methodology followed. Section 3.3 provides the results of the semiparametric regres-sions and section 3.4 presents the results of the tests for the presence of nonlinearities in the Taylor rules. Section 3.5 offers a sensitivity analysis of the estimation results using an alternative measure of the economic outlook. The last section collects all tables and estimation results.
3.2 Data and methodology
3.2.1 Data
We use quarterly OECD forecasts of inflation and real GDP growth for 8 indus-trialized countries: Australia, Japan, New Zealand, Norway, Sweden, Switzerland, United States and Canada. All data are collected from Datastream and the offi-cial Central Banks websites. The inflation and real GDP growth forecasts refer to a one-year horizon for the current year with respect to the corresponding quarter of the previous year. The data span the period from 1976 Q3 to 2010 Q4 for all countries. In all specifications the dependent variable is the key policy rate, i.e., the market rate targeted by the Central Banks. In addition, the policy rate used in the regressions accounts for the fact that its definition has changed over time for several Central Banks. The inflation and output growth forecasts are revised over time which permits to track more accurately the actual reaction function of the Central Banks.2
The appendix provides a detailed description of the variables and table 3.1 displays some summary statistics. The latter shows that the policy rate is high in the countries that have experienced high levels of inflation. New Zealand is the country with the highest average inflation rate while Japan features the lowest. The GDP
2We have also estimated the policy rules with the real-time Consensus Economics monthly inflation and output growth forecasts for Japan, Norway, Sweden, Switzerland, USA and Canada from December 1989 to March 2010. In the regressions we used the forecasts of inflation and real GDP growth with a fixed horizon of one-year.
growth forecasts are quite similar among the OECD countries with Australia scoring the highest average growth rate and Switzerland featuring the lowest. The forecasts for the change in the output gap exhibit a smaller variability compared to the GDP growth forecasts and are on average close to zero for all countries.3 The unit root tests performed point out that the policy rate and the inflation forecasts are likely to be nonstationary in most of the countries even though the Augmented Dickey-Fuller statistics often show evidence against the null hypothesis of unit root.
Nevertheless, we are in a relatively small sample context to fully rely on asymptotic approximations. In addition, the assumption of stationarity of the series is justified from a macroeconomic perspective given the stable monetary policy regimes in place and the firm commitment of the Central Banks to achieve price stability as their main policy objective. Moreover, there is strong evidence against the null hypothesis of unit root for the real GDP growth and output gap change forecasts in all countries.
In addition, the residuals from all estimated specifications are stationary which precludes spurious inference.
Given that the aim of this chapter is to analyze the responsiveness of Central Banks to economic fundamentals using the most accurate relevant information we do not consider the problem of estimating the level of potential output in the baseline regressions. Hence, we use forecasts of real output growth instead of the output gap in order to avoid the mismeasurement problems of potential output encountered in actual policy making along the lines of [Leitemo and Lonning, 2006] for instance4. However, in section 5 we also comment on an analysis performed with the Hodrick-Prescott (H-P) detrended output growth rate also known as a speed limit policy.
Indeed, [Walsh, 2003] shows that under discretion a Central Bank that targets the difference between the output growth rate and the growth rate of potential output (the change in the output gap) delivers the optimal pre-commitment policy outcome.
Walsh has emphasized that the implementation of a speed limit policy is particularly
3The change in the output gap is approximated by the difference between output growth and potential output growth.
4Note that there are other ways in the literature to treat the potential measurement error of the forecasts used as explanatory variables, see [Hall et al., 2015] and [Ellison and Sargent, 2012].
3.2. Data and methodology 97 relevant in the presence of measurement errors in potential output and could be welfare enhancing compared to pure discretion or to an inflation targeting regime.
We first present the analysis with the output growth forecasts and then perform the regressions with the H-P detrended change in the output gap in the robustness section. Our approach also permits to use more accurate data in the estimations and to avoid the potential end-of-period problems that arise when applying the H-P filter. The method is also in line with the approach of [Taylor, 1993] who assumes a constant level of potential GDP in his sample as well as with a recent estimation of a Taylor rule for the Fed performed in [Coibion and Gorodnichenko, 2012].
3.2.2 Methodology
We use a novel approach in the estimation procedure in monetary economics. In order to account for potential nonlinearities in the reaction function we adopt a semiparametric modeling technique.5 Based on the approach of
[Hastie and Tibshirani, 1986, Hastie and Tibshirani, 1990] we specify an additive model6 (AM) which is a regression model that contains smooth functions of covari-ates. This specification is quite flexible in capturing potential smooth nonlinearities in the policy rule while features the advantage of avoiding the curse of dimension-ality problem. The AM specifications are all estimated in the R software using penalized regression splines with the library ”mgcv” and function ”gam”. We used smoothing splines with the option s() in the function gam. The latter uses thin plate regression splines as smoother. We chose the default settings for the dimension of the basis. The degree of smoothness of the splines is estimated with Generalized Cross Validation (GCV) as outlined in [Wood, 2006]. Even though this procedure penalizes model over-fitting, we implement the GCV method using an additional
5An extensive treatment of the nonparametric and semiparametric methods is provided in [Wood, 2006], [Haerdle W., 2004] and [Fan J., 2003].
6The adopted methodology closely follows [Wood, 2006]. This additive model is a special case of a general family of semiparametric models known as Generalized Additive Models (GAM). The additive specification is obtained from the GAM using the canonical link due to the fact that the dependent variable is continuous.
penalty term with the option ”gamma”. The latter imposes a more smooth esti-mated function as explained in [Kim and Gu, 2004]. Formally we want to estimate a smooth function, say s, through:
mins (y−s)2+λP(s) (3.1)
where λ is a smoothing parameter and P(s) measures roughness of s. (3.1) can be parametrized so as to have a constrained linear problem which, can be transformed into a non-constrained linear parametric problem. The smoothing parameter is chosen through a Generalized Cross Validation procedure. Finally, inference is based on Bayesian arguments assuming that y’s are i.i.d. and normally distributed, see [Wood, 2006] for a more detailed description of the methodology.