I. Introduction

Estimates of potential output and the output gap are both difficult to produce, but key for macroeconomic policy making.² Accordingly, the subject has generated a large academic and policy literature. This paper builds on the work by Alichi and others (2017), who developed a structural multivariate filter that brings to bear a Phillips curve, a dynamic Okun’s law, and an equation that links the output gap to capacity utilization.³ They showed that the structural multivariate filter outperforms univariate filters, such as the Hodrick-Prescott (HP) filter, in that it provides more reliable real-time estimates of potential output and the output gap. The advantage of a structural multivariate filter over an HP-type filter is that the multivariate filter incorporates structural relationships between macroeconomic fundamentals, and is flexible to incorporate additional information (such as capacity utilization, inflation expectations, and in the model presented below, monetary policy).

This paper extends the multivariate filter methodology by adding a monetary policy block. This block brings in short- and long-term interest rates (i.e., the term structure), and a monetary policy reaction function to capture policy responses to relevant cyclical developments. This addition provides us with a fuller and more structural description of the dynamics of the output gap, compared to the earlier models in which the policy response was, at best, implicit. The model developed in this paper estimates a deeper and prolonged post-global-financial-crisis output gap compared with the same model without a monetary policy block.

The remainder of the paper is organized as follows. Section II summarizes the new model, called MPMOD (short for Monetary Policy Model), discusses key relationships, and reports the estimation results. Section III compares model results against a univariate filter, and against a multivariate filter without a monetary policy block. Section IV presents the impulse response analysis. Section V concludes.

II. MPMOD

MPMOD is an extension of the multivariate filter model presented in Alichi and others (2017). The original model includes a Phillips curve, a dynamic Okun’s law equation linking the unemployment gap to the output gap (Okun, 1962), and an equation linking the output gap to the Fed’s measure of capacity utilization in the manufacturing sector. The stochastic process for GDP includes a persistent cyclical component as well as two shocks that permanently change the level of potential output (a level shift, and a persistent, though not permanent, deviation from the long-term growth rate). In this paper, we extend that model to include a monetary policy reaction function for interest rate on the one-year government bond and a model for the 10-year government bond yield. This allows us to estimate and project both the short-term (one-year) equilibrium real interest rate and the 10-year government bond term premium.

The details of the model are provided in Appendix A, and only the new monetary block is discussed in this section. The list of standard macro variables includes real GDP, the unemployment rate, consumer price inflation, the Fed’s survey of manufacturing capacity utilization, the one-year government bond yield, and the 10-year government bond yield.

The output gap $\left({\hat{y}}_t\right)$ is defined as the deviation of real GDP, in log terms (y_t), from its potential level $\left({\overline{y}}_t\right)$:

The stochastic process for real GDP is defined by three equations, (2)-(4), and three types of shocks:

The level of potential output $\left({\overline{y}}_t\right)$ evolves according to a stochastic process for trend potential growth $\left(g_{\overline{y},t}\right)$ and a simple level-shock term $\left({\mathit{ϵ}}_{\overline{y},t}\right)$. Trend potential growth is subject to shocks $\left({\mathit{ϵ}}_{g_{\overline{y}},t}\right)$ whose impact fades gradually away depending on the parameter ${\rho }_{g_{\overline{y}}}$ (a lower value of which means a slower adjustment back to the steady-state growth rate following a shock). Both the shock to trend potential growth $\left({\mathit{ϵ}}_{g_{\overline{y}},t}\right)$ and the shock to the level of potential output $\left({\mathit{ϵ}}_{\overline{y},t}\right)$ result in unit roots in potential output and GDP.

Finally, the output gap $\left({\hat{y}}_t\right)$ is a function of contemporaneous and lagged values of the one-year real interest rate gap $\left({\widehat{rr}}_t^{1Y}\right)$, which in turn is the deviation of the short-term interest rate from its equilibrium level. The output gap equation also incorporates shocks to trend potential growth $\left({\mathit{ϵ}}_{g_{\overline{y}},t}\right)$ and shocks to the level of potential output $\left({\mathit{ϵ}}_{\overline{y},t}\right)$. The specification intends to replicate the insights from structural DSGE models, where anticipated increases in future productivity growth $\left({\mathit{ϵ}}_{g_{\overline{y}},t}\right)$ can, in the short-to-medium term, result in larger increases in aggregate demand than potential output as consumers and firms respond to higher expected levels of future income and demand (Juillard and others, 2007). In the same DSGE models, pure level shocks to potential output $\left({\mathit{ϵ}}_{\overline{y},t}\right)$ that raise productivity once and for all reduce the marginal costs of production and inflation in the short run. GDP is also subject to demand shocks $\left({\mathit{ϵ}}_{\hat{y},t}\right)$, which raise aggregate demand in the economy by more than potential.

Equation (5) is the monetary policy interest rate reaction function. The one-year nominal interest rate $\left(r{s}_t^{1Y}\right)$ responds to the deviation of inflation (π_t) from target $\left({\pi }_t^{Tar}\right)$ and the output gap $\left(\hat{y}\right)$. The reaction function captures the dual mandate of the Federal Reserve:

The two shocks in this equation are: ϵ_{rs^1Y, t} which captures interest rate movements unexplained by all the elements in the reaction function; and ϵ_{π^Tar, t} which is designed to explain interest rate hikes due to the transition from a high to a low inflation regime that happened in the early 1980’s and 1990’s.

The real one-year interest rate $\left(r{r}_t^{1Y}\right)$ is defined as the difference between the nominal one-year interest rate and expected inflation:

Expected inflation is modeled as a linear combination of model-consistent expected inflation and lagged actual inflation:

The equilibrium real interest rate $\left({\overline{rr}}_t^{1Y}\right)$ is modeled as a slow-moving autoregressive process (i.e., with high ${\rho }^{{\overline{rr}}^{1Y}}$), which reverts to its long-run steady-state level $\left({\overline{rr}}^{SS}\right)$ and is also subject to a shock $\left({\mathit{ϵ}}_{{\overline{rr}}^{1Y},t}\right)$:

The model allows the long-term bond yield to shed light on the estimates of the equilibrium real interest rate. Based on the expectations theory of the term structure, the interest rate on 10-year government bonds is the sum of the average of expected future one-year interest rates, a term premium $\left({\sigma }_t^{Term}\right)$, and a transitory shock to the 10-year interest rate (ϵ_rs^10Y):

The term premium follows a slow autoregressive process (i.e., with high ρ^σTerm) towards its long-run steady (σ^{Term, SS}), subject to a shock (ϵ_{σ^term, t}):

To the extent that forecasters see through the effects of temporary supply shocks on inflation and have a view about how output gaps affect inflation, information from Consensus forecasts helps us identify the structural shocks affecting the inflation equation. In addition, Consensus forecasts of future GDP growth can help us identify shocks that directly affect the output gap versus shocks that can have permanent and highly-persistent effects on the underlying trend growth rate of potential output. The Consensus forecasts are modeled as MPMOD’s forecast for the CPI and GDP plus a measurement error. For example, Consensus forecasts for CPI inflation one year ahead would be equal to the structural model’s forecast for CPI inflation plus a measurement error. The variances of the measurement errors are based on the historical forecast errors from Consensus forecasts.

The full model, including the new monetary block, is estimated with annual data covering the period from 1980 to 2016. We use annual instead of quarterly data to avoid problems with noise in high-frequency measures of inflation and also because Consensus Economics publishes annual forecasts for GDP growth and CPI inflation. We use 5-year-ahead CPI forecasts as a measure of the perceived long-term inflation target. This information helps to identify the underlying structural shocks of the model and changing perceptions of the inflation target in the monetary policy reaction function.

Figure 1 summarizes the MPMOD results for the United States from 1980 to 2026. For comparison, we have also included estimates of the output gap, potential growth, and the natural rate of unemployment (NAIRU) from the Congressional Budget Office (CBO)⁴. The CBO, using the framework of a Solow growth model, disaggregates output into five sectors: nonfarm business, households and nonprofit institutions, government, farm, and housing (see CBO papers, 2001, 2014). The output gaps from the MPMOD and the CBO are highly correlated, but the MPMOD’s estimates are larger, on average, by about 0.8 percentage points. The periods of excess demand from MPMOD typically preceded increases in inflation and hikes in the monetary policy interest rate. The two models’ estimates of potential growth are also highly correlated—both move partly with trends in actual growth, with the CBO’s estimates implying higher potential growth in the early 1980s and late 1990s.

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Comparison of MPMOD Estimates with CBO Estimates

Citation: IMF Working Papers 2018, 152; 10.5089/9781484366325.001.A001

Source: Authors’ estimates, CBO.

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Estimates of the NAIRU from both models trend down, with the CBO’s estimates being slightly lower at the end of the sample. Note that the CBO’s estimates include a temporary increase in the NAIRU during the global financial crisis. These small differences in the NAIRU estimates do not materially affect recent potential growth or gaps. To see this, Figure 2 presents estimates of the MPMOD output gap and potential growth, taking the CBO estimates of the NAIRU as an observable variable. The results are almost unchanged.

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Sensitivity Analysis

Citation: IMF Working Papers 2018, 152; 10.5089/9781484366325.001.A001

Source: Authors’ estimates, CBO.

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The MPMOD multivariate filter produces more robust real-time estimates of potential output and output gap than a univariate HP filter does. Figure 3, Panel 3a, 3b, and 3c compare real-time estimates produced by MPMOD, an HP filter with Λ = 6.25⁵, and MPMOD without the monetary policy block. Each red circle in the figure is the estimate of potential output by the relevant model using only data up to that point. By contrast, the solid black line uses the full data set to generate ex-post estimates by each of the three models. The grey bars are the revisions (differences) between the real-time and ex-post estimates: a smaller bar implies a smaller revision between the real-time and ex-post estimates. The real-time MPMOD estimates track the corresponding ex-post estimates quite closely, outperforming the HP filter. Moreover, MPMOD with the monetary policy block has generally smaller revisions compared to MPMOD without the monetary policy block.

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Real-time Estimates and Rolling Forecasts

Citation: IMF Working Papers 2018, 152; 10.5089/9781484366325.001.A001

Source: Authors’ estimates.

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Panel 3d and 3e show the real-time forecasts of MPMOD and the HP filter. Each colored circle is the real-time estimate (the same as the circles in Panel 3a, 3b and 3c), and the colored lines are the real-time forecasts at each time period (i.e., using data only up to that time).

Figure 3, Panel 3e illustrates the so-called end-point problem that plagues the HP filter, which essentially extrapolates its last (end point) estimate. Some insight into this behavior can be gleaned by examining the HP filter in a state space representation applied to the log of GDP:

The HP filter decomposes the actual GDP level into trend and gap components (equation 12). The first difference of the trend GDP, its growth rate, is assumed to be a random walk (equation 13), while the GDP gap is assumed to be white noise (equation 14). The smoothness of the trend is pinned down by explicitly specifying the signal-to-noise ratio (Λ, equation 15). From this, it is clear that whenever there is a shift in output growth, the HP filter assumes it is permanent and forecasts accordingly. Unless growth is stable, or the practitioner intervenes to adjust the results in an ad hoc way, this leads to large revisions.⁶

Figure 4 compares the estimates of the output gap and potential growth of MPMOD with the same model but without the monetary policy block or the associated interest rates and term premium. MPMOD estimates somewhat deeper and prolonged recessions, especially after the global financial crisis. The simultaneous estimation procedure implies that the reaction function and term structure equation have information about the output gap. Thus, the extraordinary easing of policy is explained by the need to fight an extraordinarily large (estimated) output gap.

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Comparison of Multivariate Filter Results with and without the Monetary Policy Block

Citation: IMF Working Papers 2018, 152; 10.5089/9781484366325.001.A001

Source: Authors’ estimates.

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III. Properties of MPMOD

Figure 5 summarizes the MPMOD detailed results for 1980 to 2026. The implicit inflation target is based on inflation expectations 5-year-ahead from Consensus Economics, available from 1989 to 2017. For 1980, the 5-year-ahead CBO inflation forecast is used. The values from 1981 to 1988 are treated as unobservable and computed from the Kalman filter. The perceived inflation target decreased in the 1980’s and 1990’s, reflecting the Volker disinflation.

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MPMOD Detailed Results

Citation: IMF Working Papers 2018, 152; 10.5089/9781484366325.001.A001

Source: Authors’ estimates.

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Before the global financial crisis, the U.S. economy was operating above potential, pushing inflation up. As a result, monetary policy tightened moderately. The global financial crisis that started in late 2008 pushed U.S. economy into a deep recession. The unemployment rate rose sharply and capacity utilization in the manufacturing sector dropped. In response, monetary (and fiscal) policy became markedly expansionary.⁷

After the great recession, GDP grew faster than potential and the (negative) output gap narrowed correspondingly. However, the inflation rate has remained below target. Under such circumstances, monetary policy was correctly accommodative. MPMOD estimates indicate that the U.S. output gap is now almost closed and the unemployment rate is almost at the NAIRU. The model suggests that for the next few years the output gap will go positive and the unemployment rate will fall below the NAIRU.

IV. Impulse Response Analysis

Figures 6, 7, 8, and 9 depict impulse responses to the shocks on: the output gap, the level of potential, the growth rate of potential, and the perceived inflation target.

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Impulse Response for a 1-Standard-Deviation Positive Shock to the Output Gap

Citation: IMF Working Papers 2018, 152; 10.5089/9781484366325.001.A001

Source: Authors’ estimates.

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Impulse Response for a 1-Standard-Deviation Positive Shock to the Level of Potential Output

Citation: IMF Working Papers 2018, 152; 10.5089/9781484366325.001.A001

Source: Author’s estimates.

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Impulse Response for a 1-Standard-Deviation Positive Shock to the Growth Rate of Potential Output

Citation: IMF Working Papers 2018, 152; 10.5089/9781484366325.001.A001

Source: Authors’ estimates.

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Impulse Response for a 1-Standard-Deviation Negative Shock to the Inflation Target

Citation: IMF Working Papers 2018, 152; 10.5089/9781484366325.001.A001

Source: Authors’ estimates.

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In Figure 6, we consider a one-standard-deviation positive demand shock to output (ϵ_y,t), which causes output to temporarily deviate from its steady-state level. There is by construction no change in potential output. The positive output gap pushes inflation up by nearly 0.2 percentage points. In response to this shock, monetary policy tightens and the short-term interest rate rises by more than 0.3 percentage points. This is transmitted to the long-term interest rate through expectations. The long-term interest rate goes up by 0.1 percentage points.

Figure 7 illustrates a one standard deviation positive shock to the level of potential output. The potential output level shock pushes demand up as economic agents start to consume and invest more, but initially supply rises more than demand and output gap becomes negative. In the Phillips curve equation, the increase in productivity lowers marginal costs, which also pushes inflation down. In response, monetary policy loosens. The short-term interest rate decreases by 0.04 percentage points.

Figure 8 illustrates a shock to the growth rate of potential output $\left({\mathit{ϵ}}_{g_{\overline{y}},t}\right)$, which causes an increase in potential output growth for several periods until the impact fades out and growth returns to its (unchanged) steady-state. As in the previous shock, consumers and firms recognize that their permanent incomes are higher and so increase consumption and investment. Unlike the previous shock, however, potential output does not rise immediately and a positive output gap opens up. Inflation rises and monetary policy tightens.

Finally, figure 9 illustrates a negative shock to the inflation target by one percentage point, which causes monetary policy to tighten. Tighter monetary policy reduces real output temporarily, but it eventually returns to (unchanged) potential. By that point, inflation has fallen to the new target.

V. Concluding Remarks

This paper extends the multivariate filter approach for measuring potential output developed by Alichi and others (2017) by adding a monetary policy block (MPMOD). This addition provides more structural description of dynamics of the output gap and allows monetary policy to interact with demand and supply shocks. The addition of the monetary block generates estimates of the post-global-financial-crisis recession that are deeper and more prolonged. As with other multivariate filters, MPMOD produces more reliable real-time estimates than an HP filter, and the addition of the monetary policy block further improves the reliability of the structural multivariate filter.

Additional extensions of the multivariate filter model are underway. Alichi and others (2018a) add a production function and disaggregate full-employment labor inputs into the participation rate, the NAIRU, and labor force population. Alichi and others (2018b) extend the basic model to allow for partial hysteresis in the labor market. Finally, Alichi and others (2018c) show how the model can be extended to incorporate information about the financial cycle.