A Utility Based Comparison of Some Models of Exchange Rate Volatility

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 441

February 1993

A UTILITY BASED COMPARISON CF SOME MODELS OF EXCHANGE RATE VOLATILITY

Kenneth D. West Hali J. Edison Dongchul Cho

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an

acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.

ABSTRACT

When estimates of variances are used to make asset allocation decisions, underestimates of population variances lead to lower expected utility than equivalent overestimates: a utility based criterion is asymmetric, unlike standard criteria such as mean squared error. To illustrate how to estimate a utility based criterion, we use five bilateral weekly dollar exchange rates, 1973-1989, and the corresponding pair of Eurodeposit rates. Of homoskedastic, GARCH, autoregressive and nonparametric models for the conditional variance of each exchange rate, GARCH models tend to produce the highest utility, on

average. A mean squared error criterion also favors GARCH, but not as sharply.

A Utility Based Comparison of Some Models of Exchange Rate Volatility

Kenneth D. West, Hali J. Edison and Dongchul Cho’ I. Introduction

This paper evaluates the out of sample performance of some univariate models for exchange rate volatility, using bilateral weekly data for the dollar versus the currencies of Canada, France, Germany, Japan and the United Kingdom, 1973-1989, and the corresponding pair of Eurodeposits, 1981-1989. The models considered include homoskedastic, GARCH, and nonparametric ones, as well as autoregressions in both the absolute value and square of exchange rate changes. The metric we use to compare the models is a utility based one: how much would an investor with a mean-variance utility function, who uses the estimates of one of these models to divide her wealth between a pair of Eurodeposits, be willing to pay to use one model rather than another?

Recent research on conditional volatility has established that for many financial variables, including exchange rates, squared changes that are large tend to be followed by squared changes that are also large (Bollerslev et al. (1990)). This empirical fact has stimulated a variety of formal statistical models. Since the relative merits of many of these models are as yet not well established, there is a need for systematic evaluation and comparison.

Some previous authors have compared the out of sample performance of univariate models applied to stock price data. Using a mean squared error criterion, Pagan and Schwert (1990), found that GARCH and ARMA models are preferred to nonparametric and Markov

switching cnes, Akgiray (1989) that GARCH dominates naive and ARMA models. Using a

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criterion based on performance in a simulated market, Engle et al. (1990) also found GARCH preferable to naive and ARMA models. Finally, Friedman and Kuttner (1988) compared multivariate GARCH and AR models, using stock and bond data. Among other statistics, they examined mean squared errors, but did not seem to find strong grounds for preferring one model to another.

One inessential sense in which the present paper differs from any of these is in its use of exchange rate data, which we study largely because such data apparently have yet to be used in a systematic comparison of volatility models. More importantly, we also depart from earlier work in how we measure performance. An appropriate measure of performance depends on the use to which one puts the estimates of volatility, and our measure is probably not the best one if one wanis to, say, study the links between observable macro variables and volatility (e.g., Schwert (1989a)). But insofar as models for volatility are motivated by reference to investment by risk averse utility maximizers--as, indeed, they often are (c.g., Engle and Bollerslev (1986), Friedman and Kuttner (1988))--a utility based measure seems quite appropriate.!

Our measure is based on the following presumption: at a given point in time, one estimate of a conditional variance is better than another if investment decisions based on it lead to higher (population) expected utility. Similarly, an estimator or model of a conditional variance is preferred if, on average, over many time periods, it leads to higher expected utility. We show that under the assumption that utility is either (a)exponential, and asset returns are jointly normal, or (b)quadratic, such a utility based criterion is fundamentally

different from statistical ones based on mean squared and mean absolute error: the utility

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criterion is: asymmetric, with underestimates of the population conditional variance-covariance matrix leading to lower expected utility than equivalent overestimates.

To illustrate the use of our measure empirically, we assume that an investor divides her wealth between two assets, weekly or quarterly Eurodeposits denominated in (1)dollars and (2)the currency of one other country (Canada, France, Germany, Japan, or the United Kingdom). We consider an investor who knows the population conditional variance of exchange rate changes, but is forced to make a wealth allocation using not the population value but one of a set of estimates. Different estimates will lead to different wealth allocations and, therefore, different levels of expected utility. We envision the investor using estimates from each of our models to produce a sequence of hypothetical wealth allocations over a number of successive periods, and ask the following: which model’s implied allocations produce the highest expected utility, on average, and how much would such an investor pay for the right to allocate wealth according to that model rather than another?

For quadratic utility, we show that one can estimate the average expected utility produced using a given volatility model, even when one does not have our hypothetical investor’s |nowledge of the time series of population conditional variances. If, for a given level of beginning of period wealth, one model produces higher expected utility, on average, than does another, then the better model will produce equal average utility with a lower beginning level. We interpret the difference in beginning wealth as the average per period fee that our hypothetical investor would be willing to pay to use the higher rather than lower

utility model.

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Although there was some variation across data sets, we find that GARCH models tend to do best. Depending on the dataset, an investor would typically be willing to pay about .05 to 2 percent, or 5 to 200 basis points of her wealth, annually, to switch to GARCH from another model. Confidence intervals around these point estimates, however, tend to be large. The t-statistics indicate that the fee is statistically significantly different from zero at conventional levels only about one fourth of the time; F-tests of the null that all six models yield the same utility are significant a little less than half the time. Under an out of sample mean squared error criterion, however, the statistical significance of differences across models is even less pronounced.

One way to gauge the economic significance of the utility based figures is to iriterpret them as a transactions fee that a professional money manager could charge an investor capable of estimating, say, homoskedastic but not GARCH models of exchange rate risk. As such, the 5 to 200 range seems to bracket what Wall Street mutual funds charge for their services (Ippolito (1989), New York Times, May 14, 1991, page F14), which seems to us a substantial figure.

While the immediate motivation for our research is the relatively recent literature on conditional volatility, our results are relevant for evaluation of any models for second moments of asset returns. Eun and Resnick (1984), for example, use a mean squared error criterion in evaluating models for correlations across share prices, motivating their study with reference to mean-variance portfolio analysis. An implication of this paper is that such a

criterion is probably not the best.

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Before turning to the body of the paper, two introductory cautions seem advisable, to set the reader’s expectations straight. First, while we have tried to make a sensible choice of models to study, we do not claim to be comprehensive, and some readers may feel that we have unwisely excluded some important models. For such readers we emphasize that we consider one of our contributions to be the technique used to produce the rankings of the models. Second, we abstract from a number of potentially important complications involved in real world investments. We ignore, for example, default risk, transactions costs such as bid-asked spreads, and issues about the timing of settlement of transactions, including that our exchange and interest rate series, which we obtained from two different sources, are sampled at slightly different times (of the same day); we also acknowledge that the very simple portfolios that we consider are not well diversified. Our aim is simply to get a rough idea of the magnitucle of the potential benefits of better volatility models, not to quantify these benefits to many decimal places.

Section II briefly outlines the motivation for our utility based measure rather than a standard statistical one, in a more general framework than is required for our empirical work. Section III describes how we apply our measure to exchange rate data, Section IV our data and models, Section V our empirical results. Section VI concludes. An Appendix contains some technical details, and an additional appendix available from the authors upon request contains sorne results omitted from the paper to save space.

II. Utility Versus Statistical Measures of Estimator Quality A basic message of our paper is that when comparing estimators of conditional

variances, rankings from a utility based criterion might differ from those from a statistical

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mean squared or mean absolute error criterion, because of a certain asymmetry in utility evaluation of estimators. A general statement is given in the proposition below.

We begin, however, with a simple numerical example. This example does not illustrate the asymmetry, but it does point out that utility and statistical measures may be dramatically different, and thus motivates our desire to estimate a utility based measure. Suppose one has an exponential utility function, U(W,,,) = -exp(-OW,,,), where 6>0 and W,,, is period t+1 wealth. Suppose that there are three assets, one riskless. Let = (y,,p,)’ and H denote the mean and covariance matrix of the (2x1) vecior of excess returns, f = (f,,f,)’ the (2x1) vector of fractions of period t wealth put in the two risky assets. As is well known, maximization of expected utility leads to f = (1/0W)H"n, where W is period t wealth. Suppose further that H is the identity matrix, and p,=p,>0. Then the optimal fraction satisfies f,=f,=(n,/8W)=(n,/8W).

Assume that an investment decision must be made using the true p and one of two noisy

estimates of H,

2 -1| It 0 | A, = | | A, = | | l-1 2I 10 1.0011 Which is the better estimate of the true H (which equals the identity matrix)? By common statistical measures such as the average of the squared differences between the nonredundant elements of H and the Hi,’s, A, is "closer" to H and therefore is better. But a routine calculation

indicates that A, leads to exactly the optimal (expected utility maximizing) fraction. The basic

presumption of this paper is that A, is therefore a preferable estimate.

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This numerical example obviously is special. We now state a proposition that illustrates that, in a very general sense, utility and standard statistical criteria are different. Let W,,, be wealth in period t+1. Assume: (1)The utility function is either (a)exponential, U(W,,,) = -exp(-O8W,,,), 9>0, and asset returns are normally distributed, or (b)quadratic, U(W,,,) = Wir - SYWis1, y>0, and asset returns have finite means and variances. (2)There are k>1 risky assets, with positive definite variance-covariance matrix H. There may or may not be a (k+1)st riskless asset. If not, k22; if so, expected returns on the risky assets are greater than those on the riskless asset. (3)There are no consiraints on short sales; the fraction of wealth put in a given asset may be less than zero or greater than one. (4) The population conditional mean of returns is used in making investment decisions.

Assumptions (1)-(3) are used to get a convenient closed form solution. Assumption (4) is used to focus on the effects of errors in estimation of H. Note that this assumption rules out a comparison of various parameterizations of GARCH-M models, for example.

Suppose we wish to compare two estimates of H, A, and f.. Let E,U;,,,, 1=1,2, denote expected utility that results when model i is used to make an investment decision, where the true variance covariance matrix H is used in computing expected utility. Our basic result is that there is an asymmecry in the utility loss from estimation error, with estimates of H that are too large being preferred to those that are too small.

Proposition: Suppose that H=H+V, A=H-V, where V is a positive semidefinite symmetric and A, is a positive definite matrix. Then E,U,,,,2E,U,,,,; equality holds if and only if use of A, and of A, result in the allocation implied by use of the population variance-covariance matrix H.

(Algebra tc derive the proposition is in the additional appendix available on request.)

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To illustrate the proposition, consider Figure 1, which plots expected utility as a function of i the estimate of H, when H is a scalar and utility is quadratic with parameters matching those in our empirical work. By assumption, highest expected utility occurs when fi =H=(.0 15)?=.000225. Expected utility declines the farther away is Bint from h,.?. What is to be noted is that, in contrast to the usual mean squared or mean absolute error criterion, this objective function is asymmetric around h,, penalizing estimates that are too small more sharply than those that are too large. As we shall see, this asymmetry plays a role in the empirical results. Ill. Estimation of Average Utility

It is helpful to begin by defining some notation. Let

e, = log difference of weekly exchange rate (dollars per unit of (1a) foreign currency);

hj = var,(e,,,) = Eet,; = (population) variance of ¢,,,, (1b) conditional on information generated by past e,, sSt;

Binns = fitted conditional variance of e,,;, according to model m (Ic) (e.g., model m is GARCH(1,1), or homoskedastic), estimated using data on past e,, SSt;

h, = hers Aig, = Bena (1d)

N= endpoint of first sample used in estimation; (le)

T= endpoint of last sample used in estimation. (1f)

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Note that in (1b) the conditional variance is equated with the raw (as opposed to central) conditional second moment of e,. This is in accord with Meese and Rogoff (1983), Diebold and Nason (1990), Meese and Rose (1991) and the findings summarized below that the conditional first moment of e, is zero. For concreteness in interpreting (1b) and (1c), it may help to note that in the tables below we report results for weekly and quarterly horizons for investment decisions, which require estimates of h,, (weekly) and h,,, h,,, ...., and h,,, (quarterly). To do so for, say, weekly horizons, we obtain for each model T-N+1 fitted values Bene t=N,...,T, for models m=1,...,M, where the number of models M in the tables below is 6. Note, finally, our dating convention: what we denote h, corresponds to what is often called h,,, or 6,,, (e.g., Engle (1982)).

We specialize the general environment described in the previous section to one in which utility is quadratic. We assume a two country world with two assets, one sold domestically, the other sold abroad. To focus on the question at hand, we assume that apart from the conditional variance of exchange rate changes, all relevant moments of the return distribution are known.

Let the utility function and wealth constraint be

utility in period t+] = W,,, - SYW%,;, (2)

Wu = WF (Riv t@e41) + (1-f)Ruil,

where W,,, and ¥ are, as in the previous section, wealth and taste for risk, f, is the fraction of wealth put in the foreign asset (possibly negative, possibly greater than one), R,,, is the gross return

on the foreign asset in terms of foreign currency and R,,, is the gross return on the domestic asset.

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For each period, use each model, one by one, to choose the fraction of wealth that maximizes expected utility, taking each model’s point estimate for the conditional variance as the correct expectation. Given the assumption that the mean return on the asset is known, in a given period the optimal fraction will vary across competing models only insofar as the estimates of the conditional variance vary. Let f,,, be the fraction that results when model m is used (the exact formula is given in equation (A-1) in the appendix), Wa, = Wilfa(Reiteny + (-fadRuid and Uj.) the resulting wealth and utility. If f,, depends only on information known at time t--as it will in an

out of sample study such as ours--it is straightforward to show that mathematically expected utility

may be written

EU = ELWreet - SYWil (3)

= Wic,+d,u(h,fi,)},

where E, is mathematical expectations, c, and d, are certain variables that depend on y and Ru i-Ris but not on h,, and u(h,f,,,) is a certain function that is linear in h, Explicit formulas are given in the Appendix. Figure 1, which was already discussed in the preceding section, plots u(h,,h,,,) as a function of fae for h, = (.015)* (approximately the sample variance of e, in our data).

We cannot use (3) directly to determine which model yields the highest mathematically expected utility, since the whole problem is that we do not know the population (mathematical) expectation h,. But since uth, fi,,) is linear in h,, we can get an estimate that is right on average by

replacing h, with the ex-post realized value e?,,. We therefore compute average utility for a given

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model by replacing h, with the ex-post realized value e?,, and averaging the result for t=N....,T,

with W, held. fixed at a constant level W: (T-N+1)'Z1_,W[c,+d,u(e2,,,fi,)] = Uz, (4)

In a large sample, this will be close to the average of the conditional mathematical expectation, (T-N+1)'ZT_xE,U,,4;- Of course the asymmetry in Figure 1 now revolves around e?,, rather than h,. Average utility depends on taste for risk. Consider fixing the coefficient of relative risk aversion (CRRA), which for quadratic utility is yW/(i-yW). In this case, the variables c, and d, in (3) do not depend on W and expected utility is linearly homogeneous in wealth: double wealth (holding the CRRA constant) and expected utility doubles. (Of course, by fixing relative risk aversion rather than Y, we are implicitly interpreting quadratic utility as an approximation to a

nonquadratic utility function, with the approximating choice of y dependent on wealth.)

By definition, the optimal model requires less wealth to yield any specified level of average utility than does any given suboptimal model. We interpret the difference in required wealth as the average per period fee that the investor would be willing to pay to switch from a suboptimal to the optimal model. For a given suboptimal model, we report the ratio of this fee to an initial level of wealth (the exact level is arbitrary, since the linear homogeneity noted in the previous paragraph means that ratio is independent of the initial level). For convenience of interpretation, we express this in annual basis points. Example: Suppose that with a horizon of one week, an optimal GARCH model with initial wealth of $9999 yields the same average utility as does a suboptimal

homoskedastic model with initial wealth of $10,000. Then we report an annualized fee of 52

12 weeks/year x [($10,000-$9999)/$10,000)] x 100 x 100 = 52, where the first 100 converts to

percentage and the second to basis points.

The appendix shows that if, say, model 1 is the optimal model, model m an arbitrary

suboptimal model, this fee may be computed as

(52/j) x 10000[1-(U,/U,)], (5)

where Jj is the horizon, j=1 or 13.

How does variation in risk aversion affect this fee? In the general framework of the previous section, the effects are ambiguous. But when there is a risk free asset, as in our empirical work, it can be shown that the expected utility benefits of a better model are lower for more risk averse investors: if, say, E,U,,,,-E,U,,4; >0 (ie., model 1 is better than model m), then O(EUy41-E,U nu1)/0(CRRA)<O. The intuition is that greater risk aversion leads to larger fractions of wealth in the safe asset and less variation in expected outcomes across models. A likely empirical implication is that for a given time series of returns and volatility estimates, the higher is the CRRA YW/(1-yW), the lower will be the estimated value of (5).° IV. Data and Models A. Data

Our exchange rates are measured as dollars per unit of foreign currency, between the U.S. and Canada, France, Germany, Japan and the United Kingdom.* The data are Wednesday, New

York noon bid rates, as published in The Federal Reserve Bulletin.

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The returns R,,, and R*,, are Eurodeposit rates. For one week maturities, the data are from the London market. Wednesday closing rates (which we believe are at noon New York time, apart from variation induced by daylight savings), average of bid and asked, were available for France, Germany. Japan, the United Kingdom and the U.S. (but not Canada). These were kindly supplied

by Karen Lewis; the ultimate source is The London Financial Times. We cleaned up some obvious

recording errors before using these data (details available on request). For one quarter maturities, the data are generally from the Zurich market, occasionally (when Zurich data were not available) from the London market. Wednesday bid rates, 10:00AM Swiss time (4:00AM New York time. again apart from variation induced by daylight savings) were available for all six countries. The source is the Bank of International Settlements. For both exchange and interest rate data, when Wednesday was a holiday we used Thursday data; when Thursday was a holiday as well we used Tuesday cata.

_ After an initial observation was lost due to differencing the exchange rate data, the exchange rate sample for each country included the 863 observations from March 14, 1973, to September 20, 1989. Plots and summary statistics on the exchange rates are presented in West and Cho (1992). To conserve space, we limit ourselves here to a brief summary of the familiar pattern: exchange rate changes appear to have zero unconditional means, be serially uncorrelated, have zero skewness and very fat tails; the squares of exchange rate changes appear to be highly serially correlated.

We arbitrarily began our out of sample exercise at the midpoint of the exchange rate data,

and the first sample for which we fit any volatility models included the 432 observations from March 14, 1973 to June 17, 1981 (N=432 in the notation of equation (le)); accordingly, the first

interest rate observations that we used were those for June 17, 1981. As we added additional

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observations, we rolled the sample, fixing the sample size at 432, and dropping what had been the initial observation as each additional observation was added on. The final week used in estimation was April 5, 1989 (T=839 in equation (1f)), which means that our final sample spanned the 432 observations from December 17, 1980 to April 5, 1989 and the number of forecasts, as well as the size of our sample of interest rate observations, was 408. An earlier version of this paper tried not only 1 and 13 week but 4 and 24 week horizons as well, and this accounts for our withholding the final 24 (instead of 12) weeks of data (i.e., accounts for T=839 instead of T=851 in (3-1f)). Results for 4 and 24 weeks are not reported since they are similar to those for 1 and 13 weeks.

Table | contains some basic statistics on the foreign - U.S. differential. For ease of interpretation, these are expressed at annualized rates; the corresponding weekly or quarterly rates were used in the empirical work. The standard errors here and in subsequent tables were computed by (l)applying the asymptotic theory in Hansen (1982) to the moment conditions used to produce the estimates, and (2)using the Newey and West (1987) technique to estimate a certain spectral density that this theory requires.

According to Table 1, the differentials have broadly similar patterns. Lines (7) and (9) indicate that they tended to stay within a band about 3 percentage points wide; lire (11), which in columns (1) to (5) gives the number of weeks in which the foreign rate is higher than the U.S. rate, reveals that the differential rarely changed sign during the sample period. With the exception of the French weekly rate, there is considerable serial correlation in the interest rate differentials (lines (2) and (3)); nonetheless, computation of a statistic not reported in the Table, T(p,-1) (where T=408

is the sample size and , is the first order autocorrelation reported in row (2)), rejects the null of a

15 unit root at the five percent level in all four weekly differentials and in the Canadian and French quarterly differentials as well.

For some brief periods in the carly part of the sample, French interest rates were rather high, at times extraordinarily so.” These temporary spikes account for the large standard deviation (line (2)) and the relatively little serial correlation in French differential (lines (3), (4)). Apart from France, the other interest rate differentials followed more stable patterns.

In computing our utility based measure, we treat each currency in isolation, and produce nine sets of estimates, four for weekly and five for quarterly rates. Under our assumptions, a U.S. resident will invest a positive amount in a bond denominated in foreign currency only if the expected return on the bond denominated in foreign currency is higher than that on the dollar bond; since we also assume that the expected change in exchange rates is zero, this happens only if the foreign nominal return is higher. The converse is true for a foreign resident dividing her portfolio between bonds denominated in her own and in U.S. currency.

It is evident from line (11) of Table 1 that there would be little grounds for comparing volatility estimates for German and Japanese exchange rates if our hypothetical investor were a U.S. resident: since weekly and quarterly Deutschemark rates, and quarterly yen rates, were lower than dollar rates for every single week in the sample, a U.S. resident dividing her wealth between deutschemark or yen bonds on the one hand and dollar bonds on the other would never put any money in the former. We would have a similar though less dramatic problem for Canadian and French data if the investor were a resident of one of those two countries. So that our utility based

measure could use all 408 estimates of conditional variances, for each model and exchange rate, we

elected to make the hypothetical investor in a given week a U.S. resident if the U.S. interest rate is

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lower, a foreign resident if the foreign interest rate is lower. For a given exchange rate, the fee that an investor would pay to switch to the best model is then interpreted as the sum of the fees paid by investors in the two countries.

B. Models and Estimation Techniques

Column (1) of Table 2 lists the models we estimated, column (3) the acronyms used in some subsequent tables. Column (2) gives the formula for the one period ahead conditional variance, except for the nonparametric estimator for which the formula for the arbitrary j period ahead forecast is given. Since all the other models are linear, multiperiod forecasts can be obtained by the usual recursive prediction formulas.

The homoskedastic model (line (1)) simply set the conditional variance at all horizons equal to the sample mean of lagged e7’s.

Two GARCH models were used (lines (2) and (3)). Both were estimated by maximum likelihood assuming conditional normality, using analytical derivatives, with presample values of h and e? set to sample means. Lee and Hansen (1991) and Lumsdaine (1989) show that the conditional normality assumption is not necessary for the consistency and asymptotic normality of the estimators. We chose GARCH(1,1) and IGARCH from a larger set of possible GARCH models after some preliminary in- and out of sample analysis suggested that these were the best GARCH models.

We also studied two autoregressive models, both of which were estimated by OLS. One autoregression used e? (line (4)). It is included because GARCH models imply ARMA processes for e (see Bollerslev (1986)); OLS estimation of such autoregressions therefore might perform

comparably to more complicated GARCH estimation (although under the GARCH null, such OLS

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estimation is asymptotically inefficient). (In practice, this model occasionally produced negative point estimates of the conditional variance, in which case we used the homoskedastic estimate.) As in Schwert (1989a, 1989b), the other autoregression used le! (line (4)). Schwert suggests the factor of (1/2) because the variance of a zero mean normally distributed random variable is (x/2) times the square of the expected value of its absolute value. For both autoregressions, the lag length of 12 was chosen because for all countries in sample results indicated that such a lag length was more than suffic.ent to produce a Q-statistic that implied white noise residuals.

Finally, we also tried a nonparametric estimator (line (6)). It can be interpreted as working off the basic definition E(e?,,le,) = Wet. f(e?,,le,)de?,,, where f(e;,ile,) is the density of e{,, conditional one, See Pagan and Ullah (1990a,1990b) for an excellent exposition. As in Pagan and Schwert (1990) we used a Gaussian kernel, defined in column (2), with the bandwidth b = 6(N-j)", 6 the sample standard deviation of e,, t=1,....N-j.. We did not try any other kernel. We did a little experimentation with some alternative fixed bandwidths and information sets, comparing out of sample mean squared errors, but found that these yielded similar results.

V. Empirical Results

For our utility based measure, we report in detail results with a CRRA of one (ie., YW/(1-yYW)=1, in the notation of section III); below, we summarize results with a CRRA of 10. Table 3 has estimates and asymptotic standard errors of (5), with Eurodeposits of one week maturity in panel A, one quarter maturity in panel B.

One’s eyes are drawn to the "0.000" entries for GARCH(1,1), which appear for five of the nine rows. IGARCH yielded the highest average utility in two other data sets (Germany, both

horizons), and was second to GARCH(1,1) in four others. The nonparametric model was best for

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France (weekly) and Canada (quarterly), but otherwise did not perform very impressively. The remaining three models generally did poorly. Note that the fine performance of the GARCH models as a class is not an artifact of the presence of two such models: had we not estimated IGARCH models, GARCH(1,1) would have been best in 6 rather than 5 datasets; had we not estimated GARCH(1,1) models, IGARCH would have been best in 6 rather than 2 datasets.

The statistical significance of differences across models is weak, however. Only five of the twenty entries in Table 3A, and seven of the twenty five entries in Table 3B are significantly different from zero at the ten percent level (two-tailed test). As indicated in the last column of each panel, the nine tests for the equality of utility levels across all models is rejected at the five percent level once and at the ten but not five percent level twice.

The economic significance of differences across models appears to be more pronounced. In the weekly data in Table 3A, the four non-GARCH models had three digit estimates more often than not, indicating that an investor would be willing to give up more than 100 basis points of her wealth, annually, to switch from using one of these models to the optimal one. At the longer horizon, performance is more similar across models: the median figure in panel B is 45, in panel A is 187. This is consistent with the well known fact that conditional heteroskedasticity in exchange rates tends to die out rapidly (Diebold (1988)).

One way to gauge these figures is to interpret them as a fee that a professional money manager could charge an investor not capable of estimating GARCH models. As such, these figures seem to be above what Wall Street mutual funds typically charge for their services (Ippolito

(1989), New York Times, May 14, 1991, page F14), which suggest to us that they are substantial.

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Table 4 summarizes some experiments we performed to see whether these results are sensitive to the sample used and to the choice of relative risk aversion. In Table 4, specification A is the one used in Table 3, and is repeated for convenience. Specification B recalculated the entries in Table 3 using each of the two halves of the sample rather than the whole sample, specification C recalculated using each of the four quarters of the sample. Specification D recalculated using the whole sample, and a higher assumed level of risk aversion.

As one can see in column 2 of panel B, GARCH models tended to perform best in all these additiona. experiments. Columns 3 and 4 indicate that statistical significance of differences between models about as strong as was suggested by Table 3. Column | indicates that the estimates of the wealth one would sacrifice to use the best model are a little lower in the later parts of the sarnple, and that increased risk aversion (specification D), which, as we noted above, will lead to a narrowing of differences across models, happens to do so rather sharply. We therefore slightly amend our summary of results, to state that our estimates imply that an investor would be willing to give up 5 to 200 basis points of her wealth, annually, to switch to GARCH from another model; even the lower bound of this range strikes us as substantial.

How do these results compare with those of the usual mean squared error criterion? Table 5 presents rankings by this criterion, for a one week horizon. (The mean squared errors underlying the rankings are available on request.) While in each country either GARCH(1,1) or IGARCH has the lowest mean squared error, the GARCH(1,1) model overall does not perform as well as it did by the ut:lity based criterion (see the entries for France and Germany). Moreover, the 7(5) Statistics in the next to last column suggest that there is little to recommend one model over

another, in the sense that for no country can one reject the null that all six mean Squared errors are

20

the same at conventional significance levels. Even more striking is that there is precious little evidence that whichever GARCH model had the lowest mean squared error is substantially better than the homoskedastic model. The last column indicates that one cannot reject the null that the mean squared error for the homoskedastic model is the same as the best GARCH model at anything close to conventional significance levels. We conclude that the mean squared error criterion also favors GARCH as a class, but not as sharply as does our utility based criterion.

We close this section with a closer look at a particular period, which suggests that it is the asymmetry in our utility based criterion that accounts for the differences between meari squared error and utility rankings. A comparison of Tables 3 and 5 indicated to us that a detailed examination of French data might be revealing in this connection, because for such data GARCH(1,1) does poorly by the mean squared error, well by the utility based criterior. Consider the 13 weeks from August 14, 1985 to November 6, 1985. The length of this interval was chosen because the relevant figures could be graphed clearly; the dating of this interval was chosen because it is centered around the Plaza Accord, which was announced on September 22, 1985, and which caused the largest weekly change in the dollar/franc exchange rate in our sample (7.7 percent).

Figure 2A plots the annualized interest rate differential, which we present simply to reassure the reader that the estimates of our utility based measure that we are about to present are not based on unusual interest rates. Figure 2B plots the absolute value of the exchange rate together with the square root of the corresponding conditional variance for the GARCH(1,1), IGARCH, and homoskedastic model. Only three models, and square roots rather than squares, were plotted to

make the figure more legible. Figure 2C plots the evolution over time of estimates of the wealth

21

an investor would sacrifice to use GARCH(1,1); the first estimate, for 8/14/85, is based on one observation, the final estimate, for 11/6/85, is based on 13.

In the: first three weeks of this period, Figure 2B suggests that GARCH(1,1) did a poorer job of fitting the realized square of the exchange rate than did the other two models, and Figure 2C bears out this impression. During the next four weeks, from 9/3 to 9/25, it is hard to tell from Figure 2B which models are tracking the exchange rate best. But Figure 2C indicates that by 9/25, the GARCH(1,1) model delivered the highest average utility, a ranking that was maintained until the end of the 13 week period that is graphed. In comparing Figures 2B and 2C, what is particularly striking is (1)the degeneration of the homoskedastic relative to the GARCH(1,1) model during the week ending 9/25 (the week of the Plaza accord), and (2)the continued domination of GARCH(1,1) after 9/25 despite its substantial overestimates of the conditional variance.® This illustrates the asymmetry of the utility based criterion: it may be seen in Figure 2B that the homoskedastic model underestimated only slightly relative to the GARCH(1,1) model for the week ending 9/25, and that the GARCH(1,1) model overestimated dramatically relative to the homoskedastic model in some subsequent weeks. But the underestimate has a much stronger effect on utility than do the overestimates.

Figure 2B suggests to the eye that the GARCH(1,1) model does poorly by a mean squared error criterion. This impression is borne out by a formal calculation. Table 6 contains wealth sacrifices and rankings by mean squared errors for all six models, for this 13 week period.

GARCH(1,1) was the best by the utility based criterion, worst by the mean squared error criterion.

22

VI. Conclusions

We conclude with some suggestions for future research. An obvious possibility is to see if other models, such as those surveyed in Bollerslev et al. (1990), dominate GARCH. Another is to apply our analysis to a portfolio of assets that is better diversified, such as one that includes equities. A third is to permit flexible use of a variety of models by allowing for weighted combinations of fitted conditional means and variances and/or implied fractions, possibly with time varying weights. Finally, it would be very desirable to compare volatility models in an

environment of dynamic rather than static utility maximization.

23

Footnotes

“ The first author is a professor at the University of Wisconsin, the third author is assistant professor at Texas A&M University; the second author is a senior economist in the Division of International Finance, Federal Reserve Board, Washington, D.C., U.S.A. We thank an anonymous referee, Buz Brock, Frank Diebold, Takatoshi Ito, Blake LeBaron, Robin Lumsdaine and participants. in various seminars for helpful comments and discussions, Karen Lewis for providing data, and John Hulbert for research assistance. West thanks the National Science Foundation, the Sloan Fourdation, and the University of Wisconsin Graduate School for financial support. This paper represents the views of the authors and not necessarily those of the Board of Governors of the Federal Reserve System or other members of its staff.

1. For a model of the conditional mean of stock returns, McCulloch and Rossi (1991) also use a utility approach, and Breen et al. (1989) aim, as do we, to estimate how much an investor would pay to use a model.

2. This scalar result does not generalize in an obvious way to higher dimensions. If H is a matrix, it is possible to have A, > A, > H (where for matrices A and B, A>B means A-B is positive definite) with EU,,,, > EU nes. 3. It is not absolutely certain that in any given sample increased risk aversion will lead to a lower fee; a sufficient condition is Uj,,;-U,,.;>0 for all t.

4. We also obtained Italian data. But in sample statistics suggest a nonzero unconditional mean. We dropped Italy rather than fit means as well as variances.

5. On at least one occasion, the high rate preceded an EMS realignment that depreciated the franc:

the interest rate differential of 306 percent (Table 1A, column 2, line 10; corresponding weekly rate

24

is about 2.7 percent), occurred on March 16, 1983, and the following week there was a realignment that depreciated the franc against the Deutschemark by about 8 percent (Edison and K.aminsky (1991)). This suggests that our assumption the change in exchange rates is never predictable is a little extreme, at least in the first part of the sample; in the empirical work we therefcre make sure that our results hold when we exclude the earlier parts of the sample.

6. Here, we are identifying the square of the exchange rate with the population conditional variance, although these in fact differ by a zero mean expectational error; note that the fact that. the sample

contains only 13 observations means that this expectational error may contribute substantially to our

estimates of average utility.

25

References

Akgiray, Vedat, 1989, "Conditional Heteroskedasticity in Time Series of Stock Returns: Evidence and Forecasts," Journal of Business 62, 55-80.

Bollerslev, Tim, 1986, "Generalized Autoregressive Conditional Heteroskedasticity,” Journal of Econometrics 31, 307-327.

Bollerslev, Tim, Chou, Ray Y., Jayaraman, Narayanan, and Kenneth F. Kroner, 1990, “ARCH Modelling in Finance: A Selective Review of the Theory and Empirical Evidence, with Suggestions for Future Research,” manuscript, Northwestern University.

Breen, William, Glosten, Lawrence R. and Ravi Jagannathan, 1989, "Economic Significance of Predictable Variations in Stock Index Returns,” Journal of Finance 44, 1177-1191.

Diebold, Francis X., 1988, Empirical Modeling of Exchange Rate Dynamics, Berlin: Springer-Verlag.

Diebold, Francis X., and James Nason, 1990, "Nonparametric Exchange Rate Prediction?", Joumal of International Economics 28, 315-332.

Edison, Hali J. and Garciela Laura Kaminsky, 1991, "Target Zones, Intervention and

Exchange Rate Volatility, France, 1979-1990," manuscript, Federal Reserve Board of Governors.

Engle, Robert F., 1982, "Autoregressive Conditional Heteroskedasticity, with Estimates of the Variance of United Kingdom Inflation,” Econometrica 50, 987-1007.

Engle, Robert F. and Tim Bollerslev, 1986, "Modelling the Persistence of Conditional Variances, " Econometric Reviews 5, 1-50.

Engle, Robert F., Che-Hsiung Hong, and Alex Kane, 1990, "Valuation of Variance Forecasts with Simulated Options Markets,” manuscript, UCSD.

Eun, Cheol S. and Bruce G. Resnick, 1984, "Estimating the Correlation Structure of International Share Prices," Journal of Finance 39, 1311-1324.

Friedman, Benjamin M., and Kenneth N. Kuttner, 1988, "Time-Varying Risk Perceptions and the Pricing of Risky Assets," NBER Working Paper No. 2694.

Hansen, Lars Peter, 1982, "Asymptotic Properties of Generalized Method of Moments Estimators,” Econometrica 50, 1029-1054.

26

Ippolito, Richard A., 1989, "Efficiency with Costly Information: A Study of Mutual Fund Performance,” Quarterly Journal of Economics 1-24.

Lee, Sang-Won and Bruce E. Hansen, 1991, "Asymptotic Properties of the Maximum Likelihood Estimator and Test of the Stability of Parameters of the GARCH and IGARCH Models," manuscript.

Lumsdaine, Robin L., 1989, "Asymptotic Properties of the Maximum Likelihood Estimator in GARCH(1,1) and IGARCH(1,1) Models,” manuscript, Harvard University.

Meese, Richard A. and Andrew K. Rose, 1991, "An Empirical Assessment of Non-Linearities in Models of Exchange Rate Determination", Review of Economic Studies 58, 603-618.

Meese, Richard A. and Kenneth S. Rogoff, 1983, "Empirical Exchange Rate Models of the Seventies: Do They Fit Out of Sample?", Journal of International Economics 14, 3-24.

McCulloch, Robert and Peter E. Rossi, 1990, "Posterior, Predictive and Utility-Based

Approaches to Testing the Arbitrage Pricing Theory,” Journal of Financial Economics 28, 7-38.

Newey, Whitney K., and Kenneth D. West, 1987, "A Simple, Positive Semidefinite,

Hetroskedasticity and Autocorrelation Consistent Covariance Matrix,” Econonictrica 55, 703-708.

Pagan, Adrian R., and G. William Schwert, 1990, "Alternative Models for Conditional Stock Volatility,” Journal of Econometrics 45, 267-290.

Pagan, Adrian R., and Aman Ullah, 1990a, "Chapter 3: Methods of Density Estimation,” manuscript, University of Rochester.

Pagan, Adrian R., and Aman Ullah, 1990b, "Chapter 4: Non-Parametric Estimation of Conditional Moments," manuscript, University of Rochester.

Schwert, G. William, 1989a, "Business Cycles, Financial Crises and Stock Volatility," 33-126 in K. Brunner and A. H. Meltzer (eds.), IMF Policy Advice, Market Volatility,

Commodity Price Rules and Other Essays, Camegie Rochester Series on Public Policy No. 31.

Schwert, G. William, 1989b, "Why Does Stock Market Volatility Change over Time?", Journal of Finance 44, 1115-1154.

West, Kenneth D. and Dongchul Cho, 1992, "The Predictive Ability of Several Models of Exchange Rate Volatility,” manuscript, University of Wisconsin.

27

Appendix Average utility is estimated as follows. For a one period (one week) horizon, use models m=1,...,.M to solve MAX ffm) [EU mest =Ei(W mesi--S YW mesi)model m used to compute Ee.) S.t. Woner = Wilfr(Resiteur) + O-fmd Real: Assume that the interest rate differential is uncorrelated with the change in exchange rates, E(Ri- Re Jeu: = 0. Let pa = Ri.-Ru; > 0, W=W,, and assume 1-yWR,,, > 0 (otherwise the

investor can reach satiation with certainty). Elementary calculus yields

(AL) fgg = [Hear -YWR)/ CWI Aa) ==> EUnu: = [c, + duch.h,)]W, c = (Rui SYWRiw) d, = (YW) (-YWRY, w(hy fm) = [zg SQ Ha) eth)

Let 5=yW/(1-yW) be the coefficient of relative risk aversion. Substitute the ex-post realized exchange rate square e¢,, for its conditional expectation h, and average over many time periods

to get average utility,

(A2) Ug = [¢ + UglW, c = (T-Nt+1)'Zi xc,

Up = (T-N#1) "Etec wea sy)

28

Suppose that model 1 turns out to be the best. Let m be an arbitrary suboptimal model.

We see from (A2) that when model 1 with wealth W-AW yields average utility equivalent to

model m with wealth W, AW satisfies (ct+u,)(W-AW) = (c+u,,)W. The corresponding fraction

of wealth is AW/W = [(c+u,)-(c+u,,)|/(c+u,) = [1-(U,/U J]. As indicated in equation (3-5), we express this ratio in basis points.

For a 13 week horizon: in (A2) replace e%,, with (€,,, +... + Gr3)"s be with Bone tit Bent 13+ The implicit timing assumption is that investors are using weekly data to make investment decisions every quarter (every thirteen weeks). One thirteenth are investing the first week in the quarter, ... , one thirteenth the last week of the quarter. The figure for average utility that we

compute is the average of average utility for each of the thirteen groups of investors.

29

Figure I

Graph of u(h,sfi,,,) as a Function of finns h,=0.000225

600

500

0.000000 0.000225 0.000450 0.000675

30

Figure Il . Fines and Wealth Sacrifice for France, Weekly

A. Interest Rate Differential (percent)

2.2

;

B/14/85 68/28 9/11 9/25 To7o 10/23 11/6/85

8/14/85 8/28 9/11 9/25 10/9 10/25 11/6/85

1.0

B. le, and Square Root of Bact

oO

OQ - D - " N S a © homo “ S a (1,1) vi tg

oO S oO

S So 4 fo] _— a = = = S = o ws, Se |

8/14/85 6/28 9/11 9/25 10/3 10/25 11/6/85

31

Table 1

Summary Statistics on Annualized Interest Rate Differentials

A. Weekly (2) (3) (4) (5) (6) France Germany Japan U.K. U.S. (1)Mean 3.76 -3.59 -3.61 1.65 9.32 (1.21) (0.22) (0.43) (0.43) (0.52) (2) Standard Deviation 17.94 1.32 2.47 2.53 2.89 (7.90) (0.14) (0.39) (0.27) (0.45) (3), 0.50 0.90 0.95 0.94 0.96 (0.08) (0.036) (0.02) (0.03) (0.02) (4) p, 0.07 0.86 0.92 0.89 0.94 (0.03) (0.04) (0.02) (0.04) (0.03) (5)correlation with 0.08 -0.70 -0.91 -0.76 1.00 U.S. rate (2.78) (0.32) (0.58) (0.45) (6)Minimum -2.51 -7.56 -12.87 -7.37 5.75 (7)Q1 0.63 -4.13 -4.50 -0.06 7.13 (8)Median 1.63 -3.44 -3.00 1.94 8.63 (9) Q3 2.75 -2.75 -2.06 3.57 9.88 (10)Maximum 306.00 -0.65 0.50 6.25 19.63 (11)No. of obs > 0 341 0 7 300 408 B. Quarterly (1) (2) (3) (4) (5) (6) Canada France Germany Japan U.K. U.S. (1)Mean 1.19 2.87 -3.75 -3.82 1.50 9.57 (0.18) (0.54) (0.23) (0.43) (0.43) (0.55) (2) Standard deviation 0.99 3.53 1.27 2.38 2.42 2.98 (0.10) (0.63) (0.15) (0.38) (0.22) (0.45) (3) p; 0.95 0.89 0.97 0.98 0.98 0.98 (0.02) (0.02) (0.01) (0.01) (0.01) (0.01) (4) p, 0.90 0.77 0.93 0.96 0.95 0.97 (0.03) (0.04) (0.02) (0.03) (0.02) (0.02) (5)correlation with -0.11 0.21 -0.77 -0.93 -0.79 1.00 U.S. rate (0.14) (0.25) (0.32) (0.55) (0.40) (6) Minimum -1.06 -1.69 -7.06 -12.13 -5.62 5.63 (7) Q1 0.56 0.75 -4.31 -4.75 -0.43 7.44 (8)Median 1.25 1.82 -3.56 -3.19 1.75 8.94 (9) Q3 1.75 4.00 ~3.00 -2.25 3.50 10.31 (10) Maxirum 4.19 25.75 -1.12 0.00 5.37 19.38 (11)No. cf obs > 0 344 346 0 0 281 408 Notes:

1. The sample includes 408 observations from 6/17/81 to 4/5/89; both quarterly and weekly rates are sampled weekly. Non-U.S. interest rates are expressed as an excess over the U.S. rate. Data are described in the text.

2. In rows (3) and (4), Pl and p2 are the first and second autocorrelation coefficients.

3. Asymptotic standard errors in parentheses.

32

Table 2 Models (1) (2) (3) Model Formula for h. Acronym Homoskedastic Model 1. Homoskedastic h, = @ homo GARCH Models 2. GARCH (1,1) h, = @ + ae? + Bh,_, (1,1) 3. IGARCH(1,1) h, = ae? + (1-@)h,-_; ig Autoregressive models 4. AR(12) in e& h. = @ + 2i2,0,e7...: e2AR 5. AR(12) in fe. h, = (W/2) (E,le...1)77 |e|AR BE. les:| = @ + Zi2,a.}e..s.41 Nonparametric Model 6. Gaussian kernel h..; = E(ezsler); nonp he = Deri Wey, jez 650 Weng = Cony J LNiCen yr

Cy, = exp(-.5 (ey-e.)7/b*], b= bandwidth defined in text

33

Table 3

Wealth Sacrifice for Right to Highest Utility Model

A. Weekly Horizon

Model homo (1,1) ig e2AR leJAR nonp x7 (5) Country France 862.6 78.7 144.5 245.2 . 108.5 0.0 3.174 (973.0) (267.0) (394.8) (441.7) (187.4) [0.673] Germany 42.5 47.1 0.0 319.1° 208.9"" 306.7 12.246" . (31.5) (38.7) (177.1) (76.5) (230.7) [0.032] Japan 37.3 0.0 40.2 62.6 169.0° 687.7 9.912° (171.7) (39.1) (181.8) (90.3) (473.3) [0.078] U.K. 204.77" 0.0 40.2 412.3" 234.6 482.2 8.517 (102.9) (30.9) (229.1) (153.2) (366.0) [0.130] B. Quarterly Horizon Model homo (1,1) ig e2AR |e|AR nonp x (5) Country Canada 1.7 56.8°° 1031.3 28.97" 94.4 0.0 9.480° (9.0) (28.8) (736.8) (11.1) (62.2) [0.091] France 181.3 0.0 8.4 148.5 210.4° 180.2° 7.893 (116.0) (20.9) (95.1) (120.4) (102.6) [0.162] Germany 29.4 4.9 0.0 29.7 48.5 44.7 5.128 (46.6) (25.1) (46.4) (48.9) (52.7) [0.400] Japan 120.1° 0.0 25.2 40.8 75.1 153.6" 7.838 (71.9) (34.8) (34.6) (51.4) (80.4) {0.165] U.K. 33.0 0.0 81.8" 0.1 8.5 32.5 5.643 (71.3) (46.8) (29.3) (22.8) (68.7) {0.343] Notes:

1. An investor is assumed to divide her wealth between Eurodeposits in dollars and those in the currency of the indicated country, 6/17/81-4/5/89. Each row reports estimates of (5), the wealth that the investor would give up to use the model that yielded the highest average utility (the model with the "0.0" entry). The units are annual basis points. Smaller numbers mean better performance. Table 2 describes the acronyms for the models. Relative risk aversion is set to l.

2. Asymptotic standard errors are in parentheses; "*" indicates Significance at 10 percent level, "**" at five percent level (two-tailed test).

3. The x?(5) column reports a test of the null that all five nonzero entries in a given row are equal to zero, with asymptotic p-value in brackets.

34 Table 4 Effects: of Alternative Specifications

A. Description of Specifications

Sample period CRRA Description A 6/17/81-4/5/89 1 Table 3 specification Bl 6/17/81-5/8/85 1 first half of Table 3 sample B2 5/15/85-4/5/89 1 last half of Table 3 sample Cl 6/17/81-5/25/83 “1 first quarter of Table 3: sample |. C2 6/1/83-5/8/85 1 second quarter of Table 3 sample C3 5/15/85-4/22/87 1 third quarter of Table’3 sample c4 4/29/87-4/5/89 1 fourth quarter of Table 3 sample ; D 6/17/81-4/5/89 10 Table 3 sample, with higher CRRA B. Summary of Empirical Results (1) (2) (3) (4) Median Estimate Number of Countries Number of Number of x‘ (5) of (5), Wealth for which best t-statistics Statistics Sacrifice model is: significant at: significant at: (1,1) ig. .10 ..05 -10 = .05 Weekly: A 186.9 2 1 5 2 2 1 Bl 297.0 2 1 4 1 2 0 B2 54.5 1 2 2 0 0 0 cl 321.4 2 “1 3 1 1 1 C2 74.2 1 1 3 2 2 2 C3 63.7 0 2 5 2 Oo. 0 C4 10.0 1 3 4 2 2 0 D 4.6 2 1 5. 2 2: 0 z Quarterly: A 44.7 3 1 7 2 1: 0 Bl 100.4 2 2 6 1 2,1 B2 23.2 1 0 6 3 2 2 Cl 55.4 1 3 5 2 2.8 1 C2 ; 49.0 1 3 6 1 3 2 C3 53.4 1 1 8 6 3 3 c4 17.4 0 2 il 5 3 2 D 0.7 2 1 5 1 -1 1 Notes:

1. Specification A is the one reported in detail in Table 3, and is repeated here for convenience. : ; :

‘2. Panel B is based on estimates for the 4 (weekly) or 5 (quarterly) countries and 6 models listed in Table 3A. Since, for a-given country, the estimates of the best model (the "0.0" model) do not figure into the computation of the number of the panel B values, the total number of values underlying each weekly row is 20 for columns (1) and (3), 4 for columns (2) and (4); the corresponding quarterly figures are 25 and 5.

35 Table 5

Rankings by Out of Sample Mean Squared Error, Weekly Horizon

Model homo (1,1) ig e2AR |e|AR nonp x7 (5) x' (1)

Country

Canada 5 1 3 4 2 6 7.244 1.243 [0.203] [0.265]

France 2 6 1 5 3 4 8.911 0.011 [0.113] [0.918]

Germany 2 5 1 6 . 4 3 8.147 0.012 [0.148] [0.912]

Japan 3 1 2 5 4 6 6.414 0.770 [0.268] [0.380]

U.K. 4 2 1 5 3 6 3.779 1.521 {0.582} {0.217}

Notes:

1. In each row, "1" indicates best (smallest) mean squared error for the indicated country, "2" second best, ... , “6" worst.

2. The x*(5!) column reports a test of the null of the equality of the six mean squared errors underlying the ranking in a given row, with asymptotic p-value in brackets.

3. The x?(1\) column reports a test of the null of the equality of mean squared error for the homoskedastic and best model (either GARCH(1,1), or IGARCH, as indicated), with asymptotic p-value in brackets.

36

Table 6

Results for Weekly Horizon, France, 8/14/85-11/6/85

Model homo (1,1) ig e2AR |e|AR nonp Estimates of (5), 77.6 0.0 8.4 48.2 16.5 105.5 wealth sacrifice to use highest utility model Rankings by out of sample 3 6 1 5 2 4

mean squared error

Notes:

1. For interpretation of the estimates of (5), see the notes to Table 3. 2. For interpretation of the rankings by mean squared error, see the notes to Table IV.

Additional Appendix pAi

West, Edison and Cho, "A Utility Based Comparison of Some Models of Exchange Rate Volatility"

This not-for-publication appendix contains results omitted from the body of the paper to save space. Following are:

I. Plots of annualized interest rate differentials Il. Proof of proposition

Il. Notes on one week interest rate data.

IV. Details of the results underlying summary of utility based results for alternative specifications. V. Details of the results underlying summary of mean squared error results.

Much additional information on the exchange rate data and on the estimates of the models is in West and Cho (1992) and the additional appendix to West and Cho (1992).

pA2

Annualized Interest Rate Differentials (percent)

A. Weekly Rates

U.S. 10

oe Sst a2 es o&+ = 6S&5S —=686 67 6a as 90

Canada

61 82 as ae as ee 87 8a as 20

61 a2 a3 a+ es a6 87 8a 69 20

Japan 8

-15 -10 -5

B. Quarterly Rates

15

5

-15 -10 -5

81 a2 es a+ es a6 87 68 as 90

Non-U.S. rates are expressed as an excess over U.S. rate. data for Canada are not available.

Weekly

Data are described in the text.

Additional Appendix, pA3 Il. Proof of Proposition

Assume exponential utility, with all assets are risky. The proof for quadratic utility is similar. For either utility function, the proof when there is a riskless asset follows as a special case. To simplify notation, all time subscripts are dropped.

Let the gross return on asset i be R,, i=1,...,.k. Let H=[H;] be the corresponding (kxk) full rank variance covariance matrix, assumed known for the moment. Let r=R,-R, be the return in excess of the return on asset i for i<k-1, r=(r,,...,..,,)’, B=Er and Q=E(r-p)(ru)’=CHQ’, where the (k-1)xk matrix Q has 1 in row i, column i, for i<k-1, -1 in all rows in column k, and O elsewhere. Let @ be the (k-1)x1 vector of covariances of r with R,, @=QHgq, where the kx1 vector q has 1 in the k’th row and zero’s elsewhere. Let f, be the fraction put in asset i, i=1,...,k-1, with 1-f,-...-f,., the fraction in asset k. The problem is to maximize E{-exp[-OW(f’r+R,)]} = -exp[-OWE(f'r+R,) +.50°W’var(f’r+R,)] = -exp[-OWf up-OWER, +.50°W7(f’ Of+2f’ w+H,,)] = -c.exp[-OWf’p +.50°W"(f’ QF+2f’ w)], c=exp(- OWER,+.56°W’H,,)>0; the first equality follows since returns are normally distributed. Then f = Q'[(p/8W)-a).

Now let At be an estimate of the variance covariance matrix, f= &'[(W/OW)-O],

6=QHQ’, G=QAq. Expected utility, then, is

(AL) -c.exp[-OW?'p + .50°W2(P OF+2P w)]

= -c.exp[.5(p-8Wo)’ O00" (p-8WO) - (p-8WO)’O(p-6Wo)].

Additional Appendix, pA4

Let V be positive semidefinite, H=H+V, AL=H-V. We wish to show that (A1) is larger when O=0,=Q48,Q’=0+QVQ’, =0+QVq=a+v than when Q=0,=QH,Q’, =0-QVg=0rv. Let Q' be a square root of Q, Q=Q'"Q'”’. Then Q, = 2'7(1+Q'?QVQ’Q'"”)Q"™, d, = 2'7(1-2'’QVQ’Q"”)Q'”, Since D'’QVQ’Q™ is symmetric and positive semidefinite, it can be written as PAP’, where PP’=I and A=diag(A,,...,4,,) is the diagonal matrix of its eigenvalues. For future reference, note that it may be shown that since V is positive semidefinite, Q is of rank k-1 and H,=H-V positive definite, 0SA,<1.

Let a,’ be the 1x(k-1) i’th row of P’Q"”. Since }, = Q!?P(1+A)P’Q"”’, we have

(A2) E(U,,,/H=H+V=H,) = -c.exp[.5(p-8WO)’Q:'00;'(p-8W) - (p-6WO)’O;'(n-8Wo)] = -c.exp{.5[p-8W(wtv))’Q" PI+A)?P’ Q"?[p-OW(wtv)] - [p-OW(a+v)]’Q' PUI+A)'P’Q'?(n-8Wo)} = -c.exp { .SZf}{[p-OW(@+v)]’a/(1+A,)}? - Lea { [p-OW(@+v)]’a}[(u-OWo)’a/(1+A,) } Similarly (A3) E(U,,,H=H-V=H,) = -c.exp { .SZfi{[y-8W(w-v)]’a/(1-A,)}? - » Xi {lp-Ow(@-v)]’a,} [(p-8Wo)’a,}/(1-2,). }.

Thus,

Additional Appendix, pA5 (A4). E(U,,,/|H=H+V) 2 E(U,,,JM=H-V) <==>

Sri {[p-OW(w+v))’a/(1+A,)}? - Te [p-eW(o+v)P'a)[(u-8Way'a/(1+%)) + SIE{(p-0Wo)’a,? < SZ {[p-OW(w-v)]’a/(1-A,)}? - Zizi {[p-OW(w-v)]’a;}((p-OWo)’a|/(1-A,) + SEL-OWo)’a,)? <s=> SZ { [p-OW(@tv)]’a/(1+A,) - (u-OWa)’a, }? <

SIE { [p-OW(w-v)]’'a/(1-A,) - (p-8Wa)’a, }”.

It is easily verified that since 0SA,<1, the inequality holds for each i and thus for the sum as well. |

That equality holds in the proposition if and only if the two estimates yield the optimal fraction follows since it may be shown that (AS) E(U,,,H=H+V) = E(U,,,IN=H-V) <==>

OWv = -QVQ’Q''(y-8Ww) <==>

(Q+QVQ’)"[(H/OW)-@-Vv] = Q"[(n/8W)-a@] = (Q-QVQ’)'[(u/OW)-w+Vv], where the three expressions on the last line are the vectors of fractions chosen if A=H+V,

=H and H=H-V. The second line follows from the first by noting that the first line requires

that the i’th term on the left hand side of the final expression in (A4) be the same as the i’th on the right hand side for all i, writing these k-1 equalities in matrix form and manipulating the resulting expression; the third line in (A5) follows from the second by straightforward

algebraic manipulation.

Additional Appendix, pA6

III. Notes on one week interest rate data.

The raw data had both bid and asked rates. Some observations had bid higher > asked. We checked all of these against microfiche copies of The London Financial Times, and corrected five errors. We then rounded off both bid and asked to two digits, and then, as noted in the text, we averaged the

two.

We had no one week interest rates for France the entire week of 10/8/84-10/ 12/84. So for

10/10/84, we simply used the quarterly rate.

IV. Details of the results underlying summary of utility based results for alternative specifications.

The format of the following tables is the same as that of Table III, except that there are no

parentheses around asymptotic standard errors. Except when otherwise note, the (CRRA is set to 1.

WEEKLY HORIZON

FR

GE

JA

UK

FR

GE

JA

UK

ER

GE JA

UK

FR GE JA

UK

FR GE

JA

homo

1751

45

&3 340

31.4 186

.137 1940.

082

477 437

-348 482

885 -219

homo

0

45

98. 61.

529 15.

756

.519 43.

197

-000

051 974

homo

3526

721. 78.

569 297

629 3834.

682

979 653

818 -299

158.673 130.092

homo 4.492 10:169

6.961 29.214

8

-474

30.253

471.158 352.324 -

o0mo 0.000

73.886 83.146

0.000

Additional Appendix, pA7 *

6/17/81 - 5/8/85

(1,1) ig e2AR 182.757 312.280 491.861 532.339 790.502 882.192

85.531 0.000 580.862 75.656 343.289 0.000 72.101 69.655 73.101 357.724 0.000 83.908 357.181 69.683 235.567 5/15/85 - 4/5/89 (1,1) ig e2AR 0.000 2.206 24.010 4.481 29.869 8.567 0.000 57.188 7.674 36.058 8.701 16.965 64.720 11.967 16.233 50.794 3.528 0.000 471.013 10.272 390.191 6/17/81 - 5/25/83

(1,1) ig e2AR

383.410 637.188 1008.032

1059.816 1578.587 1754.576 147.479 0.000 966.968 149.410 684.046 0.000 22.601 595.079 23.391 426.326 0.000 148.392 347.664 131.015 305.930

6/1/83 ~ 5/8/85

(1,1) ig e2AR 5.276 10.991 0.000

10.725 10.909

23.557 0.000 194.595

39.742 ° 92.183 461.643 583.178 5.841

632.970 787.473 77.652 0.000 19.399 366.701

‘ 40.753 - 321.381 5/15/85 - 4/22/87

(1,1) ig e2AR 1.269 7.905 48.309

31.488 33.997 56.015 2.569 0.000 108.861

10.918 62.307 5.302 33.942 25.335

je|AR 109.895 342.864

299.194 143.503

294.727 177.966

413.877 297.481

|e|AR 132.444 130.203

118.587 63.504

51.907 33.777

58.881 35.912

|e [AR 227.282 678.977

295.125 192.316

560.294 315.797

279.983 247.311

|elAR 15.379 13.190

303.265 189.164

490.556 625.298

547.822 515.428

}e{AR 265.692 258.062

217.659 110.320

53.423

nonp

0.

142 90

300.

412. 825. 700

nonp 25 19.

471 449

1083 865

141.

78.

nonp 0.

194 185

1063. 493.

102 62.

nonp

000

-211 396

874 574

967

345

299

699

.335 .778

-973 -387

971 621

000

531 282

167 083

.352

896

22.449 26.723

89.869 37.601

0.000

1549 .860 1401.215

nonp

51.034 48.855

925.882 889.855

425.173

-994 0.762

10.950 0.052

-614 087

ow

720 451

om)

xX? (5) 3.134 .679

oO

-515 356

ow

143 399

ow

5.642 0.343

x7 (5) 3.083 0.687

4.983 0.418

13.557 0.019

.173 -102

ow

x? (5) 4.162 0.526

11.727

26

0.039

-752

0.000

3.348 0.646

x7 (5) 546 0.616

Ww

- 404 269

on

UK

FR GE JA

UK

FR GE JA

UK

190 112

4 2

17 27

oo

20

23.

1 0

Oo

-507 - 633

homo -552 .063

-165 -959

-003 -008

.681 -989

homo -543 327

034 - 768

-916 - 490

-970 -703

10.

10. 19.

022

173 998

(1,1) 2.224

1

14 12

12. 22

0.

-532

-562 -401

102

-888

000

(1,1)

1 6.

1 0

QUARTERLY HORIZON (1,1)

CA

FR

GE

JA

CA

FR

GE

JA

UK

15 11

344. 218.

70. 71.

266 139

161.

103

0

17.

17

22

20.

homo -603 -870

415 848

997 493

-587 527

066 .315

homo -000

962 - 338

344 926

-000

4. 9

0 39 31.

18. 30

0

-763

435

.138 - 933

-000

-000

322

737 -000

-507

008 975

.773

-000

(1,1)

121. 61.

0

oO

10

95.

510 635

-000 -881 -700

- 468 -869

152

33 0

Additional Appendix, paé

049 .000

16

880 766

4/29/87 - 4/5/89

i 0 0 0

3 8

6/17/81 - 4/5/89, CRRA=10

g -000 -000 -000

119 -405

ig

3 9

0

ro

oo

.313 -522

.000 - 983 -014

-968 755

- 607

-053 - 484

e2AR

3

1. 5.

23

104.

98

64. 36.

-204 680

540 - 418

092 -492

941 643

e2AR

5. 10.

7

6/17/81 - 5/8/85 ig

139.

101

12.

44 0

17

10.

083 215

976 .353

-000

-000

- 428 310

722 634

728 378

547 -677

-004 -392

e2AR

18

9. 285.

172

93.

77

101. 76.

55.

44

5/15/85 - 4/5/89

1

1934.

1393

3.

6

35.

41 80

S57.

241

g 938 -196

987 997

028 116

-920 827

.086

-355 493

598 - 948

694 - 605

474 188

026 737

e2AR

51.

29

11 12

QO.

iO ©

40

618 - 767

-173 225

000 -451 -685 - 338

30.

113. 71.

358

118 327

le|AR 2.686

1

19. 16.

50. 61.

7.

16

.722

563 981

395 114

743

-290

le|AR

ro

nN &

Wu

-547 -504

104 - 988

- 142 -054

-681 -634

|e|AR

43. 34.

402

110 84

108. 100.

46.

39

080 093

- 948 215.

682

-580 -401

471 040

625

-232

|e|AR

157

17

20. 11.

70.

29

65.

- 890 122.

610

-639 17.

905

665 421

836

- 690

523

415.

275. 139.

898

791 564

nonp

3. 1.

17. 26.

1741.

1674

11 16

057 444

011 462

686

-298

-220 -302

nonp

0.

16.

11

000

931 606

931

.818

-718 022

nonp

0. 334. 187.

100. 90.

321.

154

145

000 139 619

381 123

186

527

-511 99.

987

nonp

12.

12

26.

19 23

14.

190

-508

030

418

-238 15.

728

- 683 -853

502

-414

6.154

ow O Pa ou ON ow O UISe on N

on

-292

(5)

361 096

.357 374

-378 - 496

014 075

(5)

-987 702

891 054

-218 069

-481 -187

(5)

-256 -385

-935 077

-000 849

-191 -032

525 -355

(5)

345 014

889 -565

076 299

-010 .016

CA

FR

GE

JA

UK

CA

FR

GE

JA

UK

CA

FR

GE

JA

UK

CA

FR

GE

JA

homo 60.726 86.938

712.601 40€.752

46.491 122.291

362.110 254.191

19.177 26.066

homo 0.000

0.511 27.746

92.614 °66.874

170.393 115.364

303.118 182.443

homo 2.000

35.797 30.426

116.217 48.885

0.000

homo 0.000 1.333 1.654 0.000

0.013

58.

775

(1,1)

13. 70.

Q.

5S. 58.

CO

880 346

000

421 650

.173 -338

-804 646

(1,1)

13.

21. 18

23 13.

37 56.

662

808

005

. 630

-911

978

-910

638

-000

(1,1)

198. 109.

0 46 21.

3.

6

150. 107.

341 933

.000

478

951

523

145

102 984

(1,1)

44, 21.

PR

34. 18

17.

405 496

247 -502

660

-533

407

Additional Appendix, pA9

140.

311

34.395

6/17/81 - 5/25/83

ig 0.

47 90

0

23 28

000

.365 .169

-000

-000

224 .324

e2AR 42.230 81.143

595.635 308.543

138.180 145.794

107.461 125.391

0.000

6/1/83 - 5/8/85

ig

309. 165.

QO.

16. 11.

504 964

000

-000

.000

425 009

e2AR 24.217 13.261

0.159 23.033

48.980 31.688

95.446 93.043

115.411 83.212

5/15/85 - 4/22/87

ig

3514. 2594

9 14

0. 162. 103.

370 255.

622

.820

.209 .178

000

087 114

-854

008

e2AR 85.613 52.230

22.201 22.341

54.170 24.567

4.192 6.764

53.432 64.723

4/29/87 - 4/5/89 ig

349. 211.

QO.

141.

74,

0.

608 127

000 106 200

000

e2AR 17.500 16.780

1.367 1.529

17.259 9.635

12.711

49.417

|e lAR 52.867 102.734

804.458 380.108

171.258 162.620

180.031 193.741

235 -582

sO

|e |AR 63.159 51.463

27.081 39.796

49.592 24.360

36.410 52.857

98.294 72.285

|e {AR 290.788 220.644

35.994 33.727

86.260 37.858

46.193 21.484

86.664 91.644

Je|AR 24.516 37.278

-487 150

HO

26.502 17.690

95.421

7.207

nonp 25.264 76.702

692.711 332.015

136.816 172.304

496.731 284.949

9.653 20.328

nonp 4.473 4.345

0.726 20.764

63.760 46.895

144.410 105.316

287.477 179.287

nonp 20.972 22.083

52.280 34.049

110.142 40.494

16.017 11.841

23.540 13.496

nonp 3.376 8.123

0.964 1.526

7.837 5.290

7.374

x? (5) .179 0.394

Loa)

10.102 0.072

-714 .173

26.634 0.000

-148 -928

fom =

x7 (5) .721 .172

on

17.920 0.003

11.025 0.051

24.256 0.000

-328 -649

OW

x? (5) 15.011 0.010

- 996 -221

on

foe]

396 0.136

15.135 0.010

11.184 0.048

x? (5) 0.057

3.711 0.592

17.394 0.004

12.256

IFDP NUMBER

441

440

439

438

437

436

435

434

433

432

431

430

429

A10

International Finance Discussion Papers

TITLES 1993

A Utility Based Comparison of Some Models of Exchange Rate Volatility

Cointegration Tests in the Presence of Structural Breaks

1992

Life Expectancy of International Cartels: An Empirical Analysis

Daily Bundesbank and Federal Reserve Intervention and the Conditional Variance Tale in DM/$-Returns

War and Peace: Recovering the Market's Probability Distribution of Crude Oil Futures Prices During the Gulf Crisis

Growth, Political Instability, and the Defense Burden

Foreign Exchange Policy, Monetary Policy, and Capital Market Liberalization in Korea

The Political Economy of the Won: U.S.-Korean Bilateral Negotiations on Exchange Rates

Import Demand and Supply with Relatively Few Theoretical or Empirical Puzzles

The Liquidity Premium in Average Interest Rates

The Power of Cointegration Tests

The Adequacy of the Data on U.S. International Financial Transactions: A Federal Reserve Perspective

Whom can we trust to run the Fed? Theoretical support for the founders views

AUTHOR(s)

Kenneth D. West Hali J. Edison Dongchul Cho Julia Campos

Neil R. Ericsson David F. Hendry

Jaime Marquez

Geert J. Almekinders Sylvester C.W. Eijffinger

William R. Melick Charles P. Thomas

Stephen Brock Blomberg

Deborah J. Lindner

Deborah J. Lindner

Andrew M. Warner

Wilbur John Coleman II Christian Gilles Pamela Labadie

Jeroen J.M. Kremers Neil R. Ericsson Juan J. Dolado

Lois E. Stekler

Edwin M. Truman

Jon Faust

Please address requests for copies to International Finance Discussion Papers, Division of International Finance, Stop 24, Board of Governors of the Federal

Reserve System, Washington, D.C.

20551.

IFDP NUMBER

428

427

426

425

424

423

422

421

420

All

International Finance Discussion Papers TITLES 1992

Stochastic Behavior of the World Economy under Alternative Policy Regimes

Real Exchange Rates: Measurement and Implications for Predicting U.S. External Imbalances

Central Banks’ Use in East Asia of Money Market Instruments in the Conduct of Monetary Policy

Purchasing Power Parity and Uncovered Interest Rate Parity: The United States 1974 - 1990

Fiscal Implications of the Transition from Planned to Market Economy

Does World Investment Demand Determine U.S. Exports?

The Autonomy of Trade Elasticities: Choice and Consequences

German Unification and the European Monetary

System: A Quantitative Analysis

Taxation and Inflation: A New Explanation for Current Account Balances

AUTHOR(s)

Joseph E. Gagnon Ralph W. Tryon

Jaime Marquez

Robert F. Emery

Hali J. Edison William R. Melick

R. Sean Craig Catherine L. Mann

Andrew M. Warner

Jaime Marquez

Gwyn Adams Lewis Alexander Joseph Gagnon

Tamim Bayoumi Joseph Gagnon