Pornpinun Chantapacdepong

Fractional Integration and the forward premium puzzle.

Thammasat University, 6 May 2008.

2 1. Concepts and motivation

• The forward premium puzzle is the violation of the

uncovered interest parity (UIP)

• The UIP parity states that

∆st,t+k =a + b(ft,k - st) + εt,t+k

• ∆st,t+k is expected changes in exchange rates.

• (ft,k - st) is the forward premium

• H0 : a=0, b=1

3 2. Previous literature on UIP

• Empirical work finds that b≠1 and is usually negative e.g. Frankel, 1980; Fama, 1984; Bekaert and Hodrick, 1993 (these works use a variety of exchange rates against the US dollar).

• Froot and Thaler (1990) find that the average coefficient across some 75 published estimates of “b” is -0.88. In these, a few are positive, but none of them have the coefficient, “b” statistically greater than or equal to unity.

4 3. Objectives

The objectives of this paper are to:

• Test the efficiency of forward markets for foreign

exchange in the 10 most commonly traded currencies

using daily data from 1994 to 2007.

• Attempts to explain the forward premium anomaly by

1) Statistical artefacts of the data

2) Foreign exchange risk premium

5 Statistical artefacts of the data

• This view pays attention to the time series properties of the return on spot exchange rate and the forward premium.

The explanation is based on

1) The long memory behaviour of the forward premium(while the returns on the spot exchange rate are stationary)

2) The existence of structural breaks in the forward premium

6 Foreign exchange risk premium

• The foreign exchange risk premium explanation states that the forward rates are biased predictors of actual exchange rate movements because there exists a risk premium on one country's currency relative to another's.

• Domowitz and Hakkio (1985) gives an explanation of the risk premia through the conditional variance of the market forecast error (the error in forecasting the spot rate using the forward rate).

rpt,t+k = vart (ft,k – st+k)

7 4. Methodology

• Unit root test on the expected exchange rate depreciation and the forward premium.

• Correct the structural breaks in the forward premium series.

• Estimate the long memory parameters of the forward premium series using the ARFIMA model.

• Estimate the risk premia by the conditional variance of the market forecast error using the FIGARCH model.

8 Fractional Integration (ARFIMA model)

• This model is useful for series that exhibit significant

autocorrelation ( or long term dependence) between

observations widely separated in time.

• The ARFIMA(p,d,q) model for yt is written as

Φ(L)(1-L)d (yt-μ)=θ(L)εt

where Φ(L) and θ(L) are the autoregressive polynomial

and the moving average polynomial in the lag operator L

• d is the fractional differencing parameter.

9 Fractionally Integrated GARCH model

• We estimate the risk premia by modeling the long memory in the conditional variance for the forward rate forecast error series.

• The Fractionally Integrated Generalised Autoregressive Conditional Heteroskedasticity (FIGARCH) model explains the fractional integration (the long memory) behaviour in the conditional variance of the series.

• This model is developed by Baillie, Bollerslev and Mikkelsen (1996).

10 5. Results: Forward premium

• Forward premium are non stationary and are in fact highly persistent (corresponds to Crowder, 1994; Evans and Lewis 1995; and Mark, Wu and Hai 1993).

• There is fractionally integrated behaviour in the forward premium series, 0.5 < d < 1 (corresponds to Baillie and Bollerslev, 1994; Maynard and Phillips, 2001 and Choi and Zivot, 2005).

• The spot return in all currencies are stationary and follow I(0) process. Thus, it is inappropriate to apply conventional regression analysis to test the hypothesis of the forward rate being an unbiased predictor of the future spot rate.

11

12

13

14 Results: Forward premium (cont.)

• The presence of structural breaks in the forward premium series is found to increase the persistence of the series

• Even controlling for regime shifts in the forward premium series, the fractional integrated behaviour still persists.

15

16

17

Results: Risk premia

• There is strong evidence of long memory in the conditional variance of the forecast error.

• The long memory parameter in the conditional variance specification is significantly different from zero and falls into the stationary range. Thus the propagation of shocks to the variance of forward rate forecast error is proved to occur at a slow hyperbolic rate of decay.

• This means that the conditional variance matters in determining deviation of the forward rate from the expected future spot rate.

18 6. Conclusion

• This work corroborates earlier findings that a principal reason for rejection of the forward rate unbiasedness hypothesis is differences in persistence between the forward premium series and the spot rate return and the existence of the time varying exchange rate risk premium.

• The presence of structural breaks also proves to play an important role in explaining the long memory of the forward premium data. Allowing for structural breaks reduces the persistence of the forward premium across all currencies and model specifications. Nevertheless, the forward premium still follows the (nonstationary) fractionally integrated process.

19 6. Conclusion (cont.)

• In the risk premia estimation part, this work suggests a new method in estimating the foreign exchange rate risk premium.

• In estimating the foreign exchange risk premia, the FIGARCH model is found to be econometrically superior to regular stable GARCH model as in Baillie and Bollerslev (1990) and the ARCH-in-mean model as used by Domowitz and Hakkio (1985).

• Finding that the risk premia exists proves that the Uncovered Interest Parity does not hold.