In econometrics, seemingly unrelated regression (SUR) is a technique for analyzing a model with multiple equations and correlated error terms.

An economic model may contain multiple equations which are independent of each other on the surface: they are not estimating the same dependent variable, they have different independent variables, etc. However, if the equations are using the same data, the errors may be correlated between the two equations. SUR is an extension of the linear regression model which allows correlated errors between equations.

The mathematics is very similar to computing Huber-White standard errors. Suppose we have a series of equations

y_i = x_iβ_i + ε_i

where x, β, and ε are vectors and i=1...M where M is the number of equations. Assume each equation has N observations. Let Σ be a M x M matrix representing the covariance of residuals between the equations. Then SUR is merely computing the GLS estimation for β:

where

where is the Kronecker product and V(Y) is an M x N matrix

SAS Example:

• A. Zellner, "An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias,"JASA 57(1962) pp.348-364
• J.C.G. Boot, "Investment Demand: an Empirical Contribution to the Aggregation Problem," IER 1(1960) pp.3-30.
• Y. Grunfeld, "The Determinants of Corporate Investment," Unpublished thesis, Chicago, 1958
  
  data grunfeld;
  input year ge_i ge_f ge_c wh_i wh_f wh_c;
  label ge_i = 'Gross Investment, GE'
  ge_c = 'Capital Stock Lagged, GE'
  ge_f = 'Value of Outstanding Shares Lagged, GE'
  wh_i = 'Gross Investment, WH'
  wh_c = 'Capital Stock Lagged, WH'
  wh_f = 'Value of Outstanding Shares Lagged, WH';
  cards;
  1935 33.1 1170.6 97.8 12.93 191.5 1.8
  1936 45.0 2015.8 104.4 25.90 516.0 .8
  1937 77.2 2803.3 118.0 35.05 729.0 7.4
  1938 44.6 2039.7 156.2 22.89 560.4 18.1
  1939 48.1 2256.2 172.6 18.84 519.9 23.5
  1940 74.4 2132.2 186.6 28.57 628.5 26.5
  1941 113.0 1834.1 220.9 48.51 537.1 36.2
  1942 91.9 1588.0 287.8 43.34 561.2 60.8
  1943 61.3 1749.4 319.9 37.02 617.2 84.4
  1944 56.8 1687.2 321.3 37.81 626.7 91.2
  1945 93.6 2007.7 319.6 39.27 737.2 92.4
  1946 159.9 2208.3 346.0 53.46 760.5 86.0
  1947 147.2 1656.7 456.4 55.56 581.4 111.1
  1948 146.3 1604.4 543.4 49.56 662.3 130.6
  1949 98.3 1431.8 618.3 32.04 583.8 141.8
  1950 93.5 1610.5 647.4 32.24 635.2 136.7
  1951 135.2 1819.4 671.3 54.38 723.8 129.7
  1952 157.3 2079.7 726.1 71.78 864.1 145.5
  1953 179.5 2371.6 800.3 90.08 1193.5 174.8
  1954 189.6 2759.9 888.9 68.60 1188.9 213.5
  ;
  

The following statements compute the SUR estimates for the Grunfeld model.

proc syslin data=grunfeld sur;
ge: model ge_i = ge_f ge_c;
westing: model wh_i = wh_f wh_c;
run;

[此贴子已经被作者于2006-4-23 8:29:28编辑过]

Both Limdep and Stata can do it also.

Limdep:

The command for the GLS estimator is:

SURE ; Lhs = y1, y2, ..., yM (Your list of Lhs variables)
; Eq1 = Rhs variables for first equation

; Eq2 = Rhs variables for second equation
; ...
; EqM = Rhs variables for M'th equation $

Maximum Likelihood Estimation of Constrained Systems
The following is for models in which the constraints are equalities of the parameters across (or within) equations. (For other types, use constrained FGLS.)

SURE ; Lhs = list of dependent variables

; Rhs = full list of independent variables
; Labels = labels to use for the parameters
; Pattern = the parameter matrix $

Stata:

sureg (depvar1 varlist1) (depvar2 varlist2) ... (depvarN varlistN) [ if] [in] [weight]

stata 也简单的。

sureg (price foreign length) (mpg displ = foreign weight), isure corr

或者

reg3 (y1 x1 x2 x3 x4 x5) (y2 x1 x3 x5 x6 x8), sure