Springer Texts in Business and Economics

TheAR(1) model. From (5.26), by continuous substitution just like (5.29), one could stop at ut_s to get

ut = Psut_s + ps 1£t_s+1 + ps 2©t_s+2 + .. + pet-1 + ©t for t > s.

Note that the power of p and the subscript of e always sum to t. Multiplying both sides by ut-s and taking expected value, one gets

E (utut_s) = psE (uj-s) + ps-1E (©t_s+1ut_s) + .. + pE (et_1ut_s) + E (©tut_s)
using (5.29), ut_s is a function of et_s, past values of et_s and uo. Since uo is independent of the e’s, and the e’s themselves are not serially correlated, then ut_s is independent of et, et_i,..., et_s+i. Hence, all the terms on the right hand side of E(utut_s) except the first are zero. Therefore, cov(ut, ut_s) = E(utut_s) = psau2 for t > s.

5.7 Relative Efficiency of OLS Under the AR(1) Model.

T T T T

a. "ols = J2 xtyt xt2 = " + J2 xtut xt2 with E ("ols) = " since xt

t=i t=i t=i t=i

and ut are independent. Also,

2 / T T 2 T / T 2

var ("ols) = E ("ols _ " = E xtut/£xt2 = E xt2E К xt2

t=i t=i t=i t=i

C E і EE xjxsutu, 3

using the fact that E(utus) = pjt_sj 0,2 as shown in problem 6.

Alternatively, one can use matrix algebra, see Chap. 9. For the AR(1) model, Й = E(uu') is given by Eq. (9.9) of Chap. 9. So

+ (pT_3x1xT + pT_2 x2xt + .. + xT)

collecting terms, we get

ЛЛ X2

XT

+ (1 + p2)x| - pX4X3 + .. + xT - pXT—1Xt] .

b. For Xt following itself an AR(1) process: Xt = Xxt-1 + vt, we know that к is the correlation of Xt and Xt-1, and from problem 5.6, correl(xt, Xt-s) = ks. T-1 T As T! 1, к is estimated well by Also, к2 is estimated t=1 t=1

T—2 T

well by xtXt+2 x2, etc. Hence

t=1 t=1

r( " pw)

var("ols 0 + p2 — 2pk) 0 + 2pk + 2p2X2 + ..)

(1 — p2)(1 — pk)

(1 C p2 — 2pk)(1 C pk)

where the last equality uses the fact that (1 C pk)/(1 — pk) = (1 C 2pk C 2p2k2 C..). For k = 0, or p = k, this asy eff("ols) is equal to (1 — p2)/(1 C p2).

c. The asy eff("ols) derived in part (b) is tabulated below for various values of p and k. A similar table is given in Johnston (1984, p. 312). For p > 0, loss in efficiency is big as p increases. For a fixed k, this asymptotic efficiency drops from the 90 to a 10% range as p increases from 0.2 to 0.9. Variation in k has minor effects when p > 0. For p < 0, the efficiency loss is still big as the absolute value of p increases, for a fixed k. However, now variation in k has a much stronger effect. For a fixed negative p, the loss in efficiency decreases with k. In fact, for k = 0.9, the loss in efficiency drops from 99% to 53 as p goes from —0.2 to —0.9. This is in contrast to say k = 0.2 where the loss in efficiency drops from 93 to 13% as p goes from —0.2 to —0.9.

(Asymptotic Relative Efficiency of pols) x 100

' P '

Asymptotic Relative Efficiency

100

80

60

40

20

0

-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

d. Ignoring autocorrelation, s2 x2 estimates a2 x^, but

t=i t=i

asy. var (Pols)

so the asy. bias in estimating the var("ols) is

and asy. proportionate bias = —2pX/(i + pX).

Percentage Bias in estimating var((°ols)

P

This is tabulated for various values of p and X. A similar table is given in Johnston (1984, p. 312).

For p and X positive, var ols^ is underestimated by the conventional for­mula. For p = X = 0.9, this underestimation is almost 90%. For p < 0, the var("ols) is overestimated by the conventional formula. For p = —0.9 and X = 0.9, this overestimation is of magnitude 853%. e. et = yt — yt = (" — "ols)xt + ut

Hence, p e2 = "ols — " p x2 + p u2 — 2 "ols — " p xtut and

t=i t=i t=i t=i

So that E(s2) = E ^ P e2/(T — 1)

If p = 0, then E(s2) = a2. If xt follows an AR(1) model with parameter X, then for large T, E(s2) = au2 (Т - 1-^ /(T - 1).

For T = 101, E(s2) = ctu2(101 - iC-pX /100.

This can be tabulated for various values of p and X. For example, when p = X = 0.9, E(s2) = 0.915 a2.