Price fluctuation and market integration of selected cereal in north-eastern nigeria, 2001-2010

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PRICE FLUCTUATION AND MARKET INTEGRATION OF SELECTED CEREAL IN NORTH-EASTERN NIGERIA, 2001-2010

 

ABSTRACT

 

Prices contain information crucial to maximizing the returns to production and marketing investments. At planting time, a farmer’s planting decision depends on expected profits, which invariably hinge on the anticipated prices of the crop or mix of crops that would prevail in the market at the time of sale and on the farmer’s interpretation of those prices. A trader, in search of profitable arbitrage, reads and translates price signals in deciding on what crops to buy, where to buy, and when to sell. Apart from guiding production and marketing decisions, prices govern the optimal allocation of resources among competing uses. The accuracy, reliability, and promptness of market information are therefore critical in attaining pricing efficiency. Broadly, the study attempted to analyze the price fluctuation and market integration of selected cereal grains in North-eastern Nigeria. The specific objectives of the study were to: (i) estimate the extent of the various components of price; (ii) derive the probability distribution of cereal grain price in the long-run; (iii) determine the existence and level of inter-market price dependency; (iv) examine the speed of price adjustment to long-run equilibrium and (v) examine the Granger Causality among rural and urban cereal grain markets. The study was conducted in North-eastern Nigeria. Purposive sampling technique was used to select two states, of Adamawa and Taraba, from the six states that made up the North-east geopolitical zone. Only secondary data were used in the study. Secondary data on monthly bases for the prices of 100kg of three cereal grains, maize, rice and sorghum in both rural and urban markets in the study area were obtained from Adamawa and Taraba States Agricultural Development Program offices for a period of 10 years (2001-2010). Data were analyzed using descriptive statistics such as price decomposition technique, and inferential statistics such as Markov Chain, Vector Autoregressive and Error Correction Models. The results revealed that, the trend component showed an upward movement for all the three commodities. The seasonal variation had indexes ranged from 198.15 to 52.61, 142.83 to 61.88, and 141.44 to 66.25 for maize, rice and sorghum, respectively. The random and cyclical variations had negligible and insignificant indices with the former having 0.01 all through and the later ranging from 0.93 to 1.26. Probability distribution matrices of the three cereal grains were 0.18, 0.48 and 0.34 for maize, 0.27, 0.68 and 0.05 for rice and 0.48, 0.25 and 0.27 for sorghum. The Augmented Dickey-Fuller unit roots test indicated I(0), I(1) and I(1) for maize, rice and sorghum, respectively. Null hypothesis of β = 1 was rejected against β = 0. Trace

 

statistics for rural and urban markets were not significant ( Rural and urban prices of maize responded to shocks within and between each market. The speed with which the system adjusted to shocks and restored equilibrium between the short and the long-run were -0.170725 and -0.29517 for urban and 0.592237 and 0.38034 for rural prices of rice and sorghum, respectively. Granger Causality showed that a bi-directional flow of price signals existed between rural and urban prices of maize, while rural prices of rice and sorghum did not Granger-Cause urban prices of rice and sorghum. Also, urban prices of both rice and sorghum did not Granger-cause rural prices of both rice and sorghum. Findings of the study showed an imperfect market integration for North-eastern Nigeria cereal grain markets, this indicate that there may be substantial benefits in developing better infrastructure facilities to effectively link production centers to market centers and in improving market knowledge by providing more relevant, accurate, and timely public market information.

 

 

 

 TABLE OF CONTENTS 
ContentsPages
Title i
Certificationii
Dedicationiii
Acknowledgementsiv
Table of Contentsv
List of Tablesix
List of Figuresx
Abstractxi
CHAPTER ONE: INTRODUCTION 
1.0Background of the Study1
1.2Problem Statement7
1.3Objectives of the Study11
1.4Research Hypotheses11
1.5Justification of the Study11
1.6Limitations of the Study13
CHAPTER TWO: LITERATURE REVIEW 
2.1Spatial Competition14
2.1.1Estimating Spatial Price Relationship17
2.2Theoretical Framework20
2.3Time Series21
2.3.1Stationary and Nonstationary Time Series22
2.3.2Stationarity22
2.3.3Non-stationarity23
2.4Unit Roots24
2.5Conceptual Framework25
2.5.1Measure of Integration26
2.6Markov Chain30
2.6.1Markov Processes and Random Walk Hypotheses31
2.7Analytical Framework33
2.7.1Studies on Market Integration around the World33
2.7.2Empirical Evidence from Market Integration in Nigeria45
2.8Analysis of Time Series47
2.9Markov Processes54
2.9.1Markov Chains55
2.10Price Forecasting Models59
2.11Estimation of Transitional and Initial Probability Matrix62
CHAPTER THREE: METHODOLOGY 
3.1The Study Area66
3.2Sampling Procedure67
3.3Data Collection67
3.4Data Analysis67
3.4.1Components of Time Series68
3.4.2Isolation of Time Series Components69
3.4.3Transition Probability72

 

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3.4.4Using Markov Chain72
3.4.5Limitations of Markov   Chain72
3.4.6Error Correction of Cointegration Equation or System76
CHAPTER FOUR: RESULTS AND DISCUSSION 
4.1Components of Time Series87
4.2Probability Distribution of Cereal Price in the Long-run87
4.2.1Maize87
4.2.1.1Interpretation of Maize Transition Matrix88
4.2.2Rice89
4.2.3Sorghum90
4.3Existence and Level of Inter-Market Price Dependency92
4.3.1Unit roots test92
4.3.2Impulse Response Function92
4.3.3Johansen Cointegration test92
4.4Speed of Price Adjustment to Long-run Equilibrium92
4.4.1Rice92
4.4.2Sorghum102
4.5Granger-Causality among Rural and Urban Markets105
4.5.1Maize105
4.5.2Rice105
4.5.3Sorghum106
CHAPTER FIVE: SUMMARY, CONCLUSION AND 
RECOMMENDATIONS 
5.1Summary107
5.2Conclusion107
5.3Recommendations109
5.4Major Contributions of the Study to knowledge110
5.5Suggestions for Further Studies110
References111
Appendix128

 

 

CHAPTER ONE

 

INTRODUCTION

 

1.1        Background of the Study

 

The grain sub-sector plays an important role in the economic development of Nigeria. The output of the sub-sector (Ismaila, Gana, Twanya & Dogara 2010; Okunneye, 2003) constitutes a large proportion of staple food stuffs in Nigeria. Between 1985 and 1995, cereal grain accounted for almost 50 % of the total food supply in Nigeria when expressed in grain equivalent (Akpan & Udoh, 2009; Ukoha). On the other hand, Paulino and Sarma (1988) reported that about 70% of the total food crop area harvested in Nigeria was devoted to cereals and the remaining 30% to non-cereals.

 

The most important cereal grain crops grown and marketed in Nigeria are maize, rice, sorghum, millet and wheat (Akpan & Udoh, 2009; Global Information and Early Warning System on Food and Agriculture (GIEWS), 2008; Ismaila et al., 2010; Oguntunde, 1989; Wudiri, 1992). Of these, rice, maize, millet and sorghum are the major sources of energy staple food available and affordable in Nigeria, and are the commodities that are of considerable importance for food security, expenditure and income of households in Northern Nigeria (Ismaila et al., 2010; Maziya-Dixon et al., 2004).

 

In most parts of Asia and Africa, cereal products comprise 80% or more of the average diet, in central and western Europe, as much as 50% and in the United States, between 20 – 25% (Food and Agriculture Organization, 1996). Also, the increased demand for cereals, as a result of rapid urbanization, means that food crops must increasingly be produced to meet the needs of the rural and urban population (Balarabe, 2003). According to the Central Bank of Nigeria (CBN) (2000), Okunneye (2003) and Ukoha (2005), most Nigerians depended on cereal grains for their

 

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daily dietary needs and the price of these grains is one factor that determines the extent to which Nigerians can pay for these food commodities. Cereal grains availability and prices have become a major welfare determinant for the poorest segments of the Nigerian consumers who also are least food secured (Akande, 2001). Also, CBN (2000), Akande (2001) and Akpan and Udoh (2009) have affirmed that, the nominal or producer price of cereal grains have continuously fluctuated over the past years.

 

Spatial market integration of agricultural products has been widely used to indicate overall market performance (Faminow & Benson, 1990). In spatially integrated markets, competition among arbitragers will ensure that, a unique equilibrium is achieved where local prices in regional markets differ by no more than transportation and transaction costs. Information of spatial market integration, thus, provides indication of competitiveness, the effectiveness of arbitrage, and the efficiency of pricing (Sexton, Kling & Carman, 1991).

 

If price changes in one market are fully reflected in alternative market, these markets are said to be spatially integrated (Goodwin & Schroeder, 1991). Prices in spatially integrated markets are determined simultaneously in various locations, and information of any change in price in one market is transmitted to other markets (Gonzalez-Rivera & Helfand, 2001). Markets that are not integrated may convey inaccurate price signal that might distort producers’ marketing decisions and contribute to inefficient product movement (Goodwin & Schroeder 1991), and traders may exploit the market and benefit at the cost of producers and consumers. In more integrated markets, farmers specialize in production activities in which they are comparatively proficient, consumers pay lower prices for purchased goods, and society is better able to reap increasing returns from technological innovations and economies of scale (Vollrath, 2003). Market integration of agricultural products has retained importance in developing

 

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countries due to its potential application to policy making. Based on the information of the extent of market integration, government can formulate policies of providing infrastructure and information regulatory services to avoid market exploitation.

 

In theory, spatial price determination models suggest that, if two markets are linked by trade in a free market regime, excess demand or supply shocks in one market will have an equal impact on price in both markets. Given the wide range of ways prices may be related, the concept of price transmission can be thought of as being based on three notions, or components (Balcombe & Morisson, 2002; Prakash, 1998). These are:

 

co-movement and completeness of adjustment which implies that changes in prices in one market are fully transmitted to the other at all points of time;

 

dynamics and speed of adjustment which implies the process by, and rate at which, changes in prices in one market are filtered to the other market or levels; and,

 

asymmetry of response which implies that upward and downward movements in the price in one market are symmetrically or asymmetrically to the other. Both the extent of completeness and the speed of the adjustment can be asymmetric.

 

Within this context, complete price transmission between two spatially separated markets is defined as a situation where changes in one price are completely and instantaneously transmitted to the other price, as postulated by the Law of One Price (LOP). In this case, spatially separated markets are integrated. In addition, this definition implies that if price changes are not passed-through instantaneously, but after some time, price transmission is incomplete in the short-run, but complete in the long run, as implied by spatial arbitrage condition. The distinction between short-run and long-run price transmission is important, and the speed by which prices adjust to

 

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their long-run relationship is essential in understanding the extent to which markets are integrated in the short-run. Changes in the price at one market may need some time to be transmitted to other markets for various reasons, such as policies, the number of stages in marketing and the corresponding contractual arrangements between economic agents, storage and inventory holding, delays caused in transportation or processing, or “price leveling” practices.

 

Fluctuation in prices seriously affects cereal productivity in Nigeria (Ismaila et al., 2010). For instance, the demise of poultry and poultry processing companies following the outbreak of avian influenza in Nigeria has adversely affected the demand for maize, a major component of poultry feeds across Nigeria. With last year’s stock of grains still in the market, serious concern has been raised about the impact of the abundant supplies on prices with the exception of sorghum which is commonly demanded by breweries and other drink manufacturing companies in Nigeria. Conversely, the low demand for maize has discouraged many farmers from maize production and consequently increased the price of maize in 2008.

 

The Nigerian government realizing the importance of the grain sub-sector had several times intervened in standardizing grain prices through agricultural price policy reformation. Some of the instruments used, as pointed out by Okoh and Akintola (2005) and Akpan and Udoh (2009), included input subsidies, strategic grain reserve scheme of 1976, ban on importation of rice and maize in 1985 and the liberalization of the economy in 1986 among other measures. Despite these lofty attempts, the producer prices of grains continued to fluctuate as presented in Table 1.

 

It is obvious from Table 1 that the major grain crops in Nigeria showed a broad dispersion in producer prices across the specified policy periods. For instance, between 1970 and

 

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1974, the mean producer price of rice was N301.40/ton and 17.12 % coefficient of variability in prices. In 1975 to 1979 the mean price of rice increased by more than 100 % compared to the p tfre-Operation Feed the Nation (OFN) period. The fluctuations were the increasing function of time across the specified policy periods. Similar trends were obtained in the producer prices of maize, millet and sorghum. The highest coefficient of variability was obtained during the Structural Adjustment Programme (SAP) period for all the crops. It was 67.97 % for rice, 69.39 % for maize, 83.97 % for millet and 70.64 % for sorghum.

 

 

Table 1.1: Major Grain Prices under Different Agricultural Policy Regimes in Nigeria

 

Policy Regime  Rice   Maize   Millet   Sorghum
 Mean priceCV%Mean priceCV%Mean priceCV%Mean priceCV%
 (N)/ton (N)/ton (N)/ton (N)/ton 
           
Pre-OFN301.417.12157.414.79140.039.56148.419.27
OFN604.020.12375.828.57141.019.32274.612.14
GR1423.738.85788.021.22622.735.36582.332.80
SAP7483.167.972938.369.392759.683.972689.970.64
Post-SAP39789.824.5320113.837.8919701.035.991856.233.23
Aggregate CV (%)   134.03   138.99   159.13   48.24

 

Source: Akpan and Udoh  (2009)

 

 

 

It is evident however, that farmers in Nigeria in particular and in Africa in general face dramatic fluctuations in prices of the crops they produce (Akpan & Aya, 2009; Fafchamps, 2000; Nuhu, Ani & Bawa, 2009; Nzomoi, 2008; Simister & Chanda, 2006; Williams, 2009; Zulauf & Roberts, 2008). Because of the fixed character of inputs in agriculture especially land and partly labour force as well as the nature of production, agricultural producers very often are not able to respond in the most economical way to the changes in prices of agricultural products and inputs.

 

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These factors of consequence, which is inelastic demand for most agricultural and food product lead to high fluctuation of agricultural product prices. This further reflects in fluctuation of farmers’ incomes leading to deterioration of their welfare (Abdissa & Dereje, 2001; Grega, 2002).

 

According to Grega (2002), these fluctuations in grain prices make agriculture a risky business. In the opinion of Grega (2002), even if inelastic supply of inputs is eliminated (e.g. there is increased flexibility of using agricultural inputs) still many other factors such as weather, disease and pest would be present. Consequences of the risk in agricultural production are the existence of deviations from the balanced volume of agricultural production demanded by the market leading to price instability of this production (Balarabe, Ahmed & Chikwendu, 2006; Doll & Orazem, 1984; Livingstone & Ord, 1984).

 

Though some level of price fluctuation provides information signals about market situation and may serve as an instrument for adjustment of supply to demand, high price fluctuation has a deteriorating effect on the whole economy and makes social structure unstable (Grega, 2002). Also, Fafchamps (2002) reported that fluctuation in a single commodity price affects farmers who specialize in that commodity. In practice, many resource poor farmers-though not all-have a diversified crop portfolio and should be relatively isolated against single price changes. But the welfare consequences of even a small drop in revenue might be dire. A large commercial farmer may be able to absorb a one year 50% drop in revenue while the small subsistence farmer may starve as a result of a 5 % drop in revenue. It is therefore not surprising that commercial farmer is typically more specialized than the small holder farmer. Agricultural price fluctuation may affect not only farm revenue but also the price farmers pay for the products consumed (Achanya, 2004). A rise in the world price of rice for instance, tends to raise food

 

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prices in rice importing countries not only for rice but for other food products which serve as rice substitutes. Previous studies on the marketing and pricing of staple foodstuffs in different parts of Nigeria have concluded that the marketing and pricing information transmission mechanism are inefficient although there are many buyers and sellers in the market (Anthonio, 1968; Dittoh, 1994; Jones, 1969; Okoh & Akintola, 2005; Thodey, 1969).

 

Agricultural prices greatly influence the pace and direction of agricultural development. Prices serve as market signals of the relative scarcity or abundance of a given product. Prices also serve as incentives to direct the allocation of economic resources and to a large extent they determine the structure and rate of economic growth (Ariyo, Voh & Ahmed, 2001). Information on agricultural commodity price in both developed and developing countries like Nigeria is important to both producers and consumers. Prices vary among markets and almost throughout the year, and understanding the nature and trend of such variations is essential for good planning by the producers, consumers and policy makers alike (Adegeye & Dittoh, 1985; World Bank, 2000).

 

1.2         Problem Statement

 

There have been reports about food insecurity, rising prices and vulnerable population. The World Food Programme (WFP) (2011) has said that the crisis was a silent Tsunami that was threatening to plunge more than 100 million people on every continent to hunger. The increasing commodity prices and the attendant social unrest have not been confined to the developing countries alone. Spain, Israel and South Korea have witnessed demonstrations by consumers protesting increased commodity prices. The rising global food prices pose a serious threat to political stability especially to the developing countries. There have been riots in Burkina Faso, Cameroun, Egypt, Indonesia, Cote d’Ivoire, Mauritania, Mozambique, Senegal and Zimbabwe

 

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among others. In Haiti, where food prices had risen by 65 % in the last six years, protesters recently took to the street comparing their hunger pangs to the burn of battery acid! In Nairobi, police violently dispersed demonstrators who were agitated by continuous rise in food prices in May 2008. The World Bank has identified 33 countries at risk of public disorder on account of soaring food prices.

 

According to Akpan and Udoh (2009), agricultural commodities price have experienced unprecedented fluctuations and continuous increases since 2002 until mid-2008. They argued that this has brought about price volatility, food inflation, poverty and hunger. Coupled with inadequate market price transmission, high food prices has increased the levels of food deprivation, droved millions of people into food insecurity, worsening conditions for many who were already food insecure, and threatening long term global food security. This places a tremendous pressure on achieving the Millennium Development Goals (MDGs) on hunger by year 2015 (FAO, 2008).

 

Fluctuations in food prices might not be rapid, but they create pressure on wages, lower real incomes, rising inflation, unemployment and decreasing demand for non-agricultural products. On the other hand, price decline could lead to a misleading allocation of inputs in agricultural sectors, which could seriously damage production ability and international competitiveness of this industry. Grega (2002) stated that if agricultural product prices were too low, the situation in the sector could be deteriorating by the consequent outflow of qualified labor force to other sectors of the economy and lead to migration of rural population to the urban areas and so, cause the depopulation of the rural areas. Production activities in the grain sub-sector may be retarded due to produce price uncertainty and production risk and marketing

 

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inefficiency. Resource use efficiency may also decrease because farmers may have less useful information on prices to guide them in production decision.

 

Policy formulation has failed to take cognizance of the fact that production and marketing /pricing constitute a continuum and that the absence of development in one retards progress in the other (Okoh & Akintola, 2005). Since the agricultural producer is both a seller of his produce as well as a buyer of agricultural production requisites, agricultural prices cover not only the prices “received” by farmers (“output prices”) but also the prices “paid” by farmers (“input price”). The farmer is also a buyer of consumer goods for use in his own household, while the second category of purchase can be regarded as “agricultural prices” the third category of purchase are not to be regarded as “agricultural prices”. The second and third categories of the prices are not, therefore, strictly within the coverage of this study.

 

There have been researches over the years on the integration of various combinations of Nigerian foodstuff markets, only one has been identified in the north-eastern Nigeria. The principal studies in this area include those of Anthonio (1968, 1988), Jones (1969), Gilbert (1986), Thodey (1969), Hays and McCoy (1977), Delgado (1986), Adekanye (1988), Ejiga

 

(1988), Dittoh (1994), Okoh (1999), Okoh and Akintola (1999), Nuhu et al. (2009), Obayelu and Salau (2010), and Ugwamba and Okoh (2010). These studies covered market integration, price efficiency and pricing conduct of various foodstuffs (gari, rice, cowpea, cassava roots, vegetables, sorghum, and maize) in different regions of Nigeria. The price series used for the various studies were collected weekly or fortnightly by the researchers except for Okoh (1999), and Okoh and Akintola (1999) which used monthly series collected by staff of Agricultural Development Projects (ADPs). With the exception of Dittoh (1994), Okoh (1999), Okoh and Akintola (1999) and Obayelu and Salau (2010), these studies used the classical correlation

 

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coefficients and simple static regression equation of the form (P1 = A + bP1) to draw conclusions

 

about the integration and efficiency of the markets for various foodstuffs.

 

The findings of these studies are doubtful. This is because the bivariate correlation coefficient and static regression methods are beclouded by problems of overwhelming seasonal and secular trends, as well as the possibility of autocorrelation from a static model calibrated to non-stationary time series, leading to spurious correlations and inferential errors (Blyn, 1973; Delgado, 1986; Granger & Newbold, 1974; Harris, 1979; Iyoha & Ekanem, 2004; Palaskas & Harris-White, 1993; Ravalion, 1986; Timmer, 1994).

 

Among these studies, only Nuhu et al. (2009) was undertaken in the north-eastern Nigeria and employed the static linear regression model in drawing conclusion about how markets were cointegrated and none attempted forecasting price of agricultural produce. This study has adopted a vector Autoregressive Model (VAR) and an Error Correction Model (ECM) in addressing the spatial market integration. Also, previous studies (Simister & Chanda, 2009) claimed an alarming increase and instability in staple food prices in the northern Nigeria. The various components of price and the probability distribution of price were considered to estimate each component and forecast the prices of the three commodities (maize, rice and sorghum).

 

The extent of uncertainty caused by price inefficiency and instability in the agricultural sector has made the industry a risky one. Therefore, there is the need to examine the integration of rural and urban markets in relation to competitiveness, effectiveness of arbitrage and pricing efficiency in cereal grain (maize, rice and sorghum) marketing in the north- eastern Nigeria. The salient research questions for which answers were provided for in this study are:

 

  • what is the magnitude of the various components of price?

 

  • what is the probability distribution of cereal price in the long-run?

 

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  • does inter-market price dependencies exist and at what level?

 

  • is the price adjustment in the market delayed or instantaneous? aynd

 

  • what is the nature of causality if any among rural and urban markets?

 

1.3         Objectives of the Study

 

The broad objective of this study is to analyze the price fluctuation and market integration of selected cereal grains in North-east Nigeria. Specifically the study:

 

  1. estimated the extent of the various components of price;

 

  1. derived the probability distribution of cereal price in the long-run;

 

  • determined the existence and level of inter-market price dependencies;

 

  1. examined the speed of price adjustment to long-run equilibrium; and

 

  1. examined the Granger-Causality among urban and rural markets.

 

1.4         Research Hypotheses

 

The following null hypotheses were tested to guide the study:

 

  • farmers will not receive better prices in the long-run.

 

  • cereal grain markets are spatially independent and inefficient

 

  • there is price collusion with instantaneous price adjustment,

 

  • there is price matching with delayed price adjustment, and

 

1.5         Justification of the Study

 

Agricultural prices are important economic variables in a market economy. Price relationships have a significant influence on decisions relating to the type and volume of agricultural production activity. They provide a measure for reaching judgment on policy formulation and administrative and executive action.

 

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In the short-run, an individual farmer needs output prices to determine the price and volume of his sales so as to optimize the return from his farm production. In the long run, knowledge of price trends helps a farmer to formulate the investment plan on his farm and to take decision on the structure and nature of his enterprise. An understanding of the normal differences in the prices of his products and production requisites during the year helps a farmer to react logically to the marketing situations in order to optimize the planning of the sale of his product and the purchase of his supplies. His production plan is governed by the price expectations of the various commodities he can produce.

 

Business organizations use agricultural price data in a number of ways; such as planning the size of their agricultural business enterprise, determining the time and place for purchasing agricultural production requisites, deciding on inventory expansion or contraction and hedging, selecting the market and time of sale of their produce so as to reap the best advantage and formulating credit policies. These organizations also use price information to decide on the nature and volume of storage accommodation needed for stocking goods and to determine the quantum of flow required from time to time to keep prices from fluctuating sharply.

 

Accurate and reliable agricultural price information for different crops in different areas at different times is necessary for any rational policy on prices of agricultural products. To this end, this work is hoped to be a valuable source of information to policy makers, producers and consumers. It would also be relevant in the academics for teaching and research purposes as well as a body of knowledge that would promote greater awareness on market price survey. It would also add to the volume of literature on price analysis.

 

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1.6         Limitations of the Study

 

The following were the limitations of the study:

 

  1. the study of spatial market integration requires information on prices, trade flows and transfer costs, however, the study was limited to wholesale prices, since that was the only information available. Transfer cost and other information on cost that are needed in the determination of factors responsible for cointegration are always difficult to come by especially in Africa. Therefore, the study only performed the cointegration of rural and urban markets;

 

  1. the study did not cover the entire North-eastern states due to the intense security challenges posed by insurgent Boko Haram in the geopolitical zone, hence only the two states of Adamawa and Taraba were purposively selected for the study since these were the less vulnerable states as at the time of study; and

 

  • analysis such as that of Markov Chain and decomposition of time series data, were done manually as a result of non-availability of statistical packages to handle them, hence cumbersome. Because all the analysis of Markov chain and decomposition technique were done manually and as result took a lot of time in bringing the result out.

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