CHAPTER FOUR
METHODOLOGY OF THE STUDY
This chapter is devoted to explaining the methodological approach to this thesis.
Section 3.1 deals with the model specification as was used in (Mundlak, Larson & Butzer 1997)with a bit of modification. Section 3.2 examines the approach used in applying the model to statistical problem. Data used in estimating the model, the description of variables used as well as their sources are discussed in section 3.3.
3.1 THE EMPRICAL MODEL
The point of departure in relation to this issue is that producers (that is countries in this case) are faced with a lot of options when taking a decision on the techniques needed in production. That is, the problem of how to produce. This fact put forward a choice of the combination of inputs and outputs.
Given: x = the vector of all inputs Fj = the production function of the jth technique T = a technology set comprising of all possible techniques That is, T= {Fj(x); j=1, …, j} In a case where countries choose the methods used in relation to their constraints, constrained (k) and unconstrained (v) inputs can be differentiated. Therefore, x = (k, v) with the assumption that they do not possess an alternative cost.

This leads to a Lagrangian optimization problem where the goal of the producer at this stage is to decide the method suitable with the right level of inputs in other to maximize profit. The Lagrangian equation becomes; L = ∑ pj Fj (vj, kj) - ∑ wvj – λ (∑ kj – ku ) -(1) Subject to Fj(.)€T;vj≥0;kj≥0 Where Pj = the price of the output w = the price vector of the unconstrained inputs k = the stock of constrained inputs The equation is solved using the Kuhn-Tucker necessary condition. Given s = (k, p, w, T) is the vector of the state variables of the problem and the optimal solution be vj*(s), kj*(s), λ*(s). The optimal allocation of inputs v(s) is used to show how much of the difference j have been put to use. Furthermore, in cases where optimal levels of these inputs are zero for a technique j, that technique cannot be into use. This is due to the fact that positive levels of these inputs are required. Therefore, optimal output techniques j is; Yj*= Fj (vj*, kj*) While the implemented technology (IT) would be defined by; IT(S) = {Fj(vj,kj); F(vj*, kj*)≠0,Fj€T} The use of empirical analysis can help provide estimates of production function that are in line with the implemented technology. Aggregate production function shows aggregate output that is being produced by a set of smaller production functions. These functions do not posses any unique feature because the set of micro function that shows the techniques implemented are endogenous. Let the aggregate production function be written as; ∑ pj yj *(s) ≣ F (x*, s) ≣ ø(s) (2) The aggregate production function above is condition on s, which is also condition on x* and on F (x*s). This circuitous conditionality hinders the possibility to make use of the changes in x as a base point. This means that in line with the analysis, it is impossible to get a stable production function from a mere sample of observations taken over points in time because they are subject to change in technology. In a bid to overcome this problem, empirical production function can be approximated so as to make it computable. Likewise, for the production function to become a conventional one, x should be segregated from s. this segregation can be made possible by differentiating x and x* such that the observed output becomes; ∑ pj yj ≣ F(x, s) (3) The change is in the use of F(x, s) which is not a function since x can be given to various techniques arbitrarily with an allocation rule leading to x*as the only condition for a unique solution. In order to achieve a solution F (x, s) is used as a function of s. A method in which F(x, s) could be expressed as a Cobb Douglas function was established by (Mundlak 1988) but with elasticity as a function of the state variables and the inputs. It is expressed as: ln y =⎾ (s) + B( s, x ) ln x + u (4)
Where y= the value added per worker B(s, x) = the slope ⎾(s) = the intercept u= the stochastic term of the function.
This estimation could be used in an empirical analysis. However, the data on factor shares, which would be needed, was not available. In a bid to transform the model to a more easily estimable model; further variation could be used in the function above. A change in the state variable would affect r(s) and B(s, x) directly as well as indirectly through their effect on inputs. The change in the formulation expressed by (Mundlak, Larson & Butzer 1997)was simply to alter the representation of the response of the implemented technology between state variables.
The model then becomes: Ei ≣ δT(s)/δsi + ln x [ δB (s)/δsi] (5)
With the presence of several techniques by available technology, a change in the state variable will led to a change in the mixture of techniques used in production and a change in the inputs used in any of the techniques.

3.2 THE STATISTICAL APPROACH OF THE MODEL
Overtime, state variables vary across countries. This is because technology is dependent on it. Countries possess unique factors of endowments, resource constraints and economic variables that do not change over time. For this reason results obtained from either a time series analysis or a simple cross-section always differ. Although there seem to be more cross country than time variation when examining variables, a more detailed and useful information might be obtained using a joint panel framework.
The vector matrices used to generate residuals are tagged as:
W, B(i), W(i), B(t), W(t) and W(it).
Given that: xi and xt = the average of xit over i and t respectively. That is;
Wx = (xit- x), W (i)x = (xit-xi), W (t)x =(xit -xt)
W (it)x = (xit –xi –xt +x ……)
B (i)x = (xi-x), B(t)x = (x1-x), I = 1,…, N, t = 1, …, T.
Therefor the following can be derived,
W = W (i) + B (i)
W = W (i) +W (t) – W (it)
W = B (i) +B (t) +W (it)
The different kinds of regressions that can be run include:
-Pooled (W, b); within-time and cross-country [W (it), w (it)]
-Within-country alone [W (i), W (i)]
-Within-time alone [W (t), w (t)]
-Between-time alone [B (t), b (t)]
-Cross-country alone [B (i), b (i)]
(Maddala 1971) In a bid to simplify the equations obtained previously showed that the pooled regression could also be written as a matrix-weighted average of all within and between regressions. Given that the data of W (it) that is, both country and time dummies is obtained from the between-time and between –country variations, it should be the regression with the most stable mash-up of implemented technology. The regression result denoted by W (i) shows cross-country variability and no time variability. Therefore, when W (it), B (i) and B (t) have been derived, W(I) and W(t) can be calculated by:
-W (i) = W (it) + B (t)
-W (t) = W (it) + B (i)
Practically, b (i), b (t) and w (it) can all be made use of when trying to find any differences in the between country and within time regression results. However this study will only make use of the cross-country time panel W (it) as it is known to give a stable result.
3.3 DATA
In this study data is sourced from two data banks. In view of the nature of the analysis that is, comparative analysis, it is very crucial for the integrity of the data source to be credible and consistent in order to avoid unbiased panel.
All data used in this research was obtained from the World Development Indicator 2014 (complied by ESDS) except for that of fertilizer which was gotten from the Food Agriculture and Organization (FAO) database.
The study made use of developing countries making use of panel data with a time period of 32 years that is from 1980-2012. 99 countries were included in the final analysis.
The dependent variable of choice in this study is labour productivity (labourp). Ordinarily Total Factor Productivity should be the ideal measure for agricultural productivity however, due to the unavailability of data on the micro and macro level, partial measure of agriculture(that is land and labour productivity) are made used which is in line with the works of (Weibe 2003)
Labour productivity is the most commonly used partial measure of agricultural productivity. It is measured as the agricultural value added per worker at constant US dollars with 2000 as the base year. It was calculated in line with (Mundlak, Larson & Butzer 1997). INDEPENDENT VARIABLES * Irrigated Land (irrigated_ld)
The amount of irrigated land as a percentage of total agricultural land is used as one of the measure of used to determine the level of technology a country has . It can therefore be said that countries with more of their land irrigated should be technologically superior and experience increased agricultural productivity.it is however imperative to note that this variable may indicate environment rather technical because some countries have more drier and arid climate may have more incentive to irrigate than those with sufficient annual rainfall. * Machinery (machnry)
In this case, agricultural machineries is made used as a measure of capital because countries with adequate machineries are most likely to be more productive than countries who lack machineries. It is measured as the number of tractors available per 100sq km of arable land. * Fertilizer (fertil)
The use of fertilizer can be used as a proxy for all chemical inputs made use of during the production process. The fertilizer use is measured in kilogram per hectare of arable land and it is expected that the amount of chemical input will bring about a positive impact on agricultural productivity. * Education (eductn)
Another commonly used measure of technology is education. This is so because, as more individuals in a country get educated, it is expected that the economy of such country will improve in terms of innovation, technology and productivity. It is measured as the number of school enrolment and it comprises of both primary and secondary school. * Cereal Yield (cereal_yd)
This variable was added due to the fact that the quantity of cereal used in a country’s farm should indicate the quality of seed variety used in that country. This is to say that a country that engages in the use of modern engineered seeds should have a more productive yield than a country that makes use of local varieties thereby making provision for a measure of local technology adoption in agriculture. This is measured in the amount of hectograms per hectare. * Commercial Bank (comm_banks)
In a bid to measure the availability of credit, the number of commercial banks branches per 100,000 adults was obtained. Although it may have been more ideal to make use of micro finance banks there was unfortunately insufficient data and thus it could not be used. It is expected that countries with adequate access to credit facilities are more likely to be more productive than countries that do not have access to credit facilities.

* Inflation (infltn)
This variable is used as a proxy for variation in prices. This means that an increase in price level (inflation) and variation in prices in a country when compared with other countries can denote more price volatility in that country. That is, higher level of volatility can lead to two things. First, it can trigger productivity on the side of the farmers who ensures that they keep producing as much as they can every year in other to conquer the unforeseen problem. Second, framers may choose to reduce level of productivity and decide not to produce in an unstable market environment with uncertain prices. This variable is measured by the annual growth rate of the GDP implicit deflator on Consumer Price Index. Implicit deflator is known as the ratio of GDP in current local currency to GDP in constant local currency.
I therefore estimate the following specification: irrigated_ldit = αit + β1irrigated_ldit + β2machnryit + β3fertilit + β4eductnit + β5cereal_ydit + β6comm_banks it+ β7infltnit + β8dummy_SSAmecit+β9dummy_EAmechit (6)
With i denoting countries and t denoting time.X