A flexible Sine Alpha Power Weibull ACD model

Abstract

High-frequency financial markets generate data at irregular intervals, and durations, defined as the elapsed time between successive market events play a key role in modeling market volatility and liquidity. Classical innovation distributions frequently fall short of capturing the heavy tails, skewness, and intricate conditional intensity patterns present in financial data, despite the Autoregressive Conditional Duration (ACD) model being a typical tool for modeling these temporal dependencies. Although the ACD model is frequently utilized to capture historical dependencies in financial data, classical innovation distributions often fail to adequately represent heavy tails, skewness, and complex conditional intensity patterns. To address this restriction, the Sine Alpha Power Weibull Autoregressive Conditional Duration (SAP–W–ACD) model was developed. This approach offers a flexible framework by incorporating the recently introduced Sine Alpha Power-G (SAP-G) family of distributions into a Weibull baseline. “This study offers an in-depth theoretical analysis, derived the model's PDF, CDF, Quantile function, and series expansions for raw moments.” We employed Maximum Likelihood Estimation (MLE) to determine the model parameters and confirm the numerical reliability of these estimators using Monte Carlo simulations. High-frequency exchange rate durations from Ethiopia, Kenya, and Somalia analyzed using the SAP–W–ACD model. In terms of log-likelihood and information criteria, the empirical results show that the SAP–W–ACD model performs much better than competing specifications, such as the Lomax-ACD, and Gompertz-ACD models. Tests on standardized residuals demonstrate that the model effectively accounts for duration clustering. Furthermore, conditional intensity analysis identifies a 'positive aging' effect within East African currency markets, indicating that the probability of a rate adjustment increases as the time between events increases.

1 Introduction

The analysis of high-frequency data (HFD) has gained significant momentum due to recent advances in computational technology and data storage capacity [1]. This trend is especially noticeable in financial research, driven by the increased accessibility of intraday market transaction data. Inter-trade duration, or the irregularly spaced time interval between consecutive, is a crucial feature in this context [2]. These durations provide essential information regarding market liquidity, information arrival, and volatility behavior. A variety of evidence from market microstructure literature suggests that durations exhibit “clustering,” where short durations are followed by short durations and long by long [3]. To capture these temporal dynamics [4], introduced the Autoregressive Conditional Duration (ACD) model. The choice of the innovation (error) distribution is fundamental to the ACD framework's effectiveness [5]. Financial duration data often possess a unimodal distribution with heavy right tails and high density near zero [6]. While classical baseline definitions such as the exponential and Weibull are popular, they frequently lack the flexibility to capture the skewness, kurtosis, and non-monotonic conditional intensity patterns inherent in financial series [7]. As a result, scholars have investigated strategies for extending classical distributions. The Alpha Power Transformation (APT), first proposed by Mead et al. [8], which adds a parameter to baseline distributions to improve their adaptability [9].

In recent years, there has been an increase in trigonometric generators to improve distributional flexibility. According to Ahmad et al. [10], applying trigonometric functions to probability distributions can help address complex problems in fields such as computer science, economics, and hydrology. Trigonometric transformations, such as the Sine-G and Cos-G families, give a better foundation for modeling real data since they allow parameters to oscillate and influence the behavior of the distribution curve [11]. Recent literature has reported various successful applications of these families, including the Exponentiated Sine-G family for lifespan studies [12] and Secant Kumaraswamy family's engineering and finance data [13].

The flexibility of these models is particularly evident in their conditional intensity functions. For instance Mulagala and Nagarjuna [14], a study showed that sine-based models may represent a wide range of conditional intensity shapes, including bathtub, J-shape, reversed J-shape, and non-monotone forms. Alyami et al. [15] demonstrated the usefulness of the Sine Exponentiated Weibull-H (SEW-H) family by modeling total factor productivity data from the UK food chain. These models use advanced estimate approaches beyond Maximum Likelihood, such as the Maximum Product of Spacing and Cramer-von Mises methods, to achieve robust parameter estimation.

The Sine Alpha Power-G (SAP-G) family represents a significant advancement in this field. To avoid over-parameterization, it combines the APT with a trigonometric sine generator, introducing only one additional parameter to the baseline distribution [16]. Although the SAP-Weibull (SAP-W) distribution has been used to construct a novel log-location-scale regression model for engineering applications, its potential in dynamic, time-dependent frameworks such as the ACD model has yet to be explored.

This study addresses a gap in the literature by introducing the SAP-W-ACD model. By combining the SAP-W distribution with the ACD framework, we produced a versatile solution for handling complex financial data. We defined the model's basic mathematical features, including the probability density function (PDF) and cumulative distribution function (CDF). Using real currency data from Ethiopia, Kenya, and Somalia, we discovered that our approach outperforms prior methods, such as Lomax-ACD and Gompertz-ACD, in identifying market patterns.

2 Sine Alpha Power Generated (SAP-G) family

We develop a method for construct families of distributions based on the trigonometric sine function and the alpha power transformation, following the approach of Alghamdi et al. [16]. The CDF of a baseline distribution is represented by G(x). The CDF of the SAP-G family is obtained by embedding the Alpha Power Transformation within the Sine-G generator as follows in Equation 1

By differentiating G(x) with respect to x, the PDF f(x) is given by Equation 2:

where f(x) is the baseline PDF. The Duration Function is The Conditional Intensity Function (CIF) in Equation 3

Quantile Function: The quantile function Q(p), defined for a probability p∈(0, 1), is obtained by inverting the cumulative distribution function [17] as shown in Equation 4:

2.1 The Sine Alpha Power-Weibull (SAP–W) distribution

The shape parameter k and scale parameter λ of the two-parameter Weibull distribution are used as the baseline for defining the SAP–W distribution. The baseline Weibull CDF and PDF are defined as shown in Equations 5 and 6:

2.2 SAP–W distribution

Substituting the baseline into the SAP-G generator, the CDF of the SAP–W innovation is Equation 7

and the PDF is Equation 8

The model's ability to accumulate probability at different rates is demonstrated by the CDF in Figure 1a, indicating considerable flexibility in representing both short-term and long-term duration events. Additionally, the SAP-W distribution may successfully capture unimodal, reversed-J, and heavy-tailed data structures, which are typical of high-frequency financial transactions, as shown by the PDF plots in Figure 1b.

Figure 1

Conditional Intensity Function in Equation 9:

As illustrated in Figure 2, the conditional intensity function exhibits diverse behaviors including monotonic increasing, decreasing, and non-monotonic shapes—highlighting the SAP-W model's flexibility in capturing the complex aging effects and clustering dynamics inherent in empirical trade durations.

Figure 2

Quantile Function Formula

The quantile function Q(p) for the SAP-W distribution, for 0 < p < 1, is Equation 10:

2.3 Sine Alpha-Power Weibull ACD model (SAP–W–ACD)

The SAP–W–ACD model characterizes durations as x_i = ψ_iϵ_i, where ϵ_i is a stochastic innovation and ψ_i is the conditional mean. Under this framework, the observed duration is decomposed into a dynamic component that captures temporal dependencies and a random component following the SAP–W distribution.

To capture stochastic time clustering, we specify the conditional mean using the standard ACD linear structure ψ_i = ω+γ₁x_i−1+β₁ψ_i−1. In this specification, ω represents the baseline constant, γ₁ captures the short-term impact of the previous duration (the autoregressive component), and β₁ represents the persistence of the conditional mean.

To ensure model identifiability and maintain the interpretation of ψ_i as the actual conditional expected duration, we impose the unit-mean constraint E[ϵ_i] = 1. This requirement is fundamental to the ACD framework, as it ensures that the conditional mean of the duration x_i is exactly ψ_i. For the model to be empirically applicable and mathematically stable, several implicit constraints must be satisfied. To ensure that the conditional mean ψ_i is strictly positive for all i, the following non-negativity conditions are required: ω>0, γ₁≥0, β₁≥0 [18]. To ensure the duration process is weakly stationary, the sum of the autoregressive and moving average coefficients must be less than unity γ₁+β₁ <1. If this condition holds, the unconditional mean (the long-run average time between events) exists and is given by

2.4 Standardized residuals for model diagnostic

A critical step in the empirical application of the SAP–W–ACD model is the evaluation of the standardized residuals. Once the parameters are estimated, the residuals are defined as [19]. According to the model assumptions, if the SAP–W–ACD specification is correct, the sequence should be independent and identically distributed (i.i.d.) with a mean of 1.

2.4.1 Moment properties and identification constraint

The r-th raw moment of the SAP–W distribution is essential for defining the structural properties of the innovation term. Substituting the SAP–W, PDF into the moment definition

we obtain Equation 11

To evaluate this integral analytically, we utilize the power series expansion of the cosine function via the Taylor series:

Let . Also, expanding as Equation 12

Using the binomial expansion and the Gamma integral , the r-th moment is derived as Equation 13

To satisfy the ACD requirement E[ϵ_i] = 1, we set r = 1 and define the scaling factor Λ such that

By imposing the identification constraint λ = 1/Λ, the scale parameter is restricted to a function of the shape parameters. This mapping resolves the structural competition between the intercept ω and the scale parameter, allowing ψ_i to be uniquely interpreted as the conditional mean duration.

For the numerical implementation of the double infinite series in Equation 16, we utilize truncation limits of M = 50 for the outer summation over m and N = 50 for the inner summation over n. Given the rapid decay of the series terms due to the factorials in the denominator, numerical convergence within a precision tolerance of 10⁻¹⁰ was observed for the benchmark parameters (α = 0.5, k = 1.2) at m = 18 and n = 15. The resulting computed scaling factor is Λ(0.5, 1.2) = 0.4758226. This precise value was utilized to set the restricted scale λ = 1/Λ, thereby ensuring that the innovation distribution possesses a unit mean (E[ϵ_i] = 1) as required by the ACD framework.

2.4.2 Analytical convergence of the scaling factor

To establish the mathematical validity of the scaling factor Λ defined in Equation 16, we demonstrate the absolute convergence of the underlying double infinite series. This expansion originates from the trigonometric component of the innovation density, characterized by the term: where F(x) = 1−e^−(x/λ)k represents the baseline Weibull CDF. By defining , we observe that since F(x)∈[0, 1], the argument y is strictly bounded within the interval [0, π/2].

The Taylor expansion converges absolutely for all real y. Furthermore, by substituting the power series representation , the general term T_{m, n} of the resulting double series satisfies the following asymptotic bounds characterized as in Equations 15 and 16:

uniformly in n, and

uniformly in m. Consequently, for a fixed index m, the inner series over n converges absolutely by the ratio test, while the outer series over m is dominated by the convergent series . By the Weierstrass M-test [20], the double series converges absolutely for all α>0, α≠1, and k>0.

2.4.2.1 Boundary case analysis

The robustness of the scaling factor is maintained at the limits of the parameter space:

• Case α → 1: As α approaches unity, the term (α^F(x)−1)/(α−1) converges to F(x) via L'Hôpital's rule. This indicates that the singularity at the unit boundary is removable. For α in a neighborhood of 1, the series reduces by continuity to the standard Weibull case. In empirical applications, α is typically sufficiently far from unity to capture heavy-tailed behavior, ensuring the stability of the numerical evaluation.

• Case k → ∞: In the limit k → ∞, the Gamma function Γ(1+1/k) → 1. Because the term is monotonically bounded by for all k>0, the convergence is uniform. By the Dominated Convergence Theorem, the limit exists and remains finite.

• Case k → 0⁺: As k → 0⁺, the Gamma function exhibits super-exponential growth, . However, this is precisely compensated by the exponential decay of the term . The product remains bounded, ensuring Λ = O(1) as k → 0⁺, consistent with a well-defined limit for heavy-tailed distributions.

2.4.2.2 Numerical validation

This analytical foundation ensures that Λ is a reliable scaling factor for high-frequency financial modeling. As shown in Table 1, the empirical means of the simulated SAP-W innovations are consistently close to unity across a wide range of parameter combinations. This confirms that the scaling factor effectively enforces the unit-mean condition required for the normalized innovation process.

2.4.3 Standardized innovation distribution

By substituting the identification constraint λ = 1/Λ into the general density, the PDF of the standardized innovations ϵ_i~SAP–W(k, α) is expressed as follows in Equation 17:

2.4.4 Conditional PDF of duration

Given the transformation ϵ_i = x_i/ψ_i and the Jacobian |dϵ_i/dx_i| = 1/ψ_i, the conditional PDF of the observed duration x_i given the filtration is Equation 18

where is the parameter vector.

Thus, the conditional quantile of x_i is Equation 19:

2.4.5 Conditional intensity function

The conditional intensity function provides crucial insights into market dynamics by quantifying the instantaneous probability of an event occurring given the time elapsed since the last transaction and past market information. The formula of the conditional intensity function is Equation 20

The conditional intensity function characterizes the instantaneous probability of an event, dynamically scaled by the conditional mean ψ_i to reflect varying market speeds. Its sine-alpha-power specification allows the model to capture complex, non-monotonic conditional intensity patterns such as consolidation periods and momentum building, while the standardization Λ ensures that the innovation process remains scale-invariant with a unit mean.

2.5 Model estimation

This section details the estimation procedures for the SAP-W distribution and the subsequent ACD frameworks. Frequentist estimation is conducted via MLE. The numerical properties of the SAP–W distribution allow for flexible and complex shapes in the conditional intensity of durations.

2.5.1 Frequentist estimation of SAP-W distribution

The parameter vector of the SAP–W distribution, θ = (λ, k, α)^⊤, is estimated using the MLE method. By maximizing the likelihood function associated with the observed data, the frequentist estimates of the parameters λ,k, and α are obtained.

Let x = (x₁, x₂, …, x_n) denote a random sample of n independent and i.i.d. observations drawn from the SAP–W distribution. The likelihood function L(θ∣x) is defined as the joint probability density of the sample and is given by Equation 21

The log likelihood function is Equation 22

Score Functions: The MLEs of the parameters are obtained by solving the score equations. The partial derivatives of the log-likelihood function ℓ(θ∣x) with respect to the parameters are given below in Equations 23–25

Likelihood Functions for SAP–W–ACD ModelFor observed durations x₁, x₂, …, x_n and initial values for ψ₁, the conditional likelihood is . Which become Equation 26

Conditional Log-Likelihood Function in Equation 27

Score Functions: The MLEs are obtained by solving the score equations. These score functions are defined as the partial derivatives of the log-likelihood function ℓ(θ∣x) with respect to the model parameters in Equations 28–35

Since the Fisher information matrix is complicated to derive analytically, the observed information matrix J(θ) is used to construct confidence intervals for the model parameters. The observed information matrix is given by Equation 36

2.6 Competing ACD model specifications

To assess the empirical performance of the proposed SAP–W–ACD model, we compare it with several competing specifications. To confirm model identifiability, each innovation process is normalized to have a unit mean (E[ϵ_i] = 1). The following definitions provide the conditional log-likelihood functions for the competing and proposed specifications.

SAP-Lomax-ACD mode :

Conditional Log-Likelihood for SAP-Lomax-ACD is given by Equation 37

SAP-Gompertz-ACD:

Conditional Log-Likelihood for SAP-Gompertz-ACD is given by Equation 38

2.7 Model comparison

This section outlines the frequentist metrics employed to evaluate the proposed SAP–W–ACD model against competing specifications. These metrics facilitate the identification of the optimal model for capturing the complex dynamics inherent in duration data by balancing goodness-of-fit with parsimony. Specifically, this study utilizes the Akaike Information Criterion (AIC), the Bayesian Information Criterion (BIC), and the Hannan-Quinn Information Criterion (HQIC), defined as: , , . Where denotes the maximized log-likelihood, p represents the number of estimated parameters, and n is the sample size.

2.8 Simulation study

This section assesses the numerical stability, parameter identifiability, and finite-sample performance of the proposed SAP–W distribution and its dynamic expansion, the SAP–W–ACD model. We evaluate the robustness of the suggested estimators in both static and dynamic data-generating environments using Frequentist and Bayesian frameworks.

2.8.1 Design and data-generating process

Monte Carlo simulations are conducted for sample sizes n∈{100, 200, …, 1000}. To ensure numerical stability and to properly evaluate the asymptotic properties of the Maximum Likelihood Estimators particularly given the complexity of the likelihood function involving double infinite series and trigonometric components each design is implemented with R = 1000 independent replications. The finite-sample performance of the estimators is assessed using the Average Bias (AB) and the Mean Squared Error (MSE), defined as

The data-generating process (DGP) follows the inverse transform method.To ensure the stability of the duration process and reproducibility of the results, the recursion is initialized by setting the first conditional mean to the unconditional mean of the process, defined as . Since the innovations are i.i.d., the first innovation ϵ₁ is extracted directly from a unit-mean SAP-W distribution by applying the quantile function to uniform random variables P∈(0, 1). With ψ₁ and ϵ₁ determined, the first observed duration is formed as x₁ = ψ₁ϵ₁. This initial duration provides the necessary lag to update the conditional mean ψ_i for i>1 through the standard ACD structure. Subsequently, all remaining observed durations are formed as x_i = ψ_iϵ_i, where ψ_i is governed by the recursive ACD process. This procedure ensures that each time point i is sampled directly from the conditional quantile function of the model.

2.8.2 Data-generating process for the SAP–W–ACD model

To facilitate the reproducibility of the simulation and empirical results, the sequence for generating the SAP–W–ACD duration series is summarized in the following steps:

• Computation of the Scaling Factor: Given the shape parameters (α, k), compute the scaling factor Λ using the double infinite series defined in Equation 16 (or via numerical integration). Set the restricted scale parameter λ = 1/Λ to satisfy the unit-mean constraint E[ϵ_i] = 1.

• Initialization of the Recursion: Set the initial conditional mean ψ₁ to the unconditional mean of the process to ensure steady-state initiation:

• Generation of Innovations (ϵ_i): For each time step i = 1, …, n, generate a stochastic innovation ϵ_i by applying the SAP–W quantile function Equation 10.

• Construction of Observed Durations (x_i): Form the observed duration for the current step as the product of the conditional mean and the innovation: x_i = ψ_i·ϵ_i

• Recursive Update of the Conditional Mean (ψ_i): Update the conditional mean for the subsequent period using the linear ACD transition equation: ψ_i+1 = ω+γ₁x_i+β₁ψ_i. Repeat steps 3 through 5 until the target sample size n is reached.

2.8.3 Estimation methods

To ensure a thorough understanding of model dependability and parameter uncertainty, we employ MLE to evaluate the asymptotic properties and efficiency of the frequentist estimators.

2.8.4 Simulation design and model identifiability

The simulation is structured to assess the model performance and parameter identifiability at two distinct levels:

• 1. Set 1: Baseline SAP–W Distribution

In the static case, we focus on the baseline innovation parameters. The parameter vector is defined as θ = (α, k, λ). This set serves as a mathematical benchmark to evaluate the MLE performance for the unconstrained distribution, where the scale parameter λ is treated as an independent free parameter to ensure the numerical recovery of the baseline Weibull characteristics. To evaluate the MLE across increasing sample sizes, the true values are set at α = 0.5, k = 1.2, and λ = 0.8.

• 2. Set 2: SAP–W–ACD Model

In the dynamic case, we evaluate the model's ability to capture ACD clustering. The dynamic parameter vector is defined as θ = (ω, γ₁, β₁, α, k). Crucially, this set evaluates the performance of the scaled unit-mean innovation by imposing the identification constraint λ = 1/Λ. In this constrained framework, the scale parameter is no longer estimated independently but is strictly restricted to ensure E[ϵ_i] = 1. This restriction is fundamental for theoretically justifying the interpretation of ψ_i as the conditional expected duration, as it ensures the innovation term is properly normalized across the entire parameter space, satisfying the core requirement of the ACD framework. The true values are fixed at ω = 0.2, γ₁ = 0.15, β₁ = 0.7, α = 0.5, and k = 1.2.

2.9 Results of frequentist simulation

The consistency and numerical stability of the MLE are confirmed by the results presented in Tables 2, 3. As the sample size n increases, AB and MSE for all parameters systematically decrease toward zero, demonstrating monotonic convergence and asymptotic consistency. The results illustrate the distinction between the two models and the successful application of the unit-mean identification constraint. While the baseline distribution in Table 2 treats the scale parameter λ as a free parameter, the SAP–W–ACD model in Table 3 eliminates λ by imposing the restriction λ = 1/Λ. This resolves the structural identification conflict between the intercept ω and the innovation scale, ensuring the duration process is properly normalized and that ψ_i is uniquely identified as the conditional mean.

As the sample size n increases, the AB and MSE for all parameters systematically decrease toward zero, indicating asymptotic consistency. The visual representation of these convergence trends is provided in Figures 3, 4. The robust convergence observed in the constrained ACD specification Table 3 demonstrates that the scaling factor Λ is numerically stable and that the unit-mean property is effectively maintained. This convergence validates the structural integrity of the model and justifies the interpretation of ψ_i as the conditional mean duration in the subsequent empirical analysis.

Figure 3

Figure 4

2.10 Empirical results

The empirical analysis is based on high-frequency exchange rate duration data for three East African economies: Ethiopia, Kenya, and Somalia. The data consists of the time intervals (measured in seconds) between consecutive exchange rate adjustments. These datasets were obtained from financial market records and official central bank databases. To facilitate reproducibility, the raw exchange rate data and the derived duration series are provided as a Supplementary File accompanying this manuscript. This specific type of data is crucial for understanding market liquidity and the intensity of information arrival in developing financial markets.

The descriptive statistics for the exchange rate durations are presented in Table 4. All three countries exhibit positive skewness and high kurtosis, indicating that the distributions are leptokurtic and heavy-tailed. Ethiopia displays the most extreme values, with a kurtosis of 20.30. Notably, the minimum duration for all countries is 86,400 s (corresponding to 24 h), and the maximum is 172,800 s (48 h), suggesting that exchange rate updates are tied to daily or bi-daily reporting cycles.

The visual analysis of the durations supports the statistical results. The density plots in Figure 5 confirm the statistical findings, showing a sharp primary peak at the 24-h mark and a secondary smaller peak at 48 h. This bimodal structure suggests that while most updates occur daily, there is a consistent subset of data where updates take two days, necessitating a flexible model like the SAP-W to capture these nuances. Furthermore, Figure 6 displays the ACF for each country. Significant spikes that exceed the confidence intervals (dashed lines) are observed across multiple lags. This presence of significant autocorrelation indicates “duration clustering,” where periods of high market activity (short durations) tend to be followed by high activity, and periods of low activity (long durations) follow low activity. The existence of this temporal dependence justifies the use of the ACD framework for these datasets.

Figure 5

Figure 6

2.11 Frequentist results

Table 5 displays the parameter estimates for the three competing models in Ethiopia, Kenya, and Somalia. Strong duration clustering is confirmed by the results for the SAP–W–ACD and SAP-Lomax-ACD models, which demonstrate considerable persistence in exchange rate durations as demonstrated by β₁ values greater than 0.91 and large γ₁ coefficients. The majority of parameters have minimal standard errors and are accurately estimated, indicating statistical dependability. Notably, the SAP-Gompertz-ACD model produces small values for the autoregressive components (ω, γ₁, β₁), indicating that the SAP-W-ACD offers a more reliable and stable representation of the temporal dynamics in these financial markets, whereas the SAP–W and SAP-Lomax frameworks successfully capture the dynamic dependencies.

2.12 Model comparison and adequacy

The empirical results for the model comparison are summarized in Table 6. Based on the information criteria, the SAP-W-ACD model is identified as the optimal specification for all three countries. Notably, the SAP-W-ACD demonstrates a substantially higher log-likelihood (LL) compared to the SAP-Lomax and SAP-Gompertz specifications. This significant discrepancy particularly the large gap observed in the Ethiopian market is attributed to the structural suitability of the Weibull baseline in capturing the specific peakedness (k = 4.8452) and monotonic hazard dynamics of these exchange rate durations, which the competing baselines fail to mathematically accommodate. To ensure the reliability of these results, all models were cross-verified using multiple starting values for global optimization. Diagnostic results confirm model adequacy for Ethiopia and Somalia, where corrected Ljung-Box p-values exceed the 0.05 threshold (0.2086 and 0.0958, respectively). However, the model remains statistically inadequate for the Kenyan market (p < 0.05), indicating significant remaining serial correlation in the residuals.

2.12.1 Residual analysis

To confirm that the SAP–W–ACD model has successfully captured the temporal dependence in the exchange rate durations, we examine the standardized residuals defined as . According to the model assumptions, if the specification is correct, the residuals should be independent and i.i.d.

In contrast to Figure 6, which showed significant autocorrelation in the raw duration data, Figure 7 displays the ACF of the SAP–W–ACD residuals. The absence of significant lags in it proves that the duration clustering has been effectively eliminated, making the residuals approximately white noise.

Figure 7

2.12.2 Goodness-of-fit and fitted density

Figure 8 shows the distributional flexibility of the SAP–W innovation process. The empirical histogram of standardized durations appears with the fitted SAP–W density curve (red line). For all three countries, the model shows a high degree of accuracy in capturing the heavy right-tail behavior and the peaked mode of the exchange rate adjustments.

Figure 8

2.12.3 Conditional intensity analysis

A key feature of the SAP–W–ACD model is its ability to characterize market risk through the conditional intensity function. Figure 9 presents the estimated Conditional Intensity. For Ethiopia, Kenya, and Somalia, the conditional intensity is monotonically increasing (k>1).

Figure 9

This indicates a “positive aging” effect: as the time elapsed since the last exchange rate adjustment grows, the conditional probability of a new adjustment increases. This pattern reflects the mounting market pressure and the urgency for rate adjustments in these East African economies.

2.12.4 Diagnostic plots: Q-Q plots and PIT histograms for model validation

Figure 10 displays the distributional diagnostics used to verify the fit of the SAP-W-ACD model for Ethiopia, Kenya, and Somalia. To account for the daily reporting of exchange rates, a stochastic jitter was applied to ensure the data aligns with the continuous theoretical model.

• Quantile-Quantile (Q-Q) Plots: As seen in Q-Q Plots in Figure 10, the blue dots follow the 45-degree reference line very closely, which proves that the SAP-W distribution is accurate. The slight curve at the top of the line is normal and represents the rare, long gaps found in frontier currency markets.

• Probability Integral Transform(PIT) Histograms: PIT Histograms in Figure 10 shows that the histograms are mostly flat and stay near the red dashed line, indicating that the probability predictions are reliable. Although a minor U-shape appears due to extreme outliers, this level of uniformity confirms the model is highly adequate.

Figure 10

3 Conclusion

The SAP–W–ACD model, which expands the Sine Alpha Power-G family of distributions into a dynamic time-dependent framework, was created and evaluated in this study.The model offers a more reliable method for analyzing irregularly spaced financial data by eliminating the limitations of traditional ACD innovation distributions. This paper's theoretical focus involves a complete description of the SAP–W distribution and the creation of a scaling factor to meet the standard unit-mean needs of ACD models. Through simulation, we show that the Maximum Likelihood Estimators for the innovation and autoregressive parts of the model are both reliable and statistically consistent.

An empirical analysis of foreign exchange markets in Ethiopia, Kenya, and Somalia provides several key findings. First, the SAP–W–ACD model outperforms the Lomax-ACD, and Gompertz-ACD frameworks, offering a superior fit for the skewed and heavy-tailed duration data characteristic of these markets. Second, residual diagnostics confirm that the SAP–W–ACD model successfully accounts for the duration clustering (serial correlation) found in the raw data, particularly within the highly persistent markets of Kenya and Ethiopia.

Finally, the conditional intensity analysis shows monotonically increasing conditional intensity in all three countries (k>1), suggesting that the conditional probability of an exchange rate adjustment rises with the duration of time since the last event.

In practical terms, the SAP–W–ACD model provides East African central banks and financial authorities with a more accurate tool for tracking liquidity risks and currency market pressure. Subsequent research could build upon this framework by integrating exogenous variables, employing Markov-switching models to identify structural shifts, or utilizing Bayesian inference to better estimate parameter uncertainty in limited datasets.

3.1 Limitations and future research

The primary limitation of this study is the inability of the standard ACD(1,1) specification to fully filter the durations in highly active markets like Kenya. The residual autocorrelation suggests that the linear (1,1) lag structure is insufficient for capturing the volatile clustering and high-frequency updates unique to the Kenyan exchange rate. Consequently, future research should explore higher-order specifications, such as ACD(2,1) or ACD(2,2), as well as advanced frameworks like Log-ACD and Fractionally Integrated ACD (FIACD) models to better account for long-memory persistence.

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