The Rationality and Heterogeneity of Survey Forecasts of the
Yen-Dollar Exchange Rate: A Reexamination
Richard Cohen, Shigeyuki Abe, and Carl Bonham
September 15, 2005 DRAFT Do Not Quote
Abstract This paper examines the rationality and diversity of industry-level forecasts of
the yen-dollar exchange rate collected by the Japan Center for International Finance. In
serveral ways we update and extend the seminal work by Ito (1990). By using partially
disaggregated forecasts to test for unbiasedness, weak efficiency, and micro-homogeneity of
coefficients across industry groups, we minimize the effects of inconsistent inference due to
aggregation or pooling. We undertake a two-way comparison between univariate (simple)
and bivariate (joint) tests of unbiasedness, on the one hand, and differenced and level specifications on the other. In our systems estimation and testing, we use an innovative GMM
technique (Bonham and Cohen, 2001) that allows for forecaster cross-correlation due to the
existence of common shocks and/or herd effects. Micro-homogeneity test results for correct
unbiasedness specifications are uniformly negative. There does not, however, appear to be
a close relationship between patterns of individual deviations from unbiasedness and microhomogeneity test results. We find that integration accounting and regression specification
matter for rationality testing. We argue that joint (bivariate) tests of unbiasedness and efficiency should be abandoned in favor of separate tests for each. Using correct specifications,
we find that simple unbiasedness deteriorates with forecast horizon, and weak efficiency is
widely rejected at all horizons.
Keywords: Rational Expectations, Heterogeneity, Exchange Rate, Survey.
Richard Cohen
College of Business and Public Policy
University of Alaska Anchorage
3211 Providence Drive
Anchorage, AK 99508
[email protected]
Shigeyuki Abe
Center for South East Asian Studies
Doshisha University
[email protected]
Carl S. Bonham
Department of Economics
University of Hawaii at Manoa
Honolulu, HI 96822
[email protected]
1
Overview
This paper examines the rationality of industry-level survey forecasts of the yen-dollar
exchange rate collected by the Japan Center for International Finance (JCIF).1 Because the
forecasts and targets are integrated, we test for unbiasedness using both differenced and
level forms of these variables. Although the differenced specification is most common, if
the forecast and target are integrated and cointegrated, then the differenced specification
is misspecified as it omits all long run informtion. For both differenced and level forms of
the variables, we compare test results from univariate regressions (of the forecast error on a
constant), where the null hypothesis is that the intercept equals zero, with joint test results
from the more commonly used bivariate regressions (of the target on the forecast, with an
intercept), where the null hypothesis is that the intercept equals zero and the slope equals
one. This distinction may be significant for two reasons. First, we show that the joint
hypothesis of the bivariate regression tests not only unbiasedness but also a type of weak
efficiency, i.e., lack of correlation between forecast and forecast error. To distinguish between
tests for unbiasedness and this type of weak efficiency, we also conduct a battery of other
weak efficiency tests on the forecast error. Second, the bivariate differenced specification
may also suffer from inconsistency if the forecasts are aggregated (see Figlewski and Wachtel
(1983)) or pooled (see Zarnowitz (1985)). Neither of these critiques of inconsistency apply
when the unbiasedness test is conducted either in a univariate regression or a cointegrated
framework. We are not aware of any other rationality studies which make this two-way
comparison between differenced and level specifications, on the one hand, and univariate
and bivariate specifications, on the other.
We also examine the diversity of forecaster performance by testing for micro-homogeneity
in the unbiasedness and weak efficiency equations. The extent of diversity has implications
for rationality, since rational forecasts should satisfy cross-forecaster homogeneity restric1
See Appendix use reference 1 for description of the data.
1
tions for unbiasedness and efficiency. Similarly, a failure of rationality across forecasters
should be evidenced by a failure of micro-homogeneity, unless all forecasters are equally irrational. A failure of micro-homogeneity casts doubt on the theoretical underpinnings of the
representative agent assumption used in much of macroeconomic and financial modeling. It
also has implications for the market microstructure hypotheses.
We use the most disaggregated data available for our analysis. Few other researchers have
used disaggregated data on exchange rate expectations, for either forecast evaluation or forecast modeling, in large part because so few disaggregated databases exist. Notable examples
of disaggregated analysis include Ito (1990), who uses the JCIF data,2 and MacDonald
(1992), MacDonald and Marsh (1996), Chionis and MacDonald (1997), and Benassy-Quere
et al. (2003), all of which employ the Consensus Forecasts of London. In addition, Abou and
Prat (2000) used disaggregated stock price data from the Livingston survey.
In our tests for micro-homogeneity, it is reasonable to assume the existence of crossforecaster error correlation due to the possibility of common macro shocks and/or herd
effects in expectations. To this end, we incorporate two innovations not previously used by
investigators studying survey data on exchange rate expectations. First, we use Pesaran’s
(2004) CD test to verify the statistical significance of cross-sectional dependence of forecast
errors. Second, in our micro-homogeneity tests we use a GMM system that incorporates
a variance-covariance matrix with allowance for cross-sectional as well as moving average
and heteroscedastic errors.3 By more accurately describing the panel’s residual variance2
Ito had access to individual and industry-level forecasts, whereas Bryant (1995) had access only to
aggregated data. We have access only to industry-level (as well as aggregate) forecasts. Hence, there may
be some aggregation bias in our industry level aggregation of the data. However, it is noteworthy that,
using disaggregated data of stock market expectations from the Livingston survey, Abou and Prat (2000)
were able to conclude that their expectational models exhibited more heterogeneity over time and between
industry groups than within groups. See below for further discussion of aggregation bias.
3
Keane and Runkle (1990) first accounted for cross-sectional correlation (in price level forecasts) using a
GMM estimator on pooled data. Bonham and Cohen (2001) tested the pooling specification by extending
this GMM estimator to construct a Wald statistic for testing the micro-homogeneity of individual forecaster
regression coefficients in a system. Elliott and Ito (1999) used single equation estimation that incorporated
a White correction for heteroscedasticity and a Newey-West correction for serial correlation. See section 5.5
on Ito tests.
2
covariance structure, we expect this systems approach to improve the efficiency of our microhomogeneity test statistics.
2
Why test for rational expectations of the exchange rate?
The Rational Expectations Hypothesis (REH) states that economic agents know the
true data generating process (DGP) for the forecast variable. This implies that the market’s
subjective probability distribution of the variable is identical to the objective probability
distribution, conditional on a given information set, φt−1 . Equating first moments of the
market, Em (st |φt−1 ), and objective, E(st |φt−1 ), distributions,
Em (st |Φt−1 ) = E(st |Φt−1 ).
(1)
It follows that the REH implies that forecast errors have both unconditional and conditional means equal to zero. A forecast is unbiased if its forecast error has an unconditional
mean of zero. A forecast is efficient if its error has a conditional mean of zero. The condition
that forecast errors be serially uncorrelated is a subset of the efficiency condition where the
conditioning information set consists of past values of the forecast variable.
In financial markets such as those for foreign exchange, rational expectations plays a
crucial role in the formulation of market efficiency hypotheses. Market efficiency implies that
all available information is embodied in the current market price. Hence, market efficiency
implies an efficient allocation of information, and given this information set, it is impossible
to earn above-normal risk-adjusted profits.
The simplest form of foreign exchange market efficiency is a joint hypothesis that agents
are both risk neutral and form rational expectations of the exchange rate. This form of
market efficiency is embodied in the uncovered interest rate parity (UIP) condition
sem,t,h − st = it − i∗t ,
(2)
3
where st represents the natural logarithm of the nominal spot exchange rate at time t,4 i.e.,
price of a unit of foreign currency in units of the domestic currency, sem,t,h represents the
market’s time t expectation of the h-period ahead spot rate, it is the domestic interest rate
on a nominally riskless asset that matures in h periods, and i∗t is the corresponding interest
rate on the foreign asset.5
Transactions costs aside, UIP may fail to hold because either or both parts of the joint
hypothesis do not hold. First, the assumption that foreign and domestic assets are perfect
substitutes may fail. Thus, because their positions are uncovered, domestic investors may
require a (possibly time-varying) exchange risk premium (ρt ) for investing in foreign assets
that must be converted back to the domestic currency. Therefore, the UIP condition should
be respecified as
sem,t,h − st + ρt = it − i∗t .
(3)
Fama (1984) showed that using the (relatively less stringent) covered interest parity (CIP)
condition
ft,h − st = it − i∗t ,
(4)
together with the UIP condition, allows a decomposition of the forward premium, ρt into an
expected depreciation component and a risk premium component, i.e.,
ft,h − st = sem,t,h − st + ρt .
(5)
where ft,h is the forward rate at time t on an h-period contract.
Equivalently, we can solve for ρt = ft,h − sem,t,h . That is, the risk premium is the expected
excess return (over and above the expected spot rate) from taking a forward position. Using
4
Following other authors (e.g., Frankel and Froot (1987) and Liu and Maddala (1992)), we take natural
logarithms of the exchange rate variables, so as to avoid the Siegel paradox, which arises from Jensen’s
inequality.
5
Perhaps the most cited stylized fact about exchange rates in a floating exchange rate regime is that they
are typically indistinguishable from a random walk with drift, i.e., Et st+1 − st = constant, and therefore
unpredictable. However, a random walk (with drift) in the exchange rate is only consistent with UIP if 1)
rational expectations obtains, i.e., sem,t,1 = Et st+1 , and 2) the nominal exchange rate differential is constant
(not necessarily zero).
4
this definition of the risk premium, and defining the forecast error ηt+h as st+h − sem,t,h , Froot
and Frankel (1989) showed that β ≡ 1 − βρ + βη in the regressions
st+h − st = α + β(ft,h − st ) + εt+h ,
(6)
ft,h − st+h = αρ + βρ (ft,h − st ) + ερ,t+h ,
(7)
st − sem,t,h = αη + βη (ft,h − st ) + εη,t+h .
(8)
Equation (6) is often used to test the unbiasedness of the forward rate as a predictor of the
future spot rate. The interpretation of the beta identity is that any systematic deviation of
the forward rate from the future realized spot rate would produce a β that deviates from one.
This deviation can be explained by a time-varying risk premium (represented by a nonzero
covariance of ρ with the forward premium) in (7) and/or an expectational error (represented
by a nonzero covariance of η with the forward premium) in (8). Thus, tests of rational
expectations are necessary to explain whether the deviation from uncovered interest rate
parity (given covered interest rate parity) is due to the failure of a constant risk premium
and/or the failure of rational expectations.
The ability to decompose deviations from UIP into time-varying risk premium and systematic forecast errors components also has implications for policymakers. First, according
to the portfolio balance model, if a statistically significant time-varying risk premium component is found, this means that it − i∗t is time-varying, which in turn implies that foreign
and domestic bonds are not perfect substitutes; changes in relative quantities (which are
reflected in changes in current account balances) will affect the interest rate differential. In
this way, sterilized official intervention can have significant effects on exchange rates. Second,
if a statistically significant expectational error of the destabilizing (e.g., “bandwagon”) type
is found, and policymakers are more rational than speculators, a policy of “leaning against
5
the wind” could have a stabilizing effect on exchange rate movements. (See Cavaglia et al.
1994.)
More generally, monetary models of the exchange rate (in which the UIP condition is
embedded), which assume model-consistent (i.e., rational) expectations with risk-neutrality,
generally have not performed well empirically, especially in out-of-sample forecasting. (See,
e.g., Bryant 1995.) One would like to be able to attribute the model failure to some combination of a failure of the structural assumptions or a failure of the expectational assumption.
3
Why test rational expectations with survey forecast data?
It is clear that, to test rational expectations or risk neutrality, rather than maintain one or
the other as an untested assumption, one must be able to operationalize sem,t,h , the market’s
expectation of the future spot rate. Since this variable is unobservable, one must seek
reasonable proxies. Beginning with Frankel and Froot (1987) and Froot and Frankel (1989),
much of the literature examining exchange rate rationality in general, and the decomposition
of deviations from UIP in particular, has employed the representative agent assumption to
justify using the mean or median survey forecast as a proxy for the market’s expectation.
In both studies, Frankel and Froot found significant evidence of irrationality. Subsequent
research has found mixed results.
3.1 Why test rational expectations with disaggregated survey forecast data?
Liu and Maddala (1992, p. 366) articulate the mainstream justification for using aggregated forecasts in tests of the “Although ...data on individuals are important to throw light
on how expectations are formed at the individual level, to analyze issues relating to market
efficiency, one has to resort to aggregates.” In fact, Muth’s (1961)[p. 316] original definition
of rational expectations seemed to allow for the possibility that rationality could be applied
to an aggregate (e.g., mean or median) forecast. “. . . [E]xpectations of firms (or, more gen-
6
erally, the subjective probability distribution of outcomes) tend to be distributed, for the
same information set, about the predictions of the theory (or the ’objective’ probability distribution of outcomes.)” However, if individual forecasters have different information sets,
Muth’s definition does not apply. To take the simplest example, the (current) mean forecast
is not in any forecaster’s information set, since all individuals’ forecasts must be made before
a mean can be calculated. Thus, current mean forecasts contain private information (see
MacDonald, 1992) and therefore cannot be tested for rationality. Nevertheless, until the
1990s, few researchers tested for the rationality of individual forecasts, even when that data
was available.
Using the mean forecast also produces inconsistent parameter estimates. Figlewski and
Wachtel (1983) were the first to show that, in an unbiasedness equation, the presence of
private information variables in the mean forecast error sets up a correlation with the mean
forecast. This inconsistency occurs even if all individual forecasts are rational. In addition,
Keane and Runkle (1990) pointed out that, when some forecasters are irrational, using the
mean forecast may lead to false acceptance of the unbiasedness hypothesis, in the unlikely
event that offsetting individual biases allow parameters to be consistently estimated. (See
also Bonham and Cohen (2001), who argue that, in the case of cointegrated targets and
predictions, inconsistency of estimates in rationality tests using the mean forecast can be
avoided if corresponding coefficients in the individual rationality tests pass a test for microhomogeneity.)
A large theoretical literature examines how individual expectations are mapped into an
aggregate market expectation, and whether the latter leads to market efficiency. (See, e.g.,
Figlewski 1978, 1982, 1984; Kirman 1992; Haltiwanger and Waldman 1989.) In Haltiwanger
and Waldman’s (1989) theoretical model, a group of forecasters may exhibit rational expectations collectively, even when the members of the group do not. That is, “standard
[i.e., individual] and aggregate rational expectations typically yield different equilibria and
. . . the size of the difference depends positively on the degree of synergism [i.e., correlation
7
of forecast errors, even when there is no private information]” (Haltiwanger and Waldman,
1989, p. 619).6 For example, one could question whether the equal weights used to construct
the mean forecast were biased estimates of the weights in the true market expectation. If so,
the mean forecast contains measurement error–another source of inconsistency, in addition
to the existence of private information. We discuss sources of measurement error at the
individual level in section 4 below.7 This is a special case of the more general measurement
error issue related to the size and representativeness of the sample.8
4
Rational reasons for the failure of the Rational Expectations Hypothesis using
disaggregated data
Other than a failure to process available information efficiently, there are numerous ex-
planations for a rejection of the REH. One set of reasons relates to measurement error in the
individual forecast. (Recall from above that a separate measurement error problem may occur with aggregated forecasts.) Researchers have long recognized that forecasts of economic
variables collected from public opinion surveys should be less informed than those sampled
from industry participants. However, industry participants, while relatively knowledgeable,
may not be properly motivated to devote the time and resources necessary to elicit their
best responses.9,10 The opposite is also possible. Having devoted substantial resources to
produce a forecast of the price of a widely traded asset, such as foreign exchange, forecasters
6
Our paper focuses on individual rationality but allows for the possibility of synergism by incorporating
heteroscedasticity and autocorrelation consistent standard errors in individual rationality tests and also
cross-forecaster correlation in tests of micro-homogeneity.
7
Frankel and Froot (1987) argue that, if the measurement error is random over all forecasters, then, under
the null hypothesis of rationality, it will cancel in the aggregate.
8
Prat and Uctum (2000) show that, if the exchange rate expectation is lognormally distributed, then
using an arithmetic, rather than geometric, average of individual forecasts introduces another source of
measurement error–this time with a constant bias.
9
For example, Keane and Runkle (1990) claimed that their rationality tests, using respondents from (what
is now) the Survey of Professional Forecasters, would be less contaminated by weak incentives.
10
An implication of strong incentives to produce accurate forecasts is that forecasters should tend to delay
reporting their forecasts until close to the survey’s deadline date, so as to accumulate as much relevant
information as possible.
8
may be reluctant to reveal their true forecast before they have had a chance to trade for
their own account.11 Furthermore, reported individual forecasts may not represent the mean
of the forecaster’s subjective probability distribution if that distribution is skewed and the
forecaster reports another measure of central tendency, e.g., the median.
Second, some forecasters may not have the symmetric quadratic loss function embodied
in typical measures of forecast accuracy, e.g., minimum mean squared error. (See Zellner
1986; Stockman 1987; Batchelor and Peel 1998.) In this case, the optimal forecast may
not be the MSE. In one scenario, related to the incentive aspect of the measurement error
problem, forecasters may have strategic incentives involving product differentiation. For example, consider a forecast industry in which most forecasters tend to follow the herd and
produce relatively homogeneous forecasts. In this situation, it may pay for some forecasters
with sufficient reputational capital to produce extreme forecasts, which enhance the forecasters’ brand-name recognition more than it (subsequently) damages their record for forecast
accuracy. Laster et al. (1999) called this practice “rational bias.” Lamont (2002) found evidence that the tendency of forecasters to produce extreme forecasts increased with age and
experience. Ehrbeck and Waldmann (1996) tested hypotheses in which a less able forecaster
moderates his personal forecast by weighting it with the prediction pattern of more able
forecasters (see also Batchelor and Dua 1990a,b, 1992).
In addition to strategic behavior, another scenario in which forecasters may deviate from
the symmetric quadratic loss function is simply to maximize trading profits. This requires
predicting the direction of change, regardless of MSE.12 (See Elliott and Ito 1999; Boothe and
Glassman 1987; LeBaron 2000; Leitch and Tanner 1991; Lai 1990. Goldberg and Frydman
11
To mitigate the confidentiality problem in this case, the survey typically withholds individual forecasts
until the realization is known, or (as with the JCIF) masks the individual forecast by only reporting some
aggregate forecast (at the industry or total level) to the public.
12
This type of loss function may appear to be relevant only for relatively liquid assets such as foreign
exchange, but not for macroeconomic flows. However, the directional goal is also used in models to predict
business cycle turning points. Also, trends in financial engineering may lead to the creation of derivative
contracts in macroeconomic variables, eg., CPi futures.
9
(1996) and Pilbeam (1995) generated directional forecasts from structural models of the
exchange rate.)
Third, there may be a peso problem. The expectations formation process incorporates
a rational probability of a future shift in the exchange rate regime. It is significant that,
whichever regime is reflected in the realized exchange rate, the forecast error is biased and
serially correlated as long as expectations are a probability-weighted average of exchange
rates in two different regimes.
Fourth, and analogous to the peso problem, is the phenomenon of learning. The expectations formation process incorporates a rational probability of a past shift in the exchange
rate regime. Again, while learning whether the regime has shifted, the forecast error is biased
and serially correlated (See Lewis (1995)). Both the peso problem and rational learning are
small sample problems in the sense that, once sufficient learning has occurred that forecasters
have completely adjusted to one regime or the other (perhaps the new regime), the rational
forecast error will be unbiased.
Fifth, rational speculative bubbles may exist. In contrast to the previous theories for
the failure of the REH, which assumed a single representative agent model of the forecast
generating process, rational bubbles require some diversity in the FGP. Despite the fact that
informed traders know that the current exchange rate is above the equilibrium determined
by fundamentals,13 they recognize the market significance of momentum traders (either uninformed (i.e., “noise”) traders or technical traders (chartists)), who are characterized by
“bandwagon”, rather than regressive or rational, expectations. Fundamentalists then might
believe that the directional momentum of the exchange rate will continue in the short run.
At some point, when the deviation of the exchange rate from its fundamental value reaches
a threshold, regressive expectations will dominate, the bubble will burst, and the exchange
rate will fall rapidly. The difficult issue for traders is to figure out what the threshold exchange rate is that marks the transition from short run to long run expectations. The most
13
The fundamentals are represented by the solution to a particular structural model of the exchange rate.
10
obvious implication of rational bubbles is that the larger swings in the exchange rate will
produce excess volatility, relative to what would be expected without the bubble. However,
it is also true that the asymmetric probability distribution of the bubble component of the
exchange rate (i.e., slow expansion and rapid deflation) will be reflected in the implied rational expectation forecast errors. The skewness in the forecast errors is a violation of the
normality assumption of rational expectations forecast errors (see Sarno and Taylor, 2002,
p. 27).
Peso problems, learning about past regime shifts, and rational bubbles all have in common
an attempt by the forecaster to distinguish between temporary and permanent shifts in the
DGP. Flood and Hodrick (1990) note that, because all three theories are characterized by
skewness in rational forecast errors, the theories may be difficult to distinguish empirically.
Further, these explanations for the failure of the REH assume that deviations from rational
forecasts are temporary. However, Sarno and Taylor (2002) observe that the statistically
significant deviation from uncovered interest rate parity (as measured by a β in equation (6)
that is closer to minus one than plus one) cannot be explained as a temporary phenomenon.
Since nonrejection of the unbiasedness and efficiency tests is necessary for the rationality
of each forecaster, it follows that failure of the micro-homogeneity restrictions in unbiasedness
or efficiency tests, i.e., a minimum amount of diversity, indicates irrationality on the part
of some forecasters. The converse does not necessarily hold, since a failure to reject microhomogeneity could conceivably be due to the same degree of irrationality of each individual
in the panel. In the next section we investigate the empirical evidence for these conjectures.
5
Rationality tests
As discussed above, rational expectations is an important maintained hypothesis in many
representative agent models. If tests of these models produce rejections, it is useful to be
able to determine whether the failure is due to a failure of the structural assumption (e.g.,
uncovered interest parity) alone or a failure of the rational expectations assumption or both.
11
A natural progression of rationality testing ascends from unbiasedness to (weak) efficiency
with respect to the forecast variable’s own history to (strong) efficiency with respect to
other variables. Yet, it is important to note that the result from one type of rationality test
does not have implications for the results from any other types of rationality tests. Here
we test for unbiasedness and weak efficiency, leaving the more stringent tests of publicly
available information for future analysis. We also relate the “conventional” unbiasedness
test (bivariate regression of the realization on a constant and the forecast, with a null that
the slope equals one and the intercept equals zero) to the alternative specification (univariate
regression of the forecast error on a constant, with the null that the constant equals zero).
Finally, we compare tests in levels of the variables to tests in first differences.
5.1 Unbiasedness tests
Expressing the target and forecast in first difference form,14 the simple unbiasedness
equation is specified as
ηi,t,h = st+h − sei,t,h = αi,h + ǫi,t,h ,
(9)
where ηi,t,h is the forecast error of individual i, for an h-period-ahead forecast made at time
t.15 The results are reported in Table 1A. For the one-month horizon, unbiasedness cannot
be rejected at conventional significance levels for any group. For the three-month horizon,
unbiasedness is rejected only for exporters (at a p-value of 0.03). For the six-month horizon,
unbiasedness is rejected at the 5% level only for exporters, though it rejected at the 10% level
for all groups. The direction of the bias for exporters is negative; that is, they systematically
underestimate the value of the yen, relative to the dollar. Ito (1990) found the same tendency
using only the first two years of survey data (1985-1987). He characterized this depreciation
bias as a type of “wishful thinking” on the part of exporters.
14
Over the sample period the realized yen/dollar exchange rate is I(1). Unit root and cointegration test
results are in section 5.4 below.
15
Throughout this paper, we use the Newey-West (1987) heteroscedasticity and autocorrelation consistent
(HAC) variance-covariance matrix to capture the M A(h − 1) lag structure of individual regression residuals.
12
In addition to testing the unbiasedness hypothesis at the individual level, as described
in 3.1, we are interested in the degree of heterogeneity of coefficients across forecasters. The
null of micro-homogeneity is given by H0 : αih = αjh , for all i, j 6= i.
Before testing for homogeneous intercepts in equation (9) we must specify the form for
our GMM system variance-covariance matrix. We allow for cross-forecaster correlation,
including lagged cross-correlations through lag h − 1. Previous authors have modeled crosscorrelations without explicitly testing for their existence. They typically justify an M A(h−1)
VCV structure by appealing to information lags.16 In contrast, we use Pesaran’s (2004) CD
(cross-sectional dependence) test to check for contemporaneous correlations among pairs of
forecasters.
CD =
s
N
N
−1 X
X
2T
ρ̂ij ,
N (N − 1) i=1 j=1+1
(10)
where T is the number of time periods, N = 4 is the number of individual forecasters, and
ρ̂ij is the sample correlation coefficient between forecasters i and j, i 6= j. Under the null
hypothesis of no cross-correlation, CD ∼ N (0, 1).17 See Table 6 for CD test results. All
a
contemporaneous correlations are significant at the 0% level.18
In the four-equation system used to test the micro-homogeneity restrictions, we replace
Zellner’s (1962) SUR variance-covariance matrix with a GMM counterpart that incorporates the Newey-West single equation corrections plus allowances for corresponding crosscovariances, both contemporaneous and lagged. See Appendix 2 for details. Despite having
only two failures of unbiasedness at the 5% level over all three horizons, microhomogeneity is
rejected at a level of virtually zero for all horizons. The rejection of microhomogeneity at the
16
Keane and Runkle (1990) provide some empirical support for their modeling of cross-sectional correlations, noting that the average covariance between a pair of forecasters is 58% of the average forecast
variance.
17
Unlike Breusch and Pagan’s (1980) LM test for cross-sectional dependence, Pesaran’s (2004) CD test is
robust to multiple breaks in slope coefficients and error variances, as long as the unconditional means of the
variables are stationary and the residuals are symmetrically distributed.
18
CD tests for lagged cross-sectional dependencies of specific orders are also possible. We would expect
these tests to be statistically significant for lags of h - 1 or less. In fact, this is the assumption we make in
the system variance-covariance matrix, described next.
13
one-month horizon occurs despite the failure to reject unbiasedness for any of the industry
groups. We hypothesize that the consistent rejection of micro-homogeneity regardless of the
results of individual unbiasedness tests is the result of sufficient variation in individual bias
estimates as well as precision in these estimates.
5.2 Explicit tests of weak efficiency
The literature on rational expectations exhibits even less consensus as to the definition
of efficiency than it does for unbiasedness. In general, an efficient forecast incorporates
all available information—private as well as public. It follows that there should be no
relationship between forecast error and any information variables known to the forecaster
at the time of the forecast. Weak efficiency commonly denotes the orthogonality of the
forecast error with respect to functions of the target and prediction. For example, there is no
contemporaneous relationship between forecast and forecast error which could be exploited
to reduce the error. Strong efficiency denotes orthogonality with respect to the remaining
variables in the information set. Below we perform two types of weak efficiency tests. In the
first type, we regress each group’s forecast error on three sets of weak efficiency variables.
1. Single and cumulative lags of the mean forecast error (lagged one period):
st+h − sei,t,h = αi.h +
h+7
X
βi,t+h−k (st+h−k − sem,t+h−k,h ) + ǫi,t,h
(11)
k=h+1
2. Single and cumulative lags of mean expected depreciation (lagged one period)
st+h − sei,t,h = αi.h +
h+7
X
βi,t+h−k (sem,t+h−k,h − st−k ) + ǫi,t,h
(12)
k=h+1
3. Single and cumulative lags of actual depreciation
st+h − sei,t,h = αi.h +
h+6
X
βi,t+h−k (st+h−k − st−k ) + ǫi,t,h
k=h
14
(13)
Notice that the first two sets of weak efficiency variables include the mean forecast, rather
than the individual group forecast. Our intention is to allow a given group to incorporate
information from other groups’ forecasts via the prior mean forecast. This requires an extra
lag in the information set variables, relative to a contemporaneously available variable such
as the realized exchange rate depreciation.
For each group and forecast horizon, we regress the forecast error on the most recent
seven lags of the information set variable, both singly and cumulatively. We use a Wald test
of the null hypothesis αi,h = βi,t+h−k = 0 and report chi-square test statistics, with degrees
of freedom equal to the number of micro-homogeneity restrictions. If we were to perform
only simple regressions (i.e., on each lag individually), estimates of coefficients and tests of
significance could be biased toward rejection due to the omission of relevant variables. If
we were to perform only multivariate regressions, tests for joint significance could be biased
toward nonrejection due to the inclusion of irrelevant variables. It is also possible that
joint tests are significant but individual tests are not. This will be the case when the linear
combination of (relatively uncorrelated) regressors spans the space of the dependent variable,
but individual regressors do not.
In the only reported efficiency tests on JCIF data, Ito (1990) separately regressed the
forecast error (average, group, and individual firm) on a single lagged forecast error, lagged
forward premium, and lagged actual change. He found that, for the 51 biweekly forecasts
between May 1985 and June 1987, rejections increased from a relative few at the one- or threemonth horizons to virtual unanimity at the six-month horizon. When he added a second
lagged term for actual depreciation, rejections increased “dramatically” for all horizons.
The second type of weak efficiency tests uses the Breusch-Godfrey(1978) LM test for the
null of no serial correlation of order k=h or greater, up to order k=h+6, in the residuals of
15
the forecast error regression, equation (11).19 The LM test proceeds as follows. First, use
OLS to estimate the equation
st+h − sei,t,h = αi,h +
h−1
X
βi,k (st+h−k − sei,t−k,h ) + ǫi,t,h
(14)
k=1
Second, successively regress the residuals ǫ̂i,t,h from equation (14) on all the regressors in
that equation along with k = h, ..., h + 6 lags of ǫ̂i,t,h :
ǫ̂i,t,h = αi,h +
h−1
X
βi,k (st+h−k − sei,t−k,h ) +
k=1
h+6
X
φi,l ǫ̂i,t−l,h + ηi,t,h .
(15)
l=h
Third, successively compute the F-statistic for H0 : φi,h = . . . = φi,h+6 = 0.20 Results are
presented in Tables 3A-5D. For each group, horizon, and variable, there are seven individual
tests, i.e., on a single lag, and six joint tests, i.e., on multiple lags. These 13 tests are
multiplied by four groups times three horizons times three weak efficiency variables for a
total of 468 efficiency tests.
Using approximately nine more years of data than Ito (1990), we find many rejections.
In some cases, nearly all single lag tests are rejected, yet few if any joint tests are rejected.
(See, e.g., expected depreciation at the three- and six-month horizons.) In other cases, nearly
all joint tests are rejected, but few individual tests. (See, e..g, actual depreciation at the
three-month horizon and expected depreciation at the six-month horizon.) Remarkably, all
but one LM test for serial correlation at a specified lag produces a rejection at less than
a 10% level, with most at less than a 5% level. Thus, it appears that the generality of
the alternative hypothesis in the LM test permits it to reject at a much greater rate than
the conventional weak efficiency tests, in which the variance-covariance matrix incorporates
the Newey-West correction for heteroscedasticity and serial correlation. Finally, unlike Ito
(1990), we find no strong pattern between horizon length and number of rejections.
19
This is a general test, not only because it allows for an alternative hypothesis of higher-order serial
correlation of specified order, but also because it allows for serial correlation to be generated by AR, MA or
ARMA processes.
20
We use the F-statistic because the χ2 test statistics tend to over-reject, while the F-tests have more
appropriate significance levels (see Kiviet (1987)).
16
In addition to testing the weak efficiency hypothesis at the individual level, we are interested in the degree of heterogeneity of coefficients across forecasters. Here the null of
micro-homogeneity is given by H0 : φil = φjl , for l = h, . . . h + 6, for all i, j 6= i. There
are 468 tests/4 groups = 117 micro-homogeneity tests. The null hypothesis of equal coefficients is H0 : αi,h = αj,h , βi,t+h−k = βj,t+h−k for all i, j 6= i As with the micro-homogeneity
tests for unbiasedness, our GMM variance-covariance matrix accounts for serial correlation
of order h-1 or less, generalized heteroscedasticity, and cross-sectional correlation or order
h-1 or less. We report χ2 (n) statistics, where n is the number of coefficient restrictions, with
corresponding p-values. Rather than perform all 117 micro-homogeneity tests, we choose a
sample consisting of the shortest and longest lag for which there are corresponding individual and joint tests (i.e., for the k=h+1st and k=h+6th lag). Thus, there are 4 tests (two
individual and two corresponding joint tests) times three horizons times three variables for
a total of 36 tests. Every one of the micro-homogeneity tests are rejected at the 0% level.
5.3 Joint tests of unbiasedness and weak efficiency
Many, perhaps most, empirical tests of the “unbiasedness” of survey forecasts are conducted using the bivariate regression equation21
st+h − st = αi,h + βi,h (sei,t,h − st ) + εi,t,h .
(16)
It is typical for researchers to interpret their non-rejection of the joint null (αi,h , βi,h ) = (0, 1)
as evidence of simple unbiasedness. However, Holden and Peel (1990) show that this result
is a sufficient, though not a necessary, condition for unbiasedness. The intuition for the
lack of necessity comes from interpreting the right-hand-side of the bivariate unbiasedness
regression as a linear combination of two potentially unbiased forecasts: a constant equal
to the unconditional mean forecast plus a variable forecast, i.e., st+h − st = (1 − βi,h ) ×
E(sei,t,h − st ) + βi,h (sei,t,h − st ) + εi,t,h . Then the intercept is αi,h = (1 − βi,h ) × E(sei,t,h − st ).
21
Our variance-covariance matrices for both the individual equations and the 4-equation system parallel
those of the unbiasedness test in the previous subsection. Again, see Appendix 2 for details.
17
The necessary and sufficient condition for unbiasedness is that the unconditional mean of the
subjective expectation E[sei,t,h −st ] equal the unconditional mean for the objective expectation
E[st+h − st ]. However, this equality can be satisfied without αi,h being equal to zero, i.e.,
βi,h = 1.
Figure 1 shows that an infinite number of αi,h , βi,h estimates are consistent with simple
unbiasedness. The only constraint is that the regression line intersect the 45 degree ray
from the origin where the sample mean of the forecast and target are equal. Note that, in
the case of differenced variables, this can occur at the origin, so that αi,h = 0, but βi,h is
unrestricted (see Figure 2). It is easy to see why simple unbiasedness holds: in Figures 1
and 2 the sum of all horizontal deviations from the 45 degree line to the regression line, i.e.,
forecast errors, equal zero.22 To investigate the rationality implications of different values
for αi,h and βi,h , we follow Clements and Hendry (1998) and rewrite the forecast error in the
bivariate regression framework of (16) as
ηi,t,h = st+h − sei,t,h = αi,h + (βi,h − 1)(sei,t,h − st ) + εi,t,h
(17)
A special case of weak efficiency occurs when the forecast and forecast error are uncorrelated, i.e.,
E[ηi,t,h (sei,t,h − st )] = 0
(18)
= αi,h E(sei,t,h − st ) + (βi,h − 1)E(sei,t,h − st )2 + E[εi,t,h (sei,t,h − st )]
Thus, satisfaction of the joint hypothesis (αi,h , βi,h ) = (0,1) is also sufficient for weak efficiency with respect to the current forecast. However, it should be noted that (18) may
still hold even if the joint hypothesis is rejected. Thus, satisfaction of the joint hypothesis
represents sufficient conditions for both simple unbiasedness and this type of weak efficiency,
However, when αi,h 6= 0, and αi,h 6= (1 − βi,h ) × E(sei,t,h − st ), there is bias regardless of the value of βi,h .
See Figure 3, where the bias, E(st+h − sei,t,h ), is measured as the horizontal distance from the point on the
regression line where E(sei,t,h − st ) = E(st+h − st ) to the 45 degree line, implying systematic underforecasts.
22
18
but necessary conditions for neither.23 Finally, satisfaction of the joint hypothesis is neither
necessary nor sufficient for lack of serially correlated forecast errors—one requirement for
weak efficiency. To see this, note that
E[ηi,t,h ηi,t−1,h ]
n
o
= E [αi,h + (βi,h − 1)(sei,t,h − st ) + εi,t,h ][αi,h + (βi,h − 1)(sei,t−1,h − st−1 ) + εi,t−1,h ]
= αi,h 2 + αi,h (βi,h − 1)E(sei,t−1,h − st−1 ) + αi,h (βi,h − 1)E(sei,t,h − st ) +
(19)
(βi,h − 1)2 E[(sei,t,h − st )(sei,t−1,h − st−1 )] + E[εi,t,h εi,t−1,h ]
This expression may equal zero even if αi,h 6= 0 and βi,h 6= 1. On the other hand, if αi,h = 0,
and βi,h = 1, then serially correlated residuals imply serially correlated forecast errors.24
Recall that using the univariate regression to test the simple unbiasedness hypothesis, we
found only two rejections of unbiasedness at the 5% level (for exporters at the three- and sixmonth horizons). At the same time, we found strong rejections of microhomogeneity for all
forecasts at all horizons. Turning now to tests of the joint hypothesis (αi,h , βi,h ) = (0,1) in
Table 2, we find very different results. The joint hypothesis is rejected at the 5% significance
level for all groups at the one-month horizon (indicating the possible role of inefficiency with
respect to the current forecast), but only for the exporters at the three- and six- month
horizons. This stands in contrast to the simple unbiasedness results, where performance
deteriorated with increases in the horizon. Also in contrast to the simple unbiasedness
results, there are no rejections of microhomogeneity. Clearly, the results depend on which
definition of unbiasedness-nonrejection of the simple or joint hypothesis–one adopts!
23
If βi,h = 1, then, whether or not αi,h = 0, the variance of the forecast error equals the variance of
the bivariate regression residual, since then var(ηi,t,h ) = (βi,h − 1)2 var(sei,t,h − st ) + var(εi,t,h ) + 2(βi,h −
1)cov[(sei,t,h − st ), εi,t,h ] = var(εi,t,h ). Figure 3 illustrates this point. Mincer and Zarnowitz (1969) required
only that βi,h = 1 in their definition of forecast efficiency. If in addition to βi,h = 1, αi,h = 0, then the mean
square forecast error also equals the variance of the forecast. Mincer and Zarnowitz emphasized that, as long
as the loss function is symmetric, as is the case with a minimum mean square error criterion, satisfaction of
the joint hypothesis implies optimality of forecasts.
24
In Figure 3, where αi,h 6= 0, but βi,h = 1, serial correlation in the forecast error will occur unless
2
αi,h = −E[εi,t,h εi,t−1,h ].
19
The significance of the αi,h ’s in the joint regressions (16) generally follows the same
pattern as in the unbiasedness tests (9), except with slightly more evidence against the null
that this parameter equals zero. However, the null that βi,h = 1 is rejected for all groups
at the one-month horizon, but only for the exporters at the three- and six-month horizons.
This implies that weak efficiency with respect to the current forecast fails at the one-month
horizon, but not at the longer horizons, with only two exceptions. Thus, it appears that
tests of the joint hypothesis at the one-month horizon are rejected due to failure of this type
of weak efficiency, not simple unbiasedness.25 Tests of the joint hypothesis fare better at the
longer horizons primarily because this type of weak efficiency is rarely rejected, despite the
uniform deterioration in the simple unbiasedness statistics in Table 1. This longer-horizon
result is unusual, because weak efficiency is usually taken to be a more demanding criterion
than unbiasedness. However, the longer-horizon result may not be as conclusive as the βi,h
statistics suggest. For all tests at all horizons, the null hypothesis that βi,h equals zero
also cannot be rejected. Thus, for the longer two horizons (with just the one exception for
exporters at the three-month horizon), hypothesis testing cannot distinguish between the null
hypotheses that βi,h equals one or zero. Therefore, we cannot conclude that weak efficiency
with resepect to the forecast prevails when unbiasedness does not. The failure to precisely
estimate the slope coefficient also produces R2 s that are below 0.05 in all regressions. This
lack of power is at least consistent with the failure to reject microhomogeneity at all three
horizons. The conclusion is that testing only the joint hypothesis has the potential to obscure
the difference in performance between the unbiasedness and weak efficiency tests. This
conclusion is reinforced by an examination of figures 4 - 6, the scatter plots and regression
lines for the bivariate regressions.26 All three scatter plots have a strong vertical orientation.
25
For this reason, Mincer and Zarnowitz (1969) and Holden and Peel (1990) suggest that, if one begins by
testing the joint hypothesis, rejections in this first stage should be followed by tests of the simple unbiasedness
hypothesis in a second stage. Only if unbiasedness is rejected in this second stage should one conclude that
forecasts are biased. For reasons described immediately following, our treatment reverses the order, so that
unbiasedness and weak efficiency can be separately assessed for each forecast.
26
Note that, for illustrative purposes only, we compute the expectational variable as the four-group average
percentage change in the forecast. However, recall that, despite the failure to reject micro-homogeneity at
20
With this type of data, it is easy to find the vertical midpoint and test whether it is different
from zero. Thus, (one-parameter) tests of simple unbiasedness are feasible. However, it is
difficult to fit a precisely estimated regression line to this scatter, because the small variation
in the forecast variable inflates the standard error of the slope coefficient. This explains why
the βi,h ’s are so imprecisely estimated that the null hypotheses that βi,h = 1 and 0 are
simultaneously not rejected. Because of the correlation between the slope and intercept,
the intercepts also have relatively high standard errors. For all regressions except exporters
at the three- and six-month horizons, the standard error of the intercept is larger in the
bivariate regression than in the simple unbiasedness regression. This also explains why the
R2 s are so low. Thus, examination of the scatter plots also reveal why bivariate regressions
are potentially misleading about weak efficiency as well as simple unbiasedness. That is
why, in the previous subsection, we tested weak efficiency directly, using the past history
of exchange rate changes, forecast changes, and forecast errors.27 Thus, in contrast to both
Mincer and Zarnowitz (1969) and Holden and Peel (1990), we prefer to separate tests for
simple unbiasedness from tests for (all types of) weak efficiency at the initial stage. This
obviates the need for a joint test.28
More fundamentally, the relatively vertical scatter of the regression observations around
the origin is consistent with an approximately unbiased forecast of a random walk in exchange
rate levels. Other researchers (e.g., Bryant (1995) have found similar vertical scatters for
regressions where the independent variable, e.g., the forward premium/discount ft,h − st ,
the “exchange risk premium” ft,h − st+h , or the difference between domestic and foreign
any horizon, the Figlewski-Wachtel critique implies that these parameter estimates are inconsistent in the
presence of private information. (See the last paragraph in this subsection.)
27
As Nordhaus (1987, p. 673) puts it, “A baboon could generate a series of weakly efficient forecasts by
simply wiring himself to a random-number generator, but such a series of forecasts would be completely
useless.” For these unbiasedness regressions, however, forecast efficiency, even weak form, appears difficult
to support.
28
However, in the general case of biased and/or inefficient forecasts, Mincer and Zarnowitz (1969, p. 11)
also viewed the bivariate regression “as a method of correcting the forecasts . . . to improve [their] accuracy
. . . Theil (1966, p.33) called it the ‘optimal linear correction.”’ That is, the correction would involve 1)
subtracting αi,h , then 2) multiplying by 1/βi,h . Graphically, this is a translation of the regression line
followed by a rotation, until the regression line coincides with the 45 degree line.
21
interest rates (i − i∗ ), exhibits little variation. As Bryant (1995)[p. 51] lamented in reporting
corresponding regressions using a shorter sample from the JCIF, “the regression... is...not one
to send home proudly to grandmother.” Bryant (1995) drew the conclusion that “analysts
should have little confidence in a model specification [e.g., uncovered interest parity] setting
[the average forecast] exactly equal to the next-period value of the model...model-consistent
expectations...presume a type of forward-looking behavior [e.g., weak efficiency] that is not
consistent with survey data on expectations” (p. 40).
Finally, even if the bivariate regressions were correctly interpreted as joint tests of unbiasedness and weak efficiency with respect to the current forecast, and even if the regressions
had sufficient power to reject a false null, the micro-homogeneity tests would be subject
to additional econometric problems. Successfully passing a pre-test for micro-homogeneity
does not ensure that estimated coefficients from either aggregated or pooled bivariate unbiasedness regressions will be consistent. Figlewski and Wachtel (1983) showed that, even
if all forecasters produced unbiased forecasts, the existence of private information sets up a
nonzero correlation between the consensus forecast and the consensus forecast error, leading to inconsistent coefficient estimates. Their argument also holds if all forecasters are
“equally” irrational.29 This problem with consensus tests of the REH led a number of authors to argue for the pooling of disaggregated data.30 However, Zarnowitz (1985) showed
that, because each forecaster shares the same dependent variable in a bivariate regression,
there is cross-sectional dependence between the forecast and the forecast error, which again
29
The extent to which private information influences forecasts is more controversial in the foreign exchange
market than in the equity or bond markets. On the one hand, Chionis and MacDonald (1997, p. 211) claim
that “in the foreign exchange market there is little or no inside information that would account for the
heterogeneity of expectations.” On the other hand, they do cite the early exchange rate microstructure work
of Lyons. According to the latter author (Lyons, 2002, p. 61), “That order flow explains such a large
percentage of price moves underscores the inadequacy of this public info framework. The information the
FX market is aggregating is much subtler than the textbook models assume.” To the extent that one agrees
with the market microstructure emphasis on the importance of the private information embodied in dealer
order flow, the Figlewski-Wachtel critique remains valid.
30
These authors preferred pooling individual data for reasons of increased degrees of freedom and possibly
enhanced interpretability of estimated coefficients. This rationale is especially compelling for surveys in
which individuals’ forecasts frequently suffer from short time spans and typically contain large numbers of
missing observations. (See, e.g., Keane and Runkle (1990).)
22
leads to inconsistent parameter estimates. Thus, failure to reject micro-homogeneity does
not permit a test of unbiasedness using consensus or pooled data. On the other hand,
Bonham and Cohen (2001) showed that, if the realization and forecast were integrated and
cointegrated, both the Figlewski-Wachtel and Zarnowitz critiques break down asymptotically. In this case, micro-homogeneity could serve as a useful pretest for both aggregation
and pooling. Thus, we repeat the joint tests in the bivariate regression using the levels of
the forecast and realization.
5.4 Unbiasedness Tests on Levels
We undertake unbiasedness tests on levels for three reasons. Assume that the realization
and forcasts are integrated and cointegrated. First, as mentioned above, the cointegrating
(bivariate) regressions avoid both aggregation bias (due to Figlewski and Wachtel (1983)) and
pooling bias (due to Zarnowitz (1985)) present in bivariate regressions of stationary variables
(see Bonham and Cohen (2001)). Second, the differenced version of the joint unbiasedness
regression is misspecified. The Engel-Granger (1987) representation theorem proves that a
cointegrating regression such as the levels joint regression (16) has an error correction form
that includes both differenced variables and an error correction term in levels. Under the joint
null, the error correction term is the forecast error, and the differenced specification omits
this levels information. (See also Hakkio and Rush (1989).) Third, and most importantly, we
can investigate the possibility that the imprecise coefficient estimates in the longer horizon
bivariate regressions (16) are an artifact of a misspecified bivariate regression in differences.
We test for unbiasedness and weak efficiency in levels using Liu and Maddala’s (1992)
method of “restricted cointegration.” This specification imposes the joint restriction αi,h = 0,
βi,h = 1 on the bivariate regression
st+h = αi,h + βi,h sei,t,h + εi,t,h
(20)
and tests whether the residual (the forecast error) is non-stationary. In a bivariate regression,
any cointegrating vector is unique. Therefore, if we find that the forecast errors are station23
ary, then the joint restriction is not rejected, and (0,1) must be the unique cointegrating
vector.31
To test the null hypothesis of a unit root, we estimate the augmented Dickey-Fuller(1979)
(ADF) regression
∆yt+1 = α + βyt + γt +
p
X
θt δyt+1−p + ǫt+1
(21)
t=1
where y is the level and first difference of the spot exchange rate, the level and first difference
of each group forecast, including the mean forecast, and the forecast error (i.e., the residual
from the “restricted” cointegrating equation). The number of lagged differences to include
in (21) is chosen by adding lags until a Lagrange Multiplier test fails to reject the null
hypothesis of no serial correlation (up to lag 12). We test the null hypothesis of a unit root
(i.e., β = 0) with the ADF t and z tests. We also test the joint null hypothesis of a unit
root and no linear trend (i.e.,β = 0 and γ = 0).
A stationary forecast error is a necessary condition for forecast rationality. We interpret
unbiasedness plus lack of serial correlation in the residuals of the forecast error regression
(determined by LM efficiency tests discussed in section 5.2) as the necessary condition for
forecast rationality.
At the one-month horizon, the null of a unit root in the residual of the restricted cointegrating regression (i.e., the forecast error) is rejected at the 1% level for all groups. (See
Table 6.) Hence, inference in the simple unbiasedness tests in Table 1A is valid; we do
not reject unbiasedness for any group. However, for every group, all seven LM tests (for
cumulated lags two through eight) show evidence of serial correlation at less than a 1% level.
Thus, we reject weak efficiency. This is roughly consistent with the differenced bivariate
unbiasedness test results in Table 2A, where we rejected the joint hypothesis of α = 0, β = 1
31
Note that the Holden and Peel (1990) critique does not apply in the I(1) case; because the interecept
cannot be an unbiased forecast of a nonstationary variable. Thus, the cointegrating regression line of the
level realization on the level forecast must have both α = 0 and β = 1 for unbiasedness to hold. This differs
from Figure 2, the scatterplot in differences, where αi,h = 0 but βi,h 6= 1. Intuitively, the reason for the
difference in results is that the scatterplot in levels must lie in the first quadrant, i.e., no negative values of
the forecast or realization.
24
(which is a joint test of unbiasedness and efficiency with respect to the current forecast) at
less than the 10% level for all groups.
We find nearly identical results at the three-month horizon: the forecast error is stationary
for all groups, but all seven LM tests again find evidence of serial correlation for all groups,
this time at less than the 7% level.
In contrast, at the six-month horizon, the evidence is clearly in favor of a unit root in
the forecast error, for all four groups.32 However, the tests for the unrestricted cointegrating
residual reject the null of a unit root for all groups. Thus, we reject the null that the
forecast and realization cointegrate with a slope of unity, and we also reject the null of
simple unbiasedness because a forecast error with a unit root cannot be mean zero. Given
our finding of a unit root in the forecast errors, the simple unbiasedness tests in Table 1C
are invalid. Finally, the differenced bivariate regressions in Table 2C are misspecified as they
omit the general equilibrium error (ε̂i,t,h = st − α̂i − β̂i sei,t,h ). Interestingly, these regressions
provide somewhat less evidence against the joint hypothesis of unbiasedness and lack of
serially correlated forecasts than do the unit root and LM tests. The hypothesis is rejected
at the 5% level only for exporters, at the 10% level for exporters and life insurance & import
companies. However, as discussed above, this may be due to the misspecification.
Overall, it is reassuring that the restricted cointegration tests are generally consistent
with the simple unbiasedness tests on the differenced variables. The only exception is for the
six-month forecasts, in which regression of the nonstationary forecast error on a constant is
invalid.33
There is also a cautionary note about the joint tests in differences. They confound
the separation of tests of simple unbiasedness (which by themselves are invalidated by the
32
Banks and brokers, as well as life insurance and import companies, failed to reject a unit root at the
10% level in two out of three of the unit root tests. Exporters failed to reject in all three tests.
33
So are the efficiency regressions of the forecast error on lags of the three differenced information set
variables.
25
presence of a unit root in the forecast error) with tests of weak efficiency. This is not true
of bivariate regressions in levels.
5.5 Ito’s heterogeneity tests
In Table 3, we replicate Ito’s (1990) and Elliott and Ito’s (1999) test for forecaster group
“heterogeneity,” which regresses the deviation of the individual forecast from the crosssectional average forecast on a constant.34,35
sei,t,h − sem,t,h = (αi,h − αm ) + (εi,t,h − εm,t )
(22)
To achieve consistent inference in the presence of both heteroscedasticity and serial correlation in the regression residuals, we employ a general Newey-West variance covariance matrix
with the lag window a function of the sample size.
One may view Ito’s “heterogeneity” tests as complementary to our micro-homogeneity
tests. On the one hand, one is not certain whether a single (or pair of?) individual rejection(s) of, say, the null hypothesis of a zero mean deviation in Ito’s test would result in a
rejection of micro-homogeneity overall. On the other hand, a rejection of micro-homogeneity
does not tell us which groups are the most significant violators of the null hypothesis. It
turns out that Ito’s mean deviation test produces rejections at a level of 6% or less for all
groups at all horizons except for banks and brokers at the one-month horizon and life insurance and import companies at the six-month horizon.36 Thus, it is not surprising that
micro-homogeneity tests on the four-equation system of Ito equations produce rejections at
34
Algebraically, Ito’s regression can be derived from the individual forecast error regression by substracting
the the mean forecast error regression. Recall that our group results are not entirely comparable to Ito’s
(1990), since our dataset, unlike his, combines insurance companies and trading companies into one group,
and life insurance companies and import-oriented companies into another group.
35
Chionis and MacDonald (1997) performed an Ito-type test on individual expectations data from Consensus Forecasts of London.
36
Elliott and Ito (1999), who have access to forecasts for the 42 individual firms in the survey, find that,
for virtually the same sample period as ours, the null hypothesis of a zero deviation from the mean forecast
is rejected at the 5% level by 17 firms at the one-month horizon, 13 firms for the three-month horizon, and
12 firms for the six-month horizon. These authors do not report results by industry group.
26
a level of virtually zero for all three horizons. Since Ito’s regressions have a similar form
(though not a similar economic interpretation) to the tests for simple unbiasedness in Table
1, it is not surprising that they have similar results. The intercept is estimated with sufficient
precision to allow rejection of the null hypothesis.
6
Conclusions
In this paper, we undertake a reexamination of the rationality and diversity of JCIF
forecasts of the yen-dollar exchange rate. In several ways we update and extend the seminal
paper by Ito (1990). Whereas Ito (1990) and Elliott and Ito (1999) measured diversity
as a statistically significant deviation of an individual’s forecast from the cross-sectional
average forecast, we perform a separate test of micro-homogeneity for each type of rationality
test–unbiasedness as well as weak efficiency–that we first conducted at the industry level.
Demonstrating that individual forecasters differ systematically in their forecasts (and FGPs)
has implications for the market microstructure research program. As Frankel and Froot
(1990, p. 182) noted, “the tremendous volume of foreign exchange trading is another piece
of evidence that reinforces the idea of heterogeneous expectations, since it takes differences
among market participants to explain why they trade.” As pointed out by Bryant (1995),
a finding of micro-heterogeneity in unbiasedness and weak efficiency tests also casts doubt
on the assumption of a rational representative agent commonly used in macroeconomic and
asset pricing models.
In order to conduct the systems estimation and testing required for the micro-homogeneity
test, our GMM estimation and inference makes use of an innovative variance-covariance matrix that extends the Keane and Runkle (1990) counterpart from a pooled to an SUR-type
structure. Our variance-covariance matrix takes into account not only serial correlation
and heteroscedasticity at the individual level (via a Newey-West correction), but also forecaster cross-correlation up to h-1 lags. We document the statistical significance of the crosssectional correlation using Pesaran’s (2004) CD test.
27
We find that, for each group, the ability to produce unbiased forecasts deteriorates with
horizon length: no group rejects simple unbiasedness at the one-month horizon, but all groups
reject at the six-month horizon (at less than a 10% level). Exporters consistently perform
worse than the other industry groups, with a tendency toward depreciation bias. Using
only two years of data, Ito (1990) found the same result for exporters, which he described
a a type of “wishful thinking.” We also find that, irrespective of the ability to produce
unbiased forecasts at a given horizon, micro-homogeneity is rejected at virtually a 0% level
for all horizons. We find this result to be somewhat counterintuitive, in light of our prior
belief that micro-homogeneity would be more likely to obtain if there were no rejections of
unbiasedness. Evidently, there is sufficient variation in the estimated bias coefficient across
groups and/or high precision of these estimates to make the micro-homogeneity test quite
sensitive. The unbiasedness results are almost entirely reversed when we test the hypothesis
using the conventional bivariate specification. That is, the joint hypothesis of zero intercept
and unit slope is rejected for all groups at the one-month horizon, but only for exporters and
the three- and six-month horizons. Thus, in stark contrast to the univariate unbiasedness
tests, as well as Ito’s (1990) bivariate tests, forecast performance does not deteriorate with
increases in the horizon: there are no rejections at any horizon.
We hypothesize that at least two factors explain the contrasting results of the univariate
versus bivariate unbiasedness specifications. First, Holden and Peel (1990) have emphasized
that failure to reject the joint test in the bivariate regression is a sufficient, but not necessary,
condition for unbiasedness. This is because the joint hypothesis simultaneously tests weak
efficiency with respect to the current forecast forecast errors as well as unbiasedness. Failure
to reject the univariate test is, however, necessary and sufficient for unbiasedness. Thus,
we conjecture that, in the one-month case, the joint hypothesis fails due to the nonzero
correlation between forecast and forecast error.
This conjecture is borne out by two types of explicit tests for weak efficiency. In the first
type, we regress the forecast error on single and cumulative lags of mean forecast error, mean
28
forecasted depreciation, and actual depreciation. We find many rejections of unbiasedness.
In the second type, we use the Godfrey (1978) LM test for serial correlation of order h
through h+6 in the residuals of the forecast error regression. Remarkably, all but one LM
test at a specified lag length produces a rejection at less than a 10% level, with most at less
than a 5% level.
What, then, explains the failure to reject the bivariate unbiasedness hypothesis at the
longer horizon, where forecasting performance would be expected to deteriorate, as it did
in the univariate case? The key here, we conjecture, is the lack of power of the bivariate
test. Not only can we not reject the (0,1) joint hypothesis, we also cannot reject a (0,0)
hypothesis. Visually, the scatterplot for all three horizons resembles a vertically aligned
ellipse. This is what we would expect in a relatively efficient market: the forecasted change
is nearly unrelated to (and varies much less than) the actual change. At the longer horizons,
the difficulty of forecasting exchange rate changes may more than offset the failure of weak
efficiency with respect to the current forecast.
Finally, we conduct unbiasedness tests on levels to investigate the possibility that the imprecise coefficient estimates in the three- and six- month bivariate regressions are an artifact
of differencing. We use the Liu and Maddala (1992) method of “restricted cointegration,”
i.e., setting the coefficients in the cointegrating vector equal to their hypothesized values of
zero and one, then testing the restricted specification for a unit root in the residuals. We find
that the null of a unit root is rejected for all groups at the one- and three- month horizons,
i.e., the forecast error is stationary. Thus, the inference on simple unbiasedness in Tables
1A and 1B is consistent. However, for all industry groups there is much more evidence for
a unit root in the residuals at the six-month horizon. Thus, the forecast error appears to
be nonstationary. This result by itself rejects unbiasedness at the six-month horizon, and
implies that the inference on unbiasedness in Table 1C is invalid. Interestingly, all tests for
the null of a unit root in the residuals of the unrestricted cointegration regressions fail to
29
reject. This result implies that the bivariate regression in differnces is misspecified, since it
omits an error correction term.
In conclusion, we have attempted to explore the nature of rationality tests on integrated
variables. We compared the results from two common unbiasedness specifications using
differenced variables–univariate and bivariate regressions. We showed that the bivariate
specification not only tests a type of weak efficiency along with unbiasedness, but also has
less power to reject the null than the univariate specification. The latter result may be
related to the unpredictability of price changes in an efficient market.
We also compared unbiasedness tests in differences and levels. All (unrestricted) cointegration tests rejected the null of a unit root in the regression residuals. This points to
another problem with the bivariate regression–misspecification. For all these reasons, we
recommend that simple unbiasedness and weak efficiency tests be undertaken using separate
specifications at the outset (rather than only if the joint test is rejected, as was suggested by
Mincer and Zarnowitz (1969) and Holden and Peel (1990).37 Before conducting such rationality tests, one should test the restricted cointegrated regression residuals, i.e., the forecast
error, for stationarity. Results of such tests presented here cast doubt on the validity of all of
the joint tests, perhaps obviating the need to reconcile the short and long-horizon joint test
results. Clearly, integration accounting and regression specification matter for rationality
testing.
37
Of course, should one wish to undertake a joint test in the bivariate regression, the cointegrating bivariate
regression in levels is an alternative to the misspecified differences regression. A variety of methods, such as
those due to Saikkonen (1991) or Phillips and Hansen (1990), exist that allow for inference in cointegrated
bivariate regressions.
30
st+h − st
b bbbbbbbb
b
b
b
b bbbbbbbbbbbbbbbb bbb
b
b bbb b
b
b
bbb bbbb bbbbbbbbbbbb
bb bbb
b bbbbbbbbbbbbbb
b b bb bb b b bbb
E(st+h − st )
bbbbbbb bbb bbbb bbbbbb
b
b
b
bb bbbbbbb
b b bbbb bbbbb bb
b b b bbbbbbbbbbbbbbbb bb
b b bbb
bbbb bbbbbbbbbbbbbb
b
b
b
b
bbb bb bbb
bbbbbbb b
b bbbbbb
b bb bbbbbbbb bbbb
E(sei,t,h − st )
sei,t,h − st
45o
Figure 1: Unbiasedness with α 6= 0, β 6= 1
st+h − st
0
E(st+h
45o
b bbb
b bbbbbbb
bb
b
b bbbbbbbbbbb
b
b
b bbbb b b
bbbbbbbbbbbbb b
bbb
bbbbbbbbbbbb b
b
b
bbbbbbbbbb
bbbbbbbbbbbbb
bbb bbbb
b b
bbbbbbbbbbb bbb
b
b
b
b
b bb b
− st )
bbbbbbbbbbbbb
b
b
bb bbbbb
bbbb
bbbbbbb
bbb bbbbbbbbb
b bbbbb
bb
b bbbbbbb b
bb
bb
bbb bbbbb
bbbbbbbbb
b
bb
bbb
bbbbbbbbbbb
bb
b
b
b
bbbbb
E(sei,t,h − st )
bbb bbbbbb
0
sei,t,h − st
Figure 2: Unbiasedness with α = 0, β 6= 1
31
st+h − st
E(st+h
b b b bb b
b bbbbbb
bb b
b
b b bb bbbbbb b
b
b
b
b
b
b bbb bb
b b bbb bbbbb bb b
b b bb b
b b b bbb bbbbbb b
b b
b b bbb bbb bbbbb
bbbbb bbbbb bb
b bb b bb b
bb b b
b b bb bbbb b bb
b bbb bbb bbbb b
− st )
b bbbbbbbbb bb b b bb bbbbbbb b bias
b bbbbbbbb bb
b
b bb
bbb bbbbbb bb
b
b
b bbb
b b bbb bb
b bb b bbbb b
b
b b bbbbbb
b bb bbbb b
bbb
bb
E(sei,t,h − st )
45o
sei,t,h − st
Figure 3: Bias with α 6= 0, β = 1
st+h − st
0.1
b
b
b bb
b
bbb bb
bb b b
b bb bbbbbbbbb
bb b b
b bbbbbbbbbbbbbbbbbbbbbb
b
bbbb bbbbbbbbbbbbbbbbbbbbbb
bbbbbbbbbbbbbbb b
bb bbbbbbb b
b b
b bbbbbbbbbbbbbbb b
bbb bb
bbbbb b
bb b bb
b bb b bbb bb
b b bb
bb
bb b
0.05
0
-0.05
-0.1
-0.15
-0.15
-0.1
-0.05
0
0.05
0.1
sei,t,h − st
Figure 4: Actual vs Expected Depreciation, 1 Month-Ahead Forecast
32
st+h − st
0.2
bb
bb
0.15
b
b
bb
bbb b b
b
b bbbbbb bbb b
b bbbbbbbbbbbbbbb
b bb b b
bbb bb bbbbb b
bbb bbbbbbbbbbbbbbbb
b
b bbbbbbbbbbbbbbbbbbbb
b b bb bbbbbbbbbbb
bb
bbb b bb bb
bbbbbbbb bbbb bbb b
b bbbb bbb
bb b bb
b bb b
b bb
b bb
b
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 e0.15 0.2
si,t,h − st
Figure 5: Actual vs Expected Depreciation, 3 Month-Ahead Forecast
st+h − st
0.3
0.2
b b bbb
bb
b
b bbb b
bb b bb b
b
b b bbbbbbb b
b bbb
b bbb bbb
b bbbbb b
bbbb bbbbb bbbbbbbb b b
b bbbbbbbbbbbbb
bb bbb bbbbbbbbbbb
b b
b bbbbbbbbbbbbbbbbb
b b b bb
b bbbbbbbbbb
b
bb b b bbb
bbbb bbbb b
b bb
bb
0.1
0
-0.1
-0.2
-0.3
-0.3
-0.2
-0.1
0
0.1
0.2 0.3
sei,t,h − st
Figure 6: Actual vs Expected Depreciation: 6 Month-Ahead Forecast
33
0.1
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
sei,t,h − st
st+h − st
-0.1
86/05/14
91/01/16
96/10/15
Figure 7: Actual & Expected Depreciation, 1 Month-Ahead Forecast
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
sei,t,h − st
st+h − st
-0.2
86/05/14
91/01/16
96/07/30
Figure 8: Actual & Expected Depreciation, 3 Month-Ahead Forecast
34
0.25
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
sei,t,h − st
st+h − st
-0.25
86/05/14
90/11/27
96/05/15
Figure 9: Actual & Expected Depreciation, 6 Month-Ahead Forecast
35
Table 1A: Forecast Error (FE) Tests (1 month forecasts)
Individual regresions
st+h − sei,t,h = αi,h + εi,t,h for h = 2
(9)
degrees of freedom = 261
i=1
i=2
i=3
i=4
(banks & (insurance &
(export
(life ins. &
brokers) trading cos.) industries) import cos.)
αi,h
0.000
0.001
-0.002
0.002
t (NW)
0.115
0.524
-0.809
0.720
MSL(NW)
0.909
0.600
0.418
0.472
MH tests H0 : αi,h = αj , for all i, j 6= i
χ2 (GMM)
41.643
MSL(GMM)
0.000
Table 1B: Forecast Error (FE) Tests (3 month forecasts)
Individual regresions
st+h − sei,t,h = αi,h + εi,t,h for h = 6
(9)
degrees of freedom = 257
i=1
i=2
i=3
i=4
(banks & (insurance &
(export
(life ins. &
brokers) trading cos.) industries) import cos.)
αi,h
-0.010
-0.008
-0.019
-0.010
t (NW)
-1.151
-0.929
-2.165
-1.121
MSL(NW)
0.250
0.353
0.030
0.262
MH tests H0 : αi,h = αj , for all i, j 6= i
χ2 (GMM)
40.160
MSL(GMM)
0.000
36
Table 1C: Forecast Error (FE) Tests (6 month forecasts)
Individual regresions
st+h − sei,t,h = αi,h + εi,t,h for h = 12
(9)
degrees of freedom = 257
i=1
i=2
i=3
i=4
(banks & (insurance &
(export
(life ins. &
brokers) trading cos.) industries) import cos.)
αi,h
-0.031
-0.031
-0.045
-0.034
t (NW)
-1.736
-1.779
-2.441
-1.887
MSL(NW)
0.083
0.075
0.015
0.059
MH tests H0 : αi,h = αj , for all i, j 6= i
χ2 (GMM)
24.426
MSL(GMM)
0.000
37
Table 2A: Unbiasedness Tests (1 month forecasts)
Individual unbiasedness regressions
st+h − st = αi,h + βi,h (sei,t,h − st ) + εi,t,h for h = 2
(16)
degrees of freedom = 260
i=1
i=2
i=3
i=4
(banks & (insurance &
(export
(life ins. &
brokers) trading cos.) industries) import cos.)
αi,h
-0.003
-0.004
-0.007
-0.006
t (NW)
-1.123
-1.428
-2.732
-1.903
MSL(NW)
0.262
0.153
0.006
0.057
βi,h
t (NW)
0.437
1.674
H0 : βi,h = 1, for i = 1, 2, 3, 4
χ2
4.666
MSL
0.031
0.289
1.382
-0.318
-1.237
0.008
-0.038
4.666
0.001
4.666
0.000
4.666
0.000
Unbiasedness Tests: H0 : αi,h = 0, βi,h = 1, for i = 1, 2, 3, 4
χ2 (NW)
4.696
11.682
29.546
19.561
MSL
0.096
0.003
0.000
0.000
MH tests H0 : αi,h = αj , βi,h = βj for all i, j 6= i
χ2 (GMM)
9.689
MSL
0.138
38
Table 2B: Unbiasedness Tests (3 month forecasts)
Individual unbiasedness regressions
st+h − st = αi,h + βi,h (sei,t,h − st ) + εi,t,h for h = 6
(16)
degrees of freedom = 256
i=1
i=2
i=3
i=4
(banks & (insurance &
(export
(life ins. &
brokers) trading cos.) industries) import cos.)
αi,h
-0.013
-0.011
-0.017
-0.014
t (NW)
-1.537
-1.362
-2.060
-1.517
MSL(NW)
0.124
0.173
0.039
0.129
βi,h
t (NW)
0.521
1.268
H0 : βi,h = 1, for i = 1, 2, 3, 4
χ2
1.362
MSL
0.243
0.611
1.868
0.082
0.215
0.484
1.231
1.415
0.234
5.822
0.016
1.728
0.189
Unbiasedness Tests: H0 : αi,h = 0, βi,h = 1, for i = 1, 2, 3, 4
χ2 (NW)
2.946
2.691
11.156
3.023
MSL
0.229
0.260
0.004
0.221
MH tests H0 : αi,h = αj , βi,h = βj for all i, j 6= i
χ2 (GMM)
5.783
MSL
0.448
39
Table 2C: Unbiasedness Tests (6 month forecasts)
Individual unbiasedness regressions
st+h − st = αi,h + βi,h (sei,t,h − st ) + εi,t,h for h = 12
(16)
degrees of freedom = 256
i=1
i=2
i=3
i=4
(banks & (insurance &
(export
(life ins. &
brokers) trading cos.) industries) import cos.)
αi,h
-0.032
-0.032
-0.039
-0.034
t (NW)
-1.879
-1.831
-2.099
-1.957
MSL(NW)
0.060
0.067
0.036
0.050
βi,h
t (NW)
0.413
0.761
H0 : βi,h = 1, for i = 1, 2, 3, 4
χ2
1.166
MSL
0.280
0.822
1.529
0.460
0.911
0.399
-0.168
0.110
0.740
1.147
0.284
1.564
0.211
Unbiasedness Tests: H0 : αi,h = 0, βi,h = 1, for i = 1, 2, 3, 4
χ2 (NW)
4.332
3.5
7.899
5.006
MSL
0.115
0.174
0.019
0.082
MH tests H0 : αi,h = αj , βi,h = βj for all i, j 6= i
χ2 (GMM)
7.071
MSL
0.314
40
Table 3A: Weak Efficiency Tests (1 Month Forecasts)
st+h − sei,t,h = αi,h +
Ph+6
p=h
Banks &
Brokers
Lags
χ2
Pvalue
βi,t+h−p (st+h−p − sem,t+h−p,h ) + ǫi,t,h for h = 2
Ins. &
Trading Cos.
χ2
Export
Industry
χ2
Pvalue
Pvalue
(11)
Life Ins.&
Import Cos.
χ2
Pvalue
Single
2
3
4
5
6
7
8
0.132
-0.914
0.160
0.450
0.046
0.002
0.091
0.895
0.361
0.689
0.502
0.831
0.967
0.763
0.139
1.971
0.006
0.050
0.104
0.282
0.436
0.709
0.160
0.938
0.823
0.747
0.595
0.509
1.871
0.027
1.634
1.749
0.686
0.069
0.022
0.171
0.869
0.201
0.186
0.408
0.793
0.883
0.334
0.186
0.714
1.180
0.188
0.001
0.300
0.563
0.667
0.398
0.277
0.665
0.970
0.584
Cum.
3
4
5
6
7
8
0.765
4.626
4.747
5.501
6.252
5.927
0.682
0.201
0.314
0.358
0.396
0.548
1.778
3.463
4.382
5.592
6.065
5.357
0.411
0.326
0.357
0.348
0.416
0.617
1.746
8.763
7.680
7.652
8.879
8.390
0.418
0.033
0.104
0.176
0.180
0.299
0.585
5.349
5.081
5.768
6.677
6.087
0.746
0.148
0.279
0.329
0.352
0.530
Selected micro-homogeneity tests
H0 : αi,h = αj,h , βi,t+h−p = βj,t+h−p for all i, j 6= i
χ2 (n) Pvalue
n
Single
2
8
122.522
43.338
0.000
0.000
6
6
Cum.
3
8
136.830
201.935
0.000
0.000
9
24
VCV matrix incorporates Newey-West correction for serial correlation
χ2 statistics for mean forecast error regressions (p-value underneath)
Degrees of freedom (n) represent number of regressors, excluding intercept
(n=1 for single lag, n=max. lag -2 for cumulative lags)
41
Table 3B: Weak Efficiency Tests (1 Month Forecasts)
st+h − sei,t,h = αi.h +
Ph+6
p=h
Banks &
Brokers
Lags
χ2
Pvalue
βi,t+h−p (sem,t+h−p,h − st−p ) + ǫi,t,h for h = 2
Ins. &
Trading Cos.
χ2
Export
Industry
χ2
Pvalue
(12)
Life Ins.&
Import Cos.
χ2
Pvalue
Pvalue
Single
2
3
4
5
6
7
8
-2.325
4.482
3.162
3.956
6.368
8.769
5.451
0.020
0.106
0.075
0.047
0.012
0.003
0.020
5.641
3.519
2.580
2.993
4.830
6.786
4.114
0.018
0.061
0.108
0.084
0.028
0.009
0.043
3.658
3.379
2.805
3.102
5.952
7.755
4.417
0.056
0.066
0.094
0.078
0.015
0.005
0.036
7.011
5.877
4.911
7.467
9.766
12.502
7.564
0.008
0.015
0.027
0.006
0.002
0.000
0.006
Cum.
3
4
5
6
7
8
5.592
5.638
5.189
6.025
7.044
10.093
0.061
0.131
0.268
0.304
0.317
0.183
6.138
5.896
4.964
5.068
5.746
8.494
0.046
0.117
0.291
0.408
0.452
0.291
4.116
4.283
3.784
4.847
5.940
7.919
0.128
0.232
0.436
0.435
0.430
0.340
7.508
7.888
8.009
8.401
9.434
12.530
0.023
0.048
0.091
0.136
0.151
0.084
Selected micro-homogeneity tests
H0 : αi,h = αj,h , βi,t+h−p = βj,t+h−p for all i, j 6= i
χ2 (n) Pvalue
n
Single
2
8
40.462
30.739
0.000
0.000
6
6
Cum.
3
8
42.047
46.124
0.000
0.004
6
24
VCV matrix incorporates Newey-West correction for serial correlation
χ2 statistics for mean forecast error regressions (p-value underneath)
Degrees of freedom (n) represent number of regressors, excluding intercept
(n=1 for single lag, n=max. lag -2 for cumulative lags)
42
Table 3C: Weak Efficiency Tests (1 Month Forecasts)
st+h − sei,t,h = αi.h +
Ph+6
p=h
Banks &
Brokers
Lags
χ2
Pvalue
βi,t+h−p (st+h−p − st−p ) + ǫi,t,h for h = 2
Ins. &
Trading Cos.
χ2
Export
Industry
χ2
Pvalue
Pvalue
(13)
Life Ins.&
Import Cos.
χ2
Pvalue
Single
2
3
4
5
6
7
8
-0.328
1.621
0.002
0.086
0.165
0.850
0.597
0.743
0.203
0.964
0.770
0.685
0.357
0.440
0.639
3.060
0.335
0.042
0.916
1.861
1.088
0.424
0.080
0.562
0.837
0.339
0.172
0.297
1.249
0.000
0.819
1.001
0.029
0.329
0.317
0.264
0.993
0.366
0.317
0.864
0.566
0.574
0.023
0.550
0.146
0.344
0.095
1.152
1.280
0.879
0.458
0.702
0.557
0.758
0.283
0.258
Cum.
3
4
5
6
7
8
1.978
3.304
3.781
3.651
4.493
5.619
0.372
0.347
0.436
0.601
0.610
0.585
3.169
3.501
4.248
4.646
5.609
6.907
0.205
0.321
0.373
0.461
0.468
0.439
1.940
5.567
5.806
5.756
6.608
7.907
0.379
0.135
0.214
0.331
0.359
0.341
1.132
3.318
3.598
3.819
5.040
6.521
0.568
0.345
0.463
0.576
0.539
0.480
Selected micro-homogeneity tests
H0 : αi,h = αj,h , βi,t+h−p = βj,t+h−p for all i, j 6= i
χ2 (n) Pvalue
n
Single
2
8
150.698
45.652
0.000
0.000
6
6
Cum.
3
8
161.950
214.970
0.000
0.000
9
24
VCV matrix incorporates Newey-West correction for serial correlation
χ2 statistics for mean forecast error regressions (p-value underneath)
Degrees of freedom (n) represent number of regressors, excluding intercept
(n=1 for single lag, n=max. lag -2 for cumulative lags)
43
Table 3D: LM Test for Serial Correlation (1 Month Forecasts)
H0 : βi,t−h = ... = βi,t−h−6 = 0, for h = 2
P
Ph+6
e
in ǫ̂i,t,h = αi,h + h−1
k=1 βi,k (st+h−k − si,t−k,h ) +
l=h φi,l ǫ̂i,t−l,h + ηi,t,h , (15)
where ǫ is generated from
P
e
st+h − sei,t,h = αi,h + h−1
(14)
k=1 βi,k (st+h−k − si,t−k,h ) + ǫi,t,h
Cum. lags (k)
n−k
2
219
3
205
29.415
0.000
18.339
0.000
F (k, n − k)
p-value
30.952
0.000
Insurance & Trading Cos.
19.506 15.372 11.661 9.695 8.120 7.050
0.000 0.000 0.000 0.000 0.000 0.000
F (k, n − k)
p-value
32.387
0.000
20.691
0.000
29.694
0.000
Life Ins. & Import Cos.
18.606 14.596 11.093 9.586 9.154 7.937
0.000 0.000 0.000 0.000 0.000 0.000
F (k, n − k)
p-value
F (k, n − k)
p-value
4
192
5
179
6
166
7
153
8
144
Banks & Brokers
14.264 11.180 9.699 7.922 6.640
0.000 0.000 0.000 0.000 0.000
Export Industries
16.053 12.951 10.628 9.418 7.520
0.000 0.000 0.000 0.000 0.000
44
Table 4A: Weak Efficiency Tests (3 Month Forecasts)
st+h − sei,t,h = αi,h +
Ph+6
p=h
Banks &
Brokers
Lags
χ2
Pvalue
βi,t+h−p (st+h−p − sem,t+h−p,h ) + ǫi,t,h for h = 6
Ins. &
Trading Cos.
χ2
Export
Industry
χ2
Pvalue
Pvalue
(11)
Life Ins.&
Import Cos.
χ2
Pvalue
Single
6
7
8
9
10
11
12
0.667
0.052
0.006
0.055
0.264
0.299
0.172
0.414
0.820
0.940
0.814
0.607
0.585
0.678
0.954
0.071
0.010
0.043
0.278
0.381
0.336
0.329
0.789
0.921
0.836
0.598
0.537
0.562
4.493
1.434
0.382
0.140
0.001
0.020
0.011
0.034
0.231
0.537
0.708
0.980
0.888
0.918
1.719
0.268
0.001
0.060
0.432
0.598
0.633
0.190
0.605
0.976
0.806
0.511
0.439
0.426
Cum.
7
8
9
10
11
12
8.966
12.288
11.496
8.382
11.596
11.527
0.011
0.006
0.022
0.136
0.072
0.117
11.915
16.263
15.528
12.136
18.128
15.983
0.003
0.001
0.004
0.033
0.006
0.025
19.663
23.290
22.417
16.839
23.782
21.626
0.000
0.000
0.000
0.005
0.001
0.003
12.350
15.146
14.778
12.014
15.330
13.038
0.002
0.002
0.005
0.035
0.032
0.071
Selected micro-homogeneity tests
H0 : αi,h = αj,h , βi,t+h−p = βj,t+h−p for all i, j 6= i
χ2 (n) Pvalue
n
Single
6
188.738
12
63.364
0.000
0.000
6
6
Cum.
7
12
0.000
0.000
9
24
217.574
229.567
VCV matrix incorporates Newey-West correction for serial correlation
χ2 statistics for mean forecast error regressions (p-value underneath)
Degrees of freedom (n) represent number of regressors, excluding intercept
(n=1 for single lag, n=max. lag -2 for cumulative lags)
45
Table 4B: Weak Efficiency Tests (3 Month Forecasts)
st+h − sei,t,h = αi.h +
Ph+6
p=h
Banks &
Brokers
Lags
χ2
Pvalue
βi,t+h−p (sem,t+h−p,h − st−p ) + ǫi,t,h for h = 6
Ins. &
Trading Cos.
χ2
Export
Industry
χ2
Pvalue
Pvalue
(12)
Life Ins.&
Import Cos.
χ2
Pvalue
Single
6
7
8
9
10
11
12
3.457
4.241
5.748
6.073
8.128
8.511
6.275
0.063
0.039
0.017
0.014
0.004
0.004
0.012
2.947
3.834
5.177
5.843
7.868
8.004
6.691
0.086
0.050
0.023
0.016
0.005
0.005
0.010
3.470
4.390
5.410
5.968
7.845
8.308
6.635
0.062
0.036
0.020
0.015
0.005
0.004
0.010
3.681
4.370
6.053
6.474
8.521
8.429
6.079
0.055
0.037
0.014
0.011
0.004
0.004
0.014
Cum.
7
8
9
10
11
12
4.717
5.733
5.195
7.333
8.539
8.758
0.095
0.125
0.268
0.197
0.201
0.271
4.985
5.209
5.411
9.245
6.658
6.747
0.083
0.157
0.248
0.100
0.354
0.456
4.954
5.045
5.112
9.456
7.488
7.796
0.084
0.168
0.276
0.092
0.278
0.351
4.928
6.736
6.053
7.872
7.955
8.698
0.085
0.081
0.195
0.163
0.241
0.275
Selected micro-homogeneity tests
H0 : αi,h = αj,h , βi,t+h−p = βj,t+h−p for all i, j 6= i
χ2 (n) Pvalue
n
Single
6
12
57.130
58.230
0.000
0.000
6
6
Cum.
7
12
63.917
126.560
0.000
0.000
9
24
VCV matrix incorporates Newey-West correction for serial correlation
χ2 statistics for mean forecast error regressions (p-value underneath)
Degrees of freedom (n) represent number of regressors, excluding intercept
(n=1 for single lag, n=max. lag -2 for cumulative lags)
46
Table 4C: Weak Efficiency Tests (3 Month Forecasts)
st+h − sei,t,h = αi.h +
Ph+6
p=h
Banks &
Brokers
Lags
χ2
Pvalue
βi,t+h−p (st+h−p − st−p ) + ǫi,t,h for h = 6
Ins. &
Trading Cos.
χ2
Export
Industry
χ2
Pvalue
Pvalue
(13)
Life Ins.&
Import Cos.
χ2
Pvalue
Single
6
7
8
9
10
11
12
0.268
0.055
0.331
0.513
1.038
1.335
1.184
0.604
0.814
0.565
0.474
0.308
0.248
0.276
0.450
0.037
0.305
0.482
1.077
1.532
1.620
0.502
0.848
0.581
0.488
0.299
0.216
0.203
3.657
0.599
0.029
0.022
0.318
0.563
0.616
0.056
0.439
0.864
0.883
0.573
0.453
0.433
1.065
0.003
0.230
0.577
1.344
1.872
1.979
0.302
0.957
0.632
0.448
0.246
0.171
0.159
Cum.
7
8
9
10
11
12
6.766
8.752
8.654
9.421
9.972
8.581
0.034
0.033
0.070
0.093
0.126
0.284
8.767
11.784
11.588
12.890
13.137
11.823
0.012
0.008
0.021
0.024
0.041
0.107
15.683
18.330
18.929
19.146
19.597
17.670
0.000
0.000
0.001
0.002
0.003
0.014
10.052
11.162
11.309
12.275
13.003
11.431
0.007
0.011
0.023
0.031
0.043
0.121
Selected micro-homogeneity tests
H0 : αi,h = αj,h , βi,t+h−p = βj,t+h−p for all i, j 6= i
χ2 (n) Pvalue
n
Single
6
151.889
12
66.313
0.000
0.000
6
6
Cum.
7
12
0.000
0.000
9
24
164.216
193.021
VCV matrix incorporates Newey-West correction for serial correlation
χ2 statistics for mean forecast error regressions (p-value underneath)
Degrees of freedom (n) represent number of regressors, excluding intercept
(n=1 for single lag, n=max. lag -2 for cumulative lags)
47
Table 4D: LM Test for Serial Correlation (3 Month Forecasts)
H0 : βi,t−h = ... = βi,t−h−6 = 0, for h = 6
P
Ph+6
e
in ǫ̂i,t,h = αi,h + h−1
k=1 βi,k (st+h−k − si,t−k,h ) +
l=h φi,l ǫ̂i,t−l,h + ηi,t,h ,
where ǫ is generated from
P
e
st+h − sei,t,h = αi,h + h−1
k=1 βi,k (st+h−k − si,t−k,h ) + ǫi,t,h
Cum. lags (k)
n−k
F (k, n − k)
p-value
6
126
7
117
3.452
0.003
8
108
9
99
(15)
(14)
10
94
11
89
12
84
2.856
0.009
Banks & Brokers
3.023
2.951
2.599
0.004
0.004
0.008
2.652
0.006
2.921
0.002
2.584
0.007
2.341
0.012
F (k, n − k)
p-value
3.499
0.003
2.850
0.009
Insurance & Trading Cos.
3.408
2.907
2.492
0.002
0.004
0.011
F (k, n − k)
p-value
4.687
0.000
3.956
0.001
Export Industries
4.409
3.572
2.928
0.000
0.001
0.003
2.819
0.003
2.605
0.005
2.482
0.021
Life Ins. & Import Cos.
2.501
2.168
1.866
0.016
0.031
0.060
1.794
0.067
1.811
0.059
F (k, n − k)
p-value
2.352
0.035
48
Table 5A: Weak Efficiency Tests (6 Month Forecasts)
st+h − sei,t,h = αi,h +
Ph+6
p=h
Banks &
Brokers
Lags
χ2
Pvalue
βi,t+h−p (st+h−p − sem,t+h−p,h ) + ǫi,t,h for h = 12
Ins. &
Trading Cos.
χ2
Export
Industry
χ2
Pvalue
Single
12
13
14
15
16
17
18
0.597
0.612
0.811
0.944
1.550
2.192
3.020
0.440
0.434
0.811
0.331
0.213
0.139
0.082
0.787
0.761
0.906
1.017
1.556
2.254
2.920
0.375
0.383
0.341
0.313
0.212
0.133
0.088
Cum.
13
14
15
16
17
18
2.305
5.152
5.294
8.594
12.489
14.596
0.316
0.161
0.258
0.126
0.052
0.042
1.618
5.389
6.576
8.766
12.736
15.091
0.445
0.145
0.160
0.119
0.047
0.035
2.620
2.414
2.556
2.419
3.168
3.771
4.416
Pvalue
101.660
43.721
0.000
0.000
6
6
Cum.
13
18
187.021
449.252
0.000
0.000
9
24
Pvalue
0.344
0.354
0.343
0.331
0.239
0.176
0.128
3.041 0.219
6.472 0.091
7.220 0.125
11.495 0.042
17.611 0.007
21.041 0.004
1.419
3.359
4.565
8.325
10.881
12.861
0.492
0.339
0.335
0.139
0.092
0.076
VCV matrix incorporates Newey-West correction for serial correlation
χ2 statistics for mean forecast error regressions (p-value underneath)
Degrees of freedom (n) represent number of regressors, excluding intercept
(n=1 for single lag, n=max. lag -2 for cumulative lags)
49
χ2
0.894
0.858
0.899
0.945
1.385
1.827
2.318
n
Single
12
18
Life Ins.&
Import Cos.
0.105
0.120
0.110
0.120
0.075
0.052
0.036
Selected micro-homogeneity tests
H0 : αi,h = αj,h , βi,t+h−p = βj,t+h−p for all i, j 6= i
χ2 (n) Pvalue
(11)
Table 5B: Weak Efficiency Tests (6 Month Forecasts)
st+h − sei,t,h = αi.h +
Ph+6
p=h
Banks &
Brokers
Lags
χ2
Pvalue
βi,t+h−p (sem,t+h−p,h − st−p ) + ǫi,t,h for h = 12
Ins. &
Trading Cos.
χ2
Export
Industry
χ2
Pvalue
Pvalue
Single
12
13
14
15
16
17
18
3.022
2.107
1.637
0.985
1.550
0.232
0.023
0.082
0.147
0.201
0.321
0.213
0.630
0.881
3.109
2.380
1.989
1.391
0.973
0.530
0.168
0.078
0.123
0.158
0.238
0.324
0.467
0.682
3.601
3.196
2.665
1.894
1.577
1.068
0.471
0.058
0.074
0.103
0.169
0.209
0.301
0.492
Cum.
13
14
15
16
17
18
4.364
9.598
10.787
8.594
15.909
13.066
0.013
0.022
0.029
0.126
0.014
0.071
6.684
8.138
8.641
8.971
11.663
13.804
0.035
0.043
0.071
0.110
0.070
0.055
6.140
8.395
9.152
10.073
12.624
14.644
0.046
0.039
0.057
0.073
0.049
0.041
Selected micro-homogeneity tests
H0 : αi,h = αj,h , βi,t+h−p = βj,t+h−p for all i, j 6= i
χ2 (n)
Pvalue
n
Single
12
18
38.198
55.915
0.000
6
6
Cum.
13
18
46.234
98.324
0.000
0.000
9
24
VCV matrix incorporates Newey-West correction for serial correlation
χ2 statistics for mean forecast error regressions (p-value underneath)
Degrees of freedom (n) represent number of regressors, excluding intercept
(n=1 for single lag, n=max. lag -2 for cumulative lags)
50
(12)
Life Ins.&
Import Cos.
χ2
2.830
2.131
1.700
1.064
0.710
0.387
0.109
Pvalue
0.093
0.144
0.192
0.302
0.198
0.534
0.741
7.389 0.025
8.839 0.032
10.537 0.032
10.917 0.053
15.762 0.015
12.714 0.014
Table 5C: Weak Efficiency Tests (6 Month Forecasts)
st+h − sei,t,h = αi.h +
Ph+6
p=h
Banks &
Brokers
Lags
χ2
Pvalue
βi,t+h−p (st+h−p − st−p ) + ǫi,t,h for h = 12
Ins. &
Trading Cos.
χ2
Export
Industry
χ2
Pvalue
Pvalue
(13)
Life Ins.&
Import Cos.
χ2
Pvalue
Single
12
13
14
15
16
17
18
0.103
0.149
0.292
0.654
0.575
2.519
4.130
0.748
0.699
0.589
0.419
0.448
0.113
0.042
0.198
0.203
0.319
0.625
1.136
2.243
3.536
0.656
0.652
0.572
0.429
0.286
0.134
0.060
1.231
1.131
1.310
1.794
2.398
3.422
4.705
0.267
0.287
0.252
0.180
0.121
0.064
0.030
0.256
0.250
0.326
0.585
1.042
1.865
2.778
0.613
0.617
0.568
0.444
0.307
0.172
0.096
Cum.
13
14
15
16
17
18
1.445
4.226
7.182
9.752
11.554
12.322
0.486
0.238
0.127
0.083
0.073
0.090
0.548
4.048
6.226
7.180
11.194
11.062
0.761
0.256
0.183
0.208
0.083
0.136
1.140
5.345
7.801
8.608
13.429
14.223
0.566
0.148
0.099
0.126
0.037
0.047
0.531
2.398
5.732
6.435
7.984
7.741
0.767
0.494
0.220
0.266
0.239
0.356
Selected micro-homogeneity tests
H0 : αi,h = αj,h , βi,t+h−p = βj,t+h−p for all i, j 6= i
χ2 (n) Pvalue
n
Single
12
18
113.567
37.223
0.000
0.000
6
6
Cum.
13
18
260.006
409.126
0.000
0.000
9
24
VCV matrix incorporates Newey-West correction for serial correlation
χ2 statistics for mean forecast error regressions (p-value underneath)
Degrees of freedom (n) represent number of regressors, excluding intercept
(n=1 for single lag, n=max. lag -2 for cumulative lags)
51
Table 5D: LM Test for Serial Correlation (6 Month Forecasts)
H0 : βi,t−h = ... = βi,t−h−6 = 0, for h = 12
P
Ph+6
e
in ǫ̂i,t,h = αi,h + h−1
k=1 βi,k (st+h−k − si,t−k,h ) +
l=h φi,l ǫ̂i,t−l,h + ηi,t,h ,
where ǫ is generated from
P
e
st+h − sei,t,h = αi,h + h−1
k=1 βi,k (st+h−k − si,t−k,h ) + ǫi,t,h
Cum. lags (k)
n−k
12
54
13
52
2.798
0.000
3.618
0.000
F (k, n − k)
p-value
2.321
0.018
Insurance & Trading Cos.
2.446
2.112
2.243
1.667
0.011
0.027
0.018
0.089
F (k, n − k)
p-value
2.747
0.006
3.134
0.002
1.687
0.096
F (k, n − k)
p-value
F (k, n − k)
p-value
14
50
15
48
(15)
(14)
16
46
17
44
18
42
2.578
0.006
2.671
0.005
2.762
0.003
1.674
0.086
1.672
0.085
Export Industries
2.685
2.709
1.818
0.005
0.005
0.052
1.844
0.052
1.871
0.048
Life Ins. & Import Cos.
2.068
0.393
2.739
1.749
0.033
0.971
0.004
0.070
1.839
0.053
1.987
0.034
Banks & Brokers
3.462
3.926
0.001
0.000
52
Table 6A: Unit Root Tests (1 Month Forecasts)
∆yt = α + βyt−1 + γt +
Pp
k=1 θt ∆yt−k
Log of spot rate (n=276)
H th difference of log of spot rate
+ ǫt for h = 2
(21)
Lags
ADF t test
ADF z test
Joint test
0
12
-2.828
-4.306***
-9.660
-167.473***
6.820**
9.311***
Group 1 (Banks & brokers)
Log of forecast
H th difference of log of forecast
Forecast error
Restriced CI eq.
Unrestriced CI eq.
0
0
-2.274*
-7.752***
-5.072
-96.639***
6.327**
30.085***
1
1
-11.325***
65.581***
-249.389***
64.676***
Group 2 (Insurance & trading cos.)
Log of forecast
H th difference of log of forecast
Forecast error
Restriced CI eq.
Unrestriced CI eq.
0
0
-2.735*
-7.895***
-5.149
-94.986***
6.705***
31.252***
1
1
-11.624***
-11.750***
-270.302***
68.053***
Group 3 (Export industries)
Log of forecast
H th difference of log of forecast
Forecast error
Restriced CI eq.
Unrestriced CI eq.
1
0
-2.372
-8.346***
-4.806
-111.632***
5.045**
34.889***
1
1
-10.324***
-10.392***
-211.475***
53.757***
Group 4 (Life ins. & import cos.)
Log of forecast
H th difference of log of forecast
Forecast error
Restriced CI eq.
Unrestriced CI eq.
* rejection at 10% level
** rejection at 5% level
*** rejection at 1% level
0
1
-2.726*
-5.216***
-5.009
-52.911***
6.438**
3.630***
1
1
-10.977***
-10.979***
-231.837***
60.820***
53
Table 6B: Unit Root Tests (3 Month Forecasts)
∆yt = α + βyt−1 + γt +
Pp
k=1 θt ∆yt−k
Log of spot rate (n=276)
H th difference of log of spot rate
+ ǫt for h = 6
Lags
0
2
ADF t test
-2.828
-4.760***
(21)
ADF z test
Joint test
-9.660
-49.769***
6.820**
11.351***
Group 1 (Banks & brokers)
Log of forecast
H th difference of log of forecast
Forecast error
Restriced CI eq.
Unrestriced CI eq.
0 -2.840*
0 -5.092***
-4.852
-48.707***
7.610***
12.990***
6
6
-29.429***
4.673**
-3.022**
-3.343*
Group 2 (Insurance & trading cos.)
Log of forecast
H th difference of log of forecast
Forecast error
Restriced CI eq.
Unrestriced CI eq.
0 -2.778*
0 -6.514***
-4.533
-71.931***
8.858***
21.588***
6
2
-31.038***
4.956**
-3.068**
-4.539***
Group 3 (Export industries)
Log of forecast
H th difference of log of forecast
Forecast error
Restriced CI eq.
Unrestriced CI eq.
0 -3.105**
1 -4.677***
-4.549
-41.524***
9.090***
10.944***
6
5
-31.207***
5.659**
-3.317**
-5.115***
Group 4 (Life ins. & import cos.)
Log of forecast
H th difference of log of forecast
Forecast error
Restriced CI eq.
Unrestriced CI eq.
* rejection at 10% level
** rejection at 5% level
*** rejection at 1% level
0 -2.863*
1 -4.324***
5
4
54
-4.825***
-5.123***
-4.400
-39.870***
8.161***
9.352***
-118.586***
11.679***
Table 6C: Unit Root Tests (6 Month Forecasts)
∆yt = α + βyt−1 + γt +
Pp
k=1 θt ∆yt−k
Log of spot rate (n=276)
H th difference of log of spot rate
+ ǫt for h = 12
(21)
Lags ADF t test
ADF z test
Joint test
0 -2.828
17 -3.189**
-9.660
-26.210***
6.820**
5.500**
Group 1 (Banks & brokers)
Log of forecast
H th difference of log of forecast
Forecast error
Restriced CI eq.
Unrestriced CI eq.
0 -2.947**
0 -4.772***
-4.254
-44.018***
1 -2.373
7 -3.285**
-13.577*
9.131***
11.389***
2.947
Group 2 (Insurance & trading cos.)
Log of forecast
H th difference of log of forecast
Forecast error
Restriced CI eq.
Unrestriced CI eq.
0 -2.933**
0 -6.007***
-4.004
-64.923***
1 -2.114
1 -2.684*
-11.464*
9.531***
18.044***
2.399
Group 3 (Export industries)
Log of forecast
H th difference of log of forecast
Forecast error
Restriced CI eq.
Unrestriced CI eq.
0 -3.246**
12 -4.961***
12 -1.515
0 -2.931
-4.059
-44.532***
-5.601
10.704***
12.331***
1.466
Group 4 (Life ins. & import cos.)
Log of forecast
H th difference of log of forecast
Forecast error
Restriced CI eq.
Unrestriced CI eq.
* rejection at 10% level
** rejection at 5% level
*** rejection at 1% level
0 -3.133**
0 -4.795***
-4.196
-44.062***
2 -2.508
1 -2.851*
-14.535**
55
9.549***
11.537***
3.148
Table 7: CD Tests for Contemporaneous Cross-Sectional
Dependendence of Forecast Errors
st+h − sei,t,h = αi.h + ǫi,t,h
(9)
Horizon (h)
q
CD
1 month
MSL
37.945
0.000
3 months
MSL
31.272
0.000
6 months
MSL
25.571
0.000
a
N
N −1
2T
CD = N (N
j=1+1 ρ̂ij ∼ N (0, 1)
i=1
−1)
N = 24, T = 276, ρ̂ij is the sample correlation coefficient
between forecaster i and j, i 6= j.
P
P
56
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60
Appendix 1: Data Description and Variable Definitions
Every two weeks, the JCIF in Tokyo conducts telephone surveys of yen/dollar exchange
rate expectations from 44 firms. The forecasts are for the future spot rate at horizons
of one month, three months, and six months. Our data cover the period May 1985 to
March 1996. This data set has very few missing observations, making it close to a true
panel. For reporting purposes, the JCIF currently groups individual firms into four industry
categories: 1) banks and brokers, 2) insurance and trading companies, 3) exporters, and 4)
life insurance companies and importers. On the day after the survey, the JCIF announces
overall and industry average forecasts. (For further details concerning the JCIF database,
see the descriptions in Ito (1990, 1994), Bryant (1995), and Elliott and Ito (1999).)
61
Appendix 2: Testing Micro-homogeneity with Survey Forecasts
The null hypothesis of micro-homogeneity is that the slope and intercept coefficients in the
equation of interest are equal across individuals. This paper considers the case of individual
unbiasedness regressions such as equation (16) in the text, repeated here for convenience.
st+h − st = αi,h + βi,h (sei,t,h − st ) + εi,t,h
(A2.1)
and tests H0 : α1 = α2 = . . . = αN , and β1 = β2 = . . . = βN .
Stack all N individual regressions into the Seemingly Unrelated Regression system
S = Fθ + ε
(A2.2)
where S is the N T × 1 stacked vector of realizations, st+h , and F is an N T × 2N block
diagonal data matrix

F1

F=


...
FN

.

(A2.3)
Each Fi = [ι sei,t,h ] is a T ×2 matrix of ones and individual i’s forecasts, θ = [α1 β1 . . . αN βN ]′ ,
and ε is an N T × 1 vector of stacked residuals. The vector of restrictions, Rθ = r,
corresponding to the null hypothesis of micro-homogeneity is normally distributed, with
Rθ − r ∼ N [0, R(F′ F)−1 F′ ΩF(F′ F)−1 R′ ], where R is the 2(N − 1) × 2N matrix



R=



1
0
−1
0
...

0
.. 
. 

0 1
0 −1 0
,
..
... ... ...

0 
. 0
0 ... 0
1
0 −1
(A2.4)
and r is a 2(N − 1) × 1 vector of zeros. The corresponding Wald test statistic,
(Rθ̂ − r)′ [R(F′ F)−1 F′ Ω̂F(F′ F)−1 R′ ](Rθ̂ − r), is asymptotically distributed as a chi-square
random variable with degrees of freedom equal to the number of restrictions, 2(N − 1).
For most surveys, there are a large number of missing observations. Keane and Runkle
(1990), Davies and Lahiri (1995), Bonham and Cohen (1995, 2001), and to the best of our
knowledge all other papers which make use of pooled regressions in tests of the REH have
dealt with the missing observations using the same approach. The pooled or individual regression is estimated by eliminating the missing data points in both the forecasts and the
realization. The regression residuals are then padded with zeros in place of missing observations to allow for the calculation of own and cross-covariances. As a result, many individual
variances and cross-covariances are calculated with relatively few pairs of residuals. These
individual cross-covariances are then averaged. In Keane and Runkle (1990) and Bonham
and Cohen (1995, 2001) the assumption of 2(k + 1) second moments, which are common
to all forecasters, is made for analytical tractability and for increased reliability. In contrast to the forecasts from the Survey of Professional Forecasters used in Keane and Runkle
(1990) and Bonham and Cohen (1995, 2001), the JCIF data set contains virtually no missing observations. As a result, it is possible to estimate each individual’s variance covariance
62
matrix (and cross-covariance matrix) rather than average over all individual variances and
cross-covariance pairs as in the aforementioned papers.
We assume that for each forecast group i,
2
E[εi,t,h εi,t,h ] = σi,0
for all i, t,
2
E[εi,t,h εi,t+k ] = σi,k for all i, t, k such that 0 < k ≤ h,
E[εi,t,h εi,t+k ] = 0 for all i, t, k such that k > h,
(A2.5)
Similarly, for each pair of forecasters i and j we assume
E[εi,t,h εj,t ] = δi,j (0) ∀i, j, t,
E[εi,t,h εj,t+k ] = δi,j (k) ∀i, j, t, k such that k 6= 0, and −h ≤ k ≤ h.
E[εi,t,h εj,t+k ] =
0
∀i, j, t, k such that k > |h|.
(A2.6)
Thus, each pair of forecasters has a different T × T cross-covariance matrix,

Pi,j




=



δi,j (0) δi,j (−1)
...
δi,j (−h)
0
δi,j (1) δi,j (0) δi,j (−1)
...
0
..
..
...
...
...
.
.
...
δi,j (1)
δi,j (0) δi,j (−1)
0
δi,j (h)
...
δi,j (1)
δi,j (0)





,



(A2.7)
′
Finally, note that Pi,j 6= Pj,i , rather Pi,j
= Pj,i . The complete variance-covariance matrix,
denoted Ω, has dimension N T × N T , with matrices Qi on the main diagonal and Pi,j off
the diagonal.
The individual Qi , variance covariances matrices are calculated using the Newey and West
(1987) heteroscedasticity-consistent, MA(j) corrected form. The Pi,j matrices are estimated
in an analogous manner,
63