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)

j-о

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...

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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...

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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...

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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,

A

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...

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