1 Introduction

As various recent episodes of financial and real turmoil (like the 2007–2010 crisis or the current pandemic shock) illustrate, financial markets might be subjected to rather significant and recurrent fluctuations possibly connected to monetary and fiscal policies, alongside the macroeconomic outlook. Among the key factors behind such fluctuations, the nature and the shifts in investors’ expectations mechanisms play a preeminent role. As information newly arrived on the market in regard to economic conditions and policies is incorporated into trading decisions, the prices and returns of financial assets are correspondingly adjusted. However, such information is frequently incomplete, costly and it is asymmetrically distributed among investors. In addition, these have access to different tools for gathering and integrating this information into their decision-making mechanisms. Equally important, such mechanisms may be significantly different from those used by public authorities in building up and implementing their policies. Hence, an information asymmetry problem may occur between these authorities and agents with important consequences for the formation of financial assets’ prices.

Currently, a growing body of literature focuses on this issue. For instance, by involving the economic policy uncertainty (EPU) index, Adjei and Adjei (2017) finds that, after controlling for business cycle effects, economic policy uncertainty is inversely related to contemporaneous market returns. Nevertheless, in that study, such effect subsists only during recessions or recessionary states of the economy, and it has no discernible effects during expansionary periods. For Chinese equity markets, Chiang (2019) shows that stock returns are negatively correlated with both local and global uncertainty innovations, even when controlling for changes in sentiments and liquidity. Meanwhile, Chiang (2020) provides supporting evidences that responses of stock prices to changes in US policy uncertainty do not only display a negative effect in the current period, but also have at least a one-month time-lag.

As for the more specific effect of monetary policy uncertainty, Bekaert et al. (2013) reveals that VIX strongly co-moves with measures of monetary policy stance. In these results, a lax monetary policy decreases both risk aversion and uncertainty, with the former effect being stronger. While the current VIX is positively associated with future (real) Fed funds rates, the relationship turns negative and significant after 13 months and high VIX levels are correlated with expansionary monetary policy on medium term. Meanwhile, Bernanke and Kuttner (2005) concludes that, on average, a hypothetical unanticipated 25-basis-point cut in the Federal funds rate target is associated with about a 1% increase in broad stock indexes. Generally, they argue that the effects of unanticipated monetary policy actions on expected excess returns account for the largest part of stock prices’ response. Bekaert et al. (2009) provides a key argument, by showing that, while the variation in price–dividend ratios and equity risk premium is primarily driven by risk aversion, uncertainty plays a large role in the term structure and it is the driver of asset returns’ countercyclical volatility.

Usually, this literature explains the impact of monetary policy uncertainty on financial asset prices, via its effects on risk aversion (for a model with counter-cyclical risk aversion accounting for a large equity premium, substantial variation in returns and price-dividend ratios and long-horizon predictability of returns, check, for instance, Campbell and Cochrane 1999). Also, it includes among the explanations: changes in private portfolios’ values (the wealth effect); changes in the cost of capital, which are associated with the shifts in monetary policy; as well as various types of information affecting both monetary policy and stock prices (Bernanke and Kuttner 2005).

On the other hand, there is a stream of literature providing evidences that investors’ expectations on financial markets are not fully rational and largely dependent of investor sentiments (Bloomfield and Hales 2002; Hommes et al. 2005; Su 2010; Corredor et al. 2013; Dergiades 2012). These evidences emphasize the existence of positive/negative biases in expectations’ mechanisms. In addition, heterogeneous agents acting on markets may use different heuristic techniques to form their expectations in a time-depending manner. Also, as Dosi et al. (2020: 1487) asserts: “fast and frugal robust heuristics may not be a second-best option but rather rational responses in complex and changing macroeconomic environments”. All the same, even such fast and frugal heuristic techniques may be swapped between them, according to their past performance. More generally, for each individual investor on market, there is a specific learning curve for acquiring and developing prediction tools and methods. Such curve occurs with different speeds and outcomes. Hence, in each moment, a certain heterogeneity in investors’ expectations with a direct impact on market prices mechanisms may ensue.

Finally, the information associated with monetary policy shocks is not the only type incorporated in the formation of investors’ expectations (other sources of information may include the business cycle and the macroeconomic conditions).

Summing up all these aspects, we argue that some of the results in the literature on market reactions might be biased by the neglect of possible nonlinear effects associated with adjustments in monetary policy. Unfortunately, the number of studies accounting for a nonlinear explanatory framework on this topic is still low. As Borio and Hofmann (2017:15) notes: “There is very limited analysis of nonlinearities in monetary transmission linked to the level of interest rates. The empirical literature is scant for both nonlinearities in aggregate relationships and in specific channels”. While several studies find nonlinearities, for instance, in the relationship between bank profitability and interest rates (Borio et al. 2017; Claessens et al. 2016) or between monetary policy and financial stress (Saldías 2017), Caggiano et al. (2017) asks if the stabilizing power of systematic monetary policy in response to uncertainty shocks is actually a state-contingent one. This last study uses a Smooth Transition Vector Auto-Regression (STVAR) model in order to capture the possibly different macroeconomic responses to an uncertainty shock occurring along the phases of the business cycle and it provides evidences of a reduced ability by monetary policymakers to influence output in presence of high uncertainty.

Yet, these pieces of the monetary policy-uncertainty puzzle are not currently integrated in a single unified big picture. More research needed in order to fully understand how investors react to monetary policy shocks and how they form their expectations, under different monetary policy regimes. The importance of the topic cannot be underestimated. If the monetary authorities would go beyond their traditional mandate to control inflation and place their monetary policy in the broader context of financial stability, by leaning against wind and reacting in a counter-cyclical fashion to the evolutions of financial system’s components, then it is of paramount importance to better understand the potential impact exercised by monetary policy on these components (including the financial markets).

Thus, the present study advances a two-fold contribution on the topic. First, we explore the possibilities for modelling the impact of monetary policy stance on uncertainty shifts, in the framework of distributed lag non-linear models (DLNM). Such framework provides a great amount of flexibility and it reflects the time-depending structure of the associated effects. In addition, it allows us to explore some possible answers to the underlying research question: how is the perceived economic uncertainty shifting under the influence of different asymmetric monetary policy regimes? Second, we consider the case of the Federal Reserve System monetary policy and its impact on heterogeneous uncertainty evolutions, as are these captured by the latent volatility of CBOE Volatility Index (VIX). Since this case is amongst the well-documented ones, this choice allows us to place our findings with respect to the recent literature.

The remainder of the paper is organized as follows. Section 2 provides details about the key variables, including the measurement of uncertainty and the involved proxy for monetary policy, as well as about the methodological issues related to DLNM. First, this section describes the adjustments in investors’ heterogeneous uncertainty perceptions via the evolution of the latent volatility of VIX itself. Such volatility is estimated based on a stochastic volatility model and its interpretation is straightforward: if heterogeneous investors adjust their expectations about the future evolutions of the economic conditions in an environment characterised by a higher degree of uncertainty, then an increased VIX latent volatility should be observed. However, it may take a while until such adjustments in expectations are translated into price changes and so they are not necessarily reflected by the current realized volatility. To the best of our knowledge, this is novel attempt to involve VIX latent volatility, as a dependent variable, in an empirical model seeking to capture the (nonlinear) effects of monetary policy on investors’ heterogeneous expectations in regard to the economic environment. Second, we show how we link such measure with the monetary policy, in this particular version of distributed lag models framework. Section 3 presents the United States data, with a special focus on the structural breakpoints identification. Section 4 discusses the results from a baseline model as well as from different DLNM specifications within Generalized Additive Models (GAM) family, whereas the last section concludes. Our main finding is that monetary policy impacts nonlinearly the adjustments in investors’ predictions. While a tighter monetary policy does generally contribute to an increase in VIX latent volatility, the shape of such effect varies across different specifications.

2 Methodology

2.1 Key variables: VIX stochastic volatility and federal funds rate

In order to capture uncertainty for the United States economy, we involve the VIX index which measures the ‘risk-neutral’ expected stock market variance for S&P500 index. More exactly, this index is based on the S&P 500 index options, and it is used to derive expected volatility by averaging the weighted portfolio of out-of-the-money European-style S&P500 call and put option prices that straddle a 30-day maturity (22 trading days) and cover a wide range of strikes. Usually this index is called the ‘investor fear gauge’ (Whaley 2000) and it displays a significant predictive power for stock market returns, economic activity and financial instability. Also, it reflects both stock market uncertainty and a variance risk premium (Bekaert et al. 2013; Bekaert and Hoerova 2014). We use the VIX index and not the similar CBOE VXO index (as Bloom 2009 does). Even if the latter index is similar to VIX, the benchmark index from which its value derives is the narrower S&P 100 Index (OEX), the VIX is more widely tracked by traders, analysts, and risk managers.

Since we are primary interested in the heterogeneous adjustments occurring in the degree of uncertainty (and not necessarily in the level of uncertainty per se), under the impact of different monetary policy regimes, we do not use the levels of VIX, but rather the index’s latent volatility. This is done in the framework of a stochastic volatility model. As Jacquier et al. (2004) argues, such models offer a natural alternative to the GARCH family for capturing the time-varying volatility. Perhaps one of the most relevant differences arising between these two estimators lies in the fact that while in GARCH-type models the conditional volatility is driven by shocks in the observed time series, in stochastic volatility models, it is driven by volatility-specific shocks. In addition, there are evidences that volatility predictions by GARCH-type models may fail to capture the true variation in volatility when there are regime changes in the volatility dynamics (Bauwens et al. 2014). Nevertheless, for the purpose of robustness assessment, we also consider a Markov-switching GARCH estimation of the latent volatility.

Stochastic volatility models’ main advantage is represented by their ability to separate error processes for the conditional mean and, respectively, conditional variance. Also, these are able to reflect the volatility clustering processes and to account for heavy-tailed dynamics. In addition, stochastic volatility models have the advantage of highlighting the volatility’s qualitative properties that would otherwise be more difficult to establish (Berestycki et al. 2004).

Let yt be the vector of VIX levels. In the centered parameterization form, the stochastic volatility model is given by:

$$y_{t} |h_{t} \sim N\left( {{\text{0,exp}}\left( {h_{t} } \right)} \right)$$
(1)
$$h_{t} |h_{t - 1} ,\mu ,\theta ,\sigma_{\eta } \sim N\left( {\mu + \theta \left( {h_{t - 1} - \mu } \right),\sigma^{2}_{\eta } } \right)$$
(2)
$$h_{0} |\mu ,\theta ,\sigma_{\eta } \sim N\left( {\mu ,\frac{{\sigma^{2}_{\eta } }}{{\left( {1 - \theta^{2} } \right)}}} \right)$$
(3)

Here N(μ,σ2η) denotes the normal distribution with mean μ and variance σ2η while \(\Theta = \left( {\mu ,\theta ,\sigma_{\eta } } \right)^{ \top }\) is the vector of parameters: the level of log-variance μ, the persistence of log-variance θ and the volatility of log-variance ση. The process h = (h0,h1,…, hn) is usually interpreted as the latent time-varying volatility process (the log-variance process).

In order to complete the model specifications, a prior distribution for the parameter vector Θ needs to be specified (for instance, with independent components for each parameter, i.e., p(Θ) = p(μ)p(θ)p(ση)). Therefore, we consider the following settings that account for the core properties of VIX index:

$$\begin{aligned} & \mu \sim N\left( {20,1} \right); \\ & \theta \sim {\text{Beta}}\left( {{0}{\text{.86,0}}{.11}} \right); \\ & \sigma^{2}_{\eta } \sim {\text{Gamma}}\left( {{0}{\text{.5,2}}} \right) \\ \end{aligned}$$

Finally, an MCMC algorithm provides draws from the posterior distribution of the latent log-variances h and the parameter vector Θ (see Kastner 2016, for more details on the implementation of stochastic volatility models in R package ‘stochvol’ and Kastner; and see Frühwirth-Schnatter 2014, for more methodological details).

We argue that this measure is able to better capture the outcomes of changes in the investors’ heterogeneous expectations mechanisms under the impact of various monetary policy regimes, than the level of VIX itself.

As a monetary policy proxy, we consider the federal funds rate (the interest rate that financial institutions charge each other for loans in the overnight market for reserves), as this is the primary instrument through which Fed implements its decisions. Since we are interested in the effects of monetary policy stance to be directly observed by the investors, we use the nominal levels of this rate and no other alternative measures (such as a natural real Fed funds rate to be consistent with full employment). The advantage is that no model of how investors extract the information from the observed status of monetary policy is required. Therefore, our analysis does not depend on particular assumptions about the heuristics used by investors in order to incorporate information related to the monetary policy into their expectations formation.

2.2 Distributed lag non-linear models framework

In order to model the connection between VIX latent volatility and federal funds rate, we involve the framework of the distributed lag non-linear models (DLNM). The methodology of distributed lag models (DLM) was originally formulated by Almon (1965) in time series analysis and applied, for instance, in various fields of epidemiology (Schwartz 2000). Lately, this framework was re-evaluated (Gasparrini 2014; Gasparrini et al. 2010, 2017) and largely applied in different studies’ design, especially in health and environmental modelling (Armstrong 2006; Xia et al. 2019; Li et al. 2020; Chang et al. 2020).

The key argument for involving DLNM is that the response of the uncertainty emerging on financial markets, as monetary policy shocks occur, depends on both intensity and timing of past such shocks. Nevertheless, the structure of the latent association is complex and not necessarily linear. In this context, DLNM captures the exposure-responses and the lag structure of this association.

More exactly, the response in the stochastic volatility (sv) is described at time t = 1,..., m in terms of lagged occurrences of a predictor xt (here, the effective federal funds rate), represented by the vector \({{\varvec{q}}}_{{\varvec{t}}}={\left[{x}_{\text{t} - {\text{l}}_{0}}\text{,...,}{\text{x}}_{\text{t-L}}\right]}^{\top }\) with l0 and L being the minimum and maximum lags. In our analysis involving monthly data, L is set a priori to be equal to 12 months.

The association between response and predictor is represented through a function s, which is termed as an cross-basis function and it is defined by:

$$s\left( {\boldsymbol{q}_{\boldsymbol{t}} } \right) = s\left( {x_{{t - l_{0} }} , \ldots ,x_{t - L} } \right) = \mathop \sum \limits_{{l = l_{0} }}^{L} f \cdot w\left( {x_{t - l} ,l} \right)$$
(4)

In this description, the bi-dimensional dose-lag-response function f·w(xt-l,l) is composed of two marginal functions: the standard ‘dose–response’ function f(x), and the additional lag-response function w(l), which models the lag structure in the space \(l={\left[{l}_{0}\text{,...,L}\right]}^{\top }\). Furthermore, the parameterization of f and w is obtained by applying known basis transformations to the vectors qt and l, producing marginal basis matrices Rt, C with dimensions \(\left({L} - {{l}}_{0}+ {1} \right)\times {\upsilon }_{x},\left({L} - {{l}}_{0}+ {1} \right)\times {\upsilon }_{l}\).

The function s is parameterized by coefficients η and it is constructed by:

$$s\left( {x_{{t - l_{0} }} ,...,x_{t - L} ;{\varvec{\eta}}} \right) = \left( {{\mathbf{1}}^{ \top }_{{L - l_{0} + 1}} {\varvec{A}}_{t} } \right){\varvec{\eta}} = {\varvec{w}}^{ \top }_{{\varvec{\eta}}} {\varvec{\eta}}$$
(5)

Here W is a set of known transformations derived from At, which in turn is computed by a row-wise Kronecker product between the two basis matrices:

$${\varvec{A}}_{t} = \left( {{\varvec{R}}_{t} \otimes 1^{T}_{{\upsilon_{l} }} } \right) \odot \left( {1^{T}_{{\upsilon_{x} }} \otimes {\varvec{C}}} \right)$$
(6)

The \(n\times \left({\upsilon }_{x}\cdot {\upsilon }_{l}\right)\) cross-basis matrix W, obtained by applying (4)–(6) to the full set of n observations, can be included in the design matrix of standard regression models, such as generalized linear models (GLM). Meanwhile, the ‘dose-lag-response’ surface can be recovered by predicting effects \({\hat{\beta }}_{{\text{x,l}}}\) on a grid of predictor values x and lag l. In order to ease the interpretation, \({\hat{\beta }}_{{\text{x,l}}}\) are defined as specific contrasts of f·w(xt-l,l) by centering the ‘dose–response’ function f(x) a reference value of the predictor x. These effects \({\hat{\beta }}_{{\text{x,l}}}\) can be interpreted in the scale of risk ratio or difference, for instance, by focusing on estimated lag-response associations, at a given predictor value, or the overall dose–response association obtained by cumulating the risk across the lag period (see Gasparrini et al. 2017 for more details).

A penalized extension of DLNM can be described within the family Generalized Additive Models (GAM) (Wood 2006a). This extends the strong parametric form of GLM by allowing the linear predictor to include flexible smooth functions of the covariates. The smooth components can be, for instance, defined through penalized regression splines, using low-rank basis terms and a simple form of penalized likelihood (Wood 2006b).

In this framework, the relationship between the stochastic volatility of VIX svt at month t and federal funds rate xt-1 (accounting for up to 12 lags) is estimated based on:

$$\log \left[ {E\left( {sv_{t} } \right)} \right] = \alpha + s\left( {x_{{t - l_{0} }} , \ldots ,x_{t - L} ;\eta } \right) + \beta_{1} brent_{t - 1} + \beta_{2} \log \left( {confidence_{t - 1} } \right)$$
(7)

Here, two additional sources of uncertainty surrounding the economic environment, that might impact financial markets returns and volatilities, are considered jointly with the non-linear term for federal funds rate: the global price of Brent crude (U.S. Dollars per barrel) (Alsalman 2016; Arouri et al. 2011) and consumers’ confidence, as this is captured by the confidence composite indicator of OECD for United States (Khan and Upadhayaya 2020).

3 The United States’ data

Monthly adjusted closing levels of VIX index are collected from Yahoo Finance (https://finance.yahoo.com/), for a period between 2000-01-01 and 2020-03-01, by using R package ‘BatchGetSymbols’ (Perlin 2020). We remove the non-available data ending with 243 observations.

Moreover, by using R package ‘alfred’ (Kleen 2020), we collect from the ALFRED database (Archival FRED) of the St. Louis Fed's Economic Research Division (2020) same frequency and period data related to: effective federal funds rate (not seasonally adjusted) (series ‘FEDFUNDS’), Global price of Brent Crude (series ‘POILBREUSDM’) and, respectively, Consumer Opinion Surveys:—OECD Indicator for the United States (series ‘CSCICP03USM665S’).

Figure 1 depicts the evolution of stochastic volatility and federal funds rate for the analysis period. The figure suggests that there are several distinctive sub-periods in which both VIX latent volatility and monetary policy display different profiles: the pre-2007 period; the 2007–2010 financial and real turmoil (with Federal Reserve System setting federal funds target in the range 0–0.25%—as of December 2008—and turning to unconventional monetary policies, such as large-scale asset purchases); the 2011–2013 recovering period and the return of volatility on stock markets during 2015–2017 and, respectively, in 2019 (with pandemic shock’s effects in 2020).

Fig. 1
figure 1

VIX latent volatility and federal funds rate. Notes The stochastic volatility is based on the implementation from Kastner (2016). This implementation uses a joint sampling of all instantaneous volatilities “all without a loop” (AWOL). For MCMC sampler, the number of draws is 50,000 with burn-ins equal with 10,000

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Additionally, as Table 1 shows the Zivot and Andrews (1992) test’s null hypothesis of a unit root process with drift is rejected in favour of the alternative hypothesis of a trend-stationary process (with one time break in both level and trend) for both federal funds rate and VIX’ stochastic volatility, if a higher number of lags is included in the test regression.

Table 1 Zivot and Andrews (1992) unit root with structural break test
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Moreover, as reported in Table 2, the Bai and Perron (1998; 2003) test identifies at least five significant structural breaks in both federal funds rate and VIX’ stochastic volatility series. These breaks occur during both 2007–2010 turmoil and post-2010 recovery period.

Table 2 Bai and Perron (1998; 2003) multiple breaks test
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We also account for the fact that when short and low-frequency time series are involved, there is a risk of taking their noisy components as signals (Gershunov et al. 2001). Hence, it might be useful to test the significance of signals before applying the DLM methodology. Thus, in order to check the existence of signal against noise, we apply the Monte Carlo test approach based on the standard deviations of rolling correlations, as proposed by Gershunov et al. (2001). This test verifies if the correlation patterns within some pre-defined time windows are significantly different from those occurring in white noise series. The test is implemented in R package ‘dLagM’ (Demirhan 2020). The main results are synthetized in Fig. 2. It reveals that high rolling correlations, significantly different from those arising between two independent white noise series, can be observed for almost the entire analysis period (with some possible exceptions between 2009 and 2010). Overall, the test suggests that the signal between VIX’ stochastic volatility and federal funds rate series is significant, especially for greater window lengths (starting with a 6 months’ length).

Fig. 2
figure 2

Gershunov et al. (2001) test for the existence of signal against noise for stochastic volatility and federal funds rate. Notes The dashed red lines show the limits of the 99% confidence interval for the mean of the average rolling correlations over the time points, which is shown by the horizontal solid line. The bold, solid line shows the average rolling correlation over the widths. A line is plotted to show the rolling correlations for each width

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4 Results and comments

4.1 Preliminary ARDL bounds test

In order to check for the existence of a long-run cointegration relationship between VIX volatility and monetary policy instrument, we start with an ARDL bounds test developed by Pesaran et al. (2001). Such test implies a conditional error correction form as:

$$\Delta sv_{t} = \mu_{0} + \alpha_{o} sv_{t - 1} + \alpha_{1} x_{t - 1} + \mathop \sum \limits_{i = 1}^{q} \gamma_{i} \Delta sv_{t - i} + \mathop \sum \limits_{j = 0}^{p} \beta_{j} \Delta x_{t - j} + e_{t}$$
(8)

The error correction part of the model, ECt-1 is given by:

$$EC_{t - 1} = sv_{t - 1} - \frac{{\alpha_{1} }}{{\alpha_{0} }}x_{t - 1}$$
(9)

Here μ0 is the intercept and Δ is the first difference of the series, while sv and x are once again the stochastic volatility and federal funds rate.

The hypothesis of cointegration is written over the coefficients of the conditional error correction model given in (8) and the test is applied with the following null:

$$H_{0} :\alpha_{0} = \alpha_{1} = 0$$
(10)

A Wald test statistic is computed and it is compared to the asymptotic limits given by Pesaran et al. (2001). More exactly, two sets of critical values for a given significance level are determined: one is computed under the assumption that all variables included in the ARDL model are integrated of order zero, while the second one is provided based on the assumption that the variables are integrated of order one. If the test statistic is lower than the given lower limit, H0 is not rejected and no significant cointegration relation is found. Opposite, if the test statistic is above the upper threshold, then H0 is rejected and the significance of cointegration between variables is pointed out (for details of implementation, see Demirhan 2020). Table 3 shows the results.

Table 3 ARDL bounds test
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Since the F-statistic of the test is greater than the 1% upper bound critical value, one can reject the null of no cointegration and accept that there is a long-term equilibrium relationship between VIX volatility and federal funds rate. Consistently with the ARDL bounds test, the error correction coefficient is negative and significant at 1%.

Shapiro–Wilk, Ljung-Box and Breusch-Godfrey tests provide evidences of close to normal residuals distribution and no significant autocorrelations for them. Meanwhile, Breusch-Pagan test shows that the residuals are not significantly affected by heteroskedasticity. Ramsey’s RESET test suggests that there are no particular model specification issues.

4.2 Main results

Reassured by the results of the ARDL bounds test, we further turn to the framework of DLNM analysis. As a preliminary step, we test various specifications for f(x) and w(l) in order to identify a baseline model. Table 4 reports the AIC values for different models with cross-basis functions provided by such combinations. A model with cubic regression splines with penalties on the second derivatives and degrees of freedom, i.e. the dimension of the basis matrix, equal to 10 for both f(x) and w(l) achieves the best performances in terms of AIC.

Table 4 Various combinations of f(x) and w(l) for the selection of the baseline model based on AIC
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As Fig. 3 shows, this baseline model clearly captures the non-linear impact of federal funds rate on VIX latent volatility along lag months as well as along different values of the monetary policy instrument. The VIX market volatility appears to be more sensitive to extreme values of federal funds rate compared to its response to mild values of this. Conversely, a zero rates policy appears to exercise only a small amplitude impact on market uncertainty dynamics. Nevertheless, regardless the level of the rate, the instantaneous effect is weaker than that occurring approximately 3 to 6 months later. Such effect is incompletely absorbed under a horizon of 3 to 4 quarters (with a second peak after 8–9 month). In addition, the effects associated with higher values of federal funds rate can be estimated only with higher confidence intervals.

Fig. 3
figure 3

Relative risk of an increase in VIX volatility associated with an increase in federal funds rate for GLM baseline model. Notes Baseline GLM selected in Table 4 and implemented by using R packages ‘mgcv’ (Wood 2004, 2011, 2017) and ‘dlnm’ (Gasparrini 2011). Figure 3.1 illustrates the lag-response curves specific to mild and extreme values of federal funds rates equal with 1%, 4% and 5% (with reference at 3%). Figure 3.2 (left column) depicts exposure–response relationship specific to lag 0 and lag 12 while Fig. 3.3. (right column) shows lag-response relationship for levels of federal funds rates equal with 0.2% and 6.4%

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Besides this baseline model, we also test a penalized DLNM within a GAM by involving two smoothers: one based on P-splines (Eilers and Marx 1996) (labelled as ‘PS’) and the other based on cubic regression splines with penalties on second derivatives (labelled as ‘CR’) (Wood 2006a). Here, penalization for f(x) is enforced through a default second-order difference penalty, while a double varying penalty is applied to w(l) using a second-order difference and a ridge penalty (Obermeier et al. 2015; Gasparrini et al. 2017).

Finally, the PS and CR smoothers are replaced with an unpenalized double-threshold function, i.e., linear splines which model a straight relationship below 2% and above 4% levels of federal funds rate, and a flat region in between. The simulations results are displayed in Fig. 4.

Fig. 4
figure 4

Dose-lag-response, overall cumulative dose–response, and lag-response at 4% (by column)—various GAM specifications. Notes Various GAM estimates implemented by using R packages ‘mgcv’ (Wood 2004, 2011, 2017) and ‘dlnm’ (Gasparrini 2011). The columns show the dose-lag response, an overall cumulative dose–response and, respectively, the lag-response at a level of federal funds rate equal with 4%. For GAM with doubly varying penalty-PS smoother: Deviance explained = 81.4%, -REML =  − 392.22 and scale estimation = 0.009128 while for GAM with doubly varying penalty-CR smoother: Deviance explained = 82.1%, -REML =  − 393.29 and scale estimation = 0.0088263. For both of them, the dimension of the basis used to represent the smooth term is 10 and REML estimation method, including of unknown scale. For GAM with partial penalization: Deviance explained = 65.8%, -REML =  − 351.55 and scale estimation = 0.015683

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The 3-D plots from the first column of this figure provide a comprehensive summary of the bi-dimensional exposure-lag-response association. Still, these are limited in their capacity to provide information on associations at specific values of predictor or lags, as these have only limited utility for inferential purposes, and the uncertainty of the estimated association is not reported here. Therefore, a more detailed analysis is delivered by plotting slices of the effect surface for specific predictor and lag values (as in second and third columns).

As Table 5 shows, all the linear terms are significant at 1% across GAM specifications, while the F-tests reject the null of equality to zero for all the coefficients making up the spline terms. However, it is interesting to note that the lagged values for both global price of Brent crude and confidence composite indicator of OECD for United States appears in GAM specifications to contribute to a reduction in VIX volatility.

Table 5 Coefficients and significance for GAM models
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Overall, the GAM estimates suggest both a shorter term association of monetary policy shocks with market uncertainty adjustments (and with higher amplitude compared to the GLM baseline model) and a more delayed association with lower values of funds rate. It is also interesting to note that in the double penalty with PS smoother in the estimated lag-response at 4%, the curve is shrinking toward the null at lags higher than 5–6 months, but such effect does not occur when a CR smoother is instead used. Also, if the PS and CR smoothers are replaced with an unpenalized double-threshold function, a straight relationship between volatility of VIX and federal funds rate ensues for different values of this rate. Additionally, for such function, the shape of lag-response at 4% shows a reduction in the amplitude of monetary policy impact on volatility after a short time-frame of 2–3 months.

Globally, the involved type of used penalization seems to significantly influence the shape of exposure-lag-response as well as the amplitude of estimated effects. Still, in all cases, the presence of nonlinearities is clearly detected, although the revealed time-structure is different. Equally important, these various estimates suggest that the effects associated with various monetary policy stances are not instantaneously absorbed by the market. Instead, there is a sort of persistence for these effects and some adjustments need to take place in investors’ expectations before they start to dissipate.

4.3 Robustness check

Several questions can be raised in regard to the robustness of these results. For instance, it can be argued that the financial markets’ dynamics is characterised by several ‘stylised facts’ that reflect the placement of their trajectories ‘far from equilibrium’. These facts may include: the high persistence in the autocorrelation of squared observations and leptokurtosis, heavy tails, gain or loss asymmetry, volatility clustering, asymmetry in time scales or ‘leverage effect’ (with the volatility of an asset being negatively correlated with the returns of that particular asset) (Taylor 1986; Cont 2001; Sewell 2011). The ability of any volatility estimator to capture as accurately as possible these ‘stylised facts’ is a criterion for assessing the quality of the underlying model (see for this argument Hafner and Preminger 2010).

The GARCH class of models and, respectively, the stochastic volatility models are two competing approaches widely used in the literature in order to estimate financial asset price volatility. However, there are some critical differences between them. The stochastic volatility models assume two error processes among such differences, while the GARCH model allows for a single error term. Therefore, the SV models can be viewed as more flexible and better able to capture some of the mentioned ‘stylised facts’. For instance, Carnero et al. (2004) shows that the relationship between kurtosis, persistence of shocks to volatility, and first-order autocorrelation of squares differs in GARCH (1,1) and Gaussian Auto-Regressive SV (1) models. This difference can explain why, when these models are fitted to the same series, the persistence estimated is usually higher in GARCH, than in SV models and why this last type of models is usually more adequate regarding data fitting. In contrast, GARCH models often require leptokurtic conditional distributions. Furthermore, even if asymmetric GARCH (such as QGARCH and EGARCH) and asymmetric SV models (i.e. the one proposed by Harvey and Shephard 1996) are used in the study of the asymmetric response of volatility to positive and negative changes in prices, the results in any of these asymmetric conditionally heteroskedastic models do not substantially differ in regard to the corresponding symmetric models (Carnero et al. 2004).

Nevertheless, by using a formal selection procedure between the GARCH and the SV models, based on their ability to represent the observed data properties (such as persistence, variance or kurtosis), Hafner and Preminger (2010) concludes that some SV models (although being more flexible and providing a better fit to the data) might suffer from the high variability of sample moments, especially in the second moments. Thus, a preference for SV models would only become apparent for larger sample sizes. In addition, the estimation of stochastic volatility models is more challenging due to their latent variable structure and the difficulty of evaluating their exact likelihood function.

Finally, GARCH models have a more straightforward implementation and are more prevalent in the literature and among practitioners. Consequently, if economic agents assess the impact of monetary policy on the uncertainty surrounding their decisions, a GARCH framework will be (arguable) implied rather than a stochastic volatility model.

Another issue is linked to the potential significant changes in results due to a selection of a GARCH class model instead of a stochastic volatility one. To address this topic, a Markov-switching GARCH model may be considered. This model combines the ability of the GARCH class to model a stochastic process with conditional variance, having the advantage of regime‐switching models that can divide the observed stochastic behaviour of a time series into several separate phases with different underlying stochastic processes. Therefore, the Markov-switching GARCH model can provide some interpretations for the nonlinearities associated with the time-varying volatility of financial market’s prices and returns, and it can deliver good forecasting performances (Haas et al. 2004; Luo et al. 2010; Ardia et al. 2018; Haas and Liu 2018).

Furthermore, economic agents may be concerned with both near-term effects of changes in monetary policy and longer-term effects of such changes. Therefore, more than the information synthesised by the current federal funds rate might be required to guide their decisions under different time frames. Policy uncertainty related to financial regulations and entitlement programs, concerns related to inflationary pressures, sovereign debt and currency crises should also be accounted for.

Broadly, we consider that there is a significant degree of uncertainty about the monetary policy, beyond interest rate fluctuations. Such uncertainty can raise credit spreads and reduce output, while the magnitude of such effects can sometimes be comparable with those exercised by conventional monetary policy shocks (Husted et al. 2020). Meanwhile, monetary policy’s uncertainty can affect the path of medium- and long-term interest rates. As De Pooter et al. (2021, p.2) argues: “investors are more complacent when monetary policy uncertainty is low. If so, they will be more willing to take larger and / or riskier (e.g., more duration risk) positions in interest rates. When subsequently confronted with a monetary policy surprise, they may need to make large and abrupt adjustments to ‘cut losses’ or to scale down risk-taking, which moves risk premiums. Consistent with this explanation, we find evidence that uncertainty affects how investor risk positions respond to surprises: in response to a tightening monetary policy shock, primary dealers…significantly reduce their net long positions in Treasury securities when prevailing monetary policy uncertainty is low; in contrast, position adjustments to the same shock are not statistically significant when uncertainty is high”. Therefore, the monetary authority can affect medium- and long-term interest rates, even if the federal funds rate is at the zero lower bound by managing expectations of future monetary policy and conducting large-scale purchases of longer-term bonds (Swanson and Williams 2014). Meanwhile, the literature shows that the causality may also run from monetary policy surprises to monetary policy uncertainty. For instance, Funashima (2022) finds that monetary easing shocks can contribute to increased monetary policy uncertainty, while monetary tightening shocks have insignificant effects on uncertainty.

Based on this literature, it can be argued that the effects associated with the present decisions of the monetary authority (as are these captured by the current stance of the federal funds rate) and the monetary policy’s uncertainty are associated concepts displaying bi-univocal linkages with the latter encompassing a broader content.

Finally, there is a certain inertia in the process of investors’ adjustment behaviour in response to changes in monetary policy. Hence, it would be interesting to find out how the results may change if longer time horizons are selected to account for such hysteresis effect, by increasing the value of L beyond the considered timeframe of 12 months.

Therefore, to perform a robustness assessment, we consider two alternative approaches. First, we replace the stochastic volatility estimation of VIX latent volatility with one based on a Markov-switching GARCH (MS-GARCH). This model allows a time-variation for the parameters of the GARCH model, according to a latent discrete Markov process (see for an application of such model, Zhang et al. 2014).

If Ξt-1 is the information set observed up to time t-1 and once again yt the vector of VIX levels (Ξt-1 ≡ {yt-i, i > 1}) then the general Markov-switching GARCH specification can then be formulated as:

$$y_{t} |\left( {s_{t} = k,\Xi_{t - 1} } \right) \sim D_{k} \left( {0,h_{k,t} ,\zeta_{t} } \right)$$
(11)
$$h_{k,t} \equiv h\left( {y_{t - 1} ,h_{k,t - 1} ,\Psi_{k} } \right)$$
(12)

Here \(D_{k} \left( {0,h_{k,t} ,\zeta_{t} } \right)\) is a continuous distribution with zero mean, time-varying variance hk,t and additional shape parameters reflected by \({\upzeta }_{{\text{t}}}\). The integer-valued stochastic variable st, which is defined on the discrete space {1,…,K}, is describing the Markov-switching GARCH model. Conditionally on regime st = k, hk,t is a function of the past observation, yt-1, past variance \({\text{h}}_{{\text{k,t - 1}}}\), and some additional regime-dependent vector of parameters \({\Psi }_{{\text{k}}}\). We follow the implementation of this model from Ardia et al. (2019) and we consider a three-state MS-GARCH model, where each regime is characterized by the same conditional volatility (Threshold GARCH Model) and by the same conditional distribution (Skewed GED). The results of applying the DLNM framework with this alternative measure of VIX volatility are reported by Fig. 5.

Fig. 5
figure 5

Dose-lag-response, overall cumulative dose–response, and lag-response at 4% (by column) for Markov-switching GARCH estimation of VIX latent volatility- various GAM specifications. Notes The VIX latent volatility is estimated via a three-state MS-GARCH model, where each regime is characterized by the same conditional volatility (Threshold GARCH Model) and by the same conditional distribution (Skewed GED). The columns show the ‘dose-lag’ response, an overall cumulative ‘dose–response’ and, respectively, the lag-response at a level of federal funds rate equal with 4%. For GAM with doubly varying penalty-PS smoother: Deviance explained = 83.2%, -REML = 98.19 and scale estimation = 0.66902 while for GAM with doubly varying penalty-CR smoother: Deviance explained = 82.9%, -REML = 100.59 and scale estimation = 0.67937. For both of them, the dimension of the basis used to represent the smooth term is 10 and REML estimation method, including of unknown scale. For GAM with partial penalization: Deviance explained = 68.1%, -REML = 139.24 and scale estimation = 1.1857

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The correlation coefficient between these two measures is equal to 0.72 and the MS-GARCH based volatility estimation peaks mainly during 2009–2010 (and, with a lower amplitude, during the first months of 2012). As Fig. 5 shows, the nonlinear impact of monetary policy stance is reflected once again with the use of this alternative estimator. Still, the heft of the estimated impact is more substantial now than in Fig. 4. As well, the curve looks somehow different, especially for the double penalty with CR smoother estimation.

Second, we replicate the analysis by setting a greater value of L (18 months). In addition, we replace the levels of federal funds rate with the monetary policy uncertainty index, proposed by Baker et al. (2016), as a component of the broader economic uncertainty index (EPU). The data are collected from Baker et al. (2020). The construction of the monthly EPU index and its category-specific indices for United States relies on 10 leading newspapers and a predefined set of search terms. The raw counts are scaled by the total number of articles in the same newspaper and month and each monthly newspaper-level series is standardized to unit standard deviation and then averaged across the 10 papers by month. Finally, the 10-paper series is normalized to a mean of 100.

Of course, this variable is meant to capture the degree of uncertainty associated by agents with monetary policy’s current and future evolutions, rather than its stance, like the federal funds rate. Hence, the results should be interpreted accordingly. In addition, as a news-based index of monetary policy uncertainty, it captures the degree of uncertainty perceived by the public (including financial market investors) in regard to monetary authority’s behaviour and its consequences on the macroeconomic context in general and on the financial stability in particular. For that reason, it is different (in terms of information content and sphere of inclusion) from the market-based proxies of uncertainty (such as implied volatility computed from interest rate option prices and realized volatility computed from intraday prices of interest rate futures) (see Husted et al. 2020 for a comparison between market-based and news-based measures of uncertainty).

Figure 6 reports these results for GAM specifications with PS and CR smoothers. For both of them, the estimations appear to be smoothers compared to those provided by the involvement of federal funds rates. Meanwhile, the effects associated with extremely higher values of monetary policy uncertainty (MPU) index are more clearly evidenced with a steep increase in VIX latent volatility, at a level of index (natural logarithm) around 6. In addition, the amplitude of these effects is significantly higher at such levels of MPU (although for lower level of uncertainty index, they are comparable with those produced by the monetary policy instrument).

Fig. 6
figure 6

Robustness check: overall cumulative dose–response and lag-response at (natural logarithm) MPU index = 4 (by column)—GAM specifications with PS and CR smoothers and stochastic volatility estimation of VIX latent volatility. Notes The columns show the overall cumulative dose–response and the lag-response at a level of (natural logarithm) monetary policy uncertainty index equal with 4. For GAM with doubly varying penalty-PS smoother: Deviance explained = 89%, -REML =  − 377.18 and scale estimation = 0.0058415 while for GAM with doubly varying penalty-CR smoother: Deviance explained = 89.1%, -REML =  − 381.99 and scale estimation = 0.0057228. For both of them, the dimension of the basis used to represent the smooth term is 10 and REML estimation method, including of unknown scale

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4.4 Comments

We find that the stance of monetary policy is associated, in a significant nonlinear fashion, with the heterogeneity of perceived economic uncertainty by investors.

In details, we first show that a stochastic volatility model of VIX latent volatility is able to capture some key events, which might be directly associated with a higher and more heterogeneous economic uncertainty. Second, an ARDL bounds test allows us to document a non-spurious long-term equilibrium relationship between VIX stochastic volatility and federal funds rate, as a proxy for the monetary policy. Third, our baseline GLM estimated reveals the sensitivity of VIX volatility to extreme values of monetary policy instrument as well as a time frame of 3 to 4 quarters for the transmission of the associated effects. Fourth, the GAM estimates involving two smoothers suggests that this time frame might be in fact shorter (starting from 1–2 quarters), while the positive impact of an increase in federal funds rate might have a higher impact than the baseline specifications. Fifth, our results reveals that the information signals associated with lagged values for global price of Brent crude and, respectively, confidence composite indicator of OECD for United States are both used in decision-making processes by investors and both of them contribute to a reduction in uncertainty. Finally, we find that the nonlinear positive effect is preserved when the VIX latent volatility is estimated with a three-state MS-GARCH model as well as when, instead of the monetary policy stance, the uncertainty surrounding its future design is involved.

But, then again, how plausible are such findings? Several issues and caveats might be considered here. First, we associate the VIX latent volatility with the heterogeneous shifts occurring in the mechanisms through which the investors form their expectations about macroeconomic conditions. Nevertheless, it should be recalled that VIX is a measure of expected price fluctuations in the S&P 500 index options over the next short-run period of 30 days. Therefore, such ‘latent volatility of an implied volatility measure’ approach covers at best the heterogeneity of investors’ expectations over some short-run specific periods and not over all possible prediction spans. Yet more, the latent volatility of VIX can be driven by various other factors and not solely by the endogenous heterogeneity of expectations. Consequently, the interpretation of the results should account for such limitations of the chosen proxy for the adjustments in expectations.

Second, we don’t explicitly explore here the sources of the non-linear impact exercised by monetary policy regimes. For instance, some of the evidences provided for United States stock markets (e.g. Chen 2007; Jansen and Tsai 2010; Chauvet and Sun 2014) show that monetary policy has larger effects on stock returns in bear markets, while a tight monetary policy leads to a higher probability of switching to a bear-market regime. For that reason, one may ask if the more striking impact that appears to be exercised, in our results, by a contractionary monetary policy on VIX latent volatility is merely the expression of the effects of such policy on S&P 500 components or there are also other specific factors?

Third, the identification of the exact shape of monetary policy’s impact displays certain sensitivity to model specifications. Additional research is required in order to clarify the causes of such sensitivity and to capture more precisely the time-frame and amplitude of the effects associated with a change in monetary policy stance.

Fourth, the literature provides evidences that the effects induced by monetary policy on stock markets are, in fact, time-varying and with, for instance, a particularly low reactivity of stock response to monetary policy shocks before the beginning of 2000s compared to the response of output to such shocks (but with an increased responsivity during more recent periods) and with important differences between stock and bonds markets (Galí and Gambetti 2015; Jansen and Zervou 2017; Pascal 2020). Therefore, further research need to account for the evolving reactions of markets to monetary policy stance over time, as well as for the non-uniform response of different financial markets’ segments.

Fifth, the changes taking place when monetary policy stance is replaced by monetary policy uncertainty should be examined in greater details. The subsequent analysis should be placed in the context of a model able to explain how agents incorporate monetary policy related information into their decisions-making processes and how they asses the likelihood of a change in monetary policy regimes.

Sixth, in our model we control for two additional sources of uncertainty. Nevertheless, there are other potential covariates that are not considered here. Among them, perhaps one of the most relevant is fiscal policy. Indeed, the current literature provides evidences of a bi-directional relationship between fiscal policy (and its interactions with monetary policy, for instance, via the impact of the government inter-temporal budgetary constraint on monetary policy parameters) and stock markets evolutions that might be used in order to improve the explanatory framework (Darrat 1990; Arin et al. 2009; Chatziantoniou et al. 2013; Mbanga and Darrat 2016; Bui et al. 2018). Nevertheless, even if we don’t consider these variables, more details are needed about the transmission channels associated with the global price of Brent crude and confidence composite indicator (as well as a scrutiny of their possible interactions with the design of monetary policy).

5 Final remarks

The main message of this paper is that a more realistic view of the effects associated with monetary policy stance on the mechanisms through which the investors form and change their perceptions about the economic uncertainty should incorporate nonlinear specifications and asymmetric impact.

Despite some limitations and although more research is required in order to clarify the implied mechanisms, this message is preserved across the different DLNM specifications considered as well as when the status of monetary policy is replaced with the uncertainty surrounding its current and future architecture.

Perhaps one of the most important implications, in terms of monetary policy design, is that non-linear Taylor rules might be preferable in the design of monetary policy, not only due to asymmetric central bank preferences, but more importantly due to the non-linear response of financial markets, as a key transmission channel for this policy.