Isaac2, Acquah Oware Nana Emmanuel3

Volume 4 Issue 1, Year 2025
ISSN: 2992-927X (Online)

DOI: https://doi.org/10.54536/ajase.v4i1.5751
https://journals.e-palli.com/home/index.php/ajase

Article Information ABSTRACT

Received: July 24, 2025

Accepted: August 25, 2025

Published: October 06, 2025

This study applies an ARIMA (1,1,0) model to analyze the Private Sector Pension Assets 
Under Management (AUM) in Ghana. The model’s parameters and performance metrics 
were evaluated using SARIMAX results and the Dickey-Fuller Test for stationarity. The 
SARIMAX model demonstrated a significant autoregressive term (ar. L1 = 0.9693) and 
acceptable performance metrics (MAE = 5.99, RMSE = 13.89, MAPE = 23.97%), indicating 
a strong influence of  past values on current AUM. The diagnostic tests suggested that 
residuals were not autocorrelated and approximately normally distributed. The Dickey-Fuller 
Test further confirmed the stationarity of  the time series, with a test statistic of  -5.3314 and 
a p-value of  4.7116e-06, allowing us to reject the null hypothesis of  a unit root. Overall, 
the ARIMA (1,1,0) model provides a reliable framework for forecasting and analyzing the 
Private Sector Pension AUM in Ghana, supported by robust statistical validation.

Keywords

ARIMA, Asset Under 
Management (AUM), Pensions 
Contribution in Ghana, Private 
Sector

1 Department of  Statistics, University of  Cape Coast, Ghana
2 Financial Engineering, WorldQuant University, USA
3 Department of  Economics and Finance, Youngstown State University, USA
* Corresponding author’s e-mail: emmanuelchinton7@gmail.com

INTRODUCTION
Retirement planning plays a critical role in ensuring 
financial security during old age. Without adequate savings 
or income-generating assets, many individuals face severe 
financial challenges after leaving active employment 
(Diaw, 2017). To address this, most countries have 
adopted social security and pension systems that provide 
stable income for retirees and reduce old-age poverty.
In Ghana, the enactment of  the National Pensions 
Act, 2008 (Act 766), marked a major reform of  the 
pension system. The Act replaced the Social Security and 
National Insurance Law (PNDCL 247) and introduced 
a contributory three-tier pension scheme. These tiers 
comprise: (i) a mandatory basic national social security 
scheme managed by SSNIT, (ii) a mandatory occupational 
pension scheme managed by private trustees, and (iii) a 
voluntary provident and personal pension scheme. The 
Act also established the National Pensions Regulatory 
Authority (NPRA) to regulate and supervise pension 
administration. A key innovation of  Act 766 was the 
extension of  pension coverage to informal sector 
and self-employed workers, alongside those in formal 
employment (Abebrese, 2011).
The scheme requires a total monthly contribution of  
18.5% of  basic salary, with 13.5% allocated to Tier 1 
and 5% to Tier 2. Tier 3 remains voluntary. The reform 
aimed to ensure income stability for retirees, harmonize 
pension provisions across the public and private sectors, 
and mobilize long-term funds for national development.

LITERATURE REVIEW
Theoretical Review
Life-Cycle Consumption Theory
Modigliani and Brumberg’s (1954) life-cycle hypothesis 
provides the theoretical foundation for pension systems. 
It posits that individuals plan consumption and savings 
over their lifetime to smooth income across working and 
retirement years. Without adequate savings, retirees may 
face income insecurity. In Ghana, where extended family 
support systems are weakening, the theory underscores 
the need for deliberate retirement planning.

Positive Theory of  Social Security
According to Sala-i-Martin (1996) and Tabellini (2000), 
public pensions improve economic efficiency by enabling 
older workers to retire, thereby creating employment 
opportunities for younger and more productive workers. 
Verbon (2012) further argues that pension systems act as 
retirement incentives where significant productivity gaps 
exist between older and younger generations.

Pooling Theory
Allen and Santomero (1998) highlight the efficiency of  
pension schemes in pooling risks, reducing transaction 
costs, and enhancing diversification. By mobilizing 
contributions, pension funds achieve economies of  scale 
and improved investment outcomes (Matheson et al., 
2004; Bridgen & Meyer, 2008).



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Three-Tier Pension Scheme
Act 766 structures pensions into three tiers: the basic 
mandatory scheme (Tier 1), the mandatory occupational 
scheme (Tier 2), and the voluntary provident/
personal pension scheme (Tier 3). Of  the 18.5% 
total contribution, SSNIT retains 11% for retirement 
benefits and transfers 2.5% to the National Health 
Insurance Scheme. The remaining 5% is invested 
by Tier 2 trustees. Self-employed individuals may 
voluntarily participate in Tier 3, though participation 
levels remain low.

Empirical Review
Ghana’s pension system has evolved since the colonial era, 
beginning with the Workmen’s Compensation Ordinance 
of  1940 and the non-contributory Pension Ordinance 
of  1950 for civil servants (Darkwa, 2007; NPRA, 2010). 
Over time, reforms have sought to address sustainability, 
adequacy, and coverage gaps.
Kpessa (2011) observes that pensions in Africa play a 
crucial role in alleviating poverty among the elderly and 
supporting households under demographic pressure. 
Similarly, Agnew (2013) notes that pensions provide 
stable income for the aged, disabled, and unemployed. 

However, Fiiwe (2020) highlights shortcomings in benefit 
packages, particularly the absence of  post-retirement 
healthcare, housing, and entrepreneurial support, which 
limit retirees’ welfare.
International evidence shows similar trends. In the United 
States, private pension schemes date back to 1857, with 
American Express pioneering corporate pensions in 1878 
(Bond, 2017). Pension benefits gained popularity during 
World War II as firms used them to retain workers amidst 
wage freezes (Pradmin, n.d.).

Informal Sector Participation
A major challenge in Ghana is extending pension coverage 
to the large informal sector. Although Act 766 permits 
voluntary participation through Tier 3, awareness and 
enrollment remain limited. A survey of  self-employed 
workers revealed that over 70% were unaware of  the 
scheme, while many who had knowledge of  it contributed 
irregularly due to unstable incomes. This highlights the 
need for greater education, flexible contribution options, 
and innovative pension products tailored to informal 
sector workers.

Conceptual Framework of  the Ghana Pension System

Figure 1: Conceptual Framework: Pension Theories and Ghana’s Three-Tier Scheme

MATERIALS AND METHODS
Data Collection
The study utilized secondary data for the predictive 
modeling of  the contributory pension assets under 
management of  private sector in Ghana. For this study, we 
collected a time series of  AUM of  Private Sector pensions 
industry from the National Pensions and Regulatory 
Authority (NPRA) Annual reports from 2012 to 2023.

Statistical Analysis Tool
The study employed Time Series Statistical technique 
to analyze trends over time and forecast future pension 
AUM growth in Ghana’s private sector and the analysis 
was done using Python Programming.
The study employed Autoregressive Integrated Moving 
Average (ARIMA) and SARIMAX to analyze the 
contributory trend over the period (2012-2023).

Sample Size 
The sample size for the study comprised of  817 Pensions 
Trustees in Ghana 

Autoregressive Integrated Moving Average (ARIMA) 
The Auto-regressive integrated moving average (ARIMA) 
model is one of  the most common prediction models, 
which is a time series analysis tool raised in the 1970s. It 
is a time series prediction model based on the fitting value 
of  the past data sequence to extrapolate into future. It has 
5 expressions: AR(P), MA(q), ARMA (p, q), ARIMA (p, 
d, q), ARIMA (p, d, q) × (P, D, Q)s
The Autoregressive Integrated Moving Average (ARIMA) 
model is a combination of  the differenced autoregressive 
model with the moving average model. ARIMA model 
is said to be a unit-root non stationary because its 
AR polynomial has a unit-root and a conventional 



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Table 1: Asset Under Management of  the Private Sector 
Pensions in Ghana (2012 -2022)
Years Private Sector Pension 

AUM (GHS’Billion)
2023 46.5
2022 35.3
2021 28.0
2020 22.0
2019 17.3
2018 13.0
2017 9.8
2016 8.9
2015 8.8
2014 7.4
2013 4.8
2012 4.0

Source: National Pensions Authority (NPRA) Annual Report 
(2012- 2022).

approach for handling unit-root non-stationary is to 
use differencing (Tsay, 2010). If  the differencing Wt   = 
Yt   – Y(t-1)  =  (1 − B ) Yt  or higher-order differencing  
Wt = (1-B)d Yt of  non- stationary time series then we 
call Yt  an ARIMA (p, d, q) process with order p of  AR 
process, d the number of  differences made for a series 
to become stationary and q is the order of  MA  process. 
It is expressed as:
Y’t=I+∝1Y’(t-1)+∝2Y’(t-2)+....+∝pY’(t-p)+et+θ1e(t-1)+θ2e(t-

2)+....+θq e(t-q)             (1.0)
Φp(B)(1-B)dYt=θq(B)∝t~ARIMA (p,d,q)                                  (1.1)

Multiplicative Seasonal ARIMA (SARIMAX)
The seasonal ARIMA model incorporates both non- 
seasonal and seasonal factors in a multiplicative model: 
SARIMA (p, d, q) (P, D, Q) S. Box & Jenkins proposed 
the following model when dealing with a time series that 
contains seasonal fluctuations:
Φp(B

S)Φp(B)(1-B)d(1-BS)DYt=θq(B)ε(Q)(B
S)∝t             (1.2)

Where Yt   is the observed value at time t, ∝t  is the value 
at time t of  white noise, d is order of  differencing,  is Φp 
(B) ordinary autoregressive component of  order p and  
θq(B) and is the ordinary moving average component of  
order q, Sis number of  seasons in a year and D is order 
of  the seasonal differencing, Φp (B

S ) and ε(Q)(B
S ) are the 

seasonal autoregressive and moving average difference 
of  orders P and Q at lag s. According to Box & Jenkins 
(1976), the operator polynomials are:
Φp(B)=(1-∅1B-…∅pB

p)                                                                                                  (1.3)
θq(B)=(1+∅1B-…θqB

p)                                                                                                             (1.4)    
Φp(B

S)=(1-ΦBS-…-ΦpB
sp)                                                                                              (1.5) 

Box-Jenkins (ARIMA) Model
When performing a time series analysis using ARIMA 
models, three iterative steps must be used: diagnostic 
checking by examining residuals to assess the model’s 
adequacy, parameter estimation by estimating the model’s 
unknown parameters, and model identification by 
analyzing historical data.

Model Identification
Identification of  the appropriate and suitable ARIMA 
model requires skills obtained by experience. Box & 
Jenkins postulates the following summary table on how 
to identify the model.

Table 2: Model identification (Box & Jenkins, 1976)
Model ACF PACF
ARIMA (p, d, 0) Infinite. Tails off Finite Cuts off  after p lags
ARIMA (0, d, p) Finite Cuts off  after Infinite. Tails off
ARIMA (p, d, q) Infinite. Tails off Infinite. Tails off

Model Identification
For ARIMA modeling, the autoregressive order 
(p) is usually determined by examining the partial 
autocorrelation function (PACF) of  a stationary time 
series. If  the PACF cuts off  after a certain lag, the highest 
significant lag suggests the value of  p. Conversely, if  the 
PACF does not cut off, then p is often set to zero (Box 
& Jenkins, 1976). Similarly, the moving average order (q) 
is inferred from the autocorrelation function (ACF). A 
cutoff  in the ACF after a few lags indicates the potential 
value of  q, with the last significant lag serving as an 
estimate. In ARIMA (p, d, q) models, the autocorrelation 
patterns typically show exponential decay or damped 
sine-wave behavior after the first q–p lags.

Parameter Estimation
Once a tentative model structure is identified, parameter 

estimation follows. Box and Jenkins (1976) propose 
several approaches, including the method of  moments, 
least squares, and maximum likelihood estimation 
(MLE). Given the non-linear nature of  many ARIMA 
specifications, MLE is often preferred for its efficiency 
and robustness. When estimating residuals, backcasting 
may also be applied to obtain initial values for the error 
terms.

Diagnostic Checking
After estimation, diagnostic checks are essential to confirm 
model adequacy. Residuals should resemble white noise, 
meaning they are uncorrelated, normally distributed, 
and exhibit constant variance. A residual scatter plot 
should appear structureless, without systematic trends or 
patterns. Similarly, the residual autocorrelation function 
should not display significant spikes. Statistical tests, 



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such as the Ljung-Box test or chi-square-based adequacy 
tests, are typically used to confirm that no significant 
autocorrelation remains in the residuals. Once these 
conditions are satisfied, the fitted ARIMA model can be 
considered adequate and used for forecasting.

RESULTS AND DISCUSSIONS
This section presents the outcome of  the estimation of  
the model of  this study. This begins with the forecast of  
private sector AUM using the ARIMA. For this study, 
the result presented in Chart 1 proves that private sector 

Figure 2: ARIMA Forecast of  Private Sector Pension AUM (GHS’ Billion) to 2040

Figure 3: SARIMAX Results

pensions AUM is projected to grow steeply by 2040.
Figure 3 presents the estimtaed results of  the SARIMAX 
method for  Ghana’s private pensions AUM.  The AR 
(AutoRegressive) term has a coefficient of  0.9693, which is 
significant (p-value 0.000). This indicates a strong influence 
of  past values on the current value of  the dependent variable.
The variance of  the error term is relatively high with a 
coefficient of  3.2070 and a marginally significant p-value 
(0.060), suggesting some level of  uncertainty in the model.
The diagnostic tests suggest that the residuals are not 
auto-correlated (Ljung-Box test) and are normally 

distributed (Jarque-Bera test). The heteroskedasticity test 
indicates no significant heteroskedasticity.
The performance metrics indicate that the model’s 
predictions have a mean absolute error of  5.99, root mean 
squared error of  13.89, and mean absolute percentage 
error of  23.97%. 
Overall, the SARIMAX model seems to fit the data well with 
significant AR term and acceptable performance metrics. 
However, the high variance of  the error term suggests that 
there is some uncertainty in the model’s predictions.
Figure 3 shows diagnostic plots for a statistical model. 

These plots are essential for diagnosing and validating the 
model’s assumptions and fit.

Standardized Residuals for “P” 
This plot displays the standardized residuals (differences 
between observed and predicted values) for a variable 

labeled “P” across different observations. The residuals 
fluctuate around the zero line, indicating how well the 
model’s predictions match the actual data.

Histogram Plus Estimated Density
This plot combines a histogram of  the residuals with 



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Figure 4: Diagnostic Plots for A Statistical Model

estimated density curves: -Histogram Bars: Show the 
frequency of  residuals, Orange Line: Kernel Density 
Estimate (KDE) - a smoothed version of  the histogram 
and Green Line: Represents the standard normal 
distribution (N(0,1)). The alignment of  the orange and 
green lines with the histogram bars suggests whether the 
residuals follow a normal distribution.

Normal Q-Q Plot
A Quantile-Quantile (Q-Q) plot compares the sample 
quantiles of  the residuals to the theoretical quantiles 
of  a standard normal distribution:Red Line: Represents 
the expected line for normally distributed residuals and 
Points: Represent the actual residuals. The closer the 
points are to the red line, the more normally distributed 
the residuals are.

Correlogram
This plot shows the autocorrelation of  the residuals 
at different lags:  Points with Error Bars: Indicate 
the correlation values at various lags  and the Shaded 
Area:Represents the confidence interval. Values within 
the shaded area suggest no significant autocorrelation, 
indicating the residuals are independent over time.
Table 3 shows the results of  a Dickey-Fuller test, which is 
used to test for the presence of  a unit root in a time series 
sample. The test statistic of  -5.3314 is more negative than 
all the critical values at the 1%, 5%, and 10% significance 
levels. Combined with the very low p-value, this provides 
strong evidence to reject the null hypothesis. This 
suggests that the time series is stationary and does not 
have a unit root.
Chart 5 present the times series plots of  the private sector 

Table 3: Dickey-Fuller Test Result
Metric Value
Test Statistic -5.331440528065261
p-value 4.711563618849688e-06
# Lags Used 2.0
Number of  Observations Used 9.0
Critical Value (1%) -4.473135048010974
Critical Value (5%) -3.28988060356653
Critical Value (10%) -2.7723823456790124

over the last 12 years indicating that the AUM of  private 
pensions has always been on the upward trajectory.
Table 4 presents on the forcasted times series values of  

the private sector AUM from 2012  to 2040 taking into 
account a 95% confidence intervals for both the lower 
and upper bound.



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Figure 5: Time Series Plot

Table 4: Forcasted times series values of  the private sector AUM from 2012  to 2040
Year Forecasted Values 

(GHS'Billion)
Lower Bound (95% CI) 
(GHS'Billion)

Upper Bound (95% CI) 
(GHS'Billion)

2013 5.423536852 3.275484696 7.571589008
2014 6.720661869 -0.319161914 13.76048565
2015 7.247450909 -6.312625896 20.80752771
2016 8.354215719 -13.60046814 30.30889958
2017 11.15636763 -22.46023611 44.77297137
2018 15.24192579 -34.48246896 64.96632053
2019 19.61959785 -50.78297042 90.02216611
2020 24.0972536 -71.21520887 119.4097161
2021 29.28331686 -95.25346974 153.8201035
2022 35.6109406 -123.0317088 194.25359
2023 42.86059712 -155.3054794 241.0266737
2024 50.61826353 -192.6798141 293.9163411
2025 58.83731515 -235.2233702 352.8980005
2026 67.78828048 -282.8182577 418.3948187
2027 77.63626747 -335.6172463 490.8897813
2028 88.27024515 -394.0358105 570.5763008
2029 99.51904349 -458.4250597 657.4631466
2030 111.3777465 -528.9098723 751.6653653
2031 123.9652102 -605.5382018 853.4686221
2032 137.3429456 -688.4720041 963.1578953
2033 151.4567871 -777.979251 1080.892825
2034 166.2367155 -874.2954576 1206.768889
2035 181.6869161 -977.5598355 1340.933668
2036 197.8590336 -1087.881322 1483.599389
2037 214.7753175 -1205.417149 1634.967784
2038 232.4100458 -1330.365886 1795.185977
2039 250.7348986 -1462.908715 1964.378512
2040 269.7542712 -1603.185231 2142.693774



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Figure 6: Stationarity Check: Original data and rolling statistics

CONCLUSION 
The ARIMA (1,1,0) model provides a reasonably good fit 
for the Private Sector Pension AUM series. The statistically 
significant coefficient of  the autoregressive term AR (1) 
indicates that current values are strongly influenced by 
their immediate past observations. Model diagnostics 
further show that the residuals are free from serious 
autocorrelation, as confirmed by the Ljung-Box test, and 
are approximately normally distributed according to the 
Jarque-Bera test. Nonetheless, signs of  heteroskedasticity 
were detected, which suggests the need for additional 
adjustments to enhance the model’s robustness.
In terms of  accuracy, the model yields acceptable forecast 
error measures, including MAE, RMSE, and MAPE, 
making it suitable for short-term predictions. Moreover, 
the Augmented Dickey-Fuller test confirms the 
stationarity of  the series, meaning its statistical properties 
such as mean and variance remain stable over time. This 
stationarity is particularly important, as it provides a 
strong foundation for reliable time series modeling and 
forecasting.
Finally, we suggest extensions to the model (ARIMA with 
GARCH, SARIMA with GARCH) to specifically tackle 
the heteroskedasticity problem for future research work

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