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Practical Guide ForCross Section Data AnalysisUsing EViews

I Gusti Ngurah Agung

Graduate School of ManagementFaculty of Economics University Of Indonesia

Ph.D. in Biostatistics andMSc.in Mathematical Statistics fromThe University of North Carolina at Chapel Hill

______________________________________________________________________Cross Section and Experimental Data Analysis Using EViews I Gusti Ngurah Agung 1© 2011 John Wiley & Sons (Asia) Pte Ltd

1Data Transformation

It is recognized that the very basic analysis based on any cross-section data set is to study the differences between the studied objects, either as the individuals or groups of individuals, which should have been known to have different or distinct characteristics.

For this reason, the classification analysis should be a very important part in the cross-section data analysis, with the very basic data analysis is the data analysis based on a zero-one problem indicator. For this reason, this chapter presents how to generate a dummy variable, since the dummy variables are widely used in various statistical models based on all cross-section data sets.

By selecting Quick/Generate Series… , the a dummy variable as well as a categorical variable can easily be generated using the block-copy-paste method of the equation specifications presented below to the dialog or window.

1.1 Generating Dummy Variables

1.1.1 Dummy Variables For A Single Categorical Variable

The dummy variable of k-th category of a variable V, namely DVk, for k=1,…,K, can be generated using the equation as follows:

DVk=1*(V=k) or Dk=1*(V=k) (1.1)

1.1.2 Ordinal Categorical Variables Based On A Numerical Variables

An categorical variable should be generated using the alternative methods as presented in Chapter 2, and then the dummy variables can easily be generated using the formula (1.1). For example, the ordinal categorical variable having k-categories can be generated based on a numerical variable X, namely CkX, using the following equation specification.

C2X = 1 + 1*(X>=a)C3X = 1 + 1*(X>=a) + 1*(X>=b)C4X = 1 + 1*(X>=a) + 1*(X>=b) + 1*(X>=c)C5X = 1 + 1*(X>=a) + 1*(X>=b) + 1*(X>=c) + 1*(X>=d) (1.2)

where a < b < b < c < d should be intentionally selected numbers, by using the following alternative statistics.


1). Using nonparametric statistics, such as the quantiles of the variable X, by using the function @quantile(X,q).

For an example, a = @quantile(X,0.30), and b=@quantile(X,070), to generate C3X.2). Using parametric statistics, such as the mean and standard deviation of the variable X,

which can be computed as m = @mean(X) and sd = @stdev(X). For an example, to generate C4X, a =m -1.5*sd, b = m, and c=m+1.5*sd3). On the other hand, one may used the Z-score of X, namely ZX, having m=0 and sd=1.

For an example, to generate C4ZX, a = -1.5*sd, b = 0, and c=+1.5*sd4). Using personal judgment of the researcher based on the data set available.

1.1.3 Dummy Variables For Two-Way Tabulation

Based on a two-way tabulation, say an IxJ- table of the factors A and B, it is suggested the dummy variables should be presented as the dummy-cells, namely DCij, for i=1,…,I and j=1,…,J by using the following equation, which can easily be extended to N-way tabulation.

DCij =1*(A=i and B=j) (1.3)

And based on a IxJxK-table of the factors A, B and C, it is suggested to generate the dummy variables using the following equation.

DCijk =1*(A=i and B=j and C=k) (1.4)

1.2 Generating a Cell-Factor (CF)

Since EViews 6 provides a function, namely @Expand(CF), which can directly transform the cell factor CF into its dummy variables, then it is suggested to generate a cell-factor in doing analysis based on a N-way tabulation. For the illustration find the following cell-factors, which can easily be extended for IxJ and IxJxK tabulations, as well as for N > 3.

1.2.1 Based on a 2x2-Tabulation of the factors A and B

CF=11*(A=1 and B=1)+12*(A=1 and B=2)+21*(A=2 and B=1)+22*(A=2 and B=2) (1.5)

1.2.2 Based on a 3x2-Tabulation of the factors A and B

CF=11*(A=1 and B=1)+12*(A=1 and B=2)+21*(A=2 and B=1) +22*(A=2 and B=2)+31*(A=3 and B=1)+32*(A=3 and B=2) (1.6)


1.2.3 Based on a 2x2x2-Tabulation of the factors A, B and C

CF=111*(A=1 and B=1 and C=1)+112*(A=1 and B=1 and C=2)+121*(A=1 and B=2 and C=1)+122*(A=1 and B=2 and C=2)+211*(A=2 and B=1 and C=1)+212*(A=2 and B=1 and C=2)

+221*(A=2 and B=2 and C=1)+222*(A=2 and B=2 and C=2) (1.7)

2Single-Factorial Regression Models

In this case, for the data analysis, it is considered a cell-factor CF which can be a single factor or generated based on two or more categorical factors, a set of exogenous variables or covariates, namely X1, X2, … , XK, where Xk for each k=1,…,K; can be a main factor, two- or three-way interactions, and Y is an endogenous numerical, zero-one or ordinal variable.

As an extension, if there is a classification factor generated based on an exogenous numerical covariate, then the regression would be a piecewise regression model. Each of the equation specification can be used to conduct the analysis based on several estimation settings as follows:

1). For the numerical variable Y, the estimation setting is the LS-Least Squares.2). For the zero-one variable Y, the estimation setting is the BINARY Choice.3). For the ordinal variable Y, the estimation setting is the ORDERED Choice, with a

small modification on the intercept parameter.4). For the censored observation Y, the estimation setting is the CENSORED.

Note that everyone can easily modify each equation specification by using the transformed of the endogenous or exogenous numerical variables in order to have alternative models, such as polynomial, semi-logarithmic, and trans-log linear or quadratic models, as well as the bounded regression models by using an independent variable log((Y-L)/(U-Y)) where L and U, respectively, are the subjectively selected lower and upper bounds of the variable Y.

The data analysis can easily be conducted by selecting Quick/Estimate Equation…, and then using the block-copy-paste method to insert each of the following equation specification to the dialog or window. Finally, having the result on the screen, various corresponding hypotheses can easily be tested using the Wald test, as well as additional analysis, the residual analysis in particular.

2.1 One-Way ANOVA Models ______________________________________________________________________Cross Section and Experimental Data Analysis Using EViews I Gusti Ngurah Agung 4© 2011 John Wiley & Sons (Asia) Pte Ltd

The main objectives of an ANOVA model are to test various hypotheses on the mean differences between the levels of a single factor or the cells generated by two or more (nominal) categorical factors. In order to write the statistical hypotheses based on each model, a table of the model parameters should be constructed. Refer to the corresponding table presented in the main book!

2.1.1 One-Way ANOVA Model without Intercept or “C”

Y @Expand(CF) (2.1)

2.1.2 One-Way ANOVA Model with an Intercept or “C”

Y C @Expand(CF,@Dropfirst) (2.2)

Y C @Expand(CF,@Droplast) (2.3)

Y C @Expand(CF,@Drop(*)) (2.4)

where (*) indicates a level of the cell-factor CF. For examples, an integer for a single factor, namely (k), a pair of integers for a two-way tabulation, namely (i,j), and (i,j,k) for the cell-factor of a three-way tabulation.

2.1.3 Alternative Methods

1). Having the data of the variable Y on the screen, various statistics and testing hypotheses can be derived, such as follows:

1.1 By selecting View/Descriptive Statistics &Tests, Figure 2.1(a) shown on the screen, which shows eight options, namely the graph with several alternative options, six options for the Descriptive Statistics & Test, and One-Way Tabulation, for the cross-section data. Everyone can easily obtain each output. Note that descriptive statistical summaries are very important part of all evaluation studies. Refer to Chapter 2.

Figure 2.1 The Options for the Data Analysis based on a Single Variable


1.2 In addition, numerical variable(s) also can be inserting as the series for classify. In this case the “Max # of bins:” should be taken into account.

2). Having the data of both Y and CF, specifically if Y is a zero-one or ordinal variables, on the screen various statistics and testing hypotheses can be derived, such as follows:

2.1 By selecting View, Figure 2.2 shown on the screen, and then by selecting N-Way Tabulation, and the output options…OK, the statistical results, specifically the descriptive statistical summary and the Chi-square statistics, are obtained. Refer to Chapter 2.

Figure 2.2 The Options for the Data Analysis based on a Set of Variablesw

Figure 2.3 The Options for the Data Analysis based on a Pair of Variables

2.2 Specifically for CF is an ordinal variable, by selecting Covariance Analysis…, the option in Figure 2.3(a) shown on the screen, and then by selecting the Kendall’s tau method, Figure 2.2(b) shown on the screen.

2.3 Finally by selecting the options Kendall’s Score S/Probability |S|=0 …OK, the nonparametric statistical results are obtained for testing a numeric-ordinal variables association. In order words, to test the effect of an ordinal variable CF on a numerical


Y. In fact, the Kendall’s tau also can be applied for the bivariate numerical variables. Note that the Spearman rank-order also can applied.

2.2 One-Way ANCOVA Models

The main objectives of an ANCOVA model, or an homogeneous regression model, are to test various hypotheses on the adjusted-means differences between the levels of a single factor or the cells generated by two or more (nominal) categorical factors. In order to write the statistical hypotheses based on each model, a table of the model parameters should be constructed. Refer to the corresponding table presented in the main book! Then everyone should be able to write the equation of the regression within all levels or cells.

In addition, the hypotheses on various types of the effects of the covariate(s) on the dependent variable should be theoretically defined, and then can be tested. 2.2.1 One-Way ANCOVA Model without Intercept or “C”

Y X1 . . . XK @Expand(CF) (2.5)

Note that the list of the variables, specifically the covariates, is presented exactly the same as the list in the output. For a comparison, even though “Y @Expand(CF) X1 … XK” is used as the equation specification, the output will present the model parameters as C(1), C(2), … in the ordering of the list in the equation specification (2.5).

2.2.2 One-Way ANCOVA Model with an Intercept or “C”

Y X1 . . . XK C @Expand(CF,@Drop(*)) (2.6)

Note that to generalize, a covariate Xk can be either a main factor, two- or three-way interaction-factors. For an example, for the covariate X1 and X2, the following ANCOVA model would be considered. In this case, it is theoretically defined that the effect of X1 (X2) on Y depends on X2 (X1).

Y X1 X2 X1*X2 C @Expand(CF,@Droplast) (2.7a)

For an illustration Table 2.1 presents the parameters of the model in (2.7a) for a CF having four levels.

Table 2.1 The Parameters of the Model in (2.7a) for a CF having four levelsX1 X2 X1*X2 CF=1 CF=2 CF=3 CF=4

C(1) C(2) C(3) C(4)+C(5) C(4)+C(6) C(4)+C(7) C(4)

For a comparison Table 2.2 presents the parameters of the following equation specification for a CF having four levels.


Y X1 X2 X1*X2 C @Expand(CF,@Dropfirst) (2.7b)

Table 2.2 The Parameters of the Model in (2.7b) for a CF having four levelsX1 X2 X1*X2 CF=1 CF=2 CF=3 CF=4

C(1) C(2) C(3) C(4) C(4)+C(5) C(4)+C(6) C(4)+C(7)

Note that in practice, a nonhierarchical ANCOVA model could be a good fit model, or an additive model having only the main factor X1 and X2, which is highly dependent on the data set used.

Furthermore, for the covariate X1, X2, and X3, the hierarchical ANCOVA model would be as follows, where X1, X2 and X3 are defined to have a complete association. The data analysis based on a reduced model can easily be done by using the block-copy-paste method of this equation.

Y X1 X2 X3 X1*X2 X1*X2 X2*X3 X1*X2*X3 C @Expand(CF,@Drop(*)) (2.8)

2.2.3 Suggested Additional Analysis

By having a set of numerical variables of Y and covariates, it is suggested to conduct the bivariate correlation analysis, namely Covariance Analysis in EViews, using the options presented in Figure 2.3.

Note that the impact of the multicollinearity between the independent variables of any model is unpredictable, and the output can present unexpected parameter estimate(s). Refer to the special notes and comments in Agung (2009, Section 2.14.2).

Finally, for a more advanced data analysis, everyone may consider in conducting the residual analysis, and find the following special illustrations or findings. Refer to various examples in Agung (2009).

Example 2.1 (A Special Illustration). It is recognized that the following three alternative analyses based on a pair of numerical variables X and Y, which will show exactly the same values of the t-statistics.

1). The correlation of (X,Y) by using EViews 6, 2). The OLS regression of Y on X, using the equation specification “Y C X, and 3). The OLS regression of X on Y, using the equation specification “X C Y”.

These findings indicate that the correlation analysis is sufficient for testing the causal linear effect of X and Y, including their simultaneous causal linear effects, which should be theoretically defined. In other words, the testing hypothesis should not be used to prove that X and Y have causal effects.

Example 2.2 (Another Special Illustration). It is recognized that the following two ANCOVA model of Y on X, and ANCOVA model of X on Y, also give the same t-


statistics for testing the hypothesis on the effect of the independent numerical variable on the dependent variable, adjusted for the cell-factor CF.

Y X @Expand(CF) or Y C X @Expand(CF,@Drop(*)) (2.9a)

X Y @Expand(CF) or X C Y @Expand(CF,@Drop(*)) (2.9b)

2.3 Heterogeneous Regression Models

The main objectives of an heterogeneous regression model are to test various hypotheses on the slopes differences between the levels of a single factor or the cells generated by two or more (nominal) categorical factors. In other words, to test various hypotheses on the differences of the effect of a covariate or numerical independent variable, adjusted for the other independent variables. In order to write the statistical hypotheses based on each model, a table of the model parameters should be constructed. Refer to the corresponding table presented in the main book! Then everyone should be able to write the equation of the regressions within all levels or cells.

Note that the intercepts differences of the regressions should not be taken into consideration, more over for testing hypothesis on their differences.

2.3.1 Heterogeneous Regression Model without Intercept

Y @Expand(CF) X1*@Expand(CF) . . . XK*@Expand(CF) (2.10)

2.3.2 An Alternative Heterogeneous Regression Model

Y X1… XK @Expand(CF) X1*@Expand(CF,@Dropfirst) . . . XK*@Expand(CF,@Dropfirst) (2.11)

2.3.3 Heterogeneous Regressions Using Dummy Variables

It is recognized that an heterogeneous regressions would have different sets of independent variables within the defined levels of CF. For this reason, it is suggested to using the dummy variables if and only if the regression models or their reduced models within the levels of CF have different sets or exogenous variables. For an example, based on the following full model, everyone may have various good fit reduced models which are highly dependent on the data sets used. Note that to generalize, a covariate Xk = Xk

can be either a main factor, two- or three-way interaction-factors.

Y = (C(10)+C(11)*X1 +C(12)*X2+C(13)*X3+C(14)*X4+C(15)*X5)*D1+(C(20)+C(21)*X1 +C(22)*X2+C(23)*X3+C(24)*X4+C(25)*X5)*D2+(C(30)+C(31)*X1 +C(32)*X2+C(33)*X3+C(34)*X4+C(35)*X5)*D3+(C(40)+C(41)*X1 +C(42)*X2+C(43)*X3+C(44)*X4+C(45)*X5)*D4 (2.12)


Note that this equation specification can easily be modified for a cell-factor CF having any number of levels, as well as any sets of independent variables, by using the block-copy-paste method. Furthermore, it is suggested to save the full model and all its reduced models should be using the same symbols of the parameters. In other words, an independent variable Xk will have the same symbol C(ik), i = 1,2,3 or 4, in the full and reduced models.

2.4 Alternative Regression Models

As the extension of the models presented above, several alternative regression models, either linear or nonlinear models, can easily be defined based on a set of numerical variables, as presented in Table 2.3. Corresponding to these models the following remarks are presented.

Table 2.3 Equations Specifications of Alternative Regression Models based on a Set of Numerical Variables

Dept.Var No. Independent Variables

Y, Log(Y),Log(Y-L),Log(U-Y), orLog((Y-L)/(U-Y))

Single Exogenous Variable X1 C X X^2 … X^k2 C(1) + C(2)*(X-a)^2*(X-b)3 C log(X)4 C log(X) log(X)^2 … log(X)^k

Two Exogenous Variables X1 and X25 C X1 X2 X1*X26 C log(X1) log(X2)7 C log(X1) log(X2) log L(X1)^2 log(X1)*log(X2) log(X2)^28 C log(X1) log(X2) (log(X1) - log(X2))^2

Nonlinear Models3a1 = C(1)*X^C(2)3a2 = C(1)+C(2)*X^C(3)6a1 = C(1)*X1^C(2)*X2^C(3)6a2 = C(1)*X1^C(2)*X2^C(3) + C(4)7a1 = C(1)*(C(2)*X1^C(3)+(1-C(2))*X2^C(3))^(-r/C(3))7a2 = C(1)*(C(2)*X1^C(3)+(1-C(2))*X2^C(3))^(-r/C(3))+C(4)

1). The regression models in Table 2.3 can be viewed as the models without the cell-factor as an independent variable, which can easily be extended to the models having any multivariate exogenous variables. However, everyone should be aware on the unpredictable impact of the multicollinearity of the independent variables on the parameter estimates. Refer to Section 2.14.2 in Agung (2009). Even in some cases, an error message could be obtained, especially for nonlinear models.


2). Note that L and U, respectively are the lower and upper bounds of Y, which are subjectively selected. On the other hand, a and b are specific selected numbers which are related to the predicted extreme values of the corresponding independent variable.

3). The nonlinear models 3a1 and 3a2 are related to the trans-log linear model-3, which could be the same as the Cobb-Douglas production function with an input variable. Similarly, the non-linear models 6a1 and 6a2 are related to the Cobb-Douglas production function with two input variables. Refer to Chapter 10 in Agung (2009a)

3). The nonlinear models 7a1 and 7a2 are related to the CES (Constance Elasticity of Substitution) production function, which can be approximated by using the trans-log quadratic model-7.

4). The trans-log linear model-8 is a special case of the model-7.5). By using cell-factor CF as an additional independent variable, then various ANCOVA

and Heterogeneous Regression Models, can easily be defined or derived, as the models previously presented.

6). If the cell-factor is generated based on one or two exogenous numerical variables, then the discontinuous regression models, either peace-wise or step regressions, would be obtained.

7). By following the equation specification in (2.12), everyone can easily applied various sets of exogenous variables as well as models within each levels of a defined cell-factor.

8). As a further extension of the models, specific for peace-wise or step regression models, different transformed variable of Y can be applied within each peace or level of the cell-factor CF, where CF is generated based on one or two numerical exogenous variable. For an example, Y_New = log(Y) for CF=1, and Y_New = log((Y-L)/(U-Y)) for CF = 2. Refer to Agung (2009a).

3Bi-Factorial Regression Models

In this case, it is considered two categorical factors A and B, a set of covariates, namely X1, X2, … , XK, where Xk for each k=1,…,K; can be a main factor, two- or three-way interactions, and Y is an endogenous numerical, zero-one or ordinal variable. In fact, these models can be viewed as the one-way regression models. Thence, all notes presented above are valid for these models.

The objective of this chapter is to present alternative equation specifications, which should be considered as more complex equations. However, they have advantages for doing the testing hypotheses, since their statistical results present the test-statistic, such as the t-statistic or Z-statistic, for the hypotheses based on each selected model. Refer to the main book!


3.1 Two-Way ANOVA Models

The alternative equation specifications are as follows:

3.1.1 The Simplest Two-Way ANOVA Model without Intercept “C”

Y @Expand(A,B) (3.1)

3.1.2 Two-Way ANOVA Model with an Intercept

Y C @Expand(A,B,@Drop(*,*)) (3.2)

where (*,*) indicates a cell generated by the two factors A and B.

3.1.3 Alternative Two-Way ANOVA Models

Y C @Expand(A,@Dropfirst) @Expand(A)*@Expand(B,@Droplast) (3.3)

Table 3.1 The Parameters of the Model in (3.3) for a 4x2 TabulationB=1 B=2 Diff. B(1-2)

A=1 C(1)+C(5) C(1) C(5)A=2 C(1)+C(2)+C(6) C(1)+C(2) C(6)A=3 C(1)+C(3)+C(7) C(1)+C(3) C(7)A=4 C(1)+C(4)+C(8) C(1)+C(4) C(8)

3.2 Two-Way ANCOVA Models

The alternative equation specifications are as follows: 3.2.1 The Simplest Two-Way ANCOVA Model without Intercept

Y X1 . . . XK @Expand(A,B) (3.4) 3.2.2 A Two-Way ANCOVA Model with an Intercept

Y X1 . . . XK C @Expand(A,B,@Drop(*,*)) (3.5)

3.2.3 An Alternative Two-Way ANCOVA Model

Y X1 … XK C @Expand(A,@Dropfirst) @Expand(A)*@Expand(B,@Droplast) (3.6)


Note that to generalize, a covariate Xk can be either a main factor, two- or three-way interaction-factors. For an example, for the covariate X1 and X2, the following hierarchical ANCOVA model would be considered.

Y X1 X2 X1*X2 C @Expand(A,@Dropfirst) @Expand(A)*@Expand(B,@Droplast) (3.7)

Table 3.2 The Parameters of the Model in (3.7) for a 4x2 TabulationX1 X2 X1*X2 B=1 B=2 Diff. B(1-2)

C(1) C(2) C(3)A=1 C(4)+C(5) C(4) C(5)A=2 C(4)+C(6)+C(9) C(4)+C(6) C(9)A=3 C(4)+C(7)+C(10) C(4)+C(7) C(10)A=4 C(4)+C(8)+C(11) C(4)+C(8) C(11)


The alternative equation specifications are as follows:

3.3.1 Heterogeneous Regression Model without Intercept

Y @Expand(A,B) X1*@Expand(A,B) . . . XK*@Expand(A,B) (3.8)

For an example, corresponding to the ANCOVA model (3.7), the heterogeneous regression model has the following equation specification.

Y @Expand(A,B) X1*@Expand(A,B) X2*@Expand(A,B)X1*X2*@Expand(A,B) (3.9)

Table 3.3 The Parameters of the Model in (3.9) for a 2x2 TabulationA B Intercept X1 X2 X1*X21 1 C(1) C(5) C(9) C(13)1 2 C(2) C(6) C(10) C(14)2 1 C(3) C(7) C(11) C(15)2 2 C(4) C(8) C(12) C(16)

3.3.2 An Alternative Heterogeneous Regression Model without Intercept

Y X1 … XK @Expand(A,B) X1*@Expand(A,B,@Drop(*,*)) . . . XK*@Expand(A,B,@Drop(*,*)) (3.10)

For an example, corresponding to the model (3.9), the heterogeneous regression model has the following alternative equation specification.


Y X1 X2 X1*X2 @Expand(A,B) X1*@Expand(A,B,@Drop(2,2)) X2*@Expand(A,B,@Drop(2,2)) X1*X2*@Expand(A,B,@Drop(2,2)) (3.11)

Table 3.4 The Parameters of the Model in (3.11) for a 2x2 TabulationA B Intercept X1 X2 X1*X21 1 C(4) C(1)+C(8) C(2)+C(11) C(3)+C(14)1 2 C(5) C(1)+C(9) C(2)+C(12) C(3)+C(15)2 1 C(6) C(1)+C(10 C(2)+C(13) C(3)+C(16)2 2 C(7) C(1) C(2) C(3)

3.3.2 Heterogeneous Regressions Using Dummy Cells

It is recognized that an heterogeneous regressions would have different sets of independent variables within the defined cells. For this reason, it is suggested to using the dummy variables, namely the dummy-cell DCij , if and only if the regression models or their reduced models should have different sets or exogenous variables, within the cells (A=i,B=j).

For an example, based on the following 2x2 factorial heterogeneous regression model, everyone may have various good fit reduced models which are highly dependent on the data sets used. Note that to generalize, a covariate Xk can be either a main factor, two- or three-way interaction-factors.

Y = (C(10)+C(11)*X1 +C(12)*X2+C(13)*X3+C(14)*X4+C(15)*X5)*D11+(C(20)+C(21)*X1 +C(22)*X2+C(23)*X3+C(24)*X4+C(25)*X5)*D12+(C(30)+C(31)*X1 +C(32)*X2+C(33)*X3+C(34)*X4+C(35)*X5)*D21+(C(40)+C(41)*X1 +C(42)*X2+C(43)*X3+C(44)*X4+C(45)*X5)*D22 (3.12)

For a comparison, note that this equation specification, in fact, has the same form as the equation specification in (2.12). Thence, this equation specification also can easily be modified for a cell-factor generated by any sets of categorical factors, as well as any sets of independent variables, by using the copy-paste method.

Following the subscript of DCij, the equation specification can be presented as in (3.13) using C(ijk), k=0,…,5 as the symbol of the model parameters. Thence, it can be easier to identify in writing any hypotheses within a selected cell or between cells.

Y = (C(110)+C(111)*X1 +C(112)*X2+…+C(115)*X5)*D11+(C(120)+C(121)*X1 +C(122)*X2+…+C(125)*X5)*D12+(C(210)+C(211)*X1 +C(212)*X2+…+C(215)*X5)*D21+(C(220)+C(221)*X1 +C(222)*X2+…+C(225)*X5)*D22 (3.13)

4Multi-Factorial Regression Models______________________________________________________________________Cross Section and Experimental Data Analysis Using EViews I Gusti Ngurah Agung 14© 2011 John Wiley & Sons (Asia) Pte Ltd

4.1 Suggested Equation Specification

For any multi-factorial regression models, it is suggested to apply an equation specification, where SF indicates a single selected factor between the set of factors considered, and CF is the cell-factor generated by the other factors. Thence all equation specifications for the models with the two factors A and B presented in Chapter 3 can easily be applied, by replacing the two factors A and B, respectively, with CF and SF. As an example, having the factors A, B, E and D, the cell-factor CF could be generated based on the three factors A, B and E, and SF= D.

In this case, the main objectives of the model would be to study the differences between the levels of SF by the cell-factor CF, such as the means differences of Y based on an ANOVA model, the adjusted means differences of Y based on an ANCOVA model, and the effects differences of the independent variable(s) on Y based on an heterogeneous regression model.

4.2 Alternative Equation Specifications

Corresponding to each bi-factorial regression presented in Chapter 3, the equation specification of a multi-factorial regression can easily be written by replacing @Expand(A,B) with @Expand(A,B,C) for a three-way regression model, and for a four-way regression model the function used is @Expand(A,B,C,D). For the illustration only the simplest equation specifications will be presented, such as follows:

4.2.1 The Simplest Four-Way ANOVA Model without Intercept “C”

Y @Expand(A,B,C,D) (4.1)

Note that this equation specification can easily be derived from the equation (3.1). Then everyone can easily test all hypotheses on the means differences of the numerical variable Y, and the logits or odds differences of the zero-one or ordinal variable Y, between any cells generated by the four factors A, B, C and D, using the Wald test.

However, it is suggested to test the hypotheses on the differences between the levels of a certain factor, say the levels of D, conditional for the other three factors, say A=i, B=j and C=k. Similarly, based on the four-way ANCOVA and heterogeneous regression models presented below.

4.2.2 The Simplest Four-Way ANCOVA Model without Intercept “C”

Y X1 X2…XK @Expand(A,B,C,D) (4.2)

Note that this equation specification can easily be derived from the equation (3.4). Then everyone can easily test all hypotheses on the adjusted means differences of the numerical variable Y, and the adjusted logits or odds differences of the zero-one or ordinal variable Y, between any cells generated by the four factors A, B, C and D, using ______________________________________________________________________Cross Section and Experimental Data Analysis Using EViews I Gusti Ngurah Agung 15© 2011 John Wiley & Sons (Asia) Pte Ltd

the Wald test, under the assumption that the covariates X1, X2, …, and XK, have the same effects on Y within all the cells.

In addition, every one can easily conduct the testing hypotheses on the adjusted effect of each Xk, for k=1,…,K on Y, as well as the joint adjusted effects of any subsets of the covariates.

4.2.3 The Simplest Four-Way Heterogeneous Regression Model without Intercept “C”

Y @Expand(A,B,C,D) X1* @Expand(A,B,C,D)…… XK* @Expand(A,B,C,D) (4.3)

This equation specification can easily be derived from the equation (3.8). Note that the main objective of this model is to testing various hypotheses on the effects differences of one up to all covariates on Y between any cells generated by the four factors A, B, C and D.

5 Single-Factorial Multivariate Regression Models

As an extension of the single-factorial regression models presented in Chapter 2, this chapter presents various single-factorial multivariate regression models having an endogenous multivariate (Y1,…,Yg,…,YG). However, for the multivariate models, we should be using the explicit system equations, instead of the function @Expand(CF) 5.1 MANOVA Model

Having the endogenous bivariate (Y1,Y2), and a cell-factor CF having 4-levels or cells, then the system equations for the MANOVA models can be written as follows:

5.1.1 MANOVA Model without an Intercept

Y1 = C(11)*DC1+C(12)*DC2+C(13)*DC3+C(14)*DC4Y2 = C(21)*DC1+C(22)*DC2+C(23*DC3+C(24)*DC4 (5.1)

where DCk =DCk is the dummy variable or zero-one indicator of the cell CF=k. However, for a multi-factorial model, it is suggested to use a multiple index, such as DCij = DCij is the zero-one indicator of the cell CF=ij for the bi-factorial model.

5.1.2 MANOVA Model with an Intercept

Y1 = C(11)*DC1+C(12)*DC2+C(13)*DC3+C(14)Y2 = C(21)*DC1+C(22)*DC2+C(23*DC3+C(24) (5.2)


Note that this model has the cell CF=4 as a reference group, and the other three alternative MANOVA models can easily be written. Thence, for a MANOVA model, we may have five alternative system equations. In practice, it is suggested to apply one of the four alternative models with the intercepts.

For the data analysis, the system equations in (5.1) can easily be modified or extended to any endogenous multivariate (Y1,…,Yg,…,YG), as well as any cell-factors, and then copied to the object “System”.

5.2 MANCOVA Model

5.2.1 MONCOVA Model with a Covariate

For an illustration, having the endogenous bivariate (Y1,Y2), a covariate X1 and a cell-factor CF having 4-levels or cells, then the system equations for the MANCOVA models can be written as follows:

5.2.1.1 MANCOVA Model without an Intercept

Y1 = C(11)*DC1+C(12)*DC2+C(13)*DC3+C(14)*DC4+C(15)*X1Y2 = C(21)*DC1+C(22)*DC2+C(23*DC3+C(24)*DC4+C(25)*X1 (5.3)

For the data analysis, the system equations in (5.3) can easily be modified or extended to any endogenous multivariate as well as any cell-factors, and then copied to the object “System”.

5.2.1.2 MANCOVA Model with an Intercept

Y1 = C(11)*DC1+C(12)*DC2+C(13)*DC3+C(14)+C(15)*X1Y2 = C(21)*DC1+C(22)*DC2+C(23*DC3+C(24)+C(25)*X1 (5.4)

Note that this model has the cell CF=4 as a reference group, and the other three alternative MANCOVA models can easily be written. Thence, for a MANCOVA model, we may have five alternative system equations. In practice, it is suggested to apply one of the four alternative models with the intercepts. 5.2.2 MONCOVA Model with two Covariates

For an illustration, having the endogenous bivariate (Y1,Y2), two covariates X1 and X2, and a cell-factor CF having 4-levels or cells, then the system equations for the MANCOVA models can be written as follows:

5.2.2.1 Additive MANCOVA Model with an Intercept

Y1 = C(11)*DC1+C(12)*DC2+C(13)*DC3+C(14)______________________________________________________________________Cross Section and Experimental Data Analysis Using EViews I Gusti Ngurah Agung 17© 2011 John Wiley & Sons (Asia) Pte Ltd

+C(15)*X1+C(16)*X2Y2 = C(21)*DC1+C(22)*DC2+C(23*DC3+C(24)

+C(25)*X1+C(26)*X2 (5.5)

For the data analysis, the system equations in (5.5) can easily be modified or extended to any endogenous multivariate, any number of covariates as well as any cell-factors, and then copied to the object “System”.

5.2.2.2 Interaction MANCOVA Model with an Intercept

If it is defined that the effect of X1 (X2) on each of the endogenous variable depends on X2 (X1), then we have an interaction MANCOVA model with the following system equations.

Y1 = C(11)*DC1+C(12)*DC2+C(13)*DC3+C(14)+C(15)*X1+C(16)*X2+C(17)*X1*X2

Y2 = C(21)*DC1+C(22)*DC2+C(23*DC3+C(24)*DC4+C(25)*X1+C(26)*X2+C(27)*X1*X2 (5.6)


5.3.1 Heterogeneous Regression Models with a Covariate

For an illustration, having the endogenous bivariate (Y1,Y2), a covariate X1, and a cell-factor CF having 4-levels or cells, then the system equations for the heterogeneous regression models can be presented as follows:

5.3.1.1 Heterogeneous Regression Model without a Reference Group

Y1 = C(11)*DC1+C(12)*DC2+C(13)*DC3+C(14)*DC4 +(C(15)*DC1+C(16)*DC2+C(17)*DC3+C(18)*DC4)*X1

Y2 = C(21)*DC1+C(22)*DC2+C(23*DC3+C(24)*DC4 +(C(25)*DC1+C(26)*DC2+C(27)*DC3+C(28)*DC4)*X1 (5.7)

To generalize, for Yg, g=1,…,G, the parameters C(gj), j = 5, 6, 7 and 8 are the slope parameters of X1 on Yg within the four cells considered. Table 5.1 presents the parameters of the multivariate general model for g =1,…,G.

Table 5.1 The Parameters of the Model in (5.7) by CF CF=1 CF=2 CF=3 CF=4

Intercept C(g1) C(g2) C(g3) C(g4)Slopes of X1 C(g5) C(g6) C(g7) C(g8)


For the data analysis, the system equations in (5.7) can easily be modified or extended to any endogenous multivariate as well as any cell-factors, and then copied to the object “System”.

5.3.1.2 Heterogeneous Regression Model with a Reference Group

Corresponding to the model in (5.7), several alternative system equations could be used to representing exactly the same model. One of the suggested system equations is as follows:

Y1 = C(11)*DC1+C(12)*DC2+C(13)*DC3+C(14)*DC4 +(C(15)*DC1+C(16)*DC2+C(17)*DC3+C(18))*X1

Y2 = C(21)*DC1+C(22)*DC2+C(23*DC3+C(24)*DC4 +(C(25)*DC1+C(26)*DC2+C(27)*DC3+C(28))*X1 (5.8)

To generalize, for Yg, g=1,…,G, the parameters C(gj), j = 5, 6, 7 and 8 are indicating the slope parameters of X1 on Yg within the four cells considered. Table 5.2 presents the parameters of the multivariate general model for g =1,…,G. Note that this table shows that the cell CF=4 is the reference cell for the slopes of X1 on Yg, and the other three alternative heterogeneous regression models can easily be written.

Table 5.2 The Parameters of the Model in (5.8) by CFCF=1 CF=2 CF=3 CF=4

Intercept C(g1) C(g2) C(g3) C(g4)Slopes C(g5)+C(g8) C(g6)+C(g8) C(g7)+C(g8) C(g8)

5.3.2 Heterogeneous Regression Models with two Covariates

For an illustration, having the endogenous bivariate (Y1,Y2), two covariates X1 and X2, a cell-factor CF having 4-levels or cells, then the system equations for the models can be written as follows: 5.3.2.1 Additive Heterogeneous Regression Model with a Reference Group

As an extension of the model in (5.8), one of four alternative models with a reference group of the slope parameters has the following system equations, with the model parameters presented in Table 5.3, for g = 1 and 2.

Y1 = C(11)*DC1+C(12)*DC2+C(13)*DC3+C(14)*DC4 +(C(15)*DC1+C(16)*DC2+C(17)* DC3+C(18))*X1

+(C(19)*DC1+C(110)*DC2+C(111)*DC3+C(112))*X2

Y2 = C(21)*DC1+C(22)*DC2+C(23*DC3+C(24)*DC4______________________________________________________________________Cross Section and Experimental Data Analysis Using EViews I Gusti Ngurah Agung 19© 2011 John Wiley & Sons (Asia) Pte Ltd

+(C(25)*DC1+C(26)*DC2+C(27)*DC3+C(28))*X1+(C(29)*DC1+C(210)*DC2+C(211)*DC3+C(212))*X2 (5.9)

To generalize, for Yg, g=1,…,G, the parameters C(gj), j = 5, 6, 7 and 8 are indicating the slope parameters of X1 on Yg, and the slope parameters of X2 on Yg for j = 9, 10, 11 and 12, within the four cells considered. Table 5.3 presents the parameters of the multivariate general model for g =1,…,G.


Intercept C(g1) C(g2) C(g3) C(g4)

Slopes of X1 C(g5)+ C(g8) C(g6)+ C(g8) C(g7)+ C(g8) C(g8)


For the data analysis, the system equations (5.9) can easily be modified or extended to any endogenous multivariate as well as any cell-factors, and then copied to the object “System”.

5.3.2.2 Interaction Heterogeneous Regressions Model with a Reference Group

Y1 = C(11)*DC1+C(12)*DC2+C(13)*DC3+C(14)*DC4 +(C(15)*DC1+C(16)*DC2+C(17)* DC3+C(18))*X1 +(C(19)*DC1+C(110)*DC2+C(111)*DC3+Cs(112))*X2

+(C(113)*DC1+C(114)*DC2+C(115)*DC3+C(116))*X1*X2

Y2 = C(21)*DC1+C(22)*DC2+C(23*DC3+C(24)*DC4 +(C(25)*DC1+C(26)*DC2+C(27)*DC3+C(28))*X1 +(C(29)*DC1+C(210)*DC2+C(211)*DC3+C(212))*X2

+(C(213)*DC1+C(214)*DC2+C(215)*DC3+C(216))*X1*X2 (5.10)

To generalize, for Yg, g=1,…,G, the parameters C(gj), j = 5, 6, 7 and 8 are indicating the slope parameters of X1 on Yg, the slope parameters of X2 on Yg for j = 9, 10, 11 and 12, and the slope parameters of X1*X2 on Yg for j = 13, 14, 16 and 16, within the four cells considered. Table 5.4 presents the parameters of the multivariate general model for g =1,…,G.


Intercept C(g1) C(g2) C(g3) C(g4)



Slopes of X1*X2 C(g13)+ C(g16) C(g14)+ C(g16) C(g15)+ C(g16) C(g16)


5.3.2.3 Generalize Single-Factorial Heterogeneous Regression Model

To generalize the heterogeneous regression model in (5.10), the following system equations is presented. Corresponding to this model the following remarks are presented.

Yg = (C(g10)+C(g11)*X1 +C(g12)*X2+…+C(g1K)*XK)*DC1 +(C(g20)+C(g21)*X1 +C(g22)*X2+…+C(g2K)*XK)*DC2 +(C(g30)+C(g31)*X1 +C(g32)*X2+…+C(g3K)*Xk)*DC3

+(C(g40)+C(g41)*X1 +C(g42)*X2+…+C(g4K)*XK)*DC4 (5.11) 1). This system equations, for g = 1,…,G, is representing a multivariate heterogeneous

regression model of a G-dimensional endogenous variable with a single cell-factor having four levels or cells, and Xk, k=1,…,K are numerical exogenous variables, where each Xk could be a main numerical exogenous variable, as well as a two or three interaction-factor. For example Xk1 = Xi*Xj or Xk2 = Xh*Xi*Xj.

2). Note that the system in (5.11), in fact, is representing four systems equations having numerical endogenous as well as exogenous variables, such as follows:

2.1. For DC1=1, and DC2=DC3=DC4=0,

Yg = C(g10)+C(g11)*X1 +C(g12)*X2+…+C(g1K)*XK (5.11a) 2.2. For DC2=1, and DC1=DC3=DC4=0,

Yg = C(g20)+C(g21)*X1 +C(g22)*X2+…+C(g2K)*XK (5.11b)


Yg = C(g30)+C(g31)*X1 +C(g32)*X2+…+C(g3K)*XK (5.11c)


Yg = C(g40)+C(g41)*X1 +C(g42)*X2+…+C(g4K)*XK (5.11d)

3). Furthermore, note that the estimates based on the systems equations in (5.11a) to (5.11d) can have different sets of independent variables, which are statistically accepted. However, they are unpredictable, since they are highly dependent on the data set which happen to be selected by or available for the researchers, as well as the impact of the multicollinearity between the independent variables – Refer to Section 2.14.2 in Agung (2009). Even in some cases, for a large number of independent variables, we may obtain an error message, such as “Near Singular Matrix” or “Overflow”.


4). This system equations can easily be copied and modified for any values of G and K, as well as any types of the variable Xk, and any cell-factors. Thence, having the output on the screen, various multivariate and univariate hypotheses can easily be tested using the Wald test.

6Bi-Factorial Multivariate Regression Models

In this case, it is considered two categorical factors A and B, a set of covariates, namely X1, X2, … , XK, where Xk for each k=1,…,K; can be a main factor, two- or three-way interactions, and an endogenous numerical multivariate Yg, g=1,…,G. For writing its system equations, the two factors should be represented by their dummy variables or zero-one indicators, namely DAi and DBj, respectively, indicate the zero one indicators for the levels of A=i and B=j, for i=1,2,…,I and j=1,…,J.

6.1 MANOVA Models

For an illustration, corresponding to the ANOVA model in (3.3), the system equations of a MANOVA model considered is a 4x2-factorial model as follows:

Yg = C(g11)+C(g12)*DA2+C(g13)*DA3+C(g4)*DC4+C(g15)*DA1*DB1+C(g16)*DA2*DB1+C(g17)*DA3*DB1+C(g18)*DA4*DB1 (6.1)

Table 6.1 The Parameters of the MONAVA Model in (6.1)B=1 B=2 Diff. B(1-2)

A=1 C(g11) + C(g15) C(g11) C(g15)A=2 C(g11)+C(g12)+C(g16) C(g11) + C(g12) C(g16)A=3 C(g11)+C(g13)+C(g17) C(g11) + C(g13) C(g17)A=4 C(g11)+C(g14)+C(g18) C(g11) + C(g14) C(g18)

Compared to the parameters of the ANOVA model in Table 3.1, the parameters of the MANOVA model (6.1) are presented in Table 6.1. Note that this table shows that each parameter C(g1k), for each k=2, …,8, presents the difference of the mean of Yg between two cells, which can easily be tested using the t-statistic in the output. For the data analyses the equation (6.1) can be copied to the dialog for any number of Yg’s. Note that the estimated values within each of the eight cells, in fact, is exactly the same as the observed mean of Yg.


Various system equations of a MANOVA models can easily be presented, one of those is the equation specification (6.2) with its parameters presented in Table 6.2.

Yg = C(g11)+C(g12)*DB2+C(g13)*DA1*DB1+C(g14)*DA2*DB1+C(g15)*DA3*DB1+C(g16)*DA1*DB2+C(g17)*DA2*DB2+C(g18)*DA3*DB2 (6.2)

Table 6.2 The Parameters of the MONO VA Model in (6.2) B=1 B=2

A=1 C(g11)+C(g13) C(g11)+C(g12)+C(g16)A=2 C(g11)+C(g14) C(g11)+C(g12)+C(g17)A=3 C(g11)+C(g15) C(g11)+C(g12)+C(g18)A=4 C(g11) C(g11)+C(g12)

A(1-4) C(g13) C(g16) A(2-4) C(g14) C(g17) A(3-4) C(g15) C(g18)

6.2 MANCOVA Models

It is well known that the main objectives of the application of a MANCOVA model are to study the adjusted means differences of Yg between the cells generated by the two factors A and B, under the assumption that the covariates have the same effects or slopes within all cells.

Furthermore, it is recognize that the adjusted means differences are represented using the differences of the intercepts of the eight multiple regressions. For this reason, the right-hand-side (RHS) of a MANOVA model can be used to representing the intercepts of a MANCOVA model. Thence, various system equations of MANCOVA models can easily be presented. However, corresponding to the system equations (6.1) and (6.2), the following system equations of the MANCOVA models obtained.

6.2.1 MONCOVA Model using the MANOVA Model (6.1)

By using the right-hand-side of the MANOVA model (6.1), namely RHS(6.1), the data analysis based on a MANCOVA model can be done by using the following general system equations, with its parameters presented in Table 5.8.

Yg = RHS(6.1)+C(g19)*X1+C(g110)*X2+C(g111)*X3+… (6.3)


Table 6.3 The Parameters of the MONAVA Model in (6.1)Intercepts Slopes in all cells

B=1 B=2 Diff. B(1-2) X1 X2 X3A=1 C(g11)+C(g15) C(g11) C(g15)

C(g19) C(g110) C(g111)A=2 C(g11)+C(g12)+C(g16) C(g11) + C(g12) C(g16)A=3 C(g11)+C(g13)+C(g17) C(g11) + C(g13) C(g17)A=4 C(g11)+C(g14)+C(g18) C(g11) + C(g14) C(g18)

Table 6.4 The Parameters of the MONCOVA Model in (6.4) Intercepts Slopes in all cells

B=1 B=2 X1 X2 X3A=1 C(g11)+C(g13) C(g11)+C(g12)+C(g16)

C(g19) C(110) C(g111)A=2 C(g11)+C(g14) C(g11)+C(g12)+C(g17)A=3 C(g11)+C(g15) C(g11)+C(g12)+C(g18)A=4 C(g11) C(g11)+C(g12)

A(1-4) C(g13) C(g16) A(2-4) C(g14) C(g17) A(3-4) C(g15) C(g18)

6.2.2 MONCOVA Model using the MANOVA Model (6.2)

By using the right-hand-side of the MANOVA model (6.2), namely RHS(6.2), the data analysis based on a MANCOVA model can be done by using the following general system equations, with its parameters presented in Table 6.4.

Yg = RHS(6.2)+C(g19)*X1+C(g110)*X2+C(g111)*X3+… (6.4)


Various system equations of a bi-factorial heterogeneous regression model could be presented, however, the simplest system equations will be presented based on the model in (5.11). Having the dummy variables DAi and DBj, for i =1,…,I, and j=1,…,J, the simplest system equations can easily be derived from (5.11) using the dummy variable DAi*DBj, instead of zero-one indicator DCk of the level CF=k. Thence, the following simplest system equations obtained, with its parameters presented in Table 5.6, for I = 4 and J = 2.

Yg = (C(g10)+C(g11)*X1 +C(g12)*X2+…+C(g1K)*XK)*DA1*DB1


+(C(g20)+C(g21)*X1 +C(g22)*X2+…+C(g2K)*XK)*DA1*DB2……………………………………………………………………………………………………………………………………………………

+(C(g30)+C(g31)*X1 +C(g32)*X2+…+C(g3K)*Xk)*DAI*DBJ (6.5)

Table 6.5 The Parameters of an Heterogeneous Regression Model in (6.5) Slopes

A B CF Intercept X1 …. XK1 1 11 C(g10) C(g11) …. C(g1K)1 2 12 C(g20) C(g21) …. C(g2K)2 1 21 C(g30) C(g31) …. C(g3K)2 2 22 C(g40) C(g41) …. C(g4K)3 1 31 C(g50) C(g51) …. C(g5K)3 2 32 C(g60) C(g61) …. C(g6K)4 1 41 C(g70) C(g71) …. C(g7K)4 2 42 C(g80) C(g81) …. C(g8K)

7Multi-Factorial Multivariate Regression Models

For any multi-factorial regression models, it is suggested to apply a system equations using the cell-factor as presented in Chapter 5. The second alternative is the bi-factorial model as presented in Chapter 6, where everyone should select a certain or single-factor, namely SF, and a cell-factor CF generated by the other factors. Thence all equation specifications for the models with the two factors A and B presented in Chapter 6 can easily be applied, by replacing the two factors A and B, respectively, with CF and SF. For an example, having the factors A, B, C and D, the cell-factor CF could be generated based on the three factors A, B and C, and SF= D.

8Seemingly Causal Models


8.1 Unidirectional SCM

For the seemingly causal models (SCMs) based on a set of numerical variables, either observed or measured or latent variables, the system specifications of alternative models will be presented only based on a path diagram in Figure 9.40 in the main text as well as in Agung (2009), such as follows:

Figure 8.1 The Path Diagram from Figure 4.32 in Agung (2009)

8.1.1 An Additive SCM

Its system specification is as follow:

Y1=C(10)+c(11)*Y2+C(12)*X1Y2=C(20)+C(21)*X1+C(22)*X2X1=C(30)+C(31)*X2+C(32)*X3X3=C(40)+C(41)*X2 (8.1)

Note that each independent variable in the system equations has a direct effect on the corresponding dependent variables. This type of system equations can easily be written based on any defined path diagrams.

8.1.2 A Two-Way Interaction SCM


Y1=C(10)+C(11)*Y2+C(12)*X1+C(13)*Y2*X1+C(14)*Y2*X2 +C(15)*X1*X2+C(16)*X1*X3 Y2=C(20)+C(21)*X1+C(22)*X2+C(23)*X1*X2+C(24)*X1*X3X1=C(30)+C(31)*X2+C(32)*X3+C(33)*X2*X3X3=C(40)+C(41)*X2 (8.2)

An alternative specification can be written as follows:

Y1=C(10)+(C(11)+C(12)*X1+C(13)*X2)*Y2+______________________________________________________________________Cross Section and Experimental Data Analysis Using EViews I Gusti Ngurah Agung 26© 2011 John Wiley & Sons (Asia) Pte Ltd

X2

X3

X1

Y2

Y1

+(C(14)+C(15)*X2+C(16)*X3)*X1 Y2=C(20)+(C(21)+C(22)*X2+C(23)*X3)*X1+C(24)*X2X1=C(30)+C(31)*X2+(C(32)+C(33)*X2)*X3X3=C(40)+C(41)*X2 (8.3)

Note that the term (C(11)+C(12)*X1+C(13)*X2) in the first regression indicate that the direct effect of Y2 on Y1 depends on X1 and X2, which is corresponding to the indirect effects of X1 and X2 on Y1 through Y2.

Similarly, the term (C(14)+C(15)*X2+C(16)*X3) indicates that the direct effect of X1 on Y1 depends on X2 and X3, the term (C(21)+C(22)*X2+C(23)*X3) in the second regression indicates that the direct effect of X1 on Y2 depends on X2 and X3, and the term (C(32)+C(33)*X2) in the third regression indicates the direct effect of X3 on X1 depends on X2.

8.1.3 A Three-Way Interaction SCM

Under the assumption that the three variables X1, X2, and X3 are completely correlated, as well as the three variables X1, X2, and Y2, then the three-way interactions X1*X2*X3 and X1*X2*Y2 should be used as the additional independent variables of the SCM (8.2) or (8.3), specifically for the first two regressions. Thence, the three-way interaction SCM will have the following system specification, which can easily be written based on the system specification (8.2).

Y1=C(10)+c(11)*Y2+C(12)*X1+C(13)*Y2*X1+C(14)*Y2*X2 +C(15)*X1*X2+C(16)*X1*X3 +C(17)*Y2*X1*X2+C(18)*X1*X2*X3Y2=C(20)+C(21)*X1+C(22)*X2+C(23)*X1*X2+C(24)*X1*X3 +C(25)*X1*X2*X3X1=C(30)+C(31)*X2+C(32)*X3+C(33)*X2*X3X3=C(40)+C(41)*X2 (8.4)

8.2 Simultaneous SCM

As an extension of the path diagram in Figure 8.1, Figure 8.2 presents the path diagram of the simultaneous SCMs, where the endogenous variables Y1 and Y2 are theoretically defined to have a simultaneous causal effect, as well as the exogenous variables X1 and X2. Then we will have the additive, two-way and three-way interaction SCMs as follows:


Figure 8.2 The Path Diagram from Figure 4.32 in Agung (2009)

8.2.1 An Additive Simultaneous SCM


Y1=C(10)+c(11)*Y2+C(12)*X1Y2=C(20)+C(21)*Y1+C(22)*X1+C(23)*X2X1=C(30)+C(31)*X2+C(32)*X3X3=C(40)+C(41)*X1+C(42)*X2 (8.5)

Note that each independent variable in the system equations has a direct effect on the corresponding dependent variables. This type of system equations can easily be written based on any defined path diagrams.

8.2.2 A Two-Way Interaction Simultaneous SCM

Its system specification can easily be derived from the system equation (8.2) by inserting additional independent variables, so that the following equation obtained.

Y1=C(10)+C(11)*Y2+C(12)*X1+C(13)*Y2*X1+C(14)*Y2*X2 +C(15)*X1*X2+C(16)*X1*X3 Y2=C(20)+C(21)*X1+C(22)*X2+C(23)*X1*X2+C(24)*X1*X3 +C(25)*Y1+C(26)*X1*Y1X1=C(30)+C(31)*X2+C(32)*X3+C(33)*X2*X3X3=C(40)+C(41)*X2 +C(42)*X1+C(43)*X1*X2 (8.6)

8.2.3 A Three-Way Interaction Simultaneous SCM

Under the assumption that the three variables X1, X2, and X3 are completely correlated, as well as the three variables X1, X2, and Y2, then the three-way interactions X1*X2*X3 should be used as the additional independent variables of the first two regressions in SCM (8.6), and the interaction X1*X2*Y2 should be used as the additional


X2

X3

X1

Y2

Y1

independent variables of the first and the fourth regressions. Thence, the three-way interaction SCM will have the following system specification.

Y1=C(10)+C(11)*Y2+C(12)*X1+C(13)*Y2*X1+C(14)*Y2*X2 +C(15)*X1*X2+C(16)*X1*X3 +C(17)* X1*X2*X3+C(18)* X1*X2*Y2Y2=C(20)+C(21)*X1+C(22)*X2+C(23)*X1*X2+C(24)*X1*X3 +C(25)*Y1+C(26)*X1*Y1+C(27)* X1*X2*X3X1=C(30)+C(31)*X2+C(32)*X3+C(33)*X2*X3X3=C(40)+C(41)*X2 +C(42)*X1+C(43)*X1*X2+C(44)* X1*X2*Y2 (8.7)

8.3 Discontinuous SCM with Special Notes

The extension of the SCMs in (8.1) to (8.7) are their corresponding discontinuous or peace-wise SCMs, since, in general, the individuals in the sample could easily be classified into at least two groups of individuals having specific or distinct characteristics, which lead to differential types or forms of relationships or causal effects between the set of the main variables considered. For this reason, the SCMs with dummy variables should be applied.

Thence, the system specifications of the Multivariate GLM presented in Chapter 5 could easily be used as a guide to deriving the system equations of the discontinuous or peace-wise SCMs. For example, suppose it is defined two groups of individuals, then based on the SCM (8.1) can easily be derived the following two-peace SCM, by inserting two dummy variables, namely D1 and D2, which are generated for the two groups..

Y1=(C(101)+C(111)*Y2+C(121)*X1)*D1+(C(102)+C(112)*Y2+C(122)*X1)*D2 Y2=(C(201)+C(211)*X1+C(221)*X2)*D1+(C(201)+C(212)*X1+C(222)*X2)*D2 X1=(C(301)+C(311)*X2+C(321)*X3)*D1+(C(302)+C(312)*X2+C(322)*X3)*D2 X3=(C(401)+C(411)*X2)*D1+(C(402)+C(412)*X2)*D2 (8.8)

By using the same method, the system specifications of the other two-peace SCMs can easily be written based on the SCMs in (8.2) to (8.7), which can be extended to using a cell-factor CF having any number of levels or cells. Refer the system specifications in Chapter 5, using four dummy variables.


1€¦ · web viewunder the assumption that the three variables x1, x2, and x3 are completely...

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