The essential assumption in these relationships is that remittances are motivated primarily by altruism and hence will most often exhibit countercyclical characteristics. In line with Chami et al (2008) as discussed in section 4.2, a positive productivity shock in the recipient economy will give rise to an increase in domestic output and help transfer some of the resulting benefits to the remitter by inducing him or her to reduce the amount remitted periodically and to increase his or her own consumption and vice versa.
The assumption of altruistically motivated remittances is adequately captured within a system of equations characterized by three endogenous variables in three equations namely: growth rate of output (YGR), workers‘ remittances (WR), and per capita income (PCI). The first equation is a neoclassical production function of the Cobb-Douglas form in which output (GDP) is specified as a function of labour (L), capital (K), workers‘ remittances, and a technological factor or efficiency parameter (A). Two basic assumptions of the neoclassical production function of the Cobb-Douglas form are:
1. Positive and diminishing returns to private inputs. For all K
> 0 and L > 0, the production function exhibits positive and diminishing marginal products with respect to each input such that:
2 2 2
2 2
, 0; 0; , 0
( )
dF dF d F d F d F
dK dL d LK and dK dL
Thus, the neoclassical technology assumes that, holding constant the levels of technology and labour, each additional unit of capital delivers positive additions to output, but these additions decrease as the number of machines rises. The same property is assumed for labour.
93 2. Inada conditions. The second defining characteristic of the neoclassical production function is that the marginal product of capital (or labour) approaches infinity as capital (or labour) goes to 0 and approaches 0 as capital (or labour) goes to infinity:
0 0
lim lim lim lim 0
K L K L
F F F F
K L and K L
This equation can be written explicitly as follows:
( , , , , ) (1 )
GDP f A L K WR PCI a
Where A is the technological factor of the efficiency factor within the system and relation (1a) can be re-written even in more explicit terms as:
(1 )
, (0 1)
GDP AL K WR PCI
Where α is the relative share of labour in total output and (1-α) is the relative share of capital in total output.
On a priori ground, the following are expected:
, , , 0
GDP GDP GDP PCI
L K WR WR
In turn, the second equation endogenizes PCI as a function of REER, INF, INV and the one period lag values of growth ( as follows:
( t1, , , ) (1 )
PCI f YGR REER INF INV b
The a priori expectations are:
1
, 0 , 0
t
PCI PCI PCI PCI
YGR INV and INF REER
94 The structural forms of Equations (1a – 1b) are rewritten in their linear forms as shown below.
The labour and capital input variables in equation (2a) are now in their log forms. Model 1 is intended to capture the role played by remittances in the economic growth of the remittances recipient economy as well as the distributional effect of previous period growth levels on the economy. By substituting equation (2b) into equation (2a), a single equation of the linear dynamic panel data model type is obtained in equation (3) as follows:
Substitute equation (2b) into equation (2a) to obtain the following:
(2c) Equation (2c) may be expanded to obtain the following:
(2d) Rearrange equation (2d) to obtain the following:
(2e) From equation (2e), the following dynamic panel data model may be obtained
(3)
95
(4)
is a vector of strictly exogenous covariates which include the following variables:
on the other hand, is a vector of endogenous and predetermined covariates which include the following variables:
are vectors of parameters to be estimated.
The assumption of altruistically motivated remittances is thus adequately captured within the resulting linear dynamic panel data model in equation (3).
is the usual error component decomposition of the error term;
are unobserved individual-specific effects;
are the observation-specific (idiosyncratic) errors;
are vectors of parameters to be estimated.
The individual-specific effects, are assumed to be uncorrelated across individuals, and with the disturbance of any individual at all leads and lags , but may be correlated with the explanatory variables . The mean of is zero and its variance may differ across individuals. The observation-specific disturbance has mean zero and is
uncorrelated across individuals and In general,
its variance may differ across both individuals and periods. The initial
96 observation is uncorrelated with the disturbance of any individual for all periods but may be correlated with the individual
effects . The autoregressive parameter
satisfies (dynamic stability). The vector xit may include lags of explanatory variables. It may also include covariates that are fixed over time for a given individual, and/or covariates that vary over time but are shared by all individuals.
All variables are defined as follows:
variables on the other hand are defined as follows:
In order to get a consistent estimate of δ as N →∞ with T fixed, equation (3) may be rewritten in first differenced notations. This also eliminates the individual effects as follows:
(5) The implication of transforming equation (3) into (5) is that the unobserved individual-level effects, has disappeared from the differenced equation (5)
97 because it does not vary over time. In this way, differencing has successfully dealt with the issue of country or individual specific effect also known as fixed effect. The Ds are the first difference operators.