A SETAR Model for the French franc / German
mark Exchange Rate
Background:
• During the 1990s, European countries which were part of the
ERM (Exchange Rate Mechanism) of the EMS (European
Monetary System) were required to constrain their currencies
to remain within prescribed bands relative to other ERM
currencies.
• Currencies are allowed to move up to ± 2.25% either side of
their central parity in the ERM.
• Central banks are forced to intervene in the markets (by
appreciating or depreciating) when its currency to close to its
boundary.
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Model:
• The idea is to use a SETAR model to allow for different types
of behaviour according to whether the exchange rate is close
to the ERM boundary.
• Close to the boundary, central banks will have to intervene to
maintain the parity: such interventions may affect the usual
market dynamics that ensure fast reaction to news and the
absence of arbitrage opportunities.
• Central banks are expected to intervene when either boundary
(the ceiling and the floor boundary) is hit:
• This suggests the use of the 2-threshold (3-state) SETAR.
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Data and reconsideration of the model:
• Study from Chappell et al. (1996, Journal of Forecasting)
uses daily data from 1/5/90 to 30/3/92.
• For the considered sample, DEM is never a weak currency: as
a result, the FRF-DEM exchange rate is either at the top or in
the center of the band, and never close to the bottom.
• A model with 2-threshold (3 states) SETAR is NOT
appropriate (spurious detection of the 2nd threshold).
• A model with 1-threshold (2-state) SETAR is more appropriate.
• The model orders for each regime are determined using AIC.
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Estimation results and Comments:
Model
Êt = 0.0222 + 0.9962Et−1
(0.0458) (0.0079)
For regime
Et−1 < 5.8306
Number of
observations
344
Êt = 0.3486 + 0.4394Et−1 + 0.3057Et−2 + 0.1951Et−3
(0.2391) (0.0889)
(0.1098)
(0.0866)
Et−1 ≥ 5.8306
103
• Estimation is performed using the first 450 observations
(which leaves the last 50 observations for out-of-sample
forecasting).
• Ceiling in the ERM corresponded to 5.8376 (log of FRF per
100 DEM).
• The estimated threshold is 5.8306, just below the ceiling as
expected.
• This confirms the expectation that the central bank is likely to
intervene before the exchange rate actually hits the ceiling.
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Out-of-sample forecasts using 4 competitive models:
Steps ahead
1
Panel A: mean
Random walk
1.84E-07
AR(2)
3.96E-07
One-threshold SETAR 1.80E-07
Two-threshold SETAR 1.80E-07
2
3
squared forecast error
3.49E-07 4.33E-07
1.19E-06 2.33E-06
2.96E-07 3.63E-07
2.96E-07 3.63E-07
Panel B: Median squared forecast error
Random walk
7.80E-08 1.04E-07 2.21E-07
AR(2)
2.29E-07 9.00E-07 1.77E-06
One-threshold SETAR 9.33E-08 1.22E-07 1.57E-07
Two-threshold SETAR 1.02E-07 1.22E-07 1.87E-07
5
10
8.03E-07
6.15E-06
5.41E-07
5.74E-07
1.83E-06
2.19E-05
5.34E-07
5.61E-07
2.49E-07
5.34E-06
2.42E-07
2.57E-07
1.00E-06
1.37E-05
2.34E-07
2.45E-07
Source: Chappell et al. (1996). Reprinted with permission of John Wiley and Sons.
• Overall the 1-threshold SETAR model seems to perform best:
• 1-threshold SETAR performs better than the other competitors
according to the MSE.
• The random walk model marginally outperforms 1-threshold
SETAR at short horizons according to the Median SE, but not
at longer horizons.
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