1 Example 8 (Reduced form estimation of the exchange rate, a

1
Example 8 (Reduced form estimation of the exchange rate, a problem of estimation)
Aufgabe / Problem
Ein Teilmodell mehrerer Gleichungen, das aus einer einzigen Schätzgleichung
bestimmt wird (Eine Anwendung für den Wechselkurs ) /
A submodel of several equations, which can be estimated by one single equation, the
example of the monetary submodel of exchange
Ein lineares System /a linear system
Quellen/sources:
J.A. Frankel, On the mark: A Theory of Floating Exchange Rates Based on
Real Interest Rate Differentials, American Economic Review 69, 1979,
610-519
J.A. Frankel, On the mark: Reply, American Economic Review 71, 1981
No.5, 1075-1082
J.A. Frankel, The mystery of the multiplying marks: a modification to the
monetary model, The Review of Economics and Statistics 64, 1982, 515-519
Journal of Economic Dynamics and Control, noch einsetzen
a) Leiten Sie eine geeignete einzelne Reduzierte Form Gleichung ab, aus der Sie die
Parameter der Struktur herleiten können.
Hinweis: Das Modell besteht aus einigen Gleichungen, sowie einigen strengen
Annahmen über die Größen.
b) Bestimmen Sie für ein Land Ihrer Wahl, z.B. Österreich, die Schweiz oder
Deutschland für monatliche Zahlen die numerischen Werte der Strukturkoeffizienten
Ganz erstaunlich, wie man es umformen kann, ist die ursprüngliche Hypothese noch
erkennbar? [das ist die Frage der Identifikation]
Wie steht es mit der Fehlerstruktur, die Frankel schließlich am Ende daranhängt?
Was ergibt eine Simulation mit deutschen Zahlen?
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Die endogenen und exogenen Variablen:
Endogenous Variables / Exogenous Variables:
d
forward discount = log of the forward rate - log of the spot rate
e
log of the spot rate
e
equilibrium exchange rate
e-e
deviation from the trend (exchange differential)
m
log of the domestic money supply
m*
log of the foreign money supply
m
log of the domestic equilibrium money supply
m*
log of the foreign equilibrium money supply
π
p
p*
p
π*
p*
π -π*
r
r*
t
y
y*
current rate of domestic inflation
log of the domestic price level
log of the foreign price level
log of the domestic equilibrium price level
current rate of foreign inflation
log of the foreign equilibrium price level
inflation differential
log of one plus the domestic interest rate
log of one plus the foreign interest rate
time
log of the domestic production
log of the foreign production
A linear structure / eine lineare Struktur.
Covered (closed ) interest parity:
(1)
dt = rt - r*t if d is the actual value (actual rate of depreciation)
Uncovered (open) interest parity:
(1)'
dt ≡ rt - r*t if d is the expected value (expected rate of depreciation)
Rate of depreciation
(2)
dt = -θ(et - et ) + (π -π*) [+ u2t ]
Real interest [(1)-(2) ⇔ (i)]
(i)
(et - et ) = -[(rt - π t ) - (r*t - π∗ t )]/θ = -[(rt - r*t ) - (π t - π∗ t )]/θ
(i)'
(et → et ) ⇒ [(rt → rt ) ∧ (r* t → r∗t )] ⇒ [ (r - r∗) ≡ (π -π*) ]
(Long run) purchasing power parity
(3)
e ≡ p - p*
A (conventional) domestic money demand function
(4)
m t = pt + φyt - λrt [+ u 4t ]
A (conventional) foreign money demand function
(5)
m * t = p* t + φy* t - λr* t [+ u 5t ]
Money demand differential [(4)-(5) ⇒ (ii)]
(ii)
(m t - m* t ) = (pt - p*t ) + φ(yt - y* t ) - λ(r* t - r*t ) [+ (u 4t - u 5t )]
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(ii) ∧ (i)' ⇒
(ii)'
(m t - m* t ) = (pt - p*t ) + φ(yt - y* t ) - λ(π t - π* t )
(ii)' ∧ (3)' ⇒
(iii)
(pt - p*t ) = (mt - m* t ) - φ(yt - y* t ) + λ(π t - π* t )
(iii) ∧ [(pt →pt )∧(p* t → p* t )]∧[(m t →m t )∧(m* t → m * t )]∧[(yt → yt )∧(y* y* t )] ⇒
(iii)'
(pt - p*t ) = (m t - m * t ) - φ(yt - y* t ) + λ(π t - π* t )
(iii) ∧ (3) ⇒
(iv) et = (m t - m * t ) - φ(yt - y* t ) + λ(π t - π* t )
The equilibrium relation (iv) (for e) is substituted back into (i) and assuming, that the
actual values are the equilibrium values (?) then observable relation is:
(et - et ) = -[(rt - r*t ) - (π t - π∗ t )]/θ ⇔ et = (mt - m* t ) - φ(yt - y* t ) + λ(π t - π* t )
∴
(v)
et = (mt - m* t ) - φ(yt - y* t ) + [1/θ + λ ] (π t - π∗ t ) + ut
From equation (v) the original coefficients can be identified.
In fact Frankel estimates the parameters of a single equation by monthly data 7.19742.1978 (p. 615)
et = β 1 + (r t - r*t )β 2 + (m t - m* t )β 3 + (yt - y* t )β 4 + (π t - π∗ t )β 5 + ut
(SUBMODEL, IDENTIFICATION, SELS)