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Board of Governors of the Federal Reserve System

International Finance Discussion Papers

Number 963, March 2009 --- Screen Reader
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NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at http://www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at http://www.ssrn.com/.

Abstract:

This paper attacks the Meese-Rogoff puzzle from a different perspective: out-of-sample interval forecasting. Most studies in the literature focus on point forecasts. In this paper, we apply Robust Semiparametric (RS) interval forecasting to a group of Taylor rule models. Forecast intervals for twelve OECD exchange rates are generated and modified tests of Giacomini and White (2006) are conducted to compare the performance of Taylor rule models and the random walk. Our contribution is twofold. First, we find that in general, Taylor rule models generate tighter forecast intervals than the random walk, given that their intervals cover out-of-sample exchange rate realizations equally well. This result is more pronounced at longer horizons. Our results suggest a connection between exchange rates and economic fundamentals: economic variables contain information useful in forecasting the distributions of exchange rates. The benchmark Taylor rule model is also found to perform better than the monetary and PPP models. Second, the inference framework proposed in this paper for forecast-interval evaluation can be applied in a broader context, such as inflation forecasting, not just to the models and interval forecasting methods used in this paper.

Keywords: The exchange rate disconnect puzzle, exchange rate forecast, interval forecasting

JEL classification: F31, C14, C53

Recent studies explore the role of monetary policy rules, such as Taylor rules, in exchange rate determination. They find empirical support in these models for the linkage between exchange rates and economic fundamentals. Our paper extends this literature from a different perspective: interval forecasting. We find that the Taylor rule models can outperform the random walk, especially at long horizons, in forecasting twelve OECD exchange rates based on relevant out-of-sample interval forecasting criteria. The benchmark Taylor rule model is also found to perform relatively better than the standard monetary model and the purchasing power parity (PPP) model.

In a seminal paper, Meese and Rogoff (1983) find that economic fundamentals - such as the money supply, trade balance and national income - are of little use in forecasting exchange rates. They show that existing models cannot forecast exchange rates better than the random walk in terms of out-of-sample forecasting accuracy. This finding suggests that exchange rates may be determined by something purely random rather than economic fundamentals. Meese and Rogoff's (1983) finding has been named the Meese-Rogoff puzzle in the literature.

In defending fundamental-based exchange rate models, various
combinations of economic variables and econometric methods have
been used in attempts to overturn Meese and Rogoff's finding. For
instance, Mark (1995) finds greater exchange rate predictability at
longer horizons.^{4} Groen (2000) and Mark and Sul (2001)
detect exchange rate predictability by using panel data. Kilian and
Taylor (2003) find that exchange rates can be predicted from
economic models at horizons of 2 to 3 years, after taking into
account the possibility of nonlinear exchange rate dynamics. Faust,
Rogers, and Wright (2003) find that the economic models
consistently perform better using real-time data than revised data,
though they do not perform better than the random walk.

Recently, there is a growing strand of literature that uses Taylor rules to model exchange rate determination. Engel and West (2005) derive the exchange rate as a present-value asset price from a Taylor rule model. They also find a positive correlation between the model-based exchange rate and the actual real exchange rate between the US dollar and the Deutschmark (Engel and West, 2006). Mark (2007) examines the role of Taylor-rule fundamentals for exchange rate determination in a model with learning. In his model, agents use least-square learning rules to acquire information about the numerical values of the model's coefficients. He finds that the model is able to capture six major swings of the real Deutschmark-Dollar exchange rate from 1973 to 2005. Molodtsova and Papell (2009) find significant short-term out-of-sample predictability of exchange rates with Taylor-rule fundamentals for 11 out of 12 currencies vis-á-vis the U.S. dollar over the post-Bretton Woods period. Molodtsova, Nikolsko-Rzhevskyy, and Papell (2008a, 2008b) find evidence of out-of-sample predictability for the dollar/mark nominal exchange rate with forecasts based on Taylor rule fundamentals using real-time data, but not revised data. Chinn (forthcoming) also finds that Taylor rule fundamentals do better than other models at the one year horizon. With a present-value asset pricing model as discussed in Engel and West (2005), Chen and Tsang (2009) find that information contained in the cross-country yield curves are useful in predicting exchange rates.

Our paper joins the above literature of Taylor-rule exchange rate models. However, we address the Meese and Rogoff puzzle from a different perspective: interval forecasting. A forecast interval captures a range in which the exchange rate may lie with a certain probability, given a set of predictors available at the time of forecast. Our contribution to the literature is twofold. First, we find that for twelve OECD exchange rates, the Taylor rule models in general generate tighter forecast intervals than the random walk, given that their intervals cover the realized exchange rates (statistically) equally well. This finding suggests an intuitive connection between exchange rates and economic fundamentals beyond point forecasting: the use of economic variables as predictors helps narrow down the range in which future exchange rates may lie, compared to random walk forecast intervals. Second, we propose an inference framework for cross-model comparison of out-of-sample forecast intervals. The proposed framework can be used for forecast-interval evaluation in a broader context, not just for the models and methods used in this paper. For instance, the framework can also be used to evaluate out-of-sample inflation forecasting.

As we will discuss later, we in fact derive forecast intervals from estimates of the distribution of changes in the exchange rate. Hence, in principle, evaluations across models can be done based on distributions instead of forecast intervals. However, focusing on interval forecasting performance allows us to compare models in two dimensions that are more relevant to practitioners: empirical coverage and length.

While the literature on interval forecasting for exchange rates is sparse, several authors have studied out-of-sample exchange rate density (distribution) forecasts, from which interval forecasts can be derived. Diebold, Hahn and Tay (1999) use the RiskMetrics model of JP Morgan (1996) to compute half-hour-ahead density forecasts for Deutschmark/Dollar and Yen/Dollar returns. Christoffersen and Mazzotta (2005) provide option-implied density and interval forecasts for four major exchange rates. Boero and Marrocu (2004) obtain one-day-ahead density forecasts for the Euro nominal effective exchange rate using self-exciting threshold autoregressive (SETAR) models. Sarno and Valente (2005) evaluate the exchange rate density forecasting performance of the Markov-switching vector equilibrium correction model that is developed by Clarida, Sarno, Taylor and Valente (2003). They find that information from the term structure of forward premia help the model to outperform the random walk in forecasting out-of-sample densities of the spot exchange rate. More recently, Hong, Li, and Zhao (2007) construct half-hour-ahead density forecasts for Euro/Dollar and Yen/Dollar exchange rates using a comprehensive set of univariate time series models that capture fat tails, time-varying volatility and regime switches.

There are several common features across the studies listed above, which make them different from our paper. First, the focus of the above studies is not to make connections between the exchange rate and economic fundamentals. These studies use high frequency data, which are not available for most conventional economic fundamentals. For instance, Diebold, Hahn, and Tay (1999) and Hong, Li, and Zhao (2007) use intra-day data. With the exception of Sarno and Valente (2005), all the studies focus only on univariate time series models. Second, these studies do not consider multi-horizon-ahead forecasts, perhaps due to the fact that their models are often highly nonlinear. Iterating nonlinear density models multiple horizons ahead is analytically difficult, if not infeasible. Lastly, the above studies assume that the densities are analytically defined for a given model. The semiparametric method used in this paper does not impose such restrictions.

Our choice of the semiparametric method is motivated by the
difficulty of using macroeconomic models in exchange rate interval
forecasting: these models typically do not describe the future
*distributions* of exchange rates. For instance, the Taylor
rule models considered in this paper do not describe any features
of the data beyond the conditional means of future exchange rates.
We address this difficulty by applying *Robust Semiparametric*
forecast intervals (hereon *RS* forecast intervals) of Wu
(2009).^{5} This method is useful since it does
not require the model be correctly specified, or contain parametric
assumptions about the future distribution of exchange rates.

We apply RS forecast intervals to a set of Taylor rule models that differ in terms of the assumptions on policy and interest rate smoothing rules. Following Molodtsova and Papell (2009), we include twelve OECD exchange rates (relative to the US dollar) over the post-Bretton Woods period in our dataset. For these twelve exchange rates, the out-of-sample RS forecast intervals at different forecast horizons are generated from the Taylor rule models and then compared with those of the random walk. The empirical coverages and lengths of forecast intervals are used as the evaluation criteria. Our empirical coverage and length tests are modified from Giacomini and White's (2006) predictive accuracy tests in the case of rolling, but fixed-size, estimation samples.

For a given nominal coverage (probability), the empirical coverage of forecast intervals derived from a forecasting model is the probability that the out-of-sample realizations (exchange rates) lie in the intervals. The length of the intervals is a measure of its tightness: the distance between its upper and lower bound. In general, the empirical coverage is not the same as its nominal coverage. Significantly missing the nominal coverage indicates poor quality of the model and intervals. One certainly wants the forecast intervals to contain out-of-sample realizations as close as possible to the probability they target. Most evaluation methods in the literature focus on comparing empirical coverages across models, following the seminal work of Christoffersen (1998). Following this literature, we first test whether forecast intervals of the Taylor rule models and the random walk have equally accurate empirical coverages. The model with more accurate coverages is considered the better model. In the cases where equal coverage accuracy cannot be rejected, we further test whether the lengths of forecast intervals are the same. The model with tighter forecast intervals provides more useful information about future values of the data, and hence is considered as a more useful forecasting model.

It is also important to establish what this paper is *not*
attempting. First, the inference procedure does not carry the
purpose of finding the *correct* model specification. Rather,
inference is on how useful models are in generating forecast
intervals, measured in terms of empirical coverages and lengths.
Second, this paper does not consider the possibility that there
might be alternatives to RS forecast intervals for the exchange
rate models we consider. Some models might perform better if
parametric distribution assumptions (e.g. the forecast errors are
conditionally heteroskedastic and distributed)
or other assumptions (e.g. the forecast errors are independent of
the predictors) are added. One could presumably estimate the
forecast intervals differently based on the same models, and then
compare those with the RS forecast intervals, but this is out of
the scope of this paper. As we described, we choose the RS method
for the robustness and flexibility achieved by the semiparametric
approach.

First, the models considered in this paper are standard macroeconomic models for exchange rate determination. These models do not contain theory about the underlying distributions. Using RS forecast intervals allows us to refrain from having to impose subjective additional structures on the models. Our goal is to use existing models to generate forecast intervals, instead of developing new models. Second, RS forecast intervals are good candidates as they always consistently estimates their population counterparts, independent of the properties of the models and error terms ( ). With the sample sizes we have, RS forecast intervals are deemed good approximations to the truth.

Our benchmark Taylor rule model is from Engel and West (2005) and Engel, Wang, and Wu (2008). For the purpose of comparison, several alternative Taylor rule models are also considered. These setups have been studied by Molodtsova and Papell (2009) and Engel, Mark, and West (2007). In general, we find that the Taylor rule models perform better than the random walk model, especially at long horizons: the models either have more accurate empirical coverages than the random walk, or in cases of equal coverage accuracy, the models have tighter forecast intervals than the random walk. The evidence of exchange rate predictability is much weaker in coverage tests than in length tests. In most cases, the Taylor rule models and the random walk have statistically equally accurate empirical coverages. So, under the conventional coverage test, the random walk model and the Taylor rule models perform equally well. However, the results of length tests suggest that Taylor rule fundamentals are useful in generating tighter forecasts intervals without losing accuracy in empirical coverages.

We also consider two other popular models in the literature: the monetary model and the model of purchasing power parity (PPP). Based on the same criteria, both models are found to perform better than the random walk in interval forecasting. As with the Taylor rule models, most evidence of exchange rate predictability comes from the length test: economic models have tighter forecast intervals than the random walk given statistically equivalent coverage accuracy. The PPP model performs worse than the benchmark Taylor rule model and the monetary model at short horizons. The benchmark Taylor rule model performs slightly better than the monetary model at most horizons.

Our findings suggest that exchange rate movements are linked to economic fundamentals. However, we acknowledge that the Meese-Rogoff puzzle remains difficult to understand. Although Taylor rule models offer statistically significant length reductions over the random walk, the reduction of length is quantitatively small, especially at short horizons. Forecasting exchange rates remains a difficult task in practice. There are some impressive advances in the literature, but most empirical findings remain fragile. As mentioned in Cheung, Chinn, and Pascual (2005), forecasts from economic fundamentals may work well for some currencies during certain sample periods but not for other currencies or sample periods. Engel, Mark, and West (2007) recently show that a relatively robust finding is that exchange rates are more predictable at longer horizons, especially when using panel data. We find greater predictability at longer horizons in our exercise. It would be of interest to investigate connections between our findings and theirs.

Several recent studies have attacked the puzzle from a different angle: there are reasons that economic fundamentals cannot forecast the exchange rate, even if the exchange rate is determined by these fundamentals. Engel and West (2005) show that existing exchange rate models can be written in a present-value asset-pricing format. In these models, exchange rates are determined not only by current fundamentals but also by expectations of what the fundamentals will be in the future. When the discount factor is large (close to one), current fundamentals receive very little weight in determining the exchange rate. Not surprisingly, the fundamentals are not very useful in forecasting. Nason and Rogers (2008) generalize the Engel-West theorem to a class of open-economy dynamic stochastic general equilibrium (DSGE) models. Other factors such as parameter instability and mis-specification (for instance, Rossi 2005) may also play important roles in understanding the puzzle. It is interesting to investigate conditions under which we can reconcile our findings with these studies.

The remainder of this paper is organized as follows. Section two describes the forecasting models we use, as well as the data. In section three, we illustrate how the RS forecast intervals are constructed from a given model. We also propose loss criteria to evaluate the quality of the forecast intervals and test statistics that are based on Giacomini and White (2006). Section four presents results of out-of-sample forecast evaluation. Finally, section five contains concluding remarks.

Seven models are considered in this paper. Let be the index of these models and the first model be the benchmark model. A general setup of the models takes the form of:

(1) |

where is -period changes of the (log) exchange rate, and contains economic variables that are used in model . Following the literature of long-horizon regressions, both short- and long-horizon forecasts are considered. Models differ in economic variables that are included in matrix . In the benchmark model,

where ( ) is the inflation rate, and ( ) is the output gap in the home (foreign) country. The real exchange rate is defined as , where ( ) is the (log) consumer price index in the home (foreign) country. This setup is motivated by the Taylor rule model in Engel and West (2005) and Engel, Wang, and Wu (2008). The next subsection describes this benchmark Taylor rule model in detail.

We also consider the following models that have been studied in the literature:

Model 2: | |

Model 3: | , |

where ( ) is the short-term interest rate in the home (foreign) country. | |

Model 4: | |

Model 5: | |

Model 6: | , |

where ( ) is the money supply and ( ) is total output in the home (foreign) country. | |

Model 7: |

Models 2-4 are the Taylor rule models studied in Molodtsova and
Papell (2009). Model 2 can be considered as the constrained
benchmark model in which PPP always holds. Molodtsova and Papell
(2009) include interest rate lags in models 3 and 4 to take into
account potential interest rate smoothing rules of the central
bank.^{6} Model 5 is the purchasing power parity
(PPP) model and model 6 is the monetary model. Both models have
been widely used in the literature. See Molodtsova and Papell
(2009) for the PPP model and Mark (1995) for the monetary model.
Model 7 is the driftless random walk model (
).^{7} Given a date
and horizon , the
objective is to estimate the forecast distribution of
conditional on
, and subsequently build
forecast intervals from the estimated forecast distribution. Before
moving to the econometric method, we first describe the Taylor rule
model that motivates the setup of our benchmark model.

Our benchmark model is the Taylor rule model that is derived in
Engel and West (2005) and Engel, Wang, and Wu (2008). Following
Molodtsova and Papell (2009), we focus on models that depend only
on *current* levels of inflation and the output gap.^{8} The
Taylor rule in the home country takes the form of:

(2) |

where is the central bank's target for the short-term interest rate at time , is the equilibrium long-term rate, is the inflation rate, is the target inflation rate, and is the output gap. The foreign country is assumed to follow a similar Taylor rule. In addition, we follow Engel and West (2005) to assume that the foreign country targets the exchange rate in its Taylor rule:

(3) |

where is the targeted exchange rate. Assume that the foreign country targets the PPP level of the exchange rate: , where and are logarithms of home and foreign aggregate prices. In equation (3), we assume that the policy parameters take the same values in the home and foreign countries. Molodtsova and Papell (2009) denote this case as "homogeneous Taylor rules". Our empirical results also hold in the case of heterogenous Taylor rules. To simplify our presentation, we assume that the home and foreign countries have the same long-term inflation and interest rates. Such restrictions have been relaxed in our econometric model after we include a constant term in estimations.

We do not consider interest rate smoothing in our benchmark model. That is, the actual interest rate () is identical to the target rate in the benchmark model:

(4) |

Molodtsova and Papell (2009) consider the following interest rate smoothing rule:

(5) |

where is the interest rate smoothing parameter. We include these setups in models 3 and 4. Note that our estimation methods do not require the monetary policy shock and the interest rate smoothing shock to satisfy any assumptions, aside from smoothness of their distributions when conditioned on predictors.

Substituting the difference of equations (2) and (3) into Uncovered Interest-rate Parity (UIP), we have:

(6) |

where the discount factor
. Under some
conditions, the present value asset pricing format in equation
(6) can be written
into an error-correction form:^{9}

(7) |

where the deviation of the exchange rate from its equilibrium level is defined as:

(8) |

We use equation (7)
as our benchmark setup in calculating h-horizon-ahead out-of-sample
forecasting intervals. According to equation (8), the matrix
in equation (1) includes
economic variables
,
, and
.^{10}

The forecasting models and the corresponding forecast intervals are estimated using monthly data for twelve OECD countries. The United States is treated as the foreign country in all cases. For each country we synchronize the beginning and end dates of the data across all models estimated. The twelve countries and periods considered are: Australia (73:03-06:6), Canada (75:01-06:6), Denmark (73:03-06:6), France (77:12-98:12), Germany (73:03-98:12), Italy (74:12-98:12), Japan (73:03-06:6), Netherlands (73:03-98:12), Portugal (83:01-98:12), Sweden (73:03-04:11), Switzerland (75:09-06:6), and the United Kingdom (73:03-06:4).

The data is taken from Molodtsova and Papell (2009).^{11} With
the exception of interest rates, the data is transformed by taking
natural logs and then multiplying by 100. The nominal exchange
rates are end-of-month rates taken from the Federal Reserve Bank of
St. Louis database. Output data () are proxied by
Industrial Production (IP) from the International Financial
Statistics (IFS) database. IP data for Australia and Switzerland
are only available at a quarterly frequency, and hence are
transformed from quarterly to monthly observations using the
quadratic-match average option in Eviews 4.0 by Molodtsova and
Papell (2009). Following Engel and West (2006), the output gap (
) is calculated by quadratically
de-trending the industrial production for each country.

Prices data () are proxied by Consumer Price Index (CPI) from the IFS database. Again, CPI for Australia is only available at a quarterly frequency and the quadratic-match average is used to impute monthly observations. Inflation rates are calculated by taking the first differences of the logs of CPIs. The money market rate from IFS (or "call money rate") is used as a measure of the short-term interest rate set by the central bank. Finally, M1 is used to measure the money supply for most countries. M0 for the UK and M2 for Italy and Netherlands is used due to the unavailability of M1 data.

For a given
model , the objective is to estimate from equation
(1)
the distribution of
conditional on data
that is observed up to
time . This is the -horizon-ahead
*forecast distribution* of the exchange rate, from which the
corresponding *forecast interval* can be derived. For a given
, the forecast interval of coverage
is an interval in which
is supposed to lie with a
probability of .

Models in equation (1) provide only point forecasts of . In order to construct forecast intervals for a given model, we apply robust semiparametric (RS) forecast intervals to all models. The nominal -coverage forecast interval of conditional on can be obtained by the following three-step procedure:

*Step 1.*- Estimate model by OLS and obtain residuals , for .
*Step 2.*- For a range of values of
(sorted residuals
), estimate
the conditional distribution of
by:
(9) where , is a multivariate Gaussian kernel with a dimension the same as that of , and is the smoothing parameter or bandwidth.

^{12} *Step 3.*- Find the
and
quantiles of the estimated
distribution, which are denoted by
and
,
repectively. The estimate of the -coverage
forecast interval for
conditional on
is:
(10)

For each model , the above method uses the
forecast models in equation (1) to estimate
the location of the forecast distribution, while the nonparametric
kernel distribution estimate is used to estimate the shape. As a
result, the interval obtained from this method is
*semiparametric*. Wu (2009) shows that under some weak
regularity conditions, this method consistently estimates the
forecast distribution,^{13} and hence the forecast intervals of
conditional on
, regardless of the
quality of model . That is, the forecast intervals
are *robust*. Stationarity of economic variables is one of
those regularity conditions. In our models, exchange rate
differences, interest rates and inflation rates are well-known to
be stationary, while empirical tests for real exchange rates and
output gaps generate mixed results. These results may be driven by
the difficulty of distinguishing a stationary, but persistent,
variable from a non-stationary one. In this paper, we take the
stationarity of these variables as given.

Model 7 is the random walk model. The estimator in equation (9) becomes the Empirical Distribution Function (EDF) of the exchange rate innovations. Under regularity conditions, equation (9) consistently estimates the unconditional distribution of , and can be used to form forecast intervals for . The forecast intervals of economic models and the random walk are compared. Our interest is to test whether RS forecast intervals based on economic models are more accurate than those based on the random walk model. We focus on the empirical coverage and the length of forecast intervals in our tests.

Following Christoffersen (1998) and related work, the first standard we use is the empirical coverage. The empirical coverage should be as close as possible to the nominal coverage (). Significantly missing the nominal coverage indicates the inadequacy of the model and predictors for the given sample size. For instance, if 90% forecast intervals calculated from a model contain only 50% of out-of-sample observations, the model can hardly be identified as useful for interval forecasting. This case is called under-coverage. In contrast, over-coverage implies that the intervals could be reduced in length (or improved in tightness), but the forecast interval method and model are unable to do that for the given sample size. An economic model is said to outperform the random walk if its empirical coverage is more accurate than that of the random walk.

On the other hand, the empirical coverage of an economic model may be equally accurate as that of the random walk model, but the economic model has tighter forecast intervals than the random walk. We argue that the lengths of forecast intervals signify the informativeness of the intervals given that these intervals have equally accurate empirical coverages. In this case, the economic model is also considered to outperform the random walk in forecasting exchange rates. The empirical coverage and length tests are conducted at both short and long horizons for the six economic models relative to the random walk for each of the twelve OECD exchange rates.

We use tests that are applications of the unconditional
predictive accuracy inference framework of Giacomini and White
(2006). Unlike the tests of Diebold and Mariano (1995) and West
(1996), our forecast evaluation tests do not focus on the
asymptotic features of the forecasts. Rather, in the spirit of
Giacomini and White (2006), we are comparing the population
features of forecasts generated by rolling samples of fixed sample
size. This contrasts to the traditional forecast evaluation methods
in that although it uses asymptotic approximations to do the
testing, the inference is not on the asymptotic properties of
forecasts, but on their population *finite sample properties*.
We acknowledge that the philosophy of this inference framework
remains a point of contention, but it does tackle three important
evaluation difficulties in this paper. First, it allows for
evaluation of forecast intervals that are not parametrically
derived. The density evaluation methods developed in well-known
studies such as Diebold, Gunther, Tay (1998), Corradi and Swanson
(2006a) and references within Corradi and Swanson (2006b) require
that the forecast distributions be parametrically specified.
Giacomini and White's (2006) method overcomes this challenge by
allowing comparisons among parametric, semiparametric and
nonparametric forecasts. As a result, in the cases of
semiparametric and nonparametric forecasts, it also allows
comparison of models with predictors of different dimensions, as
evident in our exercise. Second, by comparing the finite sample
properties of RS forecast intervals derived from different models,
we avoid rejecting models that are mis-specified,^{14} but
are nonetheless good approximations useful for forecasting.
Finally, we can individually (though not jointly) test whether the
forecast intervals differ in terms of empirical coverages and
lengths, for the given estimation sample, and are not confined to
focus only on empirical coverages or holistic properties of
forecast distribution, such as probability integral transform.

Suppose the sample size available to the researcher is and all data are collected in a vector . Our inference procedure is based on a rolling estimation scheme, with the size of the rolling window fixed while . Let and be the size of the rolling window. For each horizon and model , a sequence of -coverage forecast intervals are generated using rolling data: for forecast for date , for forecast for date , and so on, until forecast for date is generated using .

Under this fixed-sample-size rolling scheme, for each finite we have observations to compare the empirical coverages and lengths across models ( ). By fixing , we allow the finite sample properties of the forecast intervals to be preserved as . Thus, the forecast intervals and the associated forecast losses are simply functions of a finite and fixed number of random variables. We are interested in approximating the population moments of these objects by taking . A loose analogy would be finding the finite-sample properties of a certain parameter estimator when the sample size is fixed at , by a bootstrap with an arbitrarily large number of bootstrap replications.

We conduct individual tests for the empirical coverages and
lengths. In each test, we define a corresponding forecast loss,
propose a test statistic and derive its asymptotic distribution. As
defined in equation (10),
let
be the
horizon ahead RS forecast interval of
model with a nominal coverage of . For out-of-sample forecast evaluation, we require
to be
constructed using data from
to . The
*coverage accuracy loss* is defined as:

(11) |

For economic models (), the goal is to compare the coverage accuracy loss of RS forecast intervals of model with that of the random walk (). The null and alternative hypotheses are:

Define the sample analog of the coverage accuracy loss in equation (11):

where is an index function that equals one when , and equals zero otherwise. Applying the asymptotic test of Giacomini and White (2006) to the sequence and applying the Delta method, we can show that

(12) |

where denotes convergence in distribution, and is the long-run covariance matrix between and . The matrix is defined as:

can be estimated consistently
by its sample analog
, while
can be estimated by some HAC
estimator
, such as Newey and
West (1987).^{15}The test statistic for coverage test
is defined as:

(13) |

Define the *length loss* as:

(14) |

where is the Lesbesgue measure. To compare the length loss of RS forecast intervals of economic models with that of the random walk (, the null and alternative hypotheses are:

The sample analog of the length loss for model is defined as:

Directly applying the test of Giacomini and White (2006), we have

(15) |

where is the long-run variance of . Let be the HAC estimator of . The test statistic for empirical length is defined as:

(16) |

The coverage accuracy loss function is symmetric in our paper.
In practice, an asymmetric loss function may be better when looking
for an exchange rate forecast model to help make policy or business
decisions. Under-coverage is arguably a more severe problem than
over-coverage in practical situations. However, the focus of this
paper is the disconnect between economic fundamentals and the
exchange rate. Our goal is to investigate which model comes closer
to the data: the random walk or fundamental-based models. It is not
critical in this case whether coverage inaccuracy comes from the
under- or over-coverage. We acknowledge that the use of symmetric
coverage loss remains a caveat, especially since we are using the
coverage accuracy test as a pre-test for the tests of length.
Clearly, there is a tradeoff between the empirical coverage and the
length of forecast intervals. Given the same center,^{16}
intervals with under-coverage have shorter lengths than intervals
with over-coverage. In this case, the length test is in favor of
models that systematically under-cover the targeted nominal
coverage when compared to a model that systematically over-covers.
This problem cannot be detected by the coverage accuracy test with
symmetric loss function because over- and under-coverage are
treated equally. However, our results in section 4 show that there
is no evidence of systematic under-coverage for the economic models
considered in this paper. For instance, in Table 1,
one-month-ahead () forecast intervals over-cover
the nominal coverage (90%) for eight out of twelve exchange
rates.^{17} Note that under-coverage does not
guarantee shorter intervals either in our paper, because forecast
intervals of different models usually have different
centers.^{18} In addition, we also compare the
coverage of economic models and the random walk directly in an
exercise not reported in this paper. There is no evidence that the
coverage of economic models is systematically smaller than that of
the random walk.^{19}

As we have mentioned, comparisons across models can also be done at the distribution level. We choose interval forecasts for two reasons. First, interval forecasts have been widely used and reported by the practitioners. For instance, the Bank of England calculates forecast intervals of inflation in its inflation reports. Second, compared to evaluation metrics for density forecasts, the empirical coverage and length loss functions of interval forecasts, and the subsequent interpretations of test rejection/acceptance are more intuitive.

We apply RS
forecast intervals for each model for a given nominal coverage of
. There is no particular reason
why we chose 0.9 as the nominal coverage. Some auxiliary results
show that our qualitative findings do not depend on the choice of
. Due to different sample sizes across
countries, we choose different sizes for the rolling window
() for different countries. Our rule is very
simple: for countries with , we choose
, otherwise we set .^{20} Again, from our experience,
tampering with does not change the qualitative
results, unless is chosen to be unusually big or
small.

For time horizons and models , we construct a sequence of 90% forecast intervals for the -horizon change of the exchange rate . Then we compare economic models and the random walk by computing empirical coverages, lengths and test statistics and as described in section 3. We first report the results of our benchmark model. After that, results of alternative models are reported and discussed.

Table 1 shows results of the benchmark Taylor rule model. For each time horizon and exchange rate, the first column (Cov.) reports the empirical coverage for the given nominal coverage of 90%. The second column (Leng.) reports the length of forecast intervals (the distance between upper and lower bounds). The length is multiplied by 100 and therefore expressed in terms of the percentage change of the exchange rate. For instance, the empirical coverage and length of the one-month-ahead forecast interval for the Australian dollar are and , respectively. It means that on average, with a chance of 89.5%, the one-month-ahead change of AUD/USD lies in an interval with length %. We use superscripts , , and to denote that the null hypothesis of equal empirical coverage accuracy/length is rejected in favor of the Taylor rule model at a confidence level of 10%, 5%, and 1% respectively. Superscripts , , and are used for rejections in favor of the random walk analogously.

We summarize our findings in three panels. In the first panel
(*(1) Coverage Test*), the row of "Model Better" reports the
number of exchange rates that the Taylor rule model has more
accurate empirical coverages than the random walk. The row of "RW
Better" reports the number of exchange rates for which the random
walk outperforms the Taylor rule model under the same criterion. In
the second panel (*(2) Length Test Given Equal Coverage
Accuracy*), a better model is the one with tighter forecast
intervals given equal coverage accuracy. In the last panel
(*(1)+(2)*), a better model is the one with either more
accurate coverages, or tighter forecast intervals given equal
coverage accuracy.

For most exchange rates and time horizons, the Taylor rule model and the random walk model have statistically equally accurate empirical coverages. The null hypothesis of equal coverage accuracy is rejected in only six out of sixty tests (two rejections each at horizons 6, 9, and 12). Five out of six rejections are in favor of the Taylor rule model. That is, the empirical coverage of the Taylor rule model is closer to the nominal coverage than those of the random walk. However, based on the number of rejections (5) in a total of sixty tests, there is no strong evidence that the Taylor rule model can generate more accurate empirical coverages than the random walk.

In cases where the Taylor rule model and the random walk have equally accurate empirical coverages, the Taylor rule model generally has equal or significantly tighter forecast intervals than the random walk. In forty-two out of fifty-four cases, the null hypothesis of equally tight forecast intervals is rejected in favor of the Taylor rule model. In contrast, the null hypothesis is rejected in only three cases in favor of the random walk. The evidence of exchange rate predictability is more pronounced at longer horizons. At horizon twelve (), for all cases where empirical coverage accuracies between the random walk and the Taylor rule model are statistically equivalent, the Taylor rule model has significantly tighter forecast intervals than the random walk.

As for each individual exchange rate, the benchmark Taylor rule model works best for the French Franc, the Deutschmark, the Dutch Guilder, the Swedish Krona, and the British Pound: for all time horizons, the model has tighter forecast intervals than the random walk, while their empirical coverages are statistically equally accurate. The Taylor rule model performs better than the random walk in most horizons for the remaining exchange rates except the Portuguese Escudo, for which the Taylor rule model outperforms the random walk only at long horizons.

Five alternative economic models are also compared with the random walk: three alternative Taylor rule models that are studied in Molodtsova and Papell (2009), the PPP model, and the monetary model. Tables 2-6 report results of these alternative models.

In general, results of coverage tests do not show strong evidence that economic models can generate more accurate coverages than the random walk at either short or long horizons. However, after considering length tests, we find that economic models perform better than the random walk, especially at long horizons. Taylor rule model 4 (the benchmark model with interest rate smoothing Table 4) and the PPP model (Table 5) perform the best among alternative models. Results of these two models are very similar to that of the benchmark Taylor rule model. At horizon twelve, both models outperform the random walk for most exchange rates under our out-of-sample forecast interval evaluation criteria. The performance of Taylor rule model 2 (Table 2) and 3 (Table 3) is relatively less impressive than other models, but for more than half of the exchange rates, the economic models outperform the random walk at several horizons in out-of-sample interval forecasts.

Comparing the benchmark Taylor rule model, the PPP model and the monetary model, the performance of the PPP model (Table 5) is worse than the other two models at short horizons. Compared to the Taylor rule and PPP models, the monetary model outperforms the random walk for a smaller number of exchange rates at horizons 6, 9, and 12. Overall, the benchmark Taylor rule model seems to perform slightly better than the monetary and PPP models. Molodtsova and Papell (2009) find similar results in their point forecasts.

Table 7 shows results
with heterogeneous Taylor rules.^{21} In this model, we
relaxed the assumption that the Taylor rule coefficients are the
same in the home and foreign countries. We replace
and
in matrix
of the benchmark model with
and
, where
,
,
, and
are Taylor rule
coefficients estimated from the data of home and foreign countries.
The main findings in the benchmark model also hold in Table 7.

After Mark (1995) first documents exchange rate predictability at long horizons, long-horizon exchange rate predictability has become a very active area in the literature. With panel data, Engel, Mark, and West (2007) recently show that the long-horizon predictability of the exchange rate is relatively robust in the exchange rate forecasting literature. We find similar results in our interval forecasts. The evidence of long-horizon predictability seems robust across different models and currencies when both empirical coverage and length tests are used. At horizon twelve, all economic models outperform the random walk for six exchange rates: the Australian Dollar, French Franc, Italian Lira, Japanese Yen, Swedish Krona, and the British Pound in the sense that interval lengths of economic models are smaller than those of the random walk, given equivalent coverage accuracy. This is true only for the Danish Kroner and Swiss Franc at horizon one. We also notice that there is no clear evidence of long-horizon predictability based on the tests of empirical coverage accuracy only.

Molodtsova and Papell (2009) find strong out-of-sample exchange
rate predictability for Taylor rule models even at the short
horizon. In our paper, the evidence for exchange rate
predictability at short horizons is not very strong. This finding
may be a result of some assumptions we have used to simplify our
computation. For instance, an -coverage
forecast interval in our paper is always constructed using the
and
quantiles. Alternatively, we
can choose quantiles that minimize the length of intervals, given
the nominal coverage.^{22} In addition, the development of more
powerful testing methods may also be helpful. The evidence of
exchange rate predictability in Molodtsova and Papell (2009) is
partly driven by the testing method recently developed by Clark and
West (2006, 2007). We acknowledge that whether or not short-horizon
results can be improved remains an interesting question, but do not
pursue this in the current paper. The purpose of this paper is to
show the connection between the exchange rate and economic
fundamentals from an interval forecasting perspective.
Predictability either at short- or long-horizons will serve this
purpose.

Though we find that economic fundamentals are helpful for
forecasting exchange rates, we acknowledge that exchange rate
forecasting in practice is still a difficult task. The forecast
intervals from economic models are statistically tighter than those
of the random walk, but they remain fairly wide. For instance, the
distance between the upper and lower bound of three-month-ahead
forecast intervals is usually a 20% change of the exchange rates.
Figures 1-3
show the length of forecast intervals generated by the benchmark
Taylor rule model and the random walk for the British Pound, the
Deutschmark, and the Japanese Yen at different horizons.^{23} At
the horizon of 12 months, the length of forecast intervals in the
Taylor rule model is usually smaller than that in the random walk.
However, at shorter horizons, such as 1 month, the difference is
quantitatively small.

There is a growing strand of literature that uses Taylor rules to model exchange rate movements. Our paper contributes to the literature by showing that Taylor rule fundamentals are useful in forecasting the distribution of exchange rates. We apply Robust Semiparametric forecast intervals of Wu (2009) to a group of Taylor rule models for twelve OECD exchange rates. The forecast intervals generated by the Taylor rule models are in general tighter than those of the random walk, given that these intervals cover the realized exchange rates equally well. The evidence of exchange rate predictability is more pronounced at longer horizons, a result that echoes previous long-horizon studies such as Mark (1995). The benchmark Taylor rule model is also found to perform better than the monetary and PPP models based on out-of-sample interval forecasts.

Though we find some empirical support for the connection between
the exchange rate and economic fundamentals, we acknowledge that
the detected connection is weak. The reductions of the lengths of
forecast intervals are quantitatively small, though they are
statistically significant. Forecasting exchange rates remains a
difficult task in practice. Engel and West (2005) argue that as the
discount factor gets closer to one, present value asset pricing
models place greater weight on future fundamentals. Consequently,
current fundamentals have very weak forecasting power and exchange
rates appear to follow approximately a random walk. Under standard
assumptions in Engel and West (2005), the Engel-West theorem does
not imply that exchange rates are more predictable at longer
horizons or that economic models can outperform the random walk in
forecasting exchange rates based on out-of-sample interval
forecasts. However, modifications to these assumptions may be able
to reconcile the Engel-West explanation with empirical findings of
exchange rate predictability. For instance, Engel, Wang, and Wu
(2008) find that when there exist stationary, but persistent,
unobservable fundamentals, for example risk premium, the Engel-West
explanation predicts long-horizon exchange rate predictability in
*point forecasts*, though the exchange rate still
approximately follows a random walk at short horizons. It would
also be of interest to study conditions under which our findings in
*interval forecasts* can be reconciled with the Engel-West
theorem.

We believe other issues, such as parameter instability (Rossi, 2005), nonlinearity (Kilian and Taylor, 2003), real time data (Faust, Rogers, and Wright, 2003, Molodtsova, Nikolsko-Rzhevskyy, and Papell, 2008a, 2008b), and model selection (Sarno and Valente, forthcoming) are all contributing to the Meese-Rogoff puzzle. Panel data are also found helpful in detecting exchange rate predictability, especially at long horizons. For instance, see Mark and Sul (2001), Engel, Mark, and West (2007), and Rogoff and Stavrakeva (2008). It would be interesting to incorporate these studies into interval forecasting. We leave these extensions for future research.

Table 1.-Panel 1: Results of Benchmark Taylor Rule Model

Cov. h=1 | Leng. h=1 | Cov. h=3 | Leng. h=3 | Cov. h=6 | Leng. h=6 | Cov. h=9 | Leng. h=9 | Cov. h=12 | Leng. h=12 | |
---|---|---|---|---|---|---|---|---|---|---|

Australian Dollar | 0.895 | 7.114 | 0.888 | 14.209^{c} | 0.959 | 21.140 | 0.942 | 26.613 | 0.963 | 29.175^{c} |

Canadian Dollar | 0.814 | 3.480 | 0.794 | 6.440^{c} | 0.738 | 8.483^{c} | 0.675 | 8.669^{c} | 0.596^{x} | 9.707^{c} |

Danish Kroner | 0.920 | 8.676^{c} | 0.939 | 17.415^{c} | 0.954 | 26.198 | 0.922 | 28.712^{c} | 0.968 | 37.123^{c} |

French Franc | 0.912 | 8.921^{c} | 0.860 | 15.728^{c} | 0.928^{c} | 26.007^{c} | 0.957 | 29.924^{c} | 0.934 | 36.883^{c} |

Deutschmark | 0.927 | 8.327^{c} | 0.879 | 18.634^{c} | 0.894 | 27.923^{c} | 0.960^{a} | 33.734^{c} | 0.969 | 39.618^{c} |

Italian Lira | 0.899 | 8.291^{c} | 0.875 | 18.305 | 0.910 | 26.788^{c} | 0.846 | 32.545^{c} | 0.874 | 37.151^{c} |

Japanese Yen | 0.915 | 9.633^{z} | 0.909 | 19.762 | 0.892 | 28.451^{c} | 0.932 | 33.793^{c} | 0.883 | 37.728^{c} |

Dutch Guilder | 0.917 | 8.726^{c} | 0.907 | 18.615^{c} | 0.933 | 27.458^{c} | 0.941^{b} | 30.902^{c} | 0.959^{a} | 40.177^{c} |

Portuguese Escudo | 0.901 | 8.580^{z} | 0.928 | 18.758^{z} | 0.894^{c} | 23.552^{c} | 0.825 | 27.086 | 0.867 | 32.092^{c} |

Swedish Krona | 0.839 | 7.360^{c} | 0.860 | 15.413^{c} | 0.874 | 23.930^{c} | 0.820 | 28.090^{c} | 0.834 | 37.432^{c} |

Swiss Franc | 0.947 | 9.358^{c} | 0.916 | 19.655 | 0.963 | 26.553^{c} | 0.963 | 30.780^{c} | 0.899 | 35.008^{c} |

British Pound | 0.919 | 8.413^{a} | 0.923 | 16.592^{c} | 0.912 | 23.317^{c} | 0.900 | 26.942^{c} | 0.855 | 25.905^{c} |

Note: –h denotes forecast horizons for monthly data. –For each horizon (h), the first column (Cov.) reports empirical coverages given a nominal coverage of 90%. The second column (Leng.) reports the length of forecast intervals in terms of percentage change of the exchange rate. Empirical coverages and lengths are averages across N(h) out-of-sample trials. –Superscripts a, b, c in the column of Cov. (Leng.) denote rejections of equal coverage accuracy (equal length) in favor of the economic model at a 10%, 5% and 1% confidence level respectively. Superscripts x, y, z are defined analogously for rejections in favor of the random walk

Table 1.-Panel 2: Results of Benchmark Taylor Rule Model, Coverage Test

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 0 | 0 | 2 | 2 | 1 |

RW Better | 0 | 0 | 0 | 0 | 1 |

Note: In this panel, a better model is the one with more accurate empirical coverages. RW is the abbreviation of Random Walk.

Table 1.-Panel 3: Results of Benchmark Taylor Rule Model, Length Test Given Equal Coverage Accuracy

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 8 | 8 | 8 | 8 | 10 |

RW Better | 2 | 1 | 0 | 0 | 0 |

Note: In this panel, a better model is the one with tighter forecast intervals given equal coverage accuracy.

Table 1.-Panel 4: Results of Benchmark Taylor Rule Model, Coverage Test and Length Test Given Equal Coverage Accuracy

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 8 | 8 | 10 | 10 | 11 |

RW Better | 2 | 1 | 0 | 0 | 1 |

Note: In this panel, a better model is the one with either more accurate coverages or tighter forecast intervals given equal coverage accuracy.

Table 2.-Panel 1: Results of Benchmark Taylor Rule Model Two

Cov. h=1 | Leng. h=1 | Cov. h=3 | Leng. h=3 | Cov. h=6 | Leng. h=6 | Cov. h=9 | Leng. h=9 | Cov. h=12 | Leng. h=12 | |
---|---|---|---|---|---|---|---|---|---|---|

Australian Dollar | 0.884 | 7.146^{y} | 0.899 | 15.086^{c} | 0.928 | 21.327 | 0.901 | 27.329 | 0.872 | 30.815^{b} |

Canadian Dollar | 0.825 | 3.442^{c} | 0.783 | 6.321^{c} | 0.814 | 8.490^{c} | 0.858 | 10.034^{c} | 0.825 | 11.921^{c} |

Danish Kroner | 0.915 | 8.753^{a} | 0.939 | 17.764^{c} | 0.954 | 27.479^{z} | 0.953 | 33.426^{y} | 0.942 | 40.717 |

French Franc | 0.951 | 9.042^{c} | 0.930 | 18.783 | 0.949^{c} | 29.161^{c} | 0.936 | 34.994^{c} | 0.868 | 42.081^{c} |

Deutschmark | 0.917 | 9.090 | 0.869 | 19.217 | 0.952 | 29.746 | 0.941^{a} | 39.093^{a} | 0.980 | 44.571^{z} |

Italian Lira | 0.928 | 9.196 | 0.875 | 18.322 | 0.895 | 26.926^{c} | 0.869 | 35.883^{c} | 0.898^{a} | 41.235^{z} |

Japanese Yen | 0.915 | 9.568^{x} | 0.914 | 19.734 | 0.912 | 29.344^{c} | 0.937 | 36.834 | 0.942 | 44.385^{c} |

Dutch Guilder | 0.908 | 8.586^{c} | 0.888 | 18.782^{c} | 0.962 | 29.777 | 0.990 | 39.507^{z} | 0.990 | 47.514^{z} |

Portuguese Escudo | 0.916 | 8.005 | 0.957 | 17.924^{z} | 0.909^{c} | 24.270^{b} | 0.889 | 28.533^{z} | 0.883^{a} | 35.338^{c} |

Swedish Krona | 0.867 | 7.624 | 0.860 | 16.132 | 0.857 | 24.500^{c} | 0.837 | 32.825 | 0.811 | 37.772^{c} |

Swiss Franc | 0.941 | 9.953^{b} | 0.928 | 20.105 | 0.982 | 29.758^{c} | 0.994 | 38.267^{z} | 0.962 | 45.965^{z} |

British Pound | 0.919 | 8.627^{z} | 0.933 | 17.334^{c} | 0.922 | 26.227^{c} | 0.937 | 31.044^{c} | 0.957 | 36.397^{c} |

Note: –h denotes forecast horizons for monthly data. –For each horizon (h), the first column (Cov.) reports empirical coverages given a nominal coverage of 90%. The second column (Leng.) reports the length of forecast intervals in terms of percentage change of the exchange rate. Empirical coverages and lengths are averages across N(h) out-of-sample trials. –Superscripts a, b, c in the column of Cov. (Leng.) denote rejections of equal coverage accuracy (equal length) in favor of the economic model at a 10%, 5% and 1% confidence level respectively. Superscripts x, y, z are defined analogously for rejections in favor of the random walk.

Table 2.-Panel 2: Results of Benchmark Taylor Rule Model Two, Coverage Test

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 0 | 0 | 2 | 1 | 2 |

RW Better | 0 | 0 | 0 | 0 | 0 |

Note: In this panel, a better model is the one with more accurate empirical coverages. RW is the abbreviation of Random Walk.

Table 2.-Panel 3: Results of Benchmark Taylor Rule Model Two, Length Test Given Equal Coverage Accuracy

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 5 | 5 | 6 | 4 | 6 |

RW Better | 3 | 1 | 1 | 4 | 3 |

Note: In this panel, a better model is the one with tighter forecast intervals given equal coverage accuracy.

Table 2.-Panel 4: Results of Benchmark Taylor Rule Model Two, Coverage Test and Length Test Given Equal Coverage Accuracy

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 5 | 5 | 8 | 5 | 8 |

RW Better | 3 | 1 | 1 | 4 | 3 |

Note: In this panel, a better model is the one with either more accurate coverages or tighter forecast intervals given equal coverage accuracy.

Table 3.-Panel 1: Results of Benchmark Taylor Rule Model Three

Cov. h=1 | Leng. h=1 | Cov. h=3 | Leng. h=3 | Cov. h=6 | Leng. h=6 | Cov. h=9 | Leng. h=9 | Cov. h=12 | Leng. h=12 | |
---|---|---|---|---|---|---|---|---|---|---|

Australian Dollar | 0.884 | 7.229^{z} | 0.899 | 15.055^{c} | 0.881 | 21.055 | 0.885 | 26.359^{c} | 0.867 | 30.234^{c} |

Canadian Dollar | 0.831 | 3.453^{b} | 0.789 | 6.408^{c} | 0.814 | 8.629^{c} | 0.864 | 10.220^{c} | 0.819 | 11.971^{c} |

Danish Kroner | 0.920 | 8.753^{b} | 0.934 | 17.649^{c} | 0.949 | 27.523^{z} | 0.948 | 33.307 | 0.936 | 40.070 |

French Franc | 0.951 | 9.171 | 0.740 | 14.488^{c} | 0.722 | 20.313^{c} | 0.915 | 35.562^{a} | 0.813 | 41.350^{c} |

Deutschmark | 0.908 | 9.020 | 0.897 | 19.303 | 0.914 | 29.676 | 0.901^{c} | 37.291^{c} | 0.878 | 44.761^{z} |

Italian Lira | 0.928 | 8.900^{a} | 0.875 | 17.206^{c} | 0.872 | 26.674^{c} | 0.839 | 34.819^{c} | 0.787 | 39.569^{c} |

Japanese Yen | 0.905 | 9.179^{c} | 0.878 | 18.907^{c} | 0.892 | 25.883^{c} | 0.927 | 31.259^{c} | 0.894 | 37.049^{c} |

Dutch Guilder | 0.927 | 8.910 | 0.907 | 19.204^{a} | 0.952 | 29.426^{a} | 0.951^{a} | 36.896^{c} | 0.959 | 46.321^{z} |

Portuguese Escudo | 0.930 | 7.961 | 0.942 | 16.883^{c} | 0.955^{a} | 23.786 | 0.905 | 26.620^{c} | 0.850 | 33.745^{c} |

Swedish Krona | 0.867 | 7.316^{c} | 0.848 | 15.017^{c} | 0.840 | 23.241^{c} | 0.791 | 29.265^{c} | 0.757 | 33.751^{c} |

Swiss Franc | 0.929 | 9.761^{b} | 0.922 | 19.517^{c} | 0.939 | 28.437^{b} | 0.926^{c} | 37.519 | 0.911 | 45.619 |

British Pound | 0.929 | 8.239^{a} | 0.939 | 16.213^{c} | 0.927 | 23.951^{c} | 0.905 | 28.720^{c} | 0.952 | 34.900^{c} |

Note: –h denotes forecast horizons for monthly data. –For each horizon (h), the first column (Cov.) reports empirical coverages given a nominal coverage of 90%. The second column (Leng.) reports the length of forecast intervals in terms of percentage change of the exchange rate. Empirical coverages and lengths are averages across N(h) out-of-sample trials. –Superscripts a, b, c in the column of Cov. (Leng.) denote rejections of equal coverage accuracy (equal length) in favor of the economic model at a 10%, 5% and 1% confidence level respectively. Superscripts x, y, z are defined analogously for rejections in favor of the random walk.

Table 3.-Panel 2: Results of Benchmark Taylor Rule Model Three, Coverage Test

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 0 | 0 | 1 | 3 | 0 |

RW Better | 0 | 0 | 0 | 0 | 0 |

Note: In this panel, a better model is the one with more accurate empirical coverages. RW is the abbreviation of Random Walk.

Table 3.-Panel 3: Results of Benchmark Taylor Rule Model Three, Length Test Given Equal Coverage Accuracy

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 7 | 11 | 8 | 8 | 8 |

RW Better | 1 | 0 | 1 | 0 | 2 |

Note: In this panel, a better model is the one with tighter forecast intervals given equal coverage accuracy.

Table 3.-Panel 4: Results of Benchmark Taylor Rule Model Three, Coverage Test and Length Test Given Equal Coverage Accuracy

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 7 | 11 | 9 | 11 | 8 |

RW Better | 1 | 0 | 1 | 0 | 2 |

Note: In this panel, a better model is the one with either more accurate coverages or tighter forecast intervals given equal coverage accuracy.

Table 4.-Panel 1: Results of Benchmark Taylor Rule Model Four

Cov. h=1 | Leng. h=1 | Cov. h=3 | Leng. h=3 | Cov. h=6 | Leng. h=6 | Cov. h=9 | Leng. h=9 | Cov. h=12 | Leng. h=12 | |
---|---|---|---|---|---|---|---|---|---|---|

Australian Dollar | 0.895 | 7.119 | 0.888 | 14.424^{c} | 0.928 | 20.966 | 0.927 | 25.304^{c} | 0.872 | 27.492^{c} |

Canadian Dollar | 0.814 | 3.425^{c} | 0.771 | 6.366^{c} | 0.698 | 8.019^{c} | 0.651 | 8.631^{c} | 0.494y | 7.693^{c} |

Danish Kroner | 0.920 | 8.703^{c} | 0.929 | 17.536^{c} | 0.964 | 26.025 | 0.984^{x} | 30.891^{c} | 0.963 | 36.545^{c} |

French Franc | 0.892 | 8.361^{c} | 0.870 | 16.079^{c} | 0.938^{c} | 25.950^{c} | 0.883 | 30.016^{c} | 0.791 | 35.755^{c} |

Deutschmark | 0.927 | 8.314^{c} | 0.879 | 18.652^{c} | 0.894 | 26.803^{c} | 0.931^{c} | 33.350^{c} | 0.969 | 36.393^{c} |

Italian Lira | 0.891 | 8.663^{c} | 0.838 | 17.575^{c} | 0.865 | 26.387^{c} | 0.746 | 32.270^{c} | 0.724 | 36.422^{c} |

Japanese Yen | 0.905 | 9.157^{c} | 0.863 | 18.708^{c} | 0.866 | 24.417^{c} | 0.869 | 28.730^{c} | 0.851 | 31.470^{c} |

Dutch Guilder | 0.936 | 8.815 | 0.897 | 18.368^{c} | 0.914 | 26.700^{c} | 0.931^{c} | 30.036^{c} | 0.796 | 29.462^{c} |

Portuguese Escudo | 0.901 | 8.525^{z} | 0.913^{a} | 17.110 | 0.939^{c} | 23.461^{c} | 0.889 | 27.096^{a} | 0.917 | 28.778^{c} |

Swedish Krona | 0.861 | 7.289^{c} | 0.860 | 15.321^{c} | 0.869 | 23.340^{c} | 0.773 | 27.198^{c} | 0.728 | 31.843^{c} |

Swiss Franc | 0.947 | 9.149^{c} | 0.940 | 19.782^{a} | 0.811 | 22.796^{c} | 0.808 | 26.148^{c} | 0.671 | 26.683^{c} |

British Pound | 0.919 | 8.113^{a} | 0.913 | 15.765^{c} | 0.875 | 21.679 | 0.825 | 27.312^{c} | 0.839 | 29.081^{c} |

Note: –h denotes forecast horizons for monthly data. –For each horizon (h), the first column (Cov.) reports empirical coverages given a nominal coverage of 90%. The second column (Leng.) reports the length of forecast intervals in terms of percentage change of the exchange rate. Empirical coverages and lengths are averages across N(h) out-of-sample trials. –Superscripts a, b, c in the column of Cov. (Leng.) denote rejections of equal coverage accuracy (equal length) in favor of the economic model at a 10%, 5% and 1% confidence level respectively. Superscripts x, y, z are defined analogously for rejections in favor of the random walk.

Table 4.-Panel 2: Results of Benchmark Taylor Rule Model Four, Coverage Test

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 0 | 1 | 2 | 2 | 0 |

RW Better | 0 | 0 | 0 | 1 | 1 |

Table 4.-Panel 3: Results of Benchmark Taylor Rule Model Four, Length Test Given Equal Coverage Accuracy

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 9 | 11 | 8 | 9 | 11 |

RW Better | 1 | 0 | 0 | 0 | 0 |

Table 4.-Panel 4: Results of Benchmark Taylor Rule Model Four, Coverage Test and Length Test Given Equal Coverage Accuracy

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 9 | 12 | 10 | 11 | 11 |

RW Better | 1 | 0 | 0 | 1 | 1 |

Table 5.-Panel 1: Results of Purchasing Power Parity Model

Cov. h=1 | Leng. h=1 | Cov. h=3 | Leng. h=3 | Cov. h=6 | Leng. h=6 | Cov. h=9 | Leng. h=9 | Cov. h=12 | Leng. h=12 | |
---|---|---|---|---|---|---|---|---|---|---|

Australian Dollar | 0.895 | 7.114^{z} | 0.883 | 15.558 | 0.912 | 21.311 | 0.880 | 26.120^{c} | 0.856 | 30.316^{c} |

Canadian Dollar | 0.819 | 3.570^{z} | 0.806 | 6.872 | 0.767 | 9.615 | 0.728 | 11.078^{c} | 0.615 | 12.306^{c} |

Danish Kroner | 0.925 | 8.697^{c} | 0.939 | 18.333 | 0.938 | 25.887^{c} | 0.937 | 31.673^{c} | 0.957 | 37.447^{c} |

French Franc | 0.922 | 8.904^{c} | 0.940 | 18.029^{c} | 0.918^{c} | 25.786^{c} | 0.904 | 29.789^{c} | 0.802 | 34.209^{c} |

Deutschmark | 0.936 | 9.079 | 0.935 | 18.797^{c} | 0.942 | 27.677^{c} | 1.000 | 33.585^{c} | 0.990 | 40.570^{c} |

Italian Lira | 0.913 | 8.780^{c} | 0.868 | 17.767^{c} | 0.827 | 25.044^{c} | 0.769 | 30.190^{c} | 0.772 | 34.806^{c} |

Japanese Yen | 0.920 | 9.662^{z} | 0.899 | 19.903 | 0.912 | 28.689^{c} | 0.932 | 33.973^{c} | 0.899 | 38.568^{c} |

Dutch Guilder | 0.936 | 8.862^{y} | 0.935 | 18.904^{c} | 0.952 | 27.928^{c} | 1.000 | 33.468^{c} | 0.990 | 41.812^{c} |

Portuguese Escudo | 0.916 | 8.421^{y} | 0.928 | 19.027^{y} | 0.924^{c} | 23.918 | 0.857 | 27.450 | 0.867 | 32.467^{c} |

Swedish Krona | 0.861 | 7.541^{c} | 0.876 | 16.089 | 0.886 | 24.345^{c} | 0.855 | 31.744^{b} | 0.799 | 37.943^{c} |

Swiss Franc | 0.941 | 9.708^{c} | 0.946 | 19.694^{b} | 0.976 | 27.197^{c} | 0.950^{b} | 31.725^{c} | 0.880 | 36.235^{c} |

British Pound | 0.934 | 8.571^{y} | 0.933 | 16.954^{c} | 0.932 | 24.064^{c} | 0.947 | 28.761^{c} | 0.925^{a} | 31.372^{c} |

Table 5.-Panel 2: Results of Purchasing Power Parity Model, Coverage Test

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 0 | 0 | 2 | 1 | 1 |

RW Better | 0 | 0 | 0 | 0 | 0 |

Table 5.-Panel 3: Results of Purchasing Power Parity Model, Length Test Given Equal Coverage Accuracy

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 5 | 6 | 8 | 10 | 11 |

RW Better | 6 | 1 | 0 | 0 | 0 |

Table 5.-Panel 4: Results of Purchasing Power Parity Model, Coverage Test and Length Test Given Equal Coverage Accuracy

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 5 | 6 | 10 | 11 | 12 |

RW Better | 6 | 1 | 0 | 0 | 0 |

Table 6.-Panel 1: Results of Monetary Model

Cov. h=1 | Leng. h=1 | Cov. h=3 | Leng. h=3 | Cov. h=6 | Leng. h=6 | Cov. h=9 | Leng. h=9 | Cov. h=12 | Leng. h=12 | |
---|---|---|---|---|---|---|---|---|---|---|

Australian Dollar | 0.879 | 7.108 | 0.848 | 15.151 | 0.830 | 20.090 | 0.770 | 24.642^{c} | 0.745 | 30.099^{b} |

Canadian Dollar | 0.842 | 4.027^{x} | 0.829 | 7.492 | 0.744 | 10.518 | 0.645 | 10.689^{b} | 0.675 | 12.993^{c} |

Danish Kroner | 0.905 | 8.770^{b} | 0.893 | 17.943 | 0.897 | 25.017^{c} | 0.853 | 28.581^{c} | 0.809 | 32.504^{c} |

French Franc | 0.922 | 8.791^{c} | 0.910 | 18.237^{c} | 0.949^{b} | 26.322^{c} | 0.957 | 31.032^{c} | 0.956 | 35.971^{c} |

Deutschmark | 0.908 | 8.595 | 0.841 | 17.436^{c} | 0.808 | 24.622^{c} | 0.772 | 28.052^{c} | 0.704 | 31.364^{c} |

Italian Lira | 0.913 | 8.858^{c} | 0.882 | 18.439^{b} | 0.925 | 26.585^{c} | 0.931 | 34.857^{c} | 0.913 | 40.885^{c} |

Japanese Yen | 0.930 | 9.556 | 0.919 | 19.374^{c} | 0.887 | 28.614^{c} | 0.864 | 33.401^{c} | 0.809 | 36.520^{c} |

Dutch Guilder | 0.917 | 8.753^{a} | 0.916 | 19.408 | 0.962 | 29.149^{b} | 0.970 | 38.173 | 0.898^{c} | 41.716^{c} |

Portuguese Escudo | 0.901 | 8.086 | 0.986 | 18.484 | 0.985 | 24.744 | 0.984 | 27.230 | 1.000 | 34.222 |

Swedish Krona | 0.850 | 7.504^{a} | 0.848 | 17.097^{x} | 0.811 | 23.878^{c} | 0.826 | 31.287 | 0.805 | 34.710^{c} |

Swiss Franc | 0.905 | 9.078^{c} | 0.820 | 17.020^{c} | 0.732 | 21.212^{c} | 0.609 | 22.741^{c} | 0.513^{x} | 23.225^{c} |

British Pound | 0.909 | 7.811^{c} | 0.882 | 14.945^{c} | 0.787 | 20.788^{c} | 0.677 | 24.311^{c} | 0.656 | 26374^{c} |

Table 6.-Panel 2: Results of Monetary Model, Coverage Test

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 0 | 0 | 1 | 0 | 1 |

RW Better | 0 | 0 | 0 | 0 | 1 |

Table 6.-Panel 3: Results of Monetary Model, Length Test Given Equal Coverage Accuracy

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 7 | 6 | 8 | 9 | 9 |

RW Better | 1 | 1 | 0 | 0 | 0 |

Table 6.-Panel 4: Results of Monetary Model, Coverage Test and Length Test Given Equal Coverage Accuracy

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 7 | 6 | 9 | 9 | 10 |

RW Better | 1 | 1 | 0 | 0 | 1 |

Table 7.-Panel 1: Results of Heterogenous Taylor Rules

Cov. h=1 | Leng. h=1 | Cov. h=3 | Leng. h=3 | Cov. h=6 | Leng. h=6 | Cov. h=9 | Leng. h=9 | Cov. h=12 | Leng. h=12 | |
---|---|---|---|---|---|---|---|---|---|---|

Australian Dollar | 0.915 | 7.155 | 0.909 | 14.690^{c} | 0.959 | 20.547 | 0.963^{y} | 26.364 | 0.947 | 29.583^{c} |

Canadian Dollar | 0.825 | 3.526 | 0.794 | 6.525^{c} | 0.797 | 9.040^{c} | 0.787 | 10.370^{c} | 0.693 | 11.352^{c} |

Danish Kroner | 0.915 | 8.548^{c} | 0.929 | 17.930^{c} | 0.938 | 25.328^{c} | 0.890 | 30.504^{c} | 0.904 | 36.624^{c} |

French Franc | 0.912 | 8.864^{c} | 0.880 | 15.970^{c} | 0.845 | 20.693^{c} | 0.968 | 30.436^{c} | 0.714 | 22.789^{c} |

Deutschmark | 0.917 | 8.605^{c} | 0.907 | 18.356^{c} | 0.894 | 28.121^{c} | 0.911^{c} | 31.378^{c} | 0.939 | 33.779^{c} |

Italian Lira | 0.913 | 8.659^{c} | 0.890 | 18.664 | 0.887 | 25.840^{c} | 0.831 | 32.037^{c} | 0.693 | 32.236^{c} |

Japanese Yen | 0.920 | 9.637^{z} | 0.888 | 19.352^{b} | 0.871 | 28.018^{c} | 0.932 | 33.388^{c} | 0.878 | 36.859^{c} |

Dutch Guilder | 0.936 | 8.851 | 0.916 | 18.822^{c} | 0.942 | 27.259^{c} | 0.970 | 31.410^{c} | 0.990 | 39.882^{c} |

Portuguese Escudo | 0.916 | 8.881^{z} | 0.870 | 17.651 | 0.758 | 18.730^{c} | 0.746 | 23.852^{c} | 0.600 | 20.593^{c} |

Swedish Krona | 0.828 | 7.428^{c} | 0.854 | 15.658^{c} | 0.903 | 24.315^{c} | 0.861 | 29.866^{c} | 0.876 | 36.235^{c} |

Swiss Franc | 0.935 | 9.731^{c} | 0.940 | 19.797 | 0.970 | 27.227^{c} | 0.969 | 32.179^{c} | 0.937 | 36.567^{c} |

British Pound | 0.919 | 8.350 | 0.908 | 16.774^{c} | 0.828 | 20.811^{c} | 0.783 | 23.105^{c} | 0.720 | 23.286^{c} |

Table 7.-Panel 2: Results of Heterogenous Taylor Rules, Coverage Test

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 0 | 0 | 0 | 1 | 0 |

RW Better | 0 | 0 | 0 | 1 | 0 |

Table 7.-Panel 3: Results of Heterogenous Taylor Rules, Length Test Given Equal Coverage Accuracy

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 6 | 9 | 11 | 10 | 12 |

RW Better | 2 | 0 | 0 | 0 | 0 |

Table 7.-Panel 4: Results of Heterogenous Taylor Rules, Coverage Test and Length Test Given Equal Coverage Accuracy

h=1 | h=3 | h=6 | h=9 | h=12 | |
---|---|---|---|---|---|

Model Better | 6 | 9 | 11 | 11 | 12 |

RW Better | 2 | 0 | 0 | 1 | 0 |

Figure 1a. Length of Forecast Intervals for Benchmark Taylor Rule and Random Walk Models (British Pound), 1-Month-Ahead Forecast

Note: To facilitate graphical comparisons, the 6- and 12-month-ahead forecast intervals of the random walk have been relocated such that they have the same center as the intervals of the Taylor rule model.

Data for Figure 1a

Date | Taylor Rule | RW |
---|---|---|

1989M12 | 1.0000 | 1.0190 |

1990M1 | 0.9848 | 1.0019 |

1990M2 | 0.9543 | 0.9714 |

1990M3 | 0.9257 | 0.9448 |

1990M4 | 0.9695 | 0.9886 |

1990M5 | 0.9600 | 0.9810 |

1990M6 | 0.9390 | 0.9562 |

1990M7 | 0.9238 | 0.9390 |

1990M8 | 0.9124 | 0.8876 |

1990M9 | 0.8743 | 0.8495 |

1990M10 | 0.8838 | 0.8610 |

1990M11 | 0.8495 | 0.8305 |

1990M12 | 0.8400 | 0.8229 |

1991M1 | 0.8629 | 0.8400 |

1991M2 | 0.8495 | 0.8362 |

1991M3 | 0.8400 | 0.8229 |

1991M4 | 0.9086 | 0.8952 |

1991M5 | 0.9467 | 0.9314 |

1991M6 | 0.9524 | 0.9467 |

1991M7 | 1.0019 | 0.9905 |

1991M8 | 1.0019 | 0.9886 |

1991M9 | 0.9848 | 0.9695 |

1991M10 | 0.9676 | 0.9467 |

1991M11 | 0.9714 | 0.9467 |

1991M12 | 0.9486 | 0.9162 |

1992M1 | 0.9143 | 0.8933 |

1992M2 | 0.9257 | 0.9029 |

1992M3 | 0.9352 | 0.9181 |

1992M4 | 0.9638 | 0.9486 |

1992M5 | 0.9467 | 0.9295 |

1992M6 | 0.9257 | 0.9029 |

1992M7 | 0.9010 | 0.8800 |

1992M8 | 0.8743 | 0.8514 |

1992M9 | 0.8648 | 0.8400 |

1992M10 | 0.9200 | 0.8895 |

1992M11 | 1.0152 | 1.0190 |

1992M12 | 1.0648 | 1.1029 |

1993M1 | 1.0800 | 1.0857 |

1993M2 | 1.0590 | 1.0971 |

1993M3 | 1.1448 | 1.2019 |

1993M4 | 1.1314 | 1.1829 |

1993M5 | 1.0781 | 1.1314 |

1993M6 | 1.0743 | 1.0971 |

1993M7 | 1.1029 | 1.1257 |

1993M8 | 1.0914 | 1.1352 |

1993M9 | 1.0933 | 1.1390 |

1993M10 | 1.0895 | 1.1143 |

1993M11 | 1.0895 | 1.1314 |

1993M12 | 1.1048 | 1.1486 |

1994M1 | 1.0971 | 1.1390 |

1994M2 | 1.1010 | 1.1371 |

1994M3 | 1.1086 | 1.1486 |

1994M4 | 1.0990 | 1.1390 |

1994M5 | 1.1010 | 1.1467 |

1994M6 | 1.0990 | 1.1295 |

1994M7 | 1.0895 | 1.1124 |

1994M8 | 1.0781 | 1.0990 |

1994M9 | 1.0571 | 1.0914 |

1994M10 | 1.0419 | 1.0743 |

1994M11 | 1.0362 | 1.0476 |

1994M12 | 1.0324 | 1.0590 |

1995M1 | 1.0419 | 1.0800 |

1995M2 | 1.0362 | 1.0686 |

1995M3 | 1.0324 | 1.0705 |

1995M4 | 1.0133 | 1.0514 |

1995M5 | 1.0324 | 1.0476 |

1995M6 | 1.0171 | 1.0610 |

1995M7 | 1.0152 | 1.0533 |

1995M8 | 1.0171 | 1.0552 |

1995M9 | 1.0305 | 1.0743 |

1995M10 | 1.0381 | 1.0781 |

1995M11 | 1.0324 | 1.0648 |

1995M12 | 1.0324 | 1.0762 |

1996M1 | 1.0400 | 1.0933 |

1996M2 | 1.0552 | 1.1010 |

1996M3 | 1.0438 | 1.0876 |

1996M4 | 1.0495 | 1.0952 |

1996M5 | 1.0324 | 1.1029 |

1996M6 | 1.0324 | 1.1048 |

1996M7 | 1.0438 | 1.0857 |

1996M8 | 1.0381 | 1.0781 |

1996M9 | 1.0305 | 1.0781 |

1996M10 | 1.0248 | 1.0724 |

1996M11 | 1.0133 | 1.0533 |

1996M12 | 1.0019 | 1.0114 |

1997M1 | 0.9981 | 1.0114 |

1997M2 | 0.9733 | 1.0152 |

1997M3 | 0.9943 | 1.0362 |

1997M4 | 1.0038 | 1.0457 |

1997M5 | 0.9905 | 1.0324 |

1997M6 | 0.9981 | 1.0305 |

1997M7 | 0.9924 | 1.0229 |

1997M8 | 0.9810 | 1.0076 |

1997M9 | 1.0305 | 1.0476 |

1997M10 | 1.0324 | 1.0248 |

1997M11 | 1.0152 | 1.0038 |

1997M12 | 0.9962 | 0.9714 |

1998M1 | 1.0095 | 0.9886 |

1998M2 | 1.0095 | 0.9981 |

1998M3 | 0.9962 | 0.9924 |

1998M4 | 0.9790 | 0.9810 |

1998M5 | 0.9695 | 0.9752 |

1998M6 | 1.0095 | 0.9962 |

1998M7 | 1.0038 | 0.9886 |

1998M8 | 0.9981 | 0.9924 |

1998M9 | 1.0038 | 0.9981 |

1998M10 | 0.9771 | 0.9695 |

1998M11 | 0.9657 | 0.9619 |

1998M12 | 0.9848 | 0.9829 |

1999M1 | 0.9771 | 0.9752 |

1999M2 | 0.9867 | 0.9867 |

1999M3 | 0.9962 | 1.0019 |

1999M4 | 0.9962 | 1.0057 |

1999M5 | 1.0038 | 1.0133 |

1999M6 | 0.9962 | 1.0095 |

1999M7 | 1.0114 | 1.0229 |

1999M8 | 1.0286 | 1.0343 |

1999M9 | 1.0000 | 1.0152 |

1999M10 | 0.9886 | 1.0038 |

1999M11 | 0.9752 | 0.9848 |

1999M12 | 0.9924 | 1.0057 |

2000M1 | 0.9962 | 1.0114 |

2000M2 | 0.9867 | 0.9943 |

2000M3 | 1.0076 | 1.0190 |

2000M4 | 1.0210 | 1.0324 |

2000M5 | 1.0114 | 1.0305 |

2000M6 | 1.0610 | 1.0819 |

2000M7 | 1.0667 | 1.0819 |

2000M8 | 1.0705 | 1.0838 |

2000M9 | 1.0781 | 1.0971 |

2000M10 | 1.1219 | 1.1390 |

2000M11 | 1.1105 | 1.1257 |

2000M12 | 1.1219 | 1.1467 |

2001M1 | 1.0914 | 1.1162 |

2001M2 | 1.0819 | 1.1067 |

2001M3 | 1.0990 | 1.1257 |

2001M4 | 1.0876 | 1.1295 |

2001M5 | 1.0876 | 1.1371 |

2001M6 | 1.0952 | 1.1429 |

2001M7 | 1.1124 | 1.1638 |

2001M8 | 1.0857 | 1.1429 |

2001M9 | 1.0610 | 1.1105 |

2001M10 | 1.0533 | 1.0914 |

2001M11 | 1.0610 | 1.1010 |

2001M12 | 1.0133 | 1.1067 |

2002M1 | 1.0019 | 1.1029 |

2002M2 | 1.0133 | 1.1086 |

2002M3 | 1.0057 | 1.1143 |

2002M4 | 1.0038 | 1.1143 |

2002M5 | 0.9905 | 1.0990 |

2002M6 | 0.9752 | 1.0648 |

2002M7 | 0.9619 | 1.0476 |

2002M8 | 0.9867 | 1.0190 |

2002M9 | 1.0000 | 1.0343 |

2002M10 | 0.9867 | 1.0210 |

2002M11 | 0.9886 | 1.0210 |

2002M12 | 0.9771 | 1.0095 |

2003M1 | 0.9676 | 1.0000 |

2003M2 | 0.9581 | 0.9810 |

2003M3 | 0.9619 | 0.9886 |

2003M4 | 0.9829 | 1.0038 |

2003M5 | 0.9829 | 1.0095 |

2003M6 | 0.9314 | 0.9790 |

2003M7 | 0.9124 | 0.9543 |

2003M8 | 0.9257 | 0.9790 |

2003M9 | 0.9486 | 0.9752 |

2003M10 | 0.9352 | 0.9600 |

2003M11 | 0.9086 | 0.9181 |

2003M12 | 0.9048 | 0.9124 |

2004M1 | 0.8990 | 0.8800 |

2004M2 | 0.8857 | 0.8495 |

2004M3 | 0.8533 | 0.8324 |

2004M4 | 0.8686 | 0.8514 |

2004M5 | 0.8762 | 0.8629 |

2004M6 | 0.8724 | 0.8705 |

2004M7 | 0.8552 | 0.8438 |

2004M8 | 0.8476 | 0.8362 |

2004M9 | 0.8590 | 0.8476 |

2004M10 | 0.8705 | 0.8610 |

2004M11 | 0.8438 | 0.8229 |

2004M12 | 0.8438 | 0.8000 |

2005M1 | 0.8267 | 0.7733 |

2005M2 | 0.8419 | 0.7848 |

2005M3 | 0.8343 | 0.7714 |

2005M4 | 0.8267 | 0.7638 |

2005M5 | 0.8267 | 0.7676 |

2005M6 | 0.8419 | 0.7848 |

2005M7 | 0.8514 | 0.8000 |

2005M8 | 0.8724 | 0.8305 |

2005M9 | 0.8438 | 0.8095 |

2005M10 | 0.8400 | 0.8057 |

2005M11 | 0.8552 | 0.8229 |

2005M12 | 0.8667 | 0.8362 |

2006M1 | 0.8419 | 0.8305 |

2006M2 | 0.8400 | 0.7943 |

2006M3 | 0.8305 | 0.7905 |

2006M4 | 0.8324 | 0.7924 |

Figure 1b. Length of Forecast Intervals for Benchmark Taylor Rule and Random Walk Models (British Pound), 6-month-ahead forecast

Note: To facilitate graphical comparisons, the 6- and 12-month-ahead forecast intervals of the random walk have been relocated such that they have the same center as the intervals of the Taylor rule model.

Data for Figure 1b

Date | Taylor Rule | RW |
---|---|---|

1990M5 | 1.0000 | 1.0687 |

1990M6 | 0.9804 | 1.0528 |

1990M7 | 0.9442 | 1.0184 |

1990M8 | 0.9282 | 0.9914 |

1990M9 | 0.9515 | 1.0350 |

1990M10 | 0.9902 | 1.0270 |

1990M11 | 0.9368 | 1.0018 |

1990M12 | 0.9080 | 0.9822 |

1991M1 | 0.8564 | 0.9288 |

1991M2 | 0.8583 | 0.8840 |

1991M3 | 0.8650 | 0.9000 |

1991M4 | 0.8509 | 0.8730 |

1991M5 | 0.8479 | 0.8933 |

1991M6 | 0.8540 | 0.9135 |

1991M7 | 0.8362 | 0.9074 |

1991M8 | 0.8337 | 0.8939 |

1991M9 | 0.8920 | 0.9638 |

1991M10 | 0.9497 | 1.0031 |

1991M11 | 0.9509 | 1.0184 |

1991M12 | 1.0104 | 1.0656 |

1992M1 | 1.0227 | 1.0798 |

1992M2 | 0.9896 | 1.0595 |

1992M3 | 0.9785 | 1.0331 |

1992M4 | 0.9761 | 1.0350 |

1992M5 | 0.9429 | 1.0031 |

1992M6 | 0.9239 | 0.9767 |

1992M7 | 0.9368 | 0.9859 |

1992M8 | 0.9460 | 1.0037 |

1992M9 | 0.9779 | 1.0350 |

1992M10 | 0.9712 | 1.0153 |

1992M11 | 0.9577 | 0.9859 |

1992M12 | 0.9362 | 0.9613 |

1993M1 | 0.9025 | 0.9301 |

1993M2 | 0.8883 | 0.9178 |

1993M3 | 0.9135 | 0.9663 |

1993M4 | 0.9939 | 1.0798 |

1993M5 | 1.0546 | 1.1724 |

1993M6 | 1.0460 | 1.1736 |

1993M7 | 1.0577 | 1.1890 |

1993M8 | 1.0877 | 1.2840 |

1993M9 | 1.0656 | 1.2908 |

1993M10 | 1.0031 | 1.2209 |

1993M11 | 1.0160 | 1.2184 |

1993M12 | 1.0442 | 1.2503 |

1994M1 | 1.0460 | 1.2613 |

1994M2 | 1.0399 | 1.2650 |

1994M3 | 1.0086 | 1.2374 |

1994M4 | 1.0301 | 1.2558 |

1994M5 | 1.0313 | 1.2736 |

1994M6 | 1.0209 | 1.2644 |

1994M7 | 1.0319 | 1.2638 |

1994M8 | 1.0252 | 1.2755 |

1994M9 | 1.0270 | 1.2644 |

1994M10 | 1.0135 | 1.2724 |

1994M11 | 1.0123 | 1.2540 |

1994M12 | 1.0215 | 1.2362 |

1995M1 | 1.0129 | 1.2190 |

1995M2 | 1.0018 | 1.2233 |

1995M3 | 0.9859 | 1.2049 |

1995M4 | 0.9669 | 1.1742 |

1995M5 | 0.9595 | 1.1871 |

1995M6 | 0.9693 | 1.2098 |

1995M7 | 0.9896 | 1.1982 |

1995M8 | 0.9816 | 1.2000 |

1995M9 | 0.9687 | 1.1785 |

1995M10 | 0.9491 | 1.1736 |

1995M11 | 0.9656 | 1.1883 |

1995M12 | 0.9693 | 1.1828 |

1996M1 | 0.9693 | 1.1822 |

1996M2 | 0.9785 | 1.2037 |

1996M3 | 0.9761 | 1.2098 |

1996M4 | 0.9730 | 1.1951 |

1996M5 | 0.9613 | 1.2074 |

1996M6 | 0.9613 | 1.2245 |

1996M7 | 0.9871 | 1.2337 |

1996M8 | 0.9693 | 1.2276 |

1996M9 | 0.9736 | 1.2350 |

1996M10 | 0.9785 | 1.2442 |

1996M11 | 0.9834 | 1.2448 |

1996M12 | 0.9798 | 1.2239 |

1997M1 | 0.9834 | 1.2147 |

1997M2 | 0.9638 | 1.2166 |

1997M3 | 0.9583 | 1.2098 |

1997M4 | 0.9356 | 1.1896 |

1997M5 | 0.9067 | 1.1350 |

1997M6 | 0.9000 | 1.1337 |

1997M7 | 0.8969 | 1.1374 |

1997M8 | 0.9129 | 1.1601 |

1997M9 | 0.9294 | 1.1724 |

1997M10 | 0.9190 | 1.1577 |

1997M11 | 0.9233 | 1.1558 |

1997M12 | 0.9319 | 1.1466 |

1998M1 | 0.9209 | 1.1294 |

1998M2 | 0.9583 | 1.1528 |

1998M3 | 0.9730 | 1.1380 |

1998M4 | 0.9638 | 1.1147 |

1998M5 | 0.9393 | 1.0601 |

1998M6 | 0.9399 | 1.0748 |

1998M7 | 0.9724 | 1.0914 |

1998M8 | 0.9595 | 1.0871 |

1998M9 | 0.9460 | 1.0736 |

1998M10 | 0.9288 | 1.0669 |

1998M11 | 0.9706 | 1.0890 |

1998M12 | 0.9589 | 1.0810 |

1999M1 | 0.9460 | 1.0853 |

1999M2 | 0.9656 | 1.0920 |

1999M3 | 0.9337 | 1.0601 |

1999M4 | 0.9466 | 1.0528 |

1999M5 | 0.9521 | 1.0742 |

1999M6 | 0.9460 | 1.0675 |

1999M7 | 0.9669 | 1.0810 |

1999M8 | 0.9540 | 1.0963 |

1999M9 | 0.9466 | 1.1006 |

1999M10 | 0.9528 | 1.1086 |

1999M11 | 0.9368 | 1.1043 |

1999M12 | 0.9417 | 1.1184 |

2000M1 | 0.9607 | 1.1325 |

2000M2 | 0.9288 | 1.1110 |

2000M3 | 0.9147 | 1.0982 |

2000M4 | 0.9092 | 1.0761 |

2000M5 | 0.9209 | 1.1012 |

2000M6 | 0.9356 | 1.1061 |

2000M7 | 0.9374 | 1.0871 |

2000M8 | 0.9331 | 1.1153 |

2000M9 | 0.9411 | 1.1294 |

2000M10 | 0.9270 | 1.1276 |

2000M11 | 0.9687 | 1.1822 |

2000M12 | 0.9773 | 1.1822 |

2001M1 | 0.9828 | 1.1834 |

2001M2 | 0.9669 | 1.1982 |

2001M3 | 1.0356 | 1.2448 |

2001M4 | 0.9834 | 1.2294 |

2001M5 | 0.9748 | 1.2515 |

2001M6 | 0.9325 | 1.2190 |

2001M7 | 0.9202 | 1.2061 |

2001M8 | 0.8840 | 1.2270 |

2001M9 | 0.8368 | 1.1258 |

2001M10 | 0.8417 | 1.0356 |

2001M11 | 0.8294 | 1.0221 |

2001M12 | 0.8301 | 1.0399 |

2002M1 | 0.8294 | 1.0307 |

2002M2 | 0.8018 | 1.0147 |

2002M3 | 0.8037 | 0.9571 |

2002M4 | 0.8031 | 0.9613 |

2002M5 | 0.8141 | 0.9638 |

2002M6 | 0.8061 | 0.9454 |

2002M7 | 0.8147 | 0.9491 |

2002M8 | 0.8166 | 0.9485 |

2002M9 | 0.8172 | 0.9485 |

2002M10 | 0.8043 | 0.9356 |

2002M11 | 0.7945 | 0.9245 |

2002M12 | 0.8080 | 0.9092 |

2003M1 | 0.7810 | 0.8669 |

2003M2 | 0.7798 | 0.8779 |

2003M3 | 0.7644 | 0.8675 |

2003M4 | 0.7748 | 0.8663 |

2003M5 | 0.7816 | 0.8589 |

2003M6 | 0.7503 | 0.8503 |

2003M7 | 0.7429 | 0.8344 |

2003M8 | 0.7239 | 0.8393 |

2003M9 | 0.7362 | 0.8528 |

2003M10 | 0.7196 | 0.8571 |

2003M11 | 0.7276 | 0.8313 |

2003M12 | 0.7252 | 0.8074 |

2004M1 | 0.7319 | 0.8129 |

2004M2 | 0.7393 | 0.8055 |

2004M3 | 0.7288 | 0.7945 |

2004M4 | 0.7117 | 0.7644 |

2004M5 | 0.7110 | 0.7595 |

2004M6 | 0.7092 | 0.7325 |

2004M7 | 0.7074 | 0.7221 |

2004M8 | 0.7190 | 0.7055 |

2004M9 | 0.7067 | 0.7221 |

2004M10 | 0.7123 | 0.7307 |

2004M11 | 0.7319 | 0.7184 |

2004M12 | 0.7294 | 0.6736 |

2005M1 | 0.7282 | 0.6669 |

2005M2 | 0.7344 | 0.6761 |

2005M3 | 0.7399 | 0.6865 |

2005M4 | 0.7436 | 0.6804 |

2005M5 | 0.7190 | 0.6613 |

2005M6 | 0.6945 | 0.6380 |

2005M7 | 0.7313 | 0.6546 |

2005M8 | 0.7221 | 0.6521 |

2005M9 | 0.7215 | 0.6466 |

2005M10 | 0.7215 | 0.6491 |

2005M11 | 0.7288 | 0.6632 |

2005M12 | 0.7319 | 0.6773 |

2006M1 | 0.7632 | 0.7031 |

2006M2 | 0.7387 | 0.6521 |

2006M3 | 0.7393 | 0.6472 |

2006M4 | 0.6871 | 0.6626 |

Figure 1c. Length of Forecast Intervals for Benchmark Taylor Rule and Random Walk Models (British Pound), 12-month-ahead forecast

Note: To facilitate graphical comparisons, the 6- and 12-month-ahead forecast intervals of the random walk have been relocated such that they have the same center as the intervals of the Taylor rule model.

Data for Figure 1c

Date | Taylor Rule | RW |
---|---|---|

1990M11 | 1.0000 | 1.1366 |

1990M12 | 0.9668 | 1.1196 |

1991M1 | 0.8647 | 1.0826 |

1991M2 | 0.8136 | 1.0540 |

1991M3 | 0.8987 | 1.1005 |

1991M4 | 0.9066 | 1.0917 |

1991M5 | 0.7659 | 1.0656 |

1991M6 | 0.6750 | 1.0448 |

1991M7 | 0.6364 | 0.9875 |

1991M8 | 0.8788 | 0.9402 |

1991M9 | 0.7572 | 0.9510 |

1991M10 | 0.8672 | 0.9402 |

1991M11 | 0.8406 | 0.9415 |

1991M12 | 0.8780 | 0.9622 |

1992M1 | 0.8589 | 0.9560 |

1992M2 | 0.8709 | 0.9419 |

1992M3 | 0.6945 | 1.0154 |

1992M4 | 0.6783 | 1.0569 |

1992M5 | 0.6733 | 1.0731 |

1992M6 | 0.7597 | 1.1208 |

1992M7 | 0.7547 | 1.1200 |

1992M8 | 0.6949 | 1.0984 |

1992M9 | 0.6932 | 1.0714 |

1992M10 | 0.6849 | 1.0735 |

1992M11 | 0.6795 | 1.0394 |

1992M12 | 0.6148 | 0.9938 |

1993M1 | 0.6177 | 0.9983 |

1993M2 | 0.6534 | 1.0029 |

1993M3 | 0.6833 | 1.0345 |

1993M4 | 0.6459 | 1.0149 |

1993M5 | 0.6513 | 0.9855 |

1993M6 | 0.6613 | 0.9577 |

1993M7 | 0.6081 | 0.9116 |

1993M8 | 0.5994 | 0.8991 |

1993M9 | 0.6094 | 0.9465 |

1993M10 | 0.7073 | 1.0573 |

1993M11 | 0.8447 | 1.1445 |

1993M12 | 0.7866 | 1.1266 |

1994M1 | 0.8643 | 1.1403 |

1994M2 | 0.9290 | 1.2138 |

1994M3 | 0.9261 | 1.1955 |

1994M4 | 0.8418 | 1.1312 |

1994M5 | 0.8277 | 1.1287 |

1994M6 | 0.8738 | 1.1586 |

1994M7 | 0.8738 | 1.1876 |

1994M8 | 0.8726 | 1.2171 |

1994M9 | 0.8423 | 1.1905 |

1994M10 | 0.8771 | 1.2084 |

1994M11 | 0.8871 | 1.2262 |

1994M12 | 0.8751 | 1.2171 |

1995M1 | 0.8829 | 1.2163 |

1995M2 | 0.9041 | 1.2275 |

1995M3 | 0.8734 | 1.2167 |

1995M4 | 0.8605 | 1.2246 |

1995M5 | 0.8418 | 1.2067 |

1995M6 | 0.8439 | 1.1897 |

1995M7 | 0.8149 | 1.1739 |

1995M8 | 0.8057 | 1.1773 |

1995M9 | 0.7410 | 1.1594 |

1995M10 | 0.7140 | 1.1299 |

1995M11 | 0.7323 | 1.1424 |

1995M12 | 0.7430 | 1.1648 |

1996M1 | 0.7733 | 1.1532 |

1996M2 | 0.7543 | 1.1548 |

1996M3 | 0.7401 | 1.1345 |

1996M4 | 0.7235 | 1.1295 |

1996M5 | 0.7352 | 1.1436 |

1996M6 | 0.7372 | 1.1386 |

1996M7 | 0.7020 | 1.1378 |

1996M8 | 0.7302 | 1.1590 |

1996M9 | 0.7260 | 1.1644 |

1996M10 | 0.6974 | 1.1507 |

1996M11 | 0.6966 | 1.1615 |

1996M12 | 0.7194 | 1.1781 |

1997M1 | 0.7954 | 1.1876 |

1997M2 | 0.7522 | 1.1818 |

1997M3 | 0.7630 | 1.1889 |

1997M4 | 0.7924 | 1.1972 |

1997M5 | 0.7954 | 1.1980 |

1997M6 | 0.7493 | 1.1777 |

1997M7 | 0.7530 | 1.1689 |

1997M8 | 0.7534 | 1.1714 |

1997M9 | 0.7078 | 1.1640 |

1997M10 | 0.6812 | 1.1445 |

1997M11 | 0.6878 | 1.0922 |

1997M12 | 0.6804 | 1.0909 |

1998M1 | 0.6808 | 1.0946 |

1998M2 | 0.6870 | 1.1171 |

1998M3 | 0.7049 | 1.1034 |

1998M4 | 0.6957 | 1.0722 |

1998M5 | 0.6995 | 1.0565 |

1998M6 | 0.7049 | 1.0448 |

1998M7 | 0.6895 | 1.0232 |

1998M8 | 0.6837 | 1.0589 |

1998M9 | 0.6966 | 1.0544 |

1998M10 | 0.7177 | 1.0328 |

1998M11 | 0.7302 | 0.9597 |

1998M12 | 0.7327 | 0.9751 |

1999M1 | 0.6895 | 0.9900 |

1999M2 | 0.6816 | 0.9863 |

1999M3 | 0.7165 | 0.9738 |

1999M4 | 0.7194 | 0.9676 |

1999M5 | 0.7036 | 0.9880 |

1999M6 | 0.7119 | 0.9805 |

1999M7 | 0.6920 | 0.9851 |

1999M8 | 0.7094 | 0.9905 |

1999M9 | 0.7397 | 0.9622 |

1999M10 | 0.7352 | 0.9552 |

1999M11 | 0.7169 | 0.9689 |

1999M12 | 0.6712 | 0.9635 |

2000M1 | 0.6642 | 0.9755 |

2000M2 | 0.6189 | 0.9892 |

2000M3 | 0.6330 | 0.9929 |

2000M4 | 0.5936 | 1.0004 |

2000M5 | 0.5973 | 0.9967 |

2000M6 | 0.5845 | 1.0091 |

2000M7 | 0.6027 | 1.0216 |

2000M8 | 0.5795 | 1.0025 |

2000M9 | 0.6098 | 0.9905 |

2000M10 | 0.5878 | 0.9714 |

2000M11 | 0.5832 | 0.9929 |

2000M12 | 0.5575 | 0.9975 |

2001M1 | 0.5546 | 0.9813 |

2001M2 | 0.5712 | 1.0062 |

2001M3 | 0.5637 | 1.0191 |

2001M4 | 0.5313 | 1.0170 |

2001M5 | 0.5367 | 1.0668 |

2001M6 | 0.5363 | 1.0191 |

2001M7 | 0.5330 | 0.9896 |

2001M8 | 0.5297 | 0.9988 |

2001M9 | 0.5160 | 1.0340 |

2001M10 | 0.5077 | 0.9963 |

2001M11 | 0.4948 | 1.0029 |

2001M12 | 0.5081 | 0.9780 |

2002M1 | 0.5006 | 0.9676 |

2002M2 | 0.4736 | 0.9846 |

2002M3 | 0.4745 | 0.9900 |

2002M4 | 0.4587 | 0.9971 |

2002M5 | 0.4653 | 1.0025 |

2002M6 | 0.4620 | 1.0203 |

2002M7 | 0.4570 | 1.0108 |

2002M8 | 0.4620 | 0.9809 |

2002M9 | 0.4741 | 0.9498 |

2002M10 | 0.4653 | 0.9435 |

2002M11 | 0.4662 | 0.9527 |

2002M12 | 0.4633 | 0.9411 |

2003M1 | 0.4666 | 0.9170 |

2003M2 | 0.4608 | 0.9103 |

2003M3 | 0.4608 | 0.9103 |

2003M4 | 0.4728 | 0.8979 |

2003M5 | 0.4736 | 0.8871 |

2003M6 | 0.4633 | 0.8730 |

2003M7 | 0.5035 | 0.8323 |

2003M8 | 0.4981 | 0.8427 |

2003M9 | 0.5430 | 0.8323 |

2003M10 | 0.5575 | 0.8315 |

2003M11 | 0.5687 | 0.8248 |

2003M12 | 0.5799 | 0.8165 |

2004M1 | 0.5770 | 0.8007 |

2004M2 | 0.5820 | 0.8057 |

2004M3 | 0.5645 | 0.8186 |

2004M4 | 0.5687 | 0.8227 |

2004M5 | 0.5890 | 0.7983 |

2004M6 | 0.6102 | 0.7796 |

2004M7 | 0.5907 | 0.7792 |

2004M8 | 0.5758 | 0.7862 |

2004M9 | 0.5753 | 0.7684 |

2004M10 | 0.6351 | 0.7389 |

2004M11 | 0.5953 | 0.7347 |

2004M12 | 0.6185 | 0.7082 |

2005M1 | 0.5367 | 0.6800 |

2005M2 | 0.4720 | 0.6712 |

2005M3 | 0.5330 | 0.6866 |

2005M4 | 0.6040 | 0.6953 |

2005M5 | 0.6198 | 0.7020 |

2005M6 | 0.6040 | 0.6858 |

2005M7 | 0.5351 | 0.6800 |

2005M8 | 0.6110 | 0.6887 |

2005M9 | 0.6173 | 0.6990 |

2005M10 | 0.6210 | 0.6932 |

2005M11 | 0.5372 | 0.6737 |

2005M12 | 0.4670 | 0.6501 |

2006M1 | 0.6359 | 0.6671 |

2006M2 | 0.6214 | 0.6646 |

2006M3 | 0.6430 | 0.6584 |

2006M4 | 0.6260 | 0.6613 |

Figure 2a. Length of Forecast Intervals for Benchmark Taylor Rule and Random Walk Models (Deutschmark), 1-month-ahead forecast

Data for Figure 2a

Date | Taylor Rule | RW |
---|---|---|

1989M12 | 1.0000 | 1.1060 |

1990M1 | 1.0471 | 1.0497 |

1990M2 | 0.9522 | 1.0190 |

1990M3 | 0.9156 | 0.9614 |

1990M4 | 0.9594 | 0.9777 |

1990M5 | 0.9385 | 0.9673 |

1990M6 | 0.9045 | 0.9535 |

1990M7 | 0.8678 | 0.9627 |

1990M8 | 0.8344 | 0.9372 |

1990M9 | 0.7729 | 0.8920 |

1990M10 | 0.7723 | 0.8920 |

1990M11 | 0.7520 | 0.8658 |

1990M12 | 0.7160 | 0.8442 |

1991M1 | 0.7212 | 0.8514 |

1991M2 | 0.6185 | 0.8580 |

1991M3 | 0.6198 | 0.8416 |

1991M4 | 0.8691 | 0.9228 |

1991M5 | 1.0576 | 0.9771 |

1991M6 | 0.9797 | 0.9869 |

1991M7 | 1.0569 | 1.0229 |

1991M8 | 1.0098 | 1.0242 |

1991M9 | 0.9326 | 1.0000 |

1991M10 | 0.9025 | 0.9719 |

1991M11 | 0.9732 | 0.9686 |

1991M12 | 1.0308 | 0.9300 |

1992M1 | 1.0366 | 0.8966 |

1992M2 | 1.0406 | 0.9058 |

1992M3 | 1.0301 | 0.9260 |

1992M4 | 0.9509 | 0.9509 |

1992M5 | 0.8822 | 0.9437 |

1992M6 | 0.8665 | 0.9287 |

1992M7 | 0.8266 | 0.8999 |

1992M8 | 0.8606 | 0.8966 |

1992M9 | 0.9326 | 0.8698 |

1992M10 | 0.8338 | 0.8724 |

1992M11 | 0.8384 | 0.8927 |

1992M12 | 0.9195 | 0.9562 |

1993M1 | 0.9005 | 0.9529 |

1993M2 | 0.9116 | 0.9725 |

1993M3 | 0.9188 | 0.9889 |

1993M4 | 0.9182 | 0.9915 |

1993M5 | 0.8940 | 0.9620 |

1993M6 | 0.8986 | 0.9679 |

1993M7 | 0.9260 | 0.9967 |

1993M8 | 0.9849 | 1.0334 |

1993M9 | 0.9411 | 1.0203 |

1993M10 | 0.9018 | 0.9764 |

1993M11 | 0.9110 | 0.9882 |

1993M12 | 0.9483 | 1.0242 |

1994M1 | 0.9568 | 1.0308 |

1994M2 | 0.9974 | 1.0497 |

1994M3 | 0.9712 | 1.0452 |

1994M4 | 0.9424 | 1.0183 |

1994M5 | 0.9483 | 1.0236 |

1994M6 | 0.9274 | 0.9980 |

1994M7 | 0.9084 | 0.9804 |

1994M8 | 0.8783 | 0.9437 |

1994M9 | 0.8763 | 0.9424 |

1994M10 | 0.8645 | 0.9332 |

1994M11 | 0.8547 | 0.9149 |

1994M12 | 0.8639 | 0.9274 |

1995M1 | 0.8815 | 0.9470 |

1995M2 | 0.8501 | 0.9215 |

1995M3 | 0.8370 | 0.9045 |

1995M4 | 0.8403 | 0.8495 |

1995M5 | 0.8253 | 0.8344 |

1995M6 | 0.7317 | 0.8488 |

1995M7 | 0.7251 | 0.8436 |

1995M8 | 0.7016 | 0.8364 |

1995M9 | 0.7493 | 0.8711 |

1995M10 | 0.7853 | 0.8796 |

1995M11 | 0.7238 | 0.8514 |

1995M12 | 0.7264 | 0.8541 |

1996M1 | 0.7487 | 0.8678 |

1996M2 | 0.7853 | 0.8815 |

1996M3 | 0.7624 | 0.8835 |

1996M4 | 0.7945 | 0.8901 |

1996M5 | 0.8109 | 0.9064 |

1996M6 | 0.8266 | 0.9234 |

1996M7 | 0.8043 | 0.9208 |

1996M8 | 0.7919 | 0.9045 |

1996M9 | 0.7801 | 0.8927 |

1996M10 | 0.7925 | 0.9084 |

1996M11 | 0.7873 | 0.9182 |

1996M12 | 0.7808 | 0.9084 |

1997M1 | 0.7788 | 0.9332 |

1997M2 | 0.8240 | 0.9640 |

1997M3 | 0.9077 | 1.0085 |

1997M4 | 0.9424 | 1.0209 |

1997M5 | 0.9607 | 1.0314 |

1997M6 | 0.9450 | 1.0268 |

1997M7 | 0.9601 | 1.0406 |

1997M8 | 0.9974 | 1.0805 |

1997M9 | 1.0445 | 1.1086 |

1997M10 | 0.9823 | 1.0687 |

1997M11 | 0.9660 | 1.0510 |

1997M12 | 0.9562 | 1.0360 |

1998M1 | 0.9385 | 1.0615 |

1998M2 | 0.9476 | 1.0838 |

1998M3 | 0.9463 | 1.0818 |

1998M4 | 1.0092 | 1.0903 |

1998M5 | 0.9202 | 1.0164 |

1998M6 | 0.9012 | 0.9954 |

1998M7 | 0.9116 | 1.0052 |

1998M8 | 0.9378 | 1.0085 |

1998M9 | 0.9038 | 1.0020 |

1998M10 | 0.8671 | 1.0013 |

1998M11 | 0.8325 | 0.9653 |

1998M12 | 0.8613 | 0.9921 |

Figure 2b. Length of Forecast Intervals for Benchmark Taylor Rule and Random Walk Models (Deutschmark), 6-month-ahead forecast

Data for Figure 2b

Date | Taylor Rule | RW |
---|---|---|

1990M5 | 1.0000 | 1.0091 |

1990M6 | 0.9614 | 0.9553 |

1990M7 | 0.9059 | 0.9300 |

1990M8 | 0.9235 | 0.9212 |

1990M9 | 0.9110 | 0.9171 |

1990M10 | 0.8822 | 0.9070 |

1990M11 | 0.8491 | 0.8944 |

1990M12 | 0.8296 | 0.9053 |

1991M1 | 0.8184 | 0.8807 |

1991M2 | 0.7747 | 0.8444 |

1991M3 | 0.7777 | 0.8446 |

1991M4 | 0.7844 | 0.8196 |

1991M5 | 0.7338 | 0.7991 |

1991M6 | 0.7651 | 0.8058 |

1991M7 | 0.7531 | 0.8116 |

1991M8 | 0.7647 | 0.7963 |

1991M9 | 0.7893 | 0.8671 |

1991M10 | 0.8610 | 0.9158 |

1991M11 | 0.9214 | 0.9457 |

1991M12 | 0.9714 | 1.0032 |

1992M1 | 1.0490 | 1.0085 |

1992M2 | 0.9406 | 1.0121 |

1992M3 | 0.9110 | 0.9830 |

1992M4 | 1.0257 | 0.9807 |

1992M5 | 0.9063 | 0.9409 |

1992M6 | 0.8749 | 0.9072 |

1992M7 | 0.9017 | 0.9165 |

1992M8 | 0.9114 | 0.9396 |

1992M9 | 0.9150 | 0.9644 |

1992M10 | 0.9354 | 0.9576 |

1992M11 | 0.9131 | 0.9419 |

1992M12 | 0.8813 | 0.9129 |

1993M1 | 0.8552 | 0.8656 |

1993M2 | 0.8082 | 0.8402 |

1993M3 | 0.8008 | 0.8425 |

1993M4 | 0.8285 | 0.8622 |

1993M5 | 0.9053 | 0.9216 |

1993M6 | 0.9034 | 0.9184 |

1993M7 | 0.9820 | 0.9371 |

1993M8 | 0.9650 | 0.9527 |

1993M9 | 0.9392 | 0.9559 |

1993M10 | 0.9252 | 0.9267 |

1993M11 | 0.9239 | 0.9328 |

1993M12 | 0.9619 | 0.9604 |

1994M1 | 1.0093 | 0.9960 |

1994M2 | 0.9521 | 0.9835 |

1994M3 | 0.9161 | 0.9415 |

1994M4 | 0.9201 | 0.9523 |

1994M5 | 0.9746 | 0.9871 |

1994M6 | 0.9866 | 0.9930 |

1994M7 | 1.0168 | 1.0115 |

1994M8 | 1.0040 | 1.0076 |

1994M9 | 0.9519 | 0.9816 |

1994M10 | 0.9585 | 0.9860 |

1994M11 | 0.9506 | 0.9616 |

1994M12 | 0.9027 | 0.9445 |

1995M1 | 0.8936 | 0.9099 |

1995M2 | 0.8856 | 0.9082 |

1995M3 | 0.8618 | 0.8993 |

1995M4 | 0.8603 | 0.8821 |

1995M5 | 0.8762 | 0.8938 |

1995M6 | 0.8917 | 0.9123 |

1995M7 | 0.8730 | 0.8883 |

1995M8 | 0.8546 | 0.8720 |

1995M9 | 0.7931 | 0.8162 |

1995M10 | 0.7827 | 0.8018 |

1995M11 | 0.7950 | 0.8183 |

1995M12 | 0.7959 | 0.8135 |

1996M1 | 0.7974 | 0.8061 |

1996M2 | 0.8192 | 0.8393 |

1996M3 | 0.8234 | 0.8476 |

1996M4 | 0.8006 | 0.8209 |

1996M5 | 0.8145 | 0.8228 |

1996M6 | 0.8326 | 0.8364 |

1996M7 | 0.8190 | 0.8497 |

1996M8 | 0.8364 | 0.8516 |

1996M9 | 0.8224 | 0.8576 |

1996M10 | 0.8351 | 0.8733 |

1996M11 | 0.8535 | 0.8896 |

1996M12 | 0.8548 | 0.8870 |

1997M1 | 0.8423 | 0.8722 |

1997M2 | 0.8243 | 0.8607 |

1997M3 | 0.8319 | 0.8754 |

1997M4 | 0.8449 | 0.8868 |

1997M5 | 0.8336 | 0.8777 |

1997M6 | 0.8614 | 0.9012 |

1997M7 | 0.8906 | 0.9315 |

1997M8 | 0.9192 | 0.9722 |

1997M9 | 0.9188 | 0.9835 |

1997M10 | 0.9300 | 0.9671 |

1997M11 | 0.9568 | 0.9593 |

1997M12 | 0.9432 | 0.9292 |

1998M1 | 0.9599 | 0.9307 |

1998M2 | 0.9417 | 0.9527 |

1998M3 | 0.8851 | 0.9220 |

1998M4 | 0.8646 | 0.8953 |

1998M5 | 0.8643 | 0.8826 |

1998M6 | 0.8896 | 0.9061 |

1998M7 | 0.8900 | 0.9254 |

1998M8 | 0.8930 | 0.9233 |

1998M9 | 0.8877 | 0.9309 |

1998M10 | 0.8904 | 0.9237 |

1998M11 | 0.8809 | 0.9044 |

1998M12 | 0.8800 | 0.9135 |

Figure 2c. Length of Forecast Intervals for Benchmark Taylor Rule and Random Walk Models (Deutschmark), 12-month-ahead forecast

Data for Figure 2c

Date | Taylor Rule | RW |
---|---|---|

1990M11 | 1.0000 | 1.0730 |

1990M12 | 0.9630 | 1.0191 |

1991M1 | 0.9260 | 0.9919 |

1991M2 | 0.9336 | 0.9827 |

1991M3 | 0.9319 | 0.9999 |

1991M4 | 0.9477 | 0.9887 |

1991M5 | 0.9277 | 0.9752 |

1991M6 | 0.9195 | 0.9870 |

1991M7 | 0.8944 | 0.9604 |

1991M8 | 0.8508 | 0.9207 |

1991M9 | 0.8457 | 0.9206 |

1991M10 | 0.8409 | 0.8935 |

1991M11 | 0.7862 | 0.8712 |

1991M12 | 0.8084 | 0.8786 |

1992M1 | 0.7919 | 0.8850 |

1992M2 | 0.7852 | 0.8681 |

1992M3 | 0.8335 | 0.9454 |

1992M4 | 0.8828 | 0.9985 |

1992M5 | 0.8946 | 1.0087 |

1992M6 | 0.9213 | 1.0455 |

1992M7 | 0.9676 | 1.0467 |

1992M8 | 0.9170 | 1.0223 |

1992M9 | 0.9105 | 0.9929 |

1992M10 | 0.9506 | 0.9905 |

1992M11 | 0.8925 | 0.9505 |

1992M12 | 0.8665 | 0.9166 |

1993M1 | 0.8812 | 0.9259 |

1993M2 | 0.8987 | 0.9492 |

1993M3 | 0.9136 | 0.9743 |

1993M4 | 0.9205 | 0.9672 |

1993M5 | 0.9048 | 0.9514 |

1993M6 | 0.8814 | 0.9221 |

1993M7 | 0.8546 | 0.8745 |

1993M8 | 0.8238 | 0.8487 |

1993M9 | 0.8180 | 0.8511 |

1993M10 | 0.8425 | 0.8708 |

1993M11 | 0.9015 | 0.9311 |

1993M12 | 0.9033 | 0.9277 |

1994M1 | 0.9425 | 0.9467 |

1994M2 | 0.9467 | 0.9625 |

1994M3 | 0.9424 | 0.9655 |

1994M4 | 0.9248 | 0.9362 |

1994M5 | 0.9260 | 0.9424 |

1994M6 | 0.9538 | 0.9703 |

1994M7 | 0.9853 | 1.0060 |

1994M8 | 0.9548 | 0.9936 |

1994M9 | 0.9235 | 0.9510 |

1994M10 | 0.9322 | 0.9619 |

1994M11 | 0.9680 | 0.9971 |

1994M12 | 0.9750 | 1.0031 |

1995M1 | 0.9979 | 1.0219 |

1995M2 | 0.9918 | 1.0177 |

1995M3 | 0.9619 | 0.9915 |

1995M4 | 0.9675 | 0.9960 |

1995M5 | 0.9583 | 0.9714 |

1995M6 | 0.9369 | 0.9541 |

1995M7 | 0.9152 | 0.9192 |

1995M8 | 0.9145 | 0.9174 |

1995M9 | 0.8927 | 0.9085 |

1995M10 | 0.8867 | 0.8912 |

1995M11 | 0.8951 | 0.9029 |

1995M12 | 0.9138 | 0.9216 |

1996M1 | 0.8984 | 0.8973 |

1996M2 | 0.8864 | 0.8808 |

1996M3 | 0.8469 | 0.8246 |

1996M4 | 0.8543 | 0.8099 |

1996M5 | 0.8581 | 0.8265 |

1996M6 | 0.8585 | 0.8217 |

1996M7 | 0.8501 | 0.8143 |

1996M8 | 0.8857 | 0.8476 |

1996M9 | 0.8892 | 0.8561 |

1996M10 | 0.8789 | 0.8293 |

1996M11 | 0.8752 | 0.8311 |

1996M12 | 0.8825 | 0.8448 |

1997M1 | 0.8914 | 0.8582 |

1997M2 | 0.8959 | 0.8602 |

1997M3 | 0.9019 | 0.8665 |

1997M4 | 0.9177 | 0.8821 |

1997M5 | 0.9171 | 0.8985 |

1997M6 | 0.9212 | 0.8962 |

1997M7 | 0.9188 | 0.8810 |

1997M8 | 0.9055 | 0.8694 |

1997M9 | 0.9125 | 0.8843 |

1997M10 | 0.9206 | 0.8913 |

1997M11 | 0.9076 | 0.8821 |

1997M12 | 0.9128 | 0.9058 |

1998M1 | 0.9347 | 0.9231 |

1998M2 | 0.9601 | 0.9595 |

1998M3 | 0.9460 | 0.9533 |

1998M4 | 0.9359 | 0.9365 |

1998M5 | 0.9237 | 0.9295 |

1998M6 | 0.9066 | 0.9366 |

1998M7 | 0.9347 | 0.9725 |

1998M8 | 0.9940 | 1.0035 |

1998M9 | 0.9495 | 0.9771 |

1998M10 | 0.9350 | 0.9615 |

1998M11 | 0.9332 | 0.9477 |

1998M12 | 0.9456 | 0.9731 |

Figure 3a. Length of Forecast Intervals for Benchmark Taylor Rule and Random Walk Models (Japanese Yen), 1-month-ahead forecast

Data for Figure 3a

Date | Taylor rule | RW |
---|---|---|

1989M12 | 1.0000 | 0.9695 |

1990M1 | 0.9991 | 0.9706 |

1990M2 | 1.0087 | 0.9793 |

1990M3 | 1.0061 | 0.9841 |

1990M4 | 1.0695 | 1.0666 |

1990M5 | 1.1049 | 1.1024 |

1990M6 | 1.0753 | 1.0716 |

1990M7 | 1.0710 | 1.0382 |

1990M8 | 1.0340 | 1.0067 |

1990M9 | 1.0495 | 0.9950 |

1990M10 | 0.9858 | 0.9510 |

1990M11 | 0.9925 | 0.8902 |

1990M12 | 0.9930 | 0.8877 |

1991M1 | 1.0346 | 0.9197 |

1991M2 | 1.0305 | 0.9185 |

1991M3 | 1.0002 | 0.8968 |

1991M4 | 1.0998 | 0.9448 |

1991M5 | 1.0970 | 0.9429 |

1991M6 | 1.1097 | 0.9505 |

1991M7 | 1.1078 | 0.9611 |

1991M8 | 1.0886 | 0.9478 |

1991M9 | 1.0784 | 0.9409 |

1991M10 | 0.9780 | 0.9236 |

1991M11 | 0.9529 | 0.8993 |

1991M12 | 0.9462 | 0.8915 |

1992M1 | 0.9373 | 0.8805 |

1992M2 | 0.9158 | 0.8628 |

1992M3 | 0.9161 | 0.8782 |

1992M4 | 0.9502 | 0.9137 |

1992M5 | 0.9507 | 0.9184 |

1992M6 | 0.9347 | 0.8993 |

1992M7 | 0.9094 | 0.8723 |

1992M8 | 0.9030 | 0.8657 |

1992M9 | 0.9027 | 0.8681 |

1992M10 | 0.8747 | 0.8431 |

1992M11 | 0.8661 | 0.8333 |

1992M12 | 0.8844 | 0.8519 |

1993M1 | 0.8870 | 0.8530 |

1993M2 | 0.8951 | 0.8595 |

1993M3 | 0.8640 | 0.8305 |

1993M4 | 0.8160 | 0.8047 |

1993M5 | 0.7733 | 0.7731 |

1993M6 | 0.7599 | 0.7588 |

1993M7 | 0.6571 | 0.7387 |

1993M8 | 0.6414 | 0.7406 |

1993M9 | 0.6116 | 0.7136 |

1993M10 | 0.6299 | 0.7260 |

1993M11 | 0.6388 | 0.7360 |

1993M12 | 0.6651 | 0.7419 |

1994M1 | 0.7432 | 0.7559 |

1994M2 | 0.7707 | 0.7664 |

1994M3 | 0.6653 | 0.7311 |

1994M4 | 0.6419 | 0.7228 |

1994M5 | 0.6223 | 0.7117 |

1994M6 | 0.6367 | 0.7135 |

1994M7 | 0.6287 | 0.7051 |

1994M8 | 0.5732 | 0.6770 |

1994M9 | 0.5788 | 0.6873 |

1994M10 | 0.5437 | 0.6792 |

1994M11 | 0.5076 | 0.6763 |

1994M12 | 0.5075 | 0.6742 |

1995M1 | 0.5765 | 0.6890 |

1995M2 | 0.5552 | 0.6861 |

1995M3 | 0.5173 | 0.6636 |

1995M4 | 0.4977 | 0.6115 |

1995M5 | 0.6096 | 0.5772 |

1995M6 | 0.6199 | 0.5870 |

1995M7 | 0.6166 | 0.5832 |

1995M8 | 0.6907 | 0.6022 |

1995M9 | 0.7835 | 0.6534 |

1995M10 | 0.7001 | 0.7139 |

1995M11 | 0.7031 | 0.7159 |

1995M12 | 0.7041 | 0.7031 |

1996M1 | 0.7020 | 0.7025 |

1996M2 | 0.7208 | 0.7294 |

1996M3 | 0.7221 | 0.7297 |

1996M4 | 0.7230 | 0.7307 |

1996M5 | 0.7389 | 0.7394 |

1996M6 | 0.7229 | 0.7334 |

1996M7 | 0.7409 | 0.7507 |

1996M8 | 0.7394 | 0.7523 |

1996M9 | 0.7295 | 0.7432 |

1996M10 | 0.7460 | 0.7574 |

1996M11 | 0.7686 | 0.7745 |

1996M12 | 0.7708 | 0.7738 |

1997M1 | 0.7838 | 0.7690 |

1997M2 | 0.8110 | 0.7955 |

1997M3 | 0.8599 | 0.8306 |

1997M4 | 0.8580 | 0.8293 |

1997M5 | 0.8719 | 0.8486 |

1997M6 | 0.8300 | 0.8051 |

1997M7 | 0.7843 | 0.7720 |

1997M8 | 0.7936 | 0.7793 |

1997M9 | 0.8098 | 0.7966 |

1997M10 | 0.8272 | 0.8166 |

1997M11 | 0.8304 | 0.8177 |

1997M12 | 0.8632 | 0.8469 |

1998M1 | 0.9077 | 0.8763 |

1998M2 | 0.9062 | 0.8751 |

1998M3 | 0.8611 | 0.8501 |

1998M4 | 0.8824 | 0.8719 |

1998M5 | 0.9043 | 0.8899 |

1998M6 | 0.9280 | 0.9112 |

1998M7 | 0.9761 | 0.9479 |

1998M8 | 0.9736 | 0.9510 |

1998M9 | 0.9949 | 0.9773 |

1998M10 | 0.9842 | 0.9265 |

1998M11 | 0.8384 | 0.8431 |

1998M12 | 0.8210 | 0.8378 |

1999M1 | 0.7927 | 0.8154 |

1999M2 | 0.7644 | 0.7874 |

1999M3 | 0.7904 | 0.8108 |

1999M4 | 0.8114 | 0.8303 |

1999M5 | 0.8129 | 0.8324 |

1999M6 | 0.8303 | 0.8478 |

1999M7 | 0.8464 | 0.8390 |

1999M8 | 0.8224 | 0.8204 |

1999M9 | 0.7600 | 0.7784 |

1999M10 | 0.7040 | 0.7406 |

1999M11 | 0.6917 | 0.7343 |

1999M12 | 0.6671 | 0.7251 |

2000M1 | 0.6497 | 0.7108 |

2000M2 | 0.6701 | 0.7296 |

2000M3 | 0.7260 | 0.7580 |

2000M4 | 0.7084 | 0.7367 |

2000M5 | 0.6806 | 0.7319 |

2000M6 | 0.7185 | 0.7506 |

2000M7 | 0.7078 | 0.7354 |

2000M8 | 0.7187 | 0.7498 |

2000M9 | 0.7119 | 0.7489 |

2000M10 | 0.7100 | 0.7403 |

2000M11 | 0.7171 | 0.7514 |

2000M12 | 0.7209 | 0.7553 |

2001M1 | 0.7429 | 0.7775 |

2001M2 | 0.8222 | 0.8084 |

2001M3 | 0.8069 | 0.8045 |

2001M4 | 0.8809 | 0.8420 |

2001M5 | 0.8924 | 0.8576 |

2001M6 | 0.8857 | 0.8437 |

2001M7 | 0.8906 | 0.8478 |

2001M8 | 0.9435 | 0.8627 |

2001M9 | 0.8830 | 0.8410 |

2001M10 | 0.8547 | 0.8218 |

2001M11 | 0.8877 | 0.8415 |

2001M12 | 0.8999 | 0.8482 |

2002M1 | 0.9544 | 0.8869 |

2002M2 | 1.0358 | 0.9223 |

2002M3 | 1.0490 | 0.9290 |

2002M4 | 1.0152 | 0.9111 |

2002M5 | 1.0111 | 0.9090 |

2002M6 | 0.9612 | 0.8717 |

2002M7 | 0.8995 | 0.8503 |

2002M8 | 0.8490 | 0.8132 |

2002M9 | 0.8544 | 0.8207 |

2002M10 | 0.8644 | 0.8179 |

2002M11 | 0.8908 | 0.8370 |

2002M12 | 0.8647 | 0.8214 |

2003M1 | 0.8651 | 0.8233 |

2003M2 | 0.8493 | 0.8025 |

2003M3 | 0.8372 | 0.8021 |

2003M4 | 0.8265 | 0.7977 |

2003M5 | 0.8293 | 0.8058 |

2003M6 | 0.8153 | 0.7888 |

2003M7 | 0.8273 | 0.7926 |

2003M8 | 0.8298 | 0.7951 |

2003M9 | 0.8290 | 0.7948 |

2003M10 | 0.7912 | 0.7690 |

2003M11 | 0.7533 | 0.7334 |

2003M12 | 0.7120 | 0.7194 |

2004M1 | 0.6662 | 0.7099 |

2004M2 | 0.6629 | 0.7003 |

2004M3 | 0.7001 | 0.7031 |

2004M4 | 0.6752 | 0.7150 |

2004M5 | 0.6682 | 0.7094 |

2004M6 | 0.6981 | 0.7536 |

2004M7 | 0.6776 | 0.7335 |

2004M8 | 0.6453 | 0.6950 |

2004M9 | 0.6478 | 0.6997 |

2004M10 | 0.6451 | 0.6988 |

2004M11 | 0.6382 | 0.6905 |

2004M12 | 0.6180 | 0.6646 |

2005M1 | 0.6125 | 0.6590 |

2005M2 | 0.6112 | 0.6559 |

2005M3 | 0.6232 | 0.6528 |

2005M4 | 0.6226 | 0.6547 |

2005M5 | 0.6331 | 0.6668 |

2005M6 | 0.6221 | 0.6631 |

2005M7 | 0.6373 | 0.6765 |

2005M8 | 0.6546 | 0.6964 |

2005M9 | 0.6483 | 0.6880 |

2005M10 | 0.6544 | 0.6920 |

2005M11 | 0.6616 | 0.7145 |

2005M12 | 0.6750 | 0.7368 |

2006M1 | 0.6603 | 0.7360 |

2006M2 | 0.6201 | 0.7140 |

2006M3 | 0.6267 | 0.7287 |

2006M4 | 0.6238 | 0.7252 |

2006M5 | 0.6264 | 0.7239 |

2006M6 | 0.6703 | 0.6908 |

Figure 3b. Length of Forecast Intervals for Benchmark Taylor Rule and Random Walk Models (Japanese Yen), 6-month-ahead forecast

Data for Figure 3b

Date | Taylor Rule | RW |
---|---|---|

1990M5 | 1.0000 | 1.0812 |

1990M6 | 0.9872 | 1.0825 |

1990M7 | 1.0120 | 1.0922 |

1990M8 | 1.0282 | 1.0975 |

1990M9 | 1.2311 | 1.1483 |

1990M10 | 1.2928 | 1.1869 |

1990M11 | 1.1366 | 1.1537 |

1990M12 | 1.1536 | 1.1512 |

1991M1 | 0.8790 | 1.1163 |

1991M2 | 0.8542 | 1.1045 |

1991M3 | 0.7677 | 1.0369 |

1991M4 | 0.7039 | 0.9940 |

1991M5 | 0.7292 | 0.9911 |

1991M6 | 0.8556 | 1.0269 |

1991M7 | 0.8509 | 1.0255 |

1991M8 | 0.7721 | 1.0012 |

1991M9 | 0.9407 | 1.0537 |

1991M10 | 0.8734 | 1.0516 |

1991M11 | 0.8330 | 1.0601 |

1991M12 | 0.9249 | 1.0719 |

1992M1 | 0.8268 | 1.0571 |

1992M2 | 0.8198 | 1.0494 |

1992M3 | 0.7834 | 1.0301 |

1992M4 | 0.7062 | 1.0029 |

1992M5 | 0.6896 | 0.9943 |

1992M6 | 0.6842 | 0.9820 |

1992M7 | 0.6296 | 0.9623 |

1992M8 | 0.6932 | 0.9795 |

1992M9 | 0.7836 | 1.0190 |

1992M10 | 0.7882 | 1.0242 |

1992M11 | 0.7933 | 1.0029 |

1992M12 | 0.7222 | 0.9728 |

1993M1 | 0.7394 | 0.9655 |

1993M2 | 0.7675 | 0.9682 |

1993M3 | 0.6309 | 0.9403 |

1993M4 | 0.7411 | 0.9294 |

1993M5 | 0.7627 | 0.9501 |

1993M6 | 0.7502 | 0.9514 |

1993M7 | 0.7616 | 0.9586 |

1993M8 | 0.6755 | 0.9262 |

1993M9 | 0.7527 | 0.8975 |

1993M10 | 0.6862 | 0.8622 |

1993M11 | 0.6972 | 0.8463 |

1993M12 | 0.6603 | 0.8239 |

1994M1 | 0.7295 | 0.8260 |

1994M2 | 0.7172 | 0.7959 |

1994M3 | 0.7254 | 0.8097 |

1994M4 | 0.7523 | 0.8208 |

1994M5 | 0.7751 | 0.8274 |

1994M6 | 0.7863 | 0.8430 |

1994M7 | 0.8050 | 0.8547 |

1994M8 | 0.6317 | 0.8153 |

1994M9 | 0.5649 | 0.8061 |

1994M10 | 0.5571 | 0.7937 |

1994M11 | 0.5647 | 0.7958 |

1994M12 | 0.5284 | 0.7864 |

1995M1 | 0.5120 | 0.7551 |

1995M2 | 0.5352 | 0.7665 |

1995M3 | 0.4009 | 0.7576 |

1995M4 | 0.3106 | 0.7366 |

1995M5 | 0.3105 | 0.7304 |

1995M6 | 0.4579 | 0.7431 |

1995M7 | 0.3974 | 0.7401 |

1995M8 | 0.3495 | 0.7287 |

1995M9 | 0.3129 | 0.6715 |

1995M10 | 0.3725 | 0.6208 |

1995M11 | 0.3767 | 0.6313 |

1995M12 | 0.4034 | 0.6239 |

1996M1 | 0.4188 | 0.6420 |

1996M2 | 0.4821 | 0.6954 |

1996M3 | 0.5311 | 0.7381 |

1996M4 | 0.6490 | 0.7407 |

1996M5 | 0.6547 | 0.7562 |

1996M6 | 0.6503 | 0.7599 |

1996M7 | 0.6880 | 0.7890 |

1996M8 | 0.6911 | 0.7893 |

1996M9 | 0.6845 | 0.7904 |

1996M10 | 0.6811 | 0.7952 |

1996M11 | 0.6859 | 0.7872 |

1996M12 | 0.6904 | 0.8067 |

1997M1 | 0.6921 | 0.8084 |

1997M2 | 0.6779 | 0.7986 |

1997M3 | 0.6913 | 0.8138 |

1997M4 | 0.6999 | 0.8322 |

1997M5 | 0.7018 | 0.8314 |

1997M6 | 0.7101 | 0.8438 |

1997M7 | 0.7227 | 0.8729 |

1997M8 | 0.7548 | 0.9174 |

1997M9 | 0.7529 | 0.9159 |

1997M10 | 0.7990 | 0.9374 |

1997M11 | 0.7743 | 0.8892 |

1997M12 | 0.7389 | 0.8527 |

1998M1 | 0.7481 | 0.8608 |

1998M2 | 0.7672 | 0.8799 |

1998M3 | 0.7826 | 0.8978 |

1998M4 | 0.7818 | 0.8966 |

1998M5 | 0.7019 | 0.9285 |

1998M6 | 0.6785 | 0.9634 |

1998M7 | 0.7418 | 0.9666 |

1998M8 | 0.7406 | 0.9389 |

1998M9 | 0.7297 | 0.9630 |

1998M10 | 0.7945 | 0.9829 |

1998M11 | 0.7727 | 1.0064 |

1998M12 | 0.8549 | 1.0470 |

1999M1 | 0.8992 | 1.0504 |

1999M2 | 1.2164 | 1.0794 |

1999M3 | 1.0059 | 0.9987 |

1999M4 | 0.8664 | 0.8989 |

1999M5 | 0.8722 | 0.8933 |

1999M6 | 0.8555 | 0.8694 |

1999M7 | 0.7513 | 0.8496 |

1999M8 | 0.8570 | 0.8978 |

1999M9 | 0.9023 | 0.9194 |

1999M10 | 0.9026 | 0.9217 |

1999M11 | 0.9275 | 0.9388 |

1999M12 | 0.9212 | 0.9290 |

2000M1 | 0.7895 | 0.9183 |

2000M2 | 0.7129 | 0.8713 |

2000M3 | 0.6827 | 0.8225 |

2000M4 | 0.6732 | 0.8155 |

2000M5 | 0.6660 | 0.8053 |

2000M6 | 0.6443 | 0.7894 |

2000M7 | 0.6557 | 0.8103 |

2000M8 | 0.6693 | 0.8418 |

2000M9 | 0.6586 | 0.8181 |

2000M10 | 0.6552 | 0.8128 |

2000M11 | 0.6662 | 0.8336 |

2000M12 | 0.6707 | 0.8167 |

2001M1 | 0.6845 | 0.8327 |

2001M2 | 0.6873 | 0.8317 |

2001M3 | 0.6650 | 0.8221 |

2001M4 | 0.6860 | 0.8345 |

2001M5 | 0.6802 | 0.8388 |

2001M6 | 0.7010 | 0.8635 |

2001M7 | 0.7259 | 0.8978 |

2001M8 | 0.7444 | 0.8945 |

2001M9 | 0.7790 | 0.9393 |

2001M10 | 0.8058 | 0.9692 |

2001M11 | 0.7901 | 0.9535 |

2001M12 | 0.7865 | 0.9581 |

2002M1 | 0.8084 | 0.9749 |

2002M2 | 0.7691 | 0.9504 |

2002M3 | 0.7546 | 0.9288 |

2002M4 | 0.7637 | 0.9510 |

2002M5 | 0.7795 | 0.9586 |

2002M6 | 0.8007 | 0.9991 |

2002M7 | 0.8570 | 1.0128 |

2002M8 | 0.8634 | 1.0105 |

2002M9 | 0.9152 | 0.9737 |

2002M10 | 0.9182 | 0.9709 |

2002M11 | 0.8767 | 0.9234 |

2002M12 | 0.8720 | 0.8818 |

2003M1 | 0.7943 | 0.8289 |

2003M2 | 0.8124 | 0.8363 |

2003M3 | 0.7950 | 0.8471 |

2003M4 | 0.8349 | 0.8451 |

2003M5 | 0.8219 | 0.8293 |

2003M6 | 0.8280 | 0.8312 |

2003M7 | 0.7946 | 0.8102 |

2003M8 | 0.7986 | 0.8139 |

2003M9 | 0.7961 | 0.8094 |

2003M10 | 0.7687 | 0.8177 |

2003M11 | 0.7415 | 0.8004 |

2003M12 | 0.7171 | 0.8070 |

2004M1 | 0.7217 | 0.8095 |

2004M2 | 0.7240 | 0.8093 |

2004M3 | 0.7027 | 0.7829 |

2004M4 | 0.7050 | 0.7467 |

2004M5 | 0.6797 | 0.7446 |

2004M6 | 0.6938 | 0.7347 |

2004M7 | 0.6534 | 0.7247 |

2004M8 | 0.6766 | 0.7277 |

2004M9 | 0.6571 | 0.7357 |

2004M10 | 0.6474 | 0.7299 |

2004M11 | 0.6553 | 0.7607 |

2004M12 | 0.6410 | 0.7419 |

2005M1 | 0.6428 | 0.7423 |

2005M2 | 0.6277 | 0.7474 |

2005M3 | 0.6283 | 0.7464 |

2005M4 | 0.5976 | 0.7375 |

2005M5 | 0.5734 | 0.7099 |

2005M6 | 0.5708 | 0.7039 |

2005M7 | 0.5681 | 0.7006 |

2005M8 | 0.5649 | 0.7115 |

2005M9 | 0.5456 | 0.7136 |

2005M10 | 0.5890 | 0.7267 |

2005M11 | 0.5824 | 0.7228 |

2005M12 | 0.5930 | 0.7374 |

2006M1 | 0.6185 | 0.7591 |

2006M2 | 0.6135 | 0.7349 |

2006M3 | 0.6108 | 0.7391 |

2006M4 | 0.6672 | 0.7633 |

2006M5 | 0.7075 | 0.7871 |

2006M6 | 0.7570 | 0.7871 |

Figure 3c. Length of Forecast Intervals for Benchmark Taylor Rule and Random Walk Models (Japanese Yen), 12-month-ahead forecast

Data for Figure 3c

Date | Taylor Rule | RW |
---|---|---|

1990M11 | 1.0000 | 1.6165 |

1990M12 | 0.9901 | 1.6184 |

1991M1 | 1.1805 | 1.6329 |

1991M2 | 1.2577 | 1.6409 |

1991M3 | 1.4300 | 1.7498 |

1991M4 | 1.4912 | 1.8168 |

1991M5 | 1.4311 | 1.7661 |

1991M6 | 1.3602 | 1.7622 |

1991M7 | 1.1490 | 1.7088 |

1991M8 | 1.0786 | 1.6907 |

1991M9 | 0.9631 | 1.5872 |

1991M10 | 0.9354 | 1.4858 |

1991M11 | 1.0058 | 1.4815 |

1991M12 | 1.0488 | 1.5351 |

1992M1 | 0.8936 | 1.5329 |

1992M2 | 1.0013 | 1.4967 |

1992M3 | 0.9108 | 1.5752 |

1992M4 | 0.8965 | 1.5720 |

1992M5 | 0.8858 | 1.5847 |

1992M6 | 0.8041 | 1.6023 |

1992M7 | 0.7701 | 1.5802 |

1992M8 | 0.8993 | 1.5687 |

1992M9 | 0.9008 | 1.5398 |

1992M10 | 1.0123 | 1.4992 |

1992M11 | 1.0034 | 1.4863 |

1992M12 | 1.0202 | 1.4679 |

1993M1 | 0.9500 | 1.4384 |

1993M2 | 0.8499 | 1.4641 |

1993M3 | 0.9587 | 1.5233 |

1993M4 | 0.9606 | 1.5311 |

1993M5 | 1.0074 | 1.4992 |

1993M6 | 1.0563 | 1.4542 |

1993M7 | 1.0868 | 1.4432 |

1993M8 | 1.0865 | 1.4473 |

1993M9 | 1.0314 | 1.4056 |

1993M10 | 1.0272 | 1.3893 |

1993M11 | 1.0561 | 1.4203 |

1993M12 | 1.0505 | 1.4221 |

1994M1 | 1.0048 | 1.4330 |

1994M2 | 0.9223 | 1.3845 |

1994M3 | 0.9084 | 1.3416 |

1994M4 | 0.6862 | 1.2889 |

1994M5 | 0.5370 | 1.2651 |

1994M6 | 0.5297 | 1.2315 |

1994M7 | 0.5353 | 1.2347 |

1994M8 | 0.5219 | 1.1898 |

1994M9 | 0.5319 | 1.2104 |

1994M10 | 0.7222 | 1.2270 |

1994M11 | 0.7433 | 1.2368 |

1994M12 | 0.7589 | 1.2602 |

1995M1 | 0.9129 | 1.2777 |

1995M2 | 0.7351 | 1.2188 |

1995M3 | 0.7096 | 1.2050 |

1995M4 | 0.6967 | 1.1806 |

1995M5 | 0.7094 | 1.1607 |

1995M6 | 0.7212 | 1.1267 |

1995M7 | 0.6559 | 1.0818 |

1995M8 | 0.6451 | 1.0982 |

1995M9 | 0.5786 | 1.0853 |

1995M10 | 0.5788 | 1.0807 |

1995M11 | 0.5567 | 1.0773 |

1995M12 | 0.7131 | 1.1009 |

1996M1 | 0.6654 | 1.0964 |

1996M2 | 0.5964 | 1.0795 |

1996M3 | 0.2780 | 0.9947 |

1996M4 | 0.5406 | 0.9196 |

1996M5 | 0.5687 | 0.9352 |

1996M6 | 0.5834 | 0.9129 |

1996M7 | 0.3546 | 0.9374 |

1996M8 | 0.3770 | 1.0036 |

1996M9 | 0.7629 | 1.0609 |

1996M10 | 0.8200 | 1.0438 |

1996M11 | 0.8723 | 1.0489 |

1996M12 | 0.8593 | 1.0453 |

1997M1 | 0.9256 | 1.0854 |

1997M2 | 0.9018 | 1.0858 |

1997M3 | 0.8647 | 1.0900 |

1997M4 | 0.8324 | 1.1097 |

1997M5 | 0.8588 | 1.1219 |

1997M6 | 0.8560 | 1.1687 |

1997M7 | 0.8477 | 1.1777 |

1997M8 | 0.8487 | 1.1635 |

1997M9 | 0.8076 | 1.1857 |

1997M10 | 0.8792 | 1.2125 |

1997M11 | 0.8852 | 1.2113 |

1997M12 | 0.8906 | 1.2294 |

1998M1 | 0.9182 | 1.2718 |

1998M2 | 0.8876 | 1.3263 |

1998M3 | 0.8812 | 1.3242 |

1998M4 | 0.8928 | 1.3718 |

1998M5 | 0.9179 | 1.3014 |

1998M6 | 0.8911 | 1.2479 |

1998M7 | 0.8975 | 1.2598 |

1998M8 | 0.9038 | 1.2877 |

1998M9 | 0.8904 | 1.3200 |

1998M10 | 0.8606 | 1.3219 |

1998M11 | 0.8238 | 1.3689 |

1998M12 | 0.5344 | 1.4166 |

1999M1 | 0.8450 | 1.4146 |

1999M2 | 0.8870 | 1.3742 |

1999M3 | 0.8328 | 1.4094 |

1999M4 | 0.7842 | 1.4386 |

1999M5 | 0.8177 | 1.4729 |

1999M6 | 0.8436 | 1.5323 |

1999M7 | 0.8426 | 1.5373 |

1999M8 | 1.0542 | 1.5856 |

1999M9 | 0.8648 | 1.4738 |

1999M10 | 1.0621 | 1.3266 |

1999M11 | 0.8915 | 1.3183 |

1999M12 | 1.0623 | 1.2831 |

2000M1 | 1.0571 | 1.2417 |

2000M2 | 0.8477 | 1.2786 |

2000M3 | 0.8451 | 1.3094 |

2000M4 | 0.7853 | 1.3127 |

2000M5 | 0.7934 | 1.3370 |

2000M6 | 0.7821 | 1.3231 |

2000M7 | 0.7695 | 1.3078 |

2000M8 | 1.0103 | 1.2409 |

2000M9 | 0.9286 | 1.1713 |

2000M10 | 0.7329 | 1.1614 |

2000M11 | 0.7274 | 1.1469 |

2000M12 | 0.7153 | 1.1242 |

2001M1 | 0.7200 | 1.1540 |

2001M2 | 0.7169 | 1.1988 |

2001M3 | 0.7141 | 1.1651 |

2001M4 | 0.8048 | 1.1576 |

2001M5 | 0.7220 | 1.1871 |

2001M6 | 0.8976 | 1.1632 |

2001M7 | 0.8323 | 1.1860 |

2001M8 | 0.9516 | 1.1845 |

2001M9 | 0.7138 | 1.1709 |

2001M10 | 0.9214 | 1.1884 |

2001M11 | 0.9373 | 1.1946 |

2001M12 | 0.7969 | 1.2298 |

2002M1 | 0.6933 | 1.2786 |

2002M2 | 0.7306 | 1.2739 |

2002M3 | 0.6188 | 1.3317 |

2002M4 | 0.5480 | 1.3564 |

2002M5 | 0.5472 | 1.3345 |

2002M6 | 0.4140 | 1.3409 |

2002M7 | 0.3622 | 1.3644 |

2002M8 | 0.5346 | 1.3301 |

2002M9 | 0.5997 | 1.2998 |

2002M10 | 0.5211 | 1.1940 |

2002M11 | 0.3607 | 1.1789 |

2002M12 | 0.3331 | 1.2012 |

2003M1 | 0.3321 | 1.2379 |

2003M2 | 0.2874 | 1.2363 |

2003M3 | 0.8179 | 1.2090 |

2003M4 | 0.8381 | 1.1750 |

2003M5 | 0.8352 | 1.1125 |

2003M6 | 0.8484 | 1.0725 |

2003M7 | 0.8988 | 1.0203 |

2003M8 | 0.8980 | 1.0220 |

2003M9 | 0.8598 | 1.0378 |

2003M10 | 0.8190 | 1.0621 |

2003M11 | 0.8338 | 1.0423 |

2003M12 | 0.8399 | 1.0267 |

2004M1 | 0.8928 | 1.0008 |

2004M2 | 0.8935 | 1.0053 |

2004M3 | 0.8876 | 0.9998 |

2004M4 | 0.8756 | 1.0100 |

2004M5 | 0.9065 | 0.9886 |

2004M6 | 0.8899 | 0.9968 |

2004M7 | 0.8219 | 0.9763 |

2004M8 | 0.7525 | 0.9691 |

2004M9 | 0.7721 | 0.9371 |

2004M10 | 0.7795 | 0.8888 |

2004M11 | 0.7437 | 0.8780 |

2004M12 | 0.7726 | 0.8664 |

2005M1 | 0.7245 | 0.8546 |

2005M2 | 0.7996 | 0.8581 |

2005M3 | 0.7141 | 0.8726 |

2005M4 | 0.6893 | 0.8658 |

2005M5 | 0.5255 | 0.9023 |

2005M6 | 0.5579 | 0.8800 |

2005M7 | 0.5332 | 0.8804 |

2005M8 | 0.5097 | 0.8865 |

2005M9 | 0.5091 | 0.8853 |

2005M10 | 0.5945 | 0.8747 |

2005M11 | 0.6787 | 0.8420 |

2005M12 | 0.7025 | 0.8348 |

2006M1 | 0.6718 | 0.8310 |

2006M2 | 0.6096 | 0.8439 |

2006M3 | 0.5378 | 0.8463 |

2006M4 | 0.4247 | 0.8620 |

2006M5 | 0.4614 | 0.8573 |

2006M6 | 0.3945 | 0.8746 |

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where is the log of money supply, is the log of aggregate price, is the nominal interest rate, is the log of output, and is the money demand shock. A symmetric condition holds in the foreign country and we use an asterisk in superscript to denote variables in the foreign country. Subtracting the foreign money market clearing condition from the home, we have:

(A.1.1) |

The nominal exchange rate is equal to its purchasing power value plus the real exchange rate:

(A.1.2) |

The uncovered interest rate parity in financial market takes the form:

(A.1.3) |

where is the uncovered interest rate parity shock. Substituting equations (A.1.1) and (A.1.2) into (A.1.3), we have

(A.1.4) |

where . Solving recursively and applying the "no-bubbles" condition, we have:

(A.1.5) |

In the standard monetary model, such as Mark (1995), purchasing
power parity () and uncovered interest rate
parity hold (). Furthermore, it is assumed
that the money demand shock is zero (
) and .
Equation (A.1.5) reduces to:

We follow Engel and West (2005) to assume that both countries follow the Taylor rule and the foreign country targets the exchange rate in its Taylor rule. The interest rate differential is:

(A.1.6) |

where is the targeted exchange rate. Assume that monetary authorities target the PPP level of the exchange rate: . Substituting this condition and the interest rate differential into the UIP condition, we have:

(A.1.7) |

where
. Assuming that
uncovered interest rate parity holds () and
monetary shocks are zero, equation (A.1.7) reduces to the benchmark Taylor rule model in our paper:

In this section, we derive long-horizon regressions for the monetary model and the benchmark Taylor rule model.

In the monetary model:

where and are logarithms
of domestic money stock and output, respectively. The superscript
denotes the foreign country. Money
supplies ( and ) and
total outputs ( and ) are
usually I(1) variables. The general form considered in Engel, Wang,
and Wu(2008) is:

(A.2.1) | |||

where is the dimension of
and is an
identity matrix.
is the lag operator and
. Assume
is non-diagonal and the covariance
matrix of
is given by
. We
assume that the change of fundamentals follows a VAR(p) process in
our setup. From proposition 1 of Engel, Wang, Wu (2008), we know
that for a fixed discount factor and ,

is a correctly specified regression where the regressors and errors
do not correlate. In the case of , the
long-horizon regressions reduces to

Following the literature, for instance Mark (1995), we do not include and its lags in our long-horizon regressions. The monetary model can be written in the form of (A.2.1) by setting , . By definition, . This corresponds to , in equation (1) of section 3.

In the Taylor rule model

where , and are domestic aggregate price, output gap and inflation rate, respectively. and are coefficients of the Taylor rule model. The aggregate prices and are usually I(1) variables. Inflation and output gap are more likely to be I(0). Engel, Wang, and Wu (2008) consider a setup which includes both stationary and non-stationary variables:

(A.2.2) |

where and
() are observable (unobservable)
fundamentals.
is the first difference
of
, which contains I(1) economic
variables.^{24}From proposition 2 of Engel, Wang,
and Wu (2008), we know that for a fixed discount factor and ,

(A.2.3) |

is a correctly specified regression, where the regressors and
errors do not correlate. In the case of , the
long-horizon regressions reduces to:

The Taylor rule model can be written into the form of (A.2.2) by setting

By definition,
. This corresponds to
and
, where
is the real exchange
rate.
and
can be defined differently.
For instance,
and
. Our results do not change qualitatively under this alternative
setup.

In the benchmark model, we assume that the Taylor rule coefficients are the same in the home and foreign countries. In this appendix, we relax the assumption of homogeneous Taylor rules in the benchmark model. It is straightforward to show in this case that the benchmark model changes to:

(A.3.1) |

where the deviation of the exchange rate from its equilibrium level is defined as:

(A.3.2) |

According to equation (8), the matrix
in equation (1) includes
economic variables
,
, and
.^{25}We
first estimate the Taylor rules in the home and foreign countries
according to equations (2) and (3). Then
,
,
and
are
used in the long-horizon regressions and interval forecasts. The
results are very similar to the benchmark model and reported in
Table 7.

1. We thank Menzie Chinn, Charles Engel, Bruce Hansen, Jesper Linde, Enrique Martinez-Garcia, Tanya Moldtsova, David Papell, Mark Wynne, and Ken West for invaluable discussions. We would also like to thank seminar participants at the Dallas Fed and the University of Houston for helpful comments. All views are those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of Dallas, the Board of Governors or the Federal Reserve System. All GAUSS programs are available upon request. Return to text

2. Email: jian.wang@dal.frb.org Address: Research Department, Federal Reserve Bank of Dallas, 2200 N. Pearl Street, Dallas, TX 75201 Phone: (214) 922-6471. Return to text

3. Email: jason.j.wu@frb.gov Address: Board of Governors of the Federal Reserve System, 20th and C Streets, Washington, D.C. 20551 Phone: (202) 452-2556. Return to text

4. Chinn and Meese (1995) and MacDonald and Taylor (1994) find similar results. However, the long-horizon exchange rate predictability in Mark (1995) has been challenged by Kilian (1999) and Berkowitz and Giorgianni (2001) in subsequent studies. Return to text

5. For brevity, we omit RS and simply say forecast intervals when we believe that it causes no confusion. Return to text

6. The coefficients on lagged interest rates in the home and foreign countries can take different values in Molodtsova and Papell (2009). Return to text

7. We also tried the random walk with a drift. It does not change our results. Return to text

8. Clarida, Gali, and Gertler (1998) find empirical support for forward-looking Taylor rules. Forward-looking Taylor rules are ruled out because they require forecasts of predictors, which creates additional complications in out-of-sample forecasting. Return to text

9. See appendix for more detail. While the long-horizon regression format of the benchmark Taylor model is derived directly from the underlying Taylor rule model, this is not the case for the models with interest rate smoothing (models 3 and 4). Molodtsova and Papell (2009) only consider the short-horizon regression for the Taylor rule models. We include long-horizon regressions of these models only for the purpose of comparison. Return to text

10. We also tried the case of . Our results do not change qualitatively. Return to text

11. We thank the authors for the data, which we downloaded from David Papell's website. For the exact line numbers and sources of the data, see the data appendix of Molodtsova and Papell (2009). Return to text

12. We choose using the method of Hall, Wolff, and Yao (1999). Return to text

13. It is consistent in the sense of convergence in probability as the estimation sample size goes to infinity. Return to text

14. While RS intervals remedy mis-specifications asymptotically, it does not guarantee such corrections in a given finite sample. Return to text

15. We use Newey and West (1987) for our empirical work, with a window width of 12. Return to text

16. Center here means the half way point between the upper and lower bound for a given interval. Return to text

17. These nine exchange rates are the Danish Kroner, the French Franc, the Deutschmark, the Japanese Yen, the Dutch Guilder, the Portuguese Escudo, the Swiss Franc, and the British pound. Similar results hold at other horizons. Return to text

18. When comparing the intervals for , the random walk model builds the forecast interval around 0, while economic model builds it around . Return to text

19. Results are available upon request. Return to text

20. The only exception is Portugal, where only 192 data points were available. In this case, we choose R = 120. Return to text

21. See Appendix A.3 for details. Return to text

22. See Wu (2009) for more discussion. Return to text

23. Figures in other countries show similar patterns. Results are available upon request. Return to text

24. To incorporate I(0) economic variables, contains the levels of I(1) variables and the summation of I(0) variables from negative infinity to time . Return to text

25. Another option to incorporate heterogenous Taylor rules is to include , , , , and in . For instance, see Moldtsova and Papell (2009). However, increasing the number of regressors may cause the "curse of dimensionality" problem for our semiparametric method. To be comparable to our benchmark model, we define here such that the number of regressors is the same as in the benchmark model. Return to text

This version is optimized for use by screen readers. Descriptions for all mathematical expressions are provided in LaTex format. A printable pdf version is available. Return to text