Monetary policy rules with model and data uncertainty.

1. Introduction

In academic circles, model and data uncertainty are rarely discussed, even when the task at hand involves the construction of real-time forecasts and forecast model selection. By model uncertainty, we mean that the specification and/or parameters of a model are no longer assumed to be fixed and known. While ignoring this type of uncertainty often leads to tractable models that are easily analyzed, it may also paint a picture of the world that is oversimplified. By data uncertainty, we mean that first-released data are often noisy in the sense that incomplete and/or erroneous initial information has been used in their construction. Indeed, it may take many years of revisions before data are considered final.

From the perspective of monetary policy rules, which we here use as a vehicle to discuss model and data uncertainty, it is worth stressing that actual policy decisions are made in a real-time setting using preliminary and/or partially revised data. Thus, questions relating not only to which variables should be used but also to which data releases should be used make the process of policymaking much more complex than it is typically assumed to be in abstract models of monetary policy. In this paper, we build real-time data sets and simulate a real-time policy-setting environment in which we assume that policy is captured by movements in the actual federal funds rate, and we then assess what sorts of policy rule models and what sorts of data best explain what the Federal Reserve Board (Fed) actually did. (1) This approach allows us not only to track the performance of alternative rules over time (hence facilitating a type of model selection among competing rules), but also to more generally assess the importanc e of the data revision process in the formation of macroeconomic time series models.

The class of rules we consider are commonly referred to as Taylor's rules (see Taylor 1979, 1993a) and are motivated by the apparent existence of tradeoffs between inflation and output variability. Versions of these rules have been incorporated in and/or arise in a variety of different macroeconomic models. For example, Rotemberg and Woodford (1997, 1998) developed a rational-expectations model with intertemporally optimizing agents in which various interest rate-targeting rules arise as optimal responses of the monetary authority. This series of papers is important not only because monetary policy rules are shown to arise naturally when expected utility in a representative household is maximized, but also because the model allows for the computation of welfare measures for representative households under different monetary policy rule implementations. On the basis of their theoretical model, as well as a thorough empirical evaluation, Rotemberg and Woodford (1998) find that low and stable inflation together with stable interest rates can be achieved when Taylor's rules of the type we examine are augmented by the inclusion of lagged federal funds rates. Many other extensions and variations of Taylor's rule have been proposed in recent years. For example, policy rules that focus on exchange rates or the money supply are alternatives to rules that focus on interest rates. Indeed, the recent literature on policy rules is extensive. A partial list of relevant papers includes work by Bryant, Hooper, and Mann (1993), Henderson and McKibbin (1993), McCallum (1993, 2001), Taylor (1993b), Frankel and Chinn (1995), Fuhrer and Moore (1995), King and Wolman (1996), Fuhrer (1997), and Orphanides (2001). We take our policy rules as given and do not rationalize them with respect to any particular macroeconomic model. Thus, we do not attempt to offer new insights into the usefulness of policy rules per se (see, e.g., Taylor 1993a, b; Sargent 1999). Moreover, unlike Hansen and Sargent (2000), we do not examine the deeper issue of the effect of model uncertainty on the design of policy rules, as we do not concern ourselves with the specification of a theoretical model. Rather, our approach is to emphasize two related but different issues, namely, (i) model uncertainty viewed through the lens of parameter uncertainty and model specification and (ii) the availability and timing of data with which to examine and implement rules. Uncertainty in policy models is an issue that has recently received some attention in the literature, both from the perspective of model misspecification and from the perspective of learning. Examples of papers in this area include those of Granger and Deutsch (1992), Sargent (1999), Anderson, Hansen, and Sargent (2000), and Hansen and Sargent (2000). Related papers in the area of learning include those of Bray (1982), Marcet and Sargent (1989a, b), Woodford (1990), and Kuan and White (1994), and a review of the learning literature can be found in Marimon (1997).

With regard to data uncertainty, the importance of the timing and availability of the data that are used in the empirical evaluation of policy rules is crucial. In order to address this important issue, we use real-time data sets to replicate the information available to private agents and policymakers at any given point in time in the day-to-day process of policy setting. In this sense, we simulate a real-time policy-setting environment. Our real-time data collection strategy ensures that "future information" due to the use of information that is temporally antecedent to the date under consideration is not (accidentally) incorporated into the data set at the wrong point in time. This is particularly important for seasonally adjusted data, for example, since two-sided filters are generally used in the construction of such data, and the reestimation of the filters after date t using, say, data from t + 1 and t + 2 results in a revised seasonally adjusted figure for t that actually contains information that was available beyond period t.

Before discussing the relative merits of the use of real-time data sets, however, it is worth pointing out that within the context of timing (or availability), economic data can easily be classified into three types: (i) preliminary data, consisting of the first reported datum for each variable at each point in time; (ii) partially revised, or real-time, data, which are much more difficult to collect than preliminary data because they are made up of a full vector of observations at each point in time for each variable; and (iii) fully revised, or final, data, data which have been successively revised and to which no further revisions will be made. These are the types of data that academics often have in mind when conducting economic time series studies, perhaps simply because they are data that are not subject to revision, and it is felt that if one could adequately forecast a fully revised figure, then there would be no need for further modeling. It is quite possible, however, that true final data will never be available for many economic series. (2) Interestingly, most data sets constructed by applied economists clearly consist of a mixture of preliminary data, partially revised data, and final revised data but are clearly not real-time data sets. This poses a number of serious problems for any empirical analysis that is meant to be real-time in nature. Many of the problems associated with not using the "correct" data in the context of monetary policy rules are outlined in Orphanides (2001). Orphanides reconstructs Taylor's rule along the lines of Taylor (1993a) but for real-time data, and he demonstrates that real-time policy recommendations made on the basis of real-time data differ markedly from recommendations made on the basis of partially and fully revised data. In addition, Orphanides (2001) shows that estimated policy reaction functions based on fully revised data are very different from those based on real-time data. Although similar in many respects, our paper differs from that of Orphanides (2001) in a number of ways. For example, we use monthly data over a period of more than 20 years, examine parameter as well as model specification uncertainty, and also consider the effects of the use of seasonally adjusted versus unadjusted data. Orphanides (2001) instead uses quarterly data over the period 1987-1993 for comparison with the results of Taylor (1993a). In addition, we evaluate a large number of alternative policy rules and ascertain which one best mimics the historical record in a real-time setting in which rules are updated regularly as new information becomes available. Orphanides (2001), on the other hand, is more concerned with real-time policy recommendations that are made using Taylor's rule.

In related work, Maravall and Pierce (1983, 1986), Trivellato and Rettore (1986), Ghysels (1987), Sargent (1989), and Swanson, Ghysels, and Callan (1999) examine revision process errors, while Fair and Shiller (1990), Diebold and Rudebusch (1991), Swanson and White (1995, 1997), and Croushore and Stark (2001) point out that the comparison of econometric forecasts based on data from CITIBASE, for example, with forecasts made in real time by professional forecasters (see, e.g., Croushore 1993) is invalid, strictly speaking, because real-time data are not used in the estimation of the econometric models.

Our findings can broadly be summarized as a set of prescriptions and diagnoses that are useful not only in the context of monetary policy rule forecast model selection, but also in the context of the application of real-time data to macroeconomics in general. A partial list of our prescriptions and diagnoses is as follows.

Vintage matters. For example, it is clear that the use of only "final" data does not yield optimal forecasting models. Thus, prediction precision, and hence monetary authority credibility, is affected by the vintage (or release) of data used. Adaptive least-squares learning yields improved results. In particular, while "calibration" is better than naive estimation, both are dominated by an approach to model formation that is based on adaptive least-squares learning. Dynamic information sets are useful. Put another way, policy rules based on distributed lag polynomials of target variables outperform simpler rules. In addition, the correct application of real-time information leads to policy rule precision that is comparable to that achieved with the use of ex post data. Thus, the use of the standard (ex post) sorts of data sets routinely applied in empirical economics not only invalidates any claim that later empirical findings are representative of the real-time flow of events in the economy, but also yields no notable performance enhancement.

Data that are not seasonally adjusted are better. This may be surprising, as it is often argued that seasonal adjustment filters extract the "relevant" component of the data (see, e.g., Ghysels 1994). However, even a cursory examination of our policy simulation results reveals that rules based on data that are not seasonally adjusted have more predictive power than those based on seasonally adjusted data. Moreover, unadjusted data are directly available and avoid filtering problems that are exaggerated in real-time data sets.

Our last main finding is that patience pays off. Forecast models based solely on preliminary data do not minimize mean square forecast error risk. For example, the use of data that have been revised for nine months leads to an ~50% decrease in mean square error (MSE). This result does not suggest that one needs to throw out preliminary data (indeed, Amato and Swanson [2001] show that the "vintages" of data that yield the lowest MSE risk are dependent on the vintage of the actual data used to form forecast errors by subtracting actual realizations from forecasts). Instead, it simply underscores the importance of using preliminary data with care in empirical models.

The rest of the paper is organized as follows. In section 2, we broadly discuss monetary policy rules. Section 3 contains details of the data sets we have constructed. Empirical considerations are discussed in section 4, and our findings are presented in section 5. Conclusions are presented in the final section.

2. A Brief Background of Monetary Policy Rules

John Taylor, in his seminal 1979 paper, introduces nominal rigidities into a rational-expectations framework and derives a model of the macroeconomy in which monetary policy irrelevance does not hold. Optimal monetary policy in this setting exploits a second-order Phillips curve (i.e., a long-run tradeoff between inflation and output volatility), implying that business cycle fluctuations can be reduced by increasing the variability of inflation through accommodating monetary policies. In addition, the most important policy instrument in the United States since the 1960s has arguably been the federal funds rate. (3) Thus, it is not surprising that recent research on optimal monetary policy has focused primarily on the use of short-term interest rates as policy instruments. For example, Bryant, Hooper, and Mann (1993) report on a series of policy rule simulations in which short-term interest rates are adjusted in response to deviations (from predetermined targets) in (i) the exchange rate, (ii) the money supply , (iii) nominal output, and (iv) a combination of inflation (or the price level) and real output. The findings of these researchers suggest that rules that target inflation and output are the most successful in terms of reducing and stabilizing output and price variability. Taylor (1993a) draws on this finding and suggests a simple interest rate policy rule,

[R.sub.t] = 1 + 1.5[[pi].sub.t] + 0.5[y.sub.t], (1)

where [R.sub.t] is the federal funds rate, [[pi].sub.t] is the rate of inflation, and [y.sub.t] is the output gap (i.e., the percentage of deviation of output from its long-run trend). (4) With this rule, the monetary authority raises the federal funds rate if either inflation rises above a target rate (which is assumed to be 2) or real output rises above its long-term trend, with equal weights being applied in either case. Using quarterly data, Taylor (1979) demonstrates that this rule successfully mimics U.S. monetary policy for 1987-1992. Since our analysis is based on monthly data, we replicate Taylor's graphical evidence (see Figure 1) using real-time industrial production and the consumer price index (CPI) series. Even though Taylor's original analysis is not based on real-time data and uses quarterly data, our findings based on various versions of Taylor's rule generally agree with his observations.

In addition to Taylor's (1993a, b) evidence, other recent research that further explores rational-expectations models with sticky prices suggests that simple policy rules successfully mimic the dynamic properties of the economy (e.g., Fuhrer and Moore 1995; Fuhrer 1997; Rotemberg and Woodford 1997, 1998). Fuhrer (1997) estimates the "optimal policy frontier," which dictates the optimal tradeoff between deviations in inflation around a target and output around its potential. The magnitude of this tradeoff rises rapidly when the standard deviation of either inflation or output falls below 2%, which suggests that a balanced policy is preferable. Also, Fuhrer (1997) shows that Taylor's (1979) model lies close to this optimal frontier. Rotemberg and Woodford (1998) suggest an alternate optimal policy, one that responds positively to both the lagged funds rate itself, with a parameter larger than one, and inflation. Rotemberg and Woodford (1998, p. 52) conclude that "a simple interest-rate feedback rule of the kind proposed by Taylor (1993) can achieve outcomes nearly as good as are achievable in principle by any policy, assuming that the commitment of the monetary authority to the rule can be made sufficiently credible."

A novel view of the history of U.S. monetary policy is proposed by Sargent (1999), who introduces a new form of adaptive expectations. The methodology is applied to the hypothesized "regime shift" that followed Paul Volker's election as Chairman the Board of Governors of the Federal Reserve. Sargent suggests that the shift in policy associated with Volker's chairmanship may not have come from a sudden adherence to a rational-expectations philosophy but, rather, gradually by a learning process that was adaptive, driven by the accumulation of information on the success and/or failure of past policies. In particular, Sargent (1999, p. 141) notes that "[t]he regime shifts occur, not from a change in the government's econometric or policymaking procedures, but from a change in the beliefs created by its econometric procedure."

As mentioned above, our approach is to assess the actual performance of a variety of rules had they been followed in real-time. The rules we examine can be summarized as follows (m denotes the vintage of the data used (m is a fixed integer between 2 and 25), and t indexes the horizon over which the policy rule is implemented):

[R.sub.t] = 1 + 1.5[[pi].sub.t-1](t - m) + 0.5[y.sub.t-1](t - m), (rule 1)

[R.sub.t] = [a.sub.0] + [b.sub.1][[pi].sub.t-1](t - m) + [c.sub.1][y.sub.t-1](t - m), (rule 2)

[R.sub.t] = [a.sub.0] + [summation over ([p.sub.1]/j=0)] [b.sub.j][[pi].sub.t-1](t - j - m) + [summation over ([p.sub.2]/j=0)] [c.sub.j][y.sub.t-1](t - j - m), (rule 3)

[R.sub.t] = [a.sub.0] + [summation over ([p.sub.1](t-1)/j=0)] [b.sub.j][[pi].sub.t-1](t - j - m) + [summation over ([p.sub.2](t-1)/j=0)] [c.sub.j][y.sub.t-1](t - j - m), (rule 4)

[R.sub.t] = [a.sub.0] + [summation over ([p.sub.1]/j=1)] [summation over ([p.sub.2]/k=j-1)] [b.sub.jk][[pi].sub.t-j](t - k - m) + [summation over ([p.sub.3]/j=1)] [summation over ([p.sub.4]/k=j-1)] [c.sub.jk][y.sub.t-j](t - k - m). (rule 5)

For rules 1-5, [R.sub.t] is a short-term interest rate instrument (in the present case, either the effective or the target federal funds rate), [[pi].sub.t] is the rate of inflation, and [y.sub.t] is the output gap. Notice that for rules 1 and 2, only one vintage of data is used, and these data are always released at time t - 1 and are thus available at time t. As the vintage, m, varies between 2 and 25, 24 different versions of these rules exist, leading to 24 real-time policy simulations for each rule. Rules 3 and 4 are the same as rules 1 and 2 except that a sequence of different vintages of the target variables, which are all released at time t - 1, are incorporated in the policy rule. Recall that the different vintages correspond to updates of historical data. Rule 5 not only fixes a vintage "starting point" (as do rules 1-4), but also allows previous releases of data for the same calendar time observation to be used by policy setters. This rule thus allows for every dimension of our real-time data sets to be used in the construction of policy rules. Note that rules 3-5 nest what might be called "standard" linear rational-expectations-type forecasting models for [pi] and y, as lag dynamics and parameters are estimated on the basis of various learning criteria, which are discussed below.

Extensions to rules 2-5 that include lagged values of the short-term interest rate instrument are given as follows:

[R.sub.t] = [a.sub.0] + [b.sub.1][[pi].sub.t-1](t - m) + [c.sub.1][y.sub.t-1](t - m) + d[R.sub.t-1], (rule 6)

[R.sub.t] = [a.sub.0] + [summation over ([p.sub.1]/j=0)] [b.sub.j][[pi].sub.t-1](t - j - m) + [summation over ([p.sub.2]/j=0)] [c.sub.j][y.sub.t-1](t - j - m) + d[R.sub.t-1], (rule 7)

[R.sub.t] = [a.sub.0] + [summation over ([p.sub.1](t - 1)/j=0)] [b.sub.j][[pi].sub.t-1](t - j - m) + [summation over ([p.sub.2](t-1)/j=0)] [c.sub.j][y.sub.t-1](t - j - m) + d[R.sub.t-1], (rule 8)

[R.sub.t] = [a.sub.0] + [summation over ([p.sub.1]/j=1)] [summation over ([p.sub.2]/k=j-1)] [b.sub.jk][[pi].sub.t-j](t - k - m) + [summation over ([p.sub.3]/j=1)] [summation over ([p.sub.4]/k=j-1)] [c.sub.jk][y.sub.t-j](t - k - m) + d[R.sub.t-1]. (rule 9)

The above rules can all be interpreted as adaptations of Equation 1 to a real-time setting. For rule 1, the response coefficients are fixed. Hence, there is no parameter estimation by the agent (or the monetary authority, if it uses our schemes to parameterize its policy rules). One interpretation of rule 1 is that it is an optimal solution to some calibrated macroeconomic model. Notice that for any given simulation (across t), rule 1 involves one vintage of data (i.e., m is fixed). Thus, by varying m and comparing simulation results, we are able to assess the relevance of different vintages (or releases) of economic data for model construction. One aspect of this feature of rule 1 is that by examining the performance of this rule across vintages, we can quantify the benefits associated with waiting for more precise updates of the relevant target variables. Note also that we use monthly data, whereas Taylor (1993a) uses quarterly data. Thus, rule 1 is not the same as the Taylor (1993a) rule. Rather, the appro priate analog to Taylor's rule in our context is rule 2, as long as parameters are estimated only once at the beginning of the ex ante simulation (corresponding to our no-window case, discussed below).

As Taylor (1993a) points out, it is not clear whether the response coefficients for rule 1 are optimal. In addition, from the perspective of an agent forming policy forecast models it is reasonable to allow some flexibility in the model specification process. For these reasons, rules 2-9 are based on estimated response coefficients. (In the sequel, we use ordinary least squares (OLS) in all of our estimations.) In our analysis, we consider three different coefficient estimation schemes. The schemes are based on the amount of data used in, and the frequency of, response coefficient estimation. In the first scheme, all parameters are estimated once at the beginning of each real-time policy simulation and remain fixed thereafter. This scheme is referred to as the no-window case. In the second scheme (the fixed-window case), we use fixed rolling 50- and 100-month real-time data windows to reestimate response parameters before each new policy decision is made. The third scheme, the increasing window case, is the s ame as the fixed-window case except that we use an increasing real-time data window to estimate the coefficients, beginning with a window width of 50 months. It should be stressed that these schemes are practically feasible from the perspectives of both real-time adaptive forecast model construction and policymaking, as they entail the use of only information available in real-time.

The first rule that uses estimated coefficients is rule 2 (or rule 6 with lagged [R.sub.t]). This rule has the same policy response structure as rule 1, but with estimated coefficients. By comparing rule 1 and rule 2, we are able to assess the relative merits of using estimated rather than calibrated response coefficients. As Rotemberg and Woodford (1997, 1998) show, however, optimal policy may involve distributed lag polynomials of target variables and policy instruments. Thus, we also consider various Rotemberg-and-Woodford-type rules. First, rule 3 (or rule 7 with lagged [R.sub.t]) defines [R.sub.t] as a function of [p.sub.1] vintages of [pi] and [p.sub.2] vintages of y, where [p.sub.1] and [p.sub.2] are selected using the Schwarz Information Criterion and the maximum number of lags is 24. (5) Notice that although the use of the word lags refers to calendar time lags, by fixing the release date of our data at t - 1, we ensure that the most up-to-date revisions of all lags of our target variables are used. For rule 3, [p.sub.1] and [p.sub.2] do not change as new information becomes available but rather are based on a real-time "startup" sample of observations. Rule 4 (or rule 8 with lagged [R.sub.t]) is the same as rule 3 except that [p.sub.1] and [p.sub.2] are selected anew every time a forecast or policy decision is made. A comparison of rules 3 and 4 thus allows us to assess the impact of parameter uncertainty (which occurs in rules 3 and 4) and model specification uncertainty (which occurs only in rule 4). Note that the type of model specification uncertainty we consider is limited to lag order selection. (6) Moreover, comparison of rule 3 or 4 with rule 2 highlights the impact of including distributed lags dynamics in our policy rules, and comparison of any of rules 2-4 with rule 1 allows us to assess the impact of parameter and/or specification uncertainty on policy rules. In summary, the different ways in which the various rules can be compared allows us to disentangle, at least to some extent, the diffe rent effects that parameter, specification, and data uncertainty have on Taylor's rule.

Thus far, we have only discussed policy rules based on data that are available at time t - 1. In particular, rules 1-4 assume that optimal policy is based on a single release of data. For example, when the subscript on our target variables is t - 1, only data available in period t - 1 are used in policy formation. While these rules clearly entail real-time policy setting, they may be naive in one sense. If private agents (or policymakers) believe that different releases of data for the same calendar period contain different information, then they may want to formulate models based not only on different vintages of data released in t - 1, but also on different vintages of data released in, say, t - 2. For example, if the first and second release observations for January 1990 are formed using information sets that are nonnested, both variables may be useful. Although this sort of scenario might seem surprising, Swanson, Ghysels, and Callan (1999) find some evidence that it is indeed true. In order to allow for this eventuality, we also consider rule 5 (or rule 9 with lagged [R.sub.t]), which mixes vintages and releases of data, thereby allowing private agents or policy setters to explicitly include models of the revision process in their models.

3. Real-Time Data

In the sequel, we use four monthly U.S. time series. Two of the variables (the target and effective federal funds rates) are not subject to revision. The other two variables (industrial production and the CPI) are subject to revision, and real-time data sets for each of them (both seasonally adjusted and unadjusted) have been constructed. In particular, industrial production (IP) data for April 1950-March 1998 (1950:04-1998:03) and CPI data for January 1978-March 1998 (1978: 01-1998:03) have been gathered. It is worth noting that the data matrix for IP contains more than 170,000 nontrivial entries, while the data matrix for the CPI contains more than 29,000 nontrivial entries. As discussed above, the large number of observations in our real-time data sets is due to the nature of the data collection: At each point in time an entirely new sequence of data is collected, going back to the beginning of the sample period. In order to further illustrate these features of our data, we reproduce the matrix structure o f a generic real-time data set in Table 1. The entries in Table 1 are denoted by [X.sub.t]([tau]), where the subscript t refers to the release date of the data pertaining to period [tau], the date in parentheses. Therefore, the diagonal elements in the matrix correspond to the first released, or preliminary, data. For example, the first entry in Table 1 shows the May 1950 release of the April IP or CPI figure. Keeping [tau] fixed, the first row of Table 1 shows the series of revisions from = May 1950 up until the end of our data set at March 1998.

Both IP and CPI data are released on or around the 15th of each month, and a typical months' release of data for these variables comprises a first, or preliminary, release for the previous month and one to five months of revisions to data previously released. In addition, more comprehensive benchmark and base year revisions occur from time to time for each of these variables. Turning first to the IP data, the following details are worth noting. Seasonally adjusted and unadjusted IP figures are compiled by the Fed. The primary source for seasonally adjusted IP data is the Federal Reserve Bulletin. For unadjusted IP (before October 1995), the main source is the Bureau of Economic Analysis publication Survey of Current Business. Additional data for these series were obtained from Federal Reserve monthly statistical releases. Federal Reserve releases for IP are designated G.12.3 before May 1990 and G.17 thereafter. Recent releases and a partial real-time data set from 1972 onward for seasonally adjusted data can also be obtained from the Board of Governors of the Federal Reserve's website (http://www.bog.frb.fed.us/releases). Also, for three of the major (benchmark) revisions to IP, the Fed released separate publications: (i) Industrial Production 1957-59 Base, (ii) Industrial Production 1971, and (iii) 1976 Revision.

Aside from typical monthly revisions to recently released data, there have been various major updates to IP. Numerous updates are "benchmark" updates for which at least 10 years' data are revised. In the January 1997 benchmark update, it was announced that the primary feature of the benchmark update was to reformulate indexes on the basis of weights that are updated annually rather than every five years, as had been previously done. In general, updates involve updating seasonal adjustment weights (for seasonally adjusted data) and incorporating more complete information on important individual series, while benchmark updates additionally involve the revision of series definitions. Another type of update is the base year change. Updates of this type have occurred in July 1971, July 1985, March 1990, and January 1997. for example. Each of these base year updates coincides with a benchmark update, and in fact there are only four recent benchmark updates that do not correspond to base year updates (October 1967, July 1976, January 1994, and January 1998). Recent updates that were not benchmark updates occurred in June 1964, August 1965, October 1966, June 1972, June 1973, August 1977, July 1979, August 1980, August 1981, August 1986, September 1987, April 1993, and November 1994, for example. For IP, there are three missing entries due to two major revisions; these entries are for November and December 1953 and March 1985. We replaced each missing observation with the first available data for that period (which in each case were the data of the second release).

Our real-time data set for seasonally adjusted CPI was constructed from Federal Reserve Bank of St. Louis publications. The main source for these data is National Economic Trends. However, recent CPI releases can be obtained from http://www.stls.frb.org/fred/dataindx.html. In general, benchmark revisions to the CPI occur every 12 months, at which time revisions to the data for the preceding 12 months are reported. There is one base year revision, in January 1988.

For unadjusted CPI data, the series is compiled by the Department of Labor. The sources for the data are the Survey of Current Business and the Department of Labor Publications Monthly Labor Review and Consumer Price Index Detailed Report. There have been three recent base year updates, in January 1971, January 1988, and January 1995. These base year updates coincide with benchmark updates. Current releases and detailed information regarding this series are available at the Bureau of Labor Statistics CPI website (http://stats.bls.gov/cpifact8.htm).

The remaining variables, the target federal funds rate, and the effective federal funds rate (1979:10-1998:04) are available from the Federal Reserve Bank of St. Louis and the Board of Governors of the Federal Reserve, respectively. The target federal funds rate is the announced target of the Board of Governors for the overnight rate on interbank loans--which is currently revealed following each Federal Open Market Committee (FOMC) meeting. (7) Generally, FOMC meetings are held eight times a year, except in special circumstances. We constructed a monthly data set for this series that corresponds to the timing of our IP and CPI data sets. To do this, we assigned the appropriate federal funds rates to each month given the release dates of the IP and CPI data, which are often the 15th of each month and are almost always between the 14th and the 17th. Therefore, we align each announced change in the target federal funds rate with the latest month's release of IP and CPI data that were available to the Board of Go vernors at the time of their decision. The notation we use reflects these timing issues. If we denote some particular target federal funds rate by [r.sub.t], the preliminary data release for this observation is denoted by [X.sub.t-1]([tau]), where [tau] = t - 2, that is, at the time of the board's decision, t, the most recent information available--released at time t - 1-pertains to the calendar date t - 2. Thus, it is the availability of information that defines our "calendar time." The effective federal funds rate is the actual overnight rate on interbank loans. Again, we construct a monthly data set for this series that corresponds to our real-time data sets. In this case, for each month, we calculate the average of the effective federal funds rate for the 4 weeks following the release date of IP and CPI data.

In order to construct operational real-time measures of inflation and of the output gap, we begin by assuming that our data are "final" after 23 revisions, so that our variables are assumed not to change appreciably beyond the 24th release. In our analysis, we consider the first and twelfth differences of CPI. In particular, [[pi].sub.t]([tau] - 1) = 1200(log([CPI.sub.t]([tau] - 1)) - log([CPI.sub.t]([tau] - 2))) or [[pi].sub.t]([tau] - 1) = 100(log([CPI.sub.t]([tau] - 1)) - log([CPI.sub.t]([tau] - 13))), where [tau] = t, t - 1, t - 2, .... We consider annualized monthly inflation because it corresponds to the standard data transformation used in many empirical studies. On the other hand, twelfth-differenced CPI data are used in order to facilitate a comparison of real-time policy simulation outcomes based on either seasonally adjusted or unadjusted data. As [tau] varies, we essentially construct two new inflation series for each release of CPI data. Our output gap variable is the deviation of industrial pro duction from its log linear trend. Industrial production trend regressions are estimated using samples of six to twelve years' data, and as our empirical results were found to be qualitatively similar in all cases, we use nine years' data for output gap calculations hereinafter. In addition, we use real-time data in all trend regressions and include seasonal dummy variables when constructing gap estimates on the basis of data that are not seasonally adjusted. Thus, we construct the output gap [y.sub.t]([tau] - 1) = 100(log([IP.sub.t]([tau] - 1)) - log([IP.sub.t]([tau] - 1))), where [IP.sub.t]([tau] - 1) is the forecast of trend output based on information available at time t - 1, and [tau] = t, t - 1, t - 2,....

4. Empirical Findings

In order to facilitate the comparison of our rules, we report results for four different factors. In particular, we examine (i) ex ante mean square forecast error and mean absolute percentage error, (ii) turning point predictability, (iii) parameter estimates, and (iv) model specification. As discussed above, our approach is to simulate a real-time policy-setting environment, thereby mimicking the behavior of private agents forming expectations in real time, or the behavior of government policymakers. To do this, we consider ex ante policy evaluation periods of 50 and 100 months. In addition to our two policy evaluation periods, we examine seasonally adjusted and unadjusted data, output gap measures based on 24th-vintage data and real-time data, and the target and effective federal funds rates. Thus, there are 32 different permutations of data that can be used in the construction of tables associated with the four factors mentioned above.

In order to streamline the presentation of our findings, we omit various results that are either uninteresting or comparable to other findings that we do report. First, we report results for a 50-month evaluation period with the effective federal funds rate and a real-time output gap measure. Results for the 100-month evaluation period, the target federal funds rate, and the 24th-vintage-based output gap measure are qualitatively similar and are available on request from us. Second, we report results only for rules 1-5. Although ex ante MSE results, for example, are always better when lagged values of the policy instrument are used, our findings across rules, windows, and vintages remain qualitatively similar, and hence tabulated results for rules 6-9 are omitted for the sake of brevity. Third, we omit tabular evidence based on first-differenced data. As noted above, first differences were used only to examine seasonally adjusted data. We omit these results because our findings based on twelfth differences ar e superior to comparable results based on first differences. This finding is attributable to the fact that first-differenced real-time data are considerably noisier than real-time data based on twelfth differences. This is because the former type of data involve first and second vintages, while the latter involve first and twelfth vintages, and twelfth-vintage data are relatively more accurate than second-vintage data. (8) Finally, our evidence based on absolute percentage error loss measures does not differ from the evidence we offer based on MSE loss measures and is thus omitted for the sake of brevity.

Tables 2-9 summarize our main empirical findings. In order to get a feel for the data we use in our analysis and particularly for the importance of the revision process, we begin by providing various summary statistics that are presented in Table 2. In this table, we report on two types of data revisions, fixed- and increasing-width revisions. The fixed-width revisions are constructed as [X.sub.t+i](t) - [X.sub.t+i-1](t), and the increasing-width revisions are constructed as [X.sub.t+i](t) - [X.sub.t](t), for I = 1, ... , 11, 18, and 24. Table 2 contains three panels, corresponding to adjusted and unadjusted industrial production as well as adjusted CPI, for which mean, variance, skewness, and kurtosis figures are reported. (9)

Observe that the mean of fixed-length revisions is significantly different from zero at a 95% level of confidence for numerous vintages of data. This finding suggests that there is systematic bias in revisions of our variables, and such information could in principle be used to increase the accuracy of preliminary releases. Notice also that on the basis of increasing width-revisions, the difference between "final" (i.e., i = 24) and initial releases of data have mean biases that are significantly different from zero. This implies that a statistically significant correction could be made to all releases of the variables prior to their final release. Finally, the skewness and kurtosis statistics reported in Table 2 suggest that data revisions are characterized by clear departures from normality. This may be due to the presence of outliers in the revision process, which implies that real-time policy setting based on recent releases of data may result in policy decisions that are quite different from those which would have been made had we known the final data. Thus, the data revision process may be quite important for policy setting. A comprehensive analysis of the data revision process for our real-time data sets that includes plots of revision autocorrelation functions and discussion of the pattern of outliers in the data is given in Swanson, Ghysels, and Callan (1999). As we use inflation and output gap variables in our subsequent analysis, we also present basic statistics that are analogous to those given in Table 2 for [pi] and y (see Table 3). The results based on Table 3 are largely similar to those discussed above for the raw data, with the notable exception that the incidence of significant mean revisions given in the second column of Table 3 is reduced. However, note that the absolute magnitude of these revisions is actually quite large, suggesting that this finding may be due to the small sample sizes used in the construction of the statistics.

The rest of our empirical findings are presented in Tables 4-9. Our findings with regard to data uncertainty are reported in Tables 4 and 5 for seasonally adjusted data and in Tables 7 and 8 for data that are not seasonally adjusted. Tables 6 (adjusted data) and 9 (unadjusted data) summarize our findings with regard to parameter and model uncertainty. In Tables 4 and 7, we report MSEs associated with 50 real-time policy decisions. (10) The first columns of the tables give the vintage of the data. The calendar date of the data used for all rules is t - 1, except for rule 5, for which a mixture of calendar dates is used (see above). The second columns of the tables report MSEs for Taylor's rule (rule 1). The next three panels report MSEs for rules 2-5, for which response coefficients and/or rule specifications vary and for which the window of observations used by policy setters is fixed, increasing, or nonexistent. Tables 5 and 8 are presented in the same way, except that so-called "confusion rates" are reporte d rather than MSEs. The confusion rate indicates the proportion of times that our policy decision incorrectly predicts the direction of change in realized interest rates. Thus, a value of 0.50 corresponds to a policy rule that captures directional changes in interest rates so poorly that were we to flip a coin, we could do equally well. (11)

Finally, Tables 6 and 9 report the average and standard deviation of the response coefficients associated with the target variables when rule 2 is the policy tool. As in the previous tables, the first columns of these tables report the vintage of the data used to implement the rule. On the basis of the results presented in Tables 4-9, our findings can be summarized as follows.

Vintage Matters

It can be seen that vintage matters by noting that MSE values in Tables 4 and 7 are dependent on vintage. In Tables 4 and 7, the smallest MSE values are in boldface type for each rule. For instance, in Table 4, for rule 1, the MSE varies from 1.439 to 4.447 depending on vintage. Moreover, MSEs do not monotonically increase as vintage increases. Thus, it is not clear whether the use of only final data results in optimal forecast models. In fact, notice that the lowest MSEs in Tables 4 and 7 are associated with a fixed window of data and occur for relatively recent vintages. For example, for rule 4 with the fixed window of data, the lowest MSE vintage is t-11 for adjusted data and t-10 for unadjusted data. In addition, notice that for the same rule and the increasing window of data, the analogous lowest MSE vintages are t-2 and t-3. Thus, we have evidence that preliminary data are useful. (12) It is important to note that the "MSE-best" vintage varies across rules. This is expected, given that our rules exploit the real-time information set differently. For example, it should be expected that the MSE-best vintage for rule 2 is higher than the MSE-best vintage for rule 3. This is indeed the case for all of our results and follows because, for example, rule 3 uses all information from vintage m back, while rule 2 uses information only for a given vintage.

Adaptive Least-Squares Learning Yields Improved Results

On the basis of both seasonally adjusted and unadjusted data, the smallest MSE value for rule i is larger than the smallest MSE values for the other rules (except for rule 2) when the fixed-and/or increasing-window data are used (see Tables 4 and 7). This finding suggests that adaptive model formation is useful. Also, in the no-window case, where response coefficients are not updated, MSEs are actually worse than they are for rule 1, for which no estimation whatsoever is done. Thus, one might conclude on the basis of this finding that "calibration" (i.e., rule 1) is better than naive estimation (i.e., the no-window case) but worse than adaptive least-squares learning (i.e., the fixed- and increasing-window cases).

Dynamic Information Sets Are Useful

In Tables 4 and 7, when the fixed- and increasing-window cases are compared, rules 3 and 4 always "MSE-dominate" rule 2. In fact, MSE-best values are reduced by more than 50% in the fixed-window case when rules 2 and 3 are compared. For example, on the basis of unadjusted data (Table 7), the MSE-best value is 0.534 for rule 2 and 0.149 for rule 3, corresponding to a 72% increase in precision when dynamic information sets are used. (13) This finding provides evidence that policy rules based on distributed lag polynomials of target variables may outperform simpler rules (for further evidence see, e.g., Rotemberg and Woodford 1997, 1998) from the perspective of choosing which rule most closely mirrors what the Fed appears to have actually done.

Further evidence that dynamic information sets are useful is presented in Figure 1, in which the ex ante performance of rules 1-4 for a fixed window of data is graphically illustrated. Even casual observation of the graphs is sufficient to see that rules 3 and 4 perform better than rule 2 and that rules 2-4 perform better than rule 1. One way to check whether the graphical evidence provided in Figure 1 is indicative of the usefulness of our policy rules in practice (e.g., in the construction of forecasts and to establish monetary authority and hence monetary policy credibility) is to compare the ex ante performance of the rules with analogous ex post performance. By ex post performance, we mean that only final revised data are used in the rules. We implement our ex post analysis by using data released in April 1998. Thus, all data in our ex post sample prior to May 1996 are final, while newer data have been revised fewer than 23 times and hence are not final according to our definition. Notice that this type of ex post analysis is currently the norm rather than the exception in empirical economics, as, for example, data which have been revised many times are routinely downloaded from CITIBASE, with no regard for the fact that these data, although representative of information at time t, were actually not available until time t + i for i large. The use of data sets of this sort clearly invalidates any claim that later empirical findings are representative of the real-time flow of events in the economy. Nevertheless, a comparison of our ex ante policy simulation results with those of an ex post policy simulation should yield evidence concerning the usefulness of ex ante rules. Such evidence is given in Figure 2, in which it is apparent that the performance of our ex ante rules is essentially as good as that of an analogous set of ex post rules for which finalized data are assumed to be known. This result holds across rules and window specifications. (Results for windows other than the fixed window reported in Figur e 2 are available on request from us.) One reason for the seemingly excellent ex ante performance of the rules relative to their ex post counterparts is that adaptive least squares is used, so response coefficients are allowed to "adjust" to the type of data used.

Real-Time Specification Flexibility Pays Off

Rule 4 is the MSE-best rule based on both adjusted and unadjusted data. Thus, learning is useful in two respects. First, updating response coefficients is better than fixing coefficients (the no-window vs. the fixed- and increasing-window cases). Second, updating the order of the lag polynomial (model uncertainty) in real time is better than fixing the specification (rule 4 vs. rules 1-3 and 5).

Revision Bias is Not as Important as the Use of Real-Time Data

The finding that revision bias is not as important as the use of real-time data is based on a comparison of MSE-best values for rules 3 and in Tables 4 and 7, in which it can be clearly seen that rule 3 is MSE-best. Notice that rule 3 is closest to rule 5, as neither incorporates specification flexibility. Thus, rule 5 directly augments rule 3 by including the history of data revision in addition to different vintages of data. One caveat to this finding is that we cannot actually claim that the use of real-time data is more important than revision bias; rather, we can say that given real-time data, revision history adds nothing in terms of MSE.

Data that Are Not Seasonally Adjusted Are Better

The finding that unadjusted data are better may be surprising, as it is often argued that seasonal adjustment filters extract the "relevant" component of the data. If a time series is viewed as the sum of different unobserved components, then it is argued that the seasonal component is "irrelevant" (see, e.g., Ghysels 1994). However, what is often not recognized is that seasonal adjustment entails more frequent and larger data revisions than are associated with unadjusted data (see, e.g., Maravall and Pierce 1983). Since all of our data are annual differences, the numerical entries in Tables 4 and 7 are directly comparable. Even a cursory examination of these tables reveals that the MSE-best unadjusted rules dominate the corresponding MSE-best adjusted rules. For example, the overall MSE-best value in all of our tables is 0.124, which obtains when unadjusted data, rule 4, and the fixed window of data are used. The corresponding entry based on adjusted data is 0.244, roughly twice that based on unadjusted data . A further assessment of the noisiness of preliminary seasonally adjusted data is obtained when one examines rule 2, which does not incorporate lagged polynomial information sets and which is clearly our simplest adaptive rule. In the fixed-window case, an MSE of 1.630 obtains with vintage t - 2 when unadjusted data are used. The analogous figure based on adjusted data is 2.033. Furthermore, the MSE based on adjusted data does not decrease to 1.630 until vintage t - 8, suggesting that many revisions are necessary to smooth out the noisiness associated with seasonally adjusted data. Thus, using unadjusted data in policy decisions is MSE-preferable. This means that when constructing a real-time rule that best mimics the actual activity of the Fed, one is better off using unadjusted data. (14)

Confusion Rate Findings Based on Seasonally Adjusted Data Are Confused

Notice that all of our findings with respect to rule and data window selection and based on unadjusted data are the same when either MSE or confusion rate loss is used (see Tables 7 and 8). However, analogous findings based on adjusted data and confusion rate loss do not agree with comparable MSE results (see Tables 4 and 5). In particular, the "confusion-rates-best" models (i.e., entries in boldface type) associated with rule 1 in Table 5 are often lower than those associated with other rules in Table 5, regardless of which data window is used. This is clearly not the case with unadjusted data. If these sorts of conflicting findings occurred regardless of data adjustment, one might be led to believe that either (i) the choice of loss function is crucial or (ii) our empirical findings are not robust. Indeed, there is much evidence in the economic forecasting literature that the choice of loss function is important (see, e.g., Granger 1969; Leitch and Tanner 1991). However, given that our results are robust to evaluation period and output gap specification, it seems more likely that the problem lies in the use of seasonally adjusted data. For example, it is known that filters of various types alter the comovements between economic series (see, e.g., Canova 1998). In summary, our findings are consistent with a conjecture that turning-point prediction is more sensitive to filtering than is MSE-based prediction. This conclusion is sensible if one believes that unusual (or one-time) events drive turning points, as these types of events tend to manifest themselves as observational outliers that trigger smoothing corrections in standard adjustment programs (e.g., X-11 and X-12; see Findley et al. 1998; Ghysels, Granger, and Siklos 1996).

Response Coefficients Are Weakly Sensitive to the Choice of Vintage

Our results with respect to response coefficient sensitivity are presented in Tables 6 (adjusted data) and 9 (unadjusted data). The layout of these tables is slightly different from that of the other tables. We take rule 2 as a starting point and select the MSE-best vintage for each data window ([m.sub.opt]). We then formulate rules as follows:

[R.sub.t] = [a.sub.0] + [b.sub.1][[pi].sub.t-[m.sub.opt]+j](t - [m.sub.opt]) + [c.sub.1][y.sub.t-[m.sub.opt]+j](t - [m.sub.opt]), (2)

where j = 1,...,24. We use this rule instead of rule 2, as it allows us to separate out the impact of parameter uncertainty from that of specification uncertainty--both are mixed together if one examines parameter evolution based on rule 2. In particular, direct use of rule 2 mixes parameter and specification uncertainty because the calendar lags used in the rule depend on the vintage. The mean of the response coefficients given in Tables 6 and 9 varies smoothly, relatively monotonically, and with little variability across j. Note that output response coefficients do not depend critically on the data window for data that are not seasonally adjusted (see Table 9) but are more sensitive to the data window when adjusted data are used (see Table 6). Inflation response coefficients, however, appear to depend critically on the data window used, again with higher variability across windows being associated with seasonally adjusted data. Thus, while parameter uncertainty is not prevalent, response coefficients depend on the data window used and hence on rule design.

Patience Pays Off

Rules and models based solely on preliminary data do not minimize mean square forecast error risk. For example, our MSE-best rule (rule 4, fixed window, unadjusted data) performs much better when one waits for nine months before using new data (i.e., a 58% MSE reduction, from 0.298 to 0.124). This result does not suggest that one needs to throw out preliminary data (indeed, Amato and Swanson [2001] show that the vintages of data that yield the lowest MSE risk are dependent on the vintage of actual data used in forming forecast errors by subtracting actual realizations from forecasts). Instead, it simply underscores the importance of using preliminary data with care in empirical models.

5. Conclusions

In this paper, we have examined model and data uncertainty in the context of monetary policy rules. Particular emphasis is placed on two related but different issues, namely, (i) model uncertainty viewed through the lens of parameter and specification uncertainty and (ii) the availability and timing of data with which to examine and evaluate policy rules. In order to carry out our analysis, we built a number of large real-time data sets and carried out a series of experiments in the context of a real-time policy-setting environment. In the context of linear policy forecast models, we find that data vintage (or release) is important and that adaptive least-squares learning-based methods are preferable to simpler model formation strategies. In addition, noise produced by seasonal adjustment filtering is prohibitively extensive when measured in a variety of different ways, prompting us to conclude that the use of unadjusted data is preferable to the use of seasonally adjusted data when the objective is to form r eal-time rules that mimic what the Fed is actually going to do and what the Fed has already done. Finally, it appears to be in the best interest of forecasters to wait until some of the data uncertainty associated with preliminary data has been removed by the revision process before updating adaptive forecast models, particularly if the objective is to forecast final data.

In summary, we believe that the importance of using real-time data has not yet been fully recognized in mainstream empirical economics, and many empirical techniques rely too heavily on the presumption that economic data are final and readily available. The construction of monetary policy rules is only one example, albeit a very important one, of how data and model uncertainty become relevant when we attempt to gather empirical findings that are representative of the real-time flow of events in the economy.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Table 1

The Layout of a Generic Real-Time Data Set

                                 Release Date of Data
Date to which
Data Pertain   i = 1950:05   i = 1950:06   i = 1950:07      i = t

j = 1950:04    [X.sub.i](j)  [X.sub.i](j)  [X.sub.i](j)  [X.sub.i](j)
j = 1950:05         0        [X.sub.i](j)  [X.sub.i](j)  [X.sub.i](j)
j = 1950:06         0             0        [X.sub.i](j)  [X.sub.i](j)

j = t - 1           0             0             0        [X.sub.i](j)
j = t               0             0             0             0
j = t + 1           0             0             0             0

j = 1998:02         0             0             0             0
j = 1998:03         0             0             0             0

                                 Release Data of Data
Date to which
Data Pertain    i = t + 1     i = t + 2    i = 1998:03   i = 1998:04

j = 1950:04    [X.sub.i](j)  [X.sub.i](j)  [X.sub.i](j)  [X.sub.i](j)
j = 1950:05    [X.sub.i](j)  [X.sub.i](j)  [X.sub.i](j)  [X.sub.i](j)
j = 1950:06    [X.sub.i](j)  [X.sub.i](j)  [X.sub.i](j)  [X.sub.i](j)

j = t - 1      [X.sub.i](j)  [X.sub.i](j)  [X.sub.i](j)  [X.sub.i](j)
j = t          [X.sub.i](j)  [X.sub.i](j)  [X.sub.i](j)  [X.sub.i](j)
j = t + 1           0        [X.sub.i](j)  [X.sub.i](j)  [X.sub.i](j)

j = 1998:02         0             0        [X.sub.i](j)  [X.sub.i](j)
j = 1998:03         0             0             0        [X.sub.i](j)

Entries correspond to data pertaining to a particular calendar data
(first column) but released in revised form a various subsequent dates
("Release Date of the Data"). Thus, the diagonal of the real-time data
set corresponds to first available data (which are assumed to be made
available with a one-month lag), while the last row of data in the
matrix corresponds to the entire sample of data which would be available
in 1998:04. Thus, the last column of data corresponds to data usually
used in empirical studies. However, note that the last column of data
involves various different vintages such that some data are first
available, some have been revised once, and so on as one moves up the
column of entries.
Table 2

Real-Time Data Set Summary Statistics

Seasonally adjusted industrial production: 1979:02-1996:05

                      Fixed-Width Revisions              Increasing
                                                           -Width
                                                         Revisions
          Mean       Variance    Skewness    Kurtosis   Mean

i = 1      0.076       0.398      13.79       19.59     0.076
i = 2      0.096       0.400      13.94       19.87     0.171 *
i = 3      0.023       0.410      14.08       20.04     0.226 *
i = 4      0.015 *     0.475      14.11       20.21     0.221 *
i = 5      0.014 *     0.409      14.11       20.93     0.218 *
i = 6      0.014 *     0.469      14.10       20.69     0.224 *
i = 7      0.014 *     0.486      14.04       20.51     0.231 *
i = 8      0.015 *     0.442      14.06       20.98     0.226 *
i = 9      0.015 *     0.407      14.07       20.12     0.232 *
i = 10     0.015 *     0.429      14.08       20.33     0.227 *
i = 11     0.005       0.480      14.10       20.68     0.225 *
i = 12     0.014       0.426      14.11       20.20     0.225 *
i = 18     0.015 *     0.434      13.99       19.95     0.230 *
i = 24     0.014       0.444      13.65       19.37     0.220 *

              Increasing-Width Revisions
           Variance    Skewness    Kurtosis

i = 1        0.398     13.79       19.59
i = 2        0.848      9.776       9.846
i = 3        0.899      9.790       9.837
i = 4        0.887      9.814       9.927
i = 5        0.806      9.800       9.620
i = 6        0.781      9.763       9.147
i = 7        0.829      9.795       9.686
i = 8        0.804      9.784       9.446
i = 9        0.820      9.791       9.606
i = 10       0.827      9.786       9.543
i = 11       0.801      9.788       9.537
i = 12       0.803      9.811       9.894
i = 18       0.840      9.770       9.448
i = 24       0.821      9.718       9.853

Seasonally unadjusted industrial production: 1979:02-1996:05

                      Fixed-Width Revisions              Increasing
                                                           -Width
                                                         Revisions
          Mean       Variance    Skewness    Kurtosis   Mean

i = 1     -0.191 *     4.444     -0.145        3.454   -0.191 *
i = 2     -0.058 *     0.110      1.347       28.31    -0.207 *
i = 3     -0.016       0.078     -6.073       81.67    -0.216 *
i = 4     -0.025       0.357     -8.080      107.2     -0.146
i = 5      0.046 *     0.288      13.07      180.4     -0.184 *
i = 6      0.007       0.017      12.33      169.2     -0.177 *
i = 7      0.014 *     0.032      13.59      191.5     -0.176 *
i = 8      0.015 *     0.031      12.10      179.0     -0.180 *
i = 9      0.012 *     0.022      13.08      180.3     -0.179 *
i = 10     0.013 *     0.026      12.43      167.4     -0.182 *
i = 11     0.009       0.018      12.75      174.5     -0.183 *
i = 12     0.009       0.026      11.54      156.7     -0.181 *
i = 18    <0.001       0.001     -12.56       99.04    -0.205 *
i = 24    -0.001       0.043     -13.73      194.0     -0.220 *

              Increasing-Width Revisions
           Variance    Skewness    Kurtosis

i = 1        4.044     -0.145       3.454
i = 2        4.126     -0.142       3.404
i = 3        4.390     -0.413       4.213
i = 4        4.247     -0.075       3.547
i = 5        4.060     -0.154       3.435
i = 6        4.077     -0.153       3.417
i = 7        4.087     -0.150       3.406
i = 8        4.075     -0.150       3.421
i = 9        4.071     -0.154       3.419
i = 10       4.078     -0.161       3.435
i = 11       4.093     -0.167       3.454
i = 12       4.064     -0.151       3.437
i = 18       4.121     -0.163       3.408
i = 24       4.222     -0.232       3.596

Seasonally adjusted consumer price index: 1987:02-1996:05

                      Fixed-Width Revisions              Increasing
                                                           -Width
                                                         Revisions
          Mean       Variance    Skewness    Kurtosis   Mean

i = 1     -0.003 *    <0.001      -3.880      26.33    -0.003 *
i = 2     -0.003 *    <0.001      -2.972      21.18    -0.007 *
i = 3     -0.004 *    <0.001      -3.576      21.64    -0.006 *
i = 4     -0.003 *    <0.001      -3.839      25.53    -0.004 *
i = 5     -0.001 *    <0.001      -7.105      52.06    -0.003 *
i = 6     <0.001      <0.001      -0.551      18.39    -0.003 *
i = 7     -0.001      <0.001      -0.743      22.04    <0.000
i = 8      0.003 *    <0.001       3.904      30.62    -0.001
i = 9      0.002 *    <0.001       4.502      38.60    <0.000
i = 10     0.003 *    <0.001       4.560      36.19    -0.002
i = 11     0.001       0.001       0.241      23.66    -0.004 *
i = 12    -0.001       0.001      -1.865      25.46    -0.006 *
i = 18    -0.002 *    <0.001      -3.018      21.14    -0.006 *
i = 24    <0.001      <0.001      -5.155      22.52    -0.003 *

              Increasing-Width Revisions
           Variance    Skewness    Kurtosis

i = 1       <0.000     -3.880       26.33
i = 2        0.001     -2.391       11.36
i = 3        0.001     -2.505       12.43
i = 4        0.001     -3.274       18.60
i = 5       <0.000     -1.694       11.96
i = 6        0.001     -1.814       12.86
i = 7        0.001      0.439       15.16
i = 8        0.001     -0.169       16.37
i = 9        0.001      0.325       16.32
i = 10       0.001     -0.410       13.74
i = 11       0.001     -1.577       14.09
i = 12       0.001     -3.363       16.47
i = 18       0.001     -2.837       13.68
i = 24      <0.000     -3.880       26.33

Fixed-width revisions are constructed as ln([X.sub.t+i-1](t - 2)) -
ln([X.sub.t+i-2](t-2)), and increasing-width revisions are constructed
as ln([X.sub.t+i-1](t - 2)) - ln([X.sub.t-1](t - 2)), where X is either
IP or CPI. In these definitions, the subscript refers to the release
date of the data, while the index in parentheses denotes the date to
which the release pertains (see Table 1). For example, the "i - 2" rows
in the table correspond to ln([X.sub.t+1](t - 2)) - ln([X.sub.t](t - 2))
for the fixed-width revisions. In this case, the second release for the
period t is subtracted from the third release for period t. For
increasing-width revisions, the "i = 2" rows correspond to
ln([X.sub.t+1](t - 2)) - ln([X.sub.t-1](t - 2)), so that the first
release (or first available data) is subtracted from the third release
for the period t - 2. The sample period for which we present summary
statistics is determined by our real-time simulation experiments. The
period reported on for IP is longer than that for CPI becasue additional
IP data were needed in order to estimate trend lines for use in output
gap construction. Results for our larger sample periods (from 1950 for
IP and from 1978 for CPI) are qualitatively similar and are available on
request from us. The end period of the data is 1996:05, which
corresponds to the last interest rate observation used in our subsequent
ex ante analysis. This end date was used (as opposed to the actual end
of our sample--1998:03) in order to facilitate an ex ante-versus-ex post
comparison of policy rule performance (see discussion in section 5).

* Significantly different from zero on the basis of a 95% confidence
interval constructed using a herteroskedasticity-and
autocorrelation-consistent variance estimator.
Table 3

Summary Statistics for Vintages of the Output Gap and Inflation

                                      Fixed-Width Revisions
                                    Mean    Variance   Skewness

Seasonally adjusted output
 gap: 1990:02-1996:03
i = 1                             -8.675 *   540.7       -0.830
i = 2                             -5.631 *   384.2       -1.109
i = 3                             -4.011 *   187.0       -3.779
i = 4                             -0.192     122.2       -7.887
i = 5                             -0.389     139.0       -8.119
i = 6                             -0.873      87.56      -9.754
i = 7                              0.097      34.03      -3.703
i = 8                              0.162      54.95      -5.178
i = 9                              0.070      75.62       2.289
i = 10                            -0.323      97.71      -0.336
i = 11                             0.042     211.7        3.069
i = 12                            -1.281     151.9       -8.149
i = 18                            -0.556      28.02      -1.987
i = 24                            -2.474     527.2       -7.995

Seasonally unadjusted output
 gap: 1990:02-1996:03
i = 1                             -7.375 *   480.5        0.630
i = 2                             -3.129     587.1        3.501
i = 3                             -2.018     153.0        1.331
i = 4                             -4.046 *      3228    -10.51
i = 5                              6.848        3534     10.37
i = 6                              1.187     132.8        9.861
i = 7                              1.960     317.7        9.972
i = 8                              1.914     301.9       10.24
i = 9                              0.416     276.6        5.994
i = 10                             0.887     286.6        8.896
i = 11                            -0.123     328.4        2.660
i = 12                             0.503     337.4        7.622
i = 18                            -1.255     157.7      -10.62
i = 24                            -1.750     204.6       -8.509

Seasonally adjusted inflation:
 1990:02-1996:03
i = 1                             -0.086       1.326     -2.231
i = 2                             -0.098       1.460     -2.551
i = 3                             -0.455 *     3.049     -3.585
i = 4                             -0.266 *     1.913     -5.038
i = 5                              0.110       1.575      3.104
i = 6                             -0.110       3.284     -1.314
i = 7                             -0.105       3.101     -1.505
i = 8                              0.196       5.539      1.709
i = 9                              0.124       3.171      3.810
i = 10                             0.265       4.620      3.742
i = 11                             0.297      13.741      3.280
i = 12                             0.026       8.354     -1.008
i = 18                             0.102       1.279     10.82
i = 24                            -0.015       0.029    -10.82

                                Fixed-Widt      Increasing-Width
                                    h               Revisions
                                Revisions
                                Kurtosis        Mean     Variance

Seasonally adjusted output
 gap: 1990:02-1996:03
i = 1                              5.743       -8.675 *   540.7
i = 2                              9.198      -14.30 *   1087
i = 3                             33.96       -18.31 *   1267
i = 4                             86.90       -18.50 *   1398
i = 5                             87.28       -18.89 *   1625
i = 6                            104.1        -19.77 *   1783
i = 7                             58.49       -19.67 *   1834
i = 8                             61.66       -19.51 *   1946
i = 9                             52.15       -19.44     2087
i = 10                            35.42       -19.76     2169
i = 11                            56.22       -19.72 *   2347
i = 12                            79.83       -21.00 *   2440
i = 18                            27.74       -25.89 *   2632
i = 24                            76.15       -33.06     3513

Seasonally unadjusted output
 gap: 1990:02-1996:03
i = 1                              6.688       -7.375 *   480.5
i = 2                             27.13       -10.50 *   1095
i = 3                              9.921      -12.52 *   1165
i = 4                            114.8        -16.56 *   4571
i = 5                            112.0         -9.721 *  1243
i = 6                            103.5         -8.534 *  1313
i = 7                            105.1         -6.574 *  1639
i = 8                            109.7         -4.660    1845
i = 9                             76.21        -4.245    2054
i = 10                            97.76        -3.358    2322
i = 11                            56.88        -3.481    2729
i = 12                            84.59        -2.978    3125
i = 18                           116.0         -9.481    4296
i = 24                            77.65       -16.22     4938

Seasonally adjusted inflation:
 1990:02-1996:03
i = 1                             31.21        -0.086       1.326
i = 2                             30.02        -0.184       2.769
i = 3                             14.20        -0.639 *     5.650
i = 4                             27.37        -0.906 *     7.220
i = 5                             36.79        -0.795 *     8.997
i = 6                             17.58        -0.906 *    12.10
i = 7                             19.79        -1.011 *    15.01
i = 8                             27.36        -0.815      20.95
i = 9                             36.92        -0.692      24.32
i = 10                            27.87        -0.427      29.31
i = 11                            32.00        -0.130      43.31
i = 12                            23.69        -0.104      51.67
i = 18                           119.0         -0.432      64.37
i = 24                           119.0         -0.262      64.00

                                   Increasing-Width
                                      Revisions
                                 Skewness   Kurtosis

Seasonally adjusted output
 gap: 1990:02-1996:03
i = 1                              -0.830      5.743
i = 2                              -0.526      4.543
i = 3                              -0.646      4.452
i = 4                              -0.920      5.057
i = 5                              -1.140      5.991
i = 6                              -1.234      5.917
i = 7                              -1.193      5.645
i = 8                              -1.135      5.243
i = 9                              -1.002      4.802
i = 10                             -0.899      4.429
i = 11                             -0.763      4.058
i = 12                             -0.670      3.762
i = 18                             -0.461      3.379
i = 24                             -0.852      4.130

Seasonally unadjusted output
 gap: 1990:02-1996:03
i = 1                               0.630      6.688
i = 2                               1.293      8.607
i = 3                               1.103      7.251
i = 4                              -6.713     65.89
i = 5                               1.071      6.916
i = 6                               1.006      6.365
i = 7                               1.477      8.046
i = 8                               1.512      7.523
i = 9                               1.407      6.727
i = 10                              1.434      6.375
i = 11                              0.974      5.800
i = 12                              1.118      5.899
i = 18                              0.444      4.910
i = 24                              0.402      4.271

Seasonally adjusted inflation:
 1990:02-1996:03
i = 1                              -2.231     31.21
i = 2                              -1.552     15.14
i = 3                              -1.563      7.050
i = 4                              -1.374      5.282
i = 5                              -0.682      5.055
i = 6                              -0.374      3.949
i = 7                              -0.208      3.319
i = 8                               0.194      3.930
i = 9                               0.336      3.670
i = 10                              0.429      3.317
i = 11                              0.795      4.867
i = 12                              0.545      4.025
i = 18                              0.189      3.428
i = 24                              0.148      3.442

See notes to Table 2. Fixed-width revisions are constructed as
([X.sub.t+i-1](t - 2) - [X.sub.t+i-2](t - 2)) X 100, and
increasing-width revisions are constructed as ([X.sub.t+i-1](t - 2) -
[X.sub.t-1](t - 2)) X 100, where X is either the output gap or
inflation. The sample period reported corresponds to the period used in
the 50-month policy simulation reported in subsequent tables.
Table 4

Real-Time Mean Square Errors for Various Rules: Seasonally Adjusted Data

                            No Window                 Fixed Window
Vintage   Rule 1      Rule 2  Rule 3    Rule 5    Rule 2    Rule 3

t - 25    4.447        2.690  2.645     7.553      0.851    0.308
t - 24    4.155        2.948  2.697     5.511      0.842    0.327
t - 23    3.908        3.067  2.976     4.380      0.844    0.350
t - 22    3.625        3.252  4.583     3.065      0.822    0.500
t - 21    3.346        3.330  3.888     2.877      0.797    0.552
t - 20    2.855        3.388  5.675     3.110      0.784    0.457
t - 19    2.299        3.467  4.961     3.232      0.782    0.548
t - 18    1.809        3.555  4.700     3.417      0.809    0.445
t - 17    1.559        3.415  4.193     3.121      0.859    0.483
t - 16    1.439        3.396  2.857     3.664      0.954    0.558
t - 15    1.451        3.610  2.696     3.519      1.062    0.552
t - 14    1.485        4.355  2.333     4.210      1.127    0.474
t - 13    1.604        5.495  3.547     5.351      1.196    0.442
t - 12    1.711        7.193  3.163     6.669      1.264    0.375
t - 11    1.794        8.360  3.788     8.377      1.346    0.360
t - 10    1.896        8.258  4.372     9.596      1.430    0.356
t - 9     2.014        8.099  4.590     8.888      1.517    0.355
t - 8     2.169        7.904  5.208     8.601      1.646    0.377
t - 7     2.275        7.417  5.496     11.998     1.726    0.324
t - 6     2.391        6.557  3.941     8.013      1.808    0.343
t - 5     2.543        5.996  2.767     7.547      1.906    0.320
t - 4     2.655        5.574  2.188     4.422      1.988    0.307
t - 3     2.705        5.260  4.865     4.356      2.005    0.373
t - 2     2.793        5.042  5.097     3.519      2.033    0.355

              Fixed Window                   Increasing Window
Vintage   Rule 4    Rule 5       Rule 2     Rule 3    Rule 4    Rule 5

t - 25    0.435     0.308        1.522      1.214     1.204     1.757
t - 24    0.536     0.327        1.591      1.157     1.354     1.761
t - 23    0.441     0.350        1.635      1.187     1.328     1.580
t - 22    0.396     0.500        1.708      1.139     1.292     1.933
t - 21    0.361     0.552        1.736      1.114     1.292     1.498
t - 20    0.480     0.457        1.752      1.201     1.242     1.651
t - 19    0.441     0.591        1.772      1.303     1.190     1.722
t - 18    0.437     0.445        1.818      1.311     1.325     1.656
t - 17    0.471     0.483        1.820      1.291     1.365     1.658
t - 16    0.445     0.558        1.811      1.254     1.332     1.794
t - 15    0.283     0.632        1.765      1.191     1.357     1.758
t - 14    0.296     0.564        1.639      1.355     1.229     1.553
t - 13    0.281     0.442        1.511      1.588     1.141     1.581
t - 12    0.275     0.375        1.453      1.440     1.008     1.352
t - 11    0.214     0.360        1.425      1.318     0.977     1.395
t - 10    0.244     0.356        1.396      1.345     0.853     1.267
t - 9     0.307     0.355        1.393      1.297     0.810     1.581
t - 8     0.359     0.880        1.437      1.299     0.781     1.872
t - 7     0.373     0.418        1.451      1.285     0.676     2.261
t - 6     0.360     1.396        1.446      0.821     0.600     1.216
t - 5     0.269     0.471        1.475      0.552     0.551     1.148
t - 4     0.216     0.624        1.489      0.455     0.454     0.552
t - 3     0.231     0.535        1.457      0.594     0.404     0.521
t - 2     0.259     0.564        1.462      0.576     0.401     0.449

All rules are based on information available at time t - 1; the data in
the first column represents the vintage of data to which the data
pertain. The dependent variable in our policy rules is the effective
federal funds rate. The regressors in the rules are inflation and the
output gap. The period reported on is 1992:04- 1996:05, so mean square
errors are calculated on the basis of 50 policy decisions. Results based
on longer policy simulation periods are discussed in section 5. Entries
for rule 1 are based on Taylor's rule, for which inflation and output
gap response coefficients are fixed at 1.5 and 0.5, respectively.
Entries for the no-Window cases are based on the fixed response
coefficients. These coefficients are estimated using an initial 50-month
presample of real-time data. For the fixed-window cases, mean square
errors are constructed for policy rules that incorporate adaptive
parameter estimation schemes. In these cases, a fixed window of 50
months of real-time data (which are available to policymakers at the
time of their decisions) is used each time coefficients are revised. For
the increasing-window cases, response coefficients are coefficients are
constructed in a manner analogous to that for the fixed-window cases,
except that the windows of data used to estimate response coefficients
increase by one months each time a new policy decision is made. The
initial window in these cases is 50 months (1998:02-1992:03). The
smallest values for each rule are in boldface type.
Table 5

Real-Time Confusion Rates for Various Rules: Seasonally Adjusted Data

                            No Window                 Fixed Window
Vintage   Rule 1    Rule 2    Rule 3    Rule 5    Rule 2    Rule 3

t - 25    0.468     0.596     0.532     0.426     0.447     0.447
t - 24    0.362     0.532     0.362     0.447     0.362     0.362
t - 23    0.404     0.511     0.404     0.426     0.383     0.426
t - 22    0.362     0.447     0.404     0.383     0.362     0.383
t - 21    0.447     0.404     0.404     0.362     0.362     0.404
t - 20    0.426     0.404     0.362     0.362     0.340     0.319
t - 19    0.447     0.404     0.596     0.383     0.404     0.426
t - 18    0.426     0.383     0.617     0.404     0.383     0.511
t - 17    0.447     0.404     0.596     0.383     0.383     0.553
t - 16    0.426     0.404     0.404     0.426     0.883     0.468
t - 15    0.340     0.383     0.362     0.426     0.447     0.681
t - 14    0.489     0.468     0.383     0.362     0.468     0.553
t - 13    0.426     0.447     0.383     0.362     0.511     0.447
t - 12    0.426     0.447     0.404     0.426     0.447     0.426
t - 11    0.404     0.447     0.383     0.426     0.468     0.404
t - 10    0.383     0.447     0.447     0.426     0.489     0.404
t - 9     0.383     0.447     0.426     0.447     0.489     0.404
t - 8     0.383     0.447     0.426     0.447     0.511     0.426
t - 7     0.404     0.447     0.447     0.447     0.511     0.489
t - 6     0.426     0.447     0.383     0.468     0.489     0.447
t - 5     0.426     0.447     0.404     0.447     0.532     0.511
t - 4     0.426     0.447     0.404     0.447     0.553     0.532
t - 3     0.447     0.447     0.447     0.447     0.553     0.447
t - 2     0.426     0.447     0.447     0.447     0.574     0.511

              Fixed Window                   Increasing Window
Vintage   Rule 4    Rule 5    Rule 2        Rule 3    Rule 4    Rule 5

t - 25    0.362     0.426     0.426         0.447     0.404     0.426
t - 24    0.511     0.426     0.447         0.362     0.447     0.383
t - 23    0.532     0.362     0.426         0.362     0.404     0.426
t - 22    0.362     0.404     0.404         0.511     0.404     0.426
t - 21    0.468     0.383     0.404         0.383     0.383     0.362
t - 20    0.426     0.362     0.404         0.340     0.404     0.404
t - 19    0.426     0.426     0.362         0.404     0.447     0.383
t - 18    0.447     0.383     0.404         0.426     0.426     0.426
t - 17    0.447     0.426     0.383         0.468     0.447     0.404
t - 16    0.426     0.468     0.383         0.447     0.426     0.383
t - 15    0.511     0.404     0.404         0.383     0.447     0.383
t - 14    0.511     0.404     0.468         0.362     0.362     0.362
t - 13    0.489     0.404     0.468         0.362     0.383     0.362
t - 12    0.447     0.404     0.404         0.383     0.340     0.426
t - 11    0.383     0.447     0.489         0.426     0.383     0.426
t - 10    0.511     0.404     0.489         0.404     0.383     0.404
t - 9     0.511     0.340     0.447         0.426     0.426     0.426
t - 8     0.426     0.489     0.426         0.447     0.447     0.404
t - 7     0.468     0.574     0.340         0.447     0.447     0.468
t - 6     0.447     0.511     0.426         0.383     0.447     0.447
t - 5     0.426     0.447     0.404         0.447     0.468     0.426
t - 4     0.532     0.362     0.404         0.383     0.340     0.447
t - 3     0.362     0.426     0.426         0.383     0.340     0.383
t - 2     0.362     0.553     0.426         0.383     0.255     0.426

See notes to Table 4. Entries are confusion rates, defined as the
proportion of times (over the ex ante simulation period) for which the
direction of change in the effective federal funds rate associated with
a real-time policy decision corresponds to the ex post realized value
for the federal funds rate.
Table 6

Response Coefficient Summary Statistics Based on Rule 2: Seasonally
Adjusted Data

             No Window, [m.sub.opt] = 25     Fixed Window, [m.sub.opt]
                                                       = 19
j          [b.sub.j]       [c.sub.j]              [b.sub.j]
                                           ([[sigma].sub.[b.sub.j]])

 1           -0.411          0.735              0.084 (0.199)
 2           -0.409          0.771              0.082 (0.193)
 3           -0.406          0.797              0.068 (0.197)
 4           -0.410          0.812              0.064 (0.205)
 5           -0.410          0.810              0.069 (0.207)
 6           -0.413          0.808              0.074 (0.210)
 7           -0.413          0.805              0.077 (0.213)
 8           -0.413          0.802              0.079 (0.216)
 9           -0.419          0.797              0.081 (0.218)
10           -0.420          0.797              0.085 (0.219)
11           -0.434          0.796              0.087 (0.220)
12           -0.445          0.793              0.085 (0.229)
13           -0.457          0.791              0.090 (0.237)
14           -0.463          0.788              0.095 (0.241)
15           -0.466          0.786              0.106 (0.245)
16           -0.468          0.783              0.115 (0.248)
17           -0.465          0.783              0.119 (0.252)
18           -0.463          0.783              0.115 (0.252)
19           -0.459          0.779              0.112 (0.252)
20           -0.461          0.767              0.111 (0.251)
21           -0.466          0.758              0.107 (0.254)
22           -0.466          0.754              0.107 (0.257)
23           -0.466          0.747              0.110 (0.257)
24           -0.469          0.740              0.115 (0.259)

          Fixed Window, [m.sub.opt]     Increasing Winwow,
                    = 19                  [m.sub.opt] = 9
j                [c.sub.j]                [b.sub.j]
          ([[sigma].sub.[c.sub.j]]) ([[sigma].sub.[b.sub.j]])

 1             0.522 (0.128)            1.156 (0.339)
 2             0.519 (0.128)            1.154 (0.342)
 3             0.519 (0.132)            1.145 (0.351)
 4             0.518 (0.135)            1.145 (0.354)
 5             0.517 (0.136)            1.151 (0.357)
 6             0.516 (0.135)            1.155 (0.361)
 7             0.516 (0.136)            1.162 (0.362)
 8             0.516 (0.135)            1.168 (0.361)
 9             0.516 (0.135)            1.178 (0.362)
10             0.516 (0.135)            1.180 (0.363)
11             0.516 (0.133)            1.180 (0.363)
12             0.514 (0.133)            1.186 (0.372)
13             0.514 (0.131)            1.199 (0.380)
14             0.515 (0.131)            1.202 (0.379)
15             0.517 (0.128)            1.209 (0.380)
16             0.521 (0.126)            1.212 (0.380)
17             0.521 (0.125)            1.221 (0.378)
18             0.521 (0.126)            1.213 (0.380)
19             0.519 (0.125)            1.209 (0.381)
20             0.517 (0.124)            1.210 (0.380)
21             0.513 (0.123)            1.209 (0.383)
22             0.508 (0.123)            1.211 (0.385)
23             0.505 (0.122)            1.216 (0.384)
24             0.500 (0.122)            1.224 (0.383)

             Increasing Winwow,
               [m.sub.opt] = 9
j                [c.sub.j]
          ([[sigma].sub.[c.sub.j]])

 1             0.489 (0.020)
 2             0.483 (0.019)
 3             0.481 (0.019)
 4             0.480 (0.019)
 5             0.479 (0.019)
 6             0.478 (0.019)
 7             0.477 (0.018)
 8             0.478 (0.018)
 9             0.479 (0.018)
10             0.480 (0.018)
11             0.480 (0.018)
12             0.479 (0.018)
13             0.479 (0.019)
14             0.479 (0.019)
15             0.481 (0.020)
16             0.484 (0.021)
17             0.485 (0.021)
18             0.484 (0.021)
19             0.482 (0.020)
20             0.481 (0.020)
21             0.478 (0.020)
22             0.475 (0.019)
23             0.473 (0.019)
24             0.471 (0.019)

See notes to Table 4. [m.sub.opt] is the vintage of data for which the
mean square error is minimized for rule 2 based on the no-window,
fixed-window, and increasing-window cases. j represents the timing of
the release of data. Using the notation described in the paper, the
coefficients reported in this table are based on the regression
[R.sub.t] = [a.sub.0] + [b.sub.j] [[pi].sub.t-[m.sub.opt]+j] (t -
[m.sub.opt]) + [c.sub.j] [y.sub.t-[m.sub.opt]+j] (t - [m.sub.opt]).
Table 7

Real-Time Mean Square Errors for Various Rules: Seasonally Unadjusted
Data

                            No Window                 Fixed Window
Vintage   Rule 1     Rule 2   Rule 3    Rule 5     Rule 2    Rule 3

t - 25     5.426      2.380    4.285     5.324      1.041     0.331
t - 24     4.956      2.495    7.510     6.859      0.973     0.777
t - 23     4.566      2.520    7.532     4.525      0.888     0.799
t - 22     4.146      2.604    7.114     1.365      0.811     0.736
t - 21     3.753      2.482    4.786     5.521      0.687     0.556
t - 20     3.160      2.575    5.145     3.932      0.638     0.739
t - 19     2.456      2.597    4.472     3.319      0.562     0.367
t - 18     1.858      2.584    2.576     2.454      0.534     0.578
t - 17     1.449      2.252    2.856     2.467      0.534     0.466
t - 16     1.136      2.007    3.221     5.712      0.559     0.356
t - 15     1.043      1.975    2.203     3.698      0.615     0.341
t - 14     0.973      2.152    2.338     3.726      0.636     0.278
t - 13     0.974      3.637    2.060     2.262      0.661     0.339
t - 12     0.959      4.802    2.023     2.275      0.700     0.148
t - 11     0.903      5.577    2.619     2.870      0.737     0.340
t - 10     0.891      5.743    2.423     3.082      0.809     0.329
t - 9      0.897      6.360    3.653     3.988      0.898     0.167
t - 8      0.929      6.473    3.090     3.044      0.993     0.245
t - 7      0.963      6.310    3.111     3.000      1.117     0.225
t - 6      1.014      5.713    2.932     2.999      1.237     0.253
t - 5      1.105      5.354    3.558     4.800      1.364     0.280
t - 4      1.167      4.908    3.848     3.306      1.482     0.257
t - 3      1.206      4.523    3.653     3.536      1.572     0.271
t - 2      1.309      4.418    2.742     3.848      1.630     0.349

              Fixed Window                  Increasing Window
Vintage   Rule 4    Rule 5     Rule 2       Rule 3  Rule 4    Rule 5

t - 25     0.510     0.453      1.330        0.842   1.102     0.952
t - 24     0.496     0.836      1.311        1.253   0.954     1.235
t - 23     0.403     0.799      1.267        1.211   0.979     0.899
t - 22     0.415     0.736      1.261        1.208   0.936     0.771
t - 21     0.444     0.634      0.219        0.832   0.975     1.063
t - 20     0.410     0.679      1.217        1.093   1.044     1.024
t - 19     0.468     0.513      1.250        0.829   1.063     1.045
t - 18     0.396     0.531      1.317        1.035   1.002     1.036
t - 17     0.347     0.551      1.350        1.233   1.001     1.155
t - 16     0.383     0.376      1.316        1.105   1.109     1.054
t - 15     0.264     0.285      1.314        1.251   1.111     1.277
t - 14     0.253     0.326      1.198        1.115   1.105     1.033
t - 13     0.271     0.408      1.173        1.103   1.076     0.920
t - 12     0.174     0.293      1.128        1.063   1.050     1.175
t - 11     0.174     0.376      1.060        1.184   1.071     1.206
t - 10     0.124     0.345      1.035        1.105   1.004     1.010
t - 9      0.133     0.195      1.056        1.044   0.950     1.462
t - 8      0.225     0.281      1.066        1.057   0.812     1.161
t - 7      0.180     0.312      1.119        1.113   0.630     0.992
t - 6      0.315     0.308      1.139        0.696   0.571     0.674
t - 5      0.264     0.352      1.179        0.459   0.467     0.531
t - 4      0.245     0.281      1.180        0.638   0.322     0.559
t - 3      0.237     0.329      1.172        0.562   0.306     0.516
t - 2      0.298     0.392      1.230        0.650   0.458     0.464

See notes to Table 4.
Table 8

Real-Time Confusion Rates for Various Rules: Seasonally Unadjusted Data

                            No Window                 Fixed Window
Vintage   Rule 1    Rule 2    Rule 3    Rule 5     Rule 2    Rule 3

t - 25     0.595     0.617     0.446     0.468      0.595     0.510
t - 24     0.638     0.595     0.446     0.446      0.553     0.468
t - 23     0.595     0.617     0.446     0.404      0.595     0.446
t - 22     0.595     0.595     0.446     0.425      0.553     0.446
t - 21     0.595     0.595     0.404     0.404      0.510     0.361
t - 20     0.617     0.574     0.446     0.425      0.319     0.404
t - 19     0.553     0.617     0.404     0.383      0.468     0.489
t - 18     0.446     0.595     0.404     0.361      0.446     0.340
t - 17     0.489     0.446     0.383     0.617      0.361     0.361
t - 16     0.489     0.383     0.595     0.574      0.425     0.446
t - 15     0.468     0.340     0.531     0.595      0.468     0.468
t - 14     0.489     0.489     0.617     0.425      0.510     0.446
t - 13     0.595     0.446     0.574     0.404      0.510     0.425
t - 12     0.553     0.446     0.446     0.383      0.489     0.510
t - 11     0.468     0.446     0.425     0.425      0.553     0.383
t - 10     0.468     0.446     0.383     0.404      0.489     0.340
t - 9      0.425     0.446     0.383     0.595      0.446     0.361
t - 8      0.383     0.446     0.489     0.553      0.553     0.383
t - 7      0.383     0.446     0.425     0.425      0.489     0.425
t - 6      0.404     0.446     0.446     0.425      0.531     0.383
t - 5      0.425     0.446     0.446     0.425      0.468     0.404
t - 4      0.425     0.446     0.468     0.446      0.531     0.468
t - 3      0.425     0.446     0.425     0.404      0.574     0.383
t - 2      0.404     0.446     0.489     0.617      0.574     0.404

              Fixed Window                  Increasing Window
Vintage   Rule 4    Rule 5     Rule 2       Rule 3  Rule 4    Rule 5

t - 25     0.553     0.489      0.617        0.404   0.468     0.489
t - 24     0.574     0.489      0.553        0.446   0.425     0.446
t - 23     0.404     0.531      0.531        0.425   0.468     0.446
t - 22     0.404     0.404      0.531        0.446   0.404     0.468
t - 21     0.489     0.383      0.510        0.383   0.404     0.425
t - 20     0.361     0.446      0.383        0.340   0.404     0.383
t - 19     0.425     0.361      0.404        0.340   0.383     0.340
t - 18     0.404     0.361      0.404        0.383   0.361     0.383
t - 17     0.425     0.425      0.383        0.319   0.361     0.383
t - 16     0.468     0.553      0.383        0.383   0.383     0.361
t - 15     0.383     0.553      0.425        0.425   0.446     0.383
t - 14     0.468     0.468      0.510        0.425   0.404     0.404
t - 13     0.425     0.425      0.510        0.404   0.361     0.446
t - 12     0.404     0.595      0.446        0.425   0.383     0.383
t - 11     0.383     0.531      0.489        0.383   0.404     0.446
t - 10     0.404     0.510      0.468        0.383   0.446     0.446
t - 9      0.404     0.446      0.404        0.383   0.425     0.468
t - 8      0.425     0.425      0.446        0.425   0.446     0.468
t - 7      0.425     0.425      0.425        0.383   0.468     0.468
t - 6      0.489     0.489      0.510        0.404   0.510     0.446
t - 5      0.297     0.468      0.446        0.446   0.425     0.510
t - 4      0.404     0.446      0.425        0.425   0.425     0.425
t - 3      0.383     0.531      0.425        0.425   0.425     0.404
t - 2      0.404     0.531      0.404        0.468   0.361     0.383

See notes to Table 5.
Table 9

Response Coefficient Summary Statistics Based on Rule 2: Seasonally
Unadjusted Data

             No Window, [m.sub.opt] = 15     Fixed Window, [m.sub.opt] =
                                                         17
 j         [b.sub.j]        [c.sub.j]              [b.sub.j]
                                           ([[sigma].sub.[b.sub.j]])

 1           0.052            0.502              0.152 (0.264)
 2           0.255            0.596              0.159 (0.177)
 3           0.257            0.624              0.124 (0.164)
 4           0.260            0.636              0.113 (0.166)
 5           0.162            0.528              0.100 (0.171)
 6           0.259            0.636              0.105 (0.161)
 7           0.263            0.639              0.102 (0.155)
 8           0.266            0.640              0.094 (0.149)
 9           0.276            0.638              0.096 (0.139)
10           0.266            0.640              0.080 (0.134)
11           0.271            0.641              0.074 (0.127)
12           0.248            0.634              0.052 (0.128)
13           0.254            0.635              0.046 (0.121)
14           0.249            0.634              0.035 (0.119)
15           0.243            0.631              0.034 (0.120)
16           0.240            0.628              0.035 (0.121)
17           0.239            0.627              0.036 (0.120)
18           0.237            0.625              0.037 (0.119)
19           0.237            0.624              0.037 (0.119)
20           0.233            0.618              0.040 (0.118)
21           0.236            0.622              0.039 (0.118)
22           0.237            0.620              0.042 (0.116)
23           0.237            0.620              0.041 (0.116)
24           0.237            0.620              0.041 (0.116)

          Fixed Window, [m.sub.opt] =      Increasing Window,
                      17                    [m.sub.opt] = 10
 j                [c.sub.j]                  [b.sub.j]
          ([[sigma].sub.[c.sub.j]])  ([[sigma].sub.[b.sub.j]])

 1              0.468 (0.069)              0.903 (0.308)
 2              0.505 (0.086)              0.888 (0.275)
 3              0.518 (0.096)              0.851 (0.270)
 4              0.515 (0.098)              0.839 (0.266)
 5              0.504 (0.085)              0.838 (0.262)
 6              0.517 (0.096)              0.832 (0.259)
 7              0.518 (0.096)              0.826 (0.256)
 8              0.517 (0.097)              0.816 (0.252)
 9              0.520 (0.097)              0.819 (0.246)
10              0.521 (0.096)              0.802 (0.246)
11              0.520 (0.097)              0.796 (0.243)
12              0.517 (0.095)              0.773 (0.246)
13              0.515 (0.097)              0.768 (0.244)
14              0.511 (0.099)              0.756 (0.244)
15              0.510 (0.097)              0.754 (0.244)
16              0.509 (0.096)              0.754 (0.243)
17              0.508 (0.096)              0.754 (0.242)
18              0.507 (0.095)              0.755 (0.240)
19              0.507 (0.095)              0.755 (0.240)
20              0.504 (0.093)              0.758 (0.237)
21              0.506 (0.094)              0.757 (0.238)
22              0.505 (0.093)              0.760 (0.235)
23              0.505 (0.093)              0.760 (0.236)
24              0.505 (0.093)              0.760 (0.236)

              Increasing Window,
               [m.sub.opt] = 10
 j                [c.sub.j]
          ([[sigma].sub.[c.sub.j]])

 1              0.483 (0.029)
 2              0.487 (0.026)
 3              0.495 (0.025)
 4              0.498 (0.025)
 5              0.498 (0.025)
 6              0.498 (0.025)
 7              0.498 (0.025)
 8              0.496 (0.024)
 9              0.496 (0.024)
10              0.494 (0.024)
11              0.492 (0.023)
12              0.488 (0.023)
13              0.486 (0.022)
14              0.483 (0.021)
15              0.482 (0.021)
16              0.482 (0.021)
17              0.481 (0.021)
18              0.481 (0.021)
19              0.481 (0.021)
20              0.480 (0.021)
21              0.480 (0.021)
22              0.480 (0.021)
23              0.480 (0.021)
24              0.480 (0.021)

See notes on Table 6.

Received June 2000; accepted December 2001.

(1.) Our policy rule models are specialized in a number of respects. For example, they are all linear. In addition, we include only inflation, industrial production, and past funds rates as explanatory variables.

(2.) This is because benchmark and definitional changes, for example, are ongoing and may continue into the indefinite future. Moreover, seasonal adjustment filters involve two-sided filters that in principle have infinite leads and lags.

(3.) See Bemanke and Blinder (1992) and Bemanke and Mihov (1998) for related discussion and for a detailed account of recent U.S. monetary policy operations.

(4.) "It should be stressed that Taylor does not advocate this particular rule and notes that "... simple, algebraic formulations of such rules cannot and should not be mechanically followed by policymakers" (Taylor 1993a, p. 213). However, we view Equation 1 as his "best" specification.

(5.) Recall that when referring to vintages of data, we are referring to data available at some calendar date that have been revised. Hence, the mth vintage of data available at time t - 1 is the revised datum for period t - m.

(6.) For example, although nonlinear rules might be relevant, they are not investigated.

(7.) The FOMC's current practice is to reveal its fund rate target right after each meeting, although, as pointed out by an anonymous referee, this did not used to be the case.

(8.) This observation has been made previously by Maravall and Pierce (1983) in the context of the noisiness of preliminary seasonally adjusted data.

(9.) Tables 2 and 3 do not contain summary statistics for revisions in seasonally unadjusted CPI data, as revisions occur infrequently and primarily in conjunction with base year benchmark revisions.

(10.) In the tables, the end of our sample is May 1996 so that we do not exhaust our entire data set, which ends in March 1998. The reason for this is that we also carried out ex post policy simulations (assuming that finalized data were known) in order to assess the accuracy of our ex ante results (see below).

(11.) A case could be made for using the target rather than the effective federal funds rate when simulating confusion rates. Unlike the federal funds rate, the target rate contains a small number of discrete level shifts that are directly associated with policy decisions, while the effective rate is a smoothed version of the target rate that might also be impacted by market forces. Thus, if we were to use the target rate, we would need to include a "no-change" case in the contingency tables from which our confusion rates are taken. In addition, if the funds rate is determined only by expectations or Fed policy, then real-time data and data revision will matter if the expectations of the Fed are formed at least in part on the basis of historical data that are subject to revision. This idea corresponds to the notion that if the true state of the economy affects the funds rate, then data revision is important, although in this case dependence on data revision enters directly through the state of the economy rat her than indirectly through expectation formation on the part of the Fed.

(12.) This does not mean that final data are not useful (except in the case of rule 2), but rather that the data set that is MSE-best must contain preliminary data, while it may also contain earlier vintages, including final data.

(13.) Note also that the lower MSE in this case is obtained with the use of an earlier vintage of data.

(14.) It is also worth noting that Sims's (1974) and Wallis's (1974) results on estimation with filtered data do not apply in our context, as interest rates are unfiltered so that seasonal adjustment impacts response coefficient estimates.

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Eric Ghysels, * Norman R. Swanson, + and Myles Callan ++

* Department of Economics, University of North Carolina at Chapel Hilt and Centre Interuniversitaire de Recherche et Analyse des Organisations, Gardner Hall. CB 3305, Chapel Hill, NC 27599-3305, USA; E-mail [e-mail].

+ Department of Economics, Rutgers University, 75 Hamilton Street, New Brunswick, NJ 08901-1248, USA; E-mail [e-mail]; corresponding author.

++ Department of Economics. Clark University, 950 Main Street, Worcester, MA 01610, USA; E-mail [e-mail].

We thank Dean Croushore, Lars Hansen, Glenn Rudehuach, and Thomas Sargent for stimulating conversations during the writing of this paper and Brian Preslopsky for supplying us with target federal funds rate data. Also, comments made by seminar participants at the Federal Reserve Board, the Federal Reserve Bank of Kansas, the Federal Reserve Bank of St. Louis, Pennsylvania State University, Texas A&M University, the University of Kansas, the University of North Carolina, the University of Pennsylvania, an anonymous referee, and the editor, Kent Kimbrough, are greatly appreciated.