A model for ordinal responses with heterogeneous status quo outcomes

2.1 Relation and contribution to the literature

From a macroeconomic point of view, the proposed CronOP model is related to a vast literature on monetary policy regime switching, inspired by the seminal work of Hamilton (1989). Oscillating switches in policy regimes are typically modelled by a stochastic Markov-chain process, which is exogenous to the rest of the model. The monetary policy rule in this literature is typically represented by a model for continuous outcomes. In the real world, the major central banks move policy rates by discrete increments, and the policy regime changes are endogenous to the economic conditions, which are actively monitored by the central banks. To the best of my knowledge, the CronOP model is the first implementation of discrete-choice monetary policy rule with (observationally driven) regime switching. The existing applications of discrete-choice approach to monetary policy rules (Hamilton and Jorda, 2002; Hu and Phillips, 2004; Dolado, Maria-Dolores and Naveira, 2005; Piazzesi, 2005; Basu and de Jong, 2007; Kauppi, 2012; Van den Hauwe and Dijk, 2013) do not allow for a regime-switching behavior.

The CronOP model can be described as a cross-nested generalization of a two-level nested ordered probit (NOP) model with three nests (see the top right panel of Figure 2). In the case of unordered categorical data, when choices can be grouped into the nests of similar options, the nested logit model is a popular method. Various types of multinomial logit models with overlapping nests are proposed (Vovsha, 1997; Wen and Koppelman, 2001). The (cross)-nested models, specifically designed for the ordered alternatives, are scarce.^[1] The difference between the decision trees of the NOP and CronOP models is that all three nests of the latter overlap at a status-quo response. In the former, the decisions at both levels are observable – it is always known to which nest the observed outcome belongs, whereas in the latter, the zeros are observationally equivalent – it is never known from which of the three regimes the status quo outcomes originate.

The CronOP model can also be considered as a three-part zero-inflated model. The two-part (or hurdle) model introduced by Cragg (1971) combines a binary choice model for the probability of crossing the hurdle (the regime decision) with a truncated-at-zero model for the outcomes above the hurdle (the amount decision). An example of the two-part model for time series data (such as discrete-valued changes to the federal funds rate target) is the autoregressive conditional hazard (ACH) model of Hamilton and Jorda (2002) (see the right bottom panel of Figure 2). The zero-inflated models – the natural extensions of the two-part models – are developed to address the unobserved heterogeneity of (typically) abundant zero outcomes. The examples are the zero-inflated ordered probit (ZIOP) model of Harris and Zhao (2007) for non-negative ordinal responses, where abundant zeros are situated at one end of the ordered scale, and the middle-inflated ordered probit (MIOP) model of Bagozzi and Mukherjee (2012) and Brooks, Harris, and Spencer (2012) for ordinal outcomes that range from negative to positive responses, and where abundant zeros are situated in the middle of the choice spectrum (see the left bottom panel of Figure 2). The difference between the ACH and MIOP models is that the two parts are separately estimated in the former and the zero observations are excluded from the second part; thus, the discrimination among the different types of zeros is not accommodated. In the latter, the two nests overlap assuming the two types of zeros; thus, the probability of a zero is “inflated”.

The three-part CronOP model is a natural extension of the two-part ACH, ZIOP and MIOP models. A trichotomous participation decision (“an increase or no change”, “no change”, or “a decrease or no change”) seems to be more realistic than a binary decision (“a change” or “no change”) if applied to ordinal data that assume negative, zero and positive values. The policymakers, who are willing to adjust the rate, have already decided in which direction they wish to move it. As a practical matter, the decision to lower or increase the rate can be influenced by the direction-specific determinants and differently by the same ones. The empirical rejection of the MIOP in favor of the CronOP model suggests that the effects of the explanatory variables on the amount decisions are asymmetric; combining these two distinct decisions (“an increase” or “a decrease”) into one branch of the decision tree, as implemented in the two-part models, may seriously distort an inference.

2.2 The cross-nested ordered probit model

I describe the econometric framework of the CronOP model in a panel-data context using a double subscript, where the index i denotes one of N cross-sectional units and the index t denotes one of T time periods. An application to cross-sectional or time-series data is straightforward by setting T or N to one. For ease of exposition and without loss of generality, the observed dependent variable Δy_it is assumed to take on a finite number of integer values j coded as { − J , … , 0 , … , J } , where J ≥ 1 and a heterogeneous (inflated) response is coded as zero. The inflated outcome does not have to be in the very middle of ordered categories. However, if it is located at the end of the ordered scale, the CronOP model reduces to the ZIOP model of Harris and Zhao (2007).

I interpret the model in the context of interest rate decisions that are taken repeatedly at the policy-making meetings by each member of a monetary policy committee. The model assumes three regimes. The regime decision – the upper level of the decision tree – is determined by the continuous latent variable r i t ∗ representing the degree of the policymaker i’s policy stance. It is set at the meeting t in response to the observed data according to a regime equation

where x _it is the t ^th row of the observed T i × K β data matrix X _i , which in addition to predetermined explanatory variables may also include the lags of observed policy choice Δy_it and individual dummies; T_i is the number of observations available for the individual i; 𝜷 is a K β × 1 vector of unknown coefficients; and ν _it is an error term that is independently and identically distributed (iid) across i and t.

A regime-setting decision r_it is coded as −1, 0, or 1, if the policymaker i’s policy stance is, respectively loose, neutral or tight. The correspondence between r i t ∗ and r_it is given by

where − ∞ < α 1 ≤ α 2 < ∞ are the unknown threshold parameters.

Under the assumption that ν _it is distributed according to the cumulative distribution function (cdf) Φ, the probabilities of each possible outcome of r_it are given by

At the lower level of the decision tree, three latent regimes exist. Conditional on being in the regime r_it = 0, no further actions are taken and the rate remains unchanged:

Thus, the conditional probability of the outcome j in the neutral policy stance is

Conditional on being in the regime r i t = − 1 or r_it = 1, the continuous latent variable Δ y i t ∗ representing the desired change to the rate, is determined by the direction-specific amount equations

where 𝜸 ⁻ and 𝜸 ⁺ are the K − × 1 and K + × 1 vectors of unknown coefficients; z i t − and z i t + are the t ^th rows of the observed T i × K − and T i × K + data matrices Z i − and Z i + , which may include the lags of Δy_it and individual dummies; and ε i t − and ε i t + are the iid error terms with the cdf Φ⁻ and Φ⁺, respectively.

The discrete change to the rate Δy_it is determined by

where − ∞ ≡ μ − J − 1 − ≤ μ − J − ≤ … ≤ μ 0 − ≡ ∞ and − ∞ ≡ μ 0 + ≤ μ 1 + ≤ … ≤ μ J + 1 + ≡ ∞ are the 2J unknown thresholds.

The conditional probability of the outcome j can be computed as

Assuming that the error terms ν _it , ε i t − and ε i t + are mutually independent, the full probabilities of the outcome j are given by combining probabilities in (2), (3) and (5):

where I _{j ≥ 0} is an indicator function such that I j ≥ 0 = 1 if j ≥ 0, and I j ≥ 0 = 0 if j < 0 (analogously for I_j=0 and I _{j ≤ 0}).

Each latent equation (which is a function of stationary exogenous variables, lags of the choice variable and a stationary shock) can be consistently estimated by asymptotically normal ML estimator under fairly general conditions (Basu and de Jong, 2007). To jointly estimate the three OP equations, as is common in the identification of discrete choice models, Φ, Φ⁻ and Φ⁺ are assumed to be standard normal, and the intercept components of 𝜷, 𝜸 ⁻ and 𝜸 ⁺ (which are identified up to scale and location, that is only jointly with the corresponding threshold parameters 𝜶, 𝝁 ⁻ , and 𝝁 ⁺ and variances of 𝝂, ε ⁻ and ε ⁺ ) are fixed to zero. As long as X, Z ⁻ or Z ⁺ contains a single regressor that varies over i or t, the probabilities in (6) are absolutely estimable. They are invariant to the identifying assumptions, and can be estimated using a pooled ML estimator of the vector of the parameters θ = ( α ′ , β ′ , μ − ′ , γ − ′ , μ + ′ , γ + ′ ) ′ that solves

where q_{it j} is an indicator function such that q_{it j} = 1 if Δy_it = j and q_{it j} = 0 otherwise.

All parameters in 𝜽 are separately identified (up to scale and location) via the functional form due to the nonlinearity of OP equations (Wilde 2000). In practice, however, the standard normal cdf is often an approximately linear function over an extensive range of its argument. Thus, the simultaneous estimation of the three equations may be subject to a weak identification problem, if X, Z ⁻ and Z ⁺ have all variables in common. In this case, exclusion restrictions may be necessary to avoid weak identification. To account for the individual heterogeneity, x _it , z i t − and z i t + may include the individual fixed effect dummies. The fixed effects are more appropriate in the policy-rate-setting context than the random effects because instead of a random sample of individuals we have a complete set of all MPC members.

The starting values for 𝜽 can be obtained using the independent OP estimations of each latent equation. The asymptotic standard errors of θ ^ can be computed from the Hessian matrix. The conducted Monte Carlo simulations (see Section 3) demonstrate that the proposed estimator (written using GAUSS programming language) does a good job in estimating parameters and their standard errors.

2.3 Allowing for correlation among regime and amount decisions

The CronOP model can be extended by relaxing the assumption that the mechanisms generating the regime and amount decisions are independent, i.e. that the error terms 𝝂, ε ⁻ and ε ⁺ are uncorrelated. To obtain the CronCOP model – a correlated version of the CronOP model – I assume that ( ν , ε − ) and ( ν , ε + ) follow the standardized bivariate normal distributions with the correlation coefficients ρ ⁻ and ρ ⁺ , respectively. This correlation may emerge, for instance, from common but unobserved covariates.

The probabilities of the outcome j for the CronCOP model are given by

where Φ is the cdf of the standard normal distribution, and Φ 2 ( ϕ 1 ; ϕ 2 ; ξ ) is the cdf of the standardized bivariate normal distribution of the two random variables ϕ ₁ and ϕ ₂ with the correlation coefficient ξ. To estimate the CronCOP model by ML, we have to solve (7) by replacing the probabilities in (6) with the probabilities in (8), and redefining the parameter vector as θ = ( α ′ , β ′ , μ − ′ , γ − ′ , μ + ′ , γ + ′ , ρ − , ρ + ) ′ . The starting values for ρ ⁻ and ρ ⁺ can be obtained using the grid search from −1 to 1 and maximizing the logarithm of the likelihood function holding the other parameters fixed at their CronOP estimates.

2.4 Marginal effects

The marginal effect (ME) of a continuous covariate on the probability of each discrete choice is computed as the partial derivative with respect to this covariate, holding all other covariates fixed. For a discrete-valued covariate, the ME is computed as the change in the probabilities when this covariate changes by one increment and all other covariates are fixed. To facilitate the ME derivation, the matrices of the covariates (where the subscript i is omitted for the sake of brevity) and the corresponding vectors of the parameters can be partitioned as

where A only includes the variables common to X, Z ⁻ and Z ⁺; P only includes the variables common to both X and Z ⁺, which are not in Z ⁻; M only includes the variables common to both X and Z ⁻, but not in Z ⁺; Z ± ~ only includes the variables common to both Z ⁻ and Z ⁺, but not in X; and X ~ , Z − ~ and Z + ~ only include the unique variables that only appear in one of the latent equations.

The matrix of all explanatory variables X ^∗ and the vectors of the parameters for X ^∗ can be written as

The MEs of the row vector x i t ∗ on the probabilities in (8) can be computed for the CronCOP model as

where f is the probability density function of the standard normal distribution. The MEs for the CronOP model are obtained by setting ρ − = ρ + = 0 . The asymptotic standard errors of the MEs are computed using the Delta method as the square roots of the diagonal elements of

4.1 Data and model specification

After the adoption of direct inflation targeting in 1998, the NBP policy rate – the reference rate^[2] – may be undoubtedly treated as a principal instrument of Polish monetary policy. The reference rate is administratively set by the MPC, which consists of ten members who make policy rate decisions by formal voting once per month (since 2010, 11 times per year). The MPC moves policy rates by discrete adjustments – multiples of 25 basis points (bp), i.e. a quarter of a percent. At a policy meeting, each MPC member can express his preferred adjustment to the rate and make a proposition to be voted on. The individual policy preferences (reported in Table B1 in Online Appendix B) are consolidated into three categories: “an increase”, “no change” and “a decrease”. Such classification reflects the higher volatility of policy interest rate and inflation before 2002, making a 250-bp change made before 2002 comparable with a 50-bp change made after 2002. The sample contains the desired policy actions expressed by 31 policymakers at 190 MPC meetings. Among the 1719 observations employed in the estimations, the policymakers preferred to leave the rate unchanged 1125 times (65%), to increase the rate 253 times (15%) and to reduce it 341 times (20%).

Table B2 in Online Appendix B provides the definitions and sources of all variables. The sample descriptive statistics are summarized in Table B3 in Online Appendix B. All employed macroeconomic variables are stationary at the 0.01 significance level according to the augmented Dickey-Fuller unit root test, as shown in Table B4 in Online Appendix B.

Given the NBP strategy of direct inflation targeting, the policy regime decision in the CronOP model is assumed to be driven by a direct reaction to the changes in the economic conditions controlled by: (i) Δcpi_t – the recent monthly change to the current rate of inflation; (ii) Δ c p i t t a r , which is equivalent to Δcpi_t if the inflation is above the target, and zero otherwise (to allow for an asymmetric reaction to inflation changes depending on whether the inflation rate is above or below the target); (iii) Δecbr_t – the change to the European Central Bank (ECB) policy rate made at the last policy meeting (as a proxy for the recent economic trends in the European Union); (iv-v) Δ c p i t t a r I t 2010 and Δ e c b r t I t 2010 , where I t 2010 is an indicator variable, which is one since February 2010, and zero otherwise (to allow for a different policy reaction in the post-crisis period during the third MPC term); (vi) spread_t – the spread between the long- and short-term market interest rates (as a low-dimension market-based aggregator of publicly available information on inflationary expectations that are not reflected in the current inflation rate); (vii) Δ r a t e i , t − 1 – the original (unconsolidated) change to the policy rate proposed by the MPC member i at the previous meeting (sequential decisions are not independent – the recent policy choice affects subsequent actions); and (viii) bias _t−1 – an indicator of the “policy bias” or “balance of risks” statements of the MPC at the previous meeting (to address the policymakers’ concerns about the competence and credibility of central bank communication). The expected sign of the coefficients on these variables is positive – the larger is the value of a covariate, the larger is the probability of a tight policy stance and the smaller is the probability of a loose stance.

The amount decisions, which fine-tune and smooth the rate, are conditional on the tight or loose policy stance and controlled by (i) Δ r a t e i , t − 1 (the larger is the hike/cut at the previous meeting, the lower is the probability of the second hike/cut in a row); (ii) bias _t−1 (the tightening/easing bias is expected to increase/decrease the probability of a higher rate); (iii) spread_t (the rate hike is much more likely if the 12-month interbank rate is above the 1-week rate, rather than vice versa); (iv) the indicator variable I t 2002 , which is one since April 2002, and zero otherwise (to account for higher levels and stronger moves in the inflation and the policy rate prior to April 2002); and I t 2010 (to allow for a different policy reaction during the third MPC term). The expected sign of the coefficients is positive for spread_t and bias _t−1, negative for Δ r a t e i , t − 1 , and should be opposite in the tight and loose regimes for I t 2002 and I t 2010 : a positive sign in the tight stance but a negative sign in the loose stance enable rate moves to be triggered by smaller changes to the explanatory variables after April 2002 and February 2010, respectively.

There are no interindividual differences in the values of the macroeconomic explanatory variables. To better account for the individual heterogeneity of policy preferences (not fully controlled by Δ r a t e i , t − 1 ), I augment this specification by including the individual fixed effect (FE) dummies. For parsimony reasons, I only allow for variation in the intercepts – thirty individual dummy variables are included in each latent equation.^[3] Slope heterogeneity is not a concern because our interest is in the estimation of the average effects of the explanatory variables, not of the individual policy reactions. Under the assumption that the slope coefficients randomly differ across the individuals, the pooled ML estimator yields consistent estimates of these aggregate effects while simultaneously providing a greater statistical power and a more reliable inference. We should not expect any significant fixed T asymptotic bias of our estimator – with our temporal size (T_i is 55 on average) we are in the realm of a time-series analysis.^[4]

The fixed effects are more appropriate than the random effects because we do not have a sample of individuals who were randomly obtained from a large population but instead possess a full set of the MPC members. However, the FE specification with its 110 parameters is likely subject to a weak identification problem: there are fewer than 16 observations per parameter in the sample, and the sets of the covariates in the three latent equations overlap substantially. To prevent an overparameterization and obtain more reliable estimates, I also estimated a more parsimonious specification. I constructed the individual-specific variable dissent_i,t , which indicates a direction of member i’s dissent at a meeting t: it is equal to 1/0/-1, if the member prefers the higher/same/lower rate than the MPC. The lags of dissent_i,t reflect the dynamics of the deviation of member i’s desired rate from the rate set by the MPC at the previous meetings. I included three lags in the regime equation, two lags in the amount equation under the loose regime, and three lags under the tight regime (with the expected positive sign of all coefficients – if a member preferred a higher/lower policy rate at the previous meeting, he is likely to be more hawkish/dovish at the subsequent meeting).^[5]

4.2 Estimation results

The parsimonious specification with three lags of dissent_i,t (see Table 2) adequately captures the heterogeneity of policy preferences: it has a slightly lower log likelihood than the FE specification (−629.2 vs. −626.2), but far fewer parameters (30 vs. 110), and is strongly preferred by the information criteria (the AIC is 1318 vs. 1472, the BIC is 1482 vs. 2072).

The FE specification is heavily overparameterized: the coefficients on 40 individual dummies are not significantly different from zero at the 0.05 level, as reported in Table B6 in Online Appendix B. Therefore, the specification with the lags of dissent_i,t in Table 2 (henceforth the baseline specification) was employed in the further analysis. It saves 80 degrees of freedom, has an advantage of a greater statistical power, and can produce more efficient estimates of interest. Importantly, the estimated policy reactions to economic conditions are robust to different ways of accounting for individual heterogeneity: the coefficients on all common variables in the regime equation and on all but three variables in the amount equations are remarkably similar in both specifications. All coefficients in the baseline specification have an expected sign and are significant at the 0.01 level, with the exception of the coefficient on d i s s e n t i , t − 2 (the p-value is 0.17) and Δ c p i t (the p-value is 0.35) in the regime equation. Our expectation that the policy reaction to changes in inflation depends on whether inflation level is above or below the target is confirmed: the reaction is not significant if inflation is below the target.

Observing a large fraction of zeros does not always indicate that existing models are unsuitable. We can test which alternative model is favored by the real-world data: (i) the conventional OP model including all covariates from the CronOP model (see Table B7 in Online Appendix B); (ii) the MIOP model in which its dichotomous regime equation includes all covariates in the CronOP dichotomous amount equations and its trichotomous amount equation includes all covariates in the CronOP trichotomous regime equation (see Table B7 in Online Appendix B); (iii) the CronOP model with the baseline specification (see Table 2); and (iv) the CronCOP model with the baseline specification (see Table B8 in Online Appendix B). The NOP model is not listed because, in the case of the three outcome categories, it reduces to the usual OP model.

Table 3 compares the five competing models. The two- and three-equation models demonstrate a significant increase in the likelihood compared to the single-equation OP model. The CronOP and CronCOP models are overwhelmingly superior to the OP and MIOP models according to both information criteria and are favored by the Vuong tests (the p-value is 10⁻²⁰). There is no significant difference between the likelihoods of the CronOP and CronCOP models according to the LR test (the p-value is 0.999). The CronOP model is preferred by both information criteria.

I also estimated it with the same set of variables in both amount equations (by including dissent _i,t−3 under the loose regime) to test whether the rate hikes and cuts are generated by different processes. In our case with only three outcome categories of the dependent variable, the CronOP nests the MIOP model under “symmetrical” restrictions on the parameters in the amount equations. The LR test strongly rejects the symmetrical restrictions and prefers the CronOP model (the p-value is 10⁻³⁷).

The CronOP model also demonstrates a substantial improvement in the percentage of correct predictions (for rate cuts and hikes) and noise-to-signal ratios (for cuts and zeros), as shown in Table 4. The noise-to-signal ratios for hikes and the hit rates for zeros are similar across the three models, although slightly better in the CronOP model. Interestingly, the OP and MOP models predict more zeros (1224 and 1228) than the CronOP model (1171), but they correctly predict only 977 and 1004 zeros, respectively, whereas the CronOP model correctly predicts 1005 zeros.

To give the MIOP model additional chances, I also estimated it (a) including all CronOP covariates in both parts (the log likelihood is −725.1; see Table B9 in Online Appendix B) and (b) taking additionally all covariates in the regime equation by their absolute values to account for the binary (“a change” or “no change”) nature of the regime decision (the log likelihood is −713.6; see Table B10 in Online Appendix B). In both cases, the CronOP remains overwhelmingly superior to the MIOP model according to the information criteria^[6] and Vuong tests (the p-value is 10⁻⁹).

The model comparison relies heavily on statistical criteria. Are we simply fine-tuning the OP and MIOP models or are the resulting improvements economically meaningful? The three models produce a conflicting inference and have incompatible and opposite estimates of the MEs of some explanatory variables on choice probabilities. The most striking differences across the models are in the effects of the previous change to the rate Δ r a t e i , t − 1 (see Table 5).

We expect a positive coefficient on Δ r a t e i , t − 1 in the OP model. In the case of a rate hike, the probability of a hike/cut at the next meeting should be larger/smaller than in the case of a cut. The coefficient has a negative sign in the OP model, which nonsensically implies that the larger is the hike at the last meeting, the more likely is a cut at the next meeting. The CronOP model assumes that the rate change is the combined result of the two distinct decisions, on which a given variable may have different and even opposite effects. We expect a high level of persistency in the latent policy regime due to the slow cyclical fluctuations of macroeconomic indicators, which exogenously drive the policy stance. Central banks are typically conservative and dislike frequent reversals in the direction of interest rate movements. Therefore, we expect a positive coefficient on Δ r a t e i , t − 1 in the regime equation. Policymakers are cautious and tend to wait and see after they have moved the rate; an adjustment is typically followed by a status quo decision. The CronOP amount decisions are unidirectional, either non-positive or non-negative, if the policy stance is loose or tight, respectively. Thus, we expect a negative coefficient on Δ r a t e i , t − 1 in the amount equations. Indeed, the coefficient has a positive sign in the regime equation but the negative signs in the amount equations. It implies that the larger is the hike at the last meeting, the larger is the probability of a tight regime at the next meeting and, conditional on the tight/loose stance, the smaller is the probability of a hike/cut and the larger is the probability of no change.

The differences in the MEs of Δ r a t e i , t − 1 on the choice probabilities obtained across the three models are intriguing: the CronOP model has the opposite signs of the MEs compared with the OP and MIOP models.^[7] According to the CronOP model, if Δ r a t e i , t − 1 changes from −25 bp to 0 bp, holding all other variables fixed, the probability of a rate cut decreases by 0.084, the probability of a hike increases insignificantly, and the probability of no change increases by 0.084. By contrast, the OP and MIOP models produce a conflicting and misleading inference: the probability of a cut increases by 0.025 and 0.009, the probability of a hike decreases by 0.003 and 0.001, and the probability of no change decreases by 0.021 and 0.009, respectively.

The impact of Δ r a t e i , t − 1 on choice probabilities from the three models is also graphically compared in Figure 3. The predicted probabilities exhibit sharp contradictions. For example, if the policy bias is easing and Δ r a t e i , t − 1 increases, the probability of no change increases in the CronOP model but decreases in the OP and MIOP models. Similarly, the three models make a conflicting inference regarding the probability of a rate reduction. The OP and MIOP models fail to provide an accurate assessment of the relationship between the explanatory variables and outcome probabilities and produce an absurd inference. In contrast, the capability of the CronOP model to disentangle the opposite directions of the effect of Δ r a t e i , t − 1 on the regime and amount decisions produces an economically more reasonable inference.

Figure 3:

Comparison of competing models: the CronOP model provides the economically more reasonable estimates of choice probabilities.

Notes: 1719 observations. The probabilities are shown for the range of the preferred change to the rate Δrate i , t − 1 and two values of Δbias t − 1 (easing and neutral) at the last MPC meeting if the inflation rate is above the target and the other variables are fixed at their sample median values.

Figure 4 shows the estimated probabilities of latent policy regimes, which are averaged for each meeting across all MPC members. The probability profiles differ considerably in the periods of policy easing, maintaining and tightening, as demonstrated in Figure 5. Averaged over all meetings, the probabilities of the loose, neutral and tight policy stances are 0.33, 0.41 and 0.26, respectively, whereas the observed frequencies of the cuts, status quo decisions and hikes are 0.20, 0.65 and 0.15, respectively. Not all zeros are generated by a neutral policy stance.

Figure 4:

Actual policy decisions and estimated probabilities of latent policy regimes.

Figure 5:

Probabilities of latent regimes in different policy periods remarkably differ.

Notes: 1719 observations. The estimates are obtained from the baseline CronOP model. For the definitions of the policy periods, refer to Figure 1.

These findings are refined in Figure 6, which reports the decomposition of unconditional probability of no change into three parts, Pr ( Δ y i , t = 0 | r i , t = − 1 ) , Pr ( Δ y i , t = 0 | r i , t = 0 ) and Pr ( Δ y i , t = 0 | r i , t = 1 ) , conditional on the loose, neutral and tight zeros, respectively. This decomposition substantially varies and, as we hypothesized, the identified three types of zeros are unproportionally distributed across different policy periods. During policy easing and tightening, the fractions of neutral zeros are 0.47 and 0.63, respectively. The fraction of neutral zeros is only 0.70 even among the zeros that are clustered between the rate reversals during policy maintaining. For the entire sample, the portions of the loose, neutral and tight zeros are 0.20, 0.62 and 0.18, respectively. The average predicted probability of a no-change decision during the observed no-change outcomes is decomposed similar as 0.19/0.64/0.17 (see Table B12 in Online Appendix B). Less than two-thirds of the status quo decisions appear to be generated by the neutral policy regime. The policymaking process in the NBP appears to be inertial by choice: 40% and 44% of all outcomes in the loose and tight regimes, respectively, are the status quo decisions.

Figure 6:

The decomposition of Pr ( Δ y i , t = 0 ) into the probabilities conditional on the loose, neutral or tight regimes remarkably differs in different policy periods.

Notes: 1719 observations. The estimates are obtained from the baseline CronOP model. For the definitions of the policy periods, refer to Figure 1.

The MEs on the unconditional probability of no change can also be decomposed into three components. For example, the 0.084 (with the 0.024 robust standard error) ME of Δ r a t e i , t − 1 on Pr ( Δ y i , t = 0 ) is the combined result of the −0.098 (0.025), 0.178 (0.040) and 0.003 (0.002) effects conditional on the loose, neutral and tight policy regimes, respectively (see Table B13 in Online Appendix B).

To graphically illustrate how the decomposition of Pr ( Δ y i , t = 0 ) depends on data, it can be plotted as a function of two explanatory variables, holding all others fixed. For example, Figure 7 shows that if the individual policy choice at the last MPC meeting was to leave the rate unchanged, the inflation is above the target, the last ECB policy decision was a 25-bp cut, and the other variables are fixed at their sample median values, then Pr ( Δ y i , t = 0 ) is composed, on average, of 88% of loose and 12% of neutral zeros. However, if the ECB left the policy rate unchanged, then it is composed of 6% of loose and 94% of neutral zeros. If the ECB decision was a 25-bp hike, then Pr ( Δ y i , t = 0 ) is composed of 49% of neutral and 51% of tight zeros.

Figure 7:

The decomposition of Pr ( Δ y i , t = 0 ) into three components conditional on the loose, neutral and tight policy regimes as a function of policy rate choice at the last MPC meeting Δ r a t e i , t − 1 and recent ECB policy decision Δecbr_t .

Notes: 1719 observations. The probabilities are computed for the range of Δrate i , t − 1 and three values of Δecbr _t if the inflation rate is above the target and all other variables are held at their sample median values. The estimates are obtained from the baseline CronOP model. For the definitions of the policy periods, refer to Figure 1.

The sensitivity of the obtained empirical results is examined along several dimensions (see Online Appendix C for details). All key empirical findings – the parameter estimates from the CronOP model (see Table C1 in Online Appendix C), the comparison of the ME estimates from the OP, MIOP and CronOP models (see Table C2 in Online Appendix C) and the model performance comparison (see Table C3 in Online Appendix C) – are highly robust with respect to various modifications of the baseline specification such as: the alternative definitions of the individual policy rate preferences and the previous policy choices, the different measures of explanatory variables, the inclusion of other potentially influential explanatory variables, and use of different subsamples.