Forward-Looking Taylor Rules

Following Clarida, Galí, and Gertler (1998, 1999, 2000), assume that within each operating period the central bank has a target for the nominal short-term interest rate r*_t, which is a function of the state of the economy represented by the gaps between expected inflation and output from their respective targets:

where r* is the desired nominal interest rate when both inflation and output are at their target levels; $E\left({\pi }_{t+k_{\pi }}|{\Omega }_t\right)$ and $E\left(x_{t+k_y}|{\Omega }_t\right)$ denote the expectations of inflation at time t + k_π and the output gap at time t + k_y; π* is the level of inflation (implicitly or explicitly) targeted by the central bank; the output gap $x_{t+k_y}\equiv y_{t+k_y}-y_{t+k_y}^{*},$, where y is output and y* is the efficient level of output.²

Consider the implied target for the ex ante real interest rate.³ Central banks will pursue monetary policy aimed at stabilizing inflation if the parameter β > 1, while rules characterized by β ≤ 1 will tend to be destabilizing. Clearly, the parameter γ is consistent with stabilizing output directly only if γ > 0 (Clarida, Galí, and Gertler, 2000).

Equation (1) may be, however, too restrictive to describe the behavior of short-term rates. As documented in earlier contributions (Goodfriend, 1987), central banks may operate smoothing changes in interest rates for different reasons (e.g., effects on capital markets, loss of reputation, private consensus building). To account for this behavior we can specify equation (1) assuming that the current short-term interest rate r_t adjusts to the target rate $r_t^{*}$ according to a partial adjustment mechanism of the form $r_t=[1-\rho \left(1\right)]r_t^{*}+\rho \left(L\right)r_{t-1}+v_t,$ where ρ(L) is an n - lag polynomial, L is the lag operator, and υ_t is a zero-mean exogenous interest rate shock. Allowing for smoothing in the above fashion in equation (1) results in an interest rate rule where in each period central banks act on the short-term interest rate, reducing the gap between the current target rate and its past value:

where we have eliminated the unobserved conditional expectations by rewriting equation (1) in terms of realized variables; α = [1 - ρ(1)] (r* - βπ*); λ = [1 - ρ(1)]β; υ = [1 - ρ(1)]γ; and the error term ${\epsilon }_t={\upsilon }_t-\left[1-\rho \left(1\right)\right]\left\{\beta \left[{\pi }_{t+k_{\pi }}-E\left({\pi }_{t+k_{\pi }}|{\Omega }_t\right)\right]+\gamma \left[x_{t+k_y}-E\left(x_{t+k_y}|{\Omega }_t\right)\right]\right\}.$ is the sum of a zero-mean exogenous shock and a linear combination of forecast errors that are orthogonal to the variables considered in the information set Ω_t. Given estimates of the parameters in the forward-looking Taylor rule given in equation (2), one can therefore recover the implied estimates of β and γ.

Augmenting Forward-Looking Taylor Rules with Asset Prices and Exchange Rates

Equation (2) does not explicitly consider the role of asset prices and exchange rates as monetary policy targets (or instruments). It is under debate whether and how asset prices and exchange rates should be taken into account in formulating monetary policy (Taylor, 2001; Clarida, 2001). While asset prices may be used as indicator variables, in the last decade or so several major central banks have started taking into consideration the apparent increase in financial instability, of which one important dimension is represented by the volatility of asset prices. In fact, as recorded in several contributions, asset booms and busts have been important factors in macroeconomic fluctuations in both industrial and developing countries (Borio, Kennedy, and Prowse, 1994). Similar arguments apply to the role of exchange rates in central banks’ reaction functions (Clarida, Galí, and Gertler, 1998; Taylor, 2001).

Thus, we consider a baseline equation—which may be termed “augmented Taylor rule”—of the form

where e_t-1 and s_t-1 denote the lagged real exchange rate and stock price; a = [1 - ρ(1)](r* - βπ* - ϕs* - δe*), where s^* and e^* denote the equilibrium levels of asset prices and exchange rates, respectively; λ = [1 - ρ(1)]β; υ = [1 − ρ(1)]γ; χ = [1 -ρ(1)]δ and κ = [1 - ρ(1)]ϕ, where δ and ϕ are the parameters determining the central bank’s response to exchange rates and asset prices disequilibria; and ${\epsilon }_t={\upsilon }_t-\left[1-\rho \left(1\right)\right]\left\{\beta \left[{\pi }_{t+k_{\pi }}-E\left({\pi }_{t+k_{\pi }}|{\Omega }_t\right)\right]+\gamma \left[x_{t+k_y}-E\left(x_{t+k_y}|{\Omega }_t\right)\right]\right\}.$

Equation (3) generalizes the standard forward-looking Taylor rule to account for the possibility that both asset prices and exchange rates enter the interest rate rule. Clearly, equation (3) nests not only the standard forward-looking Taylor rule estimated, for example, by Clarida, Galí, and Gertler (2000) but also its variants, which allow only for either exchange rates (Taylor, 2001) or asset prices (Bernanke and Gertler, 1999; Gilchrist and Leahy, 2002), respectively.

Let z_t denote a vector of instruments comprising the central bank’s information set at the time the instrument rate r_t is chosen (i.e., z_t ∈ Ω_t). Equation (3) implies a set of orthogonality conditions because the elements of z_t include lagged variables that help forecast inflation, output as well as contemporaneous variables that are uncorrelated with the interest rate shock υ_t. More formally, since E[ε_t ∣ z_t ] = 0:

This set of orthogonality conditions is the basis of the estimation of the parameter vector {β, γ, ϕ, δ, ρ}, using the Generalized Method of Moments (GMM), with an optimal weighting matrix that accounts for possible serial correlation in ε_t. To the extent that the dimension of the vector z_t exceeds the number of parameters being estimated, equation (3) implies overidentifying restrictions that can be tested to assess the validity of the set of instruments used (Hansen, 1982). The results of this test can be interpreted as follows: under the null hypothesis the central bank adjusts the interest rate according to equation (3), with the expectations of future inflation and the output gap based on the relevant information available to policymakers at time _t. Under general assumptions this implies that there exists a set of parameters $\{\tilde{\beta },\tilde{\gamma },\tilde{\phi },\tilde{\delta },\tilde{\rho }\}$ such that the residuals obtained from estimating (3) are orthogonal to the information set Ω_t). Under the alternative hypothesis the violation of the orthogonality conditions will lead to the statistical rejection of the model.

In line with some recent empirical literature, we assume that the real exchange rate follows a persistent but stationary process, implying that purchasing power parity (PPP) holds in the long run (Sarno and Taylor, 2002, and the references therein). The real exchange rate equilibrium level can then be captured by a constant included in the intercept term of the reaction function so that effectively the exchange rate variable entering (3) is the lagged deviation from PPP. Indeed, it is often assumed in international macroeconomic models that the instrument rate is affected by past values of exchange rates disequilibria (Obstfeld and Rogoff, 1995; Taylor, 2001; Hunt and Rebucci, 2003). Defining the real exchange rate as the domestic price of foreign currency, the resulting monetary policy rule acts to stabilize the exchange rate if δ > 0. In other words, an appreciation of the real exchange rate will require a cut in short-term interest rates.

Even if equation (3) denotes a forward-looking Taylor rule, it has been assumed, as in Bernanke and Gertler (1999), that only past (once-lagged) stock price disequilibria affect the instrument rate, r_t. This assumption implies that central banks might intervene only when asset prices have been observed to deviate from equilibrium, rather than anticipating potential misalignments. Given our formulation, to stabilize asset prices we must have ϕ > 0. This implies that whenever asset prices positively (negatively) deviate from their equilibrium level—for example, because of poor regulatory practice or “irrational exuberance” (Shiller, 2000)—central bankers will increase (decrease) the instrument rate to offset the anomalous price dynamics.

A Subsidiary Issue: The Role of Output-Gap Measures

In the previous subsections we have discussed how central banks may act by setting a target for the nominal short-term interest rate as a function of the state of the economy, which includes, inter alia, the output gap. In the empirical literature on central banks’ reaction functions, the output gap has generally been measured by detrending output using quadratic deterministic trends or smoothed trends representing the equilibrium level of output that would be obtained under perfectly flexible prices (Hodrick and Prescott, 1997; Baxter and King, 1999, and the references therein). In practice, the efficient level of output is likely to behave differently in response to different real shocks (e.g., shocks related to changes in technology, government expenditure plans, productivity, or consumers’ preferences). As a result the output-gap measure that is relevant for monetary policy purposes may be different from the one captured by detrended output.

A series of recent contributions (Galí and Gertler, 1999; Galí, Gertler, and López-Salido, 2001) emphasize this point in the context of a structural derivation of the Phillips curve. They show that under certain restrictions on technology and labor market structure (Rotemberg and Woodford, 1999) within a local neighborhood of the steady state, real marginal costs are proportionately related to the output gap. A measure of real marginal costs might therefore be considered as an appropriate proxy for the output gap (Woodford, 2001).

The most common measure of marginal cost used in the literature is represented by the cost of increasing labor input. If output is a differentiable function of the labor input and firms are wage takers, then the (real) marginal cost is equal to the (real) wage divided by the marginal product of labor. More formally, assume for simplicity that the aggregate production function is given by a Cobb-Douglas specification $Y_t=A_t{N}_t^{1-\alpha }$, where Y_t is aggregate output, N_t is employment at time t, A_t is a common technology factor, and 1 > α > 0. The real marginal cost (RMC) can be defined as

where W_t is the nominal wage, and P_t is the aggregate price level. Real marginal costs move over time according to observed fluctuations in economic activity and employment. Indeed, there is a large literature supporting the view that marginal costs increase more than prices during expansions, inducing real marginal costs to increase. This is motivated on the basis of two arguments. First, holding constant other determinants of labor supply, real wages must rise during expansions to induce more people to work. Second, if one makes the standard assumption that the production function is concave, for a given level of technology the marginal product of labor is a decreasing function of labor input (Rotemberg and Woodford, 1999). This behavior is therefore consistent with the procyclical dynamics of real marginal costs.

However, variations in real unit labor costs have often been found to be negatively (instead of positively) correlated with detrended GDP (Rotemberg and Woodford, 1999; Woodford, 2001; Galí, Gertler, and López-Salido, 2001; Sbordone, 2002). To shed light on this countercyclical behavior, it is instructive to decompose movements in real unit labor costs in order to isolate the factors driving this variable. Following Galí, Gertler, and López-Salido (2001), real marginal costs are equal to the measure of distance of output from its efficient level:

where c_t is consumption, ${\mu }_t^w$ is the gross wage markup, and lowercase variables are defined in logarithms. The expression in brackets [(c_t - φn_t) -(y_t- n_t)] is the (log-linearized) measure of distance of output from its efficiency level, or, using the Galí, Gertler, and López-Salido (2001) terminology, the “inefficiency wedge.” In equation (6), (c_t − φn_t) is the marginal cost of labor supply, and φ is the inverse of the elasticity of labor supply. If we assume for simplicity an economy with just consumption goods (c_t = y_t), then we can derive an expression that explicitly links the “inefficiency wedge” to the output gap:

where Θ is an indicator of steady state distortions as a result of the existence of market power in labor and goods markets, and $y_t^{*}$ is the level of output that would be obtained under perfectly flexible prices and wages.

The countercyclical behavior of real marginal costs can be explained by using equations (6)–(7) to obtain

Equation (8) tells us that the presence of frictions in the labor market may drive the dynamics of real marginal costs. In fact, as documented in Woodford (2001) and Galí, Gertler, and López-Salido (2001) and as suggested by equation (8), while the “inefficiency wedge” moves closely with the business cycle over time, the wage markup should behave in a countercyclical fashion, suggesting the likelihood of temporary wage rigidities. By removing this markup component from real marginal costs, it should be possible to get a representative measure of the output gap, which might be used as an indicator of stabilization.

Figure 1 shows our time series for real unit labor costs after adjusting for wage markup and detrended output gap for the United States, the United Kingdom, and Japan. The calculation of the adjusted labor costs follows exactly our description of the markup adjustment given above, assuming, as is standard in this literature, the labor supply elasticity ϕ = 1 (Galí, Gertler, and López-Salido, 2001, p. 1263). The graphs in Figure 1 suggest that the prediction of the simple decomposition outlined above is confirmed, as evidenced by the fact that the adjusted real marginal cost measure appears to be positively correlated with the business cycle.

Output Gaps and Labor Shares

Citation: IMF Staff Papers 2004, 003; 10.5089/9781589063518.024.A006

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Output Gaps and Labor Shares

Citation: IMF Staff Papers 2004, 003; 10.5089/9781589063518.024.A006

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Output Gaps and Labor Shares

Citation: IMF Staff Papers 2004, 003; 10.5089/9781589063518.024.A006

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We now turn to a description of our data set.

Forward-Looking Taylor Rules

Table 1 reports GMM estimates of the interest rate rule parameters $\{\tilde{\beta },\tilde{\gamma },\tilde{\rho }\}$ in a standard forward-looking Taylor rule for the United States, the United Kingdom, and Japan. We begin with an interest rate rule—equation (2)—where only expected inflation and expected output gap are considered as explanatory variables, also experimenting with various measures of these two arguments. Following previous research (Clarida, Galí, and Gertler, 1998, 2000), we consider a specification of the interest rate rule with two lags of the interest rate for the United States and only one lag for the United Kingdom and Japan. We allow for a maximum of four lags of the variables in the instrument set. The first four specifications we estimate consider two measures of inflation, namely the CPI and the GDP deflator, and two conventional measures of the output gap, namely quadratically and HP-detrended GDP.

Table 1.

Standard Forward-Looking Taylor Rule: The Role of the Output Gap

Notes: The estimated parameters refer to equation (2). The interest rate targets used are the U.S. federal funds rate, the Bank of England base rate, and the call money rate for Japan, respectively. The sample period is September 1979 to December 2000. The instruments set includes a constant, lagged output gap, inflation, the interest rate, and the log-difference of a world commodity price index. Estimates are obtained by GMM with a correction for moving average autocorrelation obtained by two-step nonlinear two-stage least square (Hansen, 1982). The optimal weighting matrix is obtained from the first step of two-stage least square parameter estimates. ${\pi }_t^{CPI}$ and ${\pi }_t^{GDP}$ are inflation rates measured using the CPI and GDP deflator, respectively. $x_t^Q$ and $x_t^{HP}$ are measures of the output gap (calculated over the period 1960–2000) obtained by means of quadratic-trend regression and Hodrick-Prescott filter (smoothing parameter set to 1,600), respectively.LS_t and LS_t are the labor share (calculated as in the text) and the labor share adjusted for the wage markup, respectively. J-test is the test for overidentifying restrictions (Hansen, 1982), which is distributed as χ² under the null. For this test only p-values are reported. $\tilde{\rho }$ denotes the sum of the autoregressive parameters associated with the lagged interest rate instrument. Standard errors are reported in parentheses.

Table 1.

Standard Forward-Looking Taylor Rule: The Role of the Output Gap

Panel A: United States	$\tilde{\beta }$	$\tilde{\gamma }$	$\tilde{\rho }$	J-test
1. ${\pi }_t={\pi }_t^{CPI},x_t=x_t^Q$	1.688	0.021	0.827	0.253
	(0.591)	(0.328)	(0.267)
2. ${\pi }_t={\pi }_t^{CPI},x_t=x_t^{HP}$	1.764	0.624	0.816	0.192
	(0.456)	(1.887)	(0.285)
3. ${\pi }_t={\pi }_t^{GPD},x_t=x_t^Q$	1.720	−0.021	0.746	0.394
	(0.426)	(0.211)	(0.257)
4. ${\pi }_t={\pi }_t^{GPD},x_t=x_t^{HP}$	1.582	−1.169	0.663	0.998
	(0.324)	(2.943)	(0.215)
5. ${\pi }_t={\pi }_t^{CPI},x_t=L{S}_t$	1.925	−0.855	0.851	0.085
	(0.285)	(0.332)	(0.08)
6. ${\pi }_t={\pi }_t^{GPD},x_t=L{S}_t$	2.036	−0.723	−0.830	0.258
	(0.125)	(0.143)	(0.05)
7. ${\pi }_t={\pi }_t^{CPI},x_t=L{S}_t$	2.205	0.869	0.852	0.471
	(0.267)	(0.387)	(0.126)
8. ${\pi }_t={\pi }_t^{GPD},x_t=L{S}_t$	2.392	1.073	0.844	0.714
	(0.389)	(0.492)	(0.228)

Panel B: United Kingdom	$\tilde{\beta }$	$\tilde{\gamma }$	$\tilde{\rho }$	J-test
1.${\pi }_t={\pi }_t^{CPI},x_t=x_t^Q$	1.039	0.241	0.786	0.067
	(0.203)	(0.185)	(0.038)
2.${\pi }_t={\pi }_t^{CPI},x_t=x_t^{HP}$	0.984	0.232	0.906	0.412
	(0.155)	(1.653)	(0.017)
3.${\pi }_t={\pi }_t^{GPD},x_t=x_t^Q$	1.054	0.315	0.738	0.079
	(0.228)	(0.208)	(0.097)
4.${\pi }_t={\pi }_t^{GPD},x_t=x_t^{HP}$	1.019	0.360	0.814	0.354
	(0.253)	(1.518)	(0.071)
5. ${\pi }_t={\pi }_t^{CPI},x_t=L{S}_t$	1.457	−0.586	0.790	0.117
	(0.459)	(0.299)	(0.026)
6. ${\pi }_t={\pi }_t^{GPD},x_t=L{S}_t$	1.271	−0.613	0.846	0.163
	(0.124)	(0.281)	(0.024)
7. ${\pi }_t={\pi }_t^{CPI},x_t=L{S}_t$	1.086	0.448	0.894	0.999
	(0.100)	(0.174)	(0.007)
8. ${\pi }_t={\pi }_t^{GPD},x_t=L{S}_t$	1.208	0.596	0.866	0.999
	(0.080)	(0.213)	(0.012)

Panel C: Japan	$\tilde{\beta }$	$\tilde{\gamma }$	$\tilde{\rho }$	J-test
1.${\pi }_t={\pi }_t^{CPI},x_t=x_t^Q$	1.737	0.050	0.972	0.991
	(0.259)	(0.020)	(0.003)
2. ${\pi }_t={\pi }_t^{CPI},x_t=x_t^{HP}$	1.504	0.025	0.970	0.986
	(0.182)	(0.008)	(0.003)
3. ${\pi }_t={\pi }_t^{GPD},x_t=x_t^Q$	1.328	0.005	0.892	0.853
	(0.174)	(0.001)	(0.028)
4. ${\pi }_t={\pi }_t^{GPD},x_t=x_t^{HP}$	1.102	0.014	0.963	0.993
5. ${\pi }_t={\pi }_t^{CPI},x_t=L{S}_t$	(0.082)	(0.005)	(0.003)
	1.473	−0.023	0.903	0.998
6. ${\pi }_t={\pi }_t^{GPD},x_t=L{S}_t$	(0.549)	(0.006)	(0.035)
	1.259	−0.009	0.850	0.769
7. ${\pi }_t={\pi }_t^{CPI},x_t=L{S}_t$	(0.196)	(0.003)	(0.029)
	2.366	0.034	0.946	0.828
8. ${\pi }_t={\pi }_t^{GPD},x_t=L{S}_t$	(0.389)	(0.015)	(0.019)
	1.854	0.018	0.877	0.921
	(0.186)	(0.008)	(0.037)

Notes: The estimated parameters refer to equation (2). The interest rate targets used are the U.S. federal funds rate, the Bank of England base rate, and the call money rate for Japan, respectively. The sample period is September 1979 to December 2000. The instruments set includes a constant, lagged output gap, inflation, the interest rate, and the log-difference of a world commodity price index. Estimates are obtained by GMM with a correction for moving average autocorrelation obtained by two-step nonlinear two-stage least square (Hansen, 1982). The optimal weighting matrix is obtained from the first step of two-stage least square parameter estimates. ${\pi }_t^{CPI}$ and ${\pi }_t^{GDP}$ are inflation rates measured using the CPI and GDP deflator, respectively. $x_t^Q$ and $x_t^{HP}$ are measures of the output gap (calculated over the period 1960–2000) obtained by means of quadratic-trend regression and Hodrick-Prescott filter (smoothing parameter set to 1,600), respectively.LS_t and LS_t are the labor share (calculated as in the text) and the labor share adjusted for the wage markup, respectively. J-test is the test for overidentifying restrictions (Hansen, 1982), which is distributed as χ² under the null. For this test only p-values are reported. $\tilde{\rho }$ denotes the sum of the autoregressive parameters associated with the lagged interest rate instrument. Standard errors are reported in parentheses.

View Table

Table 1.

Standard Forward-Looking Taylor Rule: The Role of the Output Gap

Panel A: United States	$\tilde{\beta }$	$\tilde{\gamma }$	$\tilde{\rho }$	J-test
1. ${\pi }_t={\pi }_t^{CPI},x_t=x_t^Q$	1.688	0.021	0.827	0.253
	(0.591)	(0.328)	(0.267)
2. ${\pi }_t={\pi }_t^{CPI},x_t=x_t^{HP}$	1.764	0.624	0.816	0.192
	(0.456)	(1.887)	(0.285)
3. ${\pi }_t={\pi }_t^{GPD},x_t=x_t^Q$	1.720	−0.021	0.746	0.394
	(0.426)	(0.211)	(0.257)
4. ${\pi }_t={\pi }_t^{GPD},x_t=x_t^{HP}$	1.582	−1.169	0.663	0.998
	(0.324)	(2.943)	(0.215)
5. ${\pi }_t={\pi }_t^{CPI},x_t=L{S}_t$	1.925	−0.855	0.851	0.085
	(0.285)	(0.332)	(0.08)
6. ${\pi }_t={\pi }_t^{GPD},x_t=L{S}_t$	2.036	−0.723	−0.830	0.258
	(0.125)	(0.143)	(0.05)
7. ${\pi }_t={\pi }_t^{CPI},x_t=L{S}_t$	2.205	0.869	0.852	0.471
	(0.267)	(0.387)	(0.126)
8. ${\pi }_t={\pi }_t^{GPD},x_t=L{S}_t$	2.392	1.073	0.844	0.714
	(0.389)	(0.492)	(0.228)

Panel B: United Kingdom	$\tilde{\beta }$	$\tilde{\gamma }$	$\tilde{\rho }$	J-test
1.${\pi }_t={\pi }_t^{CPI},x_t=x_t^Q$	1.039	0.241	0.786	0.067
	(0.203)	(0.185)	(0.038)
2.${\pi }_t={\pi }_t^{CPI},x_t=x_t^{HP}$	0.984	0.232	0.906	0.412
	(0.155)	(1.653)	(0.017)
3.${\pi }_t={\pi }_t^{GPD},x_t=x_t^Q$	1.054	0.315	0.738	0.079
	(0.228)	(0.208)	(0.097)
4.${\pi }_t={\pi }_t^{GPD},x_t=x_t^{HP}$	1.019	0.360	0.814	0.354
	(0.253)	(1.518)	(0.071)
5. ${\pi }_t={\pi }_t^{CPI},x_t=L{S}_t$	1.457	−0.586	0.790	0.117
	(0.459)	(0.299)	(0.026)
6. ${\pi }_t={\pi }_t^{GPD},x_t=L{S}_t$	1.271	−0.613	0.846	0.163
	(0.124)	(0.281)	(0.024)
7. ${\pi }_t={\pi }_t^{CPI},x_t=L{S}_t$	1.086	0.448	0.894	0.999
	(0.100)	(0.174)	(0.007)
8. ${\pi }_t={\pi }_t^{GPD},x_t=L{S}_t$	1.208	0.596	0.866	0.999
	(0.080)	(0.213)	(0.012)

Panel C: Japan	$\tilde{\beta }$	$\tilde{\gamma }$	$\tilde{\rho }$	J-test
1.${\pi }_t={\pi }_t^{CPI},x_t=x_t^Q$	1.737	0.050	0.972	0.991
	(0.259)	(0.020)	(0.003)
2. ${\pi }_t={\pi }_t^{CPI},x_t=x_t^{HP}$	1.504	0.025	0.970	0.986
	(0.182)	(0.008)	(0.003)
3. ${\pi }_t={\pi }_t^{GPD},x_t=x_t^Q$	1.328	0.005	0.892	0.853
	(0.174)	(0.001)	(0.028)
4. ${\pi }_t={\pi }_t^{GPD},x_t=x_t^{HP}$	1.102	0.014	0.963	0.993
5. ${\pi }_t={\pi }_t^{CPI},x_t=L{S}_t$	(0.082)	(0.005)	(0.003)
	1.473	−0.023	0.903	0.998
6. ${\pi }_t={\pi }_t^{GPD},x_t=L{S}_t$	(0.549)	(0.006)	(0.035)
	1.259	−0.009	0.850	0.769
7. ${\pi }_t={\pi }_t^{CPI},x_t=L{S}_t$	(0.196)	(0.003)	(0.029)
	2.366	0.034	0.946	0.828
8. ${\pi }_t={\pi }_t^{GPD},x_t=L{S}_t$	(0.389)	(0.015)	(0.019)
	1.854	0.018	0.877	0.921
	(0.186)	(0.008)	(0.037)

Notes: The estimated parameters refer to equation (2). The interest rate targets used are the U.S. federal funds rate, the Bank of England base rate, and the call money rate for Japan, respectively. The sample period is September 1979 to December 2000. The instruments set includes a constant, lagged output gap, inflation, the interest rate, and the log-difference of a world commodity price index. Estimates are obtained by GMM with a correction for moving average autocorrelation obtained by two-step nonlinear two-stage least square (Hansen, 1982). The optimal weighting matrix is obtained from the first step of two-stage least square parameter estimates. ${\pi }_t^{CPI}$ and ${\pi }_t^{GDP}$ are inflation rates measured using the CPI and GDP deflator, respectively. $x_t^Q$ and $x_t^{HP}$ are measures of the output gap (calculated over the period 1960–2000) obtained by means of quadratic-trend regression and Hodrick-Prescott filter (smoothing parameter set to 1,600), respectively.LS_t and LS_t are the labor share (calculated as in the text) and the labor share adjusted for the wage markup, respectively. J-test is the test for overidentifying restrictions (Hansen, 1982), which is distributed as χ² under the null. For this test only p-values are reported. $\tilde{\rho }$ denotes the sum of the autoregressive parameters associated with the lagged interest rate instrument. Standard errors are reported in parentheses.

The estimation results reported in Table 1 (specifications 1 to 4) yield parameter values that are consistent with the results recorded in the literature (Clarida, Galí, and Gertler, 1998, 2000). In particular, for the United States, for each specification considered, the estimate of β is always correctly signed, strongly statistically significant, and greater than unity, while the estimate of γ is not statistically different from zero at conventional levels of significance (see Panel A of Table 1). In turn, this implies that the Federal Reserve has responded only to deviations of expected inflation from the implicit inflation target, not to the expected output gap, over the sample examined. For the United Kingdom (see Panel B of Table 1), the estimate of β is also correctly signed and strongly statistically significant for each of specifications 1 to 4; the estimated β is only slightly greater than unity and generally lower than for the U.S. reaction function. Estimates of γ obtained using the HP-detrended GDP series are not statistically significant at conventional significance levels (Kozicki, 1999). For Japan (see Panel C of Table 1), the estimate of β is again correctly signed and significant, and its size is somewhere in between the estimates for the United Kingdom and the ones of the U.S. reaction function. Estimates of γ are statistically significant but very small in magnitude. Also, for all of these specifications and for each country, the overidentifying restrictions cannot be rejected, and hence the Hansen test supports the validity of the instrument set used.

The last four specifications in Panels A to C of Table 1 show the results from estimating the forward-looking Taylor rule with arguments in expected inflation and output gap where the relevant measure of the output gap is given by a proxy of real marginal costs. Woodford (2001) argues that this output-gap measure may be more appropriate in this context. Specifications 5 and 6 are estimated using the real marginal cost as a proxy for the output gap, while specifications 7 and 8 consider the real marginal cost adjusted for the wage markup, following the procedure described in “A Subsidiary Issue: The Role of Output-Gap Measures” in Section I. The parameters estimates yield, also in this case, fairly clear results. Whenever we consider real unit labor cost into the interest rate rule, the estimated parameter γ is found to be statistically significant for each country examined, but with the wrong sign. This finding can be heuristically explained by the documented evidence of negative correlation between general measures of output gap and real marginal costs (Galí, Gertler, and López-Salido, 2001; Woodford, 2001).

Specifications 7 and 8 are estimated using real marginal costs adjusted for wage markup. As discussed in Section I’s “Role of Output-Gap Measures,” this correction circumvents the problem of the negative correlation between real marginal costs and the business cycle. Interestingly, using this measure of real marginal costs, the estimated parameter γ is, for each country, correctly signed (positive) and statistically significant, providing evidence in favor of the adjusted real marginal costs measure (Woodford, 2001).

An Augmented Forward-Looking Taylor Rule

Next, we estimate the interest rate rule parameters $\{\tilde{\beta },\tilde{\gamma },\tilde{\phi },\tilde{\delta },\tilde{\rho }\}$ for the United States and the United Kingdom in an augmented forward-looking Taylor rule of the form (3) using CPI inflation and the adjusted real marginal costs as proxies for inflation and output gap, respectively, and using the dividend-price ratio and the real effective exchange rate as proxies for stock market and exchange rate disequilibria, respectively. The target horizon for the forward-looking variables is assumed to be one quarter for both inflation and the output gap (i.e., k_π = k_y = 1 in equation (3)).

The results of the GMM estimation of equation (3), reported in Table 2, produce estimates of the parameters that are all statistically significant and exhibit the expected sign for both the United States and the United Kingdom. For Japan, the only insignificant parameter is the one associated with asset prices. More precisely, the estimated parameters ϕ and δ, associated with asset price and exchange rate disequilibria, respectively, are positive for each country examined, although they are small in absolute value. Interestingly, δ is always statistically significant for each country, and ϕ is statistically significant for the United States and the United Kingdom but insignificantly different from zero for Japan.⁷ Further the large p-values of the test for overidentifying restrictions suggest that the instrument set used is valid.

Table 2.

Augmented Forward-Looking Taylor Rule: Baseline Estimates

Notes: See notes to Table 1. The estimated parameters refer to equation (3). The sample period is September 1979 to December 2000. The baseline specification was estimated using the CPI to measure inflation and the adjusted labor share to measure the output gap. The forward-looking horizon for inflation and the output gap is one quarter (k_π= 1, k_x = 1). The instruments set includes a constant plus four lags of output gap, inflation, the interest rate, log difference of a world commodity price index, real effective exchange rate, and dividend-price ratio. Values in brackets are p-values from executing the Hansen (1992) test statistic for the null hypothesis of parameter stability.

Table 2.

Augmented Forward-Looking Taylor Rule: Baseline Estimates

	$\tilde{\beta }$	$\tilde{\gamma }$	$\tilde{\delta }$	$\tilde{\phi }$	$\tilde{\rho }$	J-test
United States	1.252	0.719	0.073	0.015	0.767	0.935
	(0.195)	(0.229)	(0.009)	(0.005)	(0.040)
	[0.246]	[0.192]	[0.221]	[0.188]	[0.147]
United Kingdom	1.034	0.542	0.056	0.007	0.889	0.999
	(0.026)	(0.037)	(0.006)	(0.003)	(0.001)
	[0.160]	[0.104]	[0.175]	[0.173]	[0.136]
Japan	1.115	0.013	0.002	0.0002	0.895	0.745
	(0.440)	(0.003)	(0.0008)	(0.0005)	(0.034)
	[0.122]	[0.386]	[0.160]	[0.179]	[0.332]

Notes: See notes to Table 1. The estimated parameters refer to equation (3). The sample period is September 1979 to December 2000. The baseline specification was estimated using the CPI to measure inflation and the adjusted labor share to measure the output gap. The forward-looking horizon for inflation and the output gap is one quarter (k_π= 1, k_x = 1). The instruments set includes a constant plus four lags of output gap, inflation, the interest rate, log difference of a world commodity price index, real effective exchange rate, and dividend-price ratio. Values in brackets are p-values from executing the Hansen (1992) test statistic for the null hypothesis of parameter stability.

View Table

Table 2.

Augmented Forward-Looking Taylor Rule: Baseline Estimates

	$\tilde{\beta }$	$\tilde{\gamma }$	$\tilde{\delta }$	$\tilde{\phi }$	$\tilde{\rho }$	J-test
United States	1.252	0.719	0.073	0.015	0.767	0.935
	(0.195)	(0.229)	(0.009)	(0.005)	(0.040)
	[0.246]	[0.192]	[0.221]	[0.188]	[0.147]
United Kingdom	1.034	0.542	0.056	0.007	0.889	0.999
	(0.026)	(0.037)	(0.006)	(0.003)	(0.001)
	[0.160]	[0.104]	[0.175]	[0.173]	[0.136]
Japan	1.115	0.013	0.002	0.0002	0.895	0.745
	(0.440)	(0.003)	(0.0008)	(0.0005)	(0.034)
	[0.122]	[0.386]	[0.160]	[0.179]	[0.332]

Notes: See notes to Table 1. The estimated parameters refer to equation (3). The sample period is September 1979 to December 2000. The baseline specification was estimated using the CPI to measure inflation and the adjusted labor share to measure the output gap. The forward-looking horizon for inflation and the output gap is one quarter (k_π= 1, k_x = 1). The instruments set includes a constant plus four lags of output gap, inflation, the interest rate, log difference of a world commodity price index, real effective exchange rate, and dividend-price ratio. Values in brackets are p-values from executing the Hansen (1992) test statistic for the null hypothesis of parameter stability.

The results in Table 2 also confirm the presence of interest rate inertia (detected in all of our previous specifications and well documented in the relevant literature), as captured by the lagged values of interest rates in equation (3). These findings support the view that the Federal Reserve, the Bank of England, and the Bank of Japan smooth the adjustment of interest rates toward their target values (Clarida, Galí, and Gertler, 2000; Eijffinger and Huizinga, 1999).

Some recent contributions point out that monetary policy rules estimated over long sample periods may be affected by parameter instability (Oliner, Rudebusch, and Sichel, 1996; Rudebusch, 1998, 2001). To test for parameter instability, we carried out a Hansen (1992) test for the null hypothesis of parameter stability, applied individually to each of the estimated parameters reported in Table 2. The results, reported in square brackets in Table 2, show that for each country examined the null hypothesis of parameter stability cannot be rejected at conventional statistical significance levels.

Before discussing and further interpreting our empirical results, we report some robustness checks carried out on our baseline estimation results.

Robustness

To investigate the robustness of the results in Table 2, we reestimated equation (3) using alternative measures of inflation, asset price disequilibria, and alternative target horizons for each of the forward-looking variables. As discussed below, in each country and over different specifications, this robustness analysis confirms, at least in a qualitative sense, the findings of our baseline estimates reported in Table 2. Using a different measure of inflation based on the GDP deflator (see specification 1 of Table 3), the estimated parameters are not qualitatively different from the estimates recorded in Table 2.

Table 3.

Augmented Forward-Looking Taylor Rule: Robustness Analysis

Notes: See notes to Table 1. The estimated parameters refer to equation (3). k_π and k_x denote the forward-looking horizon for inflation and the output gap, respectively. Standard errors are reported in parentheses. $s_t^C$ denotes the transitory component calculated as in Cochrane (1994) using Data-stream stock price indices. $s_t^{BG}$ denotes the log-differenced changes in the Datastream stock price indices (Bernanke and Gertler, 1999); under $s_t^{BG}$, six lags of the log-differences in the stock prices have been used.

Table 3.

Augmented Forward-Looking Taylor Rule: Robustness Analysis

Panel A: United States	$\tilde{\beta }$	$\tilde{\gamma }$	$\tilde{\delta }$	$\tilde{\phi }$	$\tilde{\rho }$	J-test
1. ${\pi }_t={\pi }_t^{GPD},k_{\pi }=1,k_x=1$	1.226	0.817	0.065	0.011	0.721	0.961
	(0.209)	(0.135)	(0.010)	(0.005)	(0.069)
2. ${\pi }_t={\pi }_t^{CPI},k_{\pi }=4,k_x=1$	1.289	0.981	0.075	0.017	0.785	0.946
	(0.215)	(0.337)	(0.013)	(0.007)	(0.064)
3.${\pi }_t={\pi }_t^{GPD},k_{\pi }=4,k_x=1$	1.562	0.824	0.064	0.008	0.719	0.935
	(0.267)	(0.385)	(0.014)	(0.004)	(0.078)
4.${\pi }_t={\pi }_t^{CPI},k_{\pi }=4,k_x=2$	1.048	0.907	0.049	0.022	0.788	0.995
	(0.210)	(0.278)	(0.017)	(0.006)	(0.069)
5.${\pi }_t={\pi }_t^{GPD},k_{\pi }=4,k_x=2$	1.752	0.890	0.046	0.008	0.779	0.999
	(0.116)	(0.109)	(0.005)	(0.004)	(0.042)
6.${\pi }_t={\pi }_t^{CPI},k_{\pi }=1,k_x=1,s_t=s_t^C$	2.035	0.796	0.076	0.037	0.765	0.994
	(0.049)	(0.106)	(0.008)	(0.015)	(0.052)
7.${\pi }_t={\pi }_t^{CPI},k_{\pi }=1,k_x=1,s_t=s_t^{BG}$	1.561	0.593	0.081	0.036	0.822	0.999
	(0.099)	(0.139)	(0.009)	(0.019)	(0.037)

Panel B: United Kingdom	$\tilde{\beta }$	$\tilde{\gamma }$	$\tilde{\delta }$	$\tilde{\phi }$	$\tilde{\rho }$	J-test
1.${\pi }_t={\pi }_t^{GPD},k_{\pi }=1,k_x=1$	0.925	0.778	0.075	0.008	0.886	0.998
	(0.013)	(0.040)	(0.006)	(0.002)	(0.002)
2.${\pi }_t={\pi }_t^{CPI},k_{\pi }=4,k_x=1$	1.439	0.423	0.098	0.007	0.892	0.867
	(0.211)	(0.09)	(0.013)	(0.003)	(0.004)
3.${\pi }_t={\pi }_t^{GPD},k_{\pi }=4,k_x=1$	1.553	0.462	0.034	0.006	0.813	0.913
	(0.388)	(0.014)	(0.008)	(0.003)	(0.003)
4.${\pi }_t={\pi }_t^{CPI},k_{\pi }=4,k_x=2$	1.462	0.284	0.036	0.009	0.863	0.999
	(0.343)	(0.046)	(0.006)	(0.005)	(0.002)
5.${\pi }_t={\pi }_t^{GPD},k_{\pi }=4,k_x=2$	1.531	0.269	0.032	0.011	0.816	0.905
	(0.355)	(0.017)	(0.008)	(0.003)	(0.003)
6.${\pi }_t={\pi }_t^{CPI},k_{\pi }=1,k_x=1,s_t=s_t^C$	1.041	0.535	0.064	0.023	0.891	0.999
	(0.020)	(0.034)	(0.004)	(0.001)	(0.001)
7.${\pi }_t={\pi }_t^{CPI},k_{\pi }=1,k_x=1,s_t=s_t^{BG}$	1.079	0.713	0.028	0.018	0.903	0.998
	(0.044)	(0.093)	(0.011)	(0.002)	(0.003)

Panel C: Japan	$\tilde{\beta }$	$\tilde{\gamma }$	$\tilde{\delta }$	$\tilde{\phi }$	$\tilde{\rho }$	J-test
1.${\pi }_t={\pi }_t^{GPD},k_{\pi }=1,k_x=1$	1.094	0.014	0.0012	0.0001	0.901	0.811
	(0.434)	(0.001)	(0.0005)	(0.0006)	(0.051)
2.${\pi }_t={\pi }_t^{CPI},k_{\pi }=4,k_x=1$	1.378	0.012	0.0015	0.0001	0.940	0.751
	(0.541)	(0.002)	(0.0006)	(0.0005)	(0.049)
3.${\pi }_t={\pi }_t^{GPD},k_{\pi }=4,k_x=1$	1.548	0.011	0.003	0.0001	0.859	0.765
	(0.543)	(0.002)	(0.0012)	(0.0004)	(0.058)
4.${\pi }_t={\pi }_t^{CPI},k_{\pi }=4,k_x=2$	1.277	0.010	0.007	0.0002	0.926	0.825
	(0.374)	(0.002)	(0.0017)	(0.0008)	(0.051)
5.${\pi }_t={\pi }_t^{GPD},k_{\pi }=4,k_x=2$	1.670	0.010	0.007	0.0001	0.894	0.788
	(0.318)	(0.001)	(0.0019)	(0.0006)	(0.032)
6.${\pi }_t={\pi }_t^{CPI},k_{\pi }=1,k_x=1,s_t=s_t^C$	1.564	0.012	0.0012	0.0003	0.928	0.825
	(0.227)	(0.001)	(0.0005)	(0.0014)	(0.038)
7.${\pi }_t={\pi }_t^{CPI},k_{\pi }=1,k_x=1,s_t=s_t^{BG}$	1.343	0.011	0.009	0.0003	0.967	0.827
	(0.441)	(0.001)	(0.0035)	(0.0018)	(0.029)

Notes: See notes to Table 1. The estimated parameters refer to equation (3). k_π and k_x denote the forward-looking horizon for inflation and the output gap, respectively. Standard errors are reported in parentheses. $s_t^C$ denotes the transitory component calculated as in Cochrane (1994) using Data-stream stock price indices. $s_t^{BG}$ denotes the log-differenced changes in the Datastream stock price indices (Bernanke and Gertler, 1999); under $s_t^{BG}$, six lags of the log-differences in the stock prices have been used.

View Table

Table 3.

Augmented Forward-Looking Taylor Rule: Robustness Analysis

Panel A: United States	$\tilde{\beta }$	$\tilde{\gamma }$	$\tilde{\delta }$	$\tilde{\phi }$	$\tilde{\rho }$	J-test
1. ${\pi }_t={\pi }_t^{GPD},k_{\pi }=1,k_x=1$	1.226	0.817	0.065	0.011	0.721	0.961
	(0.209)	(0.135)	(0.010)	(0.005)	(0.069)
2. ${\pi }_t={\pi }_t^{CPI},k_{\pi }=4,k_x=1$	1.289	0.981	0.075	0.017	0.785	0.946
	(0.215)	(0.337)	(0.013)	(0.007)	(0.064)
3.${\pi }_t={\pi }_t^{GPD},k_{\pi }=4,k_x=1$	1.562	0.824	0.064	0.008	0.719	0.935
	(0.267)	(0.385)	(0.014)	(0.004)	(0.078)
4.${\pi }_t={\pi }_t^{CPI},k_{\pi }=4,k_x=2$	1.048	0.907	0.049	0.022	0.788	0.995
	(0.210)	(0.278)	(0.017)	(0.006)	(0.069)
5.${\pi }_t={\pi }_t^{GPD},k_{\pi }=4,k_x=2$	1.752	0.890	0.046	0.008	0.779	0.999
	(0.116)	(0.109)	(0.005)	(0.004)	(0.042)
6.${\pi }_t={\pi }_t^{CPI},k_{\pi }=1,k_x=1,s_t=s_t^C$	2.035	0.796	0.076	0.037	0.765	0.994
	(0.049)	(0.106)	(0.008)	(0.015)	(0.052)
7.${\pi }_t={\pi }_t^{CPI},k_{\pi }=1,k_x=1,s_t=s_t^{BG}$	1.561	0.593	0.081	0.036	0.822	0.999
	(0.099)	(0.139)	(0.009)	(0.019)	(0.037)

Panel B: United Kingdom	$\tilde{\beta }$	$\tilde{\gamma }$	$\tilde{\delta }$	$\tilde{\phi }$	$\tilde{\rho }$	J-test
1.${\pi }_t={\pi }_t^{GPD},k_{\pi }=1,k_x=1$	0.925	0.778	0.075	0.008	0.886	0.998
	(0.013)	(0.040)	(0.006)	(0.002)	(0.002)
2.${\pi }_t={\pi }_t^{CPI},k_{\pi }=4,k_x=1$	1.439	0.423	0.098	0.007	0.892	0.867
	(0.211)	(0.09)	(0.013)	(0.003)	(0.004)
3.${\pi }_t={\pi }_t^{GPD},k_{\pi }=4,k_x=1$	1.553	0.462	0.034	0.006	0.813	0.913
	(0.388)	(0.014)	(0.008)	(0.003)	(0.003)
4.${\pi }_t={\pi }_t^{CPI},k_{\pi }=4,k_x=2$	1.462	0.284	0.036	0.009	0.863	0.999
	(0.343)	(0.046)	(0.006)	(0.005)	(0.002)
5.${\pi }_t={\pi }_t^{GPD},k_{\pi }=4,k_x=2$	1.531	0.269	0.032	0.011	0.816	0.905
	(0.355)	(0.017)	(0.008)	(0.003)	(0.003)
6.${\pi }_t={\pi }_t^{CPI},k_{\pi }=1,k_x=1,s_t=s_t^C$	1.041	0.535	0.064	0.023	0.891	0.999
	(0.020)	(0.034)	(0.004)	(0.001)	(0.001)
7.${\pi }_t={\pi }_t^{CPI},k_{\pi }=1,k_x=1,s_t=s_t^{BG}$	1.079	0.713	0.028	0.018	0.903	0.998
	(0.044)	(0.093)	(0.011)	(0.002)	(0.003)

Panel C: Japan	$\tilde{\beta }$	$\tilde{\gamma }$	$\tilde{\delta }$	$\tilde{\phi }$	$\tilde{\rho }$	J-test
1.${\pi }_t={\pi }_t^{GPD},k_{\pi }=1,k_x=1$	1.094	0.014	0.0012	0.0001	0.901	0.811
	(0.434)	(0.001)	(0.0005)	(0.0006)	(0.051)
2.${\pi }_t={\pi }_t^{CPI},k_{\pi }=4,k_x=1$	1.378	0.012	0.0015	0.0001	0.940	0.751
	(0.541)	(0.002)	(0.0006)	(0.0005)	(0.049)
3.${\pi }_t={\pi }_t^{GPD},k_{\pi }=4,k_x=1$	1.548	0.011	0.003	0.0001	0.859	0.765
	(0.543)	(0.002)	(0.0012)	(0.0004)	(0.058)
4.${\pi }_t={\pi }_t^{CPI},k_{\pi }=4,k_x=2$	1.277	0.010	0.007	0.0002	0.926	0.825
	(0.374)	(0.002)	(0.0017)	(0.0008)	(0.051)
5.${\pi }_t={\pi }_t^{GPD},k_{\pi }=4,k_x=2$	1.670	0.010	0.007	0.0001	0.894	0.788
	(0.318)	(0.001)	(0.0019)	(0.0006)	(0.032)
6.${\pi }_t={\pi }_t^{CPI},k_{\pi }=1,k_x=1,s_t=s_t^C$	1.564	0.012	0.0012	0.0003	0.928	0.825
	(0.227)	(0.001)	(0.0005)	(0.0014)	(0.038)
7.${\pi }_t={\pi }_t^{CPI},k_{\pi }=1,k_x=1,s_t=s_t^{BG}$	1.343	0.011	0.009	0.0003	0.967	0.827
	(0.441)	(0.001)	(0.0035)	(0.0018)	(0.029)

Notes: See notes to Table 1. The estimated parameters refer to equation (3). k_π and k_x denote the forward-looking horizon for inflation and the output gap, respectively. Standard errors are reported in parentheses. $s_t^C$ denotes the transitory component calculated as in Cochrane (1994) using Data-stream stock price indices. $s_t^{BG}$ denotes the log-differenced changes in the Datastream stock price indices (Bernanke and Gertler, 1999); under $s_t^{BG}$, six lags of the log-differences in the stock prices have been used.