Estimation

Remember that the model is

APt = bp1Et APt+1 + bp1 APt —1 + bp2Xt + Zpt, which can be rewritten as

n = jEtnt+1 + SxH + vpt.

The model is usually estimated by means of instrumental variables, using the ‘errors in variables’ method (evm)—where expected values are replaced by
actual values and the expectational errors:

Подпись: bp2Подпись: Ext+iint = Y^t+i + 5xt + vpt – YVt+i – (A.22)

The implications of estimating the model by means of the ‘errors in variables’ method is to induce moving average errors. Following Blake (1991), this can be readily seen using the expectational errors as follows.

image376
1. Lead (A.15) one period and subtract the expectation to find the RE error:

3. Finally, re-express in terms of original variables, again using Apt = n + aApt-i:

Apt – aiApt-i = (— ) (Apt+i – aiApt) + ( ) xt + ( f— ) £pt

Подпись: Apt image378

W bpia2) biia. p

where we have exploited the two well-known relationships between the roots:

Подпись: ai + a2Подпись: ai a21

f ’

bpi

bpi

bf ‘

bpi

So even though the original model has white noise errors, the estimated model will have first-order moving average errors.

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