4.5.1 Likelihood Ratio and Related Tests
Let Ux, 0) be the joint density of a Г-vector of random variables x = (Xi, x2,. . . , xTY characterized by a ЛГ-vector of parameters 6. We assume all the conditions used to prove the asymptotic normality (4.2.23) of the maximum likelihood estimator 6. In this section we shall discuss the asymptotic tests of the hypothesis
h(0) = O, (4.5.1)
where h is a ^-vector valued differentiable function with q<K. We assume that (4.5.1) can be equivalently written as
В = r(a),
where a is a p-vector of parameters such that p = K— q. We denote the constrained maximum likelihood estimator subject to (4.5.1) or (4.5.2) as в = Ha).
Three asymptotic tests of (4.5... Read More
In this section we shall define the least squares estimator of the parameter P in Model 1 and shall show that it is the best linear unbiased estimator. We shall also discuss estimation of the error variance a2.
1.1.2 Definition of Least Squares Estimators of p and a2
The least squares (LS) estimator ft of the regression parameter fi in Model 1 is defined to be the value of P that minimizes the sum of squared residuals2
S(fi) = ( у-ХД)'(у-ХД) (1.2.1)
= y’y-2y ‘Xfi + fi’X’Xfi.
Putting the derivatives of S(fi) with respect to fi equal to 0, we have
^ = -2X’y + 2X’X0 = O, (1.2.2)
where dS/dfi denotes the AT-vector the Ah element of which is dS/dfih fit being the zth element of fi. Solving (1.2.2) for fi gives
)§ = (X’X)-1X’y. (1.2.3)
Clearly, S(fl) attains the global minimum at fi.
L... Read More
Let. . • , yrbe independent observations from a symmetric distribu
tion function F[(y — fi)la such that F(0) = Thus ц is both the population mean and the median. Here a represents a scale parameter that may not necessarily be the standard deviation. Let the order statistics be y(1) ё Уз, S • • • S ym. We define the sample median fi to be y(( т+т) if Tis odd and any arbitrarily determined point between y(T/2) and У«г/2)+і) if Tis even. It has long been known that fi would be a better estimator of (i than the sample mean /2 = Г-1 2£.i y, if F has heavier tails than the normal distribution. Intuitively speaking, this is because fi is much less sensitive to the effect of a few wild observations than fi is.
It can be shown that fi is asymptotically normally distributed wi... Read More
A stationary second-order autoregressive model, abbreviated as AR(2), is defined by
У, = РіУ,-і +р2У,-г + е» t = 0, ±1,±2……………. (5.2.16)
where we assume Assumptions A, C, and
Assumption B’. The roots of z2 — pxz — p2 — 0 lie inside the unit circle. Using the lag operator defined in Section 5.2.1, we can write (5.2.16) as (1 – PlL-p2L2)y, = et. (5.2.17)
(1 – цхЩ – p2L)yt = e„
where Ці and ц2 are the roots of z2 – pxz – p2 = 0. Premultiplying (5.2.18) by (1 —p1L)~l( 1 —fi2L)~l, we obtain
Convergence in the mean-square sense of (5.2.19) is ensured by Assumption B’. Note that even if px and p2 are complex, the coefficients on the є,_, are always real.
The values of px and p2 for which the condition |^,|, p2 < 1 is satisfied correspond to the ... Read More