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rates which are between 2% and 4%. For the remaining range of the forecasts, its inflation responsiveness is softer, suggesting that the Bank of Canada might have prioritized other policy goals or has responded to unexpected shocks. Contrarily to the Fed, the Bank of Canada has responded quite aggressively to high infla-tion forecasts by sharply raising the overnight rate. As regards the reacinfla-tion to the economic outlook, the estimated univariate function is linear. The Central Bank raises the policy rate in the event of an increase in projected output growth thus implementing a stabilizing policy for the economy. The estimated interaction term shows evidence for a sizable dependence between the inflation and output growth forecasts featuring 11.79 estimated degrees of freedom. The estimated surface of the bivariate function points to a stronger BoC’s reaction to inflation than to the economic outlook consistently with its mandate.

To sum up, one should emphasize that the augmented semiparametric specifica-tion presented in this secspecifica-tion features an important characteristic compared to the standard semiparametric specification. The augmented model accounts for the po-tential interaction between the inflation and output growth forecasts (non-zero cross-derivatives) which seems to drive the nonlinearities in the reaction functions of most Central Banks. The interaction better captures the trade-off monetary authorities face in the event of a supply (cost-push) shock. The latter helps to rationalize a nonlinear policy response for most of the Central Banks. Lastly, the section illus-trates how easy it is to interpret the semiparametric policy rules and to relate them to the actual monetary policy framework.

3.4 Nonlinearity tests

The previous section studied the performance of the augmented semiparametric and standard policy rules in terms of fit independently of each other. In this sec-tion, we compare the different policy rules with some specification tests derived

in the nonparametric framework. For this purpose, we adopt the approach of [Gozalo and Linton, 2001]. Although their approach is proposed in a kernel context, the contrasts they suggest can be used in our setting. In addition, we do not rely on the asymptotic approximation of the tests as their performance can be very far from accurate in small and moderate sample sizes. This approach also depends on some quantities that need to be estimated. Instead, we will use the wild bootstrap as our inference tool in this section. The latter outperforms the asymptotic approximation in both nominal size and power, see [Gozalo and Linton, 2001]. Moreover, as noted by [Haerdle W., 2004] the wild bootstrap is less sensitive to the choice of tuning parameters. Moreover, the time span we cover features large changes in inflation and output growth for most countries, which might probably cause heteroscedas-ticity in the error term. Thus making the wild bootstrap a natural inference tool for our setup. There are, however, some critiques of this approach that need some attention. The bootstrap routine we use might suffer from a problem in the choice of the smoothing parameter, especially when setting a semiparametric Taylor Rule as the null hypothesis of the test. Indeed, in this specific case we might need to use subsampling methods instead, see [Gonzalez-Manteiga and Crujeiras, 2013] and [Sperlich, 2014]. This criticism does not apply when we have a parametric Taylor Rule as the null hypothesis in any of the three contrasts.

We have the following relationships between the 4 rules we consider: PSPASP and PAPASP. Hence, P is a special case of any of the other policy rules studied and ASP nests all the other policy rules. For the comparison of the policy rules we use three different contrasts:

τ1 = T1

where ˆit and ˜it are the predicted values at time t of the full and restricted models respectively, and ˆt and ˜t are the corresponding residuals of the full and reduced

3.4. Nonlinearity tests 111 models. The null hypothesis is H0 : it ∈RM against the alternative Ha : it ∈FM, where RM and FM stand for reduced and full model respectively. The following steps explain how we apply the wild bootstrap in our context:

- We simulate ∗,(b)t from N(0,˜σ2t), where ˜σ2t = ˜2t. Then, we compute the boot-strapped samples (b), for b = 1, ...1000, as follows:

i∗,(b)t = ˜it+∗,(b)t

- We compute the bootstrapped contrasts denoted as τj(b) for j = 1,2,3.

- We then use the empirical distribution of the bootstrapped tests, that is:

Fˆτj(z) = 1 1000

1000

X

i=1

I(τj(i) < z),

to compute the p-value for the observed contrasts:

p-valueτj = 1 1000

1000

X

i=1

I(τj < τj(i))

where τj are the observed test statistics for j = 1,2,3.

Tables 3.10 and 3.11 report the p-values of the tests between two different specifica-tions at a time for the three contrasts. The tables show that the ASP better fits it than SP for all countries except for New Zealand. The augmented semiparametric specification does a better job than the two parametric Taylor rules for all countries.

SP performs better than the P for all countries except for Switzerland and the U.S.

In view of the results of the tests, we propose the following policy rule that best fits the actual policy rate: the augmented semiparametric rule for all countries except for New Zealand where the standard semiparametric rule fits the best. Regarding

the specification with the speed limit policy, the results are qualitatively similar except for Norway and Sweden where the standard semiparametric specification seems to perform best. Note that for all countries the estimation results are in line with the nonlinearities tests except for Switzerland. We claim that the asymptotic theory provides a poor approximation in the estimation for Switzerland. To sum up, the evidence shows the presence of nonlinearities in the reaction functions for most countries. This result is related to either the presence of nonlinear structure of the economy or/and asymmetric Central Bank preferences. The latter rules out symmetric Central Bank preferences along with a linear structure of the economy thus precluding risk-neutral Central Banks’ behavior.