 Springer Texts in Business and Economics

# Restricted MLE and Restricted Least Squares

Maximizing the likelihood function given in (7.16) subject to R/3 = r is equivalent to minimizing the residual sum of squares subject to R/3 = r. Forming the Lagrangian function

Ф(в, f) = (y — X0Y(y — X@) + 2g!(Rf3 — r) (7.32)

and differentiating with respect to в and f one gets

дФ(в, д)/дв = —2Xly + 2XlXp + 2R'f = 0 (7.33)

д V(e, f)/df = 2R — r) = 0 (7.34)

Solving for f, we premultiply (7.33) by R(XIX)-1 and use (7.34)

P = [R(XIX )-1Ri]-1(R^ols — r) (7.35)

Substituting (7.35) in (7.33) we get

Pels = Pols — (X 'X )-1Rl[R(X X )-1R1]-1 (RPols — r) (7.36)

The restricted least squares estimator of в differs from that of the unrestricted OLS estimator by the second term in (7.36) with the term in parentheses showing the extent to which the unrestricted OLS estimator satisfies the constraint. Problem 12 shows that eRLS is biased unless the restriction Re = r is satisfied. However, its variance is always less than that of eoLS. This brings in the trade-off between bias and variance and the MSE criteria which was discussed in Chapter 2.

The Lagrange Multiplier estimator fi is distributed N(0,a2[R(X'X)-1R']-1) under the null hypothesis. Therefore, to test f = 0, we use

P'[R(X 'X )-1R']p/a2 = (RPols — r)'[R(X'X )-1R']-1(RRols — r)/a2 (7.37)

Since f measures the cost of imposing the restriction Re = r, it is no surprise that the right hand side of (7.37) was already encountered in (7.29) and is distributed as x^.

Добавить комментарий

## Springer Texts in Business and Economics

### The General Linear Model: The Basics

7.1 Invariance of the fitted values and residuals to non-singular transformations of the independent variables. The regression model in (7.1) can be written as y = XCC-1" + u where …

### Regression Diagnostics and Specification Tests

8.1 Since H = PX is idempotent, it is positive semi-definite with b0H b > 0 for any arbitrary vector b. Specifically, for b0 = (1,0,.., 0/ we get hn …

### Generalized Least Squares

9.1 GLS Is More Efficient than OLS. a. Equation (7.5) of Chap. 7 gives "ois = " + (X'X)-1X'u so that E("ois) = " as long as X and u …

## Как с нами связаться:

Украина:
г.Александрия
тел./факс +38 05235  77193 Бухгалтерия
+38 050 512 11 94 — гл. инженер-менеджер (продажи всего оборудования)

+38 050 457 13 30 — Рашид - продажи новинок
e-mail: msd@msd.com.ua
Схема проезда к производственному офису:
Схема проезда к МСД

Партнеры МСД

## Контакты для заказов шлакоблочного оборудования:

+38 096 992 9559 Инна (вайбер, вацап, телеграм)
Эл. почта: inna@msd.com.ua