THE ECONOMETRICS OF MACROECONOMIC MODELLING

A.2 Solving and estimating rational expectations models

To make the exposition self-contained, this appendix illustrates solution and estimation of simple models with forward looking variables—the illustration being the hybrid ‘New Keynesian Phillips curve’. Finally, we comment on a problem with observational equivalence, or lack of identification within this class of models.

A sufficiently rich data generating process (DGP) to illustrate the tech­niques are

Apt = bp1EtApt+i + 6p1Apt-i + bp2xt + £pt, (A.9)

xt = bxxt-1 + £xt, (A.10)

where all coefficients are assumed to be between zero and one. All of the techniques rely on the law of iterated expectations,

EtEt+kxt+j = Etxt+j, к < j,

saying that your average revision of expectations, given more information, will be zero.

A.2.1 Repeated substitution

This method is the brute force solution, and therefore cumbersome. But since it is also instructive to see exactly what goes on, we begin with this method.

We start by using a trick to get rid of the lagged dependent variable, following Pesaran (1987, pp. 108-109), by implicitly defining nt as

Apt = nt + aApt-i, (A.11)

where a will turn out to be the backward stable root of the process of Apt. We take expectations one period ahead

EtApt+i = Etnt+i + aEtApt,

EtApt+i = Etnt+i + ant + a^Apt-i.

Next, we substitute for EtApt+i into original model: nt + aAp_i = bfpi (Et^t+i + ant + a2_1Apt_i) + bpiAp— + bp2xt + £pt

image290

bf

bpi

1 - bpia

 

image291

Etnt+i +

 

Apt-i

 

nt

 

1 - bPia

 

Подпись: +Подпись: xt +

bp2 [114]

1

1 - bpia

The parameter a is defined by

Подпись: or Подпись: a2 f a + f bfpi bfpi Подпись: 0 Подпись: (A.12)

bpia2 - a + bpi = 0

Подпись: ai a2 Подпись: (A.13)
image300

with the solutions

Подпись: ai Подпись: < 1.
image303

The model will typically have a saddle-point behaviour with one root bigger than one and one smaller than one in absolute value. In the following we will use the backward stable solution, defined by:

image304

In passing it might be noted that the restriction bbpi = 1 - bfpi often imposed in the literature implies the roots

Подпись: nt Подпись: bf bpi 1 - bfpiai Подпись: Et nt+i + Подпись: bp2 1 - bfpiai Подпись: xt + Подпись: 1 1 - bfpiai Подпись: £pt.

as given in (A.13) as before. We choose |ai| < 1 in the following. So we now have a pure forward-looking model

Finally, using the relationship

Подпись: ai + a21

bf

bpi

between the roots,1 so:

Подпись: (A.14)

Подпись: 1 - bpiai = 6pi«2,Подпись: bp1a2Подпись: 1Подпись: nt = YEtnt+1 + 5xt + Vpt.image318the model becomes

£pt (A.15)

(A.16)

Following Davidson (2000, pp. 109-110), we now derive the solution in two steps:

1. Find Etnt+i.

2. Solve for nt.

Find Etnt+i Define the expectations errors as:

Подпись: (A.17)nt+i = nt+i — Etnt+i.

We start by reducing the model to a single equation:

nt = Ynt+i + SbxXt-i + S£xt + Vpt — Ynt+i.

Solving forwards then produces:

nt = Y(Y^t+2 + Sbxxt + 5ext+i + Vpt+i - YYt+2)

+ SbxXt-i + S£xt + vpt — Y'Qt+i = (SbxXt-i + S£xt + Vpt — YVt+i)

+ Y (SbxXt + S£xt+i + vpt+i — YVt+2) + (Y )2nt+2

n

— ^~~^(Y)j (SbxXt+j-i + b£xt+j + vpt+j — YYt+j+i) + (Y)n+int+n+i. j=0

By imposing the transversality condition:

lim (Y)n+int+n+i = 0

and then taking expectations conditional at time t, we get the ‘discounted solution’:

Ж

Et nt+i = ^~~1(Y)j (SbxEtXt+j + SEt£xt+j+i + Etvpt+j+i — YEtVt+j+2) j=0

Ж

= J2(y )j (sbxEtXt+j). [115]

However, we know the process for the forcing variable, so:

Et-ixt = bxxt-,

Etxt = xt,

Etxt+1 bxxt,

Etxt+2 = Et(Et+lxt+2) = Etbxxt+1 = b“Xxt,

Etxt+j = bj xt.

We can therefore substitute in:

Ж

Etnt+i = J2(Y)j (Sbxbx xt)

j=0

Ж

= Sb^2 (Ybx)j xt j=0 Sbx

Подпись:
Подпись: nt

Подпись: Y Подпись: Sbx 1 - Ybx Подпись: xt + Sxt + vpt.

lEt nt+i + 5xt + vpt

image324

Solve for nt Finally, using (A.11) and (A.16) we get the complete solution:

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