Keywords

1 Introduction

The main interest of this paper is the seemingly unrelated regression (SUR) model, which was first publicized in Arnold Zellner’s paper in 1962. At first, the model was used to construct a system of different linear equations where all equations were estimated simultaneously by applying a two-step generalize least squares estimator [16]. Since then, SUR model has been used widely for many applications especially in the estimation of demand -or supply- system; for examples, modeling the demand system for a given commodity when the data sets come from different households and estimating a market system when the model consists of two different equations, mainly demand and supply of any commodity. Looking at its properties, the SUR model could gain more efficiency in the estimation as it can combine information from the different equations by letting errors from the equations be related .

But apart from its advantage, the SUR model comes with a strong assumption that an error term of each equation is assumed to have -or must have- a normal distribution. In spite of its strong assumption, the SUR model still has been used for many studies, e.g., Baltagi [1], Takada et al. [15], Sparks [14], Bilodeau and Duchesne [2], Drton and Richardson [4], and Lar et al. [8]. Although their results turned out to be good, and the models performed well, the fact that the models were estimated under the strong assumption of normal distribution should not be ignored. This gives us a possibility that the SUR model could be improved by getting rid of or relaxing this assumption. For instance, the Gaussian Copula gives the bivariate normal distribution if and only if the margins are normally distributed [17], which is the assumption of the conventional SUR model, otherwise not.

To relax this strong assumption, we take into account an advantage of Copula approach. It is more flexible on account of its ability to link the different marginal distributions of residuals of each equation in the model. In particular it makes the model more realistic and far from the assumption like a normal distribution. Perhaps Copula is a motivation to help us reach the goal relaxing the normal distribution assumption of SUR model. To the best of our knowledge, Copula has often been applied in the regression model for being the joint distribution function of dependent variables, which have different distributions; for examples, Kolev and Paiva [7], Masarotto and Varin [9], Noh et al. [11], and Parsa and Klugman [12]. However, we have never seen Copula applied in SUR model in the analysis of demand and supply of the Thai rice market.

The idea of this paper is that we try to relax the normal distribution assumption of error term of each equation in SUR model by applying Copula approach in the estimation. Then, the equations in SUR model are allowed to have different marginal distributions of residuals and Copula function can link them as a joint distribution. Therefore, it is the purpose of this paper to introduce the Copula-based SUR model as an alternative to the conventional SUR model. The advantage of incorporating Copula in the model will be investigated by making a comparison to the conventional SUR model.

For an overview of this paper, we begin with our intention to improve the SUR model in the introduction. To see an empirical result using this model, the real dataset of Thai rice is employed. Since the market generally works as a system, it has at least demand and supply equations which work together. Moreover, rice is one of the most important agricultural commodities of Thailand, and Thai people have always taken an interest in. According to these reasons, the Thai rice market is concerned to be an application for this paper. The following Sect. 2 is the discussion on methodology used in this paper. Then in Sect. 3, the estimation based Copula will be entirely explained in procedure. The estimated results are presented in Sect. 4.

2 Methodology

2.1 Seemingly Unrelated Regression (SUR) Model

The SUR model introduced by Zellner [16] is a generalization of system of linear regression equations or multivariate regression equations. It consists several regression equations that have their own dependent variables and can also be estimated separately. But the advantage of the SUR model is that it can gain efficiency or improve estimation by combining information from different equations. Consider the structure of SUR model, as it consists several regression equations, let us say M equations, the system of M equations can be shown as in the following.

$$\begin{aligned} Y_{i}=X_{i}\beta _{i}+\varepsilon _{i}\quad i=1,\ldots ,M \end{aligned}$$
(1)

where \(Y_{i}\) is a vector of dependent variables, \(X_{i}\) is a matrix of independent variables or regressors, and \(\beta _{i}\) is a vector of an unknown parameter called regression coefficients. \(\varepsilon _{i}\) is a vector of the error terms. The important assumption of the SUR model is that it assumes that the errors in the different equations are related. Thus we can assume that \(E\left[ \varepsilon _{ia}\varepsilon _{ib}\vert X\right] =0 \quad ; a\ne b\) whereas \(E\left[ \varepsilon _{ia}\varepsilon _{ib}\vert X\right] = \sigma _{ij}\).

That means the SUR model allows non-zero covariance between the error terms of different equations in the model. And the errors are considered to have a normal distribution, that is \(\varepsilon _{i}=\left( \varepsilon _{t,1},\ldots ,\varepsilon _{t,M}\right) \backsim N(0,\varSigma ) \) where the sigma matrix, \(\varSigma \) , is a variance-covariance matrix for M equations, such that

$$\begin{aligned} \varSigma { \left( { \varepsilon }_{ t }{ \varepsilon }_{ t } \right) =\begin{bmatrix} { \sigma }_{ 11 }&{ \sigma }_{ 12 }&.&.&{ \sigma }_{ 1M } \\ { \sigma }_{ 21 }&{ \sigma }_{ 22 }&&{ \sigma }_{ 2M } \\ .&.&. \\ .&&.&. \\ { \sigma }_{ M1 }&.&.&.&{ \sigma }_{ MM } \end{bmatrix} } \end{aligned}$$
(2)

The covariance matrix for system is

$$\begin{aligned} \varGamma { =\begin{bmatrix} { \sigma }_{ 11 }I&{ \sigma }_{ 12 }I&.&.&{ \sigma }_{ 1M }I \\ { \sigma }_{ 21 }I&{ \sigma }_{ 22 }I&&{ \sigma }_{ 2M }I \\ .&.&. \\ .&&.&. \\ { \sigma }_{ M1 }I&.&.&.&{ \sigma }_{ MM }I \end{bmatrix} }=\varSigma \otimes I \end{aligned}$$
(3)

where I is an identity matrix, this system equation can be estimated by

$$\begin{aligned} \beta _{sure}=(X'\varGamma ^{-1} X)^{-1} X'\varGamma ^{-1}Y \end{aligned}$$
(4)

2.2 Copula Approach

The most fundamental theorem, which describes the dependence in Copula, is the Sklar’s theorem. Sklar [13] has proposed the linkage between the marginal distributions which is possible to have different distributions with the same correlation, but different dependence structure. The linkage between the marginal distributions is called Copula. Formally, let H be an n-dimensional joint distribution function of the random variables \(x_{n}\) with marginal distribution functions \(F_{n}\) . So, there exist the n-Copula C such that for all \(X_{n}\).

$$\begin{aligned} H(x_{1},\ldots ,x_{n})=C(F_{1}(x_{1}),\ldots ,F_{n}(x_{n})) \end{aligned}$$
(5)

where C is Copula distribution function of a n-dimensional random variable. If the marginals are continuous, C is unique. Equation 2 defines a multivariate distribution function F. Thus, we can model the marginal distribution and joint dependence separately. If we have a continuous marginal distribution, Copula can be determined by

$$\begin{aligned} C(u_{1},\ldots ,u_{n})=C(F^{-1}_{1}(u_{1}),\ldots ,F^{-1}_{n}(u_{n})) \end{aligned}$$
(6)

where u is uniform [0,1].

2.3 Copula Families

There are two important classes of Copula, namely elliptical Copula and Archimedean Copula. The symmetric Gaussian and t-Copula are the families of Copula in Elliptical Copula while the asymmetric rank, Clayton, Gumbel, Joe, and Ali-Mikhail-Haq (AMH) are five important families in Archimedean Copula.

2.3.1 Elliptical Copula

(1) Gaussian Copula

In the case of n-dimensional Gaussian or Normal Copula, we can rewrite Eq. 3 as

$$\begin{aligned} C(u_{1},\ldots ,u_{n})=\varPhi ^{\varSigma _{n}}_{n}(\varPhi ^{-1}_{1}(u_{1}),\ldots ,\varPhi ^{-1}_{n}(u_{n}) \end{aligned}$$
(7)

where \(\varPhi ^{\varSigma _{d}}_{d}\) is n-dimensional standard normal cumulative distribution and \(\varSigma -{n}\) variance-covariance matrix. The density of the Gaussian Copula can be calculated from Eq. 4 and has the following form

$$\begin{aligned} c\left( { u }_{ 1 },\ldots ,{ u }_{ n } \right) =\left( \sqrt{ det\varSigma _{ n } } \right) ^{ -1 }\left( \frac{ 1 }{ 2 } \left( { \varPhi }_{ 1 }^{ -1 }\left( { u }_{ 1 } \right) \cdot \cdot \cdot { \varPhi }_{ n }^{ -1 }\left( { u }_{ n } \right) \right) \cdot \left( { \varSigma }_{ n }^{ -1 }-I \right) \cdot \begin{pmatrix} { \varPhi }_{ 1 }^{ -1 }\left( { u }_{ 1 } \right) \\ . \\ . \\ { \varPhi }_{ n }^{ -1 }\left( { u }_{ n } \right) \end{pmatrix} \right) \end{aligned}$$
(8)

(2) T-Copula

In the case of n-dimensional student-t or T- Copula, we can rewrite Eq. 3 as

$$\begin{aligned} c\left( { u }_{ 1 },\ldots ,{ u }_{ n } \right) =\int _{ -\infty }^{ { t }_{ v }^{ -1 }\left( { u }_{ 1 } \right) }{ ... } \int _{ -\infty }^{ { t }_{ v }^{ -1 }\left( { u }_{ n } \right) }{ f_{ { t }_{ 1 }\left( v \right) }\left( x \right) dx } \end{aligned}$$
(9)

where \(f_{ { t }_{ 1 }\left( v \right) }\left( x \right) \) n-dimensional t-density is function with degree of freedom v and \(t_{v}^{-1}\) is the quantile function of a standard univariate \(t_{v}\) distribution. In the estimation, the density of t-Copula is evaluated by

$$\begin{aligned} { c }_{ v,P }^{ t }\left( { u }_{ 1 },\ldots ,{ u }_{ n } \right) =\frac{ f_{ v,P }\left( { t }_{ v }^{ -1 }\left( { u }_{ t } \right) ...{ t }_{ v }^{ -1 }\left( { u }_{ t } \right) \right) }{ { \varPi }_{ i=1 }^{ n }f_{ v }\left( { t }_{ v }^{ -1 }\left( { u }_{ t } \right) \right) } \end{aligned}$$
(10)

where \(f_{ v,P }\) is the joint density of a \(t_{d}(v,0,P)\)-distributed random vector where P is the correlation matrix implied by the dispersion matrix \(\varSigma \).

2.3.2 Archimedean Copula

According to Genest and MacKay [5], the general form of the Archimedean Copula can be defined as

$$\begin{aligned} { c }\left( { u }_{ 1 },\ldots ,{ u }_{ n } \right) ={\left\{ \begin{array}{ll} { \varPhi }^{ -1 }\left( \varPhi \left( { u }_{ t } \right) +\cdots +\varPhi \left( { u }_{ t } \right) \right) \quad ,if\quad { \varSigma }_{ t=1 }^{ n }\varPhi \left( { u }_{ i } \right) \le \varPhi \left( 0 \right) \\ 0\qquad \quad \quad \quad \quad \quad \quad ,\quad otherwise \end{array}\right. } \end{aligned}$$
(11)

where \(\varPhi \) is a decreasing function and strictly decreasing. The density of this class Copula namely, Frank, Clayton, Gumbel, Joe, and Ali-Mikhail-Haq (AMH), as proposed in Hofert et al. [6], can be written as follows:

(1) Frank Copula

$$\begin{aligned} { { c }_{ \theta }^{ F } }\left( { u }_{ 1 },\ldots ,{ u }_{ n } \right) =\left( \frac{ \theta }{ 1-exp\left( -\theta \right) } \right) ^{ d-1 }{ Li }_{ -\left( n-1 \right) }\left( { h }_{ \theta }^{ F }\left( { u }_{ 1 },\ldots { u }_{ n } \right) \right) \frac{ exp\left( -\theta { \varSigma }_{ j=1 }^{ n }{ u }_{ j } \right) }{ { h }_{ \theta }^{ F }\left( { u }_{ 1 },\ldots ,{ u }_{ 2 } \right) } \end{aligned}$$
(12)

where \({\quad { h }_{ \theta }^{ F } }\left( { u }_{ 1 },\ldots ,{ u }_{ n } \right) =\left( 1-{ e }^{ -\theta } \right) ^{ 1-n }\prod _{ j=1 }^{ n }{ \left\{ 1-exp\left( -\theta { u }_{ j } \right) \right\} } \)

(2) Clayton Copula

$$\begin{aligned} c\left( { u }_{ 1 },\ldots ,{ u }_{ n } \right) =\prod _{ k=0 }^{ n-1 }{ \left( \theta k+1 \right) \left( \prod _{ j=1 }^{ n }{ { u }_{ i } } \right) ^{ -\left( 1+\theta \right) }\left( 1+{ t }_{ \theta }\left( { u }_{ 1 },\ldots { u }_{ n } \right) \right) ^{ -n\left( n+1/\theta \right) } } \end{aligned}$$
(13)

(3) Gumbel Copula

$$\begin{aligned} { c }_{ \theta }^{ G }\left( { u }_{ 1 },\ldots ,{ u }_{ n } \right) ={ \theta }^{ n }{ C }_{ \theta }\left( { u }_{ 1 },\ldots ,{ u }_{ n } \right) \frac{ \prod _{ j=1 }^{ n }{ \left( -log{ u }_{ i } \right) ^{ \theta -1 } } }{ { t }_{ \theta }\left( { u }_{ 1 },\ldots { u }_{ n } \right) \prod _{ j=1 }^{ n }{ { u }_{ i } } } \end{aligned}$$
(14)

where \(P^{G}_{n,\alpha }(x)=\varSigma ^{n}_{k=1}\alpha ^{G}_{nk}(\alpha )x^{k}\)

where \({ \alpha }_{ nk }^{ G }\left( \alpha \right) =\left( -1 \right) ^{ n-k }{ \varSigma }_{ n=k }^{ n }{ \alpha }^{ j }s\left( n,j \right) S\left( k,j \right) =\frac{ n! }{ k! } { \varSigma }_{ j=1 }^{ k }\begin{pmatrix} k \\ j \end{pmatrix}\begin{pmatrix} \alpha j \\ n \end{pmatrix}\left( -1 \right) ^{ n-j }\), \(k\in \) \(\left\{ 1,\ldots ,n \right\} \), s and S are the stirling numbers of the first kind and the second kind, respectively.

(4) Joe Copula

$$\begin{aligned} { c }_{ \theta }^{ J }\left( { u }_{ 1 },..,{ u }_{ n } \right) ={ \theta }^{ d-1 }\frac{ \prod _{ j=1 }^{ n }{ \left( 1-{ u }_{ j } \right) ^{ \theta -1 } } }{ { h }_{ \theta }^{ J }\left( { u }_{ 1 },..,{ u }_{ n } \right) } \left( 1-{ h }_{ \theta }^{ J }\left( u \right) \right) ^{ \alpha }{ p }_{ n,\alpha }^{ J }\left( \frac{ { h }_{ \theta }^{ J }\left( { u }_{ 1 },..,{ u }_{ n } \right) }{ 1-{ h }_{ \theta }^{ J }\left( { u }_{ 1 },..,{ u }_{ n } \right) } \right) \end{aligned}$$
(15)

where \(\quad { h }_{ \theta }^{ J }\left( { u }_{ 1 },\ldots ,{ u }_{ n } \right) =\prod _{ j=1 }^{ n }{ \left( 1-\left( 1-{ u }_{ j } \right) ^{ \theta } \right) }\)

and \(\quad { P }_{ n,\alpha }^{ J }\left( x \right) ={ \varSigma }_{ k=0 }^{ n-1 }{ \alpha }_{ nk }^{ J }\left( \alpha \right) { x }^{ k }\)

(5) Ali-Makhail-Haq Copula

$$\begin{aligned} { c }_{ \theta }^{ A }\left( { u }_{ 1 },\ldots ,{ u }_{ n } \right) =\frac{ { \left( 1-\theta \right) }^{ n+1 } }{ { \theta }^{ 2 } } \frac{ { h }_{ \theta }^{ A }\left( { u }_{ 1 },\ldots ,{ u }_{ n } \right) }{ \prod _{ j=1 }^{ n }{ { u }_{ j }^{ 2 } } } { Li }_{ -d }\left\{ { h }_{ \theta }^{ A }\left( { u }_{ 1 },\ldots ,{ u }_{ n } \right) \right\} \end{aligned}$$
(16)

where \(\quad {h }_{ \theta }^{ J }\left( { u }_{ 1 },\ldots ,{ u }_{ n } \right) =\theta \prod _{ j=1 }^{ n }{ \frac{ { u }_{ i } }{ 1-\theta \left( 1-u \right) } }\)

3 Estimation of SUR Based Copula

Data

The data used here are dataset related to Thai rice consisting of the demand for rice (\(Q_{t}^{d}\)), output of grain rice (\(Q_{t}^{s}\)), export price of Thai rice (\(P_{t}^{exp}\)), Pakistan’s exported rice price (\(P_{t}^{Paki}\)), Vietnam’s exported rice price (\(P_{t}^{Viet}\)), export price of India’s rice (\(P_{t}^{Ind}\)), producer price of rice (\(P_{t}^{Farm}\)), rainfall (\(R_{t}\)), and water storage (\(W_{t}\)). The data sets are monthly frequency data collected from M1/2006 to M9/2014, covering 105 observations. Then we transformed all series data to the log-log before moving to the estimation.

Estimation

In this study, the SUR model with two equations (M \(=\) 2) was employed to derive the demand and supply in the rice market. Thus, the bivariate Copula with continuous marginal distribution is conducted in the estimation. First of all, the Augmented Dickey–Fuller test was used to test the stationary of the data series. The estimation procedures SUR based Copula involved four steps. In the first step, the conventional SUR model was estimated using a maximum likelihood technique to obtain the initial values. In the second step, we constructed the SUR Copula likelihood using the chain rule, and we have

$$\begin{aligned} \frac{ { \partial }^{ 2 } }{ \partial { u }_{ 1 }\partial { u }_{ 2 } } F\left( { u }_{ 1 },{ u }_{ 2 } \right)&= \frac{ { \partial }^{ 2 } }{ \partial { u }_{ 1 }\partial { u }_{ 2 } } C\left( { F }_{ 1 }\left( { u }_{ 1 } \right) ,{ F }_{ 1 }\left( { u }_{ 1 } \right) \right) \nonumber \\&= f_{ 1 }\left( { u }_{ 1 } \right) f_{ 2 }\left( { u }_{ 2 } \right) c\left( f_{ 1 }\left( { u }_{ 1 } \right) f_{ 2 }\left( { u }_{ 2 } \right) \right) \end{aligned}$$
(17)

where \(u_{1}\) and \(u_{2}\) are the marginal assumption of normal or student-t distribution, \(f_{1}(u_{1})\) and \(f_{2}(u_{2})\) are normal functions of demand and supply equation and \(c(f_{1}(u_{1},f_{2}(u_{2})\) is a density function of Copula function. In this study, Gaussian, T, Frank, Clayton, Gumbel, Joe, and Ali-Mikhail-Haq (AMH) Copula are employed for constructing the joint distribution function of a bivariate random variable with the univariate marginal distribution.

Then, we took a logarithm to transform Eq. 13, we get

$$\begin{aligned} lnL={ \varSigma }_{ i=1 }^{ T }\left( lnl\left( { \theta }_{ 1 }|{ X }_{ d } \right) +lnl\left( { \theta }_{ 2 }|{ X }_{ s } \right) +ln{ f }_{ 1 }\left( { u }_{ 1 } \right) +ln{ f }_{ 2 }\left( { u }_{ 2 } \right) +lnc\left( { F }_{ 1 }\left( { u }_{ 1 } \right) ,{ F }_{ 2 }\left( { u }_{ 2 } \right) \right) \right) \end{aligned}$$
(18)

where \(lnl\left( { \theta }_{ 1 }|{ X }_{ d } \right) \) and \(lnl\left( { \theta }_{ 2 }|{ X }_{ s } \right) \) are the logarithm of the likelihood function of demand and supply equation, respectively, which can be defined as

$$\begin{aligned} lnL=-\frac{ T }{ 2 } ln\left( 2\pi \right) -\frac{ T }{ 2 } ln\left( \varSigma \right) \left( \frac{ 1 }{ 2\varGamma } \left( Y-X\beta \right) \prime \left( Y-X\beta \right) \right) \end{aligned}$$
(19)

And \(lnc(F_{1}(u_{1}),F_{2}(u_{2}))\) is a bivariate Copula density assumed for Gaussian, T, Frank, Clayton, Gumbel, Joe,and Ali-Mikhail-Haq Copula (See. Eq. 6–12).

In the third step, we used the maximum likelihood technique again to maximize the SUR Copula likelihood to obtain the final estimated parameters. And the last step involved fitting the SUR Copula likelihood by identifying the appropriate marginal distribution and Copula function. The minimum AIC and BIC were used to find the best-fit the combination between Copula function and marginal distribution.

Finally, we extended the estimated parameters which are obtained from the SUR based Copula to compute the following equations considering the intercept terms of demand and supply \(\left( { \hat{ \alpha } }_{ 1 },{ \hat{ \alpha } }_{ 2 } \right) \) and price elasticity of demand and supply, namely, \({ \hat{ \beta } }_{ 1 }\) and \({ \hat{ \delta } }_{ 1 }\), as follows:

$$ \begin{bmatrix} -{ \hat{ \beta } }_{ 1 }&1 \\ -{ \hat{ \delta } }_{ 1 }&1 \end{bmatrix}^{ -1 }\begin{bmatrix} { \hat{ \alpha } }_{ 1 } \\ { \hat{ \alpha } }_{ 2 } \end{bmatrix}=\begin{bmatrix} { \kappa }_{ 1 } \\ { \kappa }_{ 2 } \end{bmatrix} $$
$$ CS=\frac{ { \kappa }_{ 1 }\times \left( { \hat{ \beta } }_{ 1 }-{ \kappa }_{ 2 } \right) }{ 2 } $$
$$ PS=\frac{ { \kappa }_{ 1 }\times \left( { \kappa }_{ 2 }-{ \hat{ \delta } }_{ 1 } \right) }{ 2 } $$

Therefore, according to the above equations, in this paper the total social welfare (TS) can be simply measured as TS \(=\) CS \(+\) PS , where CS and PS are the consumer surplus and the producer surplus, respectively.

4 Empirical Results

Checking stationaries of the data used in this paper was done before getting started estimating results. We used Augmented Dickey–Fuller (ADF) unit root test as a tool, and the results are that all variables passed the test at level with probability equals to zero, meaning they are all stationary. Then the Copula-based SUR model were estimated.

4.1 Estimation Results for Demand and Supply

We used Copula in order to improve the SUR model. The estimation of SUR model has been improved by adding Copula density function into the likelihood and we called it SUR Copula likelihood (see Sect. 3). We found very often that the papers working on Copula usually assume any family of Copula for the joint distribution of variables before estimation, e.g., Parsa and Klugman [12], Kolev and Paiva [7], Leon and Wu [3], and Masarotto and Varin [9]. But in this paper we have tried to make things different by testing all possible Copula families and a marginal distribution of standardized residuals, then specifying the family and the marginal distributions which is best fit for the model. Therefore, the results are exhibited in the following table.

The result below is one of our motivations of this paper as we tried to not restrict the type of marginal distribution. Because, sometimes, the error or the marginal distribution of residuals might be non-normal and it might not be the same distributions when the the equations are different. For example, the SUR model which is the linear regression system and consists of several equations whose marginal distributions of residuals may be different. The advantage of Copula is that it can handle this problem, and the empirical results have already been shown as below.

Table 1 AIC and BIC criteria for model choice
Full size table

Table 1 shows AIC and BIC criteria for comparison of Copula families as well as the marginal distributions. To choose the best family and marginal distribution for our data, we will look at the minimum AIC and BIC criteria and as we can see, the minimum AIC and BIC are –1480.577 and –1480.301, respectively as underlined. So we should choose Gaussian as the most appropriate Copula for our data, and the most suitable marginal distributions are the normal distribution for demand and student-t distribution for supply.

Since we believe that Copula could improve the SUR model, we tried to compare the results between the Copula-based SUR model, which is our concern, and the conventional SUR model -estimated by a generalized least squares (GLS)- by looking at the sum of squared errors. We link the marginal distributions of demand and supply together using Gaussian Copula function (see Table 1) and the estimated results for demand and supply of the rice market using the Copula-based SUR model are shown in Table 2. The model performed very well, as we can see obviously that the results of the Copula-based SUR model are significant at level of 0.01 and the coefficient of all variables estimated from this model seem to make sense of the key point of demand and supply theory. Recall, as we know from the theory, the demand will be reduced if the price increases. On the other hand, the supply will do the opposite to the demand. It will be increased if the price increases.

Table 2 Estimation of SUR based copula for demand and supply in comparison with SUR estimated by a generalized least squares
Full size table

To compare with the conventional SUR model, we estimated the SUR model using one of the well-known estimators for SUR model [10] that is the generalized least squares. The results are shown in Table 2 beside the Copula-based SUR model. We can see that for each variable there is not much difference between the coefficients estimated by the Copula-based SUR and the conventional SUR, but the Copula-based SUR model has a lower value of standard error (SE) for all estimated parameters compared with the result of the conventional SUR. It gives us a possibility that Copula can improve the efficiency of estimation, since SE for all estimated parameters are lower. Because of the advantage of Copula, the covariance estimation is improved. The marginal distributions are not restricted anymore; it can be any distribution and not need to be the same for the different equations -here are demand and supply equations- because Copula can link them together. Unlike the case of SUR model estimated by GLS, the distribution is usually assumed to be normal. If we consider the model comparison and take into account the sum of squared errors (SSE) as a tool, we will see that the Copula-based SUR as a whole is slightly better than the conventional SUR for having lower SSE of both demand and supply equations. It may confirm our belief that Copula can provide an improvement in SUR model.

Fig. 1
figure 1

Residuals of demand (top left) and supply (top right) equations of the Copula-based SUR and Residuals of demand (bottom left) and supply (bottom right) of the conventional SUR

Full size image

A comparison of residuals of demand and supply equations between the Copula-based SUR model and the conventional SUR model are illustrated in Fig. 1. These residuals exhibit homogeneity and independence. Figure 1 shows that the residual plots of these two models have a similar structure of residuals. By the assumption of the conventional SUR model, both demand and supply equations are assumed to have a normal distribution for error terms. Figure 1 shows that the structure of residuals seem to correspond well to the assumption. The residual plot of supply equation of the Copula-based SUR model (top right) provides the same structure as the plot of the conventional SUR (bottom right) since both of them have the same distribution of residuals, which is a normal distribution, whereas the residual plot of demand equation of the Copula-based SUR (top left) is slightly different from the plot of the conventional SUR (bottom left) since the distribution of residual of demand equation of the Copula-based SUR is student-t (see Table 1)

According to the results, by using this dataset, the Copula-based SUR model provides the similar result to the conventional SUR model (see Table 2 and Fig. 1). It implies that the conventional SUR is acceptable for this dataset. Nevertheless, it would be more desirable to relax the normal assumption of error terms of the SUR model when there is an existence of non-normal distribution in error terms. Even though the estimated results of these two models are not much different, some improvement in SUR model is still be found. Considering the result shown in Table 2, standard errors for all estimated parameters of the Copula-based SUR model are slightly lower than the conventional SUR. As well as the comparison of SSE where the Copula-based SUR model is slightly better than the conventional SUR for having lower SSE. It is possible to conclude that Copula can provide an improvement in SUR model by relaxing that strong assumption.

4.2 Application: Welfare Measurement for Thai Rice Market

According to the results from the previous part, we got the demand and supply equations estimated by the Copula-based SUR model. Then we take into account equilibrium of the market to find the price at the equilibrium point or the market price. Since welfare, which is our concern, can be measured by summing up of consumer surplus (CS) and producer surplus (PS) where both surpluses are related to the equilibrium price level. The consumer surplus is the amount of the willingness to pay above the equilibrium price under the demand curve and the producer surplus is the amount of the willingness to produce or to sell below the market equilibrium price above the supply curve. By the definition, the estimated results for the total welfare of the rice market can be shown in Table 3. The table shows that the amount of overall welfare of the rice market is equal to 5.8277(\(10^{-6}\)) which is the sum of consumer surplus, 1.3682(\(10^{-6}\)) and producer surplus, 4.4595(\(10^{-6}\)).

Table 3 Welfare measurement
Full size table

5 Conclusions

The Copula-based SUR model were introduced in this paper as an alternative to the conventional SUR model, in which requires the assumption of normal distribution of error terms. The Copula density function has been applied into the likelihood to not restrict the marginal distributions, and to relax the normality assumption of errors in the model. The results have shown that Gaussian Copula is the most appropriate function for being the linkage between the marginal distributions where the normal and the student-t distributions are the best fit for the marginal distributions of demand and supply, respectively. The real dataset of Thai rice market is used for making a comparison between the conventional SUR model estimated by GLS and the Copula-based SUR model. The Copula-based SUR model seems to perform slightly better on the ground of having lower standard errors and sum of squared errors. However, it is not much different in the sum of squared errors between both models. This might imply that the conventional SUR model is adequate for this dataset. Nevertheless, our research results might confirm our belief that Copula could improve the SUR model in term of relaxing the normal distribution assumption of error terms.