International Macroeconomics and Finance: Theory and Empirical Methods Phần 2 pdf

2.1. UNRESTRICTED VECTOR AUTOREGRESSIONS 31

is labeled a in (2.12). The forecast error variance in q1t attributable to

innovations in q2t is given by the first diagonal element in the second

summation (labeled b). Similarly, the second diagonal element of a is

the forecast error variance in q2t attributable to innovations in q1t and

the second diagonal element in b is the forecast error variance in q2t

attributable to innovations in itself.

A problem you may encountered in practice is that the forecast error

decomposition and impulse responses may be sensitive to the ordering

of the variables in the orthogonalizing process, so it may be a good

idea to experiment with which variable is q1t and which one is q2t. A

second problem is that the procedures outlined above are purely of a

statistical nature and have little or no economic content. In chapter

(8.4) we will cover a popular method for using economic theory to

identify the shocks.

Potential Pitfalls of Unrestricted VARs

Cooley and LeRoy [32] criticize unrestricted VAR accounting because

the statistical concepts of Granger causality and econometric exogene￾ity are very different from standard notions of economic exogeneity.

Their point is that the unrestricted VAR is the reduced form of some

structural model from which it is not possible to discover the true rela￾tions of cause and effect. Impulse response analyses from unrestricted

VARs do not necessarily tell us anything about the effect of policy in￾terventions on the economy. In order to deduce cause and effect, you

need to make explicit assumptions about the underlying economic en￾vironment.

We present the CooleyóLeRoy critique in terms of the two-equation

model consisting of the money supply and the nominal exchange rate

m = ²1, (2.13)

s = γm + ²2, (2.14)

where the error terms are related by ²2 = λ²1 + ²3 with ²1

iid

∼ N(0, σ2

1),

²3

iid

∼ N(0, σ2

3) and E(²1²3) = 0. Then you can rewrite (2.13) and (2.14)

as

m = ²1, (2.15)

32 CHAPTER 2. SOME USEFUL TIME-SERIES METHODS

s = γm + λ²1 + ²3. (2.16)

m is exogenous in the economic sense and m = ²1 determines part of ²2.

The effect of a change of money on the exchange rate ds = (λ + γ)dm

is well defined.

A reversal of the causal link gets you into trouble because you will

not be able to unambiguously determine the effect of an m shock on

s. Suppose that instead of (2.13), the money supply is governed by

two components, ²1 = δ²2 + ²4 with ²2

iid

∼ N(0, σ2

2), ²4

iid

∼ N(0, σ2

4) and

E(²4²2) = 0. Then

m = δ²2 + ²4, (2.17)

s = γm + ²2. (2.18)

If the shock to m originates with ²4, the effect on the exchange rate

is ds = γd²4. If the m shock originates with ²2, then the effect is

ds = (1 + γδ)d²2.

Things get really confusing if the monetary authorities follow a feed￾back rule that depends on the exchange rate,

m = θs + ²1, (2.19)

s = γm + ²2, (2.20)

where E(²1²2) = 0. The reduced form is

m = ²1 + θ²2

1 − γθ , (2.21)

s = γ²1 + ²2

1 − γθ . (2.22)

Again, you cannot use the reduced form to unambiguously determine

the effect of m on s because the m shock may have originated with ²1,

²2, or some combination of the two. The best you can do in this case

is to run the regression s = βm + η, and get β = Cov(s, m)/Var(m)

which is a function of the population moments of the joint probability

distribution for m and s. If the observations are normally distributed,

then E(s|m) = βm, so you learn something about the conditional ex￾pectation of s given m. But you have not learned anything about the

effects of policy intervention.

2.1. UNRESTRICTED VECTOR AUTOREGRESSIONS 33

To relate these ideas to unrestricted VARs, consider the dynamic

model

mt = θst + β11mt−1 + β12st−1 + ²1t, (2.23)

st = γmt + β21mt−1 + β22st−1 + ²2t, (2.24)

where ²1t

iid

∼ N(0, σ2

1), ²2t

iid

∼ N(0, σ2

2), and E(²1t²2s) = 0 for all t, s.

Without additional restrictions, ²1t and ²2t are exogenous but both mt

and st are endogenous. Notice also that mt−1 and st−1 are exogenous

with respect to the current values mt and st.

If θ = 0, then mt is said to be econometrically exogenous with

respect to st. mt, mt−1, st−1 would be predetermined in the sense that

an intervention due to a shock to mt can unambiguously be attributed

to ²1t and the effect on the current exchange rate is dst = γdmt. If

β12 = θ = 0, then mt is strictly exogenous to st.

Eliminate the current value observations from the right side of (2.23)

and (2.24) to get the reduced form

mt = π11mt−1 + π12st−1 + umt, (2.25)

st = π21mt−1 + π22st−1 + ust, (2.26)

where

π11 = (β11 + θβ21)

(1 − γθ) , π12 = (β12 + θβ22)

(1 − γθ) ,

π21 = (β21 + γβ11)

(1 − γθ) , π22 = (β22 + γβ12)

(1 − γθ)

umt = (²1t + θ²2t)

(1 − γθ) , ust = (²2t + γ²1t)

(1 − γθ) ,

Var(umt) = (σ2

1 + θ2σ2

2)

(1 − γθ)2 , Var(ust) = (γ2σ2

1 + σ2

2)

(1 − γθ)2 ,

Cov(umt, ust) = (γσ2

1 + θσ2

2)

(1 − γθ)2 .

⇐(14) (last 3

If you were to apply the VAR methodology to this system, you expressions)

would estimate the π coefficients. If you determined that π12 = 0,