Category Advanced Econometrics Takeshi Amemiya

Autoregressive Models

5.1.2 First-Order Autoregressive Model

Consider a sequence of random variables {y,}, t = 0, ± 1, ±2,. . . , which follows

Уі = РУі- i + e(, (5-2.1)

where we assume

Assumption А. {є,}, t = 0, ± 1, ±2,…………. are i. i.d. with Ее, = 0 and

Ee}= a2 and independent of y,-i, yf_2,….

Assumption B. p < 1.

Assumption C. Ey, — 0 and Ey, yt+h = yh for all t. (That is, {}>,} are weakly stationary.)

Model (5.2.1) with Assumptions A, B, and C is called a stationaryfirst-order autoregressive model, abbreviated as AR(1).

From (5.2.1) we have

У, = Psy,-s + 2 fa-i – (5-2.2)


But 1іт,_ю E(psy,_s)2 = 0 because of Assumptions В and C. Therefore we have

Уг-^pbt-,, (5.2.3)

which means that the partial summation of the right-hand side converges to y, in the mean square. The model (5.2...

Read More

Constrained Least Squares Estimator (CLS)

The constrained least squares estimator (CLS) of /?, denoted by Д is defined to be the value of P that minimizes the sum of squared residuals

S(P) = (y-XP)'(y-XP) (1A2)

under the constraints (1.4.1). In Section 1.2.1 we showed that (1.4.2)Js mini­mized without constraint at the least squares estimator fi. Writing S(fi) for the sum of squares of the least squares residuals, we can rewrite (1.4.2) as

S(P) = S(P) + (P-P)’X’X(P-P). (1.4.3)

Instead of directly minimizing (1.4.2) under (1.4.1), we minimize (1.4.3) under ^1.4.1), which is mathematically simpler.

Put p — P = 6 and Q’ji—c = y. Then, because S(P) does not depend on P, the problem is equivalent to the minimization of S’X’XS under Q’S = y...

Read More

General Results

4.1.1 Consistency

Because there is no essential difference between maximization and minimiza­tion, we shall consider an estimator that maximizes a certain function of the parameters. Let us denote the function by QT(у, в), where у = (Уі, y2, • . . , Утї is a Г-vector of random variables and в is a А-vector of parameters. [We shall sometimes write it more compactly as £?r(0).] The vector в should be understood to be the set of parameters that characterize the

distribution of y. Let us denote the domain of в, or the parameter space, by 0 and the “true value” of в by 0O. The parameter space is the set of all the possible values that the true value 60 can take...

Read More

The Case of an Unknown Covariance Matrix

In the remainder of this chapter, we shall consider Model 6 assuming that 2 is unknown and therefore must be estimated. Suppose we somehow obtain an estimator 2. Then we define the feasible generalized least squares (FGLS) estimator by

Подпись: (6.2.1)fif = (X’2-1X)-1X’2_ly,


assuming 2 is nonsingular.

For fip to be a reasonably good estimator, we should at least require it to be consistent. This means that the number of the firee parameters that character­ize 2 should be either bounded or allowed to go to infinity at a slower rate than T. Thus one must impose particular structure on 2, specifying how it depends on a set of free parameters that are fewer than Tin number. In this section we shall consider five types of models in succession...

Read More