```-------------------------------------------------------------------------------
help: r2reg3                                                        dialog: r2r
> eg3
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+-------+
----+ Title +------------------------------------------------------------

r2reg3: Overall System R2 - Adjusted R2 - F Test - Chi2 Test

+--------+
----+ Syntax +-----------------------------------------------------------

r2reg3

+-------------+
----+ Description +------------------------------------------------------

r2reg3 computes overall system R-squared (R2), Adjusted R2, and the overall
significance of F-Test, and Chi2-Test, after:
- (3SLS) Three-Stage Least Squares reg3 for systems of simultaneous
equations.
- (SUR) Seemingly Unrelated Regression sureg for sets of equations.

r2reg3 used Four types of criteria and tests, as discussed in:
McElroy(1977), Judge et al(1985), Dhrymes(1974), Greene(1993), and
Berndt(1991).

see eq.12.1.33-36 in Judge et al(1985, p.477).

1- Berndt  System R2 = 1-|E'E|  / |Yb'Yb|
2- McElroy System R2 = 1-(U'W*U)/ (Y'W*Y)
3- Judge   System R2 = 1-(U'U)  / (Y'Y)
Q
4- Dhrymes System R2 = Sum [R2i (yi' Dt yi']/[Y(I(Q) # Dt)Y]
i=1
5- Greene  System R2 = 1-(Q/trace(inv(Sig))*SYs)

Judge and Dhrymes are identical results.

From each type of these system R2's, r2reg3 can calculate Adjusted R2,
F-Test, and Chi2-Test:

F-Test = R2/(1-R2)*[(QN-K)/(K-Q)]
Chi2-Test = -N*(log(1-R2))

where
|E'E| = determinant of residual matrix (NxQ)
|Yb'Yb| = determinant of dependent variables matrix in deviation from mean (Nx
> Q)
yi = dependent variable of eq. i (Nx1)
Y = stacked vector of dependent variables (QNx1)
U = stacked vector of residuals (QNx1)
W = variance-covariance matrix of residuals (W=inv(Omega) # I(N))
N = number of observations
K = Number of Parameters
Q = Number of Equations
R2i = R2 of eq. i
Dt = I(N)-JJ'/N, with J=(1,1,...,1)' (Nx1)
SYs = (Yb1*Yb2*...Ybq)/N
Sig = Sigma hat Matrix
Degrees of Freedom F-Test    = (K-Q), (QN)
Degrees of Freedom Chi2-Test = (K-Q)
Log Determinant of Sigma     = log|Sigma matrix|
Log Likelihood Function LLF  =-(N*Q/2)*(1+log(2*_pi))-(N/2*abs(log(Sigma)))

+---------------+
----+ Saved Results +----------------------------------------------------

r2reg3 saves the following in r():

Scalars
r(N)           Number of Observations
r(k)           Number of Parameters
r(k_eq)        Number of Equations
r(chi_df)      DF chi-squared
r(f_df1)       F-Test DF1 Numerator
r(f_df2)       F-Test DF2 Denominator
r(r2_b)        Berndt R-squared
r(r2_j)        Judge R-squared
r(r2_m)        McElroy R-squared
r(r2_d)        Dhrymes R-squared
r(r2_g)        Greene R-squared
r(f_b)         Berndt F Test
r(f_j)         Judge F Test
r(f_m)         McElroy F Test
r(f_d)         Dhrymes F Test
r(f_g)         Greene F Test
r(chi_b)       Berndt Chi2 Test
r(chi_j)       Judge Chi2 Test
r(chi_m)       McElroy Chi2 Test
r(chi_d)       Dhrymes Chi2 Test
r(chi_g)       Greene Chi2 Test
r(lsig2)       Log Determinant of Sigma
r(llf)         Log Likelihood Function

+----------+
----+ Examples +---------------------------------------------------------

clear all

sysuse r2reg3.dta , clear

* (1) SUR Model:

sureg (y1 y2 x1 x2) (y2 y1 x3 x4)

r2reg3

return list

* (2) 3SLS Model:

reg3 (y1 y2 x1 x2) (y2 y1 x3 x4) , exog(x1 x2 x3 x4)

r2reg3

return list

* If you want to use dialog box: Press OK to compute r2reg3

db r2reg3

. sysuse r2reg3.dta , clear
. reg3 (y1 y2 x1 x2) (y2 y1 x3 x4) , exog(x1 x2 x3 x4)
. r2reg3

========================================================
* Overall System R2 - Adjusted R2 - F Test - Chi2 Test *
========================================================
+----------------------------------------------------------------------------
> +
|     Name |       R2 |   Adj_R2 |        F |  P-Value |     Chi2 |  P-Value
> |
|----------+----------+----------+----------+----------+----------+----------
> |
|   Berndt |   0.9191 |   0.9004 |  49.2299 |   0.0000 |  42.7469 |   0.0000
> |
|  McElroy |   0.8042 |   0.7590 |  17.7985 |   0.0000 |  27.7216 |   0.0001
> |
|    Judge |   0.8228 |   0.7819 |  20.1180 |   0.0000 |  29.4159 |   0.0001
> |
|  Dhrymes |   0.8228 |   0.7819 |  20.1180 |   0.0000 |  29.4159 |   0.0001
> |
|   Greene |   0.8104 |   0.7666 |  18.5204 |   0.0000 |  28.2672 |   0.0001
> |
+----------------------------------------------------------------------------
> +
Number of Parameters         =           8
Number of Equations          =           2
Degrees of Freedom F-Test    =      (6, 34)
Degrees of Freedom Chi2-Test =           6
Log Determinant of Sigma     =      9.2772
Log Likelihood Function      =   -127.1003

+------------+
----+ References +-------------------------------------------------------

Berndt, Ernst R. (1991) "The practice of econometrics: Classical and

Dhrymes, Phoebus J. (1974) "Econometrics: Statistical Foundations and
Applications", 2ed edition Springer- Verlag New York, USA..

Greene, William (1993) "Econometric Analysis", 2nd ed., Macmillan
Publishing Company Inc., New York, USA.; 490-491.

Judge, Georege, W. E. Griffiths, R. Carter Hill, Helmut Lutkepohl, &
Tsoung-Chao Lee(1985) "The Theory and Practice of Econometrics", 2nd
ed., John Wiley & Sons, Inc., New York, USA; 477-478.

Kmenta, Jan (1986) "Elements of Econometrics", 2nd ed., Macmillan
Publishing Company, Inc., New York, USA; 645.

McElroy, Marjorie B. (1977) "Goodness of Fit for Seemingly Unrelated
Regressions: Glahn's R2y,x and Hooper's r~2", Journal of
Econometrics, 6(3), November; 381-387.

+-----------------+
----+ Acknowledgments +--------------------------------------------------

I would like to thank the authors of the following Stata modules:

- Nicholas J. Cox: for writing Stata programs about various matrix tasks matod
> d.

- Philip Ryan: for writing Stata programs about product of observations rprod.

+--------+
----+ Author +-----------------------------------------------------------

Assistant Professor
Agricultural Research Center - Agricultural Economics Research Institute - Eg
> ypt
WebPage at IDEAS:       http://ideas.repec.org/f/psh494.html
WebPage at EconPapers:  http://econpapers.repec.org/RAS/psh494.htm

+-----------------+
----+ r2reg3 Citation +--------------------------------------------------