#### Autoregressive Distributed Lag Model (ARDL)

The traditional approach to determining long run and short run relationships among variables has been to use the standard Johnson Cointegration and VECM framework, but this approach suffers from serious flaws as discussed by Pesaran et al. (2001). Johnson cointegration test will not give you a consistent way of deciding the cointegration rank. As independent variable is I(1) while dependent variables are I(0) and I(1) as explained previously so most appropriate technique is ARDL(Autoregressive Distributed Lag Model). As this type of dynamic model is part of the Koyck distribution class of models. It is used in models where adjustment does not occur immediately, but takes a number of time periods to fully adjust. W e can apply a specific restriction to a general ARDL model to determine if partial adjustment is taking place. This model has as its dependent variable a desired value or planned value. This desired value is then determined by the usual explanatory variables.

In order to obtain results, we utilize the ARDL approach to establish the existence of long-run and short-run relationships. ARDL is extremely useful because it allows us to describe the existence of an equilibrium/relationship in terms of long-run and short-run dynamics without losing long-run information.

Following Pesaran et al. (

2001), we assemble the vector autoregression (VAR) of order

*p,* denoted VAR (

*p*), for the following growth function:

${Z}_{t}=\mu +{\displaystyle \sum _{i=1}^{p}{\beta}_{i}}{z}_{t-i}+{\u03f5}_{t}$

(4)

where

*z*_{
t
} is the vector of both

*x*_{
t
} and

*y*_{
t
} , where

*y*_{
t
} is the dependent variable defined as labor force participation rate (LFPR),

*x*_{
t
} is the vector matrix which represents a set of explanatory variables and

*t* is a time or trend variable. According to Pesaran et al. (

2001),

*y*_{
t
} must be I(1) variable, but the regressor

*x*_{
t
} can be either I(0) or I(1). We further developed a vector error correction model (VECM) as follows:

$\Delta {z}_{t}=\mu +\mathit{\alpha t}+\lambda {z}_{t-1}+{\displaystyle \sum _{i=1}^{p-i}{\gamma}_{t}}\Delta {y}_{t-i}+{\displaystyle \sum _{i=1}^{p-1}{\gamma}_{t}}\Delta {x}_{t-i}+{\u03f5}_{t}$

(5)

where ∆ is the first-difference operator. The long-run multiplier matrix λ as:

$\lambda =\left[\begin{array}{l}{\lambda}_{\mathit{YY}}{\lambda}_{\mathit{YX}}\\ {\lambda}_{\mathit{XY}}{\lambda}_{\mathit{XX}}\end{array}\right]$

The diagonal elements of the matrix are unrestricted, so the selected series can be either I(0) or I(1). If λYY=0, then *Y* is I(1). In contrast, if λYY<0, then *Y* is I(0).

The VECM procedures described above are imperative in the testing of at most one cointegrating vector between dependent variable

*y*_{
t
} and a set of regressors

*x*_{
t
}. To derive model, we followed the postulations made by Pesaran et al. (

2001) in Case III, that is, unrestricted intercepts and no trends. After imposing the restrictions λ

_{YY}=0, μ≠0 and

*α=* 0, the hypothetical function can be stated as the following unrestricted error correction model (UECM):

$\begin{array}{l}\Delta {\left(\mathit{LFPR}\right)}_{t}={\beta}_{0}+{\beta}_{1}\left(\mathit{LFPR}\right){}_{t-1}+{\beta}_{2}\left(\mathit{AD}\right){}_{t-1}+{\beta}_{3}\left(\mathit{HEXP}\right){}_{t-1}\\ \phantom{\rule{5.5em}{0ex}}+{\beta}_{4}\left(\mathit{IMR}\right){}_{t-1}+{\beta}_{5}\left(\mathit{GCF}\right){}_{t-1}+{\beta}_{6}{\left(\mathit{LE}\right)}_{t-1}\\ \phantom{\rule{5.5em}{0ex}}+{\beta}_{7}\left(\mathit{PPBED}\right){}_{t-1}+{\beta}_{8}\left(\mathit{SSE}\right){}_{t-1}+{\beta}_{9}\left(\mathit{TOP}\right){}_{t-1}\\ \phantom{\rule{5.5em}{0ex}}++{\displaystyle \sum _{i=1}^{p}{\beta}_{10}}\Delta \left(\mathit{LFPR}\right){}_{\begin{array}{l}t-i\end{array}}+{\displaystyle \sum _{i=0}^{q}{\beta}_{11}}\Delta {\left(\mathit{AD}\right)}_{t-i}\\ \phantom{\rule{5.5em}{0ex}}+{\displaystyle \sum _{i=0}^{r}{\beta}_{12}}\Delta \left(\mathit{HEXP}\right){}_{t-i}+{\displaystyle \sum _{i=0}^{s}{\beta}_{13}}\Delta \left(\mathit{IMR}\right){}_{t-i}\\ \phantom{\rule{5.5em}{0ex}}+{\displaystyle \sum _{i=0}^{t}}{\mathit{\beta}}_{14}\Delta \left(\mathit{GCF}\right){}_{t-i}+{\displaystyle \sum _{i=0}^{u}}{\beta}_{15}\Delta \left(\mathit{LE}\right){}_{t-i}\\ \phantom{\rule{5.5em}{0ex}}+{\displaystyle \sum _{i=0}^{v}}{\mathit{\beta}}_{16}{\left(\mathit{PPBED}\right)}_{t-i}+{\displaystyle \sum _{i=0}^{w}}{\mathit{\beta}}_{17}\Delta \left(\mathit{SSE}\right){}_{t-i}\\ \phantom{\rule{5.5em}{0ex}}+{\displaystyle \sum _{i=0}^{x}}{\mathit{\beta}}_{18}\Delta \left(\mathit{TOP}\right){}_{t-i}+{u}_{t}.\end{array}$

(6)

Where ∆ is the first-difference operator and *u*_{
t
} is a white-noise disturbance term. Equation (6) also can be viewed as an ARDL of order (p, *q*, *r, s, t, u, v, w, x*). Equation (6) indicates that agriculture expenditure tends to be influenced and explained by its past values. The structural lags are established by using minimum Akaike’s information criteria (AIC). From the estimation of UECMs, the long-run elasticities are the coefficient of one lagged explanatory variable (multiplied by a negative sign) divided by the coefficient of one lagged dependent variable (Bardsen, 1989). For example, in Equation (6), the long-run inequality, investment and growth elasticities are (*β*_{2}/*β*_{1}), (*β*_{3}/*β*_{1}), *β*_{4}/*β*_{1} etc. The short-run effects are captured by the coefficients of the first-differenced variables in Equation (6).

After regression of Equation (6), the Wald test (*F*-statistic) was computed to differentiate the long-run relationship between the concerned variables. The Wald test can be carry out by imposing restrictions on the estimated long-run coefficients of economic growth, inequality, investment and public expenditure. The null and alternative hypotheses are as follows:

*H*_{0} : *β*_{1} = *β*_{2} = *β*_{3} = *β*_{4} = *β*_{5} + *β*_{6} + *β*_{7} = *β*_{8} = *β*_{9} = 0 (no long-run relationship)

Against the alternative hypothesis

*H*_{0} : *β*_{1} ≠ *β*_{2} ≠ *β*_{3} ≠ *β*_{4} ≠ *β*_{5} ≠ *β*_{6} ≠ *β*_{7} ≠ *β*_{8} ≠ *β*_{9} ≠ 0 (long-run relationship exists)

The computed *F*-statistic value will be evaluated with the critical values tabulated in Table CI (iii) of Pesaran et al. (2001). According to these authors, the lower bound critical values assumed that the explanatory variables *x*_{
t
} are integrated of order zero, or I(0), while the upper bound critical values assumed that *x*_{
t
} are integrated of order one, or I(1). Therefore, if the computed *F*-statistic is smaller than the lower bound value, then the null hypothesis is not rejected and we conclude that there is no long-run relationship between poverty and its determinants. Conversely, if the computed *F*-statistic is greater than the upper bound value, then agriculture expenditure and its determinants share a long-run level relationship. On the other hand, if the computed *F*-statistic falls between the lower and upper bound values, then the results are inconclusive.