Monetary policy rules with PID control features: evidence from the UK, USA and EU

ABSTRACT

This paper considers the extent to which the monetary policy operations of three major central banks can be regarded as an application of Proportional-Integral-Derivative (PID) control rules. The paper outlines the general PID framework and estimates a series of dynamic models to identify how interest rate policy adjustments are affected by the rate of inflation and the level of macroeconomic activity. The paper examines data for the UK, the USA and the Eurozone. The results suggest that the PID rules can provide a useful theoretical and empirical framework for estimating central bank responses to the inflation and macroeconomic activity variables by improving the explanatory power of the Taylor rule model and determining the effect of the parameters.

KEYWORDS:

• Monetary policy rules
• Taylor rules
• feedback control
• PID control rules

JEL CLASSIFICATION:

• E52
• E58

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. See, for example, Williams (2003); Levin, Wieland, and Williams (2003); and Schmitt-Grohe and Uribe (2007).

2. An interesting discussion of model uncertainty and the relevance of constructs such as the output gap for monetary policy can be found in the interchanges between members of the UK Monetary Policy Committee and the House of Lords Economic Affairs Committee (House of Lords 2005). The implications of model uncertainty are also discussed in King (2012).

3. Such uncertainties can arise partly because of ambiguities in the physics of the system and partly because of performance variations arising from factors related to design materials, and construction qualities.

4. A comprehensive discussion of PID control features can be found in Astrom and Murray (2008).

5. A theoretical rationale for the inclusion of the lagged interest rate can be found in Woodford (2003).

6. In practice, this can be countered by the application of low-frequency filtering, to retain some of the (trend) error forecast information contained in the first difference term, while removing the high-frequency noise variation.

7. Or $\mathrm{\Delta }{r}_t=k_1\cdot e_t+k_2\cdot e_{t-1}+k_3\cdot e_{t-2}$, which is often referred to as ‘velocity algorithm’ because it calculates the control action as the change in the direction of the input from its previous position (see Levin, Wieland, and Williams 2003).

8. The data were obtained from the Federal Reserve Economic Data Base (FRED) and the official statistics provided by the Bank of England and ECB websites. Although data is available beyond 2007, we impose 2007Q4 as the end date of the analysis to avoid the distorting impact of the financial crisis and the zero-bound wind-up problem.

9. See, for example, Goodfriend (2007).

10. When the smoothing parameter is set at 1600, the HP filter effectively identifies the cyclical deviations of GDP as fluctuations up to a period of around 8 or so years. It is possible to vary the cycle length by changing the smoothing parameter or by utilizing alternative high-pass or band-pass filters, such as the Butterworth filter (see, for example, Pollock 2000; Shepherd and Dixon 2008).

11. However, the GDP smoothing approach using an HP filter is strongly correlated with the potential output published by FRED (0.998***). ***Significant at the 1% level.

12. The minimum-t statistics are −5.960 (UK), −5.547 (US) and −5.188 (EU). The critical values for the minimum-t are given by Zivot and Andrews (1992); 1%: −5.57 5%: −5.08 10%: −4.82. To avoid detecting breaks closer to the two ends of the sample the data was trimmed at 10%. For the UK and EU, we detect possible trend breaks in 2003Q1 and 2005Q1; for the USA, a possible trend break occurs in 2001Q1. These dates closely coincide with the start/end of a recession in the USA and labour market reforms in Europe. Clemente et al. (1998) provide similar results regarding the order of integration of the series (the estimated t-statics are −3.905, −12.706 and −4.818; this is compared against the 5% critical value, −3.560). However, this test suggests potential beaks in 2002Q1 for the UK, 2003Q4 for EU, and 2002Q4 for the USA, although only the former one is found to be statistically significant at the 5% level. For the purpose of this paper, however, it is important to note that both tests indicate that the policy variable is probably stationary.

13. We also estimate more general models without lagged values of the dependent variable and with dummy variables incorporated, to allow different parameter values for the above target and below target deviations. The latter objective was to determine whether the policy preferences of the central banks are symmetric with respect to deviations of inflation and macroeconomic activity from their target values or whether they exhibit, for example, a greater aversion to above-target than to below-target inflation. However, since the exclusion of the lagged dependent variable did not provide further insights and the coefficients of the dummy variables were generally statistically insignificant the results for these models are not reported.

14. The contemporaneous correlations between the output gap and inflation deviation are −0.081, 0.015 and 0.003 for the UK, the US and the EU, respectively, and 0.176, −0.248** and −0.099 for the inflation and unemployment deviations. **Significant at the 5% level.

15. Under normality, OLS estimates, which by definition minimize a sum of squared residuals, are also maximum likelihood estimates. Comparing Model 2A with Model 1A, we obtain x²(2) = 0.07 (p-value = 0.965). Similarly, Model 1A is selected over Model 3A (x²(4) = 1.66 and p-value = 0.798).

16. We also estimate the model without the lagged interest rate term to contrast the estimates with the ones that appeared in column 1 of Table 2 of Hawkins, Speakes, and Hamilton (2015). We find the inflation gap and output gap responses of 1.104 and 0.349, respectively. These estimates are closer to the ones reported by Hawkins, Speakes, and Hamilton (2015), and Taylor (1993). However, allowing for interest rate smoothing in the specification increases the (adjusted) R-squared by 0.819. Also, the LR-test between the two models confirms that the model with the lagged interest rate term has more explanatory power (LR x²(1) = 198.42; statistically significant at the 1% level). We therefore proceed with our analysis including the lagged interest rate as an explanatory variable in the PI and PID models.

17. However, the F-test does not reject the null hypothesis of the equality of the coefficients in the two regressions (F-value = 0.58 and p-value = 0.451).

18. Model 1B against Model 3B: x² (4) = 9.99 and p-value = 0.041; Model 2B against Model 3B: x² (2) = 7.76 and p-value = 0.021; Model 4B against Model 6B: x² (4) = 29.26 and p-value = 0.000; and Model 5B against Model 6B: x²(2) = 10.40 and p-value = 0.006.

19. Furthermore, the PI-rule model is compared to PID-rule model using the LR-test, but this model identification approach provides conflicting evidence with the AIC criterion (x² (2) = 11.11 and p-value = 0.004).