Correlated discrete and continuous outcomes with endogeneity and lagged effects: past season yield impact on improved corn seed adoption

Abstract

Farmers in Sub-Saharan Africa have lower agricultural technology adoption rates compared to the rest of the world. It is believed that the past season yield affects a farmer's capacity to take on the riskier improved seed variety; but this effect has not been studied. We quantify the effect of past season yield on improved corn seed use in future seasons while addressing the impact of the seed variety on yield. We develop a maximum likelihood method that addresses the fact that farmers self-select into a technology resulting in its effect on yield being endogenous. The method is unique since it models both lagged and endogenous effects in correlated discrete and continuous outcomes simultaneously. Due to the prescence of the lagged effect in a three year dataset, we also propose a solution to the initial conditions problem and demonstrate with simulations its effectiveness. We used survey longitudinal data collected from Kenyan corn farmers for three years. Our results show that higher past season yield increased the likelihood of adoption in future seasons. The simulation and empirical studies indicate that ignoring the self selection of improved seed use biases the results; we obtain a different sign in the covariance.

1. Introduction

The success of the Asian Green revolution that enabled the continent to feed its hungry can provide a blue print for reducing hunger in Africa. The key driver of this revolution was the adoption of improved grain varieties of rice and wheat, where adoption rates were and have remained quite high [22]. However in Africa, the adoption rates are significantly lower and vary from region to region. Corn is an important staple food in East, Central and Southern Africa, accounting for much of the daily caloric requirement [2]. Increasing its productivity will reduce the hunger levels in Africa. We therefore study adoption of the seed using survey data from Kenya in particular the impact of lagged crop yield on improved seed adoption decision. In this paper, we define improved seed as corn seed variety that is purchased from sellers or open pollinated varieties, while local are those indigenous to the country Kenya and have been in place before the first variety was ever introduced in 1955 [9].

In studies of dynamics of adoption in East Africa [1,17,26], farmers adopted agricultural technologies in a cyclical pattern. This means a significant portion of farmers did not adopt improved agricultural technology continuously across all seasons, for example they adopted in season 1, failed to adopt in season 2 and adopted again in season 3 or adopted in the first two seasons only. One reason for this could be the risk aversion nature of farmers. Including the lagged yield effect was proposed by [8] as a measure of access to cash that influences farmers' capacity to take on riskier technology in the next season; there has been no study so far that has implemented this. This is because quantifying the lagged yield effect presents three estimation challenges. First, improved seed variety affects the expected yield and is endogenous [3,13,16,20,26]. Endogeneity is similar to selection in biological/medical field where farmers make the decision to adopt a technology non-randomly (‘self-selection’). Secondly, there is correlation between repeated observations and finally the lagged effect is not independent of the error term. In this paper, we propose a simultaneous equation system to deal with endogeneity and estimate parameters using the full maximum likelihood method in place of the common two stage method. This is because the two stage method is less efficient than maximum likelihood method [24] and cannot deal with the issue of endogeneity in this setting.

Our first contribution is to extend the method for lagged and cross-lagged effects in mixed discrete and continuous outcomes to accommodate endogeneity of the discrete outcome. Secondly, we also extend wooldridge's [28] initial conditions model for limited dependent variables to a two equation system.

Improved seed use is a binary variable while yield is continous. Statistical methods dealing with mixed discrete and continous outcomes were first introduced by Olkin and Tate [21]. These methods have evolved into two strands: one factorizes the joint distribution into a product of conditional and marginal distribution [4,6,7,14,15,21,27] while the second uses a shared random variable to capture the correlation between the two outcomes [11,18,23]. We will use the factorization method since the latent model has been shown to be equivalent to the factorization method [27]. A latent variable method for lagged and cross lagged discrete and continous data was proposed by [11] but does not address the endogeneity problem. In addition, their method is only suitable for positively correlated variables and cannot distinguish between time dependent and fixed correlations. In econometrics, models for discrete endogenous effects on a continous variable are common [5,12]. We adopt Heckman's hybrid model specification [12] to deal with endogeneity. To deal with both endogeneity and lagged effects issues, we will combine the ideas of endogeneity from [12] and lagged effects from [11] to solve the full likelihood.

We present and describe the model in the method section and demonstrate its suitability with simulation studies. In the data subsection, we present the sources of empirical data and some descriptive statistics. Finally in the results section, we present the results using the survey data from Kenya. We then conclude with a discussion section.

2. Method

Figure 1 shows the dynamic adoption and yield relationship. A farmer self selects into adoption of improved seed at time t−1, this affects the yield. The yield at time t−1 in turn influences the adoption decision at time t. The process continues over time.

Figure 1.

The dynamic adoption process. A farmer adopts improved seed or not at time t−1, this affects the yield ( ${\alpha }_2$). The yield at time t−1 in turn influences the adoption decision at time t ( ${\beta }_2$).

We define the two-equation system as:

$$\begin{array}{rl}& y_{1it}={\mu }_{1it}+{\alpha }_2{y}_{2it}^{\ast }+u_{1it}+v_i\end{array}$$ (1)

$$\begin{array}{rl}& y_{2it}^{\ast }={\mu }_{2it}+{\beta }_2{y}_{1it-1}+u_{2it}+w_i\end{array}$$ (2)

$$\begin{array}{rl}& \left(\begin{array}{c}{u}_{1it}\\ u_{2it}\end{array}\right)\sim \mathrm{N}\left[\begin{array}{c}\left(\begin{array}{c}0\\ 0\end{array}\right),\left(\begin{array}{cc}{\sigma }_t^2& {\rho }_t{\sigma }_t\\ {\rho }_t{\sigma }_t& 1\end{array}\right)\end{array}\right]\\ & \left(\begin{array}{c}{v}_i\\ w_i\end{array}\right)\sim \mathrm{N}\left[\begin{array}{c}\left(\begin{array}{c}0\\ 0\end{array}\right),\left(\begin{array}{cc}{\sigma }_v^2& {\sigma }_{vw}\\ {\sigma }_{vw}& {\sigma }_w^2\end{array}\right)\end{array}\right]\\ & y_{2it}=\{\begin{array}{ll}1& y_{2it}^{\ast }>0\\ 0& y_{2it}^{\ast }\le 0\end{array}\\ & {\rho }_t={\rho }_0+c_1{\mathrm{D}}_{t_2}+c_2{\mathrm{D}}_{t_3}\\ & {\sigma }_t={\sigma }_0+d_1{\mathrm{D}}_{t_2}+d_2{\mathrm{D}}_{t_3}\\ & {\mathrm{D}}_{t_k}=\{\begin{array}{l}1\mathrm{for}\mathrm{time}{t}_k\\ 0\mathrm{otherwise}\end{array},\mathrm{k}=2,3.\end{array}$$

where ${\mu }_{1it}={\alpha }_0+\alpha {\alpha }_{1}x{x}_{1it}$, ${\mu }_{2it}={\beta }_0+\beta {\beta }_{1}x{x}_{2it}$, $y_{1it}$ is the log of corn yield in kg/acre, $y_{2it}$ is a binary variable indicating improved seed use, i ( $i=1,2,\dots ,N$) represents a household while N is the total number of households and t ( $t=1,2,\dots ,T$) represents the year. $x{x}_{1it}$ and $x{x}_{2it}$ are vectors of control variables and $\theta \theta =({\alpha }_0,\alpha {\alpha }_1,{\alpha }_2,{\beta }_0,\beta {\beta }_1,{\beta }_2,{\sigma }_0,{\rho }_0,c_1,c_2,d_1,d_2,{\sigma }_v^2,{\sigma }_{vw},{\sigma }_w^2)$ are parameters to be estimated. In this paper we have N = 1171 and T = 3. The control variables $x{x}_{1it}$ and $x{x}_{2it}$ are described in detail in Table 1. Two versions of Equations (1) and (2) are estimated:

1. Model with no endogenous effect ( ${\alpha }_2=0$)

2. Model with endogenous effect ( ${\alpha }_2\ne 0$.).

Table 1. Description of variables in each equation.

Variable	Description
Dependent variables
Yield ( $y_1$)	log of yield measured in kg/acres
Improved seed use ( $y_2$)	Binary variable indicating whether a farmer used improved seed variety or not
Variables in yield equation
# of adult females	The number of adult females> 15 years living in the household for at least 6 months
# of adult males	Number of adult males> 15 years
Gender	Gender of the head of household
Min Temperature	Minimum growing season temperature in ${}^{\circ }$C
Age	Age of household head in years
Sale price	Sale price of corn harvest in Kshs/kg
Fertilizer price	Cost of chemical fertilizer in Kshs/kg
Farm size	Total land area allocated to corn in acres
Variables in improved seed equation
Distance to seed seller	Distance to the nearest improved seed seller in km
Tenure	Legal form of land ownership: Owned, rented or other
Education	Highest level of education of head: None; Primary (<8yrs in school),
	Secondary (9–12), College (>12)?
Death in < 3years	Death of a household member within the past three years
Marital status	Is the household head married or not?
Variables in both equations
Wage rate	Daily wage rate for general farm labor
Asset Value	The log of total value of household and agricultural assets in '000's of Kshs.
Land preparation cost	Cost of hired land preparation in Kshs/acre
Temperature	Harvest season temperature in ${}^{\circ }$C
Rainfall	Main growing season rainfall in mm
Seed price	Cost of improved seed variety in Kshs/kg
Group member	Does any member belong to a group? producer cooperatives,
	informal selhelp groups, saving and cooperatives etc
Credit	Did a farmer have access to credit or not?

Heckman [12] estimates a model with no lagged effect while Teixeira et al. [11] estimates a model with no endogeneity.

2.1. Estimation procedure described

The proposed model can be described as a general case of the lagged and cross-lagged effect model.

Considering a pair of outcomes ( $y{y}_{1i},y{y}_{2i}$), for a household i across all time periods as a cluster, the cross-lagged model involves specifying the distribution of a cluster as a product of conditionals as shown below as is standard in dynamic models:

$$f(y{y}_{1i},y{y}_{2i})=f(y_{1i1},y_{2i1})\prod _{t=2}^{T}f(y_{1it},y_{2it}|y_{1it-1},y_{2it-1}).$$

where $y{y}_{1i}^{\prime }=(y_{1i1},y_{1i2},\dots ,y_{1iT})$ and $y{y}_{2i}^{\prime }=(y_{2i1},y_{2i2},\dots ,y_{2iT})$. In contrast to the model by [11] that uses a common (random) effect to capture all correlations, we use the factorization method that retains the bivariate nature of our random variables ( $u_{1it},u_{2it}$) and ( $v_i,w_i$) allows us to distingush between the various forms of correlations in our equation system. In addition, the model also includes the endogenous effect of the binary variable on the continous outcome.

Substituting equation (2) into (1), the final equation system becomes the following:

$$\begin{array}{rl}& y_{1it}|(y_{1it-1},v_i,w_i)={\mu }_{1it}+{\alpha }_2({\mu }_{2it}+{\beta }_2{y}_{1it-1}+w_i+u_{2it})+v_i+u_{1it}\\ & y_{2it}^{\ast }|(y_{1it-1},v_i,w_i)={\mu }_{2it}+{\beta }_2{y}_{1it-1}+w_i+u_{2it}.\end{array}$$

This model is just the lagged model with a modified error structure as follows:

$$\left(\begin{array}{c}{u}_{1it}+{\alpha }_2{u}_{2it}\\ u_{2it}\end{array}\right)\sim N\left[\begin{array}{c}\left(\begin{array}{c}0\\ 0\end{array}\right),\left(\begin{array}{cc}{\sigma }_t^2+{\alpha }_2^2+2{\alpha }_2{\rho }_t{\sigma }_t& {\rho }_t{\sigma }_t+{\alpha }_2\\ {\rho }_t{\sigma }_t+{\alpha }_2& 1\end{array}\right)\end{array}\right].$$

When ${\alpha }_2=0$ this model simplifies to the cross-lagged model proposed by [11]

2.2. Specification of the joint distribution

We split the joint distribution into a product of the marginal distribution for the continuous variable and conditional distribution of the binary given the continuous variable. The joint distribution becomes:

$$\begin{array}{rl}& f(y_{1it},y_{2it}|y_{1it-1},v_i,w_i)=f(y_{1it}|y_{1it-1},v_i,w_i).P(y_{2it}=k|y_{1it},y_{1it-1},v_i,w_i)\\ & k=0,1\\ & \mathrm{where}\\ & f(y_{1it}|(y_{1it-1},v_i,w_i)=\frac{1}{\sqrt{2\pi {\sigma }_t^{2\ast }}}{\mathrm{e}}^{-(1/2{\sigma }_t^{2\ast })(y_{1it}-{\mu }_{1it}-{\alpha }_2{\mu }_{2it}-{\alpha }_2{\beta }_2{y}_{1it-1}-v_i-{\alpha }_2{w}_i{)}^2}.\\ & P(y_{2it}=1|y_{1it},y_{1it-1},v_i,w_i)=P(y_{2it}^{\ast }>0|y_{1it},y_{1it-1},v_i,w_i)\\ & =\mathrm{\Phi }\left(\frac{{\mu }_{2it}+w_i+\frac{{\rho }_t^{\ast }}{{\sigma }_t^{\ast }}(y_{1it}-{\mu }_{1it}-{\alpha }_2{\mu }_{2it}-{\alpha }_2{\beta }_2{y}_{1it-1}-v_i-{\alpha }_2{w}_i)}{\sqrt{1-{\rho }_t^{\ast 2}}}\right).\\ & P(y_{2it}=0|y_{1it},v_i,w_i)=1-P(y_{2it}=1|y_{1it},v_i,w_i).\\ & \mathrm{where}{\sigma }_t^2\ast ={\sigma }_t^2+{\alpha }_2^2+2{\alpha }_2{\rho }_t,\mathrm{and}{\rho }_t^{\ast }=\frac{{\rho }_t{\sigma }_t+{\alpha }_2}{{\sigma }_t^{\ast }}.\end{array}$$

The joint distribution conditional on ( $v_i,w_i$) is specified as:

$$f(y{y}_{1i},y{y}_{2i}|v_i,w_i)=f(y_{1i1},y_{2i1})\prod _{t=2}^{T}f(y_{1it},y_{2it}|y_{1it-1},y_{2it-1}).$$ (3)

We integrate out the random effects ( $v_i,w_i$) to get the unconditional distribution as follows:

$$f(y{y}_{1i},y{y}_{2i})={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(y{y}_{1i},y{y}_{2i}|v_i,w_i)f(v_i,w_i)\mathrm{d}{v}_i\mathrm{d}{w}_i.$$

Assuming that observations are independent across households, the likelihood is given by:

$$L(\theta \theta |y{y}_1,y{y}_2)=\prod _{i=1}^{N}f(y{y}_{1i},y{y}_{2i}).$$

We maximize the log likelihood above. Since, a closed form of the likelihood above does not exist, we approximate it using adaptive gaussian quadrature.

To ensure that the the variance-covariance matrix of the random effects (v, w) is positive definite, we employ the log-choleski decomposition of the matrix.

$$\left(\begin{array}{cc}{\sigma }_v^2& {\sigma }_{vw}\\ {\sigma }_{vw}& {\sigma }_w^2\end{array}\right)=\left(\begin{array}{cc}{\mathrm{e}}^{2s_{11}}+s_{12}^2& s_{12}{\mathrm{e}}^{s_{22}}\\ s_{12}{\mathrm{e}}^{s_{22}}& {\mathrm{e}}^{2s_{22}}\end{array}\right).$$

The parameters estimated are therefore $s_{11},s_{12}$ and $s_{22}$.

2.3. Initial conditions problem

This arises in short dynamic panels as in our case. It is the difficulty in specifying the observed first period distribution when data is not observed from the beginning in the prescence of lagged effects. We have four years of outcomes and three complete panels of data that was observed from 2004, yet the improved seed was introduced in the 1950's. Wooldridge [28] proposed a method that does not involve specification of the first period distribution but instead specifying the random effect (unobservable heterogeneity) distribution conditional on the first period outcome in limited dependent variables for a one equation system. Our method involves a two equation system with cross-lagged effects. We therefore adopt Wooldrige's [28] proposed solution to suit the cross-lagged case as show below.

$$\left(\begin{array}{c}{v}_i\\ w_i\end{array}\right)|(y_{1i1},y_{2i1})\sim N\left[\begin{array}{c}\left(\begin{array}{c}{\tau }_1^{\prime }{\overline{x}}_{1i}+{\tau }_2{y}_{1i1}\\ {\delta }_1^{\prime }{\overline{x}}_{2i}+{\delta }_2{y}_{1i1}\end{array}\right),{\mathrm{\Sigma }}_{uv}\end{array}\right].$$ (4)

The joint distribution of the outcomes in Equation (4) is redefined as follows where the 1st period distribution is no longer explicitly defined:

$$f(y{y}_{1i},y{y}_{2i},v_i,w_i|y_{1i1},y_{2i1})=f(v_i,w_i|y_{1i1},y_{2i1})\prod _{t=2}^{T}f(y_{1it},y_{2it}|y_{1it-1},y_{2it-1},v_i,w_i).$$ (5)

We conduct various simulations with different formulations of the Equation (5) as described in Section 3 next.

3. Simulation study

We generate data for 20 time periods and use observations for times 18, 19 and 20 in the estimation conditional on time 17( $t_1$) output values. We used 1000 observations in each time period and repeated the simulation 500 times. The data is generated as follows:

$$\begin{array}{rl}& y_{1it}={\alpha }_0+{\alpha }_1{x}_{1it}+{\alpha }_2{y}_{2it}^{\ast }+u_{1it}+v_i\\ & y_{2it}^{\ast }={\beta }_0+{\beta }_1{x}_{2it}+{\beta }_2{y}_{1it-1}+u_{2it}+w_i\end{array}$$

where

$$\begin{array}{rl}& y_{2it}=\{\begin{array}{ll}1& y_{2it}^{\ast }>0\\ 0& \mathrm{otherwise}\end{array}\\ & \left(\begin{array}{c}{u}_{1it}\\ u_{2it}\end{array}\right)\sim N\left(\begin{array}{c}\left(\begin{array}{c}0\\ 0\end{array}\right),\left(\begin{array}{cc}1& \rho \\ \rho & 1\end{array}\right)\end{array}\right),\left(\begin{array}{c}{v}_i\\ w_i\end{array}\right)\sim N\left(\begin{array}{c}\left(\begin{array}{c}0\\ 0\end{array}\right),\left(\begin{array}{cc}1& {\sigma }_{vw}\\ {\sigma }_{vw}& 1\end{array}\right)\end{array}\right)\\ & x_{1it}\sim N(0.5,1),x_{2it}\sim N(0,1).\end{array}$$

For t = 1 we use the long run lagged effect as:

$$\begin{array}{rl}& y_{1it-1}=({\mu }_1+{\alpha }_2{\mu }_2)/(1-{\alpha }_2{\beta }_2)\\ & \mathrm{where}{\mu }_1={\alpha }_0+{\alpha }_1{x}_{1it}\\ & \mathrm{and}{\mu }_2={\beta }_0+{\beta }_1{x}_{2it}.\end{array}$$

Our general proposed solution to the initial conditions problem simplifies to:

$$\begin{array}{rl}& y_{1it}={\alpha }_0+{\alpha }_1{x}_{1it}+{\alpha }_2{y}_{2it}^{\ast }+{\alpha }_3{y}_{1i0}+{\alpha }_4{\overline{x}}_{1i}+v_i+u_{1it}\\ & y_{2it}^{\ast }={\beta }_0+{\beta }_1{x}_{2it}+{\beta }_2{y}_{1it-1}+{\beta }_3{y}_{1i0}+{\beta }_4{\overline{x}}_{2i}+w_i+u_{2it}\\ & \left(\begin{array}{c}{v}_i\\ w_i\end{array}\right)\sim \mathrm{N}\left(\begin{array}{c}\left(\begin{array}{c}0\\ 0\end{array}\right),\left[\begin{array}{cc}{\sigma }_v^2& {\sigma }_{vw}\\ {\sigma }_{vw}& {\sigma }_w^2\end{array}\right]\end{array}\right).\end{array}$$

We estimate four variations:

• (s1)(s1) Naive estimate: ( ${\alpha }_3={\alpha }_4={\beta }_3={\beta }_4=0$).
• (s2)(s2) We dont consider the endogenous effect on initial conditions ( ${\alpha }_3={\alpha }_4=0$).
• (s3)(s3) No endogenous effect ( ${\alpha }_2=0$).
• (s4)(s4) We estimate the model as specified.

We varied parameters of the models. The exhaustive results of all simulations will require many pages but the results did not change significantly. Selected simulation results are provided in the Appendix section. The simulation results for the 4 models above are presented in Tables 2 and 3.

Table 2. Simulation results for $\rho =0.45,{\sigma }_{vw}=0.35$. SE and RMSE represent the standard error and root mean square of simulations.

		Model (s1)			Model (s2)			Model (s3)			Model (s4)
Parameter	True value	Mean	SE	RMSE	Mean	SE	RMSE	Mean	SE	RMSE	Mean	SE	RMSE
${\alpha }_0$	1	1.0697	0.0424	0.0816	1.0934	0.0399	0.1016	0.2032	0.0561	0.7988	0.6639	0.0503	0.3398
${\alpha }_1$	0.5	0.5069	0.0246	0.0255	0.5059	0.024	0.0247	0.4708	0.0253	0.0386	0.4998	0.0254	0.0255
${\alpha }_2$	0.5	0.8443	0.083	1.3469	0.7675	0.0428	0.2709				0.499	0.0295	0.0295
${\alpha }_3$	–							0.5605	0.0180		0.3066	0.0179
${\alpha }_4$	–							0.0844	0.0679		0.002	0.0598
${\beta }_0$	−0.75	−0.9039	0.0497	0.1617	−1.1418	0.0731	0.3986	-0.9546	0.0619	0.2138	−0.9953	0.0608	0.2527
${\beta }_1$	1	0.7657	0.0716	0.2450	0.8392	0.0567	0.1705	0.7152	0.0513	0.2894	1.0006	0.0536	0.0536
${\beta }_2$	0.4	0.5763	0.0304	0.9768	0.3615	0.0316	0.0498	0.2893	0.0261	0.1137	0.3998	0.0293	0.0293
${\beta }_3$	–				0.3802	0.0357		0.2983	0.0303		0.2229	0.0317
${\beta }_4$	–				−0.0139	0.0753		0.0072	0.0776		−0.0033	0.0807
${\sigma }_{vw}$	0.35	−0.3031	0.0993	0.6606	−0.3737	0.0756	0.7742	0.6857	0.0930	0.3483	0.0268	0.0559	0.3280
ρ	0.45	0.3256	0.0658	0.7784	0.1667	0.0725	0.0855	0.7347	0.0266	0.2859	0.4526	0.0492	0.0556
σ	1	0.9844	0.0289	0.0328	0.991	0.0303	0.0316	1.3179	0.0213	0.3186	0.9984	0.023	0.0231
${\sigma }_v^2$	1	0.6341	0.0815	0.1405	0.5811	0.0644	0.4238	1.2824	0.0853	0.295	0.5472	0.0516	0.4557
${\sigma }_w^2$	1	0.3475	0.1079	0.6614	1.031	0.1805	0.1831	0.9093	0.145	0.171	0.7693	0.1353	0.2674

Table 3. Simulations for 500 runs.

		Model (s1)			Model (s2)			Model (s3)			Model (s4)
Parameter	True value	Mean	SE	RMSE	Mean	SE	RMSE	Mean	SE	RMSE	Mean	SE	RMSE
${\alpha }_0$	1	0.9466	0.0461	0.0705	0.8873	0.0496	0.1231	1.3735	0.0493	0.3767	0.9528	0.0528	0.0708
${\alpha }_1$	−2	−2.0182	0.0232	0.0295	−2.0134	0.0229	0.0266	−1.8530	0.0233	0.1488	−2.0004	0.0260	0.0260
${\alpha }_2$	0.5	0.6034	0.0438	0.1123	0.6912	0.0525	0.1983				0.5005	0.0403	0.0403
${\alpha }_3$	–							0.2701	0.0136	0.2704	0.1491	0.0156	0.1499
${\alpha }_4$	–							−0.4369	0.0682	0.4422	0.0034	0.0661	0.0662
${\beta }_0$	0.5	0.4952	0.0460	0.0463	0.4773	0.0461	0.0514	0.4910	0.0460	0.0468	0.4897	0.0459	0.0470
${\beta }_1$	0.3	0.2773	0.0284	0.0363	0.2555	0.0298	0.0535	0.2906	0.0398	0.0409	0.2990	0.0325	0.0325
${\beta }_2$	0.4	0.4093	0.0236	0.0254	0.3465	0.0253	0.0591	0.3939	0.0239	0.0247	0.4011	0.0241	0.0242
${\beta }_3$	–				0.1303	0.0155	0.1313	0.0323	0.0199	0.0379	0.0283	0.0198	0.0346
${\beta }_4$	0				0.0017	0.0639	0.0639	−0.0082	0.0867	0.0871	−0.0023	0.0724	0.0725
${\sigma }_{vw}$	−0.35	−0.4723	0.0805	0.1464	−0.5866	0.0890	0.2528	0.1476	0.0562	0.5007	−0.3780	0.0729	0.0781
ρ	−0.45	−0.4396	0.0419	0.0432	−0.5155	0.0427	0.0782	0.0449	0.0410	0.4966	−0.3803	0.0441	0.0825
σ	1	1.2458	0.0361	0.2485	1.3032	0.0432	0.3063	1.1001	0.0175	0.1016	1.1855	0.0304	0.1880
${\sigma }_v^2$	1	1.0000	0.1337	0.1337	1.0558	0.1415	0.1521	0.9991	0.1336	0.1336	0.9969	0.1334	0.1334
${\sigma }_w^2$	1	0.9650	0.0778	0.0853	0.9668	0.0852	0.0914	1.0872	0.0650	0.1088	0.8410	0.0666	0.1724

Our main focus is on the lagged effect ${\beta }_2$ ( $\mathrm{actual}\mathrm{value}=0.4$) and endogenous effect ${\alpha }_2$ ( $\mathrm{actual}\mathrm{value}=0.5$). The simulation results show that to accurately estimate the lagged effect, both the initial conditions problem and endogenous effect need to be accounted for. From the results in Table 2, we found that model s4 estimates the lagged effect ${\beta }_2$ most accurately yielding a value of 0.3998 as well as the endogenous effect. The results from models s1 and s2 demonstrate that failure to accurately account for the initial conditions problem results in biased estimates of both the endogenous effect and lagged effect. In both cases, because of positive correlation between the two equations, the endogenous effect ${\alpha }_2$ is overestimated at 0.84 for model s1 and 0.77 for model s2 instead of the actual value of 0.5. The lagged effect is overestimated in model s1 at 0.58 while slightly underestimated in model 2 at 0.36. Model s3, accounts for the initial conditions problem comprehensively but does not include the endogenous effect. Here, the lagged effect is worse than in model s2 at 0.29. Model s4 which adequately accounts for both the initial conditions problem and endogeneity accurately estimates both the lagged effect at 0.3998 while the endogenous effect is 0.499. ${\alpha }_1$ is generally estimated with low bias across the four models with model s3 having the highest bias at 0.0292. ${\beta }_1$ and ρ have high bias across models s1, s2 and s3 but model s4 accurately estimates it.

All four models however, did not estimate the covariance of the random effect ${\sigma }_{vw}$ accurately. Interestingly, model s3 seems to have the least bias in estimating the random effects' variance of the continuous variable ${\sigma }_v^2$. Unfortunately, all the specifications would not yield accurate covariance ${\sigma }_{vw}$, but model s4 had the correct sign of covariance and least mean square error. Moreover, when the covariance ${\sigma }_{vw}$ is less than zero, model s4 estimated it with the least bias as shown in Table 3.

We therefore recommend model s4 to account for endogeneity, cross-lagged effect and the initial conditions problem in a system of correlated binary and continuous dependent variable model.

4. Empirical study

4.1. Data

There were two sources of data used: survey data and weather data.

4.1.1. Survey data

The data used in this study is from a National Househod Survey for the years 2004, 2007 and 2010. The data was collected by the Tegemeo Institute in Kenya together with Michigan State University in collaboration with USAID (United States Agency for International Development). It is available at: (http://www.tegemeo.org/index.php/resources/data.html). The survey design is available at: (http://www.tegemeo.org/index.php/resources/data/528-survey-instruments-and-data-documentation.html) upon request from Tegemeo Institute. The information collected included agricultural activities such as crop production at field level, household assets, non-farm income, consumption and demographic details such as education. It covered 1397 households in 2004 (of which 1374 were growing corn), 1342 in 2007 (of which 1308 were growing corn) and 1309 in 2010 (of which 1285 were growing corn). This yielded a balanced panel of 1248 households before removing missing values which were assumed to be missing at random. In 2004 we had 12 missing values of land preparation cost and 21 missing values of distance to nearest fertilizer market. In 2007 we had 16 households with missing land preparation cost, 2 of improved seed users were missing the seed price and 14 with missing education status. In 2010 we had 14 missing values of distance to seller, 1 missing credit information, 6 with missing land preparation cost and 5 with missing distance to fertilizer market. Once these were eliminated, we had 1341 households in 2004, 1276 households in 2007 and 1259 in 2010 yielding a balanced panel of 1171 households that was used in the analysis ([19] used 1167 households in their study which used the same dataset). Figure 2 shows the location of sampled locations.

Figure 2.

Map of Kenya with sample study sub-locations. A sub-location is the 2nd smallest administrative unit after the village.

A household was defined as adopting improved seed variety if it planted improved seed in at least one field. Land tenure refers to the legal form of land ownership. We selected the land tenure status of the largest corn field. There were initially five categories which were regrouped into three categories; rented and owned with title deed remained the same while ‘owned by parent’, ‘communal/government ownership’ and ‘owned without title’ were classified as ‘Other’. All households that did not sell their corn had missing selling prices. We replaced these values by the minimum district price since we assume that had they sold the would have sold at the minimum price. Fertilizer and improved seed prices were only available for those who used them, for those that did not use fertilizer or adopt improved seed variety we used the median district price.

4.1.2. Weather data

The Weather data was obtained from the Climate Research Unit(CRU) Datasets [10]. The website from which these datasets was downloaded is: (https://crudata.uea.ac.uk/cru/data/hrg/cru_ts_4.01). It comprises of monthly weather data across the global land mass collected at a longitude-latitude resolution of 0.5 degrees by 0.5 degrees. The data comes from the nearest weather stations and is interpolated in some cases. Since the weather data is available at 0.5 increments such as (−0.25, 12.25) and (0.25, 12.75), we map the actual latitude and longitude of our sub-locations to the nearest CRU latitude and longitude. A sub-location is the next smallest unit after the village from which households were sampled. We selected the following weather information: main growing season rainfall in mm and harvest period average temperature in ${}^{\circ }$C for Kenya for the years 2003/04; 2006/07 and 2009/10. The main growing and harvests seasons differ from region to region as shown in Table 4.

Table 4. Description of the main harvest and planting period.

Region	Harvest Period	Planting period
Eastern Kenya	Jan-March: Next year	Aug 01-Sept 31
Western	July/October	Mar 15- April 15
Coast	July/October	Mar 15 -April 30
Central	July/October	Mar 1-Mar 31
Rift Valley	Nov/Dec	Mar 1- April 15

4.1.3. Descriptive statistics of the data

Tables 5 and 6 provides descriptive statistics of key numeric and categorical variables used respectively. There was an increase in the percentage of improved seed adopters across the three years from: 63.36% in 2004, 69.60% in 2007 to 80.96% in 2010. Farmers who adopted improved seed variety had higher average yields than those who did not in all the years. The yield in 2010 was much lower compared to 2007 and 2004. This may be due to failed rainfall as can be seen from the much lower average rainfall and higher average temperatures in 2010 compared to other years.

Table 5. Numeric variable descriptive statistics by adoption.

	$\mathrm{Year}=2004$				$\mathrm{Year}=2007$				$\mathrm{Year}=2010$
	Improved seed use				Improved seed use				Improved seed use
	No (N = 429)		Yes (N = 742)		No (N = 356)		Yes (N = 815)		No (N = 223)		Yes (N = 948)
Variable	Mean	Std Dev	Mean	Std Dev	Mean	Std Dev	Mean	Std Dev	Mean	Std Dev	Mean	Std Dev
Yield	5.396	1.446	6.426	1.007	5.752	1.025	6.594	0.788	5.418	1.379	5.896	1.745
Distance to seed seller	4.346	6.556	2.739	4.085	3.275	3.541	3.030	3.238	4.478	6.349	4.089	4.600
Rainfall	156.241	61.638	154.064	52.544	161.903	40.902	161.378	66.319	122.149	31.225	120.223	45.224
Temperature	21.518	2.766	19.890	2.884	21.342	2.739	19.887	2.653	22.210	2.414	20.463	2.819
Wagerate	81.352	38.191	82.749	30.993	85.337	32.917	88.209	28.805	118.946	42.974	121.598	39.095
Assetval	3.696	1.332	4.435	1.324	4.413	1.197	4.657	1.236	4.154	1.281	4.901	1.231
Land preparation cost	0.550	0.826	0.904	1.031	0.600	0.810	0.961	1.053	0.773	1.114	1.139	1.130
# adult females	2.138	1.305	2.212	1.197	2.022	1.242	2.187	1.251	2.022	1.406	2.127	1.209
# adult males	2.075	1.394	2.263	1.395	1.992	1.391	2.290	1.426	2.031	1.567	2.217	1.400
Farm size	1.723	1.953	2.017	2.915	1.620	1.953	1.917	3.230	1.368	2.149	1.650	2.226
Min Temperature	15.462	3.504	13.143	3.190	15.464	3.330	13.482	3.053	16.092	3.383	13.676	3.280
Age	57.746	13.539	55.887	13.184	58.941	13.318	58.725	13.400	62.457	13.138	60.151	13.203
Sale price	5.219	4.788	8.312	5.728	10.856	2.383	10.835	2.571	15.989	5.926	16.216	5.177
Fertilizer price	29.198	4.129	26.858	3.190	37.402	4.634	34.877	3.491	51.345	9.419	52.213	9.469
Seed price	116.947	68.183	132.366	20.869	144.535	62.411	129.191	22.592	168.235	80.886	136.614	32.438

Table 6. Categorical variable descriptive statistics by adoption.

	$\mathrm{Year}=2004$		$\mathrm{Year}=2007$		$\mathrm{Year}=2010$
	Improved seed use		Improved seed use		Improved seed use
Variable	No (N = 429)	Yes (N = 742)	No (N = 356)	Yes (N = 815)	No (N = 223)	Yes (N = 948)
	Count	Count	Count	Count	Count	Count
Tenure: Owned	193	416	146	425	105	510
Rented	28	60	29	117	9	91
Other	208	266	181	273	109	347
Education: None	112	120	83	137	71	156
Primary	238	388	202	432	120	505
Secondary	66	183	54	196	24	218
College	13	51	17	50	8	69
Group member: Yes	311	576	280	610	137	695
No	118	166	76	205	86	253
Death in <3 yrs: Yes	69	86	53	101	43	95
No	360	656	303	714	180	853
Credit: Yes	96	288	169	435	105	533
No	333	454	187	380	118	415
Marital status: Married	304	608	241	627	134	690
Other	125	134	115	188	89	258
Gender: Male	311	620	250	650	136	719
Female	118	122	106	165	87	229

Across the three years, the distance to seed seller, main season average rainfall and harvest period temperatures were lower for adopters compared to non adopters. However, the average distance to seed seller increased among both adopters and non adopters from 2007 to 2010. This may be due to the fact that the new adopters in 2010 had larger distances. Younger farmers were more likely to adopt improved corn seed variety than older ones. Households that adopted improved seed varieties across the three years had higher asset values, wagerates, land preparation cost, household size and farm sizes. We also observed that the farm sizes generally decreased over time for both adopters and non adopters an indication of increased population.

Those with higher education were also more likely to be adopters than non adopters. Households with married heads, male heads and access to credit were more likely to adopt compared to unmarried and female headed households across the three years. In 2004 and 2010, those households that belonged to a group(cooperative) were more likely to adopt improved seeds while in 2007 the opposite was true. Households that experienced a death within the last three years were less likely to adopt.

4.2. Results

Maximum likelihood estimates of model 1 (no endogenous effect) and model 2 (endogenous effect) are presented in Table 7. Note that model 1 is a special case of model 2 with one less parameter.

Table 7. ML estimates yield and improved seed adoption.

	Model 1			Model 2
Parameter	Estimate	SE	Pr( $T>|t|$)	Estimate	SE	Pr( $T>|t|$)
Improved corn seed equation
Intercept	1.0183	0.5000	0.0419	0.8173	0.4696	0.0820
logyield ${}_{t-1}$	0.0332	0.0321	0.3007	0.0736	0.0234	0.0017
logyield ${}_{00}$	0.3484	0.0355	<.0001	0.3198	0.0330	<.0001
Temperature	0.5467	0.0928	<.0001	0.5370	0.0919	<.0001
Distance to seller	−0.0094	0.0844	0.9117	−0.1690	0.0604	0.0053
Wage rate	0.5202	0.1240	<.0001	0.5163	0.1223	<.0001
Asset value	0.1625	0.0487	0.0009	0.1587	0.0483	0.0011
Seed price	−0.6441	0.1231	<.0001	−0.6161	0.1223	<.0001
Tenure: Owned	0.0773	0.0801	0.3348	0.1107	0.0512	0.0308
Tenure: Rented	0.1224	0.1322	0.3547	0.2027	0.0842	0.0163
Group member	−0.1279	0.0859	0.1370	−0.1292	0.0849	0.1284
Education: >Primary	0.2728	0.1361	0.0453	0.2677	0.0928	0.0040
Primary	0.1066	0.1053	0.3119	0.1244	0.0657	0.0584
Family death	−0.0475	0.0992	0.6326	−0.1373	0.0650	0.0348
Credit	0.3047	0.0754	<.0001	0.2928	0.0746	<.0001
Marital status: Other	−0.0378	0.1008	0.7077	0.0215	0.0751	0.7752
Mean time varying variables
Temperature	−0.7490	0.0963	<.0001	−0.7299	0.0950	<.0001
Distance to seller	−0.3909	0.1677	0.0199	−0.2636	0.1231	0.0325
Wage rate	−0.1826	0.2146	0.3950	−0.1657	0.2101	0.4304
Asset value	0.1737	0.0668	0.0094	0.1761	0.0654	0.0072
Seed price	0.6472	0.2429	0.0078	0.5336	0.2367	0.0244
Yield equation
Intercept	7.2551	0.3396	<.0001	6.3045	0.4719	<.0001
Improved seed*				0.7014	0.1483	<.0001
logyield ${}_{00}$	0.2208	0.0122	<.0001	−0.0386	0.0584	0.5084
Rainfall	−0.0606	0.0662	0.3604	−0.0541	0.0669	0.4188
# of adult females	−0.0301	0.0146	0.0394	−0.0236	0.0143	0.0989
# of adult males	0.0078	0.0139	0.5753	0.0064	0.0136	0.6383
Farm size	−0.0994	0.0147	<.0001	−0.1053	0.0148	<.0001
Asset value	0.0591	0.0296	0.0464	−0.0502	0.0493	0.3087
Temperature	−0.1104	0.0875	0.2071	−0.4793	0.1320	0.0003
Age	−0.0017	0.0013	0.2113	−0.0004	0.0014	0.7603
Seed price	0.2359	0.0721	0.0011	0.6584	0.1371	<.0001
Wage rate	−0.1702	0.0844	0.0439	−0.5135	0.1377	0.0002
Corn price	0.0620	0.0042	<.0001	0.0621	0.0042	<.0001
Fertilizer price	−0.0264	0.0026	<.0001	−0.0278	0.0026	<.0001
Land preparation cost	0.0389	0.0255	0.1279	0.0360	0.0257	0.1610
Female	−0.0812	0.0463	0.0798	−0.0378	0.0600	0.5290
Group member	0.0239	0.0431	0.5804	0.0624	0.0691	0.3669
Credit	0.1060	0.0385	0.0060	−0.1230	0.0773	0.1117
Mean time varying variables
Rainfall	0.0881	0.0882	0.3180	0.0949	0.0874	0.2775
Farm size	0.0680	0.0179	0.0002	0.0861	0.0180	<.0001
Asset value	0.0123	0.0345	0.7224	−0.1370	0.0614	0.0258
Temperature	0.0394	0.0877	0.6535	0.5547	0.1535	0.0003
Seed price	−0.1400	0.1125	0.2136	−0.5593	0.2020	0.0057
Wage rate	−0.0415	0.1108	0.7084	0.0833	0.1736	0.6315
Corn price	−0.0803	0.0102	<.0001	−0.0722	0.0099	<.0001
Fertilizer price	0.0116	0.0053	0.0283	0.0118	0.0051	0.0213
Land preparation cost	0.0379	0.0353	0.2832	0.0224	0.0348	0.5202
${\sigma }_0$	0.9834	0.0238	<.0001	1.1704	0.0884	<.0001
σ: d2007 ${}_t$	−0.1905	0.0291	<.0001	−0.2980	0.0494	<.0001
σ: d2010 ${}_t$	0.6777	0.0437	<.0001	0.5697	0.0751	<.0001
${\rho }_0$	0.0497	0.0589	0.3987	−0.5363	0.0982	<.0001
ρ: d2007 ${}_t$	0.2832	0.0779	0.0003	0.0504	0.0627	0.4220
ρ: d2010 ${}_t$	0.0238	0.0909	0.7933	0.2184	0.0693	0.0017
${\sigma }_v^2$	0.0507	0.0207	0.0143	0.4337	0.2161	0.0449
${\sigma }_w^2$	1.2052	0.1559	<.0001	1.1378	0.1482	<.0001
${\sigma }_{vw}$	0.1347	0.0363	0.0002	−0.7017	0.192	0.0003
−2Loglikelihood	13764			13703
BIC	14159			14106

These estimates are obtained using panel data from Kenya described in the data section. A balanced panel data of 1171 households for the years 2004, 2007 and 2010 are used in estimation. Note that lagged yield appears on the right hand side of Equation (2) in the model. Because we have a short panel of data (T = 3 years) with a cross-lagged effect of yield but the data is not observed from the onset of technology adoption, the initial conditions problem arises. This refers to the challenge of specifying the distribution of our observed first period when lagged effects are present. To account for initial conditions problem in non-linear panels, Wooldridge [28] proposed specifying the random effect conditional on the first period outcome for a single equation system instead of specifying the period one distribution. We extend wooldrige's [28] proposed solution as described in Model s4 of the simulation since we have a two equation system.

The model s4 (with endogenous effect) includes the variable improved seed* in the yield equation while the model s3 (without the endogenous effect), does not include this effect. We found this endogenous effect to be positive and significant as expected (See Table 7). This implies that farmers with a high propensity to adopt improved corn seed had higher yields. Similar studies [3,13,16,20] in Africa also found a positive impact of the endogenous agricultural technology on crop yields. We found that when the endogenous seed effect was included, there was a change in the sign of the covariance ${\sigma }_{uv}$ from positive to negative.

Table 8 gives the overall effects of variables whose means were included as regressors.

Table 8. Overall effects of time varying variables.

Parameter	Estimate	SE	Pr ( $T>|t|$)
Improved seed equation
Temperature	−0.1998	0.0186	<.0001
Distance to seller	−0.4008	0.1040	0.0001
Wage rate	0.3439	0.1746	0.0490
Asset value	0.3297	0.0470	<.0001
Seed price	0.0155	0.2112	0.9417
Yield equation
Rainfall	0.0420	0.0539	0.4366
Farm size	−0.0186	0.0086	0.0301
Asset value	−0.2055	0.0671	0.0022
Temperature	0.0931	0.0370	0.0119
Seed price	0.0297	0.1681	0.8597
Wage rate	−0.4434	0.1511	0.0034
Corn price	−0.0088	0.0099	0.3753
Fertilizer price	−0.0155	0.0050	0.0021
Land preparation cost effect	0.0583	0.0245	0.0172

In the seed equation, we found that the lagged yield effect was positive as expected from the proposal of [8]. The distance to improved seed seller negatively affected the likelihood of adoption. Higher education, credit access and asset value as expected increased the likelihood of improved seed adoption. Similar results were found by both [19,26] who used the same dataset.

In the yield equation, we found that fertilizer price significantly reduced the log of corn yield consistent with economic theory. Land preparation cost increased the yield a result also found by [26] who used the same dataset. Farm size significantly reduced the yield values; a result found by [25] who conducted a literature review of studies conducted in Sub-Saharan Africa. High harvest temperature significantly increased the yield consistent with requirements of high corn production. We expected asset value to increase the yield but the effect was negative, this could probably be due to the fact that these farmers had larger farm sizes.

4.2.1. Marginal effects

For the probit model in Equation (2), we also calculated the marginal effects at median values for discrete and average values for the continuous variables. Recall that improved seed adoption is modelled as:

$$\begin{array}{rl}& y_{2it}^{\ast }={\mu }_{2it}+{\beta }_2{y}_{1it-1}+u_{2it}+w_i\mathrm{where}{y}_{2it}=1\mathrm{if}{y}_{2it}^{\ast }>0\mathrm{and}0\mathrm{if}{y}_{2it}^{\ast }\le 0.\\ & Pr(y_{2it}=1)=Pr(y_{2it}^{\ast }>0)=\mathrm{\Phi }\left(\frac{{\mu }_{2it}+{\beta }_2{y}_{1it-1}}{\sqrt{1+{\sigma }_w^2}}\right).\end{array}$$

When $x_{jit}$ is a continuous variable, the marginal effect is:

$$\frac{\mathrm{\partial }Pr(y_{2it}=1)}{\mathrm{\partial }{x}_{jit}}=\frac{{\beta }_j}{\sqrt{1+{\sigma }_w^2}}\varphi \left(\frac{{\mu }_{2it}+{\beta }_2{y}_{1it-1}}{\sqrt{1+{\sigma }_w^2}}\right).$$

When $x_{jit}$ is a discrete variable, the marginal effect is calculated as:

$$\mathrm{Pr}(y_{2it}=1|x_{jit}=1)-\mathrm{Pr}(y_{2it}=1|x_{jit}=0).$$

Table 9 shows the marginal effects of all the variables computed at their means for continuous variables and at the medians for the dummy variables. Note that for the continuous variables whose household specific time varying means were included as additional regressors, marginal effects are appropriately adjusted for by using the sum of the two parameters in estimation. The marginal effects of all the variables included except the seed price, wage rate, group member and marital status were significant at 5% level. A change in the past season log yield by 1 kg/acre increased the likelihood of adopting the improved corn seed by 1.62%. The probability of adoption increased by 7.36% for every additional log of yield in the year 2000. Thus, the lag yields seem to have one of the largest influences on the probability of improved seed adoption. Farmers need to be convinced that improved seed adoption results in higher expected yield despite weather or other external factors such as pests and diseases. Farmers who had access to credit had a 7.05% higher probability of adoption than farmers who did not. The probability of adoption also increased with the education level at 5.64% for those with secondary and higher education and 2.96% for those with primary education compared to those with no education. Compared to those who did not own or rent their farms, farmers that rented had the highest probality of adoption of improved corn seed with 4.45% higher chance while those that owned had a 2.58% chance of adopting the improved seed compared to those that neither owned nor rented their land. Household that experienced a death in the past 3 years had a 3.06% less chance of adopting the improved seed variety.

Table 9. Marginal effects at averages for continuous and medians for dummy variables.

Variable	Estimate	SE	Pr( $T>|t|$)	95% CI
logyield ${}_{t-1}$	0.0162	0.0053	0.0021	(0.0059, 0.02650
logyield ${}_{00}$	0.0736	0.0071	<.0001	(0.0596, 0.0876)
Temperature	−0.0445	0.0042	<.0001	(−0.0527, −0.0363)
Distance to seller	−0.0940	0.0253	0.0002	(−0.1435, −0.0444)
Wagerate	0.0758	0.0399	0.0578	(−0.0025, 0.1541)
Asset value	0.0761	0.0107	<.0001	(0.0552, 0.0971)
Seed price	−0.0012	0.0482	0.9801	(−0.0957, 0.0933)
Tenure: Owned	0.0258	0.0118	0.0290	(0.0027, 0.04900)
Rented	0.0445	0.0175	0.0108	(0.0103, 0.0788)
Group member	−0.0272	0.0187	0.1457	(−0.0639, 0.0095)
Education: > Primary	0.0564	0.0189	0.0029	(0.0194, 0.0935)
Primary	0.0296	0.0154	0.0553	(−0.0007, 0.0598)
Family death	−0.0306	0.0151	0.0433	(−0.0603, −0.0009)
Credit	0.0705	0.0182	0.0001	(0.0348, 0.1062)
Marital status: Other	0.0060	0.0167	0.7164	(−0.0266, 0.0387)

5. Discussion

We have proposed a method that captures endogeneity, lagged effects and correlation in repeated observations for a mixture of discrete and correlated data while addressing the initial conditions problem that arises in short panels with lagged effects. This model is motivated by the need to quantify the role of lagged crop yield on the decision to adopt improved seed variety in East Africa. The model is an extension of Teixeira et al. [11] model to include the endogenous effect and account for the initial conditions problem.

We conduct simulations to assess the performance of the model under different scenarios. The results showed that ignoring the endogenous effect and or initial conditions problem results in biased estimates. We applied the model to national survey data of Kenyan farmers to examine the effect of lagged yield on the decision to adopt improved seed variety. We found that lagged yield was a significant positive determinant of the probability of adoption in future seasons. In particular, higher lagged yield significantly increased the likelihood of adoption in the next season. This means that for risk averse farmers with limited resources, assistance can be granted to them in periods of lower yield to ensure that they adopt the improved seed variety in each season. Secondly, we found that when the endogenous improved seed effect was not accounted for, the covariance of the random effects ${\sigma }_{vw}$ was inaccurately estimated as negative. Other variables that influenced the probability of adoption were: seed price, wage rate, temperature, asset value and access to credit. Their signs were consistent with similar studies in East Africa. The model can be applied to other technology adoption studies in Sub-Saharan Africa as well as observational paired data with lagged effects and or endogenous effects that arises commonly in economics, sociology and the health fields.

One limitation of this study is that we used adhoc methods such as weather data to account for spatial effects due to computational difficulties. Another potential area for future study is to jointly model improved seed use and fertilizer while accounting for the lagged yield effects on their adoption. These scenarios could be explicitly modelled in a bayesian setting.

Appendix. Detailed simulation results.

Below are the simulation results. Results from model 3 and model 4 indicate accounting for initial conditions as proposed by wooldridge, results in consistent estimates of the lagged effects. When initial conditions are not accounted for as in model 2 and 3, the endogenous effect estimates have some bias as well as other parameters of the binary outcome. When it comes to covariance of the random effects, model 4 has the lowest bias and mean square error. However, model 4 performs worst in estimating the random effect variance of the continuous variables.

In summary, failure to account for the initial conditions not only affects the lagged effect but also other parameters. However, the not accounting for the endogenous effect, has little effect on the lagged effect but results in bias of variance-covariance estimates.

Table A1. Simulation results for 500 runs: Endogenous term varying.

		Model s1			Model s2			Model s3			Model s4
Parameter	True value	Mean	SD	RMSE	Mean	SD	RMSE	Mean	SD	RMSE	Mean	SD	RMSE
${\alpha }_2=0.1$
${\alpha }_0$	1	1.0165	0.0417	0.0448	1.0314	0.0433	0.0535	0.4772	0.0482	0.5250	0.5268	0.0503	0.4759
${\alpha }_1$	0.5	0.5027	0.0236	0.0238	0.5034	0.0236	0.0238	0.4929	0.0247	0.0257	0.4995	0.0254	0.0254
${\alpha }_2$	0.1	0.1617	0.0293	0.0684	0.2208	0.0502	0.1308				0.1019	0.0238	0.0239
${\alpha }_3$	0							0.3996	0.0192	0.4001	0.3857	0.0200	0.3862
${\alpha }_4$	0							0.0246	0.0579	0.0629	0.0055	0.0577	0.0580
${\beta }_0$	−0.75	−0.7747	0.0618	0.0666	−0.8711	0.0699	0.1398	−0.6335	0.0610	0.1315	−0.6296	0.0609	0.1349
${\beta }_1$	1	0.9690	0.0515	0.0601	0.9245	0.0616	0.0974	1.0254	0.0518	0.0577	0.9982	0.0528	0.0528
${\beta }_2$	0.4	0.4170	0.0322	0.0364	0.3744	0.0345	0.0430	0.4088	0.0336	0.0347	0.3986	0.0335	0.0336
${\beta }_3$	0				0.1223	0.0421	0.1293	−0.1070	0.0316	0.1116	−0.0998	0.0314	0.1047
${\beta }_4$	0				−0.0017	0.0731	0.0731	−0.0021	0.0798	0.0798	−0.0041	0.0748	0.0750
${\sigma }_{vw}$	−0.35	−0.4254	0.0770	0.1078	−0.5325	0.0935	0.2051	−0.1648	0.0536	0.1928	−0.2467	0.0594	0.1192
ρ	−0.45	−0.4160	0.0397	0.0523	−0.4682	0.0486	0.0519	−0.3021	0.0377	0.1526	−0.3815	0.0406	0.0796
σ	1	1.2206	0.0272	0.2223	1.2601	0.0413	0.2633	1.1398	0.0183	0.1410	1.1860	0.0233	0.1874
${\sigma }_v^2$	1	0.9705	0.0683	0.0744	0.9546	0.0711	0.0843	0.6331	0.0512	0.3705	0.6066	0.0527	0.3969
${\sigma }_w^2$	1	1.0171	0.1364	0.1375	1.0718	0.1450	0.1618	0.9772	0.1334	0.1353	0.9749	0.1339	0.1362
${\alpha }_2=0.2$
${\alpha }_0$	1	1.0193	0.0422	0.0464	1.0492	0.0448	0.0666	0.4268	0.0483	0.5753	0.5400	0.0495	0.4626
${\alpha }_1$	0.5	0.5040	0.0233	0.0237	0.5044	0.0232	0.0236	0.4861	0.0242	0.0279	0.4994	0.0247	0.0247
${\alpha }_2$	0.2	0.2728	0.0330	0.0800	0.3845	0.0572	0.1931				0.2016	0.0257	0.0257
${\alpha }_3$	0							0.4217	0.0191	0.4221	0.3846	0.0201	0.3852
${\alpha }_4$	0							0.0454	0.0587	0.0742	0.0043	0.0559	0.0560
${\beta }_0$	−0.75	−0.8013	0.0624	0.0808	−0.9412	0.0672	0.2027	−0.6831	0.0620	0.0912	−0.6759	0.0617	0.0965
${\beta }_1$	1	0.9597	0.0536	0.0671	0.8677	0.0659	0.1478	1.0390	0.0532	0.0660	0.9991	0.0529	0.0529
${\beta }_2$	0.4	0.4398	0.0331	0.0518	0.3563	0.0349	0.0559	0.4130	0.0346	0.0370	0.3977	0.0337	0.0338
${\beta }_3$	0				0.2020	0.0352	0.2050	−0.0718	0.0328	0.0789	−0.0624	0.0323	0.0703
${\beta }_4$	0				−0.0035	0.0683	0.0683	−0.0065	0.0815	0.0817	−0.0039	0.0750	0.0751
${\sigma }_{vw}$	−0.35	−0.4705	0.0821	0.1458	−0.6401	0.0959	0.3056	−0.1082	0.0534	0.2477	−0.2823	0.0605	0.0908
ρ	−0.45	−0.4145	0.0400	0.0535	−0.5122	0.0484	0.0788	−0.2215	0.0390	0.2318	−0.3814	0.0398	0.0793
σ	1	1.2277	0.0294	0.2296	1.3048	0.0477	0.3085	1.1137	0.0179	0.1151	1.1852	0.0238	0.1867
${\sigma }_v^2$	1	0.9668	0.0696	0.0771	0.9482	0.0749	0.0911	0.6615	0.0511	0.3423	0.6045	0.0527	0.3990
${\sigma }_w^2$	1	1.0248	0.1411	0.1433	1.1199	0.1542	0.1953	0.9924	0.1384	0.1386	0.9871	0.1377	0.1383
${\alpha }_2=0.3$
${\alpha }_0$	1	1.0237	0.0427	0.0489	1.0655	0.0460	0.0800	0.3762	0.0488	0.6257	0.5650	0.0494	0.4378
${\alpha }_1$	0.5	0.5055	0.0234	0.0240	0.5046	0.0229	0.0234	0.4789	0.0241	0.0320	0.4995	0.0245	0.0245
${\alpha }_2$	0.3	0.3897	0.0385	0.0976	0.5346	0.0573	0.2415				0.3014	0.0279	0.0279
${\alpha }_3$	0							0.4433	0.0189	0.4437	0.3736	0.0204	0.3741
${\alpha }_4$	0							0.0674	0.0600	0.0902	0.0039	0.0551	0.0553
${\beta }_0$	−0.75	−0.8300	0.0633	0.1020	−0.9783	0.0676	0.2381	−0.7319	0.0630	0.0656	−0.7259	0.0627	0.0672
${\beta }_1$	1	0.9452	0.0543	0.0772	0.8228	0.0621	0.1877	1.0364	0.0532	0.0645	0.9999	0.0523	0.0523
${\beta }_2$	0.4	0.4659	0.0344	0.0743	0.3418	0.0334	0.0671	0.4117	0.0349	0.0368	0.3978	0.0334	0.0334
${\beta }_3$	0				0.2527	0.0285	0.2543	−0.0303	0.0329	0.0448	−0.0215	0.0321	0.0386
${\beta }_4$	0				−0.0071	0.0644	0.0648	−0.0125	0.0832	0.0841	−0.0031	0.0729	0.0729
${\sigma }_{vw}$	−0.35	−0.5263	0.0895	0.1978	−0.7161	0.0982	0.3791	−0.0439	0.0553	0.3110	−0.3239	0.0640	0.0691
ρ	−0.45	−0.4127	0.0401	0.0547	−0.5439	0.0437	0.1036	−0.1366	0.0398	0.3159	−0.3812	0.0391	0.0791
σ	1	1.2377	0.0322	0.2399	1.3402	0.0482	0.3436	1.1056	0.0177	0.1071	1.1851	0.0248	0.1868
${\sigma }_v^2$	1	0.9630	0.0717	0.0807	0.9559	0.0823	0.0934	0.7062	0.0522	0.2984	0.6136	0.0547	0.3902
${\sigma }_w^2$	1	1.0196	0.1415	0.1429	1.1402	0.1579	0.2112	0.9983	0.1401	0.1401	0.9962	0.1397	0.1398
${\alpha }_2=0.5$
${\alpha }_0$	1	1.0307	0.0444	0.0540	1.0763	0.0467	0.0895	0.3251	0.0499	0.4579	0.6017	0.0499	0.4014
${\alpha }_1$	0.5	0.5081	0.0232	0.0246	0.5044	0.0227	0.0232	0.4715	0.0244	0.0375	0.4996	0.0243	0.0243
${\alpha }_2$	0.5	0.5163	0.0460	0.1251	0.6589	0.0548	0.2646				0.4013	0.0302	0.0302
${\alpha }_3$	0							0.4651	0.0188	0.4618	0.3526	0.0206	0.3532
${\alpha }_4$	0							0.0900	0.0622	0.1094	0.0034	0.0547	0.0548
${\beta }_0$	−0.75	−0.8561	0.0633	0.1236	−0.9943	0.0679	0.2536	−0.7768	0.0634	0.0688	−0.7743	0.0632	0.0677
${\beta }_1$	1	0.9219	0.0558	0.0960	0.8024	0.0567	0.2055	1.0157	0.0541	0.0563	0.9998	0.0517	0.0517
${\beta }_2$	0.4	0.4928	0.0344	0.0990	0.3344	0.0316	0.0728	0.4040	0.0344	0.0346	0.3980	0.0325	0.0326
${\beta }_3$	0				0.2795	0.0262	0.2807	0.0165	0.0333	0.0372	0.0203	0.0321	0.0379
${\beta }_4$	0				−0.0114	0.0618	0.0628	−0.0224	0.0851	0.088	−0.0029	0.0710	0.0711
${\sigma }_{vw}$	−0.35	−0.5889	0.0962	0.2576	−0.7601	0.1004	0.4222	0.0313	0.0586	0.1488	−0.3661	0.0672	0.0691
ρ	−0.45	−0.4123	0.0411	0.0558	−0.5575	0.0405	0.1149	−0.0469	0.0420	0.1642	−0.3809	0.0399	0.0798
σ	1	1.2529	0.0366	0.2555	1.3561	0.0470	0.3592	1.1148	0.0178	0.0135	1.1851	0.0265	0.1870
${\sigma }_v^2$	1	0.9602	0.0780	0.0876	0.9704	0.0901	0.0949	0.7718	0.0548	0.0551	0.6323	0.0582	0.3723
${\sigma }_w^2$	1	0.9871	0.1355	0.1361	1.1397	0.1575	0.2105	0.9972	0.1389	0.0193	0.9970	0.1388	0.1389

Results show that model four which accounts for both the endogenous and initial conditions problem is effective in estimating the lagged effect. However, the difference between model 3 and 4 shows that when initial conditions is accounted for, the lagged effect is accurately estimated.

Table A2. Simulations varying lagged effects.

		Model s1			Model s2			Model s3			Model s4
Parameter	True value	Mean	SD	RMSE	Mean	SD	RMSE	Mean	SD	RMSE	Mean	SD	RMSE
${\beta }_2=0.2$
${\alpha }_0$	1	1.0298	0.0472	0.0558	1.1118	0.0523	0.1234	0.3011	0.0483	0.7006	0.6849	0.0525	0.3194
${\alpha }_1$	0.5	0.5058	0.0230	0.0237	0.5019	0.0226	0.0227	0.4830	0.0255	0.0307	0.4996	0.0243	0.0243
${\alpha }_2$	0.5	0.5627	0.0427	0.4647	0.6995	0.0549	0.2069				0.5009	0.0351	0.0351
${\alpha }_3$	0							0.4142	0.0188	0.4147	0.3240	0.0218	0.3247
${\alpha }_4$	0							0.0557	0.0625	0.0838	0.0036	0.0555	0.0556
${\beta }_0$	−0.75	−0.8369	0.0580	0.1045	−1.0059	0.0649	0.2640	−0.8056	0.0595	0.0814	−0.8072	0.0591	0.0822
${\beta }_1$	1	0.9598	0.0534	0.0669	0.8489	0.0565	0.1613	0.9814	0.0562	0.0592	1.0019	0.0524	0.0524
${\beta }_2$	0.2	0.2924	0.0290	0.0968	0.1733	0.0249	0.0365	0.1944	0.0312	0.0317	0.1986	0.0275	0.0275
${\beta }_3$	0				0.2864	0.0256	0.2875	0.0605	0.0327	0.0688	0.0579	0.0313	0.0658
${\beta }_4$	0				0.0011	0.0615	0.0615	−0.0424	0.0851	0.0951	−0.0069	0.0679	0.0683
${\sigma }_{vw}$	−0.35	−0.5218	0.0906	0.1943	−0.7568	0.1057	0.4203	0.1130	0.0602	0.4669	−0.3966	0.0730	0.0866
ρ	−0.45	−0.3826	0.0413	0.0790	−0.5183	0.0418	0.0801	0.0465	0.0437	0.4984	−0.3817	0.0406	0.0795
σ	1	1.2133	0.0318	0.2157	1.2978	0.0430	0.3009	1.1692	0.0186	0.1702	1.1857	0.0276	0.1878
${\sigma }_v^2$	1	1.0073	0.0817	0.0821	1.0763	0.0977	0.3126	0.7194	0.0542	0.2327	0.6952	0.0658	0.3118
${\sigma }_w^2$	1	0.9528	0.1411	0.1488	1.1261	0.1639	0.2068	0.9854	0.1448	0.1455	0.9838	0.1445	0.1454
${\beta }_2=0.3$
${\alpha }_0$	1	1.0386	0.0464	0.0604	1.1039	0.0498	0.1152	0.2881	0.0497	0.7137	0.6660	0.0517	0.3380
${\alpha }_1$	0.5	0.5078	0.0232	0.0245	0.5030	0.0226	0.0228	0.4737	0.0252	0.0364	0.4997	0.0243	0.0243
${\alpha }_2$	0.5	0.6008	0.0470	0.1113	0.7336	0.0553	0.2400				0.5009	0.0340	0.0340
${\alpha }_3$	0							0.4504	0.0188	0.4508	0.3250	0.0213	0.3257
${\alpha }_4$	0							0.0836	0.0638	0.1052	0.0031	0.0551	0.0552
${\beta }_0$	−0.75	−0.8562	0.0604	0.1222	−1.0060	0.0679	0.2649	−0.8098	0.0625	0.0865	−0.8120	0.0621	0.0878
${\beta }_1$	1	0.9317	0.0549	0.0876	0.8234	0.0557	0.1852	0.9796	0.0563	0.0599	1.0011	0.0525	0.0525
${\beta }_2$	0.3	0.4056	0.0307	0.1100	0.2547	0.0273	0.0529	0.2926	0.0320	0.0328	0.2989	0.0292	0.1709
${\beta }_3$	0				0.2901	0.0261	0.2913	0.0622	0.0325	0.0702	0.0582	0.0313	0.0661
${\beta }_4$	0				−0.0052	0.0606	0.0609	−0.0379	0.0860	0.0940	−0.0047	0.0683	0.0684
${\sigma }_{vw}$	−0.35	−0.5839	0.0945	0.2523	−0.7786	0.1062	0.4416	0.1159	0.0616	0.4700	−0.4031	0.0727	0.0900
ρ	−0.45	−0.3953	0.0415	0.0687	−0.5399	0.0405	0.0986	0.0472	0.0438	0.4992	−0.3809	0.0406	0.0802
σ	1	1.2357	0.0359	0.2384	1.3275	0.0454	0.3307	1.1537	0.0184	0.1547	1.1852	0.0280	0.1873
${\sigma }_v^2$	1	0.9950	0.0826	0.0828	1.0412	0.0973	0.1056	0.7904	0.0566	0.2171	0.6779	0.0640	0.3284
${\sigma }_w^2$	1	0.9491	0.1425	0.1513	1.1331	0.1675	0.2139	0.9922	0.1483	0.1485	0.9901	0.1479	0.1482
${\beta }_2=0.5$
${\alpha }_0$	1	1.0320	0.0463	0.0563	1.0464	0.0473	0.0663	0.2556	0.0528	0.7463	0.6251	0.0508	0.3784
${\alpha }_1$	0.5	0.5100	0.0238	0.0258	0.5046	0.0226	0.0231	0.4537	0.0246	0.0524	0.4996	0.0245	0.0245
${\alpha }_2$	0.5	0.7312	0.0611	0.2391	0.7773	0.0514	0.2821				0.5010	0.0336	0.0336
${\alpha }_3$	0							0.5256	0.0184	0.5260	0.3219	0.0212	0.3226
${\alpha }_4$	0							0.1436	0.0663	0.1582	0.0030	0.0544	0.0545
${\beta }_0$	−0.75	−0.8969	0.0640	0.1603	−0.9887	0.0713	0.2491	−0.8161	0.0650	0.0927	−0.8198	0.0648	0.0953
${\beta }_1$	1	0.8299	0.0612	0.1808	0.7896	0.0527	0.2169	0.9781	0.0590	0.0629	1.0008	0.0550	0.0550
${\beta }_2$	0.5	0.6272	0.0334	0.1315	0.4116	0.0340	0.0947	0.4869	0.0369	0.0392	0.4980	0.0356	0.0356
${\beta }_3$	0				0.2898	0.0256	0.2910	0.0667	0.0327	0.0743	0.0588	0.0315	0.0667
${\beta }_4$	0				−0.0250	0.0612	0.0661	−0.0344	0.0938	0.0999	−0.0035	0.0718	0.0719
${\sigma }_{vw}$	−0.35	−0.7408	0.1051	0.4047	−0.7744	0.1074	0.4377	0.1225	0.0657	0.4770	−0.4088	0.0734	0.0940
ρ	−0.45	−0.4327	0.0419	0.0453	−0.5679	0.0371	0.1236	0.0487	0.0421	0.5004	−0.3790	0.0411	0.0821
σ	1	1.3296	0.0490	0.3332	1.3773	0.0449	0.3800	1.1270	0.0180	0.1282	1.1840	0.0287	0.1862
${\sigma }_v^2$	1	0.9052	0.0915	0.1317	0.9150	0.0983	0.1300	0.9374	0.0617	0.0879	0.6394	0.0620	0.3659
${\sigma }_w^2$	1	0.9117	0.1364	0.1625	1.1217	0.1666	0.2063	0.9902	0.1470	0.1473	0.9868	0.1464	0.1470

Table A3. Simulations for various sigmavw.

		Model s1			Model s2			Model s3			Model s4
Parameter	True valu	Mean	SD	RMSE	Mean	SD	RMSE	Mean	SD	RMSE	Mean	SD	RMSE
${\sigma }_v^2=0.2$
${\alpha }_0$	1	1.0115	0.0345	0.0363	1.0164	0.0344	0.0382	0.5405	0.0433	0.4615	0.8941	0.0436	0.1145
${\alpha }_1$	0.5	0.5021	0.0214	0.0215	0.5005	0.0213	0.0213	0.4639	0.0249	0.0438	0.4996	0.0245	0.0245
${\alpha }_2$	0.5	0.5384	0.0343	0.0515	0.5459	0.0344	0.0574				0.5011	0.0324	0.0324
${\alpha }_3$	0							0.2437	0.0193	0.2444	0.0981	0.0221	0.1005
${\alpha }_4$	0							0.1115	0.0562	0.1248	0.0020	0.0468	0.0469
${\beta }_0$	−0.75	−0.8181	0.0617	0.0919	−0.9263	0.0657	0.1881	−0.8399	0.0679	0.1127	−0.8450	0.0675	0.1166
${\beta }_1$	1	0.9730	0.0508	0.0575	0.9670	0.0521	0.0617	0.9791	0.0557	0.0595	1.0004	0.0522	0.0522
${\beta }_2$	0.4	0.4606	0.0317	0.0683	0.3874	0.0316	0.0340	0.3881	0.0337	0.0358	0.3980	0.0321	0.0322
${\beta }_3$	0				0.1698	0.0264	0.1719	0.0908	0.0351	0.0974	0.0856	0.0343	0.0922
${\beta }_4$	0				−0.0118	0.0646	0.0657	−0.0953	0.0860	0.1284	−0.0031	0.0663	0.0663
${\sigma }_{uv}$	−0.35	−0.4244	0.0673	0.1003	−0.4229	0.0679	0.0996	0.1739	0.0520	0.5265	−0.3651	0.0633	0.0651
ρ	−0.45	−0.3782	0.0402	0.0823	−0.4148	0.0389	0.0525	0.0456	0.0429	0.4974	−0.3817	0.0398	0.0791
σ	1	1.2043	0.0301	0.2065	1.2103	0.0295	0.2123	1.1396	0.0182	0.1408	1.1855	0.0277	0.1876
${\sigma }_v^2$	0.25	0.3494	0.0521	0.1123	0.3520	0.0513	0.1142	0.4679	0.0416	0.2219	0.3181	0.0473	0.0830
${\sigma }_w^2$	1	0.9678	0.1393	0.1430	1.0053	0.1448	0.1449	0.9919	0.1434	0.1437	0.9845	0.1422	0.1430
${\sigma }_v^2=0.2$
${\alpha }_0$	1	1.0057	0.0291	0.0296	1.0098	0.0292	0.0308	0.4755	0.0437	0.5263	0.9549	0.0414	0.0612
${\alpha }_1$	0.5	0.5046	0.0215	0.0220	0.5000	0.0212	0.0212	0.4639	0.0249	0.0438	0.4995	0.0245	0.0245
${\alpha }_2$	0.5	0.5310	0.0300	0.0432	0.5263	0.0283	0.0386				0.4984	0.0301	0.0301
${\alpha }_3$	0							0.3033	0.0191	0.3039	0.0414	0.0208	0.0464
${\alpha }_4$	0							0.1091	0.0528	0.1212	0.0017	0.0442	0.0443
${\beta }_0$	−0.75	−0.8821	0.0602	0.1452	−1.1066	0.0684	0.3631	−1.0775	0.0708	0.3351	−1.0869	0.0692	0.3439
${\beta }_1$	1	0.9812	0.0498	0.0532	0.9830	0.0512	0.0540	0.9795	0.0574	0.0610	1.0040	0.0519	0.0521
${\beta }_2$	0.4	0.5219	0.0342	0.1266	0.3928	0.0319	0.0327	0.3815	0.0339	0.0386	0.3992	0.0323	0.0323
${\beta }_3$	0				0.3290	0.0319	0.3306	0.3146	0.0364	0.3167	0.3051	0.0355	0.3071
${\beta }_4$	0				−0.0024	0.0697	0.0697	−0.3074	0.0776	0.3170	−0.0030	0.0705	0.0705
${\sigma }_{uv}$	0.1565	−0.0266	0.0498	0.1898	0.0128	0.0437	0.1502	0.5747	0.0599	0.4225	0.0393	0.0443	0.2104
ρ	−0.45	−0.3415	0.0386	0.1151	−0.3986	0.0358	0.0626	0.0452	0.0433	0.4971	−0.3781	0.0378	0.0812
σ	1	1.1959	0.0263	0.1976	1.1970	0.0258	0.1987	1.1396	0.0182	0.1408	1.1825	0.0261	0.1843
${\sigma }_v^2$	0.2	0.0333	0.0300	0.1694	0.0322	0.0293	0.1704	0.5662	0.0464	0.3692	0.0401	0.0307	0.1628
${\sigma }_w^2$	1	0.8063	0.1330	0.2349	0.8042	0.1275	0.2336	0.8409	0.1344	0.2083	0.7950	0.1267	0.2410
${\sigma }_v^2=0.2$
${\alpha }_0$	1	1.0062	0.0251	0.0258	1.0076	0.0249	0.0260	0.5307	0.0451	0.4714	0.9634	0.0363	0.0516
${\alpha }_1$	0.5	0.5008	0.0214	0.0214	0.5001	0.0213	0.0213	0.4719	0.0258	0.0382	0.5001	0.0258	0.0258
${\alpha }_2$	0.5	0.5498	0.0305	0.0584	0.5232	0.0263	0.0351				0.4996	0.0278	0.0278
${\alpha }_3$	0							0.2630	0.0202	0.2637	0.0337	0.0158	0.0372
${\alpha }_4$	0							0.0790	0.0516	0.0943	−0.0003	0.0468	0.0468
${\beta }_0$	−0.75	−0.8346	0.0528	0.0997	−1.0304	0.0615	0.2871	−0.9573	0.0632	0.2167	−1.0223	0.0611	0.2791
${\beta }_1$	1	0.9487	0.0477	0.0700	0.9910	0.0505	0.0513	0.7131	0.0492	0.2911	0.9979	0.0506	0.0506
${\beta }_2$	0.4	0.4905	0.0280	0.0947	0.3975	0.0286	0.0287	0.2823	0.0243	0.1201	0.3979	0.0287	0.0288
${\beta }_3$	0				0.2568	0.0300	0.2585	0.3077	0.0298	0.3092	0.2494	0.0300	0.2512
${\beta }_4$	0				0.0000	0.0759	0.0759	−0.1677	0.0666	0.1804	−0.0003	0.0763	0.0763
${\sigma }_{uv}$	0.1565	−0.0165	0.0425	0.1781	0.0228	0.0402	0.1396	0.7384	0.0763	0.5869	0.0496	0.0419	0.1148
ρ	0.45	0.4632	0.0448	0.0467	0.4329	0.0478	0.0507	0.7326	0.0258	0.2838	0.4506	0.0469	0.0469
σ	1	0.9921	0.0216	0.0230	0.9915	0.0212	0.0229	1.3179	0.0213	0.3186	0.9978	0.0215	0.0216
${\sigma }_v^2$	0.2	0.0317	0.0246	0.1701	0.0302	0.0239	0.1715	0.6589	0.0559	0.4623	0.0383	0.0258	0.1637
${\sigma }_w^2$	1	0.6834	0.1243	0.3401	0.8245	0.1287	0.2176	1.0217	0.1409	0.1426	0.8208	0.1279	0.2202

Disclosure statement

No potential conflict of interest was reported by the author(s).