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 − − − − =
Hypothesis testing If F>F refuse H the restriction(s) imposed by the theory is not valid ( statistically speaking), reject the restricted least squares regression. use the standard ols method If F<F, accept H the given restriction is valid, the rls regression is preferred to ULS
Hypothesis testing: If F>Fc , refuse H0 , the restriction(s) imposed by the theory is not valid （statistically speaking）, reject the restricted least squares regression , use the standard OLS method. If F<Fc , accept H0 , the given restriction is valid, the RLS regression is preferred to ULS
14.2 Dynamic Economic Models Autoregressive and Distributed Lag models ■1。 Definition Dynamic models/ Distributed lag models There is a non-contemporaneous, or lagged, relationship between Y and the x for the effect of a unit change in the value of the explanatory variable is spread over. or distributed over. a number of time periods
14.2 Dynamic Economic Models: Autoregressive and Distributed Lag Models ◼ 1. Definition Dynamic models/ Distributed lag models: －－There is a non-contemporaneous, or lagged, relationship between Y and the Xs , for the effect of a unit change in the value of the explanatory variable is spread over, or distributed over, a number of time periods
The reasons of the dependent variable respond to a unit change in the explanatory variable(s)with a time lag Psychological reasons Technological reasons, such as the purchase of PC, automobile Institutional reasons, such as multiyear contracts
The reasons of the dependent variable respond to a unit change in the explanatory variable(s) with a time lag. · Psychological reasons. · Technological reasons, such as the purchase of PC, automobile · Institutional reasons, such as multiyear contracts
k-period distributed lag model Y=A+B义+B义+B2X2+…+B义tA+ut(1413) Bo>B>B2 The effect of a unit change in the value of the explanatory variable is felt over k periods. B: the short-run /impact multiplier, which means “ the change in the mean value of Y following a un此t change in× in the same period” (B +B ).(B +B, +B,): interim/intermediate multipliers “… in the next, following period” ∑ B,=Bo+B,+B,+.+B, long-run/total multiplier. =n
k-period distributed lag model Yt=A+B0Xt+B1Xt-1+B2Xt-2+…+BkXt-k+ut （14.13） B0＞B1＞B2 The effect of a unit change in the value of the explanatory variable is felt over k periods. ◼ B0 : the short-run /impact multiplier, which means “the change in the mean value of Y following a unit change in X in the same period” ◼ (B0+B1 ),(B0+B1+B2 ):interim/intermediate multipliers. “ ………in the next, following period” long-run/total multiplier. = = + + + + k i n Bi B0 B1 B2 Bk
2. Problems in estimation of distributed Lag Models. The distributed lag model (14.13) does not violate any of the standard assumptions of the classical linear regression model(CLRM), but when we use the ols to estimate. there are some practical problems: (1 Economic theory does not tell us how many lagged values of the explanatory variables should be introduced
◼ 2.Problems in estimation of Distributed Lag Models: The distributed lag model （14.13） does not violate any of the standard assumptions of the classical linear regression model (CLRM), but when we use the OLS to estimate, there are some practical problems: （1）Economic theory does not tell us how many lagged values of the explanatory variables should be introduced