A. Three Sets of Output Gap Estimates

I use three different sets of estimates of fundamentals. The first may be considered ex post; the two others, real-time. The main difference among them is the way output gap estimates are constructed.

The very first specification of the rule uses 1999 vintage data for inflation and output levels. Most of these data (particularly the earlier components) have been substantially revised since their initial releases. For this particular specification I use estimates of output gap, which I call “lags and leads.” I detrend the whole path for log-output (1947:Q1—1999:Q2) using a Hodrick-Prescott filter.

Of course for most of the univariate detrending techniques (linear, quadratic, HP) as well as for multivariate ones, calculating a trend at any point in time t very much depends not only on lagged but also on lead observations. Had the economy evolved differently after t, trend estimates would have been different.

Imagine that after a long period of growth, output declines over one or two consecutive periods. At this point, we cannot draw a straightforward conclusion—whether the decline is a temporary correction, which will be followed by further growth, or is the signal of the business cycle’s turning point. Only after observing output over several more periods can we tell exactly what is happening, as this then allows for a more precise estimate of the trend at a point of interest t. Orphanides and van Norden (1999) cite lack of information about an economy’s future developments as a main cause of errors in real-time estimates of the gap.

Thus, the second specification of output gap is “lags only.” For every point in time t, I detrend lagged observations of the output levels, from 1947:Q1 till t-1. Thus, the estimates are obtained using the so-called one-sided Hodrick-Prescott filter. Also, the inflation and output data that I use are unrevised, as reported in the Philadelphia Federal Reserve’s real-time web dataset (for details see Croushore and Stark, 1999). Excluding leads of the data and ignoring revisions have a drastic effect on output gap estimates. An error term (defined as the difference between ex post lags-and-leads, and real-time lags-only estimates of output gap) ranges from -1.54 to 3.33, with a mean of 0.53, and a standard deviation of 1.45 (see Table 1). Tchaidze (2001) argues that most of these errors should be attributed to the exclusion of leads rather than to revisions.

Table 1.

Errors in Real-Time Estimates of Output Gap

Table 1.

Errors in Real-Time Estimates of Output Gap

	Lags and Leads—Lags Only	Lags and Leads—Lags and Forecasts
Mean	0.53	0.24
St.Deviation	1.45	0.87
Max	3.33	2.09
Min	-1.54	-0.97

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Table 1.

Errors in Real-Time Estimates of Output Gap

	Lags and Leads—Lags Only	Lags and Leads—Lags and Forecasts
Mean	0.53	0.24
St.Deviation	1.45	0.87
Max	3.33	2.09
Min	-1.54	-0.97

Although the data released in various statistical bulletins do not suggest much about further developments in the economy, policymakers undoubtedly know more. Even though they do not observe the leads of output and cannot make correct estimations of the trend based on observed values, they may be observing other signals which indicate approximate values of the trend and the direction which it is going to follow. Although over longer horizons these estimates become less and less precise, policymakers gain additional leverage, as they can influence future developments through the setting of policy variables.

To account for these factors, I construct the third set of estimates called “lags and forecasts.” At every point in time t, I detrend the time series, which consists of observed lagged values (from 1947:Q1 until t-1) and the forecasts for the contemporaneous as well as the four following quarters (from t until t+4). Detrending such series allows me to construct estimates which would be closer to those that policymakers used.

To construct forecasts of output, I use forecasts of output growth as reported in the “Current Economic and Financial Conditions” issues, also known as the “Greenbook.” The Greenbook is a collection of various data that are prepared by the economists at the Federal Reserve Board, which are presented to the Board of the Governors before their regular meetings (the board usually meets eight times a year).

For security reasons, the Greenbook data become publicly available only with a five-year lag. Note that data are not really forecasts in the sense that they do not reflect the policy being implemented, but rather assumptions of the economists about future policy and shocks. Obviously, had a different set of assumptions been made, in each case the forecast would have been different. However, since at the very short horizons, policy has a minimal effect (if any at all), I can assume that only the forecasts for t+3 and t+4 are not invariant with respect to the assumptions about the future policy.

As Table 1 indicates, the inclusion of forecasts significantly improves real-time gap estimates. Both the mean and standard deviations of the error term are about half of what they were before. Maximum and minimum values are much smaller in absolute terms as well. Figure 1 also shows that most of the time the lags-and-forecasts estimates lie between the lags-and-leads and the lags-only estimates, indicating an obvious improvement in estimation results.

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Output Gap Estimates

Citation: IMF Working Papers 2004, 213; 10.5089/9781451874976.001.A001

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B. Estimating the Rule

Since the lags-and-forecasts output gap estimates are much closer to the ex post lags-and-leads estimates than the estimates which are based on lags only, it should not be surprising that the difference between ex post and real-time estimates of the Taylor rule diminishes as well once the lags-and-forecasts gap estimates are used rather than those based on lags only (see Table 2). Not only is the difference between the values of response coefficient estimates smaller, the difference in fit practically disappears.⁵

Table 2.

Rules with Various Output Gap Specifications⁶

Table 2.

Rules with Various Output Gap Specifications⁶

	INFL LAG	GAP LAG	CNST	Adj R²	SSR
Lags and Leads	1.58 (0.31)	1.25 (0.13)	0.32 (1.05)	0.86	19.88
Lags and Forecasts	2.33 (0.23)	1.02 (0.14)	-1.39 (0.94)	0.82	24.58
Lags Only	2.90 (0.27)	0.83 (0.17)	-2.97 (1.11)	0.73	38.01

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Table 2.

Rules with Various Output Gap Specifications⁶

	INFL LAG	GAP LAG	CNST	Adj R²	SSR
Lags and Leads	1.58 (0.31)	1.25 (0.13)	0.32 (1.05)	0.86	19.88
Lags and Forecasts	2.33 (0.23)	1.02 (0.14)	-1.39 (0.94)	0.82	24.58
Lags Only	2.90 (0.27)	0.83 (0.17)	-2.97 (1.11)	0.73	38.01

The results still suggest a very strong inflation response of 2.3, which is higher than the values usually suggested (1.4–2.0; see Rudebusch, 2001) and a strong output gap response of 1.0 leaning to the right end of the usually suggested range (0.5–1.0; see Rudebusch, 2001). Even though fits for the lags-and-leads and lags-and-forecasts regressions are almost identical, there is a difference in the values of the estimated coefficients. The values for the lags-and-leads specification are very similar to the suggested values⁷ of 1.5, 1 and 1. The values for the lags-and-forecasts specifications are different.

Testing whether coefficients in the lags-and-forecasts specification are equal to those suggested by the lags-and-leads specification provides the following results: we cannot reject a hypothesis that an output gap coefficient for the lags-and-forecasts specification is equal to 1.25, the value suggested by the lags-and-leads specification, but we reject the hypothesis C_π = 1.58 at a 0.05 level of significance. Likewise we reject a hypothesis C = 0.32, C_π = 1.58 and C_y = 1.25 at the 0.05 significance level as well.

Thus, even though for all of the parameters, confidence intervals in the lags-and-leads and lags-and-forecasts regressions have a nonempty intersection, the tests indicate that there is a substantial difference between the two, and in particular, that the inflation coefficient is much higher than 1.5.

A. Theoretical Model

To describe the economy, I use a model which has become a somewhat standard tool for such purposes (Romer, 2000). The annual model was proposed by Ball (1999) and Svensson (1997). Orphanides (1998b) develops a similar semiannual model, and Rudebusch and Svensson (1999) derive an analogous model formulated in quarterly terms.

The model consists of the two following equations:

where a period is one year, r_t is a real interest rate set by policymakers, π_t-1 and y_t-1 refer to the newest information available to them before making a decision, and π* and r* refer to the respective long-run levels of inflation and real interest rate.

Equation (1) is an accelerationist Phillip’s curve. Equation (2) is an IS curve, assuming an inertial output gap, which is affected by a real interest rate r_t. Finally, ε_t+1 and η_t+1 are zero-mean, normally distributed random variables, reflecting supply and demand shocks.

Note that the model is completely backward looking, and thus implicitly assumes adaptive expectations. It is a common observation that alternative frameworks, assuming rational expectations, do not fit observed data as well, unless there are some agents that are backward looking to some degree (for example, Ball, 2000; Fuhrer, 1997; Roberts, 1997 and 1998).

The model assumes that policymakers can affect inflation only within two periods, as monetary policy has an effect on output gap with a one-period lag, and output gap affects inflation with a one-period lag as well. This means that when policymakers are designing monetary policy by setting an instrument variable r_t, they treat expected inflation E_tπ_t+1 as given. Furthermore, as equation (2) shows, setting the interest rate is tantamount to setting the expected output gap—for any given E_ty_t+1 we can find r_t, such that the expected output gap for the next period will be equal to E_ty_t+1.

I assume that policymakers are minimizing a weighted sum of inflation and output gap variances. As Romer (2000) shows, such an objective implies a linear response function of the following form:

where q is a parameter, determined by the weights that policymakers assign to inflation and output gap variances. Higher q is associated with a higher weight being placed on inflation variance, while lower q is associated with a higher weight being put on output gap variance.

Equation (3) shows that whenever policymakers expect inflation to be above its long-run level, they contract the economy in order to prevent it from overheating. At the same time, whenever inflation is expected to be below its long-run level, they loosen up and push the output above its potential level.

Although previous authors have assumed that shocks ε and η are completely unforecastable, I assume that policymakers do have some information about them. In particular, although unconditional expectations Eε_t+1 and Eη_t+1 are zero, policymakers’ expectations E_tε_t+1 and E_tη_t+1 as of time t are not necessarily so.

Substituting equation (1) and equation (2) into (3), we can solve for the interest rate in terms of lagged output gap and inflation, as well as expected output gap and inflation shocks:

where C_π = q/β and C_y = (λ+qα)/β. This formulation suggests that the more anti-inflationary the preferences of policymakers, the more aggressive the corresponding policy in response to deviations in both inflation and output gap lags.

Reformulating the rule in terms of a nominal rather than a real interest rate results in the following:

Note that the policymakers respond to “inertial” variables π_t and y_t as well as expected shocks E_tε_t+1 and E_tη_t+1.

Unfortunately, not all of these variables, particularly those reflecting expectations, are readily available when it comes to empirical estimations. The Greenbook does provide data on output growth and inflation forecasts, and at first glance, these might seem sufficient to solve the problem. However, the issue is more delicate.

Since inflation is predetermined for one period ahead and does not depend on the interest rate, I can use equation (1) by substituting the Greenbook forecasts in place of expected inflation E_tπ_t+1 and treat residuals as the expected inflation shock E_tε_t+1:

At the same time, as equation (2) demonstrates, expected output shock E_tη_t+1 cannot be recovered, as it depends on the assumed values of the interest rate, which are not observed.

As already mentioned, the Greenbook forecasts are calculated before the Federal Reserve Board makes a decision, and thus are not based on the true value of the policy instrument. They do not reflect implemented policy in the same manner as do forecasts produced by the Bank of England or the Bank of Canada.

The path for the interest rate assumed in these forecasts is suggested by the director of the Research Department (Michael Prell for the period covered). Very often, alternative forecasts are produced as well, and these are sometimes (though not always) included in the Greenbook. Over the course of a forecast exercise, the suggested path may well be revised if the staff (or their models) suggest that the path is unlikely actually to be realized in the economy.

Finally, apart from models, economists also use their own judgment when calculating forecasts, suggesting that these forecasts are highly subjective and do not necessarily reflect the Board’s opinions.

As mentioned earlier, the main interest in this study is whether the rule is forward looking— that is, takes into account expected shocks—or purely mechanical—that is, simply uses lags of the fundamentals. If these shocks are not directly included among the regressors, such a rule can be mistaken for a mechanical, backward-looking rule. However, when estimated as such, one implication would be a lower fit and a larger sum of squared residuals. Thus, estimating a lag-based rule, both with and without inclusion of the expected inflation shock and comparing sums of squared residuals, should indeed demonstrate whether a cook can run the Federal Reserve or not.

In addition to that, the model implies that the coefficients for lagged inflation and expected inflation shock should satisfy a restriction imposed by equation (4)—the difference between them should be equal to 1. Testing this hypothesis would provide another way of testing whether policymaking is forward looking or not.

B. Empirical Results

This subsection presents empirical results. Since the data that I use is quarterly, the notations are slightly different from those in the previous subsection. In particular, the newest pieces of information that are observed by policymakers are lagged quarterly inflation ${\pi }_{t-1}^q$, and lagged output gap $y_{t-1}^q$, measured as in Section II, on the basis of the lags-and-forecasts specification. The inflation forecast for one year ahead corresponds to the third quarter horizon forecast $E_t{\pi }_{t+3}^q$.

I start by calculating expected inflation shock variable $E_t{\epsilon }_{t+3}^q$. I estimate the forecast Phillip’s curve, analogous to equation (1), but using $E_t{\pi }_{t+3}^q$ as a dependent variable, and assume that a residual term reflects expectations of the shock, which are not related to developments in lagged fundamentals.

The estimated forecast Phillip’s curve is as follows:

Note that the coefficient on lagged inflation is very close to 1, and that the coefficient on lagged output gap is 0.34, corresponding to a sacrifice ratio of 3 (output gap of -3 percent, in absence of shocks, causes a 1 percent decrease in inflation), which is close to the estimates reported by Mankiw (1997) and Sachs (1985), 2.8 and 2.9, respectively.

Next, I define $E_t{\epsilon }_{t+3}^q$ as the difference between actual forecasts and fitted values based on equation (5):

Figure 2 demonstrates the magnitude of these shocks. Most of the time, they are small, with a standard deviation of 0.48. There are, however, several points where the expected shock reaches levels of 1 percent and higher, particularly in the second half of 1987, in 1988.2, and in 1992.2. Those moments will be discussed later on, in Section IV.

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Expected t+ 3 Inflation Shock as of t

Citation: IMF Working Papers 2004, 213; 10.5089/9781451874976.001.A001

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Estimating backward- and forward-looking specifications of the rule produces the following results:⁸

Inclusion of the expected inflation shock does improve the results, increasing the fit and lowering the sum of squared residuals. The difference between the coefficients on lagged inflation and expected inflation shock is close to 1, as the model predicts.

The presence of the expected inflation shock in the estimated rule serves as evidence that the Federal Reserve’s policy is forward looking, a factor ignored by the traditional Taylor rule. It suggests that instead of responding to events only after they have occurred, the Federal Reserve does take into account expected developments—the respective monetary policy is, in fact, designed to preempt rather than just react.

Note that the inflation coefficient is still estimated to be higher than the usually suggested values. In fact, Orphanides (1998a) reports an almost identical coefficient of 2.51 when substituting inflation and output gap forecasts for the third quarter horizon into the rule.

Estimates of the response parameters can also be used to estimate implied values for the real interest rate and inflation targets. Tchaidze (2001) calculates these according to the following formulas:

where $\widehat{C}$ and ${\widehat{C}}_{{\pi }}$ are estimates of the constant and the inflation response corresponding to the Taylor rule.

An assumption about one of the targets allows us to calculate the value for the other. Taylor (1993) suggests a 2 percent target for both variables. Judd and Rudebusch (1998) argue that under certain circumstances average values may signal policymakers’ intentions.

Assuming ${\widehat{r}}^{*}$ at a 2.00 level suggests ${\widehat{\pi }}^{*}$ equal to 3.22, while assuming ${\widehat{r}}^{*}$ equal to the sample’s average⁹ of 2.48 suggests ${\widehat{\pi }}^{*}$ equal to 3.61.

Likewise, assuming ${\widehat{\pi }}^{*}$ to be equal to 2.00 produces ${\widehat{r}}^{*}$ of 0.47, which is unrealistically low. At the same time, assuming ${\widehat{\pi }}^{*}$ to be equal to the sample’s average of 3.48 results in ${\widehat{r}}^{*}$ equal to 2.32.

These calculations suggest that, although assumption of a real interest rate target of 2.0-2.3 percent may seem sensible, an appropriate value for the inflation would be much higher, at around 3.2–3.5 percent. That transforms into a 5.2–5.8 percent target for the nominal interest rate, while an average over the sample is 5.97 percent.

A. Expected Inflation Shocks

Figure 3 shows the actual path for the federal funds rate as well as the fitted values based on the regressions with and without inflation shock included among the regressors. Most of the time, the difference between the two fitted series is not large, although inclusion of the shock seems to bring the series suggested by the rule closer to the actual path.

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Federal Funds Rate: Actual and Fitted Values

Citation: IMF Working Papers 2004, 213; 10.5089/9781451874976.001.A001

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There are, however, several points where there is a substantial difference (about 1 percent and even more) between the backward-looking rule and the actual path of the interest rate, whereas the difference between the forward-looking rule and the actual path is much lower. These divergences are caused by the high values of expected inflation shock (see Figure 2).

In particular, notice the difference between the two rules in 1987:Q3 and 1987:Q4, when the values suggested by the backward rule are 6.07 percent and 5.49 percent respectively; corresponding values for the forward-looking rule are 7.03 percent and 6.76 percent, and the actual values are 6.84 percent and 6.92 percent. Such a difference arises from a sharp increase in inflationary expectations. The 1988 Economic Report of the President (see Council of Economic Advisers) cites a “potential for greater inflation, associated in part with weakness of the dollar” and in part with the flaring up inflation expectations as the basis for such tightening (ERP, 1988, p. 38). The ERP explicitly points out that such policy was desirable in order to avoid inevitable inflationary expectations: “With output growth apparently well-maintained and inflation expectations building at times, the Federal Reserve acted to forestall a resurgence of deep-rooted inflation and to retain hard-won gains towards price stability” (ERP, 1988, pp. 39).

Less than a year later, in 1988:Q2 and 1988:Q3, there was another peak in expected inflation shock, which causes another divergence between the two rules. This peak reflects upward pressure created by the excessive liquidity pumped into the economy following the October 1987 stock market crash. Later on, the Federal Reserve had to reverse its policy and tighten as it became “evident that the stock market crash would not seriously affect spending growth” (ERP, 1989, p. 280).

Finally, the last peak in inflation expectations of 1992:Q2 seems to reflect a producer price inflation rebounce, unusually low throughout 1991 as a result of declining oil prices from their peak during the Middle East conflict in 1990:Q3 (ERP, 1993, p. 47).

B. Expected Output Gap Shocks: The Last Piece of a Puzzle?

As Figure 3 shows, the inclusion of the expected inflation shock does not explain all of the movements of the federal funds rate. In particular, from 1988:Q3 through 1989:Q1, the actual monetary policy was much tighter than the rule suggests (the differences are 1.36 percent, 1.52 percent and 1.38 percent), while in 1989:Q2 and from 1993:Q2 through 1994:Q1, the rule prescribes an interest rate about 1 percent higher than actual.

As equation (4) suggests, the difference may be explained by expected output gap shocks. As discussed earlier, these could not be retrieved from the Greenbook forecasts, and thus are omitted from empirical estimations. However, equation (4) suggests that residuals from the estimation of the forward-looking specification of the rule (as in Section III) are proportional to the expected output gap shock, as they are given by ${\beta }^{-1}{E}_t{\eta }_{t+3}^q$, where β is a parameter of the IS curve (2) that reflects the sensitivity of output gap with respect to changes in real interest rate. Ball (1999) uses β = 1 in his calibrations.

I define

and plot this variable on Figure 4.

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Expected t+3 Output Gap “Shock” as of t

Citation: IMF Working Papers 2004, 213; 10.5089/9781451874976.001.A001

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A much tighter policy than that suggested by the rule from 1988:Q3 through 1989:Q1 is translated into an excess demand, which is confirmed by the Economic Report of the President (1989, p. 275), which cites unusually rapid growth in producers’ investments in durable equipment and in the dollar’s real depreciation, which made U.S.– produced goods more competitive on the international markets.

Likewise, Figure 4 suggests expected declines in output in 1989:Q2 and from 1993:Q2 through 1994:Q1, for which the rule prescribes an interest rate about 1 percent higher than actual. I could not find, however, explicit confirmations of such beliefs, which means that other factors (such as, for example, exogenous shifts in policymakers’ preferences) may have been at play.