Statistical Approach to Examining the True Status of Long Memory and Volatility Persistence in PM10 Air Pollutant at Different Regions of Malaysia: A Methodical Methodology

doi:10.21203/rs.3.rs-3494368/v1

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Statistical Approach to Examining the True Status of Long Memory and Volatility Persistence in PM10 Air Pollutant at Different Regions of Malaysia: A Methodical Methodology

https://doi.org/10.21203/rs.3.rs-3494368/v1

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Air pollution continues to be an international problem that endangers both human health and the environment. Over the past few decades, air pollution in Malaysia has emerged as a serious potential risk due to accelerated economic expansions and seasonal transnational pollution. Particulate matter atmospheric air pollutants in Malaysia have been identified as the most rampant and dominant in the air pollution index (API) amongst other criteria pollutants. The aim of this study is to investigate the statistical issues of long memory and volatility persistence in the level of particulate matter emission from 1 January 2011 to 31 December 2021 in fourteen continuous air monitoring stations of industrial, urban, and suburban categories using the main and partitioned series before and after the regimes of break. The Ordinary Least Square Cumulative Sum (OLS-based CUSUM) test was employed to partition the original series in each monitoring station based on its estimated break dates. The long memory parameter d alongside its standard error was estimated through three techniques namely, Geweke and Porter-Hudak, Fractionally Differenced Sperio, and Exact Local Whittle estimation. The issue of volatility persistence was investigated using the hybrid of the Autoregressive Fractionally Integrated Moving Average (ARFIMA) and Generalized Autoregressive Conditional Heteroskedastic (GARCH) model. The results confirm evidence of a mean-reverting form of long memory with a higher degree of persistence in the main series and volatility persistence in both the main and partitioned series that encountered structural break. This confirms that the data-generating process of particulate matter pollutant in Malaysia possesses true long memory and volatility persistence not spurious due to neglected structural break problem. Maximum emissions in all monitoring sites were observed during the pre-break regime except for Kota Kinabalu station where it occurred during the post-break regime. Most series were characterized by higher values of kurtosis and skewness implying the significant fluctuation and non-Gaussian behavior in the affected series.

Time Series

Structural Break

ARFIMA-GARCH

ARCH Test

Long Memory

PM10

Nine out of ten people today breathe air that exceeds the acceptable threshold of pollutants approved by the World Health Organization (WHO), making air pollution an international issue responsible for roughly seven million annual global deaths through stroke, heart disease, lung cancer, diabetes, asthma, and other chronic respiratory diseases (Goshua et al. 2021; Martins et al. 2021). Air pollution can primarily be defined in terms of air quality index (AQI), consisting of six standard pollutants, including carbon monoxide (CO), ozone (O₃), particulate matter (PM₁₀ and PM_2.5), nitric oxide (NO), and nitrogen dioxide (NO₂). These pollutants exist in the form of solid, liquid droplets, or gas molecules (Jacobson, 2005; Szczurek & Maciejewska 2015). They also cause hazy and smoggy air, unpleasant smells, eutrophication, and global climate change (Bartoletti & Loperfido 2010; Mishra & Goyal 2015; Noel De Nevers, 2010).

The problem of air pollution in Malaysia on the environment and human health has been a matter of serious concern (Al-Dhurafi et al. 2018; Alyousifi et al. 2018; Fauziah et al. 2021; Koo et al. 2020; Masseran & Safari 2020; Raffee et al. 2022; Tajudin et al. 2019; Usmani et al. 2020). To adequately handle this problem, the Malaysian government through the Department of Environment (DOE) increased the number of monitoring stations to sixty-five in rural, suburban, urban, and industrial locations (EQR, 2018).

The adverse effects of particulate matter (PM₁₀ and PM_2.5) atmospheric and microscopic air pollutants have since been established and identified as the most prevalent among other criteria pollutants in Malaysia affecting human health, animals, ecosystem, and environment (Fong et al. 2018; Manga & Awang 2018; Masseran & Safari 2020; Tajudin et al. 2019; Usmani et al. 2020). The sources of these particles are both natural and human made. Natural sources include radon, fog and mist, volcanic eruptions, forest fires, soot, and salt spray. The anthropogenic emission on the other hand includes motor vehicle emission, heat and power generation, industrial activities, waste incineration, and residential cooking.

The particulate matter pollutants cause numerous medical disorders in the human body such as tuberculosis and other cardiopulmonary diseases, lung cancer, asthma, cardiovascular diseases, bronchitis, premature death and reduces life expectancy (Bo et al., 2022a; Gao et al., 2022; Lala et al., 2023; Liu et al., 2022; Lu et al., 2023; Silva et al., 2013; Wang et al., 2022). Moreover, particulate matter emission on vegetation's surface reduces the rate of photosynthesis and changes in decomposition cycles which affects animal groups and plants themselves (Correa-Ochoa et al., 2022; Grantz et al., 2003; Vincenti et al., 2022). The problem of air pollution in Malaysia, in particular the PM₁₀ caused by haze episodes and other natural and human causes, has been identified as one of the major contributing factors to the upsurge in hospital admissions for the treatment of cardiovascular and other pulmonary diseases, which mainly affects children and the elderly. According to estimates, the country's annual inpatient cost due to haze episodes is roughly USD 91,000. It is estimated to have increased the death rate to 19 percent (Latif et al., 2018).

Sansuddin et al., (2011) analyzed the concentration of PM₁₀ atmospheric pollutant at four different Malaysian air monitoring stations of industrial and residential categories over the years 2000 and 2003 using Gamma distribution, log-normal distribution, autoregressive model (AR), moving average model (MA) and the autoregressive moving average (ARMA) model. They observed high emissions of the pollutant in all the considered monitoring stations during the year 2002. The Gamma distribution best fitted the emission of the pollutant at the industrial location (Nilai and Johor) while the log-normal distribution best fitted the emission in stations (Kota Kinabalu and Kuantan) of residential category. However, PM₁₀ events for the year 2003 were also estimated and the AR (1) simple model outperforms other time series models.

Ramli et al., (2023) investigated the performance of the Bayesian Model Averaging (BMA) by predicting the next day’s PM₁₀ emission in Malaysian Peninsular using seventeen-year air quality monitoring data from nine monitoring stations. The forecast accuracy of the model was examined using five different measures including Index of Agreement (IA), Kling-Gupta efficiency (KGE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE). They recommended the BMA model to be one of the predicting instruments for forecasting air pollutants, especially the PM₁₀, having displayed impressive performance in predicting the PM10 concentration for${R}^{2}$ value starting from 0.64 to 0.75 with IA measure starting from 0.84 to 0.91 while KGE starting from 0.61 to 0.74 in all the nine monitoring stations.

Bahiyah et al., (2017) investigated the concentration of PM₁₀ pollutants in three air monitoring stations namely, Taiping, Tanjung Malim, and Pegoh in the state of Perak using the year 2015 air quality data. The highest concentration of the pollutant was observed in the urban category, followed by industrial and then sub-urban. The emission of the pollutant was observed to have reached its peak in all monitoring stations as the year approaches its end specifically in September and October and decreased sharply in all monitoring stations during November. The peak period is associated with the wind from the dry southern monsoon season usually occurring between June and October, while the decrease might be due to washing away most of the pollutants in the atmosphere due to excessive rainfall. The wind flow during the period of transboundary haze event was traced using the Hybrid-Single Practice Lagrangian Integrated Trajectory (HYSPLIT) model suggesting anthropogenic sources of the pollutant are significantly contributing to high emission of PM10 in the air.

Sentian et al., (2019) investigated the long-term temporal behavior of air pollutants including PM₁₀ in twenty air quality monitoring stations across Malaysia over the period 1997 through 2015 using the Mann-Kendall test to examine the trend of the contaminant and HYSPLIT backward trajectory model to trace the source of PM₁₀ atmospheric pollutant. The annual average concentration of PM₁₀ varied significantly between monitoring stations while the annual average for other pollutants varied less with a lower coefficient of variation. Furthermore, the concentration of PM₁₀ contaminant was found to be increasing in five monitoring stations between the years and decreasing in the remaining fifteen stations. The result of the HYSPLIT model showed that high seasonal PM₁₀ events in most monitoring sites were due to transboundary pollution from neighboring Indonesia during the period of southwest monsoon.

Alyousifi et al., (2020) applied the Spatial Markov Chain (SMC) model to the daily maximum air pollution index (API) over three years (1 January 2012 to 31 December 2014) collected from 37 Malaysian air monitoring stations. The transitional probability for the station with healthy or good emission of the pollutant to remain in a healthy state is 0.814 given that its neighboring stations are in similar condition, and 0.708 for its neighboring stations are in a moderate state. The proportion of time for the station with healthy emissions to remain in a healthy state is 0.6 if its neighboring stations are in a healthy state. This proportion decreases to 0.4, 0.01, and 0 when the neighboring stations are in states of moderate, unhealthy, and very unhealthy emissions respectively. This translates to mean stations or regions with unhealthy state of air quality possess greater chances of causing closer stations or regions to be in the same state. They maintained that the concentration of Malaysian air pollution problems is in the southwest and the central western part of Peninsular.

Mohtar et al., (2018) studied the variation of major air pollutants in Malaysia under different seasonal conditions using the hourly dataset for four continuous air monitoring stations namely, Petaling Jaya, Shah Alam, Klang, and Cheras over the period 1 January 2005 through 31 December 2015. Nitrogen dioxide happens to be a lone gaseous that shows high variation between the considered monitoring sites while other gaseous pollutants show little variation in their emissions between the study sites. The variation of O3 and PM₁₀ pollutants was observed to be seasonal and most of the time exceeded the threshold of the Malaysia Ambient Air Quality Standard (MAAQS) and the National Ambient Air Quality Standard (NAQSS). Furthermore, in all the study sites, high emissions of O3 were observed during the northeast monsoon (January to March) while peak emission of PM₁₀ occurred during the southwest monsoon (June to September).

Manga and Awang (2018) modeled and predicted the spatiotemporal PM₁₀ events for thirty-four air monitoring stations over the period 1 January to 31 December 2011 using the Bayesian autoregressive method. They observed a decrease in PM₁₀ emissions as altitude increases. With increasing altitude, they noticed a decline in PM₁₀ emissions. Their findings confirm the findings of other researchers that temperature and location affect PM₁₀ emissions.

AL-Dhurafi et al., (2018) studied the Malaysian API data as compositional data by examining the proportion of each pollutant in determining the API value to identify the most dominant pollutant in the API dataset and conducting a 12-month forecast using the best of Vector Autoregressive (VAR) models. The API dataset for Klang City in Malaysia over the period January 2005 to December 2014 was utilized to accomplish the aims of the study. Each pollutant listed in the API was regarded as an independent component and the contribution of each component to the API was measured, the PM₁₀ pollutant was found to be the most contributing and dominant pollutant among all others in the study area. However, the VAR (1) simple model happens to be the best-performing model among all other candidates in forecasting the twelve-month forecasts based on the Akaike Information, Final Prediction Error, Schwartz, and the Hannan-Quinn criteria.

The purpose of this paper is to primarily investigate the true status of long memory statistical issue and volatility persistence in the level of particulate matter of size 10 microns or less in diameter (PM₁₀) atmospheric pollutant using both the original and subseries that encountered structural break. In about two decades many research interests were on confusing long memory with occasional structural break in mean since a stationary short memory $I\left(0\right)$ the process with a structural break in the mean can show a slower rate of decay in the Autocorrelation function (ACF) and other properties of the long memory process (Cappelli & Angela 2006; Charfeddine et al. 2012; Diedold & Inoue 2001; Granger & Hyung 2004; Jensen & Liu 2006). Therefore, long memory behavior in the time series process can spuriously be detected due to the neglected problem of a structural break in the series by showing some hyperbolic decay in the Autocorrelation Function (ACF) as an observed behavior of long memory. Furthermore, our systematic approach for long memory analysis will also be extended to investigating the level of variation in PM₁₀ events by applying the Autoregressive Conditional Heteroskedasticity (ARCH) and the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models on both original and subseries that encountered structural break. The paper will also study the behavior of the probability distribution of both original and subseries concerning normal distribution and study the nature of the flow of the pollutant using the coefficient of Skewness and Kurtosis, respectively. Some time series and statistical plots are also provided as well as some descriptive statistics for the concentration of the pollutant at selected monitoring sites.

Persistence as a statistical issue is defined as the presence of significant dependence between time observations that are far apart in time. Persistence may be identified through autocorrelation function (ACF) when its plot decays hyperbolically as lag increases contrary to the exponential decay of short memory processes such as in the autoregressive moving average (ARMA) process. It could also be identified by an increase in the spectral density function without limit as the frequency tends to zero, and lastly, by the rescaled adjusted range behaving as a function${n}^{h}$ where $h>\frac{1}{2}$, $n$ is the sample size, instead of${n}^{1/2}$ feature of the short memory process. The last form is also known as the Hurst phenomenon (Hosking, 1984).

Numerous researchers investigated the degree of long memory of air pollutants all over the world, for example, Windsor and Toumi (2001) confirmed the evidence of long memory in the level of ozone, PM_10, and PM_2.5 pollutants in the United Kingdom for up to 400 days. The results of their Kurtosis statistic show non-Gaussian statistics and high intermittency in the case of PM₁₀ and PM_2.5 pollutants but nearly Gaussian and low intermittency in the case of ozone. Pan and Chen (2008) proposed a control chart for monitoring the autocorrelated long memory particulate matter (PM₁₀) dataset in Taiwan. The control chart with the autoregressive fractionally integrated moving average (ARFIMA) model performed better than the chart with the autoregressive integrated moving average (ARIMA) model as confirmed by their result. In a related development, Chelani (2012) uses ten years (2000–2009) data to confirm long-term memory in the level of three pollutants, namely, carbon monoxide (CO), nitrogen dioxide (NO₂), and ozone (O₃), at traffic area of Delhi in India using the method of detrended fluctuation analysis (DFA). However, the method of rescaled range (R/S) analysis of long memory investigation was used to investigate the degree of persistence in the case of ozone, particulate matter, nitrogen dioxide, and sulfur dioxide pollutants in Mexico City during the years 1999 to 2014 period. The degree of persistency was observed to be higher during the rainy season and lower during the dry season (Meraz et al., 2015).

In another development, Barros et al. (2016) used fourteen-year (2000–2013) monthly data for energy production to investigate the statistical issues of persistency and seasonality in Brazil. The result indicates the presence of a mean reverting form of persistency with a single structural break in two of the reviewed series. However, seasonality is found to be an essential issue in modeling long memory for energy production. Barros et al. (2016) examined five components of global carbon dioxide emissions and their per capita for the years 1751 through 2009 using the long-range dependence technique. Their results revealed that the series possesses a permanent form of persistency, and the cement production component of the pollutant’s emission has the highest degree of persistency when compared with other components of gas, liquids, solids, and gas flaring. The series was observed to have encountered structural breaks with the highest degree of persistency after the Second World War. Persistence behavior in the level of five criteria pollutants was confirmed at four major industrial cities in China, namely, Guangzhou, Shanghai, Shenzhen, and Beijing for the period September 28 for the year 2013 until December 12 for the year 2015. Observed persistence was more severe in the cities of Shenzhen and Guangzhou compared to the remaining cities under review (Chen et al. 2016). Moreover, Gil-Alana and Solarin (2018) studied the impact of the United States (US) environmental policies on nitrogen oxides (NOx) and volatile organic compounds (VOC) pollutants for fifty years (1940–2014) and investigated the degree of persistence of these pollutants during the period. Their results revealed that all the shocks in the pollutants are permanent especially in the case of nitrogen oxides by displaying orders of integration significantly greater than 1. The US environmental policies on pollutants are yet to be effective by observing negative trends within the period covered by the research. For example, the emission of these pollutants and their per capita in the year 2014 are lesser in concentration than in the year 1970.

Yaya et al. (2020) investigated air quality in the ten most polluted cities of California, USA, by looking at statistical issues of seasonality and persistence. All pollutants were found to be seasonal in the ten cities. Persistence was not found in the level of PM_2.5 but in CO and O₃ pollutants. Gil-Alana and Trani (2019) estimated the time trend and order of integration for CO₂ emission across member states of the European Union (EU) from the year 1960 to 2013. Their study also examined the average CO₂ emission of the whole EU as a case alongside those of China and the US. Their results revealed the lone evidence of a decreasing trend in the United Kingdom (UK) and a transitory form of persistence while all other members of EU states reported an increase in the trend of the contaminants with permanent shocks in the level of pollutants. Also revealed, was another transitory form of persistence in the case of China but permanent shocks for the whole EU and US. Gil-Alana et al. (2020) further confirmed the decrease in the emission of some criteria pollutants in the London metropolis but with strong evidence of persistence across the pollutants. Caporale et al. (2021) used daily records for the particulate matter (PM₁₀) emission to investigate the trend and long-range dependence using the procedure of fractional integration framework in eight European capitals, namely, Amsterdam, Berlin, Brussel, Helsinki, London, Luxembourg, Madrid, and Paris for seven years (2014–2020). Temporary shocks in the level of pollutants were confirmed with a decrease in the fluctuation of the contaminant in Berlin, Brussels, and Paris.

The method of fractional integration was also found useful in other environmental data such as temperature and climatology (Bloomfield, 1992; Bruneau et al. 2020; Franzke, 2012; Gil-Alana, 2005, 2009, 2012; Pelletier & Turcotte 1997; Rybski et al. 2006; Vyushin & Kushner 2009; Yuan et al. 2019; Yusof et al. 2013), economics, and financial time series (Abbritti et al. 2016; Backus & Zin 1993; Connolly et al. 2005; Murialdo et al. 2020). Long memory detection avails researchers the opportunity to study long memory properties of the time series process under review and gives an idea of whether the persisting effect of a particular contaminant (if present) on the environment is temporary, permanent, or explosive.

Time series structural break occurs when there is a change in the appropriate method adopted for observing and defining a time series variable over time. These breaks could also be due to a single or multiple changes in used classification, definition of the variable, coverage, or instrument during the period of measuring the time series variable of interest. A time series process that encounters one or more structural breaks is said to have one or more discontinuities in its data-generating process (DGP), and the original series will be divided into subpopulations based on the number of structural breaks in the series. The statistical properties of these subseries within a broken regime need to be investigated with the view of getting more insight into each subseries based on their respective break dates. The procedure of estimating the break dates for each break regime could be accomplished within the structure of least squares regression (Ploberger and Krämer 1992).

The motivation for this paper is to amongst other things reduce wide gaps in research on air pollution studies in Malaysia concerning human health and the environment compared to neighboring and advanced countries (Usmani et al. 2020). To the best of our knowledge, little or no has been done to investigate the long memory and volatility problems for PM₁₀ pollutants in Malaysia, hence our article is the first to investigate issues of long memory and volatility persistence of air pollutants through change point detection. The remainder of this article is structured as follows: Section 2 describes the study area, data, and methodologies employed. Section 3 gives definition of some relevant terms used in the study, while Section 4 discusses the result while concluding remarks are provided in Section 5.

3.1.1 Study Area and Data Used

The study area for the research is Malaysia. The country is in southeastern Asia with thirteen states and three federal territories parted by the China Sea from south into two regions, namely, Peninsular Malaysia and East Malaysia. Peninsular Malaysia holds territorial waters with Singapore, Vietnam, Indonesia, and Thailand on land and on the sea. Along with Brunei and Indonesia, East Malaysia also has maritime boundaries with the Philippines and Vietnam. The population of the nation, which has its national capital in Kuala Lumpur, is believed to be above 32 million, making it the 43rd most populous nation in the world. Malaysia can be found between 0° 53' 82.652" and 7° 1' 19.014" north of the equator. It is located at longitude 101° 4' 8.44" and 119° 6' 722" east of the prime meridian.

The Department of Environment Malaysia (DOE) is vested with the responsibility of monitoring the air quality across Malaysia using the air pollutant index (API). The API is calculated based on the concentration of six major pollutants obtainable in the air. These pollutants are the ground-level ozone (O₃), carbon monoxide (CO), nitrogen dioxide (NO₂), Sulphur dioxide (SO₂), particulate matter of less than 10 microns in size (PM₁₀), and particulate matter of less than 2.5 microns in size (PM_2.5). Any of these pollutants with higher concentration in the atmosphere is regarded as the API value. The Malaysian API is conceived based on the US model from the United States Environmental Protection Agency (USEPA) which is categorized into five different states of air pollution Good (0–50), Moderate (51–100), Unhealthy (101–200), Very Unhealthy (201–300), and the Hazardous state at more than 300 (Al-Dhurafi et al., 2018; Alyousifi et al., 2018).

In this paper, the eleven-year daily record for the PM₁₀ emission obtained from the DOE over the periods 1 January 2011 to 31 December 2021 is utilized. Fourteen continuous air monitoring stations of industrial, urban, and suburban categories were considered for the conduct of this research. These stations are managed by a private company Alam Sekitar Sdn. Bhd. (ASMA) on behalf of the DOE Malaysia. The company uses the Beta Attenuation Mas Monitor (BAM-1020) from Met One Instrument, Inc. to observe the PM₁₀ atmospheric pollutants across the country (Bahiyah et al., 2017). Each of these selected air monitoring stations contains 3937 available time series observations instead of 4018 observations for the eleven-year period covered by the study. The gap of 81 observations in each station was due to the nonrecording of PM₁₀ measurements from 14 April 2017 to 3 July 2017 across Malaysia because of the upgrade work on air quality monitoring stations by the Malaysian authorities. Owing to the unavailability of a recognized stochastic or mathematical technique that is recommended for the estimation of missing data when they are consecutively many, these 81 consecutive missing observations in each monitoring station were not estimated. In other words, these consecutive missing data were overlooked because their ratio to the available ones was exactly 0.02, making them statistically insignificant (Rubin, 1976; Schafer & Graham, 2002; Sterne et al., 2009). However, other missing observations in the dataset were estimated by imputation method using the average value of available series for the specific year in which the observations were missed (Sterne et al., 2009; Van Buuren & Groothuis-Oudshoorn, 2011; Little & Rubin 2019). Table 1 gives additional information for each of the fourteen monitoring stations separately, such as geographical coordinates (latitudes and longitude), duration of the dataset, and the location category of each station as to whether it is industrial, urban, or suburban.

Table 1

Location of air quality stations, duration, and category
STATIONS	LATITUDE (North)	LONGITUDE (East)	DURATION (Day)	CATEGORY
Larkin	01° 29' 40.65"	103° 44' 09.50"	01/01/2011–31/12/2021	Industrial
Port Dickson	02° 26' 28.97"	101° 52' 00.68"	01/01/2011–31/12/2021	Urban
Shah Alam	03° 06' 16.98"	101° 33' 22.39"	01/01/2011–31/12/2021	Sub Urban
Kuala Selengor	03° 19' 16.70"	101° 15' 22.47"	01/01/2011–31/12/2021	Sub Urban
Cheras	03° 06' 22.44"	101° 43' 04.50"	01/01/2011–31/12/2021	Urban
Tanjung Malim	03° 41' 15.92"	101° 31' 28.17"	01/01/2011–31/12/2021	Sub Urban
Taiping	04° 53' 55.86"	100° 40' 44.78"	01/01/2011–31/12/2021	Industrial
Indera Mahkota	03° 49' 09.18"	103° 17' 47.57"	01/01/2011–31/12/2021	Sub Urban
Kota Kinabalu	05° 52' 46.87"	116° 03' 23.13"	01/01/2011–31/12/2021	Urban
Balik Pulau	05° 20' 13.61"	100° 12' 59.21"	01/01/2011–31/12/2021	Urban
Kangar	06° 25' 47.71"	100° 12' 39.84"	01/01/2011–31/12/2021	Sub Urban
Kota Bharu	06° 08' 50.75"	102° 14' 57.24"	01/01/2011–31/12/2021	Urban
Petaling Jaya	03° 07' 59.40"	101° 36' 28.83"	01/01/2011–31/12/2021	Industrial
Seberang Jaya	05° 23' 53.41"	100° 24' 14.20"	01/01/2011–31/12/2021	Sub Urban

3.1.2 Time Series Plot

The time series plot on the PM₁₀ series from the chosen monitoring stations is being done to show the graphical pattern of the dataset at successive time intervals. The pattern could provide information about the trend's nature, including whether it is seasonal or not, possible cause, and most interestingly it can be used to presume the series is stationary or otherwise. It can also be observed from Fig. 1 that the emission of PM₁₀ pollutants in all the considered monitoring stations drastically reduced from the year 2020 upward. This reduction could be due to the COVID-19 global pandemic which brought about movement restrictions of humans, vehicles, and closure of other human activities that causes the emission of the pollutant.

The time series process is said to be stationary if its statistical properties such as mean, and variance are not time-dependent and contain no trend or seasonal components. The results of the time series plot for each station as shown in Fig. 1 signify that their respective series are not stationary in both mean and variance. This necessitates the transformation of these series to stationary ones by taking first differencing to achieve the necessary stationary requirement at least in the mean of the series. If the non-stationary series fails to be transformed after the first differencing, the second differencing becomes necessary to achieve the necessary pre-condition stationary requirement for analyzing the dataset using either mean or variance time series models. It is obvious that the non-stationary series in Fig. 1 above are successfully transformed to stationary ones after taking the first differencing as shown in Fig. 2 below. This is because the series in Fig. 2 fluctuates around the central value of the series for each monitoring station which is not the case in Fig. 1.

It can also be observed from Fig. 2 that some points in the process fluctuate significantly below and above the central value of the process which could be due to variations between observations within a given monitoring station. It may also be observed from the same figure that fluctuations clustered in a manner that low changes are followed by low changes and high changes are also followed by their corresponding high or larger changes, hence volatility is observed to be time-variant perhaps due to the presence of autocorrelation in the process of the series. This pattern is otherwise known as volatility clustering. This will be checkmated later using the autoregressive conditional heteroskedastic (ARCH$\left(p\right)$) model initiated by Engle (1982) and the generalized autoregressive conditional heteroskedastic (GARCH$\left(p,q\right)$) model created by Bollerslev (1986).

3.1.3 Kurtosis

Kurtosis is employed in this study as a separate measure to examine the nature of the flow of particulate matter pollutant in the atmosphere. It is usually denoted by k and defined as

$$k=\frac{1}{N}{\sum }_{i=0}^{N-1}{\left(\frac{y\left(u\right)-⟨y⟩}{\sigma }\right)}^{4}-3$$

The above formula calculates the kurtosis coefficient based on the fourth moment around the mean of the given dataset. It is an important measure of descriptive statistics that can be used to describe some characteristics of a given dataset. It can also be used to measure the degree of fluctuation or intermittency in a time series dataset. A positive kurtosis coefficient or small standard deviation indicates that the distribution is more pointed (leptokurtic or short-tailed) than the normal distribution curve, a negative kurtosis coefficient or large standard deviation indicates that the distribution is more flattened (platykurtic) than the normal distribution curve while a zero-kurtosis coefficient indicates that the distribution is neither too flattened nor too pointed in relation to normal distribution curve. Thus, higher kurtosis realized in Table 2 indicates a high degree of intermittency or irregular fluctuation in particulate matter emission at most monitoring stations and the series is more pointed (leptokurtic) to the normal distribution curve.

3.1.4 Skewness

Skewness is simply defined as a measure of the asymmetry of the distribution. It is an important component of the normal distribution that can be visualized on the distribution’s bell curve when data points are not distributed symmetrically to either side of the median on a bell curve. Skewness informs the direction of outliers in the series as to whether they are negatively or positively skewed. In a positive skew, the tail of the normal curve is longer on the right side but longer on the left side when skewed negatively. Positive skewness means that the outliers of the distribution curve are concentrated more on the right and closer to the mean on the left while negative skewness indicates the outliers of the distribution curve are concentrated more to the left direction and closer to the mean on the left. The coefficient of skewness can be calculated using the formula as

$$skew= \frac{n\left(n-1\right)\left(n-2\right)}{n}\sum _{j=1}^{n}\frac{{\left({X}_{j}-\stackrel{-}{X}\right)}^{n}}{{\sigma }_{x}^{3}}$$

The above formula measures the degree of skewness in relation to the data’s mean, median, and mode relative to each other. Symmetric distribution is achieved when the mean of the series is in the middle of the normal distribution curve, which means there is no skewness. Non-symmetric distribution is observed when the mean of the series is not located in the middle of the curve, thus, there is a skewed distribution to either the right or left direction.

3.1.5 Autocorrelation Function (ACF)

The ACF is a powerful tool in time series analysis that can be used to measure the correlation between ${y}_{t}$ and ${y}_{t+h}$, where $h$ is the number of lead period into the future.

Let $\left\{{X}_{t}\right\}$ be a stationary time series process of the variable $X$ at time $t$. The ACF of the time series process $\left\{{X}_{t}\right\}$ at lag $k$ is then defined as

$${\rho }_{X}\left(k\right)= \text{C}\text{o}\text{r}\left({X}_{t+k}, {X}_{t}\right)= \frac{{\gamma }_{X}\left(k\right)}{{\gamma }_{X}\left(0\right)}$$

where ${\gamma }_{X}\left(k\right)=$Cov$\left({X}_{t+k}, {X}_{t}\right)$ is the autocovariance function (ACVF) of $\left\{{X}_{t}\right\}$ at lag $k$ and Cor$\left({X}_{t+k}, {X}_{t}\right)$ is the autocorrelation of the variables ${X}_{t+k}$ and ${X}_{t}$ (Brockwell & Davis, 2016).

It is a component of the correlogram that could be used for diagnosing the time series properties of the data including stationarity or otherwise of the time series process through ACF plot. The series is said to be stationary when the ACF plot decays exponentially fast or extremely fast in a linear pattern, otherwise, it is nonstationary.

3.1.6 Time Series

Time series could be transformed to stationary series by taking first, second or higher order of differencing. The lag or backshift operator, L is defined as ${LX}_{t}= {X}_{t-1}, \text{a}\text{n}\text{d} \text{h}\text{e}\text{n}\text{c}\text{e} \left(1-L\right){X}_{t}= {Y}_{t}$. If${Y}_{t}$ is stationary, then the series ${Y}_{t}$is I$\left(0\right)$. If the number of differencing conducted to make nonstationary series to stationary is one, then the original series ${X}_{t}$ is said to be integrated of order 1 denoted as ${X}_{t} \approx I\left(1\right)$, otherwise if the number of differencing is two, then${X}_{t} \approx I\left(2\right).$

Sometimes the number of differencing needed to achieve stationarity is not necessarily integral but fractional. In such situation we say the process is integrated of order $d$ denoted as ${X}_{t} \approx I\left(d\right)$ and can be represented as

$${\left(1-L\right)}^{d}{X}_{t}= {Y}_{t}$$

where $d$ is the fractional order of integration which could be any real number.

For any real value $d$, the polynomial in the left-hand side of Eq. (4) could be expressed in terms of Binomial expansion as

$${\left(1-L\right)}^{d}= {\sum }_{j=0}^{\infty }\left(\begin{array}{c}d\\ j\end{array}\right){\left(-1\right)}^{j}{L}^{j}=1-dL+\frac{d\left(d-1\right)}{2}{L}^{2}-\dots ,$$

It is obvious from (5) that the higher the value of $d$ in the expression the higher the level of the association between the observations in distant time. Hence the parameter $d$ is most critical in determining the degree of persistency within any series of interest.

If $d=0$ in (4), then ${X}_{t}= {Y}_{t}$, indicating that the process is short memory and covariance stationary and may be weakly (ARMA) autocorrelated with the values in the autocorrelation function decaying extremely and exponentially fast. If $d$ falls within the interval $\left(0, 0.5\right)$, the process $\left\{{X}_{t}\right\}$ remains covariance stationary with its autocorrelation function taking longer to decay to previous case of covariance stationary short memory $I\left(0\right)$; if $d$ happens in the interval $[0.5, 1)$, ${X}_{t}$ is not covariance stationary but remain mean reverting due impact of the shocks will gradually disappear in the long run. Lastly, if $d\ge 1$ the process $\left\{{X}_{t}\right\}$ is both not covariance stationary and mean reverting since the adverse effects of the shocks will permanently be in place until strong converting measure are in place (Barros et al. 2016).

The process ${X}_{t}$ is said to be fractionally integrated when the $d$ value in (4) takes positive fractional values and is called ARFIMA or fractionally ARIMA model when ${Y}_{t}$ is ARMA $\left(p, q\right)$. ARFIMA processes were first introduced into the literature by Granger and Joyeux (1980) and Hosking (1981). Since then, it was found useful to describe the behavior of time series and proved to be more robust compared to ARIMA model since it can model both short-range dependence (SRD) and long-range dependence (LRD) in time series data.

3.1.7 ARCH-LM Test

There exist various methods for testing the heteroskedasticity in time series data, one of the popular is the Lagrange Multiplier Autoregressive Heteroskedastic Conditional test (ARCH-LM Test) pioneered by Engle (1982). The test is based on the autocorrelation of the ordinary least squared residuals. This test is employed to test for the presence or otherwise of heteroskedastic effect in the series of PM₁₀ air pollutants used for the conduct of this study.

The principal idea behind ARCH model was to integrate the sequence of the conditional variance (volatility) $\left\{{h}_{t}\right\}$ for the series ${Z}_{t}$ such that:

$${Z}_{t}= \sqrt{{h}_{t}}{e}_{t}$$

where the sequence of error term $\left\{{e}_{t}\right\}$ is independently and identically distributed with the Standard Normal distribution having mean equals to 0 and variance equals to one, that is, $\left\{{e}_{t}\right\}\sim$IID N$\left(0, 1\right)$, and ${h}_{t}$ associated to the previous values of ${Z}_{t}^{2}$ by means of:

$${h}_{t}= {\alpha }_{0}+ {\alpha }_{1}{Z}_{t-1}^{2}+\dots + {\alpha }_{p}{Z}_{t-p}^{2}$$

$${h}_{t}= {\alpha }_{0}+\sum _{i=1}^{p}{\alpha }_{i}{Z}_{t-i}^{2},$$

for some positive integer of order $p,$ such that ${\alpha }_{0}>0, {\alpha }_{i}\ge 0, i=1, 2, \dots , p$ (Engle, 1982).

3.1.8 GARCH (p, q) Model

The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model initiated by Bollerslev (1986) is designed to generalize the ARCH model to allow previous conditional variances in the current conditional variance equation. These models have been explored tremendously all over the globe since their creation and over time different parameter estimation techniques have also been developed.

The GARCH $\left(p, q\right)$ process as generalization of the ARCH $\left(p\right)$ is given by:

$${h}_{t}= {\alpha }_{0}+\sum _{i=1}^{p}{\alpha }_{i}{Z}_{t-i}^{2}+\sum _{i=1}^{q}{\beta }_{i}{h}_{t-i}$$

following that ${\alpha }_{0}>0,$ ${\alpha }_{i}\ge 0,$ $i=1,\dots ,p,$ and ${\beta }_{i}\ge 0,$ $i=1,\dots ,q,$ with $\sum _{i=1}^{p}{\alpha }_{i}+\sum _{i=1}^{q}{\beta }_{i}<1$, this summation is a necessary and sufficient condition for the evidence of second order stationary solution (Bollerslev, 1986).

3.1.9 ARFIMA-GARCH Hybrid Model

The presence of GARCH type innovations is permitted for fractionally integrated I(d) processes with superimposed stationary ARMA components in their mean. This can offer a reliable approach to study the conditional mean and variance of a process that demonstrates evidence of long memory and time-varying volatility.

Given a fractionally integrated process:

$$({1-B)}^{d}{X}_{t}={\mu }_{t}$$

where ${\mu }_{t}$ is the ARMA (p, q) is process, Eq. (8) can be rewritten as:

$$\varphi \left(B\right){\left(1-B\right)}^{d}{X}_{t}= \theta \left(B\right){\epsilon }_{t}$$

where B is the lag operators, $\varnothing \left(B\right)$ and $\theta \left(B\right)$ are polynomials of order p and q respectively. The residual ${\epsilon }_{t}$ from the fitted ARFIMA model in (8) is allowed to take the innovations of GARCH family models for conditional heteroskedasticity firstly extended by Baillic (1996). The variance equation of the GARCH (p, q) model can be expressed as:

$${\epsilon }_{t}={Z}_{t}{\sigma }_{t}, \text{w}\text{h}\text{e}\text{r}\text{e} {Z}_{t} \sim {\psi }_{\tau }\left(0, 1\right)$$

$${\sigma }_{t}^{2}= \omega +\sum _{i=1}^{p}{\alpha }_{i}{\epsilon }_{t-i}^{2}+\sum _{j=1}^{q}{\beta }_{j}{\sigma }_{t-j}^{2}$$

where $p\ge 0, q>0, \omega >0, {\alpha }_{i}\ge 0,$ for $i=1,\dots , p,$ and ${\beta }_{j}\ge 0,$ for$j=1, \dots ,q.$

3.2.1 Long Memory Processes

The autocorrelation function $\rho (.)$ of an ARMA process at lag $h$ converges rapidly to zero as $h⟶\infty$ in the sense there exist $r>1$ such that:

${r}^{h}\rho \left(h\right)⟶0$ as $h⟶\infty .$ (10)

Fractionally integrated ARMA processes or an ARIMA $\left(p,d,q\right)$ is said to be stationary processes when the parameter d lies within the interval $\left(0<\left|d\right|<0.5\right)$ with extremely slowly decay in the autocorrelation function satisfying the equation:

$${\left(1-B\right)}^{d}\varphi \left(B\right){X}_{t}= \theta \left(B\right){Z}_{t},$$

where $\varphi \left(z\right)$ and $\theta \left(z\right)$ are polynomials of degrees $p$ and $q,$ respectively, under the condition that:

$\varphi \left(z\right) \ne 0$ and $\theta \left(z\right) \ne 0$ for all $z$: $\left|z\right|\le 1,$

$B$ is the backward shift operator, while $\left\{{Z}_{t}\right\}$ is a sequence of white noise with mean 0 and variance ${\sigma }^{2}$ (Brockwell & Davis, 2016).

3.2.2 Long Memory Parameter Estimation

There exist various techniques available in the literature for estimating the parameter of long memory statistical analysis. For robustness, three different methods are employed to estimate the long memory parameter alongside their respective standard error estimates to check the sensitivity of one method over the others. These methods are Geweke and Porter-Hudak denoted as fdGPH (Geweke and Porter-Hudak, 1983), Fractionally differenced Sperio, fdSperio (Reisen, 1994), and Exact Local Whittle, ELW (Shimotsu and Phillips, 2005).

The GPH method is concerned with fitting a parametric model to the log power spectrum to estimate the long memory parameter under the assumption that the time series possesses a steady spectral density function. The Sperio method on the other hand incorporates fractional differencing with spectral analysis at the same time to achieve stationarity and then estimate the long memory parameter, making the method a bit more computationally complex due to the combination of methods. Lastly, the long memory parameter can be estimated using the ELW method through a likelihood-based approach and is considered more errorless compared to other methods, especially under special circumstances.

3.2.3 Geweke and Porter-Hudak Method

The procedure for estimating the degree of persistency or long memory parameter in time series data developed by Geweke and Porter-Hudak (1983) is employed in this study. The procedure is semiparametric which is based on the simple linear regression of the log periodogram on a deterministic regressor and was found efficient by numerous researchers to estimate the degree of persistence (Shimotsu 2005).

The spectral density function $f\left(\theta \right)$ of the fractionally integrated process $\left\{{Y}_{t}\right\}$ is given by:

$$f\left(\theta \right)={\left[2sin\left(\theta /2\right)\right]}^{-2d}{f}_{u}\left(\theta \right)$$

where $\theta$ is the Fourier frequency, ${f}_{u}\left(\theta \right)$ is the spectral density corresponding to ${u}_{t},$ and ${u}_{t}$ is a stationary short memory noise with 0 mean.

Consider the set of harmonic frequencies ${\theta }_{k}=\left({2\pi }_{k}/n\right)$, where $k=0, 1, ... n/2$, $n$ is the sample size. Applying the logarithm to both sides of Eq. (3) we obtain

$$\text{l}\text{n}f\left({\theta }_{i}\right) = \text{l}\text{n}{f}_{u}\left(0\right)-d\text{l}\text{n}\left[4{sin}^{2}\left({\theta }_{k}/2\right)\right].$$

Equation (4) can be seen as

$$\text{l}\text{n}f\left({\theta }_{k}\right) = \text{l}\text{n}{f}_{u}\left(0\right)-d\text{l}\text{n}\left[4{sin}^{2}\left({\theta }_{k}/2\right)\right]+\text{l}\text{n}\left[\frac{{f}_{u}\left({\theta }_{k}\right)}{{f}_{u}\left(0\right)}\right].$$

based on Wang et al. (2007).

3.2.4 Fractionally Differenced Sperio (fdSperio) Method

The principal idea behind the fdSperio approach (Reisen, 1994) is to transform a nonstationary through fractional differencing before estimating the corresponding long memory parameter value. It involves fitting a model to the log power spectrum of the time series and the inferences about long memory parameters can be derived by analyzing the rate of decay of the spectrum at various frequencies. The nonstationary time series process $\left\{{Y}_{t}\right\}$ may be transformed to stationary series as:

$${Y}_{t}= {\left(1-B\right)}^{d}{X}_{t}$$

Again, $B$ is the backshift operator while $d$ is a differencing parameter. The long memory parameter is then estimated from the slope of the regression equation based on the smoothed periodogram with Parzen lag window as:

$$\text{ln}\left\{{f}_{s}\left({w}_{j}\right)\right\}=\text{ln}\left\{{f}_{u}\left(0\right)\right\}-d\text{ln}{\left\{2sin\left(\frac{{w}_{j}}{2}\right)\right\}}^{2}+\text{ln}\left\{\frac{{f}_{s}\left({w}_{j}\right)}{f\left({w}_{j}\right)}\right\}+\text{ln}\left\{\frac{{f}_{u}\left({w}_{j}\right)}{{f}_{u}\left(0\right)}\right\}.$$

where ${f}_{s}\left(w\right)$ is a smoothed periodogram function, $f\left(w\right)$ is a spectral density of the time series process. It implies that:

$$\text{ln}\left\{{f}_{s}\left({w}_{j}\right)\right\}\approx \text{ln}\left\{{f}_{u}\left(0\right)\right\}-d\text{ln}{\left\{2sin\left(\frac{{w}_{j}}{2}\right)\right\}}^{2}+\text{ln}\left\{\frac{{f}_{s}\left({w}_{j}\right)}{f\left({w}_{j}\right)}\right\}$$

since the term $\left\{\frac{{f}_{u}\left({w}_{j}\right)}{{f}_{u}\left(0\right)}\right\}$ is negligible compared with others on the right-hand side of the equation. This equation now has the form of simple regression equation of the form:

${y}_{j}=a+b{x}_{j}+{e}_{j},$ $j=1, 2, \dots ., g\left(n\right)$ (16).

Comparing (2) with one (1), we deduce that:

${y}_{j}= \text{ln}\left\{{f}_{s}\left({w}_{j}\right)\right\},$ $b= -d,$ ${x}_{j}= \text{ln}{\left\{2sin\left(\frac{{w}_{j}}{2}\right)\right\}}^{2},$ ${e}_{j}= \text{ln}\left\{\frac{{f}_{s}\left({w}_{j}\right)}{f\left(w\right)}\right\},$ and the constant $a= \text{ln}\left\{{f}_{u}\left(0\right)\right\}.$

The long memory parameter here can be estimated from the slope of the regression line on the log plot of the spectrum (Reisen, 1994).

3.2.5 Exact Local Whittle Estimation

It is a semiparametric method for estimating long memory parameters developed by Shimotsu and Philip (2005). The ELW is believed to be one of the most sensitive measures for long memory parameters by numerous researchers and gives reliable parameter estimation for both stationary and non-stationary parts of d, especially in dealing with persistent long memory. The ELW method is based on the Whittle likelihood function derived from the spectral density of the time series analysis. It involves maximizing the likelihood function with respect to the long memory parameter d. It is proved to be consistent and with the same $N\left(0, \frac{1}{4}\right)$ for all values of d provided the optimization range lies within the interval span less than $\frac{9}{2}$ and the mean of the process is known.

Consider a time series process $\left\{{X}_{t}\right\}$, $t=1, 2, \dots , T,$ under the assumption that the process follows a fractional Gaussian noise model with parameter d, such that

$${X}_{t}= {\sigma }_{t}{\epsilon }_{t}$$

where ${\epsilon }_{t}$ is a white noise process having zero mean and constant unit variance ${\sigma }^{2},$ that is, ${\epsilon }_{t} \sim WN\left(0, {\sigma }_{t}^{2}\right)$, and ${\sigma }_{t}$ a slowly varying function.

The exact local whittle likelihood function for the memory parameter d may be defined by:

$$L\left(d\right)= \frac{1}{2\pi }{\int }_{-\pi }^{\pi }log\left[\left(\frac{1}{2}-i\lambda \right)\right]-log\left[{\Gamma }\left(\frac{1}{2}+i\lambda \right)-2i\lambda log\left[cos\left(\lambda \pi \right)\right]-2dlog\left|{\Lambda }\left(\lambda \right)\right|d\lambda \right]$$

where ${\Gamma }$ is gamma function, and ${\Lambda }\left(\lambda \right)$ is the characteristics function of the process (Shimotsu and Phillips, 2005).

Each of these three methods was found useful in literature for estimating the long memory parameter of time series data and in characterizing the order of integration of the series. However, each of the methods also gives the estimated value of standard error for estimating the univariate long memory parameter to know the precision of the estimates. The three methods were employed separately to investigate the sensitivity of each method over the other using the same series and to know the true category of the long memory parameter. By making use of these three methods, the research becomes more robust, and informative and gives more reliable results, particularly on long memory analysis than most previous related articles that use only one or two methods.

As mentioned above, the statistical issue of long memory or persistent dependency in the time series process could also be detected when its ACF plot decays hyperbolically at higher lags. The ACF plot in Fig. 3 decay with hyperbolic rates at higher lags in all the sampled monitoring stations indicating that particulate matter measurements are highly correlated and long memory is evident in the data.

The evidence of long memory in the PM₁₀ for all monitoring stations series as displayed in Fig. 3 is one of the reasons to model our series with the Autoregressive Fractionally Integrated Moving Average (ARFIMA) model instead of the Autoregressive Integrated Moving Average (ARIMA) model. Furthermore, the ARFIMA model is designed to capture both the short- and long-range dependence in time series data and to generalize the ARIMA model which was designed to capture the short-range dependence. In view of the highlighted advantages and robustness of the ARFIMA model over the ARIMA model, it is decided to employ the ARFIMA technique to model the persistency and nonstationary linear component observed in the series.

3.2.6 Ordinary Least Square (OLS)-based Cumulative Sum (CUSUM) Test

The OLS-based CUSUM test is a powerful tool for detecting structural change in time series data. Modified and suggested by Ploberger and Krämer (1992), the test is based on the cumulative sums of the common ordinary least squares residuals and empirical fluctuation process (EFP). It is defined as

$${W}_{n}^{0}\left(t\right)= \frac{1}{\widehat{\sigma }\surd n}{\sum }_{i=1}^{\left[nt\right]}{\widehat{u}}_{i}, \left(0\le t\le 1\right)$$

where ${W}_{n}^{0}\left(t\right)$ is the standard Brownian bridge as $n \infty$, ${W}_{n}^{0}\left(t\right)=W\left(t\right)-tW\left(1\right)$, $\widehat{\sigma }= \sqrt{\frac{1}{n-k}{\sum }_{t=1}^{n}{\widehat{u}}_{t}^{2}}$, residual ${\widehat{u}}_{t}= {y}_{t}-{\varvec{x}}_{t}^{T}\widehat{\varvec{\beta }}$, ${y}_{t}$ is the observation of the dependent variable at time $t$, ${\varvec{x}}_{t}^{T}={\left(1,{x}_{t2},{x}_{t3},\dots ,{x}_{tk}\right)}^{T}$ is $k\times 1$ vector of observations on the independent variable, and $\widehat{\varvec{\beta }}$ is the estimate of the $k\times 1$vector of regression coefficients when dealing with standard linear regression model.

The process starts in 0 at $t=0$ and it also returns to 0 for $t=1$. With single structural shift alternative, the statistic reaches a peak around ${t}_{0}$. The null hypothesis ${H}_{0}$ of no structural break in the series is rejected if the process goes beyond either of the boundaries $\left(-\theta , \theta \right)$. In other word the significance level can be determined when ${S}^{0}={sup}_{0\le t\le 1}\left|{W}_{n}^{0}\left(t\right)\right|$ is larger than $\theta$ (Ploberger & Krämer, 1992; Zeileis et al. 2002). The test was found to be powerful in detecting even smaller changes within the series and robust in dealing with outlier values contained in the series (Taylor, 2000).

As mentioned above, we analyzed the daily dataset of the PM₁₀ pollutant for fourteen air monitoring stations across Malaysia from 1 January 2011 to 31 December 2021 as obtained from the Malaysian Department of Environment (DOE). Fourteen continuous air monitoring stations were selected for the conduct of this research based on their respective category and location. These analyses including the descriptive statistics were conducted on the original and portioned series using an open-source R statistical free software version 2023.06.1 + 524. Descriptive statistics was employed to visualize, quantify, and describe some basic features of the PM₁₀ concentration over the eleven-year period as provided in Table 2 to summarize some vital statistical information about the PM₁₀ series and to better understand the data behavior. Thus, we looked at statistical issues of minimum, maximum, and average concentration of pollutants over the period covered by the research. Furthermore, the table also contains coefficients of kurtosis and skewness for the study of intermittency of the concentration of pollutants and how the series relates to the normal distribution curve. The standard deviation was used to examine the extent to which observed values deviate from their mean value in each monitoring station.

Between the eleven-year period covered by the study, the minimum daily emission was $1.13\mu g/{m}^{3}$ observed at the Kota Kinabalu monitoring station of the state of Sabah while the maximum daily emission stood at $447.54\mu g/{m}^{3}$ as recorded in the Port Dickson monitoring site of the state of Negeri Sembilan. Petaling Jaya air monitoring station of the state of Selangor produces the highest average emission of $43.07\mu g/{m}^{3}$, Shah Alam station in the same state has the most volatile series at $26.67\mu g/{m}^{3}$, while Kota Kinabalu of the state of Sabah has the least volatile series at $12.88\mu g/{m}^{3}$. The table also provided the exact dates on which the minimum and maximum observations occurred. The distribution of the dates on which minimum observations occurred looks random compared to the distribution of the dates on which the maximum observations occurred. Eight of the fourteen air monitoring stations recorded their maximum values in June 2013 representing 57%, five stations in October 2015 representing 36%, and only one station in March 2019 representing 7%. However, most of the periods in which the maximum PM₁₀ emission occurred coincided with the period of transboundary haze events in Malaysia and other Southeastern Asian (SEA) countries. It is of interest to note that all series of the PM₁₀ emission follow a similar statistical pattern with a standard deviation less than the mean, with high values of kurtosis and skewness.

Skewness is a measure of how the mode, median and mean relate to each other in the dataset while kurtosis measures the distance of the dataset to its mean value. It could be observed that the standard deviation is less than the kurtosis coefficient in nine monitoring sites but greater than kurtosis in the remaining five monitoring stations, this translates to an unequal distribution of leptokurtic (short-tailed) and platykurtic (long-tailed) forms between the monitoring stations. The positive skewness coefficient indicates that the mass of the normal distribution curve is concentrated more on the right but closer to the mean of the distribution on the left. The results of our kurtosis analysis indicates that the emission of PM₁₀ pollutants occurs at irregular interval or in other words there is non-Gaussian behavior in the emission of the pollutants at most monitoring sites but nearly Gaussian in the Kota Bharu monitoring site by reporting the least value.

Table 2

Results of Descriptive Statistics for PM₁₀ Main Series
Station	Minimum Date	Minimum	Maximum Date	Maximum	Mean	Stdev.	Skewness	Kurtosis
Larkin	December 22 2011	1.83	June 20 2013	348.63	38.3	20.25	4.35	39.06
Port Dickson	August 3 2016	6.88	June 23 2013	447.54	37.75	25.27	4.79	45.24
Shah Alam	April 2 2017	1.33	June 24 2013	362.38	43.03	26.67	4.06	28.64
Kuala Selengor	February 5 2015	2.50	June 24 2013	355.50	36.26	25.02	3.77	24.56
Cheras	March 26 2017	8.50	June 24 2013	318.83	40.81	23.19	3.39	22.10
Tanjung Malim	March 7 2015	1.46	June 24 2013	290.83	26.78	19.30	3.66	26.96
Taiping	April 21 2011	3.08	October 4 2015	281.00	38.66	21.93	2.45	13.30
Indera Mahkota	December 7 2013	1.67	June 23 2013	265.13	30.67	17.81	3.15	22.84
Kota Kinabalu	September 10 2011	1.13	March 15 2019	212.40	29.35	12.88	1.52	11.96
Balik Pulau	January 5 2018	5.49	October 4 2015	328.13	34.65	21.10	3.83	35.83
Kangar	October 5 2014	3.04	October 21 2015	338.08	31.06	18.32	3.56	40.84
Kota Bharu	October 24 2011	1.92	October 22 2015	193.17	37.60	19.19	1.45	4.38
Petaling Jaya	November 20 2016	1.79	June 24 2013	369.67	43.07	24.30	4.42	34.95
Seberang Jaya	July 11 2021	6.77	October 21 2015	342.67	42.49	23.51	3.10	24.41

The higher the kurtosis coefficients deviate from zero, the higher it is non-Gaussian and the lower it is closer to zero, the more it is Gaussian (Windsor and Toumi 2001). The fact that our calculated kurtosis coefficients significantly deviate from zero with positive values, shows that the emission of the particulate matter pollutant is not normally distributed within and between the considered air monitoring stations. The results from the kurtosis and skewness components of normal distribution both reveal that the emission of particulate matter pollutants is not normally distributed and confirm the presence of positive outlier values in the level of pollutants within and between the considered air monitoring stations.

Table 3 gives the result for the long memory parameter estimates for the original series for each air quality monitoring station separately. Three different approaches for estimating the long memory parameter namely, the Geweke and Porter-Hudak (fdGPH) estimate, the Sperio (fdSperio) estimate and exact local whittle estimate by Shimotsu and Phillips proven to be asymptotically normally distributed at a given optimization range.

Table 3

Results of Long Memory Parameter Estimation for PM10 Main Series
STATION	fdGPH ($d)$	s.e (fdGPH)	fdSperio$\left(d\right)$	s.e (fdSperio)	ELW(d)	s.e (ELW)
Larkin	0.4953	0.0866	0.4653	0.0344	0.5300	0.0275
Port Dickson	0.5074	0.0834	0.4882	0.0392	0.4850	0.0276
Shah Alam	0.4936	0.0838	0.4450	0.0322	0.4166	0.0276
Kuala Selengor	0.3867	0.0911	0.4052	0.0371	0.4388	0.0276
Cheras	0.4092	0.0971	0.4514	0.0341	0.4371	0.0276
Tanjung Malim	0.5025	0.0882	0.4710	0.0365	0.4266	0.0276
Taiping	0.4873	0.0856	0.4603	0.0341	0.4330	0.0276
Indera Mahkota	0.4862	0.0853	0.5060	0.0348	0.5052	0.0276
Kota Kinabalu	0.5811	0.0940	0.5455	0.0355	0.4711	0.0276
Balik Pulau	0.5245	0.0780	0.5199	0.0377	0.3897	0.0276
Kangar	0.6220	0.0935	0.5780	0.0361	0.3406	0.0276
Kota Bharu	0.4497	0.1059	0.5660	0.0236	0.4113	0.0276
Petaling Jaya	0.3638	0.1060	0.3858	0.0369	0.4153	0.0276
Seberang Jaya	0.5776	0.0985	0.5706	0.0353	0.3834	0.0276

It can be seen from Table 3 that all the PM₁₀ series were characterized in terms of long memory with parameter estimation within the range (0,1). Each estimate of the long memory parameter for all three methods has its standard error estimate in the next column on a cell corresponding to the value of the parameter. The long memory parameter estimation as contained in Table 3 reveals both stationary and nonstationary categories of the parameter but all with transitory (nonpermanent) shocks of the pollutant. It can be seen from the table that the fdGPH method is more sensitive by producing more nonstationary parameter estimates followed by the fdSperio method than the ELW method observed to have reported the least parameter estimates of long memory compared to other methods. The ELW is also observed with approximately the same constant standard error for each monitoring station. Hence, the long memory is present in all the considered monitoring stations at different levels of shock but with a common category (mean reverting or nonpermanent), that is the shock of the pollutant at various locations of these stations is not permanent and will disappear by themselves in the long run and converge to an average value. The method of long memory analysis is more robust than other conventional techniques used in most literature on air pollution studies, hence the research gives more reliable information about the time series properties of the particulate matter atmospheric pollutant.

Table 4 provides the results of the Lagrange Multiplier test for the Autoregressive Conditional Heteroskedasticity (ARCH-LM test) based on (Engle, 1982) who initiated the test. For robustness, the test is conducted at different lags of the dataset. The purpose of this test is to check whether heteroskedasticity is present in the PM₁₀ series under the null hypothesis that there is no ARCH effect at a given level of significance. The test is said to be significant if there exists an ARCH effect in the series implying the presence of many variations in the series which may be handled by the Generalized Autoregressive Heteroskedasticity (GARCH) model.

Table 4

Results of ARCH-LM Test for PM₁₀ Main Series
STATION	Lag 1		Lag 2		Lag 5
STATION	Statistic	p-value	Statistic	p-value	Statistic	p-value
Larkin	357.59	< 0.0001	1248.2	< 0.0001	1258.8	< 0.0001
Port Dickson	377.84	< 0.0001	763.67	< 0.0001	874.11	< 0.0001
Shah Alam	224.11	< 0.0001	606.19	< 0.0001	638.05	< 0.0001
Kuala Selengor	74.74	< 0.0001	632.65	< 0.0001	649.96	< 0.0001
Cheras	204.83	< 0.0001	577.14	< 0.0001	613.06	< 0.0001
Tanjung Malim	863.44	< 0.0001	1012.9	< 0.0001	1083.6	< 0.0001
Taiping	590.59	< 0.0001	723.62	< 0.0001	735.89	< 0.0001
Indera Mahkota	408.33	< 0.0001	555.5	< 0.0001	606.68	< 0.0001
Kota Kinabalu	849.55	< 0.0001	1057.5	< 0.0001	1170.9	< 0.0001
Balik Pulau	812.84	< 0.0001	966.66	< 0.0001	969.49	< 0.0001
Kangar	370.62	< 0.0001	785.42	< 0.0001	820.63	< 0.0001
Kota Bharu	104.84	< 0.0001	295.02	< 0.0001	359.85	< 0.0001
Petaling Jaya	223.55	< 0.0001	685.97	< 0.0001	705.78	< 0.0001
Seberang Jaya	833.36	< 0.0001	844.82	< 0.0001	857.56	< 0.0001

It can be observed from Table 4 that we rejected the null hypothesis of no ARCH effect in the series for all monitoring sites at different lags since the p-value is less than 0.05 level of significance. Rejection of the null hypothesis in each station confirms the significance of our ARCH-LM test, hence the alternative hypothesis for the existence of the ARCH effect in the series is accepted concluding ARCH effect is present in the series of each monitoring station at different lags. The presence of the ARCH effect in our PM₁₀ series confirms that the variation is time-variant, and this level of variation will be modeled using the GARCH model.

As mentioned above, the PM₁₀ atmospheric pollutant has been identified as the most rampant pollutant among other criteria pollutants in Malaysia, the pollutant may be assumed to be on the increase in Malaysia taking into consideration an increase in the number of human activities apart from natural causes such as industrial, power generation, motor vehicles emission, land clearing, and other developmental activities. This increase may cause the assumption of constant variance not to be satisfied making the risk of the pollutant time variant which need to be handled appropriately for effective and meaningful control measures to be in place by the authorities. In many situations, volatility in the level of the series for air pollutants tends to exhibit clustering both of high volatility during the period of high emission to be followed by low volatility during the period of low emission. This pattern of volatility clustering can be observed in Fig. 2 and confirmed empirically by the ARCH-LM test as contained in Table 4. To model the level of variation in our series and to understand its extent, Table 5 provided for the hybrid of ARFIMA-GARCH modeling for all the considered monitoring stations.

Table 5

Results of ARFIMA–GARCH Model for PM₁₀ Main Series.
STATION	Model	${ar}_{1}$(s.e)	${ma}_{1}$(s.e)	$\alpha$(s.e)	$\beta$(s.e)	$\left(\alpha +\beta \right)$	ARCH –LM [Lag 7]
Larkin	Estimate	0.477 (0.02)	-0.899 (0.012)	0.140 (0.024)	0.836 (0.029)	0.976	2.259
Larkin	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.976	0.663
Port Dickson	Estimate	0.504 (0.032)	-0.879 (0.017)	0.181 (0.028)	0.811 (0.027)	0.992	1.311
Port Dickson	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.992	0.858
Shah Alam	Estimate	0.514 (0.029)	-0.914 (0.019)	0.151 (0.024)	0.832 (0.027)	0.983	4.221
Shah Alam	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.983	0.316
Kuala Selengor	Estimate	0.533 (0.026)	-0.90 (0.013)	0.167 (0.021)	0.832 (0.018)	0.999	3.019
Kuala Selengor	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.999	0.510
Cheras	Estimate	0.560 (0.022)	-0.921 (0.012)	0.184 (0.015)	0.802 (0.014)	0.986	5.673
Cheras	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.986	0.165
Tanjung Malim	Estimate	0.599 (0.027)	-0.894 (0.014)	0.168 (0.025)	0.831 (0.025)	0.999	0.759
Tanjung Malim	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.999	0.949
Taiping	Estimate	0.541 (0.042)	-0.913 (0.027)	0.131 (0.025)	0.855 (0.032)	0.986	5.308
Taiping	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.986	0.195
Indera Mahkota	Estimate	0.622 (0.033)	-0.913 (0.022)	0.164 (0.034)	0.832 (0.033)	0.996	3.839
Indera Mahkota	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.996	0.370
Kota Kinabalu	Estimate	0.241 (0.264)	-0.691 (0.237)	0.320 (0.429)	0.679 (0.433)	0.999	0.295
Kota Kinabalu	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.999	0.993
Balik Pulau	Estimate	0.642 (0.039)	-0.917 (0.026)	0.191 (0.028)	0.796 (0.032)	0.987	5.480
Balik Pulau	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.987	0.180
Kangar	Estimate	0.683 (0.139)	-0.941 (0.093)	0.167 (0.032)	0.829 (0.031)	0.996	6.546
Kangar	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.996	0.109
Kota Bharu	Estimate	0.671 (0.015)	-0.953 (0.004)	0.125 (0.023)	0.839 (0.033)	0.964	4.013
Kota Bharu	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.964	0.345
Petaling Jaya	Estimate	0.524 (0.028)	-0.908 (0.166)	0.163 (0.028)	0.822 (0.03)	0.985	5.794
Petaling Jaya	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.985	0.156
Seberang Jaya	Estimate	0.584 (0.040)	-0.919 (0.025)	0.180 (0.030)	0.813 (0.030)	0.993	1.061
Seberang Jaya	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.993	0.903

The hybrid ARFIMA-GARCH (1, 1) model utilized in this study uses a sum of the parameter components of the GARCH model to measure how long the variance of current volatility will continue to be significant (volatility persistence). As the sum of these parameters approaches unity, the persistence of variability to the volatility becomes higher. However, the variance to volatility is permanent and the unconditional variance is infinite when the sum of $\alpha$ and $\beta$ equals 1. In such case, the process is Integrated Generalized Autoregressive Conditional Heteroskedastic model (IGARCH) implying that volatility persistence is permanent translates to means that past volatility is significant in predicting future volatility over all finite horizons (McMillan and Thupayagale, 2011). The volatility is said to be explosive if the sum of the parameters $\alpha$ and $\beta$ is greater than 1, meaning the variance to the volatility in one period will result in even greater volatility in the next period (Chou, 1988), hence, the higher the volatility the riskier the associated effect of the pollutant will be in circulation.

In Table 5 above, the autocorrelation among the observations is modeled by the ARFIMA (1,1) model while the level of variation in the observation is modeled by the symmetric GARCH (1,1) model. In all cases, the autoregressive $\left(ar\right)$ and moving average $\left(ma\right)$ components of the ARFIMA model were found to be significant also the alpha $\left(\alpha \right)$ and beta $\left(\beta \right)$ components of the standard GARCH model produced smaller p-values with far less than 0.05 level of significance. The alpha component represents the ARCH term while the beta component represents the GARCH term in the hybrid of ARFIMA-GARCH model structure. The significance of these components confirms the presence of volatility clustering earlier visualized in the time series plots and reveals that volatility is present in the considered series. Moreover, the summation of alpha and beta in all cases was found to be less than but approximately equal to one implying stationarity in the series and indicating evidence of volatility persistence in the level of PM₁₀ pollutants in the long run. It can also be observed from the table that the null hypothesis of no ARCH effect for the ARCH-LM test is not rejected in all cases, meaning the heteroskedastic effect in the series is successfully removed by the model against the results contained in Table 4. Thus, the hybrid of the ARFIMA-GARCH model fitted the series of PM₁₀ in all considered monitoring stations and successfully modeled the level and intensity of variation available in the series of PM₁₀ atmospheric air pollutants.

Figure 4 gives graphical results of the OLS-based CUSUM change point detection process for each monitoring station separately. It can be observed from the graph that the process exceeds its boundary in all the monitoring sites which signifies the evidence of a structural change in the dataset over an eleven-year period. However, the graphical results for change point detection are supplemented empirically by the OLS-based CUSUM under the null hypothesis of no structural change in the series at a given level of significance. The empirical test also estimates the day in which the structural break occurred for each series of a particular station separately.

The results of the OLS-based CUSUM test on PM₁₀ series are provided in Table 6 for each of the fourteen monitoring stations. The table also estimates the point of break for each series at which the break date is estimated alongside the structural change test (Sctest) statistic and the corresponding probability values (p-value). The null hypothesis of no break in the series at 0.05 level of significance was rejected in each case confirming the presence of structural break in each monitoring as depicted in Fig. 4. The break-date results observed in these monitoring stations vary from one station to another but are similar between some monitoring stations. It was also observed that the regime of break occurred mostly in the year 2017 corresponding to the year when the Malaysian government introduced new concessions for monitoring air quality across the country.

Table 6

Results for Structural Break Test and Estimation Break Date in PM₁₀ dataset
Station	Break Date	Position	Sctest	p-value
Larkin	April 13 2017	2295	854.9	< 0.0001
Port Dickson	April 12 2017	2294	1147.7	< 0.0001
Shah Alam	September 27 2016	2097	560.68	< 0.0001
Kuala Selengor	August 30 2016	2069	721.63	< 0.0001
Cheras	August 30 2016	2069	1025.8	< 0.0001
Tanjung Malim	August 8 2017	2331	1411.8	< 0.0001
Taiping	July 31 2017	2323	647.18	< 0.0001
Indera Mahkota	March 18 2017	2269	1347.8	< 0.0001
Kota Kinabalu	April 13 2017	2295	1823.6	< 0.0001
Balik Pulau	April 13 2017	2295	1033.2	< 0.0001
Kangar	August 23 2016	2062	1537.7	< 0.0001
Kota Bharu	July 29 2017	2321	1352.8	< 0.0001
Petaling Jaya	October 29 2015	1763	588.24	< 0.0001
Seberang Jaya	July 31 2017	2323	1854.6	< 0.0001

Each of the original series was partitioned into two according to their respective results of break dates as reported by the OLS-based CUSUM test. Descriptive statistics for subseries were provided in Table 7 to observe some statistical properties and make a comparison with descriptive statistics for the main series. Moreover, all subseries were then tested for the statistical issues of long memory and volatility persistence using the hybrid of ARFIMA-GARCH to investigate whether the subseries will report a similar pattern of long memory parameters and volatility persistence as the original series, for each air monitoring station.

Table 7

Results of Descriptive Statistics During Pre-Break and Post-Break Regimes
Station	Period	Minimum	Maximum	Mean	Stdev.	Skewness	Kurtosis
Larkin	Pre-Break	1.83	348.63	45.54	22.15	4.76	39.95
Larkin	Post Break	8.10	135.59	28.19	11.07	2.75	16.67
Port Dickson	Pre-Break	6.88	447.54	47.91	27.03	5.53	50.09
Port Dickson	Post-Break	8.351	160.18	23.56	12.83	4.75	37.29
Shah Alam	Pre-Break	1.42	362.38	51.85	31.55	3.72	22.17
Shah Alam	Post Break	1.33	156.55	32.97	14.09	2.90	17.56
Kuala Selengor	Pre-Break	2.50	355.5	45.61	29.26	3.45	19.37
Kuala Selengor	Post Break	7.10	150.10	25.89	12.90	3.41	22.61
Cheras	Pre-Break	9.29	318.83	50.83	25.68	3.51	21.35
Cheras	Post Break	8.50	151.47	29.71	13.00	3.31	21.27
Tanjung Malim	Pre-Break	1.46	290.83	35.01	20.65	3.97	28.10
Tanjung Malim	Post Break	4.37	94.68	14.83	7.33	3.78	25.78
Taiping	Pre-Break	3.08	281.00	45.53	23.85	2.40	12.73
Taiping	Post Break	5.74	154.65	28.77	13.75	2.09	9.55
Indera Mahkota	Pre-Break	1.67	265.13	38.38	17.85	4.01	31.19
Indera Mahkota	Post Break	4.41	131.44	20.18	11.16	3.89	25.36
Kota Kinabalu	Pre-Break	1.13	129.58	35.49	11.22	1.00	4.19
Kota Kinabalu	Post Break	3.34	212.40	20.78	9.81	5.68	91.71
Balik Pulau	Pre-Break	5.54	328.13	42.79	22.15	4.42	42.00
Balik Pulau	Post Break	5.49	210.84	23.28	12.65	4.28	40.99
Kangar	Pre-Break	3.04	338.08	40.32	19.28	4.59	54.47
Kangar	Post Break	4.49	97.21	20.88	9.90	1.61	5.22
Kota Bharu	Pre-Break	1.92	193.17	45.70	19.48	1.41	4.77
Kota Bharu	Post Break	5.68	98.53	25.97	11.11	1.31	3.18
Petaling Jaya	Pre-Break	5.04	369.67	52.80	30.09	4.13	26.44
Petaling Jaya	Post Break	1.79	150.01	35.18	14.02	2.42	12.66
Seberang Jaya	Pre-Break	8.13	342.67	53.58	23.80	3.92	32.11
Seberang Jaya	Post Break	6.77	130.93	26.53	10.07	1.95	10.98

The maximum emission of the pollutants in each of the monitoring stations occurred in the pre-break regime as contained in Table 7 except for Kota Kinabalu where the maximum emission of the pollutant was realized during the period of the post-break regime. It is of interest to note that other descriptive statistics contained in the table such as mean, standard deviation, skewness, and kurtosis were observed to be higher during the pre-break regime in all monitoring sites with the exception of Kota Kinabalu where skewness and kurtosis were observed to be higher during post break regime, and Kuala Selangor when the coefficient of kurtosis was also higher during the post-break period. Furthermore, the series before the broken regime in all monitoring stations was observed to be more volatile compared to the series in the post-break period. The most volatile series of $30.09\mu g/{m}^{3}$ was realized at the Petaling Jaya monitoring station while the least volatile series of $7.33\mu g/{m}^{3}$ was realized at the Tanjung Malim monitoring station during the post-break period.

Table 8 provided the results of long memory parameter estimates alongside their standard error using three different techniques using subseries for pre-break and post-break regimes in each of the fourteen stations separately. The sensitivity of estimating the long memory parameter alongside the estimate of its corresponding standard error for each method can always be compared with those of other methods using the pre-break and post-break regimes series for each station separately. It is indeed a known fact that long memory may spuriously be detected if the problem structural break is not well accounted for in the time series data since a short memory process with occasional structural break can show a slower rate of decay in the autocorrelation function and other properties of fractionally integrated $I\left(d\right)$ process. Hence the table can be used to explore whether the evidence of long memory observed in all stations using the original series is genuine or was merely spurious due to a neglected structural break, and hence a chance to compare the degree of shock in the level of pollutant between original and subseries in each station.

Table 8

Results of Long memory Parameter Estimation During Pre-Break and Post-Break Regimes.
Station	Period	fdGPH ($d)$	s.e (fdGPH)	fdSerio$\left(d\right)$	s.e(fdSperio)	ELW(d)	s.e(ELW)
Larkin	Pre-Break	0.4186	0.1044	0.4460	0.0491	0.5275	0.0333
Larkin	Post Break	0.5143	0.1212	0.4617	0.0567	0.5555	0.0374
Port Dickson	Pre-Break	0.3799	0.1359	0.4279	0.0710	0.4915	0.0333
Port Dickson	Post Break	0.4724	0.1006	0.3917	0.0475	0.6413	0.0374
Shah Alam	Pre-Break	0.5272	0.1317	0.4093	0.0557	0.4302	0.0343
Shah Alam	Post Break	0.4499	0.0881	0.4095	0.0370	0.4930	0.0360
Kuala Selangor	Pre-Break	0.1667	0.1158	0.2811	0.0549	0.4508	0.0345
Kuala Selangor	Post Break	0.4438	0.1198	0.3905	0.0452	0.5578	0.0358
Cheras	Pre-Break	0.2593	0.1042	0.3666	0.0540	0.4326	0.0345
Cheras	Post Break	0.4601	0.0808	0.4523	0.0349	0.5448	0.0358
Tanjung Malim	Pre-Break	0.3753	0.1072	0.3257	0.0522	0.4143	0.0331
Tanjung Malim	Post Break	0.4561	0.1051	0.4334	0.0393	0.5179	0.0377
Taiping	Pre-Break	0.2909	0.1164	0.4606	0.0433	0.4329	0.0311
Taiping	Post Break	0.3628	0.1323	0.3699	0.0469	0.4851	0.0376
Indera Mahkota	Pre-Break	0.3128	0.0863	0.3490	0.0493	0.4870	0.0334
Indera Mahkota	Post Break	0.4264	0.1172	0.4206	0.0650	0.6264	0.0372
Kota Kinabalu	Pre-Break	0.3225	0.0956	0.2517	0.0472	0.4322	0.0333
Kota Kinabalu	Post Break	0.5510	0.1063	0.5136	0.0432	0.4432	0.0374
Balik Pulau	Pre-Break	0.4413	0.1076	0.4414	0.0416	0.3737	0.0333
Balik Pulau	Post Break	0.3421	0.0850	0.3566	0.0428	0.4801	0.0373
Kangar	Pre-Break	0.5036	0.1343	0.3942	0.0351	0.3029	0.0346
Kangar	Post Break	0.3054	0.0941	0.3805	0.0456	0.3930	0.0357
Kota Bharu	Pre-Break	0.5962	0.1163	0.4929	0.0296	0.4083	0.0332
Kota Bharu	Post Break	0.2219	0.1344	0.2906	0.0568	0.3766	0.0376
Petaling Jaya	Pre-Break	0.4913	0.1258	0.3630	0.0504	0.4004	0.0365
Petaling Jaya	Post Break	0.3843	0.0941	0.4315	0.0421	0.4820	0.0340
Seberang Jaya	Pre-Break	0.5283	0.1126	0.4138	0.0438	0.3553	0.0331
Seberang Jaya	Post Break	0.2652	0.1159	0.2238	0.0531	0.4157	0.0376

It can be seen from the table that a transitory form of long memory is evident in each of the subseries for each monitoring station. The parameter estimates of the fdGPH and fdSperio methods are lower in eight of the fourteen air monitoring stations during pre-break periods compared to their respective post-break periods. In the same vein, the long memory parameter estimates using the ELW method are lower in thirteen of the fourteen stations during the pre-break period compared to their corresponding post-break period. The standard error values associated with each parameter estimate were observed to be lower using the ELW and fdSperio methods compared to those produced by the fdGPH method. In general, the standard error estimate using the ELW method was observed to be lower and stable compared to other methods of long memory parameter estimation.

It can be observed that all the subseries (before and after) the regime of break also exhibited evidence of long memory with the degree of persistency within the interval (0,1) as the original series for each air monitoring station. The shocks of the pollutant in the level of the subseries are also not permanent and will disappear naturally in the long run. Even though both the original and subseries reported similar statistical patterns of long memory estimation with mean-reverting form, it can be observed that the gravity of the shock in the original series for each monitoring station is higher using fdGPH and fdSperio methods than in the corresponding subseries of estimated break regimes, but the gravity of the shock in the level of pollutant is lower in most monitoring stations with the original series compared to their post-break subseries using ELW method. Hence, the results of the long memory analysis for this study confirm that all datasets for PM₁₀ emission for the eleven-year period were generated by a true long memory process since long memory is evident in both the original series and the partitioned series estimated during pre-break and post-break regimes. The higher the value of $d$ the higher the degree of persistency of pollutants at the given location of the monitoring stations.

The heteroskedasticity issue in the level of our main series has already been established both graphically and empirically but successfully modeled by the hybrid of ARFIMA-GARCH model employed in this study as contained in Table 5. However, the main series has already been divided into pre-break series and post-break series along their respective break dates as contained in Table 6. Each of these subseries is then tested for the statistical issues of heteroskedasticity and volatility persistence to investigate whether they can be influenced by the problem of structural break, in other words, whether they can be also detected spuriously as long memory due to failure to account for the structural break. Hence, Table 9 provided the result of the ARCH effect test for the pre-break and post-break series using the popular ARCH-LM test at different lags.

Table 9

ARCH-LM test for PM₁₀ Series During Pre-Break and Post-Break Regimes.
Station	Period	Lag 1		Lag 2		Lag 5
Station	Period	Statistic	p-value	Statistic	p-value	Statistic	p-value
Larkin	Pre-Break	193.45	< 0.0001	719.07	< 0.0001	725.35	< 0.0001
Larkin	Post Break	446.05	< 0.0001	474.96	< 0.0001	529.7	< 0.0001
Port Dickson	Pre-Break	211.93	< 0.0001	435.39	< 0.0001	500.11	< 0.0001
Port Dickson	Post Break	355.83	< 0.0001	411.81	< 0.0001	517.44	< 0.0001
Shah Alam	Pre-Break	109.24	< 0.0001	307.16	< 0.0001	322.23	< 0.0001
Shah Alam	Post Break	63.54	< 0.0001	179.2	< 0.0001	179.9	< 0.0001
Kuala Selangor	Pre-Break	33.77	< 0.0001	321.64	< 0.0001	330.78	< 0.0001
Kuala Selangor	Post Break	111.4	< 0.0001	298.87	< 0.0001	442.93	< 0.0001
Cheras	Pre-Break	96.16	< 0.0001	293.13	< 0.0001	310.68	< 0.0001
Cheras	Post Break	238.1	< 0.0001	239.25	< 0.0001	244.91	< 0.0001
Tanjung Malim	Pre-Break	493.26	< 0.0001	584.88	< 0.0001	629.92	< 0.0001
Tanjung Malim	Post Break	438.28	< 0.0001	473.31	< 0.0001	551.17	< 0.0001
Taiping	Pre-Break	332.56	< 0.0001	405.15	< 0.0001	411.44	< 0.0001
Taiping	Post Break	225.2	< 0.0001	295.65	< 0.0001	332.01	< 0.0001
Indera Mahkota	Pre-Break	229.91	< 0.0001	312.11	< 0.0001	342.75	< 0.0001
Indera Mahkota	Post Break	131.23	< 0.0001	194.51	< 0.0001	265.29	< 0.0001
Kota Kinabalu	Pre-Break	101.32	< 0.0001	105.44	< 0.0001	110.72	< 0.0001
Kota Kinabalu	Post Break	403.5	< 0.0001	531.53	< 0.0001	646.03	< 0.0001
Balik Pulau	Pre-Break	419.24	< 0.0001	525.24	< 0.0001	528.84	< 0.0001
Balik Pulau	Post Break	733.72	< 0.0001	737.8	< 0.0001	807.37	< 0.0001
Kangar	Pre-Break	198.42	< 0.0001	418.58	< 0.0001	438.93	< 0.0001
Kangar	Post Break	9.69	< 0.0001	62.74	< 0.0001	71.51	< 0.0001
Kota Bharu	Pre-Break	47.58	< 0.0001	157.55	< 0.0001	197.38	< 0.0001
Kota Bharu	Post Break	105.49	< 0.0001	129.75	< 0.0001	137.7	< 0.0001
Petaling Jaya	Pre-Break	91.4	< 0.0001	296.87	< 0.0001	304.83	< 0.0001
Petaling Jaya	Post Break	119.78	< 0.0001	144.35	< 0.0001	152.02	< 0.0001
Seberang Jaya	Pre-Break	478.25	< 0.0001	484.05	< 0.0001	490.46	< 0.0001
Seberang Jaya	Post Break	421.44	< 0.0001	433.29	< 0.0001	457.48	< 0.0001

The null hypothesis of no ARCH effect in the structural break series is rejected to confirm the evidence of heteroskedasticity in each of the pre-break and post-break series for each monitoring station at a 0.05 level of significance. Hence, both series before and after the regimes of break for each monitoring station are characterized by volatility meaning that volatility is evident and highly detected at all locations of our monitoring stations. For robustness, the test of the ARCH effect on each of the subseries is conducted at different lags.

To adequately model the level of these variations realized in the pre-break and post-break series, the hybrid of the ARFIMA-GARCH modeling technique is employed to handle the problem of volatility in the series.

Table 10

Results of ARFIMA-GARCH model During Pre-Break Regime.
STATION	Model	${ar}_{1}$(s.e)	${ma}_{1}$(s.e)	$\alpha$(s.e)	$\alpha$(s.e)	$\left(\alpha +\beta \right)$	ARCH-LM [Lag 7]
Larkin	Estimate	0.472 (0.033)	-0.888 (0.017)	0.211 (0.054)	0.690 (0.074)	0.901	1.972
Larkin	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.901	0.723
Port Dickson	Estimate	0.527 (0.052)	-0.876 (0.027)	0.223 (0.048)	0.751 (0.050)	0.974	1.256
Port Dickson	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.974	0.869
Shah Alam	Estimate	0.554 (0.040)	-0.911 (0.022)	0.202 (0.035)	0.772 (0.041)	0.974	4.556
Shah Alam	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.974	0.273
Kuala Selangor	Estimate	0.570 (0.037)	-0.897 (0.020)	0.196 (0.036)	0.803 (0.032)	0.999	1.715
Kuala Selangor	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.999	0.777
Cheras	Estimate	0.606 (0.047)	-0.920 (0.032)	0.236 (0.043)	0.759 (0.042)	0.995	4.748
Cheras	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.995	0.251
Tanjung Malim	Estimate	0.587 (0.039)	-0.878 (0.021)	0.210 (0.031)	0.789 (0.036)	0.999	4.807
Tanjung Malim	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.999	0.245
Taiping	Estimate	0.502 (0.052)	-0.890 (0.031)	0.179 (0.033)	0.775 (0.043)	0.954	4.972
Taiping	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.954	0.227
Indera Mahkota	Estimate	0.603 (0.033)	-0.910 (0.019)	0.174 (0.045)	0.794 (0.049)	0.968	2.290
Indera Mahkota	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.968	0.656
Kota Kinabalu	Estimate	0.551 (0.075)	-0.911 (0.047)	0.205 (0.061)	0.613 (0.124)	0.818	0.223
Kota Kinabalu	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.818	0.996
Balik Pulau	Estimate	0.655 (0.043)	-0.917 (0.026)	0.204 (0.034)	0.770 (0.039)	0.974	3.330
Balik Pulau	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.974	0.454
Kangar	Estimate	0.674 (0.048)	-0.940 (0.027)	0.157 (0.030)	0.821 (0.038)	0.978	5.976
Kangar	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.978	0.143
Kota Bharu	Estimate	0.677 (0.018)	-0.954 (0.003)	0.114 (0.027)	0.837 (0.049)	0.951	3.606
Kota Bharu	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.951	0.407
Petaling Jaya	Estimate	0.592 (0.076)	-0.933 (0.046)	0.230 (0.061)	0.756 (0.062)	0.986	6.364
Petaling Jaya	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.986	0.119
Seberang Jaya	Estimate	0.615 (0.062)	-0.926 (0.037)	0.237 (0.046)	0.729 (0.049)	0.966	0.904
Seberang Jaya	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.966	0.929

Table 10 displays the results from the hybrid ARFIMA-GARCH model using a pre-break series for all monitoring stations. It can be seen from the table that the parameters of the ARFIMA-GARCH hybrid are statistically significant by producing p-values far less than the level of significance. In particular, the autoregressive and moving average components of the ARFIMA model are significant in each case of the air monitoring station. The significance of ARCH and GARCH components confirms the evidence of heteroskedasticity and volatility in the level PM₁₀ series during the pre-break season. The ARCH component of the hybrid model was observed to be increasing with the series during the pre-break regime compared to the main series while the GARCH component was observed to be decreasing compared with the main series. Volatility persistence is evident in the pre-break regime series since the summation of the alpha and beta components of the hybrid model is fast approaching.

Table 11

Results of ARFIMA-GARCH model During Post-Break Regime
STATION	Model	${ar}_{1}$(s.e)	${ma}_{1}$(s.e)	$\alpha$(s.e)	$\beta$(s.e)	$\left(\alpha +\beta \right)$	ARCH-LM [Lag 7]
Larkin	Estimate	0.468 (0.033)	-0.909 (0.016)	0.101 (0.022)	0.862 (0.024)	0.963	9.066
Larkin	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.963	0.030
Port Dickson	Estimate	0.519 (0.042)	-0.911 (0.025)	0.111 (0.032)	0.844 (0.034)	0.955	2.372
Port Dickson	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.955	0.639
Shah Alam	Estimate	0.465 (0.038)	-0.923 (0.026)	0.056 (0.020)	0.922 (0.027)	0.978	1.209
Shah Alam	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.978	0.878
Kuala Selangor	Estimate	0.496 (0.036)	-0.910 (0.019)	0.119 (0.022)	0.842 (0.026)	0.961	1.359
Kuala Selangor	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.961	0.849
Cheras	Estimate	0.503 (0.038)	-0.926 (0.024)	0.107 (0.029)	0.834 (0.043)	0.941	2.443
Cheras	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.941	0.624
Tanjung Malim	Estimate	0.586 (0.033)	-0.914 (0.015)	0.092 (0.025)	0.888 (0.035)	0.980	0.190
Tanjung Malim	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.980	0.997
Taiping	Estimate	0.573 (0.057)	-0.927 (0.038)	0.082 (0.032)	0.896 (0.043)	0.978	1.721
Taiping	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.978	0.776
Indera Mahkota	Estimate	0.633 (0.060)	-0.913 (0.045)	0.173 (0.069)	0.788 (0.086)	0.961	2.575
Indera Mahkota	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.961	0.597
Kota Kinabalu	Estimate	-0.300 (0.131)	-0.532 (0.067)	0.773 (0.311)	0.226 (0.118)	0.999	0.151
Kota Kinabalu	p-value	0.819	< 0.0001	0.013	0.055	0.999	0.998
Balik Pulau	Estimate	0.639 (0.067)	-0.925 (0.047)	0.180 (0.043)	0.743 (0.056)	0.923	4.070
Balik Pulau	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.923	0.337
Kangar	Estimate	0.706 (0.083)	-0.946 (0.055)	0.200 (0.045)	0.717 (0.055)	0.917	0.782
Kangar	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.917	0.946
Kota Bharu	Estimate	0.681 (0.024)	-0.973 (0.001)	0.172 (0.028)	0.676 (0.048)	0.848	0.981
Kota Bharu	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.848	0.917
Petaling Jaya	Estimate	0.478 (0.033)	-0.908 (0.018)	0.084 (0.019)	0.889 (0.025)	0.973	0.539
Petaling Jaya	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.973	0.975
Seberang Jaya	Estimate	0.554 (0.043)	-0.919 (0.028)	0.131 (0.033)	0.808 (0.040)	0.939	1.381
Seberang Jaya	p-value	< 0.0001	< 0.0001	< 0.0001	< 0.0001	0.939	0.845

However, the ARCH component for the series during the post-break regime was observed to be lower compared to the main series and subseries during the pre-break period as contained in Table 11 while the GARCH component is higher in most monitoring stations compared to the pre-break and main series. In all the subseries, it is of interest to note that the ARCH-LM test component of the hybrid model successfully removed the heteroskedastic effect in the series since the null hypothesis of no ARCH effect was found to be statistically non-significant, except for Larkin station when the test was statistically significant using pre-break series at 0.05 percent level of significance.

This paper mainly examined the statistical issues of long memory, volatility problem, and some descriptive statistics in the level of PM₁₀ atmospheric pollutant in fourteen Malaysian air monitoring stations from 1 January 2011 to 31 December 2021, due to the pollutant’s effect on human health, environment, and ambient air quality. Table 1 gives information about the selected air monitoring stations such as geo-coordinates, duration, and category location for each station. Tables 2 and 7 provided the results of descriptive statistics for both the main and partitioned series to study the fluctuation of pollutants and the nature of their respective probability distribution through kurtosis and skewness respectively. Higher values of Kurtosis and Skewness were observed in most monitoring sites from both main and partitioned series indicating significant fluctuation and non-Gaussian behavior of the pollutants in the affected monitoring stations. It can be seen from the table that the most volatile series was observed at the Shah Alam monitoring station of the sub-urban category while the least volatile series was observed at the Kota Kinabalu monitoring site of the urban category. Minimum and maximum emission of the pollutants for both series over an eleven-year period were also indicated alongside their respective dates of occurrence as contained in Tables 2 and 7.

The ACF time series plot of monitoring stations utilized in this study indicates the evidence of long memory due to observance of hyperbolic decay even at higher lags, while the graphical plot of OLS-based CUSUM test of structural break presented evidence for the presence of structural break in the selected series. These issues of long memory and structural break were confirmed empirically by all methods of long memory parameter estimation used in this study and the p-value of the structural change test respectively.

The ARCH-LM test confirms the presence of heteroskedasticity in both series of PM₁₀ pollutants tested at three different lag values. These heteroskedastic effects were successfully modeled by the hybrid of the ARFIMA-GARCH model. Furthermore, volatility persistence was also evident in both series of the pollutant at all monitoring sites with the main series identified as the most volatile followed by the pre-break regime series and then post post-break regime series. This implies that the variation in the emission of PM₁₀ pollutants in Malaysia will continue to be in place for a long period of time until adequate control measures are provided by the government.

Conclusively, the PM₁₀ atmospheric air pollutants in Malaysia are characterized by true long memory and volatility persistence. Hence, the partitioned series were observed to have similar statistical properties of long memory analysis and volatility persistence with the original series confirming that the PM₁₀ datasets from the fourteen air monitoring stations were generated by a true long memory process and volatility persistence, not merely spurious due to existence of neglected structural break in the original series. Based on long memory analysis for this research, the shocks of PM₁₀ pollutant in Malaysia is transitory, and mean reverting (nonpermanent) will disappear naturally in the long run and converge to an average value in future time.

Even though the shock of the pollutant on human and environment is transient and mean reverting, the government needs to strengthen measures of converting the effects of the pollutant based on the values of long memory parameter $d$ at respective locations and neighborhood of the monitoring stations. The residents of these stations and employees at the considered industrial sites need to inculcate the habit of physical body exercise to strengthen their immune body systems to fight back the shock of the pollutants associated with them since the practice was found to be a safe health promotion strategy and reduces the risk of cardiovascular deaths from air pollutants (Bo et al. 2022).

ACKNOWLEDGEMENT

We would like to acknowledge Department of Environment (DOE) Malaysia for providing data to accomplish this study.

Funding: The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Competing interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Consent to participate (Authors’ contributions)

All authors contributed to the study conception and design. Lawan Adamu Isma’il performed analysis and writing original draft. Norhashidah Awang and Ibrahim Lawal Kane supervised, reviewed, and edited the manuscript. All authors read and approved the final manuscript.

Consent to Publish

All authors agreed with the content and gave explicit consent to submit the manuscript and obtained consent from our universities before the work is submitted.

Data availability statements: The data that support the findings of this study are available from Department of Environmental Malaysia, but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available. Data are however available from the authors upon reasonable request and with permission of Department of Environmental Malaysia.

Ethical Approval

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript. The authors have no relevant financial or non-financial interests to disclose. This study does not involve human participants or animals.

Corresponding author: Ibrahim Lawal Kane ([email protected])

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Statistical Approach to Examining the True Status of Long Memory and Volatility Persistence in PM10 Air Pollutant at Different Regions of Malaysia: A Methodical Methodology

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Abstract

Figures

Introduction

Materials and Methods

3.1.1 Study Area and Data Used

3.1.2 Time Series Plot

3.1.3 Kurtosis

3.1.4 Skewness

3.1.5 Autocorrelation Function (ACF)

3.1.6 Time Series

3.1.7 ARCH-LM Test

3.1.8 GARCH (p, q) Model

3.1.9 ARFIMA-GARCH Hybrid Model

3.2.1 Long Memory Processes

3.2.2 Long Memory Parameter Estimation

3.2.3 Geweke and Porter-Hudak Method

3.2.4 Fractionally Differenced Sperio (fdSperio) Method

3.2.5 Exact Local Whittle Estimation

3.2.6 Ordinary Least Square (OLS)-based Cumulative Sum (CUSUM) Test

Results and Discussion

Concluding Comments

Declarations

References

Additional Declarations

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