A trader who orders goods abroad that have to be paid for at some later date can settle his required payments in different ways. As an example, consider a European trader who at the end of the current month buys an amount of coffee at the price of US$100 000, to be paid by the end of next month. A first strategy to settle his account is to buy dollars now and hold these in deposit until the end of next month. This has the obvious consequence that the trader does not get the European (1 month) interest rate during this month, but the US one (assuming he holds the dollar amount in a US deposit). A second strategy is to buy dollars at the so-called forward market. There a price (exchange rate) is determined, which has to be paid for in dollars when delivered at the end of next month. This forward rateis agreed upon in the current period and has to be paid at delivery (1 month from now). Assuming that the forward contract is riskless (ignoring default risk, which is usually very small), the trader will be indifferent between the two strategies. Both possibilities are without risk, and therefore it is expected that both yield the same return at the end of next month. If not, arbitrage possibilities would generate riskless profits. The implied equality of the interest rate differential (European and US rates) and the difference between the forward rate and the spot rate is known as the covered interest rate parity(CIP) condition.
A third possibility for the trader to pay his bill in dollars is simply to wait until the end of next month and then buy US dollars at a yet unknown exchange rate. If the usual assumption is made that the trader is risk averse, it will only be attractive to take the additional exchange rate risk if it can be expected that the future spot rate (expressed in dollars per euro) is higher than the forward rate. If this is the case, we say that the market is willing to pay arisk premium. In the absence of a risk premium (the forward rate equals the expected spot rate), the covered interest rate parity implies theuncovered interest rate parity (UIP), which says that the interest rate differential between two countries equals the expected relative change in the exchange rate. In this section we consider tests for the existence of risk premia in the forward exchange market, based upon regression models.
4.11.1 Notation
For a European investor it is possible to hedge against currency risk by buying at timet the necessary amount of US dollars for delivery at timet+1 against a known rateFt, the forward exchange rate. Thus,Ftis the rate at timetagainst which dollars can be bought and sold (through a forward contract) at timet+1. The riskless interest rates for Europe and the US are given byREf,t+1 andRUSf,t+1, respectively. For the European investor, the investment in US deposits can be made riskless through hedging on the forward exchange
k k market. That is, a riskless investment for the European investor would give return
RUSf,t+1+logFt−logSt, (4.64)
whereStis the current spot (exchange) rate. To avoid riskless arbitrage opportunities (and unlimited profits for investors), this return should equal the riskless return on European deposits, that is, it should hold that
REf,t+1−RUSf,t+1=logFt−logSt. (4.65) The right-hand side of (4.65) is known as the (negative of the) forward discount, while the left-hand side is referred to as the interest differential. Condition (4.65) is known as covered interest rate parity and is a pure no-arbitrage condition that is therefore almost surely satisfied in practice (if transaction costs are negligible).
An alternative investment corresponds to an investment in US deposits without hedging the currency risk. The return on this risky investment is
RUSf,t+1+logSt+1−logSt, (4.66) the expected value of which equals (4.64) if
Et{logSt+1} =logFt or Et{st+1} =ft,
where small letters denote the log of capital letters, andEt{.} denotes the conditional expectation given all available information at timet. The equalityEt{st+1} =fttogether with covered interest rate parity implies the uncovered interest rate parity condition, which says that the interest differential between two countries equals the expected exchange rate change, that is,
REf,t+1−RUSf,t+1=Et{logSt+1} −logSt. (4.67)
Many macro-economic models employ this UIP condition. One of its consequences is that a small country cannot control both its domestic interest rate level and its exchange rates. Below, attention will be paid to the question as to whether uncovered interest rate parity holds, that is whether risk premia on the forward exchange markets exist.
The reason why the expected future spot rateEt{st+1}may differ from the forward rate ftis the existence of a risk premium. It is possible that the market is willing to pay a risk premium for taking the exchange rate risk in (4.66). In the absence of a risk premium, hedging against currency risk is free, and any investor can eliminate his or her exchange rate risk completely without costs. Because the existence of a positive risk premium for a European investor implies that a US investor can hedge exchange rate risk against the euro while receiving a discount, it is not uncommon to assume that neither investor pays a risk premium. In this case, the foreign exchange market is often referred to as being (risk-neutral) ‘efficient’ (see Taylor, 1995).
Note that the risk premium is defined as the difference between the expected log of the future spot rate and the log of the forward rate. Dropping the logarithm has the impor- tant objection that expressing exchange rates in one or the other currency is no longer irrelevant. In the logarithmic case this is irrelevant because Et{logS−1t+1} −logF−1t =
−Et{logSt+1} +logFt.
k k
ILLUSTRATION: RISK PREMIA IN FOREIGN EXCHANGE MARKETS 131
4.11.2 Tests for Risk Premia in the 1-Month Market
One approach to test for the presence of risk premia is based on a simple regression framework. In this subsection we shall discuss tests for the presence of a risk premium in the 1-month forward market using monthly data. That is, the sampling frequency cor- responds exactly to the length of the term contract. Empirical results will be presented for 1-month forwards on the US$/€ and US$/£ Sterling exchange rates, using monthly data from January 1979 to December 2001. The pre-euro exchange rates are based on the German mark. The use of monthly data to test for risk premia on the 3-month forward market is discussed in the next subsection.
The hypothesis that there is no risk premium can be written as
H0∶Et−1{st} =ft−1. (4.68) A simple way to test this hypothesis exploits the well-known result that the difference between a random variable and its conditional expectation given a certain information set is uncorrelated with any variable from this information set, that is,
E{(st−Et−1{st})xt−1} =0 (4.69) for anyxt−1that is known at timet−1. From this we can write the following regression model:
st−ft−1=xt−1 𝛽+𝜀t, (4.70) where𝜀t=st−Et−1{st}. IfH0is correct and ifxt−1is known at timet−1, it should hold that𝛽=0. Consequently,H0is easily tested by testing whether𝛽=0 for a given choice ofxt−1variables. Below we shall choose as elements inxt−1a constant and the forward discountst−1−ft−1.
Because st−1−ft−2 is observed in period t−1, 𝜀t−1 is also an element of the information set at time t−1. Therefore, (4.69) also implies that under H0 the error terms in (4.70) exhibit no autocorrelation. Autocorrelation in 𝜀t is thus an indication for the existence of a risk premium. Note that the hypothesis does not imply anything about the variance of 𝜀t, which suggests that imposing homoskedas- ticity may not be appropriate and heteroskedasticity-consistent standard errors could be employed.
The data employed are taken from Datastream and cover the period January 1979–December 2001. We use the US$/€ rate and the US$/£ rate, which are visualized in Figure 4.6. From this figure we can infer the strength of the US dollar in 1985 and in 2000/2001. In Figure 4.7 the monthly forward discount st−ft is plotted for both exchange rates. Typically, the forward discount is smaller than 1% in absolute value.
For the euro, the dollar spot rate is in almost all months below the forward rate, which implies, given the covered interest rate parity argument, that the US nominal interest rate exceeds the European one. Only during 1993–1994 and at the end of 2001 the converse appears to be the case.
Next, (4.70) is estimated by OLS taking xt−1 = (1,st−1−ft−1). The results for the US$/£ rate are given in Table 4.12. Because the forward discount has the properties of a lagged dependent variable (st−1−ft−1is correlated with𝜀t−1), the Durbin–Watson test is not appropriate. The simplest alternative is to use the Breusch–Godfrey test, which is based upon an auxiliary regression ofetuponet−1,st−1−ft−1and a constant (see above)
k k
0.5 1.0 1.5 2.0 2.5
80 82 84 86 88 90 92 94 96 98 00 US$/GBP
US$/EUR
Figure 4.6 US$/EUR and US$/GBP exchange rates, January 1979–December 2001.
–0.020 –0.015 –0.010 –0.005 0.000 0.005 0.010
80 82 84 86 88 90 92 94 96 98 00 US$/EUR US$/GBP
Figure 4.7 Forward discount, US$/EUR and US$/GBP, January 1979–December 2001.
k k
ILLUSTRATION: RISK PREMIA IN FOREIGN EXCHANGE MARKETS 133
Table 4.12 OLS results US$/£Sterling Dependent variable:st−ft−1
Variable Estimate Standard error t-ratio
constant −0.0051 0.0024 −2.162
st−1−ft−1 3.2122 0.8175 3.929
s=0.0315 R2=0.0535 R̄2=0.0501 F=15.440
and then taking10 TR2. We can test for higher-order autocorrelations by including additional lags, likeet−2andet−3. This way, the null hypothesis of no autocorrelation can be tested against the alternatives of first- and (up to) twelfth-order autocorrelation, with test statistics of 0.22 and 10.26. With 5% critical values of 3.84 and 21.0 (for an𝜒21and 𝜒212, respectively), this does not imply rejection of the null hypotheses. Thet-statistics in the regression indicate that the intercept term is significantly different from zero, while the forward discount has a significantly positive coefficient. A joint test on the two restrictions 𝛽=0 results in an F-statistic of 7.74 (p=0.0005), so that the null hypothesis of no risk premium is rejected. The numbers imply that, if the nominal UK interest rate exceeds the US interest rate such that the forward discount st−1−ft−1 exceeds 0.16% (e.g. in the early 1990s), it is found thatEt−1{st} −ft−1is positive. Thus, UK investors can sell their pounds on the forward market at a rate of, say, $1.75, while the expected spot rate is, say, $1.77. UK importers wanting to hedge against exchange rate risk for their orders in the US have to pay a risk premium. On the other hand, US traders profit from this; they can hedge against currency risk and cash (!) a risk premium at the same time.11
Thet-tests employed above are only asymptotically valid if𝜀texhibits no autocorrela- tion, which is guaranteed by (4.69), and if𝜀tis homoskedastic. The Breusch–Pagan test statistic for heteroskedasticity can be computed asTR2of an auxiliary regression ofe2t upon a constant andst−1−ft−1, which yields a value of 7.26, implying a clear rejection of the null hypothesis. The use of more appropriate heteroskedasticity-consistent standard errors does not result in qualitatively different conclusions.
In a similar way we can test for a risk premium in the US$/€ forward rate. The results of this regression are as follows:
st−ft−1= −0.0023+0.485(st−1−ft−1) +et, R2=0.0015 (0.0031) (0.766)
BG(1) =0.12, BG(12) =14.12.
HereBG(h)denotes the Breusch–Godfrey test statistic for up tohth-order autocorrelation.
For the US$/€ rate, no risk premium is found: both the regression coefficients are not significantly different from zero and the hypothesis of no autocorrelation is not rejected.
10Below we use the effective number of observations in the auxiliary regressions to determineTinTR2.
11There is no fundamental problem with the risk premium being negative. While this means that the expected return is lower than that of a riskless investment, the actual return may still exceed the riskless rate in situ- ations that are particularly interesting to the investor. For example, a fire insurance on your house typically has a negative expected return, but a large positive return in the particular case that your house burns down.
k k 4.11.3 Tests for Risk Premia Using Overlapping Samples
The previous subsection was limited to an analysis of the 1-month forward market for foreign exchange. Of course, forward markets exist with other maturities, for example 3 months or 6 months. In this subsection we shall pay attention to the question of the extent to which the techniques discussed in the previous section can be used to test for the presence of a risk premium in the 3-month forward market. The frequency of observation is, still, 1 month.
Let us denote the log price of a 3-month forward contract byft3. The null hypothesis of no risk premium can then be formulated as
H0∶Et−3{st} =ft−33 . (4.71) Using similar arguments to before, a regression model similar to (4.70) can be written as st−ft−33 =xt−3 𝛽+𝜀t, (4.72) where𝜀t=st−Et−3{st}. Ifxt−3 is observed at timet−3, the vector𝛽in (4.72) should equal zero underH0. Simply using OLS to estimate the parameters in (4.72) withxt−3= (1,st−3−ft−3)gives the following results for the US$/£ rate:
st−ft−33 = −0.014+3.135(st−3−ft−33 ) +et, R2=0.1146 (0.004) (0.529)
BG(1) =119.69, BG(12) =173.67, and for the US$/€ rate:
st−ft−33 = −0.011+0.006(st−3−ft−33 ) +et, R2=0.0000 (0.006) (0.535)
BG(1) =130.16, BG(12) =177.76.
These results seem to suggest the clear presence of a risk premium in both markets:
the Breusch–Godfrey tests for autocorrelation indicate strong autocorrelation, while the regression coefficients for the US$/£ exchange market are highly significant.These conclusions are, however, incorrect.
The assumption that the error terms exhibit no autocorrelation was based on the observation that (4.69) also holds forxt−1=𝜀t−1 such that𝜀t+1 and𝜀tare uncorrelated.
However, this result is only valid if the frequency of the data coincides with the maturity of the contract. In the present case, we have monthly data for 3-month contracts.
The analogue of (4.69) now is
E{(st−Et−3{st})xt−3} =0 for anyxt−3known at timet−3. (4.73) Consequently, this implies that𝜀tand𝜀t−j(j=3,4,5, . . .)are uncorrelated but does not imply that𝜀tand𝜀t−1or𝜀t−2are uncorrelated. On the contrary, these errors are likely to be highly correlated.
Consider an illustrative case where (log) exchange rates are generated by a so-called random walk12process, that is,st=st−1+𝜂t, where the𝜂tare independent and identically
12More details on random walk processes are provided in Chapter 8.
k k
ILLUSTRATION: RISK PREMIA IN FOREIGN EXCHANGE MARKETS 135
distributed with mean zero and variance 𝜎𝜂2 and where no risk premia exist, that is, ft−33 =Et−3{st}. Then it is easily shown that
𝜀t=st−Et−3{st} =𝜂t+𝜂t−1+𝜂t−2.
Consequently, the error term𝜀tis described by a moving average autocorrelation pattern of order 2. When log exchange rates are not random walks, the error term𝜀t will com- prise ‘news’ from periodst,t−1 andt−2, and therefore𝜀t will be a moving average even in the more general case. This autocorrelation problem is due to the so-called over- lapping samples problem, where the frequency of observation (monthly) is higher than the frequency of the data (quarterly). If we test whether the autocorrelation goes beyond the first two lags, that is, whether𝜀tis correlated with𝜀t−3up to𝜀t−12, we can do so by running a regression of the OLS residualetuponet−3, . . . ,et−12and the regressors from (4.72). This results in Breusch–Godfrey test statistics of 7.85 and 9.04, respectively, both of which are insignificant for a Chi-squared distribution with 10 degrees of freedom.
The fact that the first two autocorrelations of the error terms in the regressions above are nonzero implies that the regression results are not informative about the existence of a risk premium: standard errors are computed in an incorrect way and, moreover, the Breusch–Godfrey tests for autocorrelation may have rejected because of the first two autocorrelations being nonzero, which is not in conflict with the absence of a risk premium. Note that the OLS estimator is still consistent, even with a moving average error term.
One way to ‘solve’ the problem of autocorrelation is simply dropping two-thirds of the information by using the observations from 3-month intervals only. This is unsatisfactory, because of the loss of information and therefore the potential loss of power of the tests.
Two alternatives may come to mind: (i) using GLS (hopefully) to estimate the model more efficiently, and (ii) using OLS while computing corrected (Newey–West) standard errors. Unfortunately, the first option is not appropriate here because the transformed data will not satisfy the conditions for consistency and GLS will be inconsistent. This is due to the fact that the regressorst−3−ft−33 is correlated with lagged error terms.
We shall therefore consider the OLS estimation results again, but compute HAC stan- dard errors. Note thatH=3 is sufficient. Recall that these standard errors also allow for heteroskedasticity. The results can be summarized as follows. For the US$/£ rate we have
st−ft−33 = −0.014+3.135(st−3−ft−33 ) +et, R2 =0.1146, [0.005] [0.663]
and for the US$/€ rate
st−ft−33 = −0.011+0.006(st−3−ft−33 ) +et, R2 =0.0000, [0.008] [0.523]
where the standard errors within square brackets are the Newey–West standard errors withH=3. Qualitatively, the conclusions do not change: for the 3-month US$/£ market, uncovered interest rate parity has to be rejected. Because covered interest rate parity implies that
st−ft=R∗f,t+1−Rf,t+1,
k k where∗denotes the foreign country and the exchange rates are measured, as before, in
units of home currency for one unit of foreign currency, the results imply that, at times when the US interest rate is high relative to the UK one, UK investors pay a risk premium to US traders. For the European/US market, the existence of a risk premium was not found in the data.
Wrap-up
Heteroskedasticity is a very common violation of the Gauss–Markov conditions.
Its presence invalidates the routinely calculated standard errors for OLS, and implies that OLS is no longer the best linear unbiased estimator for the linear model. If the specification of the model is non-suspect, a convenient way to deal with het- eroskedasticity is to calculate heteroskedasticity-robust standard errors. If efficiency is an issue, one may consider the use of generalized least squares, although this comes at the cost of imposing additional assumptions about the form of heteroskedasticity.
Several tests are available to test for the presence of heteroskedasticity, including the Breusch–Pagan test and the White test. Changing the functional form of the model, for example by transforming the dependent variable in logs, may help to reduce or eliminate the heteroskedasticity problem. Serial correlation is a concern in time series applications, and is typically interpreted as a sign of misspecification.
The Durbin–Watson test provides a quick way to assess the likelihood of first-order serial correlation, but several alternative tests are available that are more generally applicable. If the serial correlation cannot be removed by changing the specification of the model, it can be dealt with by calculating Newey–West standard errors. A typical situation where this is required is when we have an overlapping samples problem.
In exceptional cases, the use of GLS can be considered. Time series models will be discussed in more detail in Chapters 8 and 9.
Exercises
Exercise 4.1 (Heteroskedasticity – Empirical)
This exercise uses data for 30 standard metropolitan statistical areas (SMSAs) in California for 1972 on the following variables:
airq indicator for air quality (the lower the better) vala value added of companies(in 1000 US$) rain amount of rain (in inches)
coas dummy variable, 1 for SMSAs at the coast; 0 for others dens population density (per square mile)
medi average income per head(in US$)
k k
EXERCISES 137
a. Estimate a linear regression model that explainsairqfrom the other variables using ordinary least squares. Interpret the coefficient estimates.
b. Test the null hypothesis that average income does not affect the air quality. Test the joint hypothesis that none of the variables has an effect upon air quality.
c. Perform a Breusch–Pagan test for heteroskedasticity related to all five explanatory variables.
d. Perform a White test for heteroskedasticity. Comment upon the appropriateness of the White test in light of the number of observations and the degrees of freedom of the test.
e. Assuming that we have multiplicative heteroskedasticity related tocoasandmedi, estimate the coefficients by running a regression of log e2i upon these two vari- ables. Test the null hypothesis of homoskedasticity on the basis of this auxiliary regression.
f. Using the results from e, compute an EGLS estimator for the linear model.
Compare your results with those obtained undera. Redo the tests fromb.
g. Comment upon the appropriateness of theR2in the regression off.
Exercise 4.2 (Autocorrelation – Empirical)
Consider the data and model of Section 4.8 (the demand for ice cream). Extend the model by including lagged consumption (rather than lagged temperature). Perform a test for first-order autocorrelation in this extended model.
Exercise 4.3 (Autocorrelation Theory)
a. Explain what is meant by the ‘inconclusive region’ of the Durbin–Watson test.
b. Explain why autocorrelation may arise as the result of an incorrect func- tional form.
c. Explain why autocorrelation may arise because of an omitted variable.
d. Explain why adding a lagged dependent variable and lagged explanatory variables to the model eliminates the problem of first-order autocorrelation. Give at least two reasons why this is not necessarily a preferred solution.
e. Explain what is meant by an ‘overlapping samples’ problem. What is the problem?
f. Give an example where first-order autocorrelation leads to an inconsistent OLS estimator.
g. Explain when you would use Newey–West standard errors.
h. Describe in steps how you would compute the feasible GLS estimator for 𝛽 in the standard model with (second-order) autocorrelation of the form𝜀t=𝜌1𝜀t−1+ 𝜌2𝜀t−2+𝑣t. (You do not have to worry about the initial observation(s).)