Chapter 14 Selected Topics in Single Equation Regression Models
Chapter 14 Selected Topics in Single Equation Regression Models
14.1 Restricted Least Squares (RLS) 1. oLS and rls (1) Unrestricted least squares(ULS) When using the ordinary least square method(ols) to estimate the parameters we do not put any prior constraint(s)or restriction (s)on the parameters So we can estimate the parameters without any restrictions. This is Uls
14.1 Restricted Least Squares（RLS） ◼ 1. OLS and RLS （1）Unrestricted least squares(ULS): When using the ordinary least square method(OLS) to estimate the parameters, we do not put any prior constraint（s）or restriction(s) on the parameters. So we can estimate the parameters without any restrictions. This is ULS
(2)Restricted least squares(RLs) inY=B1+B2×21+B×31+u1 If we put any restrictions on the parameters such as B.=2 or B.+ B.=1. we use RLs method to estimate The steps of RLS: transform the data to take into account the restrictions suggested by the relevant theory apply the least squares method (ols)
（2）Restricted least squares(RLS) In Yi=B1+B2X2i+B3X3i+ui If we put any restrictions on the parameters, such as B2 =2, or B2+ B3 =1, we use RLS method to estimate. The steps of RLS: ·transform the data to take into account the restrictions suggested by the relevant theory, ·apply the least squares method (OLS)
2.Test of the validity of the restriction (s) Le七 R2=R2 from the unrestricted regression R2-R2 from the restricted regression m =the number of linear restrictions im posed k =the number of parameters estimated in the unrestricted regression n =the number of observations
◼ 2.Test of the validity of the restriction(s): Let R2=R2 from the unrestricted regression R *2=R2 from the restricted regression m =the number of linear restrictions imposed k =the number of parameters estimated in the unrestricted regression n =the number of observations
+: the restriction(s)is valid F (R2-R2)m- Fmu-ky 14.8 (1-R2)/(n-k) Estimate the Us regression and obtain the r2 Estimate the Rls regression and obtain Find out the number of restrictions(m). Find out the coefficients estimated in the unrestricted regression (k) Compute F value
H0 : the restriction（s） is valid （14.8） ·Estimate the ULS regression and obtain the R2 ·Estimate the RLS regression and obtain R *2 ·Find out the number of restrictions(m). ·Find out the coefficients estimated in the unrestricted regression(K) ·Compute F value ( ) 2 ( ) 2 *2 ~ (1 )/( ) / Fm n k R n k R R m F − − − − =