**1. Introduction**

Food security has different dimensions and so has been defined by many authors from different angles [1]. However, food security is a situation that exists when all people, at all times, have physical, social and economic access to sufficient, safe and nutritious food that meets their dietary needs and food preference for an active and healthy life is globally accepted [2].

Estimates show that out of 7.3 billion people in the world, approximately 795 million people or 1 in 9 were suffering from chronic undernourishment in 2014– 2016 [3]. Almost all the hungry people, 780 million, live in developing countries, representing 12.9%, or 1 in 8, of the population of developing countries. Some 232 million people in Africa struggle with undernourishment daily [4]. This figure is about 29.3% of the total undernourished population and approximately 21% of the continent's population. Among all the regions of the world, sub-Saharan Africa is the only region that recorded a 10% (17.4–27.8%) increase in the number of hungry people between the periods of 1990–1992 and 2014–2016. Currently, 220 million people in sub-Saharan suffers hunger daily [4].

evidence of undernutrition among certain segments of the population [24]. There are evidence of undernutrition among certain segments of the population [25]. Inadequate nutrient intakes are often caused by household food insecurity, defined as a household's lack of access to amounts of food of the right quality to satisfy the

This study is underpinned by the Malthusian theory advanced by Thomas Malthus (1806) and the theory of food sovereignty promoted during the recent food crises of the 2007–2008 period. Malthusian theory is characterized by the views that there are too many mouths chasing too few calories as the population increases, lack of capacity to meet our food needs due to significant structural constraints, water and land degradation, distributional conflicts, and widespread, chronic food insecurity. Malthus in his prediction failed to conceive the development of important variables such as birth control and technology advancement in Agriculture. Although Malthus's theory promulgated in more than 220 years ago has been proven to be wrong, the question in the developing nation is how this growing population can be harnessed to produce enough food for all the population. The food sovereignty theory, unlike the Malthusian theory, believes that population growth is not the problem but the over-bearing power of international trade systems. Proponents of national food sovereignty movements generally favor agricultural policies that promote domestic production as an alternative to reliance on food imports. The theory of food sovereignty was first mentioned in 1996 when it became obvious that the global food organizations have no idea on how to ensure a food-secured world. Since then, the idea has gained prominence most especially in South America. Proponents of food sovereignty believe that all people have a right to healthy and culturally produced food through sustainable methods with local farmers having control over their own agricultural system [28]. The activists of food sovereignty are rallying cry against global agribusiness that stifles livelihood of smallholder farmers [29]. Food sovereignty theory rejects dependence on heavy chemical input for crop production that breed disparity in food access in the midst of growing food production [30]. It revolves around the concept of prioritizing local and household producers with opportunity of fair prices with the emphasis of community having control over productive resources like water, land and seed [31–33]. This theory believes that if they strive for a food-secured world, most go beyond the definition of the food security that revolved around sustenance of global food stock through international trade but do everything to empower the community with the right to produce for themselves rather than depending on the inter-

Finally, the trajectories by both theories speak to household food security and form a strong basis for analyzing per capita food status in South Africa. Although the Malthusian theory leans towards the far right on the outstripping tendencies of population growth based on limited resources, the theory of food sovereignty went too far to the left by opposing improved inputs most especially the chemicals and having nothing to do with international trade system. Considering the duality and capital-intensive agricultural sector of South Africa, a balanced food sovereignty theory will not only lead to local economic growth; it will also help engage the youthful population into productive farming activities, thereby improving per

dietary needs of all its members throughout the year [26]. Similarly, land requirements for food are determined by the production system, e.g. yields per hectare and efficiency in the food industry, which are also the resultant of

*Regime Switch and Effect on Per Capita Food Security Issues in South Africa*

**1.1 Theoretical framework/theory underpinning the study**

consumption patterns [27].

*DOI: http://dx.doi.org/10.5772/intechopen.86931*

national market [29].

**21**

capita food security in South Africa.

The continued population growth in Africa has rendered the per capita domestically grown food unchanged despite some improvements in agriculture with the resultant persistent hunger and poverty [5]. Although there has been tremendous growth in food production leading to a dramatic decrease in the proportion of the world's people that are hungry in the past decades, global food security situation still indicates that more than one out of seven people today still do not have access to sufficient protein and energy from their diet and even more suffer from some form of micronutrient malnourishment [6]. With the fastest population growth rate, Africa's population is projected to grow from about 796 million in 2005 to 1.8 billion by 2050 [7]. Despite urban migration, the number of rural dwellers will also continue to grow [8]. However, there are projections that parts of Africa, Asia and Central and Southern America will experience substantial declines in per capita cereal production if yields continue to grow slowly than per capita harvested areas [9]. Per capita food production in Africa declined by almost 20% between 1970 and 2000 [10].

Agriculture in African countries is widely seen to have performed worse than in Asia and Latin America. Production data per capita (of the total population) indicate that the amount of food grown on the continent per person rose slowly in the 1960s, then fell from the mid-1970s and has recently just recovered to the level of 1960 [5]. Comparatively, per capita food production increased by 102% in Asia and 63% in Latin America during the same period [5]. Studies have identified reduced investment in agricultural research, extension services and production systems by both the government and donor agents as the reasons for this [11–14]. Africa derives about 25% of its GDP from agriculture which provides jobs for 70% of the labour force, as well as a livelihood for more than 65% of the population [8]. It is however important to note that the level of local agricultural production will be determined by the amount and quality of arable land, the amount and quality of agricultural inputs (fertilizer, seeds, pesticides, etc.) as well as farm-related technology, practices and policies [9].

Interestingly, while some countries in Africa have witnessed growth in production [15], it has not necessarily improved the household food security status in the continent. South Africa produces enough food to feed its population; however, the country is increasingly experiencing worsening household food insecurity [16]. Despite the rise in employment in the country [17] and introduction of social grants by the government [18], the country has known little respite in terms of household food insecurity. About 35% of South African's population (14.3 people) experience hunger and undernutrition of which the majority are women and children [19]. Issues of ever-increasing food prices, lack of access to production resources and increased cost of electricity and oil prices are expected to make many more becoming food-insecure in South Africa [20, 21].

South African per capita land cultivation cannot be separated from the past dichotomous land ownership during the apartheid era where the white had ample access to land due to several discriminatory policies [22]. Effectively, many households in the so-called rural areas were and remain landless, while many others were left with tiny amounts of land. There are a few local black farmers that have private tenure of certain areas. Despite the abolition of the former homeland systems and subsequent redistribution of several commercial farms to emerging black farmers through leasing, access of other villagers to these lands for cultivation or collection of resources has been restricted under this arrangement [23].

South Africa is believed to have enough food supplies at a national level adequate to feed the entire population. However, a number of studies have revealed *Regime Switch and Effect on Per Capita Food Security Issues in South Africa DOI: http://dx.doi.org/10.5772/intechopen.86931*

evidence of undernutrition among certain segments of the population [24]. There are evidence of undernutrition among certain segments of the population [25]. Inadequate nutrient intakes are often caused by household food insecurity, defined as a household's lack of access to amounts of food of the right quality to satisfy the dietary needs of all its members throughout the year [26]. Similarly, land requirements for food are determined by the production system, e.g. yields per hectare and efficiency in the food industry, which are also the resultant of consumption patterns [27].

#### **1.1 Theoretical framework/theory underpinning the study**

This study is underpinned by the Malthusian theory advanced by Thomas Malthus (1806) and the theory of food sovereignty promoted during the recent food crises of the 2007–2008 period. Malthusian theory is characterized by the views that there are too many mouths chasing too few calories as the population increases, lack of capacity to meet our food needs due to significant structural constraints, water and land degradation, distributional conflicts, and widespread, chronic food insecurity. Malthus in his prediction failed to conceive the development of important variables such as birth control and technology advancement in Agriculture. Although Malthus's theory promulgated in more than 220 years ago has been proven to be wrong, the question in the developing nation is how this growing population can be harnessed to produce enough food for all the population. The food sovereignty theory, unlike the Malthusian theory, believes that population growth is not the problem but the over-bearing power of international trade systems. Proponents of national food sovereignty movements generally favor agricultural policies that promote domestic production as an alternative to reliance on food imports. The theory of food sovereignty was first mentioned in 1996 when it became obvious that the global food organizations have no idea on how to ensure a food-secured world. Since then, the idea has gained prominence most especially in South America. Proponents of food sovereignty believe that all people have a right to healthy and culturally produced food through sustainable methods with local farmers having control over their own agricultural system [28]. The activists of food sovereignty are rallying cry against global agribusiness that stifles livelihood of smallholder farmers [29]. Food sovereignty theory rejects dependence on heavy chemical input for crop production that breed disparity in food access in the midst of growing food production [30]. It revolves around the concept of prioritizing local and household producers with opportunity of fair prices with the emphasis of community having control over productive resources like water, land and seed [31–33]. This theory believes that if they strive for a food-secured world, most go beyond the definition of the food security that revolved around sustenance of global food stock through international trade but do everything to empower the community with the right to produce for themselves rather than depending on the international market [29].

Finally, the trajectories by both theories speak to household food security and form a strong basis for analyzing per capita food status in South Africa. Although the Malthusian theory leans towards the far right on the outstripping tendencies of population growth based on limited resources, the theory of food sovereignty went too far to the left by opposing improved inputs most especially the chemicals and having nothing to do with international trade system. Considering the duality and capital-intensive agricultural sector of South Africa, a balanced food sovereignty theory will not only lead to local economic growth; it will also help engage the youthful population into productive farming activities, thereby improving per capita food security in South Africa.

people between the periods of 1990–1992 and 2014–2016. Currently, 220 million

The continued population growth in Africa has rendered the per capita domestically grown food unchanged despite some improvements in agriculture with the resultant persistent hunger and poverty [5]. Although there has been tremendous growth in food production leading to a dramatic decrease in the proportion of the world's people that are hungry in the past decades, global food security situation still indicates that more than one out of seven people today still do not have access to sufficient protein and energy from their diet and even more suffer from some form of micronutrient malnourishment [6]. With the fastest population growth rate, Africa's population is projected to grow from about 796 million in 2005 to 1.8 billion by 2050 [7]. Despite urban migration, the number of rural dwellers will also continue to grow [8]. However, there are projections that parts of Africa, Asia and Central and Southern America will experience substantial declines in per capita cereal production if yields continue to grow slowly than per capita harvested areas [9]. Per capita food production in Africa declined by almost 20% between 1970 and

Agriculture in African countries is widely seen to have performed worse than in Asia and Latin America. Production data per capita (of the total population) indicate that the amount of food grown on the continent per person rose slowly in the 1960s, then fell from the mid-1970s and has recently just recovered to the level of 1960 [5]. Comparatively, per capita food production increased by 102% in Asia and 63% in Latin America during the same period [5]. Studies have identified reduced investment in agricultural research, extension services and production systems by both the government and donor agents as the reasons for this [11–14]. Africa derives about 25% of its GDP from agriculture which provides jobs for 70% of the labour force, as well as a livelihood for more than 65% of the population [8]. It is however important to note that the level of local agricultural production will be determined by the amount and quality of arable land, the amount and quality of agricultural inputs (fertilizer, seeds, pesticides, etc.) as well as farm-related tech-

Interestingly, while some countries in Africa have witnessed growth in production [15], it has not necessarily improved the household food security status in the continent. South Africa produces enough food to feed its population; however, the country is increasingly experiencing worsening household food insecurity [16]. Despite the rise in employment in the country [17] and introduction of social grants by the government [18], the country has known little respite in terms of household food insecurity. About 35% of South African's population (14.3 people) experience hunger and undernutrition of which the majority are women and children [19]. Issues of ever-increasing food prices, lack of access to production resources and increased cost of electricity and oil prices are expected to make many more becom-

South African per capita land cultivation cannot be separated from the past dichotomous land ownership during the apartheid era where the white had ample access to land due to several discriminatory policies [22]. Effectively, many households in the so-called rural areas were and remain landless, while many others were left with tiny amounts of land. There are a few local black farmers that have private tenure of certain areas. Despite the abolition of the former homeland systems and subsequent redistribution of several commercial farms to emerging black farmers through leasing, access of other villagers to these lands for cultivation or collection

South Africa is believed to have enough food supplies at a national level adequate to feed the entire population. However, a number of studies have revealed

people in sub-Saharan suffers hunger daily [4].

2000 [10].

*Food Security in Africa*

**20**

nology, practices and policies [9].

ing food-insecure in South Africa [20, 21].

of resources has been restricted under this arrangement [23].

Since the democratic dispensation in 1994, South Africa has undergone immense policy interventions aimed at improving the production capacity and food security situation of the citizenry. One of these policies, the Integrated Food Security Strategy (IFSS), was targeted mainly at increasing access to productive assets, including credit; increasing access to technologies, including food processing; supporting agriculture extension services; and improving infrastructure and trade regulations [34]. Another policy action, the Comprehensive Agriculture Support Programme, aimed at providing post-settlement support to the targeted beneficiaries of land reform and to other producers who have acquired land through private means and are engaged in value-adding enterprises for the domestic or export markets [35]. The programme was developed to benefit the hungry, subsistence and household food producers, farmers and agricultural macro-systems in the consumer environment. However, all these good policies have not really achieved the desired postapartheid South African dream as the country's Human Development Index is ranked 118 among 135 countries and Human Poverty of 13.4% and ranking of 85 amidst all policies and strategies of improving the agriculture and food security (Global Food Security Index) [36].

the biases and other limitations of researchers who conduct such research. Postpositivism holds that the goal of science is to achieve intersubjective agreement among researchers about the nature of reality rather than rely on the objective reality perceive through methods. In essence, issues should be viewed through the contributions of community of researchers rather than any individual researcher. Therefore, to address the research question, a quantitative approach was undertaken involving a time series data analysis. Similar to the approach by the FAO and USDA, a series data on Food Availability (Per Capita) Data System (FADS) include three distinct indicators which are land cultivated, price per ton and total food consumed of the selected staple food (maize). The study uses the threshold

*Regime Switch and Effect on Per Capita Food Security Issues in South Africa*

*DOI: http://dx.doi.org/10.5772/intechopen.86931*

autoregressive model which is a nonlinear approach of representing time series data as suggested by practitioners who describe the basic proponents of the model [40].

Secondary (times series) data on Production of the main staple food crops were sourced from National Agricultural Marketing Council (NAMC) from the period of 1970 to 2010. This paper focused on maize as the main staple food in South Africa. We believe the data are viable and reliable because NAMC is established by acts; it is recognized to offer advice to the government on food and trade issues. The data covered land cultivation, total production and consumption of these food crops. Corresponding national population for the same period (1970–2010) was also

We undertake series data to determine food and dietary intakes of South Africans because it provides a pattern of food and dietary evolution over time, as a result of many factors and complex interactions. The factors are historical political change, income, prices, individual preferences and beliefs, cultural traditions as well as geographical, environmental, social and economic factors which all interact in a complex manner to shape dietary consumption patterns. Data on the national availability of the main food commodities provide a valuable insight into diets and

Per capita land cultivation (ha), price index per tonne and capita consumption per tonne of the selected food crops were estimated by dividing total land cultivated and total consumption by the total population of South Africa. The percentage change in these food security indicators was also calculated to determine whether there have been positive or negative changes over the period under analysis. The estimation of past and present (1970–2010) South African food security indicators for the selected staple foods was done through the equations below

where *PLC* is the per capita land cultivated of commodity *i* at time (*t*),*TArab* is the total arable land of commodity *i* at time (*t*) and *Tpop* is the total population at

The equation is expanded and modified into a multivariate regression by including the following explanatory variables that affect cultivated per capita land,

*PLCi t*ðÞ¼ *TArabi t*ð Þ*=Tpop t*ð Þ (1)

obtained from NAMC. It is worthy to note that the study was limited to the

available and complete data on food commodity.

their evolution over time [41].

similar to the one used by the FAO:

**2.2 Data analysis**

time (*t*).

**23**

**2.1 Data collection**

This paper seeks to explore three main questions of South African food security systems. These questions are: is South African food security status sensitive to the past and the present governance regimes? Is the nationally acclaimed food sufficiency reflected in the household level? What effect does population growth in South Africa have on the food security status, is it positive or negative? These research questions are expected to generate inherent information on food security situations among South African households during the past apartheid era and the current black-dominated governance systems. This paper is set to determine per capita food security situation among South Africa households during the apartheid and post-apartheid eras. Specifically, the study seeks to determine the trend in per capita land cultivated, the price index and consumption level of maize staple foods. Various approaches have been followed to assess the world food situation. These include the development of large econometric models or the computation of technical indicators such as the population carrying capacity of the planet [37]. Within the South African context, many studies have focussed on food security status with different methodologies [20, 38, 39]; among others is a study that presents a policy impact analysis of South African food security [34]. However, this paper explores a new route, a simple-time series indicator approach. We used the indicator approach to capture food security status because it aids the process of monitoring trends and provides practical decision-making processes for enhanced policy-making processes and intervention strategies to cater for the most vulnerable individuals. We build on the theory of population growth as well as the food sovereignty theory for the comparison of per capita food security situation in apartheid and post-apartheid eras of South Africa. The analysis was undertaken to provide macro level trend information on the three main indicators: per capita land access, per capita staple food production and per capita consumption of staple food during the two important eras of South Africa. The insight and knowledge generated will be required for future policies formulation and interventions towards achieving sustainable food systems and security in South Africa.

#### **2. Methodology**

This study falls under post-positivism paradigm which believes that there is an empirical reality but that our understanding of it is limited by its complexity and by *Regime Switch and Effect on Per Capita Food Security Issues in South Africa DOI: http://dx.doi.org/10.5772/intechopen.86931*

the biases and other limitations of researchers who conduct such research. Postpositivism holds that the goal of science is to achieve intersubjective agreement among researchers about the nature of reality rather than rely on the objective reality perceive through methods. In essence, issues should be viewed through the contributions of community of researchers rather than any individual researcher. Therefore, to address the research question, a quantitative approach was undertaken involving a time series data analysis. Similar to the approach by the FAO and USDA, a series data on Food Availability (Per Capita) Data System (FADS) include three distinct indicators which are land cultivated, price per ton and total food consumed of the selected staple food (maize). The study uses the threshold autoregressive model which is a nonlinear approach of representing time series data as suggested by practitioners who describe the basic proponents of the model [40].

#### **2.1 Data collection**

Since the democratic dispensation in 1994, South Africa has undergone immense policy interventions aimed at improving the production capacity and food security situation of the citizenry. One of these policies, the Integrated Food Security Strategy (IFSS), was targeted mainly at increasing access to productive assets, including credit; increasing access to technologies, including food processing; supporting agriculture extension services; and improving infrastructure and trade regulations [34]. Another policy action, the Comprehensive Agriculture Support Programme, aimed at providing post-settlement support to the targeted beneficiaries of land reform and to other producers who have acquired land through private means and are engaged in value-adding enterprises for the domestic or export markets [35]. The programme was developed to benefit the hungry, subsistence and household food producers, farmers and agricultural macro-systems in the consumer environment. However, all these good policies have not really achieved the desired postapartheid South African dream as the country's Human Development Index is ranked 118 among 135 countries and Human Poverty of 13.4% and ranking of 85 amidst all policies and strategies of improving the agriculture and food security

This paper seeks to explore three main questions of South African food security systems. These questions are: is South African food security status sensitive to the past and the present governance regimes? Is the nationally acclaimed food sufficiency reflected in the household level? What effect does population growth in South Africa have on the food security status, is it positive or negative? These research questions are expected to generate inherent information on food security situations among South African households during the past apartheid era and the current black-dominated governance systems. This paper is set to determine per capita food security situation among South Africa households during the apartheid and post-apartheid eras. Specifically, the study seeks to determine the trend in per capita land cultivated, the price index and consumption level of maize staple foods. Various approaches have been followed to assess the world food situation. These include the development of large econometric models or the computation of technical indicators such as the population carrying capacity of the planet [37]. Within the South African context, many studies have focussed on food security status with different methodologies [20, 38, 39]; among others is a study that presents a policy impact analysis of South African food security [34]. However, this paper explores a new route, a simple-time series indicator approach. We used the indicator approach to capture food security status because it aids the process of monitoring trends and provides practical decision-making processes for enhanced policy-making processes and intervention strategies to cater for the most vulnerable individuals. We build on the theory of population growth as well as the food sovereignty theory for the comparison of per capita food security situation in apartheid and post-apartheid eras of South Africa. The analysis was undertaken to provide macro level trend information on the three main indicators: per capita land access, per capita staple food production and per capita consumption of staple food during the two important eras of South Africa. The insight and knowledge generated will be required for future policies formulation and interventions towards achieving sustainable food

This study falls under post-positivism paradigm which believes that there is an empirical reality but that our understanding of it is limited by its complexity and by

(Global Food Security Index) [36].

*Food Security in Africa*

systems and security in South Africa.

**2. Methodology**

**22**

Secondary (times series) data on Production of the main staple food crops were sourced from National Agricultural Marketing Council (NAMC) from the period of 1970 to 2010. This paper focused on maize as the main staple food in South Africa. We believe the data are viable and reliable because NAMC is established by acts; it is recognized to offer advice to the government on food and trade issues. The data covered land cultivation, total production and consumption of these food crops. Corresponding national population for the same period (1970–2010) was also obtained from NAMC. It is worthy to note that the study was limited to the available and complete data on food commodity.

We undertake series data to determine food and dietary intakes of South Africans because it provides a pattern of food and dietary evolution over time, as a result of many factors and complex interactions. The factors are historical political change, income, prices, individual preferences and beliefs, cultural traditions as well as geographical, environmental, social and economic factors which all interact in a complex manner to shape dietary consumption patterns. Data on the national availability of the main food commodities provide a valuable insight into diets and their evolution over time [41].

#### **2.2 Data analysis**

Per capita land cultivation (ha), price index per tonne and capita consumption per tonne of the selected food crops were estimated by dividing total land cultivated and total consumption by the total population of South Africa. The percentage change in these food security indicators was also calculated to determine whether there have been positive or negative changes over the period under analysis.

The estimation of past and present (1970–2010) South African food security indicators for the selected staple foods was done through the equations below similar to the one used by the FAO:

$$PLCi\ (t) = TArabi\ (t)/Tpop\ (t)\tag{1}$$

where *PLC* is the per capita land cultivated of commodity *i* at time (*t*),*TArab* is the total arable land of commodity *i* at time (*t*) and *Tpop* is the total population at time (*t*).

The equation is expanded and modified into a multivariate regression by including the following explanatory variables that affect cultivated per capita land, namely, consumption per tonne and price index of maize. The regression is as follows:

$$\text{LCUL\\_CAP}\_t = \beta\_0 + \beta\_1 \text{PRCE}\_t + \beta\_2 \text{CONSUMPTION}\_t + \varepsilon\_t \tag{2}$$

where we let *σ*^<sup>2</sup>

<sup>1</sup>*xt*�11 *Zt*�<sup>1</sup> <sup>&</sup>lt;*<sup>λ</sup>*

**Threshold effects**

the joint theory where

represented as

in relation to. *θ* ^<sup>1</sup> <sup>¼</sup> *<sup>θ</sup>* ^<sup>1</sup> ^*λ* � � and *θ*

follows:

Δ*yt* ¼ *θ* ^0 ð Þ¼ *<sup>λ</sup> <sup>T</sup>*�<sup>1</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.86931*

^<sup>2</sup> <sup>¼</sup> *<sup>θ</sup>* ^<sup>2</sup> ^*λ*

from the least squares estimation as *<sup>σ</sup>*^<sup>2</sup> <sup>¼</sup> *<sup>T</sup>*�<sup>1</sup>

standard Wald test written as: *WT* <sup>¼</sup> *<sup>T</sup> <sup>σ</sup>*^<sup>2</sup>

of the model and that edge impacts exist.

**Unit root test, asymmetry and cointegration**

Δ*yt* also implying *yt* is *I*ð Þ1 and therefore has a unit root.

difference from condition (6) and *σ*^<sup>2</sup>

hypothesis is represented as follows:

represented as *H*<sup>1</sup> : *ρ*<sup>1</sup> <0 and *ρ*<sup>2</sup> <0.

hypothesis reads as:

**25**

^ ð Þ*prce* <sup>þ</sup> *consumption* <sup>þ</sup> *<sup>θ</sup>*

Eq. (3) which can test the possible presence of nonlinearity.

∑*<sup>T</sup>*

*Regime Switch and Effect on Per Capita Food Security Issues in South Africa*

threshold estimate of the threshold value is found by minimizing *<sup>σ</sup>*2ð Þ*<sup>λ</sup>* which are

To find the least squares estimates of other parameters, a point estimate ^*λ* is used

^0

Eq. (7) shows the least squares residuals ^*et* and denotes the residual variance

study to draw standard Wald statistics and *t*-statistic inferences on parameters from

This examination utilizes the Wald test measurement to address the subject of whether the parameters of condition (3) have the nearness of limit impacts and the likelihood of general nonlinearity. This strategy is utilized in application to help the investigation of nonlinear time arrangement [40]. The limit impact vanishes under

Condition (8), is a limitation that is tested through the observation of the

OLS estimation in condition (7) of the edge model. The dismissal of the invalid theory in condition (8) implies that there is factual importance of the logical factors

*ρ*<sup>2</sup> since they control the stationarity process of *yt* in Eq. (3); as such, the null

Moreover, in a situation where *p* ¼ 1*,* the model becomes stationary if *ρ*<sup>1</sup> < 0*, ρ*2>0 and 1 þ *ρ*<sup>1</sup> ð Þ 1 þ *ρ*<sup>2</sup> ð Þ<1. Hence this suggests an alternative to *H*<sup>0</sup>

Unit root test can also be observed in a partial case where the alternative

*ρ*<sup>1</sup> ¼ 0

*<sup>H</sup>*<sup>2</sup> : *<sup>ρ</sup>*<sup>1</sup> <sup>&</sup>lt;<sup>0</sup>

(

0 *<sup>σ</sup>*^<sup>2</sup> � <sup>1</sup> � �; with *<sup>σ</sup>*^<sup>2</sup>

To test for stationarity, the test statistics are observed for the parameters *ρ*<sup>1</sup> and

Eq. (3) is then rewritten as a stationary threshold autoregression in the variable

*and or*

*ρ*<sup>2</sup> ¼ 0*, ρ*<sup>2</sup> <0*:*

∑*<sup>T</sup> <sup>t</sup>*¼<sup>1</sup>^*e*<sup>2</sup>

^*<sup>λ</sup>* <sup>¼</sup> *argmin λϵΛ <sup>σ</sup>*^<sup>2</sup>

<sup>1</sup> ^*et*ð Þ*<sup>λ</sup>* <sup>2</sup> be the LS estimate of *<sup>σ</sup>*<sup>2</sup> for a fixed *<sup>λ</sup>*. The

� �; thus the least squares estimated threshold model is as

<sup>2</sup>*xt*�<sup>11</sup> *Zt*�<sup>1</sup> <sup>≥</sup> ^ ð Þ*<sup>λ</sup> gsr* <sup>þ</sup> *prce* <sup>þ</sup> *consumption* <sup>þ</sup> ^*et*

*H*<sup>0</sup> : *θ*<sup>1</sup> ¼ *θ*<sup>2</sup> (8)

<sup>0</sup> characterized as the remaining change from

*H*<sup>0</sup> : *ρ*<sup>1</sup> ¼ *ρ*<sup>2</sup> ¼ 0 (9)

*<sup>t</sup>* , this equation is used in this

; speaking to the leftover

(7)

ð Þ*λ*

where *LLCUL\_CAPt* is the ratio of the per capita land cultivated of the maize divided by the total population which is the dependent variable, *PRCEt* represents the price index of maize, *CONSUMPTIONt* is the consumption per tonne of maize and *Ɛ<sup>t</sup>* is the residual value for the regression. The TAR model in the following regression equation was adapted from a nonlinear approach study [42] based on Turkey's debt distress status [41]. The equation therefore for the purposes of the study is inscribed with threshold variables as price per tonne and consumption per tonne, with the dependent variable as per capita land cultivated:

$$\Delta y\_t = \theta\_1^\prime \mathbf{x}\_{t-1} \mathbf{1}\_{\left(Z\_{t-1} < \lambda\right)}\\proc + consumption + \theta\_2^\prime \mathbf{x}\_{t-1} \mathbf{1}\_{\left(Z\_{t-1} \ge \lambda\right)}\\proc + consumption + e\_t \tag{3}$$

where *xt*�<sup>1</sup> <sup>¼</sup> *yt*�<sup>1</sup>*r*<sup>0</sup> *t* <sup>Δ</sup>*Bt*�<sup>1</sup>*;* …*;*Δ*yt*�*<sup>k</sup>* � �<sup>0</sup> , and *Zt*�<sup>1</sup> is the threshold variable per capita land cultivated of the maize to the total population which is the dependent variable ratio (LCUL\_CAP) and includes explanatory variables, PRCE and CON-SUMPTION for *t* ¼ 1*,* …*,T, et* is an iid error and 1ð Þ*:* is the Heaviside indicator function represented as follows:

$$\mathbf{1}\_{(.)} = \begin{cases} \mathbf{1} & \text{if } Z\_{t-1} < \lambda \\ \mathbf{0} & \text{if } Z\_{t-1} \ge \lambda \end{cases} \tag{4}$$

The threshold *λ* is given as unknown; this means the values in the interval *λ*∈*Λ* ¼ ½ � *λ*1*; λ*<sup>2</sup> where both threshold values are observed so that *P Z*ð Þ¼ *<sup>t</sup>* ≤*λ*<sup>1</sup> *π*1>0 and *P Z*ð Þ¼ *<sup>t</sup>* ≤*λ*<sup>2</sup> *π*<sup>2</sup> <1; the specification of the threshold variable *Zt*�<sup>1</sup> assists as a framework of analysis of results that the variable is predetermined, strictly stationary and ergodic with a continuous distribution function [40].

The vectors *θ*<sup>1</sup> and *θ*<sup>2</sup> are distinguished according to specific components and are discussed as follows:

$$
\theta\_1 = \begin{pmatrix} \rho\_1 \\ \beta\_1 \\ a\_1 \end{pmatrix}, \ \theta\_2 = \begin{pmatrix} \rho\_2 \\ \beta\_2 \\ a\_2 \end{pmatrix} \tag{5}
$$

With scalar quantities represented by *<sup>ρ</sup>*1and *<sup>ρ</sup>*<sup>2</sup> as the slope coefficients on *yt*�<sup>1</sup>, *β*<sup>1</sup> and *β*<sup>2</sup> which have the same dimensions as *rt* represent the slope on the deterministic components, *<sup>α</sup>*<sup>1</sup> and *<sup>α</sup>*<sup>2</sup> are the slope coefficients on <sup>Δ</sup>*yt*�<sup>1</sup>*;* ……*:;*Δ*yt*�*<sup>k</sup>* � � for the observed regimes.

The threshold estimates of the model are carried out with the use of least squares technique (more specifically, in this study we use the Huber-White covariance method in order to adjust the variance–covariance matrix of a fit from least squares, for heteroscedasticity and correlated responses). Each of the threshold value intervals *λ*∈ *Λ* is estimated by least squares (LS) as follows:

$$\Delta y\_t = \hat{\theta}\_1(\lambda)' \mathbf{x}\_{t-1} \mathbf{1}\_{\langle Z\_{t-1} < \lambda \rangle} prc + consumption + \hat{\theta}\_2(\lambda)' \mathbf{x}\_{t-1} \mathbf{1}\_{\langle Z\_{t-1} \ge \lambda \rangle} prc + consumption + \hat{\mathbf{e}}\_t(\lambda) \tag{6}$$

*Regime Switch and Effect on Per Capita Food Security Issues in South Africa DOI: http://dx.doi.org/10.5772/intechopen.86931*

where we let *σ*^<sup>2</sup> ð Þ¼ *<sup>λ</sup> <sup>T</sup>*�<sup>1</sup> ∑*<sup>T</sup>* <sup>1</sup> ^*et*ð Þ*<sup>λ</sup>* <sup>2</sup> be the LS estimate of *<sup>σ</sup>*<sup>2</sup> for a fixed *<sup>λ</sup>*. The threshold estimate of the threshold value is found by minimizing *<sup>σ</sup>*2ð Þ*<sup>λ</sup>* which are represented as

$$
\hat{\lambda} = \operatorname\*{argmin}\_{\lambda \in \Lambda} \hat{\sigma}^2(\lambda),
$$

To find the least squares estimates of other parameters, a point estimate ^*λ* is used in relation to.

*θ* ^<sup>1</sup> <sup>¼</sup> *<sup>θ</sup>* ^<sup>1</sup> ^*λ* � � and *θ* ^<sup>2</sup> <sup>¼</sup> *<sup>θ</sup>* ^<sup>2</sup> ^*λ* � �; thus the least squares estimated threshold model is as follows:

$$\Delta y\_t = \bar{\theta}^r \mathbf{x}\_{t-1} \mathbf{1}\_{\{\mathbf{Z}\_{t-1} < \bar{\lambda}\}} prce + consumption + \bar{\theta}^r \mathbf{z}\_{t-1} \mathbf{1}\_{\{\mathbf{Z}\_{t-1} \ge \bar{\lambda}\}} \mathbf{g} r + prce + consumption + \hat{e}\_t \tag{7}$$

Eq. (7) shows the least squares residuals ^*et* and denotes the residual variance from the least squares estimation as *<sup>σ</sup>*^<sup>2</sup> <sup>¼</sup> *<sup>T</sup>*�<sup>1</sup> ∑*<sup>T</sup> <sup>t</sup>*¼<sup>1</sup>^*e*<sup>2</sup> *<sup>t</sup>* , this equation is used in this study to draw standard Wald statistics and *t*-statistic inferences on parameters from Eq. (3) which can test the possible presence of nonlinearity.

#### **Threshold effects**

namely, consumption per tonne and price index of maize. The regression is

where *LLCUL\_CAPt* is the ratio of the per capita land cultivated of the maize divided by the total population which is the dependent variable, *PRCEt* represents the price index of maize, *CONSUMPTIONt* is the consumption per tonne of maize and *Ɛ<sup>t</sup>* is the residual value for the regression. The TAR model in the following regression equation was adapted from a nonlinear approach study [42] based on Turkey's debt distress status [41]. The equation therefore for the purposes of the study is inscribed with threshold variables as price per tonne and consumption per tonne, with the dependent variable as per capita land

capita land cultivated of the maize to the total population which is the dependent variable ratio (LCUL\_CAP) and includes explanatory variables, PRCE and CON-SUMPTION for *t* ¼ 1*,* …*,T, et* is an iid error and 1ð Þ*:* is the Heaviside indicator

> *Zt*�<sup>1</sup> < *λ Zt*�<sup>1</sup> ≥ *λ*

> > *ρ*2 *β*2 *α*2

1

CA (5)

*xt*�11ð Þ *Zt*�<sup>1</sup> <sup>≥</sup>*<sup>λ</sup> prce* þ *consumption* þ e^*t*ð Þ*λ*

� � for

(6)

0

B@

<sup>1</sup>ð Þ*:* <sup>¼</sup> <sup>1</sup> *if* 0 *if*

> *ρ*1 *β*1 *α*1

1

0

B@

The threshold *λ* is given as unknown; this means the values in the interval *λ*∈*Λ* ¼ ½ � *λ*1*; λ*<sup>2</sup> where both threshold values are observed so that *P Z*ð Þ¼ *<sup>t</sup>* ≤*λ*<sup>1</sup> *π*1>0 and *P Z*ð Þ¼ *<sup>t</sup>* ≤*λ*<sup>2</sup> *π*<sup>2</sup> <1; the specification of the threshold variable *Zt*�<sup>1</sup> assists as a framework of analysis of results that the variable is predetermined, strictly station-

The vectors *θ*<sup>1</sup> and *θ*<sup>2</sup> are distinguished according to specific components and are

CA*, <sup>θ</sup>*<sup>2</sup> <sup>¼</sup>

With scalar quantities represented by *<sup>ρ</sup>*1and *<sup>ρ</sup>*<sup>2</sup> as the slope coefficients on *yt*�<sup>1</sup>, *β*<sup>1</sup> and *β*<sup>2</sup> which have the same dimensions as *rt* represent the slope on the deterministic components, *<sup>α</sup>*<sup>1</sup> and *<sup>α</sup>*<sup>2</sup> are the slope coefficients on <sup>Δ</sup>*yt*�<sup>1</sup>*;* ……*:;*Δ*yt*�*<sup>k</sup>*

The threshold estimates of the model are carried out with the use of least squares

^2ð Þ*<sup>λ</sup>* <sup>0</sup> *:*

technique (more specifically, in this study we use the Huber-White covariance method in order to adjust the variance–covariance matrix of a fit from least squares, for heteroscedasticity and correlated responses). Each of the threshold value inter-

(

<sup>1</sup>*xt*�11ð Þ *Zt*�<sup>1</sup> <sup>&</sup>lt;*<sup>λ</sup> prce* þ *consumption* þ *θ*<sup>0</sup>

<sup>Δ</sup>*Bt*�<sup>1</sup>*;* …*;*Δ*yt*�*<sup>k</sup>* � �<sup>0</sup>

ary and ergodic with a continuous distribution function [40].

*θ*<sup>1</sup> ¼

vals *λ*∈ *Λ* is estimated by least squares (LS) as follows:

*:xt*�11ð Þ *Zt*�<sup>1</sup> <sup>&</sup>lt;*<sup>λ</sup> prce* þ *consumption* þ *θ*

*t*

*LCUL*\_*CAPt* ¼ *β*<sup>0</sup> þ *β*1*PRCEt* þ *β*2*CONSUMPTIONt* þ *ε<sup>t</sup>* (2)

<sup>2</sup>*xt*�11ð Þ *Zt*�<sup>1</sup> <sup>≥</sup>*<sup>λ</sup> prce* þ *consumption* þ *et*

, and *Zt*�<sup>1</sup> is the threshold variable per

(3)

(4)

as follows:

*Food Security in Africa*

cultivated:

Δ*yt* ¼ *θ*<sup>0</sup>

where *xt*�<sup>1</sup> <sup>¼</sup> *yt*�<sup>1</sup>*r*<sup>0</sup>

discussed as follows:

the observed regimes.

Δ*yt* ¼ *θ*

**24**

^1ð Þ*<sup>λ</sup>* <sup>0</sup>

function represented as follows:

This examination utilizes the Wald test measurement to address the subject of whether the parameters of condition (3) have the nearness of limit impacts and the likelihood of general nonlinearity. This strategy is utilized in application to help the investigation of nonlinear time arrangement [40]. The limit impact vanishes under the joint theory where

$$H\_0: \theta\_1 = \theta\_2\tag{8}$$

Condition (8), is a limitation that is tested through the observation of the standard Wald test written as: *WT* <sup>¼</sup> *<sup>T</sup> <sup>σ</sup>*^<sup>2</sup> 0 *<sup>σ</sup>*^<sup>2</sup> � <sup>1</sup> � �; with *<sup>σ</sup>*^<sup>2</sup> ; speaking to the leftover difference from condition (6) and *σ*^<sup>2</sup> <sup>0</sup> characterized as the remaining change from OLS estimation in condition (7) of the edge model. The dismissal of the invalid theory in condition (8) implies that there is factual importance of the logical factors of the model and that edge impacts exist.

#### **Unit root test, asymmetry and cointegration**

To test for stationarity, the test statistics are observed for the parameters *ρ*<sup>1</sup> and *ρ*<sup>2</sup> since they control the stationarity process of *yt* in Eq. (3); as such, the null hypothesis is represented as follows:

$$H\_0: \rho\_1 = \rho\_2 = \mathbf{0} \tag{9}$$

Eq. (3) is then rewritten as a stationary threshold autoregression in the variable Δ*yt* also implying *yt* is *I*ð Þ1 and therefore has a unit root.

Moreover, in a situation where *p* ¼ 1*,* the model becomes stationary if *ρ*<sup>1</sup> < 0*, ρ*2>0 and 1 þ *ρ*<sup>1</sup> ð Þ 1 þ *ρ*<sup>2</sup> ð Þ<1. Hence this suggests an alternative to *H*<sup>0</sup> represented as *H*<sup>1</sup> : *ρ*<sup>1</sup> <0 and *ρ*<sup>2</sup> <0.

Unit root test can also be observed in a partial case where the alternative hypothesis reads as:

$$H\_2: \left\{ \begin{array}{l} \rho\_1 < \mathbf{0} \text{ and } \rho\_2 = \mathbf{0}, \\\rho\_1 = \mathbf{0} \text{ or } \end{array} \right. \\
\left. \begin{array}{l} \text{and } \rho\_2 = \mathbf{0}, \\\rho\_2 < \mathbf{0}. \end{array} \right.$$

Given that *H*<sup>2</sup> holds, then *yt* will be observed as a unit root process in a single regime and a stationary process in another regime. Thus, in this case, a nonstationary process is observed, which is not a classic unit root process.

#### *2.2.1 Augmented Dickey-Fuller (ADF) test*

The ADF is the improved expansion of the condemned DF methodology and in that capacity is improved by including the slacked estimations of the reliant variable Δ*Yt*. The ADF test comprises of assessing the relapse as pursues [43]:

$$
\Delta Y\_t = \beta\_1 + \beta\_2 + \delta \mathbf{y}\_{t-1} + \sum\_{i=t}^m \alpha\_i \Delta \mathbf{y}\_{t-1} + \varepsilon\_t \tag{10}
$$

the 5% significance level is approximated to 10,000 replications. In summary, when there is no threshold effect observed, then the threshold value *λ* is not identified,

An Enders and Siklos test for cointegration and edge alteration is utilized [46] and is an expansion of the Engle-Granger test in light of the fact that it shows great power and size property over the Johansen test which accepts symmetric change in long-run balance [47], while, actually, Enders and Siklos test catches the topsy-

In the case where the threshold effect is observed, this means that a threshold value (*λ*0) is identified and thus the parameters of Eq. (3) are not equal, *θ*<sup>1</sup> 6¼ *θ*2; it is also assumed that *E*Δ*yt* ¼ 0 which is observed in model (3) given that assumption 1 by Caner and Hansen which shows *μ*1*P Z*ð Þþ *<sup>t</sup>*�<sup>1</sup> <*λ μ*2*P Z*ð Þ¼ *<sup>t</sup>*�<sup>1</sup> ≥*λ* 0 holds. However if *E*Δ*yt* 6¼ 0*,* then a time trend is included in model (3), and Δ*yt* is replaced with

; therefore, a long-run variance and long-run correlation are defined

*E* Δ*yt*

According to Theorem 6 by Caner and Hansen, if the parameters from Eq. (3)

� *<sup>N</sup>* <sup>0</sup> 0

Along these lines, this is autonomous of the negative of the traditional without pattern Dickey-Fuller t-circulation. Additionally, the Dickey-Fuller gives a preservationist bound on asymptotic dispersion, yet in addition the two-sided Wald test measurement has a valuable articulation and bound which is accounted for under hypothesis 6. In outline, when an edge impact is watched, this implies λ is distinguished, and for extensive examples, *λ* will be close to the true value of the threshold *λ*0; this means that the asymptotic distribution of *RT* is similar to the case where

The threshold adjustment for cointegration uses the Enders and Siklos test for a case when the threshold value is known. The null hypotheses P1 = 0 and P2 = 0 along with the joint hypothesis are P1 = P2 = 0, while t-Max is the maximum threshold with the largest test statistic, with F statistic denoted by *ɸ*, and thus the ɸ statistic can reflect a rejection of the null hypothesis P1 = P2 = 0 at the point when just a solitary one of the qualities is negative. In any case, on the off chance that both the p esteems are negative in measurement nature, at that point the invalid theory comes up short conceivable dismissal. Furthermore, the ɸ measurement rejects the invalid speculation of no cointegration at the 1% criticalness dimension, and t-Max measurement rejects the invalid theory at 5%, yet not the 1%, centrality level, in this manner inferring that the disseminations of ɸ and t-Max will rely upon test estimate and the quantity of factors incorporated into the cointegration relationship. Be

*Z*<sup>1</sup> þ *δ*1*DF* ≪ *DF and* � *t*2)

*Z*2  <sup>Δ</sup>*yt*�*<sup>k</sup>*

�

<sup>1</sup> � *<sup>δ</sup>*<sup>2</sup> 2 <sup>1</sup>*=*<sup>2</sup>

 *,* <sup>1</sup> *<sup>σ</sup>*<sup>21</sup> *σ*<sup>21</sup> 1

(12)

(13)

*<sup>y</sup>*>0*,* then the *t*-statistic function is

*Z*<sup>2</sup> þ *δ*2*DF* ≪ *DF*

and thus ^*λ* will remain the random sample, causing *RT* to be random.

*Regime Switch and Effect on Per Capita Food Security Issues in South Africa*

given that the Δ*yt* remains stationary and ergodic; hence, we let

*σ*2 *<sup>y</sup>* ¼ ∑ ∞ *k*¼�∞

are not equal, and if *<sup>E</sup>*Δ*yt* <sup>¼</sup> 0 and the variance *<sup>σ</sup>*<sup>2</sup>

*Given that <sup>Z</sup>*<sup>1</sup>

turvy nature of long-run equilibrium modifications.

*DOI: http://dx.doi.org/10.5772/intechopen.86931*

*2.2.4 When threshold is known*

Δ*yt* � *E*Δ*yt*

given as

**27**

�*t*1) �

<sup>1</sup> � *<sup>δ</sup>*<sup>2</sup> 1 <sup>1</sup>*=*<sup>2</sup>

the threshold value *λ*<sup>0</sup> is known.

where *<sup>ε</sup><sup>t</sup>* represents the pure white noise term, <sup>Δ</sup>*yt*�<sup>1</sup> <sup>¼</sup> *yt*�<sup>1</sup> � *yt*�<sup>2</sup> *,* which shows the number of lagged differences which are often determined empirically. ADF also tests whether the unit *δ* is equal to zero and thus largely determines the trend stationarity and nonstationarity.

#### *2.2.2 Phillips-Perron test*

The Phillips-Perron (PP) test is a non-parametric methodology with the thought of disturbance parameters and consequently takes into account heterogeneous information circulation and pitifully subordinate factors [44]. The test is depicted to be increasingly vigorous regarding unspecified sequential relationship and heteroscedasticity in the model. Other studies legitimize the utilization of nonparametric test with regard to ordinariness suspicions being disregarded and affirm that non-parametric test like the Phillips-Perron test does not accept symmetry or any fundamental conveyance and is considerably more productive and incredible than parametric techniques [45].

The regression for the PP test as proposed by Phillips and Perron is represented as follows:

$$
\rho \mathbf{y}\_t = \mathbf{a} + \rho \mathbf{y}\_{t-1} + \mathbf{e}\_t \tag{11}
$$

where *et* is the Heaviside pointer I(0) and takes into account heteroscedasticity and all things considered the PP test revises for sequential connection and heteroscedasticity mistakes in *et*. The test insights under this model are appeared as *tp*¼<sup>0</sup> and *Tp*^ which are changed and communicated as *Zt* and *Zp* measurements.

#### *2.2.3 When threshold is unknown*

For this situation, asymptotic conveyance is tried when there is no edge impact, implying that the limit esteem is obscure and hence the p esteems are additionally obscure for the given parameters. This is as indicated by hypothesis 5 by [40] where

$$\begin{aligned} \theta\_1 &= \theta\_2; \text{ therefore, } (t\_1, t\_2) \dot{\Rightarrow} (t\_1(u \ast), t\_2(u \ast)) \text{ and } R\_T \dot{\Rightarrow} R(t\_1(u \ast), t\_2(u \ast)) \le \sup\_{u \in \left[\pi\_1 \pi\_2\right]} \\ R(t\_1(u), t\_2(u)) \text{ where } u \ast &= \operatorname\*{argmax}\_{u \in \left[\pi\_1 \dots \pi\_N\right]} T(u), \end{aligned}$$

*u*∈ ½ � *π*1*; π*<sup>2</sup> *:* This means that *t*-statistics are distributed by the functions *t*1ð Þ *u* and *t*2ð Þ *u* at random argument *<sup>u</sup>*<sup>∗</sup> . The trimming range represented as [*π*1*, <sup>π</sup>*2� is free from nuisance parameters. The simulation of Monte Carlo experiment critical values of

#### *Regime Switch and Effect on Per Capita Food Security Issues in South Africa DOI: http://dx.doi.org/10.5772/intechopen.86931*

the 5% significance level is approximated to 10,000 replications. In summary, when there is no threshold effect observed, then the threshold value *λ* is not identified, and thus ^*λ* will remain the random sample, causing *RT* to be random.

An Enders and Siklos test for cointegration and edge alteration is utilized [46] and is an expansion of the Engle-Granger test in light of the fact that it shows great power and size property over the Johansen test which accepts symmetric change in long-run balance [47], while, actually, Enders and Siklos test catches the topsyturvy nature of long-run equilibrium modifications.

#### *2.2.4 When threshold is known*

Given that *H*<sup>2</sup> holds, then *yt* will be observed as a unit root process in a single

The ADF is the improved expansion of the condemned DF methodology and in that capacity is improved by including the slacked estimations of the reliant variable

> *m i*¼*t*

<sup>∝</sup>*iΔyt*�<sup>1</sup> <sup>þ</sup> *<sup>ε</sup><sup>t</sup>* (10)

*,* which

regime and a stationary process in another regime. Thus, in this case, a nonstationary process is observed, which is not a classic unit root process.

Δ*Yt*. The ADF test comprises of assessing the relapse as pursues [43]:

<sup>Δ</sup>*Yt* <sup>¼</sup> *<sup>β</sup>*<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*<sup>2</sup> <sup>þ</sup> *<sup>δ</sup>yt*�<sup>1</sup> <sup>þ</sup> <sup>∑</sup>

where *<sup>ε</sup><sup>t</sup>* represents the pure white noise term, <sup>Δ</sup>*yt*�<sup>1</sup> <sup>¼</sup> *yt*�<sup>1</sup> � *yt*�<sup>2</sup>

shows the number of lagged differences which are often determined empirically. ADF also tests whether the unit *δ* is equal to zero and thus largely determines the

The Phillips-Perron (PP) test is a non-parametric methodology with the thought

The regression for the PP test as proposed by Phillips and Perron is represented

where *et* is the Heaviside pointer I(0) and takes into account heteroscedasticity

For this situation, asymptotic conveyance is tried when there is no edge impact, implying that the limit esteem is obscure and hence the p esteems are additionally obscure for the given parameters. This is as indicated by hypothesis 5 by [40] where

�

*R t*ð Þ <sup>1</sup>ð Þ *<sup>u</sup>* <sup>∗</sup> *; <sup>t</sup>*2ð Þ *<sup>u</sup>* <sup>∗</sup> <sup>≤</sup> sup

*u* ∈½ � *π*1*π*<sup>2</sup>

ð Þ *t*1ð Þ *u* ∗ *; t*2ð Þ *u* ∗ and *RT*)

This means that *t*-statistics are distributed by the functions *t*1ð Þ *u* and *t*2ð Þ *u* at random argument *<sup>u</sup>*<sup>∗</sup> . The trimming range represented as [*π*1*, <sup>π</sup>*2� is free from nuisance parameters. The simulation of Monte Carlo experiment critical values of

*u*∈ ½ � *π*1*; π*<sup>2</sup> *:*

and all things considered the PP test revises for sequential connection and heteroscedasticity mistakes in *et*. The test insights under this model are appeared as *tp*¼<sup>0</sup> and *Tp*^ which are changed and communicated as *Zt* and *Zp* measurements.

*yt* <sup>¼</sup> *<sup>α</sup>* <sup>þ</sup> *<sup>ρ</sup>yt*�<sup>1</sup> <sup>þ</sup> *et* (11)

of disturbance parameters and consequently takes into account heterogeneous information circulation and pitifully subordinate factors [44]. The test is depicted to be increasingly vigorous regarding unspecified sequential relationship and heteroscedasticity in the model. Other studies legitimize the utilization of nonparametric test with regard to ordinariness suspicions being disregarded and affirm that non-parametric test like the Phillips-Perron test does not accept symmetry or any fundamental conveyance and is considerably more productive and incredible

*2.2.1 Augmented Dickey-Fuller (ADF) test*

trend stationarity and nonstationarity.

than parametric techniques [45].

*2.2.3 When threshold is unknown*

*θ*<sup>1</sup> ¼ *θ*2; therefore, ð Þ) *t*1*; t*<sup>2</sup>

�

*R t*ð Þ <sup>1</sup>ð Þ *<sup>u</sup> ; <sup>t</sup>*2ð Þ *<sup>u</sup>* where *<sup>u</sup>* <sup>∗</sup> <sup>¼</sup> *argmax T u*ð Þ*,*

as follows:

**26**

*2.2.2 Phillips-Perron test*

*Food Security in Africa*

In the case where the threshold effect is observed, this means that a threshold value (*λ*0) is identified and thus the parameters of Eq. (3) are not equal, *θ*<sup>1</sup> 6¼ *θ*2; it is also assumed that *E*Δ*yt* ¼ 0 which is observed in model (3) given that assumption 1 by Caner and Hansen which shows *μ*1*P Z*ð Þþ *<sup>t</sup>*�<sup>1</sup> <*λ μ*2*P Z*ð Þ¼ *<sup>t</sup>*�<sup>1</sup> ≥*λ* 0 holds. However if *E*Δ*yt* 6¼ 0*,* then a time trend is included in model (3), and Δ*yt* is replaced with Δ*yt* � *E*Δ*yt* ; therefore, a long-run variance and long-run correlation are defined given that the Δ*yt* remains stationary and ergodic; hence, we let

$$
\sigma\_y^2 = \sum\_{k=-\infty}^{\infty} E(\Delta \mathbf{y}\_t \Delta \mathbf{y}\_{t-k}) \tag{12}
$$

According to Theorem 6 by Caner and Hansen, if the parameters from Eq. (3) are not equal, and if *<sup>E</sup>*Δ*yt* <sup>¼</sup> 0 and the variance *<sup>σ</sup>*<sup>2</sup> *<sup>y</sup>*>0*,* then the *t*-statistic function is given as

$$-t\_1 \dot{\Rightarrow} \left(\mathbf{1} - \delta\_1^2\right)^{1/2} \mathbf{Z}\_1 + \delta\_1 DF \ll DF \quad \text{and} \quad -t\_2 \dot{\Rightarrow} \left(\mathbf{1} - \delta\_2^2\right)^{1/2} \mathbf{Z}\_2 + \delta\_2 DF \ll DF$$

$$\text{Given that } \begin{pmatrix} \mathbf{Z}\_1\\ \mathbf{Z}\_2 \end{pmatrix} \sim N\left( \begin{pmatrix} \mathbf{0}\\ \mathbf{0} \end{pmatrix}, \begin{pmatrix} \mathbf{1} & \sigma\_{21}\\ \sigma\_{21} & \mathbf{1} \end{pmatrix} \right) \tag{13}$$

Along these lines, this is autonomous of the negative of the traditional without pattern Dickey-Fuller t-circulation. Additionally, the Dickey-Fuller gives a preservationist bound on asymptotic dispersion, yet in addition the two-sided Wald test measurement has a valuable articulation and bound which is accounted for under hypothesis 6. In outline, when an edge impact is watched, this implies λ is distinguished, and for extensive examples, *λ* will be close to the true value of the threshold *λ*0; this means that the asymptotic distribution of *RT* is similar to the case where the threshold value *λ*<sup>0</sup> is known.

The threshold adjustment for cointegration uses the Enders and Siklos test for a case when the threshold value is known. The null hypotheses P1 = 0 and P2 = 0 along with the joint hypothesis are P1 = P2 = 0, while t-Max is the maximum threshold with the largest test statistic, with F statistic denoted by *ɸ*, and thus the ɸ statistic can reflect a rejection of the null hypothesis P1 = P2 = 0 at the point when just a solitary one of the qualities is negative. In any case, on the off chance that both the p esteems are negative in measurement nature, at that point the invalid theory comes up short conceivable dismissal. Furthermore, the ɸ measurement rejects the invalid speculation of no cointegration at the 1% criticalness dimension, and t-Max measurement rejects the invalid theory at 5%, yet not the 1%, centrality level, in this manner inferring that the disseminations of ɸ and t-Max will rely upon test estimate and the quantity of factors incorporated into the cointegration relationship. Be that as it may, the outcome is controlled by the utilization of the Enders and Siklos test approach and Monte Carlo basic qualities that additionally depend on the dynamic idea of the threshold adjustment process.
