The steps of SUR(Seemingly Unrelated Regression) Method in Eviews

The purpose of this "article" to show the steps of SUR Method in Eviews. I will show all steps of SUR method on an emprically. Suppose that we want to estimate the Phillips curve for the Denmark, Fance and Germany. So I take the data about the inflation and unemployment for 2000-2010. The datasets of variables are presented in Table 1.

Table 1: Iflation and Unemployment data sets of Germany, Denmark and France

Years	Inflation			Unemployment
	Germany	Denmark	France	Germany	Denmark	France
2000	1.471	2.925	1.6999	7.7	4.5	10.2
2001	1.978	2.35	1.63	7.8	4.2	8.6
2002	1.402	2.426	1.917	8.6	4.6	8.7
2003	1.043	2.091	2.109	9.3	5.4	8.6
2004	1.669	1.16	2.135	10.3	5.5	9.2
2005	1.557	1.809	1.736	11.1	4.8	8.9
2006	1.575	1.89	1.684	10.3	3.9	8.9
2007	2.289	1.714	1.488	8.6	3.8	8
2008	2.631	3.339	2.814	7.5	3.3	7.4
2009	0.313	1.326	0.088	7.7	6	9.1
2010	1.137	2.298	1.53	7.1	7.4	9.3

Figure 1 shows the theoretical form of the Phillips curve.

Figure1: Phillips Curve

In Figures 2, 3 and 4, the Phillips curve for Denmark, France and Germany is shown.

Figure 2: Phillips Curve of Denmark

Figure 3: Phillips Curve of France

Figure 4: Phillips Curve of Germany

The first differences are taken because all variables contain unit root.(that is, all variable are non-stationary). 

Model

Suppose that, the Phillps curve for three countries are modeled as below.

INFDEN = β₁+ β₂*UNDEN+u_1t             (1)

INFFR= β₁+ β₂*UNFR+u_2t                           (2)

INFGER = β₁+ β₂*UNGER+u_3t                 (3)

ADF test was conducted to investigate whether the variables have unit root. Since all variables have unit roots in level values, the first difference of all variables is taken and the results are shown in Table 3.

Table 2: Unit root test of variables

ADF test(trend and intercept)
Inf_Den			D(Inf_Den)
1%	-5.29	-3.51 (0.09)	-5.83	-6.23 (0.02)
5%	-4.00		-4.24
10%	-3.46		-3.59
Un_Den			D(Un_Den)
1%	-5.52	-2.1 (0.477)	-5.52	-4.75 (0.03)
5%	-4.10		-4.10
10%	-3.51		-3.51
Inf_Fr			D(Inf_Fr)
1%	-5.29	-3.02 (-0.13)	-5.52	-7.16 (0.002)
5%	-4.00		-4.10
10%	-3.46		-3.51
Un_Fr			D(Un_Fr)
1%	-5.52	-2.24 (-0.419)	-5.52	-4.82 (0.04)
5%	-4.10		-4.10
10%	-3.51		-3.51
Inf_Ger			D(Inf_Ger)
1%	-5.52	-3.43 (0.117)	-5.83	-4.91 (0.03)
5%	-4.10		-4.24
10%	-3.51		-3.59
Un_Ger			D(Un_Ger)
1%	-5.52	-2.22 (0.42)	-5.83	-5.13 (0.04)
5%	-4.10		-4.24
10%	-3.51		-3.59

Thus, the model shown by equations (1-3) is transformed into the model shown by equations (4-6).

DINFDEN = β₁+ β₂*DUNDEN+u_1t             (4)

DINFFR= β₁+ β₂*DUNFR+u_2t                           (5)

DINFGER = β₁+ β₂*DUNGER+u_3t                 (6)

If there is a relation among the error terms of these models, the models can not be estimated by the least squares method. Seemingly Unrelated Regression method developed by Zelner  are used in such cases.

Step 1: Estimate the models separtely with OLS method

All models are estimated separately by least squares method and the results are presented in Table 2.

Table 3: The results of Models

    

       *The t statistics of the coefficients are shown in parentheses.

According to Phillips' law, there is a negative relationship between inflation and unemployment. As can be seen from the coefficients of the unemployment variable, the results overlap with the economic theory. In other words, unemployment coefficients are negative in all models. But the coefficients are statistically insignificant. Because the number of observations is not as large as for estimating the model. It is useful to make SUR estimates before OLS results are used. My goal is not to achieve a perfect model from an economical and econometric point of view, but only to show the estimating stages of the SUR method. Thus, I am not interested in whether the error term of the model provides the assumptions of autocorrelation, heteroskedasticity and normality.

Step 2: Estimate the models with SUR method

Since the SUR method is a system approach, the sample model shown by equation 4-6 is solved with the system of simultaneous equations. Following the model's OLS estimate, the path shown below is followed.

Each model equation is copied and pasted into the system window. Three C (1) and C (2) coefficients will appear in the system window. If prediction is made in this way, only two coefficients will be estimated. To prevent this situation, the second and third coefficients C (1), C (2) are replaced by coefficients C (3), C (4) and C (5), C (6).

The estimated model results by the SUR method are presented in Table 4.

Table 4: The results of SUR Method.

As seen from the estimation results by the SUR method, unemployment coefficients are in line with the economic theory and  negative. Note that the unemployment coefficients in the model predicted by the SUR method are smaller than the unemployment coefficients estimated by the OLS method. However, the estimated model coefficients by the SUR method are again statistically insignificant. The most basic reason for this is that the number of observations used in model estimation is very small.

Step 3: Calculate the Correlation and Variance-Covariance Matrices

The next step in reaching these results is to choose between OLS or SUR estimation methods. Simultaneous covariance testing is required to investigate whether there are any correlations between SUR errors.  For the simultaneous covariance test, r(ij) values are calculated. Firstly variance-covariance and correlation matrices are calculated from the errors obtained from the SUR method. Correlation and variance-covariance matrices are presented in Table 5 and Table 6, respectively.

Table 5: Correlation Matrix

Table 6: Variance-Covariance Matrix

Step4: Set up hypothesis tests

    H₀: OLS method is appropriate- There is no relationship between models' errors

 H₁: SUR method is appropriate- There is a relationship between models' errors

Step 5: Calculate chi-square value

The chi-square value is calculated as follows.

Compare the calculated chi-square value with the chi-square table value. With three-degree-of-freedom  and the 95% significance chi-square-table value equals to 7.81.

According to this result, the null hypothesis can be rejected. Thus, it is concluded that there is a relation between the error terms of the models given by equations (3-6). Therefore, it is decided that these models should be estimated by the SUR method, not by OLS.

Whether the error terms have autocorrelation have been tested by the Portmanteau autocorrelation test and correlograms. The results are shown Table 7 and Table 8 respectively

Table 7: Portmanteau tests for Autocorrelations

Table 8:Correlograms of Residuals

There is no autocorrelation relation between error terms according to Portmanteau test result and correlograms. Because of, all prob values of Portmanteau test are bigger than 0.05 and correlograms showing the graphical representation of autocorrelation are in the range of no autocorrelation.

Note: There are tight economic relations between them, as they are 3 members in the European Union. This is the biggest reason why the error terms are related.