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2 Methodology

2.1 Specification of the Production Function

The production function is a function of capital and labour. While it can take various forms, for example the Leontiff form, the trans-log form and the CobbDouglas form, the Cobb-Douglas form is the form used in macroeconomic policy framework and the growth literature. Our specification of the production function therefore follows the Cobb-Douglas production function as given in Eq. (1). Constant returns to scale and positive but declining marginal productivity is assumed here.

(1)

Where Y is output, K is the stock of capital, L is labour and A is a shift parameter measuring total factor productivity.

Taking the log of Eq. (1) it can then be differentiated with respect to time to yield Eq.

(2)

The parameters α and 1 – α are the output elasticities of capital and labour respectively and
are the growth rates of output, capital, labour and total factor productivity respectively while

are the contributions of capital, labout and total factor productivity to growth of output.

Hence, information on the elasticities of capital and labour and the growth rates of output, capital and labour can be used to obtain the growth of total factor productivity. In this regard, our task is to estimate the values of α and hence 1

-α in Eq. (1) from time series data on output, capital and labour. Once these are known, using the growth rates of capital and labour for historical series, the contributions of capital and labour to growth can be obtained. With these contributions and the growth of output also computed, Eq. (2) can be used to obtain the growth of TFP (its contribution to growth) by the use of Eq. (3), which is obtained from Eq. (2).

(3)

2.2 How the Output Elasticities Are Estimated

In order to estimate the output elasticities, we express Eq. (1) in terms of output per worker (for which labour is used as a proxy). Thus Eq. (1) in terms of output per worker and capital per worker is given as in Eq.

(4).

Taking logarithm on both sides of Eq. (4) therefore gives

(5):

Thus, with data on output per worker and capital per worker, the parameter α (output elasticity of capital) and hence 1- α (output elasticity of labour) can be obtained.

2.3 Data Consideration

Data is obtained on real GDP, Labour and Gross Fixed Capital Formation for all the ECOWAS countries over the period 1980–2012 except for Liberia, which is left out due to data availability, especially on Gross fixed Capital formation (investment) over the estimation period. The data is obtained from World Bank's World Development Indicators (WDI).

To the extent that the available data is on Gross Capital formation and not capital, this data is used to generate the times series for capital stock using the perpetual inventory method.

The stock of capital is obtained for the period 1980–2012 for each country by assuming a depreciation rate (δ) of 5 % for capital and following Hall and Jones (1999) we apply Eq. (6) to obtain the initial capital stock (the capital stock for 1980initial capital stock-).

(6)

Where Ig is the growth of investment (gross fixed capital formation) from 1980 to 2012. Because investment growth is negative for some countries over some periods and the possibility of non-normality of the series for some countries, we use the median of the annual growth rates instead of the average of annual growth rates to represent the growth rate of investment over the period 1980–2012. The data for capital stock is in real form as the constant price gross capital formation was is used.

Hence, the following equation which gives the relationship between gross fixed capital formation (I) and capital stock is used to obtain the capital stock for the period 1981–2012 once the capital stock for 1980 (initial capital stock) is known.

(7)

From Eq. (7) capital stock is given as:

(8)

2.4 Estimation Technique for the Specified Model

The specified model given in Eq. (5) deals with time series data on 14 ECOWAS countries from 1980 to 2012. Hence the time dimension (T) is 33 and the number of countries (N) is 14. This is a panel data set with large T and small N. To this effect, the conventional spurious regression problems common in time series data emerges here if it is not checked for. To this end, we test for the existence of unit root in output per worker and capital per worker. That is, we apply panel unit root tests to each series. The conventional panel unit root tests are applied. That is, we apply both the homogenous panel unit root and the heterogeneous panel unit root tests. The homogeneous panel unit root tests are the Levin-Lin-Chu (LLC), Breitung and Hadri tests. The heterogenous panel tests are the Im-Pesaran and Shin (IPS), Maddala-Wu and Choi tests. The homogenous unit root tests assume that the unit rot process are the same for all the countries. That is, either the series for all the countries have unit root or they do not have while in the heterogenous case, the assumption is that some countries could have unit root in a series while the others do not have. However, it does not tell the countries that do not have unit root in case the hypothesis of the existence of unit rot is rejected.

Following the tests for unit root is the test for cointegration, as long as the variables are not stationary. This was explored in this paper. However, in panel data context when the variables are stationary, one should proceed to the estimation of the pool, fixed or random effect model while taking note of the need to test which of them is the most appropriate representation of the data. This method is applied in this paper.

Where there is cointegration a panel error correction is estimated. An alternative to the estimation of a panel error correction model is to estimate the dynamic Ordinary Least Squares (DOLS) or Fully Modified Ordinary Least Squared (FMOL) as they ensure having consistent estimators. The existence of no cointegration (long run relationship among the variables) implies that the variables must be differenced appropriately to obtain stationarity and the transformed variables should be used to estimate a fixed and a random effect model to account for country specific heterogeneity effects. Following which the Hausman test can be carried out to determine the more appropriate representation.

 
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