Money demand in the Yugoslavian hyperinflation 1991-1994
Oslo, 6 May 2005
Cagan’s economic model:
1. Money demand
2. Inflation tax
3. Continuous time model
Problems in empirical analysis of Cagan’s model:
1.
2.
3.
4.
Discrete time data (monthly)
Inability to model the data to the end
Difficulties in making congruent model
Discrepancy between “optimal” and “actual” inflation tax
Methodology:
1. Review of economic theory
2. General-to-specific analysis of data
1
Outline:
1.
2.
3.
4.
5.
Theory and past empirical results
Data and institutional background
Model 1: m, p, s
• •
Model 2: m − p, p, s
Conclusion
2
1
Theory and past empirical results
Cagan’s (1956) Continuous time economic model:
mt −¶pt = −αEt − γ,
µ
∂E
= β (Ct − Et) .
∂t t
mt = log(Mt) is log money in bank notes
pt = log(Pt) is log prices
Et adaptive expectation of Ct = ∂pt/∂t
Empirically: Discretization: Assume Ct constant within each period
Data 1920:9 to 1923:7 - but hyperinflation ends 1923:11
α̂ = 5.76
Seignorage or inflation tax:
1/α maximises revenue under additional assumptions.
Empirical results:
Germany: α̂−1 = 0.183 corresponding to exp(α̂−1) − 1 = 20%
Compare with average monthly rate of inflation, ∆Pt/Pt−1, of 322%.
3
• Sargent (1977)
• Taylor (1991):
Assume: mt, pt ∼ I(2)
•
•
•
•
so
mt − pt, ∆pt ∼ I(1)
mt − pt = −α∆pet+1 + ζ t,
∆ept+1 = ∆pt+1 − εt+1,
where ∆ept+1 is (rational/adaptive/extrapolative) expectation of inflation.
So
¡
¢
2
−1
−1
∆ pt+1 = −α (mt − pt + α∆pt) + εt+1 − α ζ t .
Frenkel (1977)
Engsted (1996)
Abel, Dornbusch, Huizinga and Marcus (1979)
Juselius and Mladenović (2002):
Vector autoregression with an explosive root and unit roots.
4
2
Data and institutional background
60
(a) log prices
3
40
2
20
1
1991
60
1992
1993
1994
(b) changes in log prices
1991
(c) log money
1992
1993
(d) changes in log money
3
40
2
20
1
0
1991
60
1992
1993
1994
1991
(e) log exchange rate
4
1993
(f) changes in log exchange rate
3
40
2
20
1
0
1991
1992
1992
1993
1994
1991
5
1992
1993
2.0
(a) log real money: m t −p t
1.5
1.0
0.5
0.0
3
1991
1992
1993
1991
1992
1993
(b) growth of log prices: ∆p t
2
1
6
2.00
Cross−plot: m t −p t against ∆p t
1.75
1.50
1.25
1.00
0.75
0.50
0.25
93:10
0.00
93:6
−0.25
0.00
0.25
0.50
0.75
1.00
1.25
1.50
7
1.75
2.00
2.25
2.50
2.75
3.00
3
Model 1: Variables in levels
Unrestricted vector autoregression:
Consider Xt = (pt, mt, st) for 1990:12 to 1993:10
Xt =
3
X
Aj Xt−j + µc + µl t + εt.
j=1
Test
p
χ2normality (2) 1.3 [0.53]
FAR(1)(1, 20) 1.8 [0.19]
FAR(3)(3, 18) 0.6 [0.62]
FARCH(3)(3, 15) 0.1 [0.94]
m
6.0 [0.05]
1.0 [0.32]
0.8 [0.53]
0.2 [0.92]
s
Test
4.5 [0.11] χ2normality (6)
0.1 [0.82] FAR(1)(9, 39)
0.3 [0.81] FAR(3)(27, 29)
0.1 [0.93]
(p, m, s)
3.1 [0.79]
1.5 [0.20]
1.1 [0.44]
Characteristic roots
Re(z) 1.21 -0.42 -0.42 0.02 0.02 0.75 0.75 -0.31 0.09
Im(z)
0 0.84 -0.84 0.90 -0.90 0.33 -0.33
0
0
|z| 1.21 0.94 0.94 0.90 0.90 0.81 0.81 0.31 0.09
8
Cointegration/Co-explosive analysis:
0
∗
∆1∆ρXt = α1β ∗0
1 ∆ρ Xt−1 + αρ β ρ ∆1 Xt−1 + ψ∆1 ∆ρ Xt−1 + µc + εt
Granger-Johansen representation:
t
t
X
X
εs + Cρ
ρt−sεs + stationary + τ c + τ l t + τ ρρt
Xt ≈ C1
s=1
s=1
β 01C1 = 0
Results:
H (r) :
H1:
Hρ:
and
β 0ρCρ = 0
¢
¡
= 1,
rank α1β ∗0
1
β ∗0
X = pt − 0.35mt − st + 0.065t,
¡ 1 0 ¢t
rank αρβ ρ = p − 1 = 2,
µ
¶
1
0
−1
β 0ρ =
.
0 1 −1
Summary:
m − p, m − s, p − s ∼ I(1)
∆p, ∆m, ∆s ∼ I(x)
9
4
Model 2: transformed variables
µ
¶
∂Pt 1
∂pt
=
by ∆pt
Cagan approximates Ct =
∂t
∂t Pt
Another measure for cost of holding money is
Pt−1
ct = 1 −
= 1 − exp (−∆pt) .
Pt
Motivation:
Nominal money stock grows as
Mt = Mt−1 + δ t,
Divide through by Pt:
When ∆pt = pt − pt−1
µ
¶
δt
Mt Mt−1 Pt−1
+
=
Pt
Pt−1
Pt
Pt
= log(Pt/Pt−1) then:
ct = 1 − exp (−∆pt) ≈ ∆pt
10
for small ∆pt.
(a) m t −p t
1.00
4
(b) c t
0.75
2
0.50
0
0.25
−2
1991
(c) m t −s t
1992
1993
1994
1.00
1991
(d) d t
1992
1993
1994
1991
1992
1993
1994
8
0.75
6
0.50
0.25
4
1991
1992
1993
1994
ct = 1 − exp (−∆pt) ≈ ∆pt for small ∆pt
dt = 1 − exp (−∆st)
Measurement error in ∆pt eliminated as ct then ≈ 1
11
Unrestricted vector autoregression:
Consider Xt = (m − s, ct, dt) for 1990:12 to 1994:1
3
X
Xt =
Aj Xt−j + µc + εt.
j=1
(a) fit: m−s
8
6
4
1991 1992 1993 1994
(e) fit: c
1.0
0.5
1991 1992 1993 1994
(i) fit: d
1.0
0.5
1991 1992 1993 1994
(m) fit: d−c
0.2
0.0
−0.2
1991 1992 1993 1994
2
1
0
−1
(b) residuals: m−s
2
1
0
−1
1991 1992 1993 1994
(f) residuals: c
2
1
0
−1
−2
1991 1992 1993 1994
(j) residuals: d
2
1
0
−1
−2
1991 1992 1993 1994
(n) residuals: d−c
2
(c) QQ plot: m−s
0
−2
−2
1991 1992 1993 1994
12
−2
(d) Chow: m−s
0.5
−2 −1 0
1
(g) QQ plot: c
2
1
0
−1
−2
−2 −1 0
1
(k) QQ plot: d
2
1
0
−1
−2
−2 −1 0
1
(o) QQ plot: d−c
2
0
1.0
2
1.0
(h) Chow: c
1994
0.5
2
1.0
(l) Chow: d
1994
0.5
2
1.0
(p) Chow: d−c
1994
0.5
−1
0
1
2
1994
Cointegration analysis:
Hypothesis H(0)
H(1)
H(2)
H(3)
Test
60.1 [0.00] 15.5 [0.20] 4.2 [0.40]
Likelihood 80.03
102.31
107.97
110.06
Cointegrating relation:
= 1 (mt − st) + 3.26ct − 10.27(dt − ct) − 8.48 .
ecm
√ t
( LR)
(2.8)
(2.0)
Seigniorage
α
b = 3.26
=⇒
α
b −1 = 36%
Average of ct is 42.6% for full period.
rather than average of ∆Pt/Pt−1.
Weak exogeneity
c − d is weakly exogeneous [p = 0.47].
13
(5.7)
(−2.7)
Parsimonious vector autoregression:
∆(m − s)t = 0.33 ecmt−1 − 0.86 ∆(m − s)t−1 + 1.1 ∆ct−2
(0.05)
(0.18)
(0.5)
−1.9 ∆(d − c)t−1 + 1.6 ∆(d − c)t−2 + 0.20ε̂t.
(0.4)
(0.3)
∆ct = −0.090ecmt−1 + 0.10 ∆(m − s)t−1 + 0.20 ∆(m − s)t−2
(0.013)
(0.04)
(0.04)
+0.60 ∆ (d − c)t−1 − 0.23 ∆ (d − c)t−2 + 0.046ε̂t,
(0.11)
∆ (d − c)t =
(0.06)
−0.25 ∆(m − s)t−1
(0.08)
−0.47 ∆ (d − c)t−1 − 0.42 ∆ (d − c)t−2 + 0.14ε̂t
(0.14)
(0.14)
14
5
Discussion
Methodology:
1. Review of economic theory
2. General-to-specific analysis of data
Addressed problems:
1. Inability to model the data to the end
2. Difficulties in making congruent models
3. Discrepancies between “estimated” and “actual” inflation tax
Solution:
1. Cagan’s continuous time economic model kept
2. Cagan’s discretization altered using ct = 1 − exp (−∆pt)
Future research:
1. Comparative analysis (Germany etc)
2. Augmenting Cagan’s model
(a) Real variables?
(b) When and how do hyperinflations end?
(c) Expectations?
Bonus: Econometric theory for explosive processes
15