Category Advanced Econometrics Takeshi Amemiya

Asymptotic Normality of the Median

Let {Yt), t = 1, 2,. . . , T, be a sequence of i. i.d. random variables with common distribution function Fand density function f. The population me­dian M is defined by

F(Af) = ^. (4.6.1)

We assume F to be such that M is uniquely determined by (4.6.1), which follows from assuming/(у) > 0 in the neighborhood of у = M. We also as­sume that f'(y) exists for у > M in a neighborhood of M. Define the binary random variable W’/a) by

Wia) = 1 if y, isa

Подпись: (4.6.2)= 0 if У,<а

for eveiy real number a. Using (4.6.2), we define the sample median ттЪу

Подпись: mT = infimage348(4.6.3)

The median as defined above is clearly unique.11

image349 Подпись: (4.6.4)

The asymptotic normality of mT can be proved in the following manner: Using (4.6.3), we have for any у

Define

P,= l-P(Yt<M+ T~l/2y).

Then, because by a Taylor expansion

P, = ~ T~ipf(M)y ...

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Least Squares as Best Linear Unbiased Estimator (BLUE)

The class of linear estimators of 0 can be defined as those estimators of the form С’ у for any TX К constant matrix C. We can further restrict the class by imposing the unbiasedness condition, namely,

EC’y=0 for all 0. (1.2.28)

Inserting (1.1.4) into (1.2.28), we obtain

C’X = I. (1.2.29)

Clearly, the LS estimator 0 is a member of this class. The following theorem proves that LS is best of all the linear unbiased estimators.

Theorem 1.2.1 (Gauss-Markov). Letf* = C’y where C isa TX К matrix of constants such that C’X = I. Then 0 is better than 0* if 0 Ф 0*.

Proof. Because 0* = 0 + C’ u because of (1.2.29), we have

V0* = EC’m’C (1.2.30)

= <r2C’C

= o2(X’X)-1 + er2[C’ – (X’X^X’HC’ – (Х’ХГ’ХТ.

The theorem follows immediately by noting that the second term of the last line of (1.2...

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Distribution Function

Definition 3.1.5. The distribution function F(x) of a random variable X(a>) is defined by

F(x) = P{o)Х(оз) < x).

Note that the distribution function can be defined for any random variable because a probability is assigned to every element of A and hence to {cojX(eo) < x) for any x. We shall write P{o)X(oj) < x} more compactly as P(X<x).

A distribution function has the properties:

(i) F(-«) = 0.

(ii) FM=1.

(iii) It is nondecreasing and continuous from the left.

[Some authors define the distribution function as F(x) = P{toX(a>) ё x}. Then it is continuous from the right.]

Using a distribution function, we can define the expected value of a random variable whether it is discrete, continuous, or a mixture of the two...

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The Almon Lag

Almon (1965) proposed a distributed-lag model

Подпись: N

yl==’ZPjX,+i-j+v»

1-І

in which 0X. . , 0N lie on the curve of a gth-order polynomial; that is,

/«, j=l,2,…,N. (5.6.6)

,Y and S=(S0,Sl,. . . , Sq)’, we

(5.6.7)

The estimation of 6 can be done by the least squares method. Let X be a TXN matrix, the /,jth element of which is Then

6 = (J’X’XJ^’J’X’y and 0 = JS.3 Note that 0 is a special case of the con­strained least squares estimator (1.4.11) where R = J and c = 0.

By choosing N and q judiciously, a researcher can hope to attain both a reasonably flexible distribution of lags and parsimony in the number of pa­rameters to estimate. Amemiya and Morimune (1974) showed that a small order of polynomials {q — 2 or 3) works surprisingly well for many economic time series.

Some researchers p...

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