## Abstract

We show that many currently popular monetary policy rules fall structurally within a class of robust industrial control known as proportional-integral-differential, or PID, control. From this identification we propose a general class of PID-based monetary policy rules that include as limiting cases the original Taylor rule as well as lagged and forward-looking extensions of thereof. The effectiveness of parsimonious extensions of the Taylor rule are consistent with the well-known effectiveness and parsimony of PID control. We find that for the same reason encountered in other PID control applications—noisy data—most monetary policy rules fall in the proportional-integral subset of PID control known as PI control. We estimate both PID and PI monetary policy rules using the historical analysis approach of Taylor and compare the performance of our PI rule to other policy rules using a recently-developed macroeconomic-model comparison methodology. A key feature of PID control is its remarkable effectiveness for systems where the equations of motion are not known. Thus, PID-based rules both link monetary policy with a tradition of practical control in the absence of known dynamical equations and provide baseline rules for monetary policy in the face of macroeconomic model uncertainty.

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## Notes

For an engaging account of the history of monetary-policy rules see Woodford (2003), chapter 1.

For an interesting and comprehensive account of the history, breadth, and current practice of feedback control in general and PID control in particular see Bennett (1993), Åström and Murray (2008). As discussed below, the vast majority of implementations of PID control in control engineering and, as we shall see in monetary policy, are in fact reduced forms of PID control such as PI control. In keeping with the tradition of control engineering, however, we use the term PID control as a general label for both PID and PI control.

Taylor (1993a) originally presented his rule in terms of average inflation and of output gap as \(r(t) = \pi (t) + 0.5 y(t) +0.5 \left( \pi (t) - 2 \right) + 2\) which can be easily rewritten in “gapped” form as \(r(t) = 4 + 0.5 y(t) +1.5 \left( \pi (t) - 2 \right) \).

\(K_0\) is the nominal Fed funds rate associated with inflation gap and GDP gap being equal to zero. \(K_P^{(\pi )}\) and \(K_P^{(y)}\) are the proportionality constants for the inflation gap and output gap respectively.

Alternatively, one could differentiate Eq. (4) with respect to time resulting in

$$\begin{aligned} \frac{dr(t_k)}{dt} = \frac{d \varvec{K}_P \varvec{\cdot }\varvec{g}(t_k)}{dt} + \varvec{K}_I \varvec{\cdot }\varvec{g}(t_k) + \frac{d^2 \varvec{K}_D \varvec{\cdot }\varvec{g}(t_k)}{dt^2} , \end{aligned}$$discretize the first-order derivatives with Eq (5) and the second-order derivatives with

$$\begin{aligned} \frac{d^2 \varvec{g}(t_k)}{dt^2} \approx \frac{\varvec{g}(t_k) + \varvec{g}(t_{k-2}) - 2 \varvec{g}(t_{k-1})}{\left( \varDelta t \right) ^2}, \end{aligned}$$set \(\varDelta t = 1\) and collect terms to obtain Eq. (7).

The velocity algorithm is formally similar to the “first-difference rules” in the monetary policy literature (cf. e.g. Levin et al. 2003). Our derivation can be viewed as a formal representation of their observation that “[a] rule with a high value [of the lagged interest-rate parameter] implicitly makes the current interest rate depend on the complete history of output and inflation, albeit in a very restricted way.” Levin et al. (2003), p. 287.

For a discussion of and references to both approaches see Taylor (1999a).

We employed ordinary least squares (OLS) analysis with Newey-West Heteroskedasticity and Autocorrelation Consistent (HAC) standard errors on quarterly (Q1 1987–Q4 1992) vintage data sourced from the ArchivaL Federal Reserve Economic Data (ALFRED) repository at the Federal Reserve Bank of St. Louis: http://alfred.stlouisfed.org/.

This was expected given that we used similar data.

A sense of the historical data that is reviewed in the FOMC meetings can be found in the material on the Board of Governors’ website (http://www.federalreserve.gov/monetarypolicy/fomccalendars.htm).

This reference and further information on The Macroeconomic Model Data Base (MMB) project can be found at http://www.macromodelbase.com. The macroeconomic models in this database are solved using Dynare (cf. Juillard 1996, 2001 and http://www.dynare.org).

In a recent paper Taylor and Wieland (2012) present the results of a novel approach to monetary-policy rule tuning that used macroeconomic simulation to obtain rule parameters that minimized the variance of the inflation, output and interest rate across three current-generation macroeconomic models (Taylor 1993b; Altig et al. 2005; Smets and Wouters 2007). With this tuning they obtained the model-averaged policy rule \(r_t = 1.06 r_{t-1} + 0.19 \pi _t + 0.67 y_t - 0.59 y_{t-1}\). Comparing the PI rule with their model-averaged rule we see first that the parameter for the lagged interest-rate (by definition unity in the PI rule) is also very close toFoornote 16 continued

unity in the Taylor and Wieland (2012) rule. This feature has often been implicated as evidence of interest-rate smoothing by central banks but, as shown in our derivation, it is a simple consequence of the velocity derivation of the PI rule. Earlier research has also linked a parameter near unity for the lagged interest rate with monetary-policy robustness (Levin et al. 2003). Comparison of the other variables is complicated by the fact that Taylor and Wieland (2012) assumed \(\beta _1^{(\pi )} = 0\): this was not a consequence of the tuning. The size and sign of the common parameters between these rules, however, suggests that the parameters of a PI rule tuned using macroeconomic simulation may be similar to those we obtained using historical analysis: a topic of future research.

A reference for each model is provided in the Appendix below. For a discussion of the MMB software see Wieland et al. (2011).

This quote is from the 2006 edition of Åström and Murray (2008).

As our derivation begins with the original Taylor rule, it is not independent of macroeconomic modelling in general. The ubiquity of the Taylor rule, however, suggests that it, like our derivation, transcends specific macroeconomic models.

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## Acknowledgments

We thank Stuart Bennett for his illuminating insights on the history of PID control. We thank Volker Wieland, Sebastian Schmidt and Elena Afanasyeva for very helpful conversations concerning the MMB software. Finally, we thank Masanao Aoki for his pioneering and inspiring work bridging control theory and practice in engineering and economics.

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Hawkins, R.J., Speakes, J.K. & Hamilton, D.E. Monetary policy and PID control.
*J Econ Interact Coord* **10**, 183–197 (2015). https://doi.org/10.1007/s11403-014-0127-3

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DOI: https://doi.org/10.1007/s11403-014-0127-3