Singlevariate and hybrid approaches

The simplest methodology to estimate potential output is the singlevariate (SV) filter. This is a purely statistical methodology, which filters the actual GDP data to extract the trend as its estimate of potential output. The most common SV filter is the Hodrick-Prescott (HP) filter. It is very easy to use the HP filter because it only requires one data series (output). However, for the same reason, the HP filter is not a reliable technique for estimating potential: it does not take advantage of information from other economic data, say inflation or labor market indicators, to guide its estimate of potential output. There are other technical problems with the HP filter too, the most import of which is the end-of sample problem, with estimates towards the end of a given sample period being subject to significant revisions as more data ultimately become available and the sample is extended.

Another technique to estimate potential output is the “hybrid” approach. It uses a SV filter to estimate trend labor and total factor productivity (TFP) and combines them with capital stock through an assumed production function to arrive at potential output. The Congressional Budget Office (2001) provides a detailed account of such a hybrid model, in which the methodology is applied to different sectors of the U.S. economy. This approach is richer than a SV filter because it allows for more detailed examination of the drivers of potential. A downside of this approach is that it assumes capital is always at its potential. The hybrid approach also suffers from the end-of-sample problems. In addition, neither the hybrid approach nor the SV filter necessarily produce estimates of potential, which are consistent with the definition of the level giving no pressure for inflation rise or fall as they fail to necessarily adjust for variations in inflation in their estimates.

Both the SV filter and the hybrid approach also suffer from the potential misspecification by assuming a deterministic trend. In other words, trend potential output is not allowed to respond to shocks that could raise or lower potential over time. This is problematic, especially when the economy is hit by a large real shock such as during the GR.

MV filter

Many contributions have adopted MV filtering methodologies to estimate potential output. Some examples are models of Laxton and Tetlow (1992), Kuttner (1994), Benes and others (2010), Fleischman and Roberts (2011), and Blagrave and others (2015). MV filtering involves separating potential output from cyclical fluctuations, through the use of data and relationships between output and other macroeconomic variables, such as inflation, labor market indicators, capital formation indicators, etc. This approach adds economic structure to estimates by conditioning them on some basic theoretical relationships (such as a Phillip’s curve relating the inflation process to the output gap). MV filtering methodologies are more complicated than SV filtering methodologies and require more data, but are at the same time more reliable because they use more information from the data for their estimates.

Shortcomings of the MV filtering approach are similar to those facing other methods—there remains an end-of-sample problem, (although we have addressed it largely in this paper using the information from growth and inflation expectations) and the estimates of potential and the output gap are only improved relative to a simple statistical filtration if the structural relationships specified in the filter are valid ones.

The MV filtering approach has the advantage of imposing well-known empirical relationships. In particular, the MV filtering approach adopted in this paper ensures that estimates of the output gap and potential are consistent with the Okun definition of potential. In addition, in its simplest form, this technique is relatively easy to implement requiring only a few variables, and it can also be augmented where data availability permits.

DSGE models

Some contributions have used DSGE models to estimate potential and the output gap (see, for example, Vetlov and others, 2011). These models have more tangible micro foundations and are very appealing. Nonetheless, these they are not so easy to interpret and remain a challenge for policymakers to use in formulation of policies.

Other models

While most contributions, including this paper, have a closed-economy model, some recent work has focused on an open economy case. For example, Alberoa and others (2013) have expanded the definition of potential output to include global imbalances. Yet, another recent strand of literature is focusing on including financial imbalances in the definition of the potential output (see Borio, Disyatat, and Juselius (2013)).

Model

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

The stochastic process for output (real GDP) is comprised of three equations, and subject to three types of shocks. First, the level of potential output $\left({\overline{Y}}_t\right)$ evolves according to potential growth (G_t) and a level-shock term $\left({\varepsilon }_t^{\overline{Y}}\right)$:

Potential growth is also subject to shocks $\left({\varepsilon }_t^G\right)$, with their impact fading gradually according to the parameter θ (with lower values entailing a slower adjustment back to the steady-state growth rate following a shock):

Finally, the output gap closes over time at a speed pinned down by parameter (ϕ), but is also subject to demand shocks $\left({\varepsilon }_t^y\right)$:

In the absence of any shocks, output would be at its steady state path and the output gap would be zero. Shocks can be threefold. They occur to: the level of potential $\left({\varepsilon }_t^{\overline{Y}}\right)$; the growth rate of potential $\left({\varepsilon }_t^G\right)$; or the output gap $\left({\varepsilon }_t^y\right)$, and can cause output to deviate from this initial steady-state path over time. Output is assumed to always return to its steady-state path following any shocks. A shock to the level of potential output in any given period will cause output to be permanently higher (or lower) than its initial steady-state path. Similarly, shocks to the growth rate of potential can cause the growth rate of output to be temporarily higher, before ultimately slowing back to the steady-state growth rate (note that this would still entail a higher level of output). And, finally, shocks to the output gap cause only a temporary deviation of output from potential.

In order to identify the three aforementioned output shock terms, a Phillips curve equation for inflation is added, which links the evolution of the output gap (an unobservable variable) to observable data on inflation according to the process. The Phillips curve equation is as follows:⁴

Equations describing the evolution of unemployment are further included to provide additional identifying information for the estimation of the output gap. The first three of these equations (6-8) are similar to the equations presented above for output. The unemployment gap (u_t) is defined in equation 6, as the difference between the actual unemployment rate $\left(U_t\right)\text{and NAIRU }\left({\overline{U}}_t\right)$:

NAIRU converges to its steady state level $\left({\overline{U}}^{ss}\right)$, which is determined outside the model, but also has a time varying trend (UG_t) and can experience shocks as well $\left({\varepsilon }_t^{\overline{U}}\right)$:

The time varying portion of NAIRU, reflects gradually increased dynamism over time, due to improved information sharing and easier mobility of the labor force, but could experience shocks $\left({\varepsilon }_t^{\overline{UG}}\right)$:

The last equation of the labor market block is an Okun’s rule equation, in which the unemployment gap depends on its past value and the output gap, but it also can experience shocks $\left({\varepsilon }_t^u\right)$:

One important aspect of the U.S. labor market is the effect of aging on labor participation and hence the unemployment rate. The following set of equations is intended to capture this important dynamic. Equation 10 defines the labor force participation gap as the difference between labor force participation (LP_t) and potential labor force participation $\left({\overline{LP}}_t\right)$:

Potential labor force participation converges to its steady state $\left({\overline{LP}}^{ss}\right)$, absent any shocks $\left({\varepsilon }_t^{\overline{LP}}\right)$:

Labor force participation gap depends on its lagged value and the unemployment gap, but can also experience shocks $\left({\varepsilon }_t^{{lp}_{-}gap}\right)$:

Next, equations describing the evolution of capacity utilization provide further identifying information. In equation 13, capacity utilization gap (capu_gap_t) is defined as the difference of capacity utilization (CAPU_t) and its potential level $\left({\overline{CAPU}}_t\right)$ at each period:

Potential capacity utilization converges to its steady state level $\left({\overline{CAPU}}^{ss}\right)$ over time, but could also experience shocks:

The last equation for the capacity utilization block is in the spirit of an Okun’s rule, but for capacity utilization. Capacity utilization gap depends on its lag, the output gap, but it can also experience shocks $\left({\varepsilon }_t^{{capu}_{-}gap}\right)$:

Next, we add equations that enable us to use information from growth and inflation expectations data. These equations capture the assumptions that expectations are formed rationally and fulfilled in the longer term, but do not materialize exactly in the short term, modeled with shocks:

Where $\left(g_{t+j}^E\right)\text{ and }\left(g_{t+j}\right)$ are expected and actual model-consistent GDP growth rates in j periods ahead, respectively. The difference between the two is captured by a shock term $\left({\varepsilon }_{t+j}^{g^E}\right)$. Likewise, $\left({\pi }_{t+j}^E\right)\text{ and }\left({\pi }_{t+j}\right)$ are expected and actual model-consistent core PCE inflation rates in j periods ahead, respectively. The difference between the two is captured by a shock term $\left({\varepsilon }_{t+j}^{g^E}\right)$.

Data for growth and inflation expectation are from Consensus Economics. The ‘strength’ of the relationship between the data on consensus and the model’s forward expectations is determined by the standard deviation of the error terms. In practice, the estimated variance of these terms allows consensus data to influence, but not completely override, the model’s expectations, particularly at the end of the sample period. The incorporation of consensus forecasts can be thought as a heuristic approach to blend forecasts from different sources and methods. The resulting impact of this information on the historical estimates of potential and the output gap is modest, as shown in the following section.

We use annual data for observables (GDP, PCE inflation, unemployment rate, labor force participation, and capacity utilization) from the U.S. authorities and other variables in the model are unobservable. We estimate the model with Bayesian estimation techniques. This is what is generally referred to as Kalman filtering techniques.⁵

To put it briefly, we apply a Bayesian Maximum Likelihood technique, in which priors are chosen for all the parameters of the model and the shocks. A truncated normal distribution for each parameter is assumed as follows

The following optimization problem is solved to find posteriors of the parameters: