# Daily Archives July 12, 2015

## An example

As an example, and in the process of illustrating different techniques, we will work out the dynamic properties of the wage-price model of Section 9.2.2. This involves evaluating the stability of the model, and the long-run and dynamic multipliers. Disregarding taxes and short-run effects, the systematic part of the model is on matrix form.

w

P

a

u

РЧ t

Steady-state properties from cointegration The long-run elasticities of the model are, from the cointegration analysis.

w = p + a — 0.1u p = 0.7(w — a)+ 0.3pi,

so the long-run multipliers of the system should be easily obtained by solving for wages and prices. For wages:

0.7(w — a) + 0.3p + a — 0.1u —0.7a + 0.3pi + a — 0.1u

0.3 0.1

— a——— u + pi

0.3 0.3 **[118]** a — 0.33u + pi.

Then for prices:

p = 0.7(w — a)+ 0...

Read More## Calculation of interim multipliers in a linear dynamic model: a general exposition

Interim multipliers provide a simple yet powerful way to describe the dynamic properties of a dynamic model. We follow Lutkepohl (1991) and derive the dynamic multipliers in a simultaneous system of n linear dynamic equations with n endogenous variables yt and m exogenous variables xt. The structural

form of the model is given by:

q q

r0yt riyt-i + – i+£t – (A.23)

i=1 i=0

To investigate the dynamic properties of the model it will be more convenient to work with the reduced form of the model:

yt=£ Aiyt-i+£ Bi xt-i+ut (A.24)

i=1 i=0

defining the n x n matrices Ai = Г-1Гі, i = 1,…,q, and the n x m matrices

Bi = r-1Di, i = 0,…,q. The reduced form residuals are given by ut= r-1et.

It is also useful to define the autoregressive final form of the model as:

yt = A(L)-1B(L)xt + A(L)-1ut (A.25)

Read More## Does the MA(1) process prove that the forward solution applies?

Assume that the true model is

Apt = bp1^pt-1 + £pt: bpl < 1

and the the following model is estimated by means of instrumental variables

Apt = bpiApt+i + £pt – What are the properties of efptl

£pt = APt – bpiAPt+i-

Assume, as is common in the literature, that we find that bfpi и 1. Then

£fpt и APt – APt+i = – A2Pt+i

= — [£pt+i + (bpi — 1)^pt + ••• ]•

So we get a model with a moving average residual, but this time the reason is not forward-looking behaviour but mis-specification.

Read More## 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:

nt = 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.

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— ) ...

Read More## Factorization

Finally, we shall take a look at this very elegant method introduced by Sargent. It consists of the following steps:

1. Write the model in terms of lead – and lag-polynomials in expectations.

2. Factor the polynomials, into one-order polynomials, deriving the roots.

3. Invert the factored one-order polynomials into the directions of converging forward polynomials of expectations.

Again, we use the simplifying definition

zt = bp2Xt + £pt:

so the model is again

Apt = bp1EtApt+i + bpiApt-i + zt.

Note that the forward, or lead, operator, F, and lag operator, L, only work on the variables and not expectations, so:

LEtzt = Etzt-i

FEtZt = Etzt+i L-i = F.

The model can then be written in terms of expectations as:

-bpiEtApt+i + EtApt – bpEApt-i = EtZt, and using the lead – and lag-operators:

(-bpiF...

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