Considering that in our sample, some countries implemented the ITR in the 1990s and others in the 2000s, for comparative purposes, the number of observations (months) for the model estimation took into account the beginning of the ITR adoption period in each of the selected countries. For example, in 1990, New Zealand was the first country to adopt ITR and, as a result, the number of observations reached 360 months; in our sample, the last country to adopt ITR was Mexico in 2001, and, therefore, the number of observations is 240 months. Even with the data discrepancy, countries present appropriate observations for estimating the VEC model, as argued by Wooldridge (2002) .³

Due to the difference in the data collected for each central bank, we sought to maintain a minimum standard in the data, with the objective of obtaining results that are compatible with the traditional theory. The analyzed variables are: interest rate (the effective interest rate of each country), CPI (monthly rate of change of price indices for each country), GPD (industrial index of physical production, as a proxy for economic activity), and exchange rate (nominal exchange rate, monthly average). Analytically, Equation (1) is the estimated regression that will be estimated for each country of the sample:

D log Interest Rate_t = α₀ + α₁DlogCPI_t−1 + α₂DlogGDP_t−1

+ α₃D log Nominal Exchange Rate_t−1 + €_t−1, (1)

where D indicates the first difference and € ~ (0, σ²).

Finally, on the one hand, based on economic literature and reinforced by Granger causality tests ( Granger, 1980 ), the price index precedes the activity proxy. On the other hand, the choice of variables was partly based on the previously mentioned empirical literature ( Modenesi & Araújo, 2013 ; Fonseca, Peres, & Araújo, 2016 ).

Once the series were properly treated, we sought to assess whether the variables in question followed a stationary stochastic pattern, thus carrying out three unit root tests for each series of data extracted from each country. The unit root tests performed were those of Phillips-Perron (PP) ( Phillips & Perron, 1988 ), Augmented Dickey & Fuller (ADF) ( Dickey & Fuller, 1979 , 1981 ), and Kwiatkowski-Phillips-Schmidt-Shin (KPSS) ( Kwiatkowski, Phillips, Schmidt, & Shin, 1992 ). The null hypothesis of the tests is that the analyzed series are non-stationary, with the exception of the KPSS test, whose null hypothesis is that the variable is stationary. The presence of constant, constant, and trend or no constant and no trend were included in the tests for each variable so that the tests could be performed as completely as possible. Considering all the tests performed, the variables of each country were integrated into order one (non-stationary).

After analyzing each variable, Johansen’s cointegration tests ( Johansen, 1988 ) were carried out in a summarized form to verify whether the linear combination of variables for each country is stationary, thus indicating that there is a long-term relationship between them. Table 2 shows the results of the cointegration tests for all analyzed countries.

The null hypothesis of the test is that there is no cointegration relationship between variables. Thus, the tests show that all countries reject the null hypothesis at a significance level of five percent, both for the trace statistics and the maximum value statistics, thus indicating the existence of at least one cointegration vector for each country. Therefore, comparative estimates are made with the adoption of the VEC model, starting with the number of lags to be included in the model for each country, using selection tests of the system of Autoregressive Vector equations (VAR).

The results obtained were heterogeneous, and most tests point to the ideal of a maximum of one, two, or three lags for each country. However, there are results that point to a greater number of lags, as are the cases of four lags for Canada and the United Kingdom, five for Brazil, Canada, and New Zealand, seven for Mexico and eight for Chile and the United Kingdom.

In addition to Johansen’s cointegration test, a test of the inverse roots of the characteristic polynomial of each country was performed. The results were significant, indicating that the estimated VEC models were stable and empirically robust. Therefore, we estimate the following models for each country, as shown in Table 3 .

Source: Software EViews 9. Elaborated by the authors.

Source: Software EViews 9. Elaborated by the authors.

Table 3 presents the following considerations. The first is the high number of gaps in developed countries, such as Canada and New Zealand. This option for high numbers of lags did not occur randomly, but rather, respecting the model’s estimation, so that the residuals remained well behaved. The second point concerns the problem of overparameterization, that is, a situation in which the number of estimated parameters exceeds the number of observations. Except for Canada, Chile, and the United Kingdom, the other countries had overparameterization problems. However, this is not especially serious since the purpose of the analysis is to compare the behavior and effectiveness of monetary policy under the ITR.

The next step is to present the results of the Granger causality tests for the block variables, which are often used to assess whether a given variable causes another variable in the Granger sense. The lag selection for this test was based on the information criterion of Akaike (1974) and Schwarz (1978) , which always adopts the smallest possible lag between the two criteria. The results are presented in Table 4 , which shows only the relationships that reject the null hypothesis of the absence of Granger causality at five percent.

Source: Software EViews 9. Elaborated by the authors.

Table 4 shows the following: 1) in general, the interest rate helps to predict the CPI—that is, the test specifically shows that there is a causal relationship between interest rates and price levels; and 2) there is a relationship between exchange rate and CPI, given that exchange rate can influence the price level via pass-through.

After presenting the Granger test for all countries, we sought to highlight the cointegration vectors for each country analyzed, as shown in Table 5 .

Source: Software EViews 9. Elaborated by the authors. Note: (*) Standard deviation in parentheses and t-statistic in brackets.

Based on Table 5 , it is possible to interpret the cointegration equations as a reaction function of each central bank for each country in the long run. The equations were normalized for the variable of interest, which could compromise the interpretation of the magnitude of the estimated coefficients for each country. However, what matters is that the interest rate is an endogenous variable and depends directly and positively on three other variables: CPI, GDP, and exchange rate. In summary, the signs of the parameters of the equations are consistent with the results.

This subsection aims to verify the effectiveness of the monetary policy, that is, in the sense of how the management of the interest rate affects the price level. To this end, we analyze the responses of the CPI variable to a positive shock (Generalized One Standard Deviation Innovations) on the interest rate, using impulse response function (IRF).

For each country, an increase in the interest rate receives a positive response to the price level. For comparative purposes, Figures 1 - 3 present the results of the countries in which the price level responds partially, moderately and explosively, respectively, when the interest rate increases.

The results for Brazil show that the price response to a restrictive monetary policy (increase in interest rates) presents a declining and negative trajectory only in period 13. Moreover, the response to a contractionary monetary policy is followed by an initial rise in inflation, with a decreasing trajectory.

There are cases in which the IRF gave results that are counterintuitive to what is expected from a restrictive monetary policy. These results were expected given the complexity of a comparative empirical exercise in which there are intrinsic differences for each country. In Chile, the effect of a restrictive monetary policy is

a permanent increase in inflation, while in New Zealand, the effect is a huge increase in inflation.

Chile, New Zealand and the United Kingdom registered very low inflation rates over almost the entire period and the increase in inflation in the last years of the sample was followed by increases in interest rates, which explains the positive relationship between inflation and interest rates.

In the case of Canada, although the relationship between interest rate and inflation is economically significant, due to the negative relationship between the two variables, the coefficient of interest rate was not significant in the long-term function and in the IRF, which cuts the zero axis at a few moments. Perhaps the low variability of interest rates in Canada is an explanation for this, as for almost 10 years, interest rates have not changed in the country.

In the case of Mexico and Brazil, interest rates and inflation are higher and more volatile and, therefore, the negative relationship between interest and inflation is confirmed.

Although this subsection has explored only the price level responds to the interest rate, the other IRFs, such as interest rate to exchange rate, CPI to interest rate, CPI to exchange rate, GDP to interest rate, GDP to exchange rate and exchange rate to interest rate, for all countries, are shown in the Appendix.

To sum up, the analysis of the empirical results presented indicates that the ITR, by itself, does not seem to be an efficient regime to guarantee price stability in the selected economies. There are, at least, two reasons: 1) In developed countries of our sample, despite the fact that these economies have more exchange rate flexibility, central bank independence, and higher level of financial development, that are some important conditions to adopt the ITR ( Petrevski, 2023 ), the relationship between inflation and interest rates was not economically significant in the estimated models; and 2) In developing countries, specifically Brazil and Mexico, despite the negative effects of interest rate shocks on inflation, the inflation rate, in the last years, has been higher than most developed and developing economies, and it also has been noted that the trajectory of the GDP, in these two countries, exhibited lower growth rate than that of the world economy, most of the time (see, for instance, IMF, 2024). Exploring the second reason, it is important to mention that, historically, the exogenous shocks played a relevant role in explaining inflation in Brazil and Mexico, namely, the commodity index and exchange rates, among others. Thus, considering that the ITR supposes that inflation rate is a consequence of a demand-pull situation, therefore increasing the interest rate to mitigate the negative impacts of exogenous shocks ends up leading to suboptimal results.