2. Related literature

In our analysis, we expand upon two main strands of literature. The first one comprises empirical work on inflation persistence in (mainly) IT countries, especially the studies that employ the FI framework to describe inflation processes. The second one stems from the theoretical analyses of alternative monetary policy strategies, mainly PLT and, more recently, AIT.

Batini and Nelson (Reference Batini and Nelson2001) offer three working definitions of inflation persistence: (a) positive serial correlation in inflation; (b) lags between systematic monetary policy actions and their peak effect on inflation; and (c) lagged response of inflation to non-systematic policy actions, i.e., policy shocks. Canarella and Miller (Reference Canarella and Miller2017b) show that inflation persistence is an important factor in determining economic outcomes from the perspective of both households and policymakers, but, at the same time, it is hard to measure and describe. A thorough overview of different types and classifications of inflation persistence is provided by Fuhrer (Reference Fuhrer2009). The author argues that a policymaker must be able to determine whether or not persistence is structural and thus may be taken as a stable feature of the economic landscape. In order to know this, she must be able to parse the sources of persistence into three types: (1) those generated by the driving process, (2) those that are a part of the inflation process intrinsic to inflation (that is, persistence that is imparted to inflation independent of the driving process), and (3) those that are induced by her own actions or communications. According to Fuhrer (Reference Fuhrer2009, p. 27), the research suggests that “central banks that are more explicit about their inflation goal and act in accordance with that commitment may enjoy less persistence in their nations’ inflation rates.” This implies that central banks’ commitment to IT is a plausible explanation for differences in inflation persistence. Williams (Reference Williams2006) indicates that commitment influences both persistence and overall dynamics of inflation through the expectations channel, by anchoring actors’ expectations to the inflation target.

While early analyses of FI go back over four decades (Granger and Joyeux, Reference Granger and Joyeux1980; Granger, Reference Granger1980), there is growing empirical support that economic time series are fractionally integrated (Parke, Reference Parke1999). Gadea and Mayoral (Reference Gadea and Mayoral2006) describe the search for a proper way to model inflation. The sticky price models of Taylor and Calvo do not capture the observed inertia of inflation well enough, nor do their modifications. The authors provide evidence corroborating that inflation is better described by FI than by a simple I(1) or I(0) dichotomy. The impact of price shocks is rarely permanent, as would be in the I(1) case, or has only a short-lasting impact, decaying to zero at an exponential rate, as would be in the I(0) case. The authors argue that even though price shocks are non-permanent, they vanish slowly in a hyperbolic rather than exponential fashion. Zagaglia (Reference Zagaglia2009) shows for 12 member countries of the Organisation for Economic Co-operation and Development (OECD) that CPI series have finite variance and are at least two standard errors below the unit root, thus confirming FI rather than integration of order one. Estimates for some $d$ coefficients are negative, but the uncertainty is too large to draw any definite conclusions. FI is also used for extremely long (8 centuries) time series by Caporale and Gil-Alana (Reference Caporale and Gil-Alana2020). According to them, the main advantage of such a framework is that it requires fewer assumptions than a simple ARMA model and, therefore, is more general.

Gadea and Mayoral (Reference Gadea and Mayoral2006) show that the FI behaviour of inflation might result from the aggregation of prices of firms that are heterogeneous in adjusting their prices to costs. Other potential explanations for long memory in inflation are aggregation in price indexes (Hassler and Wolters, Reference Hassler and Wolters1995), aggregation of heterogeneous firm production (Abadir and Talmain, Reference Abadir and Talmain2002), persistence in money supply shocks (Scacciavillani, Reference Scacciavillani1994), lack of credibility of the inflation target (Erceg and Levin, Reference Erceg and Levin2003), unanchored inflation expectations, and uncertainty about the long-term inflation objective (Orphanides and Williams, Reference Orphanides and Williams2005). Altissimo et al. (Reference Altissimo, Mojon and Zaffaroni2009) and Paya et al. (Reference Paya, Duarte and Holden2007), among others, suggest a possible connection between inflation persistence and temporal aggregation, with models estimated with higher frequency data showing lower persistence than models estimated with lower frequency data.Footnote ² Yigit (Reference Yigit2010) notes that while the empirical work assessing the relative performance of IT usually concentrates on observable variables, such as inflation and output, the true test of IT’s effectiveness could be shown by the strategy’s influence on inflation expectations. Although direct measurement of expectations is difficult, expensive, and often impossible due to a lack of data, especially before the adoption of IT, the author offers an “indirect methodology” by suggesting the link between inflation persistence and the distribution of inflation expectations. He estimates the FI parameter before and after the adoption of IT and argues that, as the fall in long memory happened exactly at the time of a regime shift, the changed nature of inflation expectations is the best explanation.

The evidence of lower inflation persistence after the adoption of IT is vast and is often used as an argument in favour of adopting the new strategy (Kuttner and Posen, Reference Kuttner and Posen2001; Levin et al. Reference Levin, Natalucci and Piger2004; Zagaglia, Reference Zagaglia2009; Bhalla et al. Reference Bhalla, Bhasin and Loungani2023). Canarella and Miller (Reference Canarella and Miller2017a) analyse shifts in inflation persistence between pre- and post-inflation targeting periods. Unlike Yigit (Reference Yigit2010), they use a modified log periodogram (MLP) to estimate inflation persistence, as this semiparametric method does not require the specification of the ARMA model. Authors confirm falling persistence after the adoption of IT. Bhalla et al. (Reference Bhalla, Bhasin and Loungani2023) argue that while this argument was clear-cut in the case of early adopters of IT, in countries that switched to the new strategy later, the benefits are not so obvious since there are a number of possible competing explanations for falling levels and persistence of inflation, such as great moderation, also among countries that did not use IT. Contrary to this approach, our goal is to use inflation persistence to show different outcomes within the IT group.

A comprehensive review of the literature on the PLT strategy can be found in Ambler (Reference Ambler2009) and Hatcher (Reference Hatcher2011). While early literature on PLT focused on its contribution to long-run price stability at the cost of higher output variability, only after a seminal paper by Svensson (Reference Svensson1999, working paper in 1996) was the influence of the expectations channel on improving the inflation-output trade-off fully grasped. Both Ambler (Reference Ambler2009) and Hatcher (Reference Hatcher2011) present the research on the validity of Svensson’s “free lunch,” list arguments in favour and against switching to PLT, and provide an exhaustive description of relevant institutional issues.

Ruge-Murcia (Reference Ruge-Murcia2014) asks whether IT central banks de facto target the price level path and shows that under certain conditions there can be an “observational equivalence between inflation and price-level targeting.” They explain that “[t]his equivalence arises from the purposeful policy action of the inflation-targeting central bank, which seeks to deliver average inflation rates close to the target rate in the short run. In principle, this equivalence may also arise as a result of symmetric shocks that take inflation sometimes above, sometimes below, its target.”

Although AIT has been present in the theoretical literature since the turn of the century (Nessén & Vestin, Reference Nessén and Vestin2005), it experienced a surge in popularity following the Fed’s announcement. AIT as a viable alternative to both PLT and IT is presented in Svensson (Reference Svensson2020), who treats “forecast targeting” as a general (and preferable) monetary policy strategy that could be crystallised as IT, AIT, PLT, temporary PLT, or nominal-GDP targeting and shows that AIT potentially dominates other proposals. Dorich et al. (Reference Dorich, Mendes and Zhang2021), who analyse alternative strategies as a potential choice for the Bank of Canada, observe that history dependence can lead to better performance in a low neutral rate environment. Since flexible IT, AIT, and PLT “differ only in the degree of history dependence they embed,” the authors find that their performance “depends critically on the importance of the effective lower bound (ELB) constraint.” In the absence of an ELB, flexible IT dominates the other options, and when the ELB is an important constraint, AIT is the preferred option. Budianto et al. (Reference Budianto, Nakata and Schmidt2023) analyse the role of the expectations channel under AIT and show, under rational expectations, the welfare-improving role of history dependence when facing low interest rates. While, according to the authors, the optimal averaging window is infinitely long, making AIT observationally equivalent to PLT, most of the benefits are to be obtained within a finite but long (e.g. a few years) window. Jia and Wu (Reference Jia and Wu2023) analyse the central bank’s incentive to deviate from the ex-ante announced AIT. They show that the optimal horizon of AIT is time-dependent and that, provided the central bank is credible, the ex-post switch from AIT back to IT might be welfare-improving.

Clarida (Reference Clarida2022) describes the Fed’s move to AIT, but his interpretation diverges from a textbook description of AIT. According to him, the Fed’s new monetary policy framework has been asymmetric from the very beginning: its goal “is to return inflation to its 2 percent longer-run goal, but not to push inflation below 2 percent.” As stated by Clarida (Reference Clarida2022, p. 9), “our framework aims ex ante for inflation to average 2 percent over time, but does not make a commitment to achieve ex post inflation outcomes that average 2 percent under any and all circumstances.” Therefore, the new strategy could be described as “temporary PLT that reverts to flexible IT (once the conditions for liftoff have been reached).” This seems to suggest that the adoption of AIT might have been a communication device aimed at escaping the ELB. It also means that real-life monetary policy strategies can easily dwell between the theoretical ideal types.

3. Empirical framework and data

This section outlines the empirical framework used in the study and describes the dataset.Footnote ³ Our main task at hand is to investigate the persistence of inflation rates at the country level. We next use this property to classify the monetary policy strategies into several categories or “shades” of inflation targeting. The empirical setting of the study relies on fractionally integrated processes. Specifically, we utilise ARFIMA models. The major feature of such models is that they encompass standard autoregressive and moving average (ARMA) components while allowing for non-integer (fractional) orders of integration in time series. One significant advantage of FI is that it breaks the dichotomous distinction between stationary (mean-reverting, I(0)) processes and nonstationary (unit-root, I(1)) processes, providing a more nuanced understanding of various degrees of persistence in inflation rates. As we discuss below, this approach adds more complexity to the modelling of inflation rates by accommodating the so-called long-memory properties of time series or the ability of the series to exhibit significant autocorrelation at long lags. Unlike models of I(0) and I(1) processes, ARFIMA models can capture intermediate levels of persistence, reflecting both short-term fluctuations and long-term dependencies. This flexibility allows ARFIMA models to represent the gradual decay of shocks’ effects over time, offering a richer depiction of the dynamics underlying the inflation rates.

A general ARFIMA( $p$ , $d$ , $q$ ) process of the inflation rate series, $\pi _{t}$ , may be expressed as:

(1) \begin{equation}\phi \left(L\right)\left(1-L\right)^{d}\pi _{t}=\theta \left(L\right)\varepsilon _{t},\end{equation}

where $\phi (L)$ is the autoregressive lag polynomial, $\theta (L)$ denotes the moving average lag polynomial, while $\varepsilon _{t}$ is the white noise error term. Our main object of interest is the fractional differencing parameter $d$ . Notice that when $d=0$ , the process in Equation (1) collapses to a standard stationary ARMA process, and the inflation rates are integrated of order zero. However, an ARFIMA process involves a range of intermediate cases.

As $d$ increases and approaches 0.5, the process remains stationary but exhibits long-term dependence, or “long memory.” In such an instance, the autocorrelation function decays more slowly over time, meaning that the effects of shocks to the inflation rate persist for a longer period than in a regular I(0) case. The higher the value of $d$ , the more pronounced the long-term dependence, indicating that it takes a considerable amount of time for the inflation rate to revert to its mean. On the other hand, when $d$ falls between −0.5 and 0, the process demonstrates negative long-term dependence, which implies a strong mean-reverting behaviour. This type of process is often referred to as “intermediate memory” or “antipersistent.” For inflation rates, this means that after experiencing a shock, the process overcompensates during its reversion to the mean, often resulting in a negative adjustment following a positive shock, and vice versa.

Figure 1. Fractional integration of inflation rates and the classification of monetary policy strategies.

Notes: The figure shows a schematic relationship between the fractional integration of inflation rates (parameter $d$ in the ARFIMA models) and monetary policy strategies.

We argue that the properties of fractionally integrated processes in inflation rates provide a valuable framework for analysing actual monetary policy strategies across countries. Our key premise is that while inflation targeting (IT) regimes are typically associated with stationary inflation rates, the degree of long-term dependence (or long memory) in inflation may vary across different inflation targets. Hence, we advance the ideas put forward by Yigit (Reference Yigit2010) and Canarella and Miller (Reference Canarella and Miller2017b) that monetary policy strategies can be investigated indirectly by scrutinising the effects of monetary policy credibility and the effectiveness of expectations management embedded in the inflation series. This approach allows for an evaluation of how well central banks establish credibility and anchor inflation expectations. Moreover, FI of inflation rates offers insights not only into the relative effectiveness of IT (in a narrow sense) as opposed to its alternatives, but also into alternative formulations of the IT frameworks in a broad sense, where some of the formulations are “observationally equivalent” to strategies conventionally treated as separate (such as AIT or PLT).

Figure 1 presents a schematic representation of the relationship between FI of inflation rates and different types of monetary policy strategies. The figure distinguishes between four main possibilities, corresponding to different values of $d$ . Around $d=0$ , we identify what we refer to as a “strict” or “orthodox” inflation-targeting regime. In this regime, inflation exhibits only short-term memory, meaning that the central bank’s interventions quickly stabilise inflation around the target without significant persistence or long-term effects.

As $d$ increases toward 0.5, the process remains stationary but exhibits longer-term dependencies, meaning that inflation shocks take longer to dissipate. This region represents a more “conventional” or “flexible” inflation-targeting regime. In this framework, the central bank allows for more flexibility in inflation management, tolerating some degree of persistence in inflationary shocks. When $d$ approaches or exceeds 0.5, inflation exhibits significant long-term dependence, implying an “uncommitted” or “ineffective” inflation-targeting regime, where central banks struggle to anchor inflation expectations effectively. Shocks to inflation are highly persistent, suggesting either weak policy interventions or a lack of credibility in the central bank’s ability to manage inflation. Conversely, moving to the left of $d=0$ , the figure shows negative values of the FI parameter, which indicate mean-reverting behaviour with negative long-term dependence. This region we link with the “average” inflation-targeting (AIT) regime. It tolerates temporary deviations from the inflation target but aims to bring the average inflation rate back to a specified level over time. Two extreme cases close the spectrum of monetary policy frameworks. Under PLT, the central bank targets a stable price level, allowing inflation to fluctuate but ensuring that the price level is corrected over time. When the inflation rate is allowed to be an I(1) process, the regime can no longer be treated as IT (non-IT). Table A.1 in the Appendix summarises the inflation persistence properties across IT regimes, highlighting a range of FI values that suggest distinct inflationary dynamics, from mean-reverting processes to highly persistent inflation.

The underlying series on the CPI price level are sourced from the IMF International Financial Statistics. The restriction we face here is the availability of monthly CPI price level data.Footnote ⁴ The CPI series are seasonally adjusted using the X-13 ARIMA-SEATS algorithm. Next, we calculate the continuously compounded, annualised inflation rates measured as month-to-month changes in the logarithm of the price index:

(2) \begin{equation}\pi _{t}=12\times 100[\log \left(P_{t}\right)-\log \left(P_{t-1}\right)].\end{equation}

This transformation is consistent with earlier applications of FI to inflation data (e.g., Canarella and Miller, Reference Canarella and Miller2017a). The resulting monthly inflation rates form the basis for our long-memory estimates and are constructed uniformly for all countries over the common estimation window.

The study covers 36 countries, both advanced and EMEs, whose monthly inflation rates are illustrated in Figure A.1 in the Appendix. Table A.2 in the Appendix reports detailed descriptive statistics of these series, which suggest a wide range of average inflation levels and volatility across countries. Our major consideration when selecting this group of economies is their status as inflation targeters, which makes the cross-country comparison of the FI in inflation rates meaningful. However, countries adopted IT at different points in time, with relatively few of them introducing this monetary policy strategy in the early 1990s. Numerous economies adopted IT in the late 1990s, at the beginning of the 2000s, or later. Hence, we begin our analysis in 2000 and end it in 2019, before the pandemic-induced global shock and the period of elevated inflation rates.Footnote ⁵ Table A.3 in the Appendix lists all the countries included in the study. The average year of IT adoption is approximately 2003 with a standard deviation of about 5.6 years, and the median year is 2001. The earliest IT adopters are Canada and the UK (in 1991 and 1992, respectively), while the most recent one is Moldova (in 2013).

Including both the Federal Reserve and the European Central Bank may be considered contentious, as their classification as inflation-targeting central banks remains ambiguous. Both institutions, however, are often characterised as de facto, rather than de jure, inflation targeters. Goodfriend (Reference Goodfriend and Goodfriend2004, p. 334) writes well before 2012, when the Fed officially adopted the IT, that “the Greenspan Fed practices inflation targeting implicitly,” arguing that “the Fed should continue to utilise the inflation-targeting procedures developed and employed implicitly by the Greenspan Fed after Chairman Greenspan retires.” Similar statements can be found in numerous authors describing both the Fed and the ECB. For example, Carare and Stone (Reference Carare and Stone2003, p. 7) treat the US as an inflation-targeting country “without a clear commitment” whereas Alichi et al. (Reference Alichi, Clinton, Freedman, Kamenik, Juillard, Laxton, Turunen and Wang2015) argue that the Fed is “functionally very close to flexible inflation targeting […] as the basis for monetary policy, although official statements of the Federal Reserve carefully avoid using such terminology.” On the other hand, despite the changes introduced in 2012 and 2020, the IMF continues to classify the US under the “other monetary framework” category rather than the “inflation-targeting framework” (see the IMF’s AREAER online database). We believe that our classification, being a de facto classification, provides a good enough reason to include both central banks in the analysis.

The selection of ARFIMA( $p,d,q$ ) models for each country’s inflation rate series follows a systematic procedure to ensure the best-fitting specifications. In the first step, we estimate a range of ARFIMA models, where the autoregressive (AR) and moving average (MA) lag lengths are chosen, and the ARFIMA parameters, including the fractional differencing parameter, are estimated. The models are selected by minimising the Akaike Information Criterion (AIC), with $p$ and q restricted to a maximum value of one.Footnote ⁶ Next, we assess the properties of the selected models. Specifically, we flag cases where (i) the estimated AR(1) coefficient is close to unity, (ii) the Ljung-Box and Breusch-Pagan diagnostic tests indicate autocorrelation in the residuals, or (iii) the confidence bands on the FI parameter are extremely wide.Footnote ⁷ If either of these issues arises, we repeat the model search with $p$ and $q$ still limited to a maximum value of 1. For models not affected by these issues, we re-evaluate the specifications to confirm their suitability. In the rare cases where satisfactory models cannot be identified, we expand the search to explore alternative combinations of lags. Whenever possible, we estimate the ARFIMA models using the maximum modified profile likelihood estimator, which has been shown to reduce bias in the presence of exogenous variables (including constants) in small samples. Only if the estimation algorithm fails to converge, we resort to the standard maximum likelihood estimator with robust standard errors.

4. Baseline results and discussion

This section presents and discusses our baseline results. The first step of our model selection procedure proved sufficient to select plausible ARFIMA specifications for 31 out of 36 countries. Hence, only a handful of cases required searching for alternative lag structures or using the maximum likelihood estimator. Full details on the estimated models are provided in Table A.4 in the Appendix. Table 1 shows the resulting point estimates of the FI parameter of the inflation rates, along with the 95-percent confidence intervals. We first notice that the point estimate of $d$ reveals substantial variation across countries, ranging from −0.28 in Norway to 0.49 in Romania, which lends support to the conjecture that there exists more than a single de facto IT regime. Second, the precision of estimates is not uniform: for countries like Hungary and Romania, the confidence interval is relatively narrow, whereas at the other end of the spectrum, e.g. in Albania and Norway, it is five times wider. Fortunately, classifying the latter countries is rather unambiguous, except for several cases, which we discuss below. Third, in only seven out of 36 countries do the confidence intervals include the integer $d$ . For other countries, the confidence intervals cover only non-integer numbers, indicating that no conventional ARMA model would be capable of correctly mirroring the dynamics of inflation. Fourth, the point estimates of $d$ reveal that in all countries the inflation rate is not only mean-reverting but also stationary. Even taking a conservative stand and looking at the upper bound of the confidence interval, we can still argue the same, but we need to admit nonstationarity in ten countries (the bound is 0.5 or more).

Table 1. Baseline estimates of fractional integration parameters of inflation rates in IT economies

Notes: The table displays the country codes, along with the point estimates and 95-percent confidence intervals of the fractional integration parameter of inflation rates obtained using ARFIMA ( $p$ , $d$ , $q$ ) models.

Table 2. “Shades” of inflation targeting across countries: baseline estimates

Notes: The table assigns one type of IT to each country, according to the baseline estimates.

To facilitate the discussion of the results, Table 2 assigns each country to one of four types of IT, while Figure 2 shows a coefficient plot, ordered from lowest to highest upper bound of the 95-percent confidence interval within the IT groups imposed. Using the time series properties of the inflation rate implied by the estimates of the fractional differencing parameter, we classify inflation-targeting countries into four categories discussed in the previous sections.

Starting with the first category of IT regimes, we find only two countries in the sample to display anti-persistence in the inflation rates, Norway and Canada. The negative values of FI imply that the autocorrelations (for lags greater than 0) of the inflation rate are negative, so the mean reversion is faster than that of the white noise. Importantly, the negative dependencies in the inflation rate alleviate or even wipe out the long-term impact of shocks on the price level, contributing to maintaining it on or close to the initial trajectory (see, e.g., Masson and Shukayev, Reference Masson and Shukayev2011; Diwan et al. Reference Diwan, Leduc and Mertens2020). This observation prompts us to label this group of countries as following the average IT.

The second category includes seven countries, in which the inflation rate displays short memory. Among them, we have Israel, an early adopter of IT, Japan and the United States, which have been long considered as pursuing an eclectic IT (Carare and Stone, Reference Carare and Stone2003), and somewhat surprisingly, countries like Armenia, Georgia, Thailand, and Peru with relatively less transparent monetary policy (Dincer et al. Reference Dincer, Eichengreen and Geraats2022; Niedźwiedzińska, Reference Niedźwiedzińska2022; Stone, Reference Stone2003). The estimate of the $d$ parameter is statistically insignificant, implying that inflation can be thought of as an I(0) process. The long-range dependencies captured by the fractional-integration parameter are non-existent, so the inflation rate reverts relatively quickly to its mean or target. Even though shocks produce no long-lasting effects on inflation, they shift the price level path, making it non-trend stationary. Letting the “bygones be bygones” is a prominent characteristic of standard IT strategy (see, e.g., Bernanke et al. Reference Bernanke, Laubach, Mishkin and Posen1999; McKibbin and Panton, Reference McKibbin and Panton2018), which we dub here strict or orthodox IT.

Figure 2. Fractional integration parameters: baseline estimates and confidence intervals.

Notes: The figure displays 95-percent confidence of the fractional integration parameters based on the ARFIMA models. The countries are sorted into four “shades” of IT, depicted by coloured bars. See Figure 1 and the main text for a further discussion of the classification.

The largest group encompasses 17 countries with a positive estimate of $d$ with the upper bound clearly below 0.5, i.e. in the range of stationarity. The typical examples of inflation targeters, such as Sweden and the United Kingdom among AEs and Chile and Hungary among EMEs, fall into this category. Some less obvious candidates are Colombia, Guatemala, and Uganda with relatively non-transparent monetary policies (Dincer et al. Reference Dincer, Eichengreen and Geraats2022; Niedźwiedzińska, Reference Niedźwiedzińska2022). Unlike the previous category, under this one, the inflation rate has a long memory and is persistent, meaning that shocks in the distant past still exhibit some influence on the dynamics of the process (Canarella and Miller, Reference Canarella and Miller2017b). Even though their effects decay slowly, at a hyperbolic rate, they dissipate fast enough to keep the variance of the inflation rate finite. The non-negligible role of long-range dependencies in the inflation rate gives rise to the conjecture that monetary authorities follow their IT strategy more flexibly than those pursuing the strict IT. For this reason, we call this type of IT flexible or conventional.

The last group, which we dub the uncommitted IT, is populated by ten EMEs often perceived as vulnerable, such as Brazil and South Africa, along with the post-transition Central and Eastern European countries, Romania and Serbia. The FI coefficient is positive and lies close to the region of nonstationarity: the upper bound of the confidence interval is at least 0.5. The effects of shocks disappear even more slowly than in the previous group of countries, making the inflation rate highly persistent. At a 5% significance level, we cannot rule out that the $d$ coefficient is 0.5 or greater. Even though the inflation rate has infinite variance in such a case, it remains a mean-reverting process (since the upper bound is well below 1) (Granger and Joyeux, Reference Granger and Joyeux1980). High persistence of inflation coupled with lengthy mean-reversion is likely to signal the ineffectively pursued IT strategy and induce us to label it as uncommitted or ineffective.

We realise that the classification is not perfect, given the variation in the precision of estimates of the FI coefficient. The issue, however, seems to be of secondary importance, since it is limited to five cases. The first two are Israel and Peru, classified into the strict IT category despite the fractional coefficient being well above 0, at the level characteristic of countries like Sweden and Switzerland, which are in the flexible IT group. Admittedly, the issue here is that the estimation is not precise enough, and the lower bound of the confidence interval is marginally below 0. Employing a 90% confidence interval would shift Israel and Peru to flexible IT (see Figure A.2 in the Appendix). The remaining three cases, the Dominican Republic, Poland, and Paraguay, fall into the uncommitted IT type due to the relatively wide confidence intervals. It is worth noting that the point estimates of $d$ for these countries are smaller than for Iceland, a country classified as following the flexible IT. Thus, marginally more precise estimates would shift these countries, enlarging the conventional IT category. Let us emphasise that, rather than jeopardising our approach, potential deficiencies in our classification encourage us to interpret the borderline cases with caution, the caveat relevant for any classification. The robustness of our results will be discussed in further detail in the section on sensitivity analyses.

To illustrate the differences among the four categories of IT regimes, in Figure 3, we plot impulse response functions (IRFs) of inflation rates to a one-point shock implied by the ARFIMA models. The diagrammatic exposition includes two representative examples of each IT type.

Figure 3. Impulse response functions of inflation rates in ARFIMA models in selected IT economies.

Notes: The figure displays impulse response functions of the inflation rate to a one-unit shock, based on the estimated ARFIMA models for selected economies. Two representative economies are shown for each IT category. The shaded bands show 95-percent confidence intervals around the base estimates.

The IRFs share a similar shape: after an initial rise, the inflation rate decreases and returns to its long-term level. It is in line with the mean-reverting property of the inflation rate, without which the monetary policy framework could not be considered any de facto IT.

The notable difference between IRFs is in the pace at which the effect of a shock peters out. The response under the AIT regime quickly becomes significantly negative and then gradually approaches zero, offsetting the initial rise in inflation. The implication is that some portion of an increase in the price level is reversed. Both in Canada and Norway, the inflation rate displays symptoms of anti-persistence.

The next two countries, Japan and Israel, belong to the strict IT group. The effect of shocks is short-lived, and there is a quick reversion to zero, which takes place just after several months. Even though the IRF does not move outside a positive territory, the response of inflation decays exponentially, in a way characteristic of non-persistent I(0) processes. Accordingly, the shock has no longer-term effects on the inflation rate, albeit the price level rises permanently.

The responses of countries in the flexible IT group resemble those in the previous case, albeit this time, shocks dissipate visibly slower, at a hyperbolic rate rather than an exponential decay. Inflation rates in the UK and Mexico exhibit greater persistence and tend to remain away from their “equilibrium” levels for extended periods. In other words, unlike the previous category, which captures short-memory processes, this one is characterised by long-memory processes.

The last two countries, Brazil and South Africa, illustrate the uncommitted IT. Similarly to the previous case, the inflation rate is a long-memory process, and its autocorrelation function decays at a slow hyperbolic rate. The main difference, however, is that the response is substantially stretched over time, and its confidence band remains above zero 15 months after the shock. This case is the only one in which the inflation rate is nonstationary, albeit it is mean-reverting.

Another way to show the difference between the IT types is by exploiting the frequency domain. The spectral density describes the relative contribution of periodic components at different frequencies to the variance of the process. Figure 4 illustrates the long-memory and short-memory spectral densities across the alternative IT categories. The former density describes the fractionally integrated series, whereas the latter portrays the fractionally differenced series.

Figure 4. Spectral density plots in selected IT economies.

Notes: The figure displays the spectral density plots of the estimated ARFIMA processes of the inflation rates in selected economies. Two representative economies are shown for each IT category. Horizontal axes represent the frequency of the inflation rate series. Solid lines denote the long-memory components, while dashed lines show the short-memory spectral density. Note that the scale on the vertical axes for BRA and ZAF differ from the remaining cases.

In theory, when the process has a short memory ( $d=0$ ), there is no difference between these spectra. This is the case of Japan and Israel, or more generally, the group of strict IT, which includes countries with the fractional coefficient close to 0.

The picture for the other groups of IT is different: the two spectra diverge at low frequencies because the inflation rate has a long memory. It aligns with the observation that the usual ARMA model can closely approximate the spectrum of the process with fractional $d$ “at all frequencies except those near zero” (Granger and Joyeux, Reference Granger and Joyeux1980). In flexible IT countries like the United Kingdom and Mexico, a part of the spectrum at low frequencies substantially contributes to the variance, signalling the persistence of the inflation rate that cannot be captured well by the conventional ARMA model with the integer $d$ . The uncommitted IT, as exemplified by the South African and Brazilian cases, is marked by the dominance of (a pole on) a low-frequency part of the spectral density. The contribution of short-term cyclical components to the variance is almost non-existent, which is in line with uncovering the elevated persistence of the inflation rate in this IT category. Long-memory and short-memory spectra are also different in the group of average IT, as exemplified by Canada and Norway. This time, however, the cyclical components at a frequency close to 0 have a negligible contribution to the variance, enabling the shocks to dissipate faster than for a short-memory process. Being anti-persistent, inflation quickly “forgets” the shocks and secures the price level stability.

5. Sensitivity analyses

This section discusses a set of sensitivity checks performed on the baseline results. We put forward four sensitivity checks. In each case, we re-estimate the ARFMIA models for each country, employing a modified sample or a different estimator, and compare FI coefficients with those obtained in the baseline case. In the sensitivity checks based on the ARFIMA models, we use specifications analogous to the baseline.Footnote ⁸ Next, we compare the alternative IT classifications with the baseline using three conventional similarity measures, i.e. accuracy, the adjusted Rand index, and Cohen’s kappa.

Figure 5. Sensitivity analysis of the baseline fractional integration parameter estimates: Part 1.

Notes: The figure displays 95-percent confidence intervals around the point estimates of the fractional integration parameters for the “Baseline” specification (as described in Section 4) and two sensitivity checks. “Winsorized data” shows the results for the CPI inflation series winsorized at the 5 and 95 percentiles. “Extended time coverage” denotes the results based on the ARFIMA models estimated on the sample that includes the post-Covid-19 period, 2000M1–2023M12.

Figure 6. Sensitivity analysis of the baseline fractional integration parameter estimates: Part 2.

Notes: The figure displays 95-percent confidence intervals around the point estimates of the fractional integration parameters for the “Baseline” specification (as described in Section 4) and two sensitivity checks. “Detrended data” shows the results based on the ARFIMA models estimated on the CPI the inflation rates with the removed linear trend. “MLP estimator” indicates the fractional integration estimates using the modified log periodogram estimator with the power of the bandwidth $T^{\alpha }$ of $\alpha =0.75$ .

First, we re-estimate the ARFIMA models using the winsorized inflation rate series. We trim and replace the extreme observations at the 5 and 95 percentiles for each economy to investigate whether our estimates are not driven by outliers, which may be especially problematic in EMEs. Even though winsorization is a mechanical procedure, it mitigates two issues. First, extreme monthly inflation values, often reflecting data anomalies, can bias persistence estimates by introducing artificial long-range dependence. Second, ARFIMA estimators can be sensitive to tail behaviour (for example, low inflation rates during the Global Financial Crisis), and small-sample distortions from rare inflation spikes may affect the estimated order of integration. The results of this sensitivity check reveal relatively stable estimates compared with the baseline, suggesting our findings are robust to such anomalies. While the winsorization reduces some of the variance in the FI estimates, particularly in emerging markets where inflation volatility is higher, the point estimates themselves remain close to the baseline. This confirms that outliers do not substantially distort the estimates, although their presence slightly inflates the confidence intervals (CIs) in the baseline. With winsorized data, the CIs for several countries narrow, particularly in economies with more volatile inflation, indicating that removing extreme values reduces the estimation uncertainty.

Second, we extend the data coverage of the inflation rates to include the post-Covid-19 period and estimate the ARFIMA model for the timespan 2000M1–2023M12.Footnote ⁹ This extension enables us to assess the stability of the classification in the presence of a major structural shift in inflation rates. The updated estimates show a noticeable increase in the FI parameters for many countries, likely reflecting the persistent inflationary effects triggered by the pandemic and its aftermath, which affected both advanced and EMEs. The inclusion of the post-Covid period also widens the confidence intervals in several cases, reflecting the heightened uncertainty during this time. The broader CIs suggest that inflation persistence during the pandemic period was more difficult to estimate precisely, particularly in economies that experienced severe shocks.

Third, we test an alternative series preparation by detrending the inflation rate series. The inflation rates used in the benchmark estimation are detrended and demeaned with a linear trend. In this sensitivity check, the point estimates remain quite close to the baseline, indicating that inflation persistence is not significantly affected by long-term trends in the data. The detrending process appears to have a minimal impact on the overall dynamics captured by the ARFIMA models. The confidence intervals are generally comparable to the baseline, with some slight narrowing in countries with stable inflation rates. It suggests that while some trends may be present in the data, they do not drive the core persistence patterns of inflation. The results confirm that short- and medium-term dynamics are more critical to understanding inflation persistence than long-term trends.

Finally, we re-estimate the models using the MLP estimator, with a bandwidth parameter of $\alpha =0.75$ , as recommended by Phillips (Reference Phillips2007). A key distinction from the estimator used to obtain the baseline results is that the MLP estimation does not require an explicit specification of the ARMA part and accommodates a broader range of values for the FI parameter, including those exceeding 0.5. The results obtained using the MLP estimator show greater deviations from the baseline, with greater spreads in the FI estimates. In several countries, MLP estimates indicate higher levels of persistence than the baseline results, suggesting that the MLP method captures longer memory processes that are not fully reflected in the ARMA-based specification. The confidence intervals are also wider, particularly in countries with more complex inflation dynamics, where the MLP estimator captures a broader range of potential values for the FI parameter.

To summarise the results of the sensitivity analyses, we weigh the classification obtained in each check against the baseline. The straightforward way to do this is to use the accuracy metric. It is a simple measure of agreement defined as the number of countries classified in the same category as in the baseline case divided by the total number of countries. The results are reported in Table 3. Neither winsorizing nor detrending data substantially alters the classification much: fewer than ten countries are reassigned to a different category compared to the baseline, and the accuracy metric remains at 75% or higher. The accuracy of the classification derived under the extended sample is somewhat lower, at 67%. Still, the consistency with the baseline results remains solid, especially given that the extended period includes the post-Covid-19 years, which introduced a major inflation shock and potential structural breaks in inflation dynamics. The classification derived by the MLP estimator has the lowest accuracy, which, however, is well above 50%. A part of the disagreement likely stems from the fact that the MLP estimator employs detrended data. Indeed, when we checked the agreement with the classification obtained on the detrended data, the accuracy metric increased to 61% (see Table A.5 in the Appendix).

Table 3. Sensitivity analysis: comparison of monetary policy regime classifications against the baseline

Notes: The table displays the pairwise comparison between classifications obtained in the baseline case and four sensitivity checks using four metrics. All kappa statistics have p-values smaller or equal than 0.01. See also Table A.5 in the Appendix.

One of the weaknesses of the accuracy metric is that it masks imbalances between categories. If, for example, we arbitrarily assigned all countries to the conventional IT, i.e., the category which dominates in the baseline classification, the accuracy would be 47%. To avoid this issue, we calculate the adjusted Rand index. It is a measure of agreement based on counting pairs of objects. In general, the index lies between 0, indicating a random agreement, and 1, denoting a perfect agreement (see, e.g., Warrens and van der Hoef, Reference Warrens and van der Hoef2022). In our hypothetical example, which assigns all countries into a single class, the adjusted Rand index is 0.

The classifications obtained under sensitivity checks are much more similar to the baseline than the uniform assignment. Using this metric, we see that classifications based on detrended or winsorized data are the least different from the baseline. When we employ the extended sample or the MLP estimator, the index deteriorates to 0.30 and 0.15, respectively. Interestingly, the low value of the index in the latter case seems to be driven by detrending data under the MLP estimator. When we compare this classification with the one based on the detrended sample, the index goes up to 0.33 (Table A.5 in the Appendix). Overall, the picture is not that different from the one based on the crude accuracy measure.

The drawback of the adjusted Rand index in our context is that it neglects the labels attached to the categories. If, for example, we arbitrarily reclassified each country one category up, i.e., from AIT to strict IT, from strict IT to conventional IT, and so on, and compared this classification with the baseline, the adjusted Rand index would be 1. Thus, to sort out this problem, we turn to Cohen’s kappa. Being the measure of the degree of agreement between two classifications, it seems well fitted to our comparisons. Cohen’s kappa values range from −1 to 1. The value of 0 indicates a random agreement between the classifications, and negative (positive) values denote less (more) agreement than random chance. For example, Cohen’s kappa for the arbitrary reclassification of all countries to the conventional IT category is −0.33.

Table 3 also reports two kappa coefficients, unweighted and weighted. The former is suitable for the case with classes corresponding to a nominal variable, whereas the latter is relevant when classes can be ordered (Sim and Wright, Reference Sim and Wright2005). The unweighted kappas obtained under four sensitivity checks tell the same story as the two other similarity metrics, although in a more reliable way. The weighted kappas are better suited to our IT classifications because they account for the ordering of categories. The kappa coefficients of more than 0.6 for the classifications under winsorized or detrended data indicate substantial agreement with the baseline.Footnote ¹⁰ For the other two classifications, the agreement is moderate. Noteworthily, the agreement between classifications derived under the MLP estimator and the detrended data is much higher, with Cohen’s kappa of almost 0.6, and is at the border of being substantial. Moreover, in Table A.5 in the Appendix, we report kappas under quadratic weighting, which strongly penalises larger discrepancies between classifications. All coefficients indicate substantial agreement with the baseline, except for the classification obtained with the MPL estimator, where the kappa value is marginally below 0.6.

In general, sensitivity checks confirm the robustness of the baseline classification while highlighting the impact of methodological variations.

6. Institutional monetary policy features and the inflation targeting classification: a cross-sectional analysis

In the previous sections, we document substantial cross-country heterogeneity in the FI parameters of inflation rates, which enables us to classify economies into four distinct monetary policy strategy categories: AIT, strict IT, flexible IT, and uncommitted IT. This section extends the analysis by focusing on the underlying factors that may explain why different countries are classified into these categories. Our main goal is to explore the institutional features of monetary policymaking that contribute to this variation. By examining these factors, we aim to understand what influences certain economies to demonstrate higher or lower persistence in inflation, and why they align with one of the four identified monetary policy strategies. This part of the analysis should also provide insights into the broader frameworks that shape a central bank’s ability to anchor inflation and respond to inflationary shocks.

Among the potential factors explaining the variation in FI of inflation rates, we first consider the overall level of economic development, proxied by GDP per capita in purchasing power parity and sourced from the World Bank WDI database. The level of development is often linked to more advanced monetary policy institutions and central bank capabilities, which may influence differences in the persistence of inflation rates across economies. Next, we consider five key factors related to the institutional monetary policy design: (i) the maturity of the IT regime, measured as the total number of years under IT until 2019, (ii) the stability of the inflation target definition, represented as the negative of the number of changes in the target during the study period, (iii) central bank independence, based on the de jure independence indices from Romelli (Reference Romelli2022), (iv) central bank transparency, based on the Dincer and Eichengreen (Reference Dincer and Eichengreen2014) index, which draws from various central bank documents (see also Dincer et al. Reference Dincer, Eichengreen and Geraats2022),Footnote ¹¹ and (v) the dual objective of internal price stability versus external exchange-rate stabilisation, measured as the share of years under a “floating” or “free floating” exchange rate regime for each economy, using the IMF classification. Note that these variables are designed to reflect characteristics typically associated with more effective inflation control, corresponding to countries classified into IT categories with lower values of the FI parameter $d$ .

Table A.6 in the Appendix summarises the baseline and alternative sets of covariates, including their definitions, sources, and major instances of missing data. Descriptive statistics for all variables are reported in Table A.7, while Figure A.3 provides boxplots illustrating their distribution. Figure 7 presents the correlation matrix between types of monetary policy strategy, FI parameters of inflation rates, and country-level variables. The results highlight several key relationships. GDP per capita shows a strong negative correlation with the type of monetary policy strategy and the parameter $d$ , suggesting that higher levels of development are associated with less persistent inflation and “lower” categories of IT (e.g., AIT or strict IT). Additionally, GDP per capita is positively correlated with institutional features of central banks, such as transparency and stability in policy targets, which further explains why more developed economies tend to exhibit lower inflation persistence. In contrast, frequent changes in the IT target are positively correlated with higher $d$ values, indicating more persistent inflation and “higher” IT categories (e.g., flexible or uncommitted IT).

Figure 7. Correlation matrix of variants of inflation targeting, fractional integration of inflation rates, and country-level covariates.

Notes: The figure plots the correlation matrix between the baseline classification of monetary policy strategies or “shades” of IT, the fractional integration parameter $d$ , and a set of country-level variables. For the definitions and sources of variables, see the discussion in Section 6.

The ordinal probit model is employed to examine the relationships between country-level variables and the “shades” of IT, with heteroscedasticity-robust standard errors used to correct for non-constant variance in residuals. This model treats the dependent variable, our four types of IT, as an ordinal variable, where lower values (AIT, strict IT) are associated with better inflation control, and higher values (flexible, uncommitted IT) correspond to greater inflation persistence. The classification reflects a degree of policy commitment to inflation targeting, but without assuming that the distances between categories are equal. The ordered probit model is thus well suited to this setting, as it captures the ordinal nature of the outcome while avoiding unjustified cardinal assumptions. It allows us to show how institutional and economic factors affect the probability of a country being classified into one of the four IT categories.

Table 4 reports the baseline results of the ordinal probit regressions. A key finding is the significant negative relationships between GDP per capita and years under IT with inflation persistence, confirming that higher levels of development and longer experience with IT are associated with a higher probability of moving from uncommitted to conventional and strict IT. From an institutional perspective, this suggests that wealthier economies tend to have stronger monetary institutions that can more effectively anchor inflation expectations and manage shocks. Longer IT maturity further reflects institutional credibility, as sustained inflation targeting builds trust in the central bank’s policies. Economically, these findings indicate that countries with more advanced economic frameworks are better equipped to contain inflationary pressures. The significant coefficient estimates on the number of changes in inflation targets and exchange-rate arrangements highlight the importance of consistent policy objectives in explaining variation in monetary policy strategies. In contrast, central bank independence, though theoretically important, does not reveal a significant relationship with the “shade” of IT, suggesting that de jure independence alone may not translate directly into effective inflation control. We similarly find no significant association between monetary policy transparency and the classification of monetary policy strategies.

Table 4. Covariates of the monetary policy regime classification: baseline ordinal probit regressions

Notes: The table shows the baseline estimation results of the ordinal probit models. Dependent variable: the IT variant under the baseline monetary policy strategy classification derived from the ARFIMA models of inflation rates. See the main text for the definitions of explanatory variables. Heteroscedasticity-robust standard errors are given in brackets. * and ^** denote statistical significance at 0.1 and 0.05 levels, respectively.

The pseudo R-squared values for the baseline specifications suggest that the covariates explain part of the variation in monetary policy classifications, while leaving room for other influences, such as external shocks, institutional nuance, or country-specific policy setups. Model specifications that include multiple covariates do not substantially improve explanatory power and raise concerns about collinearity, particularly among institution-related variables. Hence, we opted for a simpler strategy based on individual regressions, which maximises sample size and enhances the interpretability of individual covariate effects. However, to complement the baseline results, we provide additional results using alternative covariates and different estimators that employ a continuous dependent variable based directly on the FI estimates.

In Table 5, we explore alternative definitions of the country-level variables introduced in the ordered probit models. We use the log values of GDP per capita and years under IT to account for potential nonlinearities. The standard deviation of target changes replaces the number of changes to better capture the volatility of the inflation target. The measure of central bank independence is now based on the Grilli et al. (Reference Grilli, Masciandaro and Tabellini1991) classification, updated by Romelli (Reference Romelli2022).Footnote ¹² Central bank transparency is sourced from Niedźwiedzińska (Reference Niedźwiedzińska2022), and the exchange-rate regime is captured by the classification from Dąbrowski et al. (Reference Dąbrowski, Papież and Śmiech2020), which defines the share of years during which a country is classified as “float.” The results remain consistent with the baseline findings. Specifically, GDP per capita, variability of inflation targets, and the adoption of floating exchange rates continue to show significant negative relationships with the IT categories. These alternative measures reinforce the idea that economic development, stability in monetary frameworks, and consistent policy objectives0 are associated with more committed strategies. Evidence on the role of years under IT is somewhat weaker than in the baseline, but the coefficient estimate on the log of IT year is only marginally insignificant at the 0.1 level. Interestingly, we find a significant coefficient on the alternative monetary policy transparency variable, pointing to a relationship between this dimension of policymaking and stricter IT strategies.Footnote ¹³

Table 5. Covariates of the monetary policy regime classification: ordinal probit results using alternative explanatory variables

Notes: The table shows the estimation results of the ordinal probit using the alternative set of explanatory variables. Dependent variable: the IT variant under the baseline monetary policy strategy classification derived from the ARFIMA models of inflation rates. See the main text for the definitions of explanatory variables. Heteroscedasticity-robust standard errors are given in brackets. * and ^** denote statistical significance at 0.1 and 0.05 levels, respectively.

Finally, Table A.8 in the Appendix presents the weighted least squares (WLS) results, where the weights are the inverse of the standard errors of the $d$ parameter from the FI estimation.Footnote ¹⁴ The WLS regression uses the estimated $d$ values as the dependent variable, with weights based on the error variance from the ARFIMA model, and Figure A.4 in the Appendix shows the corresponding scatterplots and the fitted regression lines. The outcome of the WLS estimations supports our results. The relationships between GDP per capita, IT maturity and stability, and floating exchange-rate regimes with IT strategies remain significantly negative, although the explanatory power of each of these factors is limited. Importantly, as shown in the scatterplots, the findings are not driven by outliers, adding confidence to the robustness of these institutional and economic factors in explaining inflation persistence across countries.