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Finance and Economics Discussion Series: 2010-15 Screen Reader version

Real-time Model Uncertainty in the United States:
'Robust' Policies Put to the Test*

Robert J. Tetlow.
Division of Research and Statistics
November 19, 2009.
Federal Reserve Board

Keywords: monetary policy, model uncertainty, real-time analysis.


I study 46 vintages of FRB/US, the principal macro model used by Federal Reserve Board staff for forecasting and policy analysis, as measures of real-time model uncertainty. I also study the implications of model uncertainty for the robustness of commonly applied, simple monetary policy rules. I first document that model uncertainty poses substantial challenges for policymakers in that key model properties differ in important ways across model vintages. Then I show that the parameterization of optimized simple policy rule-rules that are intended to be robust with respect to model uncertainty-also differ substantially across model vintages. Included in the set of rules are rules that eschew feedback on the output gap, rules that target nominal income growth, and rules that allow for time variation in the equilibrium real interest rate. I find that many rules, which previous research has shown to be robust in artificial economies, would have failed to provide adequate stabilization in the real-time, real-world environment seen by the Fed staff. However, I do identify certain policy rules that would have performed relatively well, and I characterize the key features of those rules to draw more general lessons about the design of monetary policy under model uncertainty.

JEL Classifications: E37, E5, C5, C6.

1 Introduction

We have involved ourselves in a colossal muddle, having blundered in the control of a delicate machine, the working of which we do not understand.
-John Maynard Keynes, "The Great Slump of 1930" (December 1930).

Over the past decade or so, there has been an explosion of work studying the characteristics of monetary policy rules in general and interest-rate feedback rules in particular. While considerable insight has come out of this literature, so has a fundamental critique, namely that results formulated in this way may not be robust to misspecification of the underlying model. Keynes's metaphor of the economy as a "delicate machine...which we do not understand" seems as apt today as it was in 1930.

It follows that a principal concern for policy makers is uncertainty, and how to deal with it. The fast growing literature on model uncertainty seeks answers to this question; see, inter alia,Levin, Wieland and Williams (1999), Tetlow and von zur Muehlen (2001), Onatski and Williams (2003), Levin et al. (2005), Brock, Durlauf and West (2007) and Taylor and Wieland (2009).1 This strand often employs the rival models method of analysis wherein the researcher posits two or more alternative models of the economy and employs statistical or decision theoretic techniques to find a policy rule that performs "well" in each of the posited models; see, e.g., McCallum (1988). While this approach to the problem has produced interesting and useful results, it is hampered by the artificiality of the environment in which it which it has been employed. In nearly all cases, the models under consideration are either highly abstract or "toy" models that do not fit the data well, useful perhaps for making narrow points, but not to be taken seriously as tools of monetary policy design.2

Virtually absent from the above characterization of the literature is the real-time analysis of model uncertainty. At one level, this is not surprising; after all, while it is easy to conceptualize changing views about what the true model might be, it is more difficult to imagine the laboratory in which such an analysis could be conducted. That is, however, exactly what this paper provides. Our laboratory is the Federal Reserve Board staff and the FRB/US model. We examine time variation in model properties, and hence model uncertainty, as it was seen in real time by the Federal Reserve Board staff. We do this using 46 of the vintages of the Board staff's FRB/US model, or four per year, that were actually used for forecasting and policy analysis during the period from July 1996 to October 2007, examining how the model specification, coefficients, databases and stochastic shock sets changed from vintage to vintage as new and revised data came in. The advantage provided is that we can focus on those aspects of model uncertainty that are germane to policy decisions, using a model that is used to formulate advice for those decisions.

The relevance of the model is unquestionable: since its introduction in July 1996, the FRB/US model has been used continuously for communicating ideas to the Board of Governors and the Federal Open Market Committee (FOMC). All of the Greenbook's alternative scenarios focusing on domestic economic issues are conducted using the model, forecast confidence intervals are computed using FRB/US, as are optimal policy exercises that appear in the Bluebook; see Svensson and Tetlow (2005).3 In his 1987 monograph on his time as Vice Chairman of the Federal Reserve Board, Alan Blinder notes (p. 12) the important role that FRB/US simulations played in guiding his thinking; and Blinder and Yellen (2001), explain what happened in the US economy in the 1990s, using extensive simulations of the FRB/US model.

As we shall show, the US economy was buffeted by a range of economic forces over this period, including a productivity boom, a stock market boom and bust, a recession, and an abrupt change in fiscal policy. There were also 40 changes in the intended federal funds rate, 24 increases and 16 decreases.4 These events turned out to have important implications for how the Board's staff saw the economy and how they embraced those views in the model's structure. This, in turn, had important implications for what policies would, and would not, work well in such an environment.

Armed with these 46 vintages of the model, we ask whether the policy rules that have been promoted as robust in one environment or another are in fact robust in this real-world context. In other words, if the federal funds rate had followed rules that were optimized within the context of the Fed staff's FRB/US model, how would the economy have performed?

We study eight particular rules. The first is the familiar Taylor (1993) rule, although we use parameterizations that are optimal for the model vintages we are interested in. We also consider three rules that take up the argument of Orphanides (2001) and Orphanides and van Norden (2002) that the inherent difficulty in conditioning policy on unobservable constructed variables like output gaps, policy should eschew feedback on latent variables altogether. Two candidate rules follow Bennett McCallum (1988), by keying off of nominal output growth. A nominal output growth rule establishes a nominal anchor but unlike, say, an inflation targeting rule, makes no explicit call on whether shocks are real or nominal; because of this, it is arguably less susceptible to supply-side misspecifications. Two rules pick up the finding of Levin, Onatski, Williams and Williams (2005)--henceforth LOWW--to the effect that policy should respond to nominal wage inflation instead of price inflation. In this way, the policymaker pays particular attention to the labor market, arguably the part of the economy that, from a neoclassical perspective, is the most distorted.

This paper goes a number of steps beyond previous contributions to the literature. As already noted, it goes beyond the extant rival models literature through its novel and efficacious focus on models that are actually used in a policy environment. It also goes beyond the literature on parameter uncertainty. That literature assumes that parameters are random but the model is fixed over time: misspecification is simply a matter of sampling error. Model uncertainty is a thornier problem, in large part because it often does not readily lend itself to statistical methods of analysis. We explicitly allow the models to change over time in response not just to the data but to the economic issues of the day.5 Finally, as already noted, it does all this within a class of possible models that is undeniably realistic.

The analysis presented herein is, of course, based on the US economy and the FRB/US model. It should be clear, however, that the problems under study are more general than this. Uncertainty, in its various forms, is of concern for monetary authorities the world over as it is for other decision makers. Real-time data issues and data uncertainty more generally have garnered a great deal of attention in the U.K.; see, e.g., Garratt and Vahey (2006) and Garratt, Koop and Vahey (2008) and references therein. On the continent, Giannone, Reichlin and Sala (2005) study real-time uncertainty for its implications for monetary policy while Cimadomo (2008) and Giuliodori and Beetsma (2008) uncover important implications of real-time data uncertainty and, indirectly, model uncertainty for the measurement of fiscal stance and the conduct of fiscal policy for the OECD countries and the Euro area, respectively.6 The relevance of the topic for the Euroarea is particularly striking because a large part of the raison d'etre of its creation was about inducing changes in economic structure that would, as a biproduct, introduce even more model uncertainty, at least for a time.

The rest of this paper proceeds as follows. The second section begins with a discussion of the FRB/US model in generic terms, and the model's historical archives. The third section compares model properties by vintage. To do this, we document changes in real-time "model multipliers" and compare them with their ex post counterparts. The succeeding section computes optimized Taylor-type rules and compares these to commonly accepted alternative policies in a stochastic environment. The fifth section examines the stochastic performance of candidate rules for two selected vintages, the December 1998 and October 2007 models. A sixth and final section sums up and concludes.

2 Forty-six vintages of the FRB/US model and the data

2.1 The real-time data

In describing model uncertainty, it pays to start at the beginning; in present circumstances, the beginning is the data. It is the data, and the staff's view of those data back in 1996 that determined how the first vintage of FRB/US was structured. And it is the surprises from those data, and how they were interpreted as the series were revised and extended with each successive vintage, that conditioned the model's evaluation and refinement. To that end, in this subsection we examine key data series by vintage. We also provide some evidence on the model's forecast record during the period of interest. And we reflect on the events of the time, the shocks they engendered, and the revisions to the data. Our treatment of the subject is subjective--it comes, in part, from the archives of the FRB/US model--and incomplete. It is beyond the scope of this part of the paper to provide an comprehensive survey of data revisions over the period from 1996 to 2007. Fortunately, however, Anderson and Kliesen (2005) provide just such a summary.

Figure 2.1: Real-time four-quarter GDP price inflation, selected vintages
Figure 2.1 Real-time four-quarter GDP price inflation, selected vintages. The figure shows five lines, historical data for four-quarter PGDP inflation for the July 1996 (a pink solid line), February 1998 (blue dashed), August 1999 (green dot-dashed), August 2002 (red dashed) and October 2007 (orange solid) model vintages. In each case, the series ends the quarter before the vintage date. The figure shows that real-time variation in the inflation rate is within about one-half of a percentage point.

Figure 2.1 shows the four-quarter growth rate of the GDP price index, for selected vintages. (Note we show only real-time historical data because of rules restricting the publication of forecast data.) The inflation rate moves around some, but the various vintages for the most part are highly correlated. In any event, our reading of the literature is that data uncertainty, narrowly defined to include revisions of published data series, is not a first-order source of problems for monetary policy design; see, e.g., Croushore and Stark (2001). Figure 2.2 shows the more empirically important case of model measures of growth in potential non-farm business output.7

Figure 2.2: Real-time non-farm business potential growth, selected vintages
Figure 2.2 Real-time non-farm business potential growth, selected vintages. The figure shows four-quarter growth in non-farm business potential output for the same five vintages as in figure 2.1, using the same five line types. These too end the quarter prior to the vintage date.  The figure shows a wide range of estimates that vary in substantial ways in both the levels of potential output growth rates across vintages and in the volatility of those growth rates over time within a vintage.

Unlike the case of inflation, potential output growth is a latent variable the definition and interpretation of which depends on model concepts. This means is the historical measures of potential are themselves a part of the model, and so we should expect significant revisions.8 Even so, the magnitudes of the revisions shown in Figure 2.2 are remarkable. The July 1996 vintage shows growth in potential output of about 2 percent, typical of the estimates of models at the time. For the next several years, succeeding vintages show both higher potential output growth rates and more responsiveness to the economic cycle. By January 2001, growth in potential was estimated at over 5 percent for some dates, before subsequent changes resulted in a path that was lower and more variable. The very concept of potential growth had changed. Why did potential undergo such dramatic revision? Table 1 reminds us about how extraordinary the late 1990s were. The table shows selected FRB/US model forecasts for the four-quarter growth in real GDP, on the left-hand side of the table, and PCE price inflation, on the right-hand side, for the period for which public availability of the data are not restricted.9 The table shows the substantial underprediction of GDP growth over most of the period, together with underpredictions of inflation.

Table 1: Four-quarter growth in real GDP and PCE prices: selected FRB/US model forecasts

forecast date Real GDP: forecast Real GDP: data Real GDP: data - forecast* GDP price index: forecast GDP price index: data GDP price index: data - forecast*
July 1996 2.2 4.0 1.8 2.3 1.9 -0.4
July 1997 2.0 3.5 1.5 2.2 0.9 -1.3
Aug. 1998 1.7 4.1 2.4 1.8 1.3 -0.5
Aug. 1999 3.2 5.3 2.1 2.3 2.3 0.0
Aug. 2000 4.5 0.8 -3.7 2.1 2.3 0.2
Aug. 2001 2.2 3.0 0.8 2.7 0.8 -2.0
Aug 2002 3.6 3.1 -0.5 1.3 1.7 0.3
*4Q growth forecasts from the vintage of the year shown; e.g. for GDP in July 1996, forecast =100*(GDP[1997:Q3]/GDP[1996:Q3]-1), compared against the "first final" data contained in the database two forecasts hence. So for the same example, the first final is from the November 1997 vintage database.

The more recent historical measures shown in Figure 2.2 for the August 2002 and October 2007 vintages show paths that differ in two important ways from the others. First, these series are the only ones shown that are less optimistic than their predecessors. In part, this reflects the onset of the 2001 recession, particularly for the August 2002 series. Second, these two latter series show considerably more volatility over time. This is a manifestation of a change in thinking that arose in response to economic conditions of the day. In its early vintages, the modeling of potential output in FRB/US was traditional for large-scale econometric models: trend labor productivity and trend labor input, were based on exogenous split time trends. In essence, the model took the typical Keynesian view that nearly all shocks affecting aggregate output were demand-side phenomena. Then, in the late 1990s, as under-predictions of GDP growth were experienced without concomitant underpredictions in inflation, these priors were updated. The staff began adding model code to allow the supply side of the model to respond to output surprises by projecting forward revised profiles for productivity growth; what had been an essentially deterministic view of potential output was evolving into a stochastic one.10

Figure 2.3: Evolution of estimates of data and latent variables for 1996
Figure 2.3 Evolution of estimates of data and latent variables for 1996. The figure shows four-quarter growth for each of three series--real GDP, non-farm business potential output, and PCE prices--all for a single year, 1996, but as seen from the perspective of different vintages. The horizontal axis indicates the vintage database from which the point in the figure obtains.  The first few observations are forecast data, in part, because not all four observations were hard data at the time; thereafter, they are backcasts. The figure demonstrates that extensive revisions in GDP growth for 1996 were initially interpreted as increases in the output gap but as time went on became more and more associated by the model builders as representing shifts in potential output. The revisions in GDP growth were also associated, at times, with revisions in price inflation.
Figure 2.4: Real-time GDP output gaps, selected vintages
Figure 2.4 Real-time GDP output gaps, selected vintages. The figure is similar to figures 2.1 and 2.3 except it shows the GDP output gap.  The figure shows that while real-time estimates of potential output growth varied widely across vintages, as demonstrated in figure 2.2, the effect on the real-time estimates of the output gap was less extreme.  The ranges of estimates across the five vintages shown were of the order of 1.5 percentage points and the various output gaps covaried strongly over time.

Further insight on the origins and persistence of these forecast errors can be gleaned from Figure 2.3, which focuses attention on a single year, 1996, and shows forecasts and "actual" four-quarter GDP growth, non-farm business potential output growth, and PCE inflation for that year. Each date on the horizontal axis corresponds with a database, so that the first observation on the far left of the black line is what the FRB/US model database for the 1996:Q3 (July) vintage showed for four-quarter GDP growth for 1996. (The black line is broken over the first two observations to indicate that some observations for 1996 were forecast data at the time; after the receipt of the advance release of the NIPA for 1996:Q4 on January 31, 1997, the figures are treated as data.) Similarly, the last observation of the same solid black line shows what the 2006:Q4 database has for historical GDP growth in 1996. The black line shows that the data combined with the model predicted four-quarter GDP growth of 2.2 percent for 1996 as of July 1996. However when the first final data for the 1996:Q4 were released on January 31, 1997, GDP growth for the year was 3.1 percent, a sizable forecast error of 0.8 percentage points.11 The black line shows that GDP growth was revised up in small steps and large jumps right up until late in 2003 and to stand by the end of 2006 at 4.4 percent; so by the (unfair) metric of recent data, the forecast error from the July 1996 projection is 2.2 percentage points. Comparisons of the black line, measuring real GDP growth, with the red line, measuring potential output growth, show the influence that data revisions had on the FRB/US measures of potential. The slow response of the red line and the resulting gap between it and the black line in 1996 and 1997 reflects the strong Keynesian prior of the day. Then, as can be seen, in 1998 a more profound change in view was undertaken, and by 1999 nearly all real growth in 1996 was seen as emanating from supply shocks. All told, given the long climb of the black line, the revisions to potential output growth shown by the red line seem explicable, at least until about 2001. After that point, the emerging recession resulted in wholesale revisions of potential output growth going well back into history. The blue line shows that there was a revision in PCE inflation that coincided with substantial changes in both actual GDP and potential, in 1998:Q3. This reflects the annual revision of the NIPA data and with it some updates in source data.12

Despite the volatility of potential output growth, the resulting output gaps, shown in Figure 2.4, show considerable covariation, albeit with non-trivial revisions. This observation underscores the underappreciated fact that output gaps (or unemployment gaps) are not the sole driver of fluctuations in inflation; other forces are also at work, particularly trend productivity which affects unit labor costs, and relative price shocks such those affecting food, energy and non-oil import prices.

2.2 A generic description of the FRB/US model

The FRB/US model came into production in July 1996 as a replacement for the venerable MIT-Penn-SSRC (MPS) model that had been in use at the Board of Governors for many years.

The main objectives guiding the development of the model were that it be useful for both forecasting and policy analysis; that expectations be explicit; that important equations represent the decision rules of optimizing agents; that the model be estimated and have satisfactory statistical properties; and that the full-model simulation properties match the "established rules of thumb regarding economic relationships under appropriate circumstances" as Brayton and Tinsley (1996, p. 2) put it.

To address these challenges, the staff included within the FRB/US model a specific expectations block, and with it, a fundamental distinction between intrinsic model dynamics (dynamics that are immutable to policy) and expectational dynamics (which policy can affect). In most instances, the intrinsic dynamics of the model were designed around representative agents choosing optimal paths for decision variables facing adjustment costs.13

Ignoring asset pricing equations for which adjustment costs were assumed to be negligible, a generic model equation would look something like:

\displaystyle \Delta x=\alpha(L)\Delta x+E_{t}\beta(F)\Delta x^{\ast}+c(x_{t-1}% -x_{t-1}^{\ast})+u_{t}% (1)

where  \alpha(L) is a polynomial in the lag operator, i.e.,  \alpha (L)z=a_{0}+a_{1}z_{t-1}+a_{2}z_{t-2}+\ldots and  \beta(F) is a polynomial in the lead operator. The term  \Delta x^{\ast} is the expected changes in target levels of the generic decision variable,  x,  c(.) is an error-correction term, and  u is a residual. In general, the theory behind the model will involve cross-parameter restrictions on  \alpha(L),\beta(F) and  c. The point to be taken from equation (1) is that decisions today for the variable,  x, will depend in part on past values and expected future values, with an eye on bringing  x toward its desired value,  x^{\ast }, over time.

From the outset, FRB/US has been a significantly smaller model than was MPS, but it is still quite large. At inception, it contained some 300 equations and identities of which perhaps 50 were behavioral. About half of the behavioral equations in the first vintage of the model were modeled using formal specifications of optimizing behavior. Among the identities are the expectations equations.

Two versions of expectations formation were envisioned: VAR-based expectations and perfect foresight. The concept of perfect foresight is well understood, but VAR-based expectations probably requires some explanation. In part, the story has the flavor of the Phelps-Lucas "island paradigm": agents live on different islands where they have access to a limited set of core macroeconomic variables, knowledge they share with everyone in the economy. The core macroeconomic variables are the output gap,  \widetilde {y}=y-y^{\ast},the inflation rate,  \pi, and the federal funds rate,  r,as well as agents' beliefs of the long-run target rate of inflation,  \pi ^{\infty},and the equilibrium real rate of interest in the long run,  rr^{\infty}=r^{\infty}-\pi^{\infty}. These variables comprise the model's core VAR expectations block which we can write as follows:

\displaystyle \gamma(L)w=\varepsilon_{t}% (1)

\begin{displaymath} w=\left[ \begin{array}[c]{c}% \widetilde{y}-\widetilde{y}^{\infty}\ \pi-\pi^{\infty}\ r-r^{\infty}% \end{array}\right] \varepsilon=\left[ \begin{array}[c]{c}% u^{y}\ u^{\pi}\ u^{r}% \end{array}\right] . \end{displaymath}
The long-run expected value of the output gap,  \widetilde{y}^{\infty}, is zero by definition, so  w is stationary around the vector of "endpoints," \begin{displaymath}w^{\infty}=\left[ \begin{array}[c]{ccc}% 0 & \pi^{\infty} & r^{\infty}% \end{array}\right] ^{/}\end{displaymath}.

In addition to variables of this core VAR, agents have information that is germane to their island, or sector. Consumers, for example, augment their core VAR model with information about potential output growth and the ratio of household income to GDP, which forms the consumer's auxiliary VAR. Two features of this set-up are worth noting. First, the set of variables agents are assumed to use in formulating forecasts is restricted to a set that is smaller than under rational expectations. (By definition, under perfect-foresight expectations, the information set includes all the states in the model with all the cross-equation restrictions implied by the model.) Second, agents are allowed to update their beliefs, but only in a restricted way. In particular, for any given vintage, the coefficients of the VARs are taken as fixed, while agents' perceptions of long-run values for the inflation target and the equilibrium real interest rate are continually updated using simple learning rules.14

In this paper, we will be working exclusively with the VAR-based expectations version of the model. Typically it is the multipliers of this version of the model that are reported to Board members when they ask "what-if questions". This is the version that is used for forecasting and most of the policy analysis by the Fed staff, including, as Svensson and Tetlow (2005) demonstrate, policy optimization experiments. Thus, the pertinence of using this version of the model for the question at hand is unquestionable. What might be questioned, on standard Lucas-critique grounds, is the validity of the Taylor-rule optimizations carried out below. However, the period under study is one entirely under the leadership of a single Chairman, and we are aware of no evidence to suggest that there was a change in regime during this period. So as Sims and Zha (2006) have argued, it seems likely that the perturbations to policies encompassed by the range of policies studied below are not large enough to induce a change in expectations formation other than that can be captured by changes in the endpoints. Moreover, in an environment such as the one under study, where changes in the non-monetary part of the economy are likely to dwarf the monetary-policy perturbations, it seems safe to assume that private agents were no more rational with regard to their anticipations of policy than the Fed staff was about private-sector decision making.15 In their study of the evolution of the Fed beliefs over a longer period of time, Romer and Romer (2002), ascribe no role to the idea of rational expectations. Moreover, Rudebusch (2002) shows that issues of model uncertainty are often of second-order importance in linear rational expectations models. Thus the VAR-based expectations case is arguably the more quantitatively interesting one. Finally, what matters for this real-time study is that it is certainly the case that the model buiiders believe in VAR-based expectations formation and the model was, in fact, used for forecasting and policy analysis alike; see, e.g. Svensson and Tetlow (2005). Later on we will have more to say about the implications of assuming VAR-based expectations for our results and those in the rest of the literature

There is not the space here for a complete description of the model. Readers interested in detailed descriptions of the model are invited to consult papers on the subject, including Brayton and Tinsley (1996), Brayton, Levin, Tryon and Williams (1997), and Reifschneider, Tetlow and Williams (1999). However, before leaving this section it is important to note that the structure of macroeconomic models at the Fed have always responded to economic events and the different questions that those events evoke, even before FRB/US. Brayton, Levin, Tryon and Williams (1997) note, for example, how the presence of financial market regulations meant that for years a substantial portion of the MPS model dealt specifically with mortgage credit and financial markets more broadly. The repeal of Regulation Q induced the elimination of much of that detailed model code. Earlier, the oil price shocks of the 1970s and the collapse of Bretton Woods gave the model a more international flavor than it had previously. We shall see that this responsiveness of models to economic conditions and questions continued with the FRB/US model in the 1990s. The key features influencing the monetary policy transmission mechanism in the FRB/US model are the effects of changes in the funds rate on asset prices and from there to expenditures. Philosophically, the model has not changed much in this area: all vintages of the model have had expectations of future economic conditions in general, and the federal funds rate in particular, affecting long-term interest rates and inflation. From this, real interest rates are determined and this in turn affects stock prices and exchange rates, and from there, real expenditures. Similarly, the model has always had a wage-price block, with the same basic features: sticky wages and prices, expected future excess demand in the goods and labor markets influencing price and wage setting, and a channel through which productivity affects real and nominal wages. That said, as we shall see, there have been substantial changes over time in both (what we may call) the interest elasticity of aggregate demand and the effect of excess demand on inflation.

Over the years, equations have come and gone in reflection of the needs, and data, of the day. The model began with an automotive sector but this block was later dropped. Business fixed investment was originally disaggregated into just non-residential structures and producers' durable equipment, but the latter is now disaggregated into high-tech equipment and "other". The key consumer decision rules and wage-price block have undergone frequent modification over the period. On the other hand, the model has always had an equation for consumer non-durables and services, consumer durables expenditures, and housing. There has always been a trade block, with aggregate exports and non-oil and oil imports, and equations for foreign variables. The model has always had a three-factor, constant-returns-to-scale Cobb-Douglas production function with capital, labor hours and energy as factor inputs.

2.3 The model archive

Since its inception in July 1996, the FRB/US model code, the equation coefficients, the baseline forecast database, and the list of stochastic shocks with which the model would be stochastically simulated, have all been stored for each of the eight forecasts the Board staff conducts every year. Because it is releases of National Income and Product Accounts (NIPA) data that typically induce re-assessments of the model, we use four archives per year, or 46 in total, the ones immediately following NIPA preliminary releases.16

In what follows, we experiment with each vintage of model, comparing their properties in selected experiments. Consistent with the real-time philosophy of this endeavor, the experiments we choose are typical of those used to assess models by policy institutions in general and the Federal Reserve Board in particular. They fall into two broad classes. One set of experiments, model multipliers, attempts to isolate the behavior of particular parts of the model. A multiplier is the response of a key endogenous variable to an exogenous shock after a fixed period of time. An example is the response of the level of output after eight quarters to a persistent increase in the federal funds rate. The other set of experiments judge the stochastic performance of the model and are designed to capture the full-model properties under fairly general conditions. So, for example, we will compute by stochastic simulation the optimal coefficients and economic performance of simple rules, conditional on a model vintage, a baseline database, and a set of stochastic shocks.17

Model multipliers have been routinely reported to, and used by, members of the FOMC. Indeed, the model's sacrifice ratio--about which we will have more to say below--was used in the very first FOMC meeting following the model's introduction. Similarly, model simulations of alternative policies have been carried out and reported to the FOMC in a number of memos and official FOMC documents.18

The archives document model changes and provide a unique record of model uncertainty. As we shall see, the answers to questions a policy maker might ask differ depending on the vintage of the model. The seemingly generic issue of the output cost of bringing down inflation, for example, can be subdivided into several more precise questions, including: (i) what would the model say is the output cost of bringing down inflation today?; (ii) what would the model of today say the output cost of bringing down inflation would have been in December 1998?; and (iii) what would the model have said in December 1998 was the output cost of disinflation at that time? These questions introduce a time dependency to the issue that rarely appears in other contexts.

The answers to these and other related questions depend on the model vintage and everything that goes along with it: the model itself, the policy rule, the baseline database and the set of stochastic shocks.

3 Model multipliers in real time and ex post

In this section, we consider the variation in real time of selected model multipliers. In the interests of brevity, we devote space to just four multipliers. The first is the sacrifice ratio; that is, the cumulative annualized cost measured in terms of increased unemployment over five years of permanently reducing the inflation rate by one percentage point. The second is the funds rate multiplier, defined here as the percentage change in the level of real output after eight quarters that is induced by a persistent 100-basis-point increase in the nominal federal funds rate.19 In the parlance of an undergraduate textbook closed-economy model, these two multipliers represent the slope of the Phillips curve and the slope of the aggregate demand curve, respectively. To add an international element, we add an exchange-rate multiplier-specifically, the percentage change in real GDP associated with a 10-percent appreciated of the trade-weight exchange value of the US dollar, and a non-oil import price multiplier: namely, the effect on PCE inflation of a persistent 10-percent increase in the relative price of non-oil imports. The sacrifice ratio is the outcome of a five-year simulation experiment; the other multipliers are measured in terms of their effects after eight quarters, except the import-price passthrough scenario which is computed over a 12-quarter horizon.

It is easiest to show the results graphically. But before turning to specific results, it is useful to outline how these figures are constructed and how they should be interpreted. In all cases, we show two lines. The black solid line is the real-time multiplier by vintage. Each point on the line represents the outcome of the same experiment, conducted on the model vintage of that date, using the baseline database at that point in history. So at each point shown by the black line, the model, its coefficients and the baseline all differ. The red dashed line shows what we call the ex post multiplier. The ex post multiplier is computed using the most recent model vintage for each date; the only thing that changes for each point on the dashed red line is the initial conditions under which the experiment is conducted. Differences over time in the red line reveal the extent to which the model is nonlinear with respect to the phenomenon under study, because the multipliers for linear models are independent of initial conditions.

Now let us look at Figure 3.1, which shows the sacrifice ratio.20 Let us focus on the red dashed line first. It shows that for the October 2007 model, the sacrifice ratio is essentially constant over time. So if the staff were asked to assess the sacrifice ratio, or what the sacrifice ratio would have been in, say, December 1998, the answer based on the October 2007 model would be the same: about  3\frac{1}{4}% , meaning that it would take that many percentage-point-years of unemployment to bring down inflation by one percentage point. Now, however, look at the black solid line. Since each point on the line represents a different model, and the last point on the far right of the line is the October 2007 model, the red dashed line and the black solid line must meet at the right-hand side in this and all other figures in this section. But notice how much the real-time sacrifice ratio has changed over the 12-year period of study. Had the model builders been asked in December 1998 what the sacrifice ratio was, the answer based on the February 1997 model would have been about  2\frac {1}{4}. Prior to a revision in mid-2007 that was undertaken in large part expressly to reduce it, the sacrifice ratio for vintages from 2004 to 2006 was of the order of  5\frac{1}{2}, or more than double what it was in the 1990s.21

Figure 3.1: Real-time and ex post sacrifice ratios, by model vintage
Figure 3.1 Real-time and ex post sacrifice ratios, by model vintage. The figure shows two lines, the so-called ex post sacrifice ratio (a red dashed line) showing the sacrifice ratio--that is, the undiscounted, annualized, cumulative incremental unemployment cost of reducing inflation by one percentage point computed over a five-year horizon--as computed from running an experiment on the last vintage under study, the October 2007 vintage, once at each of 46 dates (quarters) between 1996:Q3 and 2007:Q4. The black solid line is the same experiment except that the computation is carried out using a different vintage and a different baseline database at each date. The latter is referred to as the real-time multiplier. The real-time multiplier rose almost steadily from about 2.25 with the July 1996 vintage to more than 5.5 in the period from about 2004 to 2006 before dropping back to about 3.3 for the last vintage in November 2007. The ex post multiplier shows very little variation, ranging from perhaps 3.3 to 3.8.

The sacrifice ratio is a crucial statistic for any central bank model. On the one hand, it describes the cost of bringing down inflation, given that one inherits a higher inflation rate than is desired because of, say, having incurred a supply shock. From this perspective, a high sacrifice ratio is a bad thing. On the other hand, however, a high sacrifice ratio reflects a flat Phillips curve, which is to say that shocks to aggregate demand of a given magnitude will manifest themselves in smaller change in inflation than would otherwise be the case. From this perspective, a high sacrifice ratio is a good thing. Which effect dominates depends on the incidence of supply and demand shocks.

The primacy of the model's sacrifice ratio to policy debates is clear from FOMC transcripts. It was, for example, a topic of discussion at the first FOMC meeting following the introduction of the model.22 Similarly, the February 1, 2000, meeting of the FOMC produced this exchange between Federal Reserve Bank of Minneapolis President Gary Stern and then FOMC Secretary (now Federal Reserve Board Vice Chairman) Donald Kohn:23

Mr. Stern: Let me ask about the Bluebook [FRB/US model simulation] sacrifice ratio. I don't know what your credibility assumption is, but it seems really high.
Mr. Kohn: It is a little higher than we've had in the past, but not much. It is consistent with the model looking out over the longer run. It is a fairly high sacrifice ratio, I think, compared to some other models, but it is not out of the bounds...

Kohn was clearly aware that the model's sacrifice ratio had undergone some change and was rightfully cognizant of how it compares against alternative models. As it happens, the increases already incurred in the sacrifice ratio were only the beginning.

The climb in the model sacrifice ratio is striking, particularly as it was incurred over such a short period of time among model vintages with substantial overlap in their estimation periods. One might be forgiven for thinking that this phenomenon is idiosyncratic to the model under study. But other work shows that this result is not a fluke.24 At the same time, as we have already noted, the model builders did incorporate shifts in the NAIRU (and in potential output), but found that leaning exclusively on this one story for macroeconomic dynamics in the late 1990s was insufficient. Thus, the revealed view of the model builders contrasts with idea advanced by Staiger, Stock and Watson (2001), among others, that changes in the Phillips curve are best accounted for entirely by shifts in the NAIRU. Toward the end of the decade, a reduction in the sacrifice ratio became an important objective of the specification and estimation of the model's wage-price block; success on this front was achieved through respecification of how long-term inflation expectations evolve over time. .

Figure 3.2: Funds rate multipliers, by model vintage
Figure 3.2 Funds rate multipliers, by model vintage. The figure is analytically identical to figure 3.1 except that the experiment is the effect on the level of GDP, in percent, of a persistent 100-basis-point increase in the nominal federal funds rate.  The real-time multiplier shown substantial variation over vintages, starting at -1.45 and rising to about -1.1 in late 1997, before falling to -2.3 in late 1999 and then rising, with some jigs and jags to -1.1 at the end in 2007.  The ex post multiplier shows significant time variation as well, beginning at -1.1 and falling continuously to about -1.65 200 and then rising again to about -1.1.

Figure 3.2 shows the funds-rate multiplier; that is, the percentage decrease in the level of real GDP after eight quarters in response to a persistent 100-basis-point increase in the funds rate. This time, the red dashed line shows important time variation: the ex post funds rate multiplier varies with initial conditions, it is largest, in absolute terms, at about 1.6 percentage point in late 2000, and lowest at the beginning and at the end of the period, at about 1 percent. The nonlinearity stems entirely from the specification of the model's stock market equation, which is written in levels, rather than in logs, a feature that makes the interest elasticity of aggregate demand an increasing function of the ratio of stock market wealth to total wealth. The mechanism is that an increase in the funds rate raises long-term bond rates, which in turn bring about a drop in stock market valuation operating through the arbitrage relationship between expected risk-adjusted bond and equity returns. The larger the stock market, the stronger the effect.25

The real-time multiplier, shown by the solid black line is harder to characterize. Two observations stand out. The first is the sheer volatility of the multiplier. In a large-scale model such as the FRB/US model, where the transmission of monetary policy operates through a number of channels, time variation in the interest elasticity of aggregate demand depends on a large variety of parameters. Second, the real-time multiplier is almost always smaller than the ex post multiplier. The gap between the two is particularly marked in 2000, when the business cycle reached a peak, as did stock prices. At the time, concerns about possible stock market bubbles were rampant. One aspect of the debate between proponents and detractors of the active approach to stock market bubbles concerns the feasibility of policy prescriptions in a world of model uncertainty.26 The considerable difference between the real-time and ex post multipliers during this period demonstrates the difficulty in carrying out historical analyses of the role of monetary policy; today's assessment of the strength of those monetary policy actions can differ substantially from what the FRB/US model implied in real time.

Figure 3.3: Real exchange-rate multipliers, by model vintage
Figure 3.3 Real exchange-rate multipliers, by model vintage. The figure is analytically identical to figure 3.1 except that the experiment is the effect on the level of GDP, in percent, of a persistent 10-percent increase in the real exchange value of the US dollar.  The real-time multiplier shows substantial time variation, beginning at about -1.0, falling to -2.1 in 1998 and then rising to about -0.8 in 2002 and then gliding downward to finish 2007 at about -1.3. The ex post multiplier is much smoother, ranging from -1.4 to -1.2.

The final two multipliers covered in this section cover real and nominal aspects of the international economy. The first is the effect of a sustained 10-percent appreciation of the real exchange value of the US dollar on real output in the United States. The second is the effect of a lasting 10-percent change in the relative price of non-oil import goods on PCE inflation.27 Without belaboring the details, the salient fact to take from these two figures is first and foremost the variability of the elasticities.28 In the case of non-oil import prices, the figure also shows quite clearly another aspect of the so-called great moderation, namely a sharp reduction over time in the influence of shocks on inflation; the phenomenon of diminished pass-through of exchange-rate shocks into inflation in particular has been documented by Campa and Goldberg (200) and Gagnon and Ihrig (2004).

Figure 3.4: Non-oil import price passthrough into PCE inflation
Figure 3.4 Non-oil import price passthrough into PCE inflation. The figure is analytically identical to figure 3.1 except that the experiment is the effect on four-quarter PCE inflation, in percentage points, of a persistent 10-percent increase in the relative price of non-oil import prices. The real-time multiplier starts out with substantial volatility and reasonably high values of the order of 0.3 to 0.8 from 1996 to 1999, but then dies out thereafter, fluctuating around zero from 2001 to 2007. The ex post multiplier fluctuates around zero for all dates.

To summarize this section, real-time multipliers show substantial variation over time, and differ considerably from what one would say ex post the multipliers would be. Moreover, the discrepancies between the two multiplier concepts have often been large at critical junctures in recent economic history. It follows that real-time model uncertainty is an important problem for policy makers. The next section quantifies this point by characterizing optimal policy, and its time variation, conditional on these model vintages.

4 Monetary policy in real time

4.1 The rules

In the current context, a monetary policy rule can be described as robust if: (i), the optimized policy coefficients do not differ in important ways across models; or (ii) the performance of the economy does not depend in an economically important way on rule parameterization. A robust policy rule can also be described as effective if (iii) it performs "well," relative to some benchmark policy rule.

A popular simple monetary policy rule is the canonical Taylor (1993) rule. One reason the Taylor rule is advocated for monetary policy because of its simplicity in that it calls for feedback on only those variables that would be central to nearly all macro models. Because of this, it is often suggested that the Taylor rule will be robust to model misspecification; see, e.g., Williams (2003) for an argument along these lines. And indeed many central banks use simple rules of one sort or another, including Taylor rules, in the assessment of monetary policy and for formulating policy advice, including in the Federal Reserve Board staff's Bluebook which describes policy options for the FOMC. In the US case, Giannone et al. (2005) show that the good fit of simple two-argument Taylor-type rules can be attributed to the small number of fundamental factors driving the US economy; that is, the two arguments that appear in Taylor rules encompass all that one needs to know to summarize monetary policy in history. We shall use the Taylor rule, appropriately parameterized, as out benchmark.

Taylor rules have their detractors. Much of the earlier work on robust policy rules has focussed on the importance of estimation and misperception of potential output and the associated mismeasurement of the output gap.29 Accordingly, some of the rules we consider are those that have been suggested as prophylactics for this problem. In other instances, it is a broader class of latent variables that have been the object of concern. For example, as we have already noted, the productivity boom in the U.S. in the second half of the 1990s, brought about misperceptions not just of the level of the output gap, but also of potential output growth going ahead; these concepts in turn have a bearing on the equilibrium real interest rate since in all but the smallest of open economies, the equilibrium real interest rate is determined, in part, by the steady-state growth rate of the economy. The two problems are related but different. Mismeasurement of the level of potential output, without corresponding errors in potential output growth, and associated  rr^{\ast} errors, are a stationary process. Missing a shift in trend growth is much more persistent and affects a wider range of variables in a fully articulated macromodel. Accordingly, some of the rules we consider stem from the addressing of the latter, more complicated problem.

Most of our analysis is restricted to the class of optimized two-parameter policy rules. This keeps the rules on equal footing in that it is to be expected that adding extra optimized parameters should improve performance, at least for a given model. It also makes keeps the already onerous computational costs to a manageable level. However, as a check against possible idiosyncrasies in results, we do consider a few three-parameter specifications.

4.1.1 Two-parameter policy rules

Our first rule is the most familiar: the Taylor rule. Formally, the Taylor rule--which for short we will often refer to as "TR"--is written:

\displaystyle r_{t}=rr_{t}^{\ast}+\widetilde{\pi}+\alpha_{Y}(y_{t}-y_{t}^{\ast})+\alpha _{\pi}(\widetilde{\pi}_{t}-\pi_{t}^{\ast})% (TR)

where  r is the quarterly average of the intended federal funds rate,  rr^{\ast} is the equilibrium real interest rate,  \pi is the inflation rate, taken to the the PCE chain-weighted price index;  \widetilde{\pi}% =\Sigma_{i=0}^{3}\pi_{t-i}/4 is the four-quarter moving average of inflation,  \pi^{\ast} is the target rate of inflation,  y is (the log of) output; and  y^{\ast} is potential output. Effectively, the rule is written as a real interest rate rule, as can be seen by taking  rr^{\ast} and  \widetilde{\pi} over to the left-hand side, leaving just output and inflation gaps on the right-hand side. 30

In our first bow to the output-gap mismeasurement problem, we also study an inflation targeting rule (ITR); that is, a rule that eschews feedback on the output gap altogether in order to avoid problems from the sort of data and conceptual revisions described in Section 2 above, as suggested by Orphanides (2001):

\displaystyle r_{t}=\alpha_{r}r_{t-1}+(1-\alpha_{r})(rr_{t}^{\ast}+\widetilde{\pi}% _{t})+\alpha_{\pi}(\widetilde{\pi}_{t}-\pi_{t}^{\ast}).% % (ITR)

For this rule and several others, we allow for instrument smoothing, with the parameter  \alpha_{r}, and allowing the term  (1-\alpha_{r})(.) to pick up the steady-state level of the real interest rate.31 In addition to the ITR, we investigate a price-level-targeting counterpart of the same specification:
\displaystyle r_{t}=\alpha_{r}r_{t-1}+(1-\alpha_{r})(rr_{t}^{\ast}+\widetilde{\pi}% _{t})+\alpha_{\pi}(p_{t}-p_{t}^{\ast}).% (PLR)

where it should be understood that  p_{t}^{\ast}need not be a fixed number; it could instead be (and is) a predetermined trending path for the (log of the) price level such that successful target renders a positive average rate of inflation. The important distinction between a price-level target and an inflation target is that in the event of an inflation surprise a price-level targeting regime is obliged not just to bring inflation back down to the target level but to bring inflation below target for a time in order to return the price level to its target path.

We will also analyze a Taylor-type rule that substitutes the change in the unemployment rate for the traditional output gap in order to allow a real variable to enter the rule while still minimizing the effects of misperceptions of potential output; see, e.g., Orphanides and Williams (2002):

\displaystyle \Delta r_{t}=\alpha_{\pi}(\widetilde{\pi}_{t}-\pi_{t}^{\ast})+\alpha_{\Delta u}\Delta u_{t}.% (URR)

Notice that this rule, designated URR, is written in the first-difference of the funds rate, a configuration that eliminates the need to condition on the equilibrium real interest rate. As such, the URR takes a step towards insulation against persistent shocks to productivity and associated mismeasurements of  rr^{\ast}.

Another much touted rule is the nominal output growth rule, along the lines suggested by Bennett McCallum (1988) and Feldstein and Stock (1994) and revisited recently by Dennis (2001) and Rudebusch (2002). Its purported advantage is that it parsimoniously includes both prices and real output growth but without taking a stand on the split between the two; for this reason it is said to be able to withstand productivity shocks. Detractors note that because output typically leads inflation, responding to the sum of the two is not as obviously beneficial as presumed. We experiment with two versions. The first is designated with the rubric "YNR I", and is written as:

\displaystyle r_{t}=\alpha_{r}r_{t-1}+(1-\alpha_{r})(rr_{t}^{\ast}+\widetilde{\pi}% _{t})+\alpha_{\Delta yn}(\Delta\widetilde{yn}_{t}-\Delta yn_{t}^{\ast })% (YNR I)

where  yn is (the log of) nominal output, and  yn^{\ast}is the target level of nominal output growth. This rendition follows the formulation of McCallum and Nelson (1999) and nests the versions studied by Rudebusch (2002). However because YNR I embodies output growth within its specification, albeit with its coefficient restricted to equal that of GDP price inflation, but not a term the level of resources utilization, we augment our analysis by including a second rendition:
\displaystyle r_{t}=(rr_{t}^{\ast}+\widetilde{\pi}_{t})+\alpha_{Y}(y_{t}-y_{t}^{\ast })+\alpha_{\Delta yn}(\Delta\widetilde{yn}_{t}-\Delta yn_{t}^{\ast}).% (YNR II)

This version, which we designate as "YNR II," has the virtue of being comparable to TR in that other than substituting nominal output growth for inflation, it is identical.

We also pick up on the finding of LOWW (2005) to the effect that a policy that responds to nominal wage inflation (WR I) instead of nominal price inflation, performs well. In this way, the policymaker pays particular attention to that part of the economy that, from a neoclassical perspective, is arguably is the most distorted. Like the nominal output growth targeting rule, because wage setting is supposed to reflect both price inflation and labor productivity, the nominal wage growth rule also has the merit of implicitly incorporating changes in trend productivity.

\displaystyle r_{t}=\alpha_{r}r_{t-1}+(1-\alpha_{r})(rr_{t}^{\ast}+\widetilde{\pi}% _{t})+\alpha_{\Delta w}(\Delta\widetilde{w}_{t}-\Delta w_{t}^{\ast})% (WR I)

where  w is (the log of) the nominal wage rate. In parallel fashion to our nominal output rules here, too, we consider a second version of the nominal wage growth rule that replaces the lagged instrument with the output gap. As a convenient short hand, we refer to this rule as "WR II":
\displaystyle r_{t}=rr_{t}^{\ast}+\widetilde{\pi}_{t}+\alpha_{Y}(y_{t}-y_{t}^{\ast}% )+\alpha_{\Delta w}(\Delta\widetilde{w}_{t}-\Delta w_{t}^{\ast})% (WR II)

4.1.2 Three-parameter policy rules

As noted, the benchmark from which all our rules are to be compared is the optimized version of the Taylor rule. There is, however, a chance that this choice is inappropriate. It is possible that the two-parameter Taylor rule is too parsimonious to adequately respond to the myriad economic disturbances to which the economy is subjected. In recognition of this possibility, we also explore a dynamic Taylor rule--let us call it "xTR" where the "x" means "extended" to add the lagged instrument as an argument:

\displaystyle r_{t}=\alpha_{r}r_{t-1}+(1-\alpha_{r})(rr_{t}^{\ast}+\widetilde{\pi}% _{t})+\alpha_{Y}(y_{t}-y_{t}^{\ast})+\alpha_{\pi}(\widetilde{\pi}_{t}-\pi _{t}^{\ast}).% (xTR)

This rule is the most commonly studied extension on the static Taylor rule; Williams (2003) argues that the inclusion of the lagged instrument can provide significant benefits in terms of economic outcomes in linearized New Keynesian models.

We also consider the same extension applied to some of the other rules so that each has a different nominal anchor but contains both the output gap and the lagged instrument, and to the URR so that it too carries a lagged instrument term plus an inflation term.

Lastly, because it is possible that concepts like the output gap cannot do justice to the real-side phenomena that buffet the economy in a world of where productivity shocks are prevalent, it seems prudent to consider conditioning policy specifically on potential output growth. At the same time, to be realistic, one should use not ex post measures of potential growth but rather the estimates that modelers were working with in real time. We can do so with the following rule, which we call the potential growth rule (Y*R):

\displaystyle r_{t}=rr_{t}^{\ast}+\widetilde{\pi}_{t}+\alpha_{\Delta Y\ast}\Delta y^{\ast }+\alpha_{Y}(y_{t}-y_{t}^{\ast})+\alpha_{\pi}(\widetilde{\pi}_{t}-\pi _{t}^{\ast})% (Y*R)

where  \Delta y^{\ast} is the vintage-consistent estimate of potential output growth. The terms  rr^{\ast}and  \alpha_{Y\ast}\Delta y^{\ast} together can be taken as a reworked estimate of the equilibrium of the equilibrium real rate, one that corrects potential output growth.

Together, these rules encompass a broad range of the rules that have been proposed as robust to model misspecification, and do so in a generic way in that their arguments do not depend on idiosyncrasies of the FRB/US model.

4.2 The policy problem

Formally, a policy rule is optimized by choosing the parameters of the rule,  \Phi=\left\{ \alpha_{i},\alpha_{j}\right\}  i,j=\{\pi,y,r,\Delta y^{\ast },\Delta yn,\Delta u,\Delta w\},i\neq j, to minimize a loss function subject to a given model vintage,  x=f(\cdot), and a given set of stochastic shocks,  \Sigma.~In our case, this is:

\displaystyle \underset{\langle\Phi\rangle}{MIN}\;% {\displaystyle\sum\limits_{i=0}^{T}} \beta^{i}\left[ \left( \pi_{t+i}-\pi_{t+i}^{\ast}\right) ^{2}+\lambda _{y}\left( u_{t+i}-u_{t+i}^{\ast}\right) ^{2}+\lambda_{\Delta r}(\Delta r_{t+i})^{2}\right] % (2)

subject to:
\displaystyle x_{t}=f(x_{t},\ldots x_{t-j},z_{t},\ldots z_{t-k},r_{t},\ldots r_{t-m}% )+v_{t}\hspace{1in}j,k,m\eqslantgtr0% (3)

\displaystyle \Sigma_{v}=v^{\prime}v% (5)

where  u is the unemployment rate,  u^{\ast} is the vintage consistent estimate of the natural rate of unemployment,  x is a vector of endogenous variables, and  z a vector of exogenous variables, both in logs, except for those variables measured in rates. Note that  \pi,y,y^{\ast},u,r,rr^{\ast },w,yn\in x while  \pi^{\ast},u^{\ast}\in z.32 In principle, the loss function, (2), could have been derived as the quadratic approximation to the true social welfare function for the FRB/US model. However, it is technically infeasible for a model the size of FRB/US. That said, with the possible exception of the term penalizing the change in the federal funds rate, the arguments to (2) are standard.33 The penalty on the change in the funds rate may be thought of as representing either a hedge against model uncertainty in order to reduce the likelihood of the fed funds rate entering ranges beyond those for which the model was estimated, or as a pure preference of the Committee. Whatever the reason for its presence, the literature confirms that some penalty is needed to explain the historical persistence of monetary policy; see, e.g., Sack and Wieland (2000).

The optimal coefficients of a given rule are a function of the model's stochastic shocks, as equation (5) indicates.34 The optimized coefficient on the output gap, for example, represents not only the fact that unemployment-rate stabilization--and hence, indirectly, output-gap stabilization--is an objective of monetary policy, but also that in economies where demand shocks play a significant role, the output gap will statistically lead changes in inflation in the data; so the output gap will appear because of its role in forecasting future inflation. However, if the shocks for which the rule is optimized turn out not to be representative of those that the economy will ultimately bear, performance will suffer. As we shall see, this dependence will turn out to be significant for our results.35

4.3 Computation

Solving a problem like this is easily done for small, linear models; FRB/US, however, is a large, non-linear model. Given the size the model, and the differences across vintages, we optimized the policy rule coefficients employing a sophisticated derivative-free optimization procedure with distributed processing. Specifically, each vintage of the model is subjected to bootstrapped shocks from its stochastic shock archive. Historical shocks from the estimation period of the key behavioral equations are drawn.36 In all, 1500 draws of 80 periods each are used for each vintage to evaluate candidate parameterizations. The target rate of inflation is taken to be two percent as measured by the annualized rate of change of the personal consumption expenditure price index.37 The algorithm is described in detail in Gray and Kolda (2004) and Kolda (2004); here we provide just a thumbnail sketch. In the first step, the rule is initialized with a starting guess; that guess and some neighboring points are evaluated. Since all our rules are two-parameter rules, we need only investigate four neighboring points: higher and lower, by some step size, for each of the two parameters, with the initial guess in the middle. The loss function is evaluated for each of the five points and one with the lowest loss becomes the center of the next cluster of five points. As the five points become less and less distinguishable from one another, the step size is reduced until the convergence criterion is satisfied.

Optimization of a two-parameter policy rule using a single Intel Xeon 2.8 GHz machine can take over ten hours, depending on the rule; distributed processing speeds things up. Because this is exercise is computationally intensive we are limited in the range of preferences we can investigate. Accordingly, we discuss only one set of preferences: equal weights on output, inflation and the change in the federal funds rate. This is the same set of preferences that have been used in optimal policy simulations carried out for the FOMC; see Svensson and Tetlow (2005).

5 Results

5.1 The Taylor rule

Let us begin with the Taylor rule (TR). In this instance, we provide a full set of results--that is, optimized parameters for each of the 46 vintages; later we will narrow our focus. The results are best summarized graphically. In Figure 5.1, the blue solid line is the optimized coefficient for the TR on inflation,  \alpha_{\pi}, while the red dashed line is feedback coefficient on the output gap,  \alpha_{Y}. Perhaps the most noteworthy observation from 5.1 is the distinct upward creep, on average, in both parameters. The inflation response coefficient never actually gets very large: it starts out quite low and only in the new century does it climb above the 0.5 percent level of the traditional Taylor rule. The rise over time in the output gap coefficient is more impressive. It too starts out low with the first vintage in July 1996 at about 0.1, but then rises more-or-less steadily thereafter-the late 1999 dip aside-reaching values generally above 1 with the later vintages.38

Figure 5.1: Optimized coefficients of the Taylor rule, by vintage
Figure 5.1 Optimized coefficients of the Taylor rule, by vintage. The figure shows two lines; both of them optimized coefficients by model vintage of the standard two-parameter Taylor rule. .The blue solid line is the feedback coefficient on four-quarter PCE inflation; the red dashed line is the feedback coefficient on the GDP output gap.  Two gray vertical bars indicate the two vintages--December 1998 and October 2007--that comprise the bulk of the quantitative analysis that follows in the paper.  Both coefficients show a significant trend upward as newer vintages succeed older ones with the inflation coefficient starting at about 0.1 and reaching a peak of nearly 0.9 in mid 2007, and the output gap coefficient starting out at about 0.3 and rising, on average, over time to nearly 1.2 in 2007.

The sharp increase in the gap coefficient in 2001 coincides with the inclusion of a new investment block which, in conjunction with changes to the supply block, tightened the relationship between supply-side disturbances and subsequent effects on aggregate demand, particularly over the longer term.39 The new investment block, in turn, was driven by two factors: the earlier inclusion by the Bureau of Economic Analysis of software in the definition of equipment spending and the capital stock, and associated enhanced appreciation on the part of the staff of the importance of the ongoing productivity and investment boom. In any case, while the upward jump in the gap coefficient stands out, it bears recognizing that the rise in the gap coefficient was a continual process.

The point to be taken from Figure 5.1 is that the time variation in model properties, described in Section 3, carries over into substantial variation in the optimized TR policy parameters. At the same time, it is clear that time variation in the multipliers is not the sole reason why optimized TR coefficients change. In fact, changes in the stochastic structure of the economy are also in play. To the extent these differences in optimized parameters, conditional on the stochastic shocks, imply significant differences in economic performance, we can say that model uncertainty is a significant problem. We can examine this question by comparing the performance of the optimized TR against other plausible parameterizations. For this exercise and nearly all that follow, we narrow our focus to just two vintages: the December 1998 vintage and the October 2007 vintage. (The optimized Taylor rule coefficients associated with these vintages are indicated in the figure by the gray bars.) These particular vintages were chosen because they were far apart in time, thereby reflecting as different views of the world as this environment allows, and because their properties are among the most different of any in the set.

In the next section we examine the implications for economic performance of the TR and the other optimized simple rules for two selected model vintages.

5.2 Optimized rules and performance

5.2.1 Two-parameter rules

To this point, we have compared model properties and optimized policies but have had nothing directly to say about performance. This section fills this void. We consider the performance, on average of the model economies under stochastic simulation. We also expand our study to encompass the wider range of simple rules introduced in sections 4.1.1 and 4.1.2. At the same time, in order to make the computational costs feasible, we focus on results for the two selected vintages. Table 2 shows the performance for the complete set two-parameter rules. The table is divided into two panels, one for each of the December 1998 and October 2007 vintages. In both panels, losses have been normalized on the performance of the optimized Taylor rule so that the efficacy of other rules can be interpreted as multiples of the TR loss.

Table 2: FRB/US model performance in stochastic simulation ^{\ast}

(2-parameter rule optimizations, selected vintages)

line policy rule parameter: anchor,  i parameter: real,  j December 1998:  \alpha_{i} December 1998:  \alpha_{j} December 1998:  Norm^{\dag } October 2007:  \alpha_{i} October 2007:  \alpha_{j} October 2007:  Norm^{\dag }
1. Taylor rule  \pi  y 0.44 0.33 1 0.53 1.17 1
2. Inflation  \pi  r 0.87 -0.32 1.44 -0.30 -0.81 4.51
3. Price level  p  r 8.14 0.46 1.03 14.7 1.14 0.96
4. U-rate  \pi  \Delta u 0.16 -2.52 1.19 0.08 -3.60 0.88
5. Nom. output I  \Delta yn  r 0.20 0.93 1.35 0.37 0.90 1.42
6. Nom output II  \Delta yn  y 0.02 0.43 1.26 0.02 1.12 1.06
7. Wage growth I  \Delta w  r 1.05 -0.41 1.46 0.63 -0.71 1.57
8. Wage growth II  \Delta w  y 0.72 0.39 0.92 0.09 1.16 1.03
 ^{\ast}~ Loss figures in the right-hand panel cannot be compared with those on the left.

 \dagAverage value for eq. (2) from 1500 stochastic simulations over 20 years, normalized so that losses are interpretable as multiples of the loss under the optimized Taylor rule.

Before delving into the numbers, it is useful to recall that the results in this table pertain to monetary authorities that understand the nature of the economy they control, including the shocks to which the economy is subject. That is, we are setting aside, for the moment, the issue of model uncertainty, which we take up in the next section. With this in mind let us focus for the moment on the optimized parameters and normalized losses for the December 1998 vintage shown in left-hand panel of Table 2. The results show, first, why the TR has been a popular specification for policy design: it renders a very good performance with losses that are lower than nearly all of the alternatives. The one rule that outperforms TR is WR II which is identical to the Taylor rule but replaces price inflation with wage inflation, shown on line 8. This rule is a version of the rule championed by LOWW (2005) on the grounds that in many models, it is wages that is the source of most nominal stickiness. It is not simply feedback on wages that is important to this result, however; the performance of WR I, on line 7, shows that a rule that replaces price inflation with wage inflation as the nominal anchor, but omits direct feedback on the output gap in favor of persistence in funds-rate setting through the presence of a lagged funds rate is the worst rule among those shown. There are other rules that are not far behind the TR in terms of performance, including the (change in) unemployment rate rule, URR, line 4, with a loss only of 19 percent more than the Taylor rule, and the price-level targeting rule, line 3, which carries a loss only slightly above that of the Taylor rule. This latter result may seem familiar to results seen elsewhere that show strong performance of price-level targets. However, to the best of our knowledge, prior results have been exclusively for linear rational expectations models where the powerful role of expectations in strengthening the error-correcting properties of such rules is paramount. That good performance arises from a price-level target in our VAR-based expectations approach is remarkable. Of related interest is the fact that the price-level rule significantly outperforms the ITR. Inflation targeting allows "bygones to be bygones" in the control of the price level, whereas price-level errors have to be reversed in price-level targeting regimes. Reversing price-level errors is a good thing when agents know that the central bank will do this, because anticipated reversals of the price level implies strongly anchored expectations for the inflation rate. When expectations are "boundedly rational," however, the conventional wisdom has been that bringing the price level back to some predetermined path will be all cost and no benefit. We see here that this is not so for the VAR-based expectations of the FRB/US model.

More generally, the performances of the other rules are not greatly different from the Taylor rule; as noted, the WR I performs the worst, but its loss is only about 1-1/2 times that of the TR, not a good performance but not disastrous either. Evidently, controlling the economy of the December 1998 vintage is a relatively straightforward task.

Let us turn now to the right-hand panel where we show parallel results for the October 2007 vintage. Here, once again, the TR does pretty well, on average, but in this instance there are two rules that do better, the price-level rule and the URR. We have already noted that parameterizations of these rules did well in the December 1998 vintage. In addition, two other rules also performed almost as well as the TR: the YNR II and WR II. These rules share two important features. First, they employ feedback on a nominal variable that attempts to correct, albeit indirectly, for trend productivity growth and errors in its measurement. Second, they maintain feedback on the output gap. Thus, notwithstanding the mismeasurement issues associated with persistent changes in productivity growth, feedback on the output gap, which is subject to errors in productivity levels, is still beneficial, as can be seen by comparing line 6 with line 5, on the one hand, and line 8 with line 7, on the other. In other words, these two rules produce good results but not entirely for the reasons for which these rules were originally advocated.

The last word on this panel concerns, once again, the ITR. Its performance controlling the October 2007 vintage could fairly be described as terrible, at 4-1/2 times the loss of the Taylor rule. Qualitatively, this is similar to the results for the December 1998 vintage, but quantitatively much worse. The reasons for this stem from the forementioned tightening of the linkages between the supply block of the model and subsequent aggregate demand fluctuations, together with the nature of the shocks that were incurred during the period over which the two rules are optimized. The rules for the December 1998 vintage are conditioned on shocks 1981 to 1995, while the October 2007 vintage is conditioned on shocks from 1988 to 2002. The former period was dominated by garden-variety demand shocks, whereas the latter had large and persistent disturbances to aggregate supply; in particular, the productivity boom of the second half of the 1990s. Moreover, many of the key shocks borne during the more recent period were larger than was the case in the earlier period.40 An implication of productivity booms is that they disrupt the "normal" time-series relationship between output (or employment) and inflation: when output fluctuations are dominated by demand shocks, and prices are sticky, output will statistically lead inflation and the optimized parameters of rules like the Taylor rule will reflect that relationship. When demand shocks are the prevalent force behind output fluctuations there is no dilemma for monetary policy: stabilizing output and stabilizing inflation are simultaneously achievable because they are one and the same. It follows that one can feedback on output (or its proxies) or inflation, and achieve good results either way. However when supply shocks drive cycles, inflation and output will tend to move in opposite directions, setting up a dilemma for the policymaker. Under these circumstances, responding to output and to inflation or no longer good substitutes for the purposes of minimizing losses and responding strictly to inflation is insufficient for controlling output.

5.2.2 Three-parameter rules

Table 3 tests the appropriateness of using the two-parameter Taylor rule as our benchmark by considering the simple extensions noted in section 4.1.2. In particular, the second row of the table shows that the performance of a Taylor extended to allow an optimized parameter on the lagged instrument renders only slightly better performance the TR itself, for either vintage. Moreover, the attempt through the use of a productivity growth term in the Y*R fares worse, as shown in the third line.41 The final two lines of the table exhibit the advantage of allowing feedback on the lagged instrument relative to the YNR II.

Table 3

FRB/US model performance in stochastic simulation ^{\ast}

(3-parameter rule optimizations, selected vintages)

line rule  \alpha_{ijk}: anchor,  i  \alpha_{ijk}: real,  j  \alpha_{ijk}: added,  k December 1998:  \alpha_{i} December 1998:  \alpha_{j} December 1998:  \alpha_{k} December 1998:  Loss^{\dag } October 2007:  \alpha_{i} October 2007:  \alpha_{j} October 2007:  \alpha_{k} October 2007:  Loss^{\dag }
1. TR  \pi  y - 0.44 0.33 - 1 0.53 1.17 - 1
2. xTR  \pi  y  r 0.33 0.29 0.33 0.98 0.22 1.07 0.22 0.98
3. Y*R  \pi  y  \Delta y^{\ast} 0.38 0.36 0.10 1.04 0.41 1.23 0.29 1.02
4. YNR II  \Delta yn  y   0.02 0.43 - 1.26 0.02 1.12 - 1.06
5. xYNR ^{\ast}  \Delta yn  y  r 0.13 0.10 0.88 1.04 0.23 0.36 0.73 0.97
 \ast The x refers to "extended," adding the lagged instrument.  \dag ~ See the notes to Table 2.

This is the one case where adding the lagged instrument to a rule that already has a nominal anchor variable and an aggregate demand term pays off in a significant way. Still, none of these rules do markedly better than the Taylor rule despite the advantage of an added parameter. We thus conclude that using the Taylor rule as our benchmark is not erecting a straw man. We also satisfied that focussing our attention, henceforth, on two-parameter policy rules is a suitable restriction.

Our goal in this paper has been to uncover policies that are robust across models. To this point, we have identified rules which when properly specified, perform well in contexts where they should perform well. The ones that do not--the inflation targeting rule, and nominal income and wage growth rules that include the lagged instrument as their second argument--are not candidates as robust performers. Whether those rules that are strong performers in their own environments are also robust is the subject of the next section.

6 Robustness

We now turn our principal issue, the robustness of optimized policies to model misspecification. The thought experiment is to imagine a policymaker that believes she is controlling the December 1998 economy model, but in half of the instances we discuss below, it turns out that it is the October 2007 vintage that is the true model. Those results are presented in Table 4. Then, in Table 5, we reverse the exercise by having that our central banker assume she is controlling the October 2007 vintage but it turns out that half of the time, it is the December 1998 vintage that is the correct model.

The same eight two-parameter rules as before are considered, with 16 parameterizations. We subject both of these models to same set of stochastic shocks as in the optimization exercise, for each candidate rule. As before, we are mostly interested in normalized losses where the normalization sets the loss under the appropriate optimized TR policy to unity (although we do show the absolute losses, for completeness). Before we proceed with the results, it is worth recalling that, at the risk of oversimplication, that the December 1998 vintage is a model that sees the US economy as being relatively stable and easy to control: rule parameterizations that are optimal for the December 1998 vintage are generally less aggressive than their October 2007 counterparts.

Table 4

Normalized model performance for optimized 2-parmeter rules under stochastic simulation*

(December 1998 model vintage)

line rule vin. anchor variables  (\alpha_{i}):  \pi anchor variables  (\alpha_{i}):  \Delta yn anchor variables  (\alpha_{i}):  \Delta w anchor variables  (\alpha_{i}):  p real variables  (\alpha_{j}):  y real variables  (\alpha_{j}):  r real variables  (\alpha_{j}):  \Delta u Dec. 98 loss: abs. Dec. 98 loss: norm.
1. Taylor D98 0.44       0.33     17.6 1
2. Taylor O07 0.53       1.17     29.2 1.66
3. Inflation D98 0.87         -0.32   25.4 1.44
4. Inflation O07 -0.30         -0.81   406 23.0
5. Price level D98       8.14 0.46     18.3 1.04
6. Price level O07       14.7 1.14     26.9 1.53
7. Nom. output I D98   0.20       0.93   23.7 1.35
8. Nom. output I O07   0.37       0.90   33.5 1.90
9. Nom. Output II D98   0.02     0.43     22.2 1.26
10. Nom. Output II O07   0.02     1.12     31.0 1.76
11. U-rate D98 0.16           -2.52 21.0 1.19
12. U-rate O07 0.08           -3.60 24.3 1.38
13. Wage growth I D98     1.05     -0.41   25.7 1.57
14. Wage growth I O07     0.63     -0.71   27.8 1.58
15. Wage growth II D98     0.72   0.39     16.2 0.92
16. Wage growth II O07     0.09   1.16     29.4 1.67
* Selected rules and model vintages. Average losses from 1500 draws of 80 periods each.

Beginning with the TR, where the normalized loss is unity by definition, we see that a policymaker who uses the October 2007 parameterization of that rule incurs losses about two-thirds higher what she could have achieved had she known the true model; the Taylor rule is not particularly robust in this sense. The inflation-targeting rule, not a particularly good performer at the best of circumstances, is disastrous when misspecified, as shown on line 4. Among the top performers--at least when the true economy turns out to be the December 1998 vintage--are the price-level rule, lines 5 and 6, the change-in-unemployment rule, lines 11 and 12, and the wage-growth rule that includes the gap, lines 15 and 16. Each of these rules performs at least as well as the Taylor rule when misspecified, and provides performance that is close to that of the Taylor when properly specified.42 The YNR II is not far off the mark set by the optimized Taylor rule.

Table 5 turns the exercise around by considering the case where the October 2007 vintage turns out to be the correct one. Misspecification of the Taylor rule is more costly here: the deterioration relative to the best policy parameterization is 80 percent. Once again, the ITR performs very poorly, while most of the rules that do include feedback on the output gap-the Taylor rule, the price-level rule, one of the nominal-output rules, and the change-in-unemployment rule, all perform well. The one notable exception to the conclusion that feedback on the output gap is always a good thing is the WR II where misspecification of the rule, as in line 16, results in large losses relative to the Taylor and most alternatives to it. Even here, though, its seems that it is feedback on wage growth that is the key to this result as the rules in lines 13 and 14, which respond to wage growth and the lagged instrument, but not the output gap, perform even worse. What this tells us is that while a wage-growth rule can turn in a very good performance, as it does when paired with the output gap on line 15, a good calibration is critical to its performance; the rule is not robust.

The PLR turns in an even stronger performance for the October 2007 vintage than it did for the December 1998 one. This result obtains notwithstanding that the parameterizations of the two rules differ significantly: the feedback parameters on the output gap are 1.14 and 0.46. As in rational expectations models, an important contribution to economic performance under this rule is that constraining the drift in the price level anchors inflation fluctuations. In both vintages of the FRB/US model, keeping inflation in check also limits cycling in long-term expected inflation. The stability of long-term inflation expectations reinforces the stabilizing force of policy making output stabilization less critical than would otherwise be the case.43

This case contrasts sharply with the URR. For this rule, feedback on inflation itself is slight at 0.08 and 0.16. But feedback on the change in the unemployment rate is vigorous: -3.60 and -2.52. Thus, aggressively tempering fluctuations in unemployment is substituting for inflation (and price-level) control. The fact that the URR is written in the first-difference of the instrument, and therefore does not depend on estimates of the equilibrium real rate of interest is also a factor; this means that the instrument can find the right level even when a productivity shock changes what that level should be. The URR is the one rule of which we are aware that was tested, by Orphanides and Williams (2002), in an environment that allowed for persistent, unobserved shocks to the "natural rate of interest," and found to execute well.

Table 5

Normalized model performance for optimized simple rules under stochastic simulation*

(October 2007 model vintage)

line rule vin. anchor variables  (\alpha_{i}):  \pi anchor variables  (\alpha_{i}):  \Delta yn anchor variables  (\alpha_{i}):  \Delta w anchor variables  (\alpha_{i}):  p real variables  (\alpha_{j}):  y real variables  (\alpha_{j}):  r real variables  (\alpha_{j}):  \Delta u Oct. 07 loss: abs. Oct. 07 loss: norm.
1. Taylor O07 0.53       1.17     17.7 1
2. Taylor D98 0.44       0.33     31.9 1.80
3. Inflation O07 -0.30         -0.81   79.9 4.51
4. Inflation D98 0.87         -0.32   134 7.57
5. Price level O07       14.7 1.14     17.0 0.96
6. Price level D98       8.14 0.46     22.8 1.29
7. Nom. output I O07   0.37       0.90   25.2 1.42
8. Nom. output I D98   0.20       0.93   29.8 1.68
9. Nom. Output II O07   0.02     1.12     18.7 1.06
10. Nom. Output II D98   0.02     0.42     24.6 1.39
11. U-rate O07 0.08           -3.60 15.6 0.88
12. U-rate D98 0.16           -2.52 18.9 1.07
13. Wage growth I O07     0.63     -0.71   104 5.89
14. Wage growth I D98     1.05     -0.41   41.9 6.38
15. Wage growth II O07     0.09   1.16     18.3 1.03
16. Wage growth II D98     0.72   0.39     57.1 3.22
* Selected rules and model vintages. 1500 draws of 80 periods each.

The results for URR is Tables 4 and 5 suggest that it could be a robust rule. We can take a closer look at the robustness of URR by computing its optimized parameters for all vintages. The results of this exercise are shown in Figure 6.1.

Figure 6.1: Optimized coefficient of the change-in-unemployment rule, by model vintage
Figure 6.1 Optimized coefficient of the change-in-unemployment rule, by model vintage. The figure is similar to figure 5.1, except for the change-in-unemployment rule, otherwise known as "URR" in the paper.  The blue dot-dashed line is the feedback coefficient on the change in the unemployment rate; the red solid line is the feedback coefficient on four-quarter PCE inflation.   As in figure 5.1, two gray vertical bars indicate the December 1998 and October 2007 vintages. The coefficient on the change in the unemployment rate varies widely and not very systematically across vintages, with values from about -1.5 to -4.4. The inflation coefficient shows very little variation across vintages and is just barely above zero.
The figure shows that the coefficient on inflation, the solid red line, is never much above zero, regardless of the vintage. By contrast, the coefficient on the change in the unemployment rate, the dashed blue line, jumps around somewhat with perhaps a slight tendency to increase, in absolute terms, over time. The range over the complete set of vintages for the coefficient on the change in the unemployment rate spans from a low of -1.4 for the November 1999 vintage, to a high of -4.2 for the August 2007 vintage, considerably wider the range encompassed by the December 1998 and October 2007 vintages, shown by the gray bars. The computations underlying 6.1 allow us to expand on the robustness analysis of Tables 4 and 5 while focussing on the unemployment rate rule. We do this it Table 6 below where we consider the performance of the most extreme parameterizations of the rule in our two benchmark vintages.

Table 6

Performance of selected parameterizations of change-in-unemployment rate rule

line rule parameterization coefficients:  \alpha_{\Delta u} coefficients:  \alpha_{\pi} losses for model, December 1998: absolute losses for model, December 1998: normalized losses for model, October 2007: absolute losses for model, October 2007: normalized
1. December 1998 -2.56 0.16 20.97 1 26.85 1.28
2. November 1999 -1.40 0.04 33.12 1.58 31.71 1.52
3. August 2007 -4.24 0.09 26.63 1.27 21.05 1.01
4. October 2007 -3.94 0.08 25.62 1.22 20.85 1

The table shows that when either of our benchmark models is subjected controlled by the most extreme parameterization of the URR, the small absolute coefficient on the (change in the) unemployment rate in the November 1999 vintage, the deterioration in control increases the loss relative to the best possible parameterization by a bit over 50 percent, as shown on line 2 of the table. The parameterization that rendered the largest coefficient on the change in the unemployment rate, the August 2007 vintage, gave coefficients that are not much different from the (chronologically close) October 2007 vintage. Thus lines 3 and 4 of the table are similar. Incremental losses, relative to the best possible URR parameterization, of 50-some percent, are not particularly large in comparison with the results in Tables 4 and 5.

7 Concluding remarks

For central banks, the appropriate design of monetary policy under uncertainty is a critical issue. Many conferences are devoted to the subject and the list of papers is lengthy and still growing. In nearly all instances, however, the articles, whether they originate from central banks themselves or from academia, have tended to be abstract applications. One posits an idealized model, or several models, of the economy and investigates, in some way, how missperceptions of, or perturbations to, the model affect outcomes. A good deal has been learned from these exercises, but results have tended to be specific to the environment of the chosen models. Moreover, the models themselves typically have not been representative of the models upon which central banks rely. It is difficult to know how serious a problem model uncertainty is if one cannot give a concrete and meaningful measure of uncertainty.

This paper has cast some light on model uncertainty and the design of policy in a much different context from the extant literature. We have examined 46 vintages of the model the Federal Reserve Board staff has used to carry out forecasts and policy analysis from 1996 to 2007. And we have done so in a real-time context that focusses on the real problems that the Fed faced over this period. Our examination looked at a number of simple policy rules that have been marketed as "robust." In the end, we uncovered a number of useful observations. First, model uncertainty is a substantial problem. Changes to the FRB/US model over the period of study were frequent and often important in their implications. The ensuing optimized policies also differed significantly in their parameterizations. Second, many simple rules that have been touted as robust, turn out to be less appealing than one might have suspected. In particular, pure inflation targeting rules and indeed nearly all rules that fail to respond to measures of aggregate demand turn out not to be robust. Third, adding an instrument smoothing term to rule that already had a nominal anchor and a real variable contributes little to the robustness and efficiency of rules. Fourth, notwithstanding problems of mismeasurement of output gaps, it generally pays for policy to feedback on some measure of excess demand. Fifth, a case can be made for designing simple rules that minimize the use of latent variables like potential output and the equilibrium real interest rate as arguments.

So why are simple rules not as reliable in the current context, when they have been in others? Levin et al. (1999) argue that simple rules do a good job of controlling economies, even in models for which their parameterizations are incorrect. The reason is because Levin et al. (1999) restricted their attention to full-information linear rational expectations models, which tend to be forgiving of wide ranges of policy rule misspecification. Loosely speaking, linear rational expectations models have loss surfaces that tend to be very flat in a large neighborhood around the optimized rule parameterization; see, e.g., Levin and Williams (2003). Economists are fond of rational expectations, and for good reason: it removes a free parameter from the model, and ensures that policy decisions are not founded on what amounts to money illusion. Nonetheless, the sense in which agents are rational is questionable. In environments such as the US economy of the 1990s and 2000s, with bubbles and financial crises, amid a broader economy that has produced fewer and milder recessions than before, it seems plausible that the economy has undergone structure shifts. To the extent that this is so, it seems reasonable to consider expectations that are somewhat less than fully rational-such models may include agents in the process of learning, as Primiceri (2005) and Milani (2007) have done, for example.


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* Contact address: Robert Tetlow, Federal Reserve Board, Washington, D.C. 20551. Email: [email protected] This and other papers can be found at I thank Flint Brayton and Dave Reifschneider for helping me interpret historical model changes as well as for useful comments. I also thank Sean Taylor and Trevor Davis for dedicated research assistance. All remaining errors are mine. The views expressed in this paper are those of the author alone and do not necessarily represent those of the Federal Reserve Board or other members of its staff. Return to Text
1. Two other aspects of uncertainty, relevant to monetary policy making are parameter uncertainty; (see e.g., Brainard (1967), Söderström (2002), Walsh (2004) and Kimura and Kurozumi (2007)) and data uncertainty (Aoki (2003), Jääskelä and Yates (2005)). These subject areas should be regarded as complementary to the study of model uncertainty. Return to Text
2. An illuminating exception to this rule is the paper of Levin et al. (2005) which uses an estimated DSGE model and finds that a nominal wage growth rule performs almost as well as the optimal policy rule. Comments on this paper by Walsh (2005) express doubts that the current generation of DSGE models is sufficiently advanced to be taken seriously for this purpose. Return to Text
3. The staff of the Federal Reserve Board prepare two documents for each FOMC meeting. The review of domestic and foreign economic conditions and projections for the future are contained in the Greenbook, so called because of its green cover. Alternative scenarios and confidence intervals also appear in the Greenbook. The review of financial conditions and policy options is in the Bluebook. Simulations using the FRB/US model appear regularly in both documents as well as a large variety of memos and reports. Fed security rules place an embargo on public release of these documents for five years. Return to Text
4. Activity for the Bank of England has been similar: over the ten years from the first meeting of the Monetary Policy Committee in June 1997 through March 2007, the bank rate has been changed 34 times. Return to Text
5. There have been a number of valuable contributions to the real-time analysis of monetary policy issues. Most are associated with data and forecasting. See, in particular, the work of Croushore and Stark (2001) and a whole conference on the subject details of which can be found at An additional, deeper layer of real-time analysis considers revisions to unobservable state variables, such as potential output; Athanasios Orphanides, alone or with co-authors, has been at the vanguard of this issue; see, e.g., Orphanides et al. (2000) . See also Giannone et al. (2005) for a sophisticated, real-time analysis of the history of FOMC behavior. Return to Text
6. Whole conferences have been organized just on the need for real-time data for the Euro area; e.g., the Center for Economic Policy Research conference "Needed: A Real Time Database for the Euro-Area," June 13-14, 2005, in Brussels Return to Text
7. More precisely we show adjusted non-farm business output adjusted to exclude owner occupied housing, and to include oil imports. This makes output conformable with the model's production function which includes oil as a factor of production. Henceforth all references to productivity or potential output are to this concept of adjusted non-farm business output. Return to Text
8. Defined in this way, data uncertainty does not include uncertainty in the measurement of latent variables, like potential output. The important conceptual distinction between the two is that eventually one knows what the final data series is-what "the truth" is-when dealing with data uncertainty. One never knows, even long after the fact, what the true values of latent variables are. Latent variables are more akin to parameter uncertainty than data uncertainty. On this, see Orphanides et al. (2000) and Orphanides (2001). Return to Text
9. A record such as the one in the table was not unusual during this period; the Survey of Professional Forecasters similarly underpredicted output growth. Tulip (2005) documents how the official Greenbook forecast exhibited a similar pattern of forecast errors. Return to Text
10. Some details on this evolution of thought are provided in an unpublished appendix to Tetlow and Ironside (2007) which can be found at Return to Text
11. It was only a month before, in December 1996, that then Chairman Alan Greenspan uttered his famous line about "irrational exhuberance." In suggesting that equity prices might have been too high to be justified by fundamentals he was reflecting the same Keynesian prior exemplified by the early data in Figure 2.3 that suggested that the strong growth of 1996 was temporary. Return to Text
12. There were methodological changes to expenditures and prices of cars and trucks; improved estimated of consumer expenditures on services; new methods of computing changes in business inventories; and some expenditures on software by businesses were removed from expenses and reclassified as business fixed investment. PCE inflation jumps again in July 2002 when the annual revisions resulted in new price index for PCE services; see Anderson and Klein (2005). Return to Text
13. The model introduced the notion of polynomial adjustment costs, a straightforward generalization of the well-known quadratic adjustment costs, which allowed, for example, the flow of investment to be costly to adjust, and not just the capital stock. This idea, controversial at the time, has subsequently been adopted in the broader academic community; see e.g., Christiano, Eichenbaum and Evans (2005). Return to Text
14. With a traditional (statationary) VAR, the endpoint of the system is the sample mean of the series. Kozicki and Tinsley (2001) show that a stationary VAR allows insufficient variability to encompass the dynamics of such variables as long-term interest rates. Return to Text
15. We might also note that working with the rational expectations vintages of the model is infeasible on many grounds. Not only do we not have a full set of rational expectations vintages, but their simulation requires very long databases for the required extended-path solution algorithms to work effectively, and optimization of parameters in large-scale non-linear rational expectations models is computationally a very daunting task. Return to Text
16. The archives are listed by the precise date of the FOMC meeting in which the forecasts were discussed. For our purposes, we do not need to be so precise so we shall describe them by month and year. Thus, the 46 vintages we use are, in 1996: July and November; then, typically thereafter the months would be January (but often February), May, August (but ocassionally July), and November (but twice October and once December). Nothing of importance is lost from the analysis by excluding every second vintage from consideration. Return to Text
17. Each vintage has a list of variables that are shocked using bootstrap methods for stochastic simulations. The list of shocks is a subset of the model's complete set of residuals since other residuals are treated not as shocks but rather as measurement error. The precise nature of the shocks will vary according to data construction and the period over which the shocks are drawn. Return to Text
18. The Board staff present their analysis of recent history, the staff forecast and alternative simulations, the latter using the FRB/US model, in the Greenbook. The FOMC also receives detailed analysis of policy options in the Bluebook Alternative policy simulations are typically carried out using the FRB/US model. In addition, for the FOMC's semi-annual two-day meetings, detailed reports are often prepared by the staff and these reports frequently involve the FRB/US model. Go to for transcipts of FOMC meetings as well as the presentations of the senior staff to the FOMC. See Svensson and Tetlow (2005) for a related discussion. Return to Text
19. These multipliers could have defined differently. The sacrifice ratio could have been cumulated over a different duration than the five years selected, or it could have been computed in terms of output instead of employment, or the cumulative losses could have been discounted. Similarly, the funds rate multiplier could have been defined in terms on unemployment instead of output, or over a different horizon. The qualitative conclusions would have been no different for any reasonable alternative. Return to Text
20. The experiment is conducted by simulation, setting the target rate of inflation in a Taylor rule to one percentage point below its baseline level. The sacrifice ratio is cumulative annualized change in the unemployment rate, undiscounted, relative to baseline, divided by the change in PCE inflation after 5 years. Other rules would produce different sacrifice ratios but the same profile over time. Return to Text
21. The sizable jump in the sacrifice ratio in late 2001 is associated with a shift to estimating the models principle wage and price equations simultaneously together with other equations to represent the rest of the economy, including a Taylor rule for policy. Among other things, this allowed expectations formation in wage and price setting decisions to reflect more recent Fed behavior than the full core VAR equations that are used in the rest of the model. Return to Text
22. transcript of the July 2-3, 1996 meeting of the FOMC, p. 42-47.

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23. Transcript, FOMC meeting, February 1, 2000, p. 41-2. Return to Text
24. In particular, the same phenomenon occurs to varying degrees in simple single-equation Phillips curves of various specifications using both real-time and ex post data. One paper along these lines is Atkeson and Ohanian (2001). Roberts (2006) shows how greater discipline in monetary policy may have contributed to the reduction in economic volatility in the period since the Volcker disinflation. Cogley and Sargent (2005) estimate three Phillips curve models simultaneously and apply Bayesian decision theory to explain why the Fed did not choose an inflation stabilizing policy before the Volcker disinflation. They too find time variation in the output cost of disinflation. Cogley and Sbordone (2008) show how the trend rate of inflation can influence the weight on marginal cost in the New Keynesian Phillips curve. Return to Text
25. The levels relationship of the stock market equation means that the wealth effect of the stock market on consumption can be measured in the familiar "cents per dollar" form (of incremental stock market wealth). Also playing a role is the log-linearity (that is, constant elasticity) of the relationship between wealth and consumption. Return to Text
26. The "active approach" to the presence of stock market bubbles argues that monetary policy should specifically respond to bubbles. See, e.g., Cecchetti et al. (2000). The passive approach argues that bubbles should affect monetary policy only insofar as they affect the forecast for inflation and possibly output. They should not be a special object of policy. See, Bernanke and Gertler (1999). Return to Text
27. The exchange-rate multiplier calculations in figure 3.3 are computed at an 8-quarter horizon while the import-price passthrough in figure 3.4 is measured at 12 quarters. The federal funds rate is held at baseline in both instances. Return to Text
28. The striking change in 1998 in figure 3.3 corresponds with a shift from a G10 aggregate of trade weights for foreign indexes to a G29 aggregate. The reversal began with a shift to chain-weighting of domestic prices indexes in 1999:Q3. Return to Text
29. See, e.g., Orphanides et. al. (2000), Orpanides (2001) and Ehrmann and Smets (2003). Return to Text
30. Our rendition of the rule differs in small ways from the Taylor (1993) original owing to operational considerations. In particular, FRB/US uses PCE inflation instead of the GDP deflator. The model also allows  rr^{\ast} to vary over time whereas Taylor kept his at a constant 2 percent. Return to Text
31. In nearly all works on optimized rules, the steady-state terms are omitted for two reasons: first, the models used are linear, so the steady state can be taken as zero; and second, no allowance is made for shifting steady states. (An exception is Orphanides and Williams (2002) who specifically consider  rr^{\ast} that shift over time.) Because we are using real models with real databases, and we are considering persistent deviations from steady state--indeed arguably this is a large part of the problem of interest--we need to retain these steady-state terms. Return to Text
32. The intercept used in the model's Taylor rule, designated  rr^{\ast}, is a medium-term proxy for the equilibrium real interest rate. It is an endogenous variable in the model. In particular,  rr_{t}^{\ast}=(1-\gamma)rr_{t-1}^{\ast }+\gamma(rn_{t}-\pi_{t}) where  r is the federal funds rate, and  \gamma =0.05. As a robustness check, we experimented with adding a constant in the optimized rules in addition to  rr^{\ast} and found that this term was virtually zero for every model vintage. Note that relative to the classic version of the Taylor rule where  rr^{\ast} is fixed, this alteration biases results in favor of good performance by this class of rules. Return to Text
33. Qualitatively speaking, our results are the same if the output gap is substituted for the unemployment gap in (2) provided the proper normalization of the weight is taken to account for the relative sized of unemployment gaps and output gaps over the business cycle. Return to Text
34. Our rules will be optimal in the relevant class, conditional on the stochastic shock set, (5), under anticipated utility as defined by Kreps (1998). Return to Text
35. The fact that the policy rule depends on the variance-covariance matrix of stochastic shocks means that the rule is not certainty equivalent. This is the case for two reasons. One is the non-linearity of the model. The other is the fact that the rule is a simple one: it does not include all the states of the model. Return to Text
36. The number of shocks used for stochastic simulations has varied with the vintage, and generally has grown. For the first vintage, 43 shocks were used, while for the November 2003 vintage, 75 were used. Return to Text
37. For these experiments any reasonable target will suffice since the stochastic simulations effectively randomize over initial conditions. Return to Text
38. There is also a sharp jump in the gap coefficient over the first two quarters of 2001. One might be tempted to think that this is related to the jump in the sacrifice ratio, shown in Figure 3.1. In fact, the increase in the optimized gap coefficient precedes the jump in the sacrifice ratio. Return to Text
39. In essence, the linkage between a disturbance to total factor productivity and the desired capital stock in the future was clarified and strengthened so that an increase in TFP that may produce excess supply in the very short run can be expected to produce an investment-led period of excess demand later on. Return to Text
40. This argument will clash with the intuition of a number of readers who may be familiar with the literature on the Great Moderation which suggests that shocks in the most recent period are smaller than they once were. The explanation is two-fold: first, the period we are dealing with here is much shorter and has smaller residuals in both datasets. Just as important perhaps is a falacy in the construction of the residuals in many studies that allege that shocks are smaller recently. The regressions from which these conclusions are drawn allow either a time trend or a free constant so that persistent supply-side shocks are mopped up in these terms. Return to Text
41. It should be the case that the addition of an added parameter cannot do worse than the best two-parameter rule. The contradictory result shown in the table is an artifact of occasional crashes in the optimization algorithm owing to the instability of the extended rule. Still, the instability of rule is, itself, a warning against such a rule. Return to Text
42. If we were to take a Bayesian perspective on this and assume that the two vintages are equally probable, the average values of the losses from the PLR, the URR and the YNR II are all less than that of the Taylor rule, for this model vintage. Return to Text
43. Taking the December 1998 vintage results as an example, price-level targeting stabilizes, long-term inflation expectations,  \pi^{\infty}from 2.1 better than does inflation targeting, or any other rule for that matter. This leads to better performance than in inflation targeting rule not so much in inflation variability but in unemployment variability. Return to Text

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