# Category Advanced Econometrics Takeshi Amemiya

## Asymptotic Tests and Related Topics

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...

## Theory of Least Squares

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)

dp

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...

## Independent and Identically Distributed Case

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...

## Second-Order Autoregressive Model

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

(5.2.19)

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 ...