2.1. Bubble derivation

Since Tirole (Reference Tirole1985), a bubble is defined as the difference between the asset price and its fundamental value, as follows:

\begin{equation*}Q_t^B=Q_t-Q^F_t\end{equation*}

where $Q_t$ denotes the price of an infinite-lived asset, $Q_t^B$ and $Q^F_t$ its bubbly and fundamental components.

A model for the asset fundamental value is required to test for the existence of bubbles.Footnote ⁷ We rely on the theory of rational asset price bubbles in a discrete-time setting [Giglio et al. (Reference Giglio, Maggiori and Stroebel2016)], such that a rational bubble is assumed to originate from the failure of the transversality condition in the rational asset pricing equation for stocks. According to this theory, a rational bubble grows at the real interest rate:

\begin{equation*}Q^B_t=E_t\left (\frac {Q^B_{t+1}}{R_t}\right )\end{equation*}

where $R_t$ is the real interest rate factor. This result implies that the price behavior is explosive. The bubble tests developed by Diba and Grossman (Reference Diba and Grossman1988), Hall et al. (Reference Hall, Psaradakis and Sola1999), and Phillips et al. (Reference Phillips, Shi and Yu2015a, Reference Phillips, Shi and Yub) (among others), indeed, ascertain the existence of bubbles from the explosive dynamics in long time series of asset prices, by using unit root and cointegration analysis on the price-dividend relationship. Other approaches, instead, rely on the present-value literature—initiated by Campbell and Shiller (Reference Campbell and Shiller1988) and followed by many others [Balke and Wohar (Reference Balke and Wohar2009), Al-Anaswah and Wilfling (Reference Al-Anaswah and Wilfling2011), and Chan and Santi (Reference Chan and Santi2021)]—for which the fundamental price component is derived under risk neutrality as the present discounted value of future dividend flows [Cochrane (Reference Cochrane2001)], as follows:

\begin{equation*}q_t^F=K+\sum _{k=0}^{\infty }\Delta ^k\left[(1-\Delta )E_t d_{t+k+1}-E_t r_{t+k}\right]\end{equation*}

where $d_t$ is the dividend stream of the asset price, $K=\log (1+Q/D)-\frac{Q/D}{1+Q/D}$ is a constant (where $Q/D$ the average price-dividend ratio), and $\Delta \lt 1$ is the mean difference between the gross rate of growth of dividends and the real rate.

We follow Galì and Gambetti (Reference Galì and Gambetti2015) for the derivation of the predicted response of the fundamental component of asset prices to an exogenous shock in the monetary policy rate $\epsilon ^m$ :

\begin{equation*}\frac {\partial q^F_{t+k}}{\partial \epsilon ^m_t}=\sum _{j=0}^{\infty } \Delta ^j\left ((1-\Delta )\frac {\partial d_{t+k+j+1}}{\partial \epsilon ^m_t}-\frac {\partial r_{t+k+j}}{\partial \epsilon ^m_t}\right )\end{equation*}

such that the response of the bubble component, $\partial q_t^B/\partial \epsilon ^m_t$ , is obtained as the asset price residual $q^B_t=q_t-q^F_t$ . Hence, the asset price response is defined as a weighted average of the two responses:

(1) \begin{equation} \frac{\partial q_{t+k}}{\partial \epsilon ^m_t}=(1-\gamma _{t-1})\frac{\partial q_{t+k}^F}{\partial \epsilon ^m_t}+\gamma _{t-1}\frac{\partial q^B_{t+k}}{\partial \epsilon ^m_t} \end{equation}

where the weight $\gamma _t\equiv Q_t^B/Q_t$ denotes the share of the bubble in the observed price. The fundamental component contributes more to the asset price response in periods where its share in asset prices is lower and vice-versa when bubbles are dominant.

A standard literature critique of the bubble theory is that risk attitudes can still explain the residual asset price component once discounted dividends are accounted for. These attitudes are reflected in risk premia, namely the covariance term between the stochastic discount factor and future payoffs. Two features of our approach mitigate this issue. First, we focus our analysis on rational bubbles, which impose restrictions on the dynamics of the bubble component. Second, we use a less restrictive model for fundamentals by allowing the whole asset price equation to switch across regimes—hence, also its fundamental component. Therefore, as argued in Chan and Santi (Reference Chan and Santi2021), the latter also embodies the role of time-varying discount rates and other latent factors driving regimes in fundamentals. Consequently, the residual—namely, our measure for the bubble—is orthogonal to asset price fundamentals and to factors generating shifts in fundamentals.Footnote ⁸

In Appendix D, we directly tackle this critique by allowing for a risk premium in the decomposition of the asset price response to the monetary shocks, as from the argument by Galì and Gambetti (Reference Galì and Gambetti2015). From this decomposition, the bubble is thus a residual risk premia component as in Jarrow and Lamichhane (Reference Jarrow and Lamichhane2022). We show that accounting for an endogenous response of the equity premium to monetary policy shocks tends to emphasize our primary evidence favoring a positive bubble response.

2.2. The MS-SVAR

The MS-SVAR is estimated with Bayesian methods using quarterly US data spanning from 1960 to 2019. Six variables are included in the model: the real GDP level $y_{t}$ , real dividends $d_{t}$ , the GDP inflation rate $\pi _{t}^{y}$ , the inflation rate for non-energy commodities $\pi _{t}^{c}$ , the federal funds rate $r_{t}$ , and the real S&P500 index $q_{t}$ . All variables enter in logs except for inflation and policy rates. For the latter, we use the shadow interest rate by Wu and Xia (Reference Wu and Xia2016) over the 2009–2016 time interval, in order to consider the zero lower bound period.Footnote ⁹

Nonlinear dynamics are introduced as Markov-switching state-dependency in the systematic and the stochastic model components. Hence, a Markov chain driving discrete coefficient states, $\xi _t^c$ , controls the former, whereas an independent Markov chain, $\xi _t^v$ , controls the latter, capturing discrete states for shocks’ heteroskedasticity. The two chains are collected under a composite process $\xi _t=\{\xi _t^c, \xi _t^v\}$ , evolving according to the transition matrix $Q=Q^{c}\otimes Q^{v}=(q_{i,j})_{(i,j)\in (H\times H)}\in \Re ^{h^{2}}$ , where $q_{i,j}$ is the transition probability from state $i$ to state $j$ and $H=\{1\ldots h\}$ is the set of possible regimes for $\xi _{t}$ .Footnote ¹⁰

The MS-SVAR model is as follows:

(2) \begin{equation} \mathbf{y}_{t}^{^{\prime }}\mathbf{A}_{0}(\xi _{t}^{c})=\mathbf{c}^{^{\prime }}(\xi _{t}^{c})+\sum _{i=1}^{\rho }\mathbf{y}_{t-i}^{^{\prime }}\mathbf{A}_{i}(\xi _{t}^{c})+\mathbf{\epsilon }_{t}^{^{\prime }}\mathbf{\Sigma }^{-1}(\xi _{t}^{v}) \end{equation}

where $\mathbf{y}_{t}^{\prime }=\left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} y_{t} & d_{t} & \pi _{t}^{y} & \pi _{t}^{c} & r_{t} & q_{t}\end{array}\right ]$ , $\mathbf{c}(\xi _{t}^{c})$ is the vector of constants, $\mathbf{A}_{0}(\xi _{t}^{c})$ is the invertible contemporaneous correlations matrix, $\mathbf{A}_{i}(\xi _{t}^{c})$ denotes the dynamic cross-correlation matrices for each lag term $\rho$ , and $\mathbf{\Sigma }$ is a diagonal matrix capturing structural shocks’ sizes. Following the standard practice in MS-VAR modeling [Sims and Zha (Reference Sims and Zha2006) and Sims et al. (Reference Sims, Waggoner and Zha2008)], we fix $\rho =5$ (i.e., to the data frequency plus one), and adopt Litterman (Reference Litterman1986)’s random walk prior, consistently with the stochastic properties of the variables.Footnote ¹¹ The calibration of priors follows Sims et al. (Reference Sims, Waggoner and Zha2008), which provide a benchmark for quarterly data MS-SVARs.Footnote ¹² A multivariate normal distribution for the orthogonal structural shocks $\epsilon _{t}$ is assumed:

(3) \begin{equation} P\left(\mathbf{\epsilon }_{t}|\mathbf{Y}^{t-1},\mathbf{\Xi }_{t},\mathbf{\theta },q\right)=\mathcal{N}\left(\mathbf{\epsilon }_{t}|\mathbf{0}_{n},I_{n}\right) \end{equation}

where the structural shocks’ standard deviations are given by the diagonal elements of $\mathbf{\Sigma }^{-1}(\xi _{t}^{v})$ , $\mathbf{\theta }$ denotes the vector of the model’s structural parameters, and $\mathbf{\Xi }_{t}$ and $\mathbf{Y}^{t-1}$ collect past information on the latent processes and data, respectively.

In this framework, we identify the monetary policy shock by imposing the (lower triangular) recursive strategy adopted by Galì and Gambetti (Reference Galì and Gambetti2015)—inherited from Christiano et al. (Reference Christiano, Eichenbaum and Evans2005)—where we order the federal funds rate after GDP, dividends, and inflation but before asset prices. Hence, we assume that monetary policy cannot respond contemporaneously to changes in asset prices and is consistent with some other key features: (i) on impact, only stock prices respond contemporaneously to the federal funds rate shock and (ii) monetary policy can react contemporaneously to output, dividends, and inflation changes. In this vein, we rely on an “information-based” identification approach, according to which slow-moving (real sector) variables do not respond contemporaneously to fast-moving (financial sector) ones, embodying a forward-looking component with considerable predictive content for the real economy.

We then test the relevance and nature of regime switches by computing the marginal data density (MDD) over differently specified competing models. As shown in Appendix B, this analysis selects a best-fit model $\left(3\xi ^v2\xi ^c_{r,q}\right)$ with a three-state Markov chain in the stochastic model component (generating variance states) and a two-state Markov chain in the asset price and interest rate systematic model component (generating monetary-financial states). Therefore, we label $\xi ^c=\xi ^{mf}$ . The resulting regimes describe a story of state-dependent monetary policy and financial interaction.

Table 1.

Descriptive statistics of states: moments

The table reports model variables’ means and standard deviations and some selected derivations, conditional on staying in each state. The equity return is computed as the growth rate of S&P500 stock prices; the real policy rate is the deviation of the federal funds rate from inflation (based on the GDP deflator); “ $q_{t}$ (bubble)” is the bubble component of S&P500 stock prices, as decomposed by Galì and Gambetti (Reference Galì and Gambetti2015), and “Equity return (bubble)” is the return rate of the bubble component of S&P500 stock prices.

2.3. Monetary-financial regimes

Monetary-financial states reflect nonlinear regularities of structural relations and shocks’ transmission, describing the interaction of monetary and financial phenomena. Their occurrence defines two states: the “high finance state” (H-fin) and the “low finance state” (L-fin). As displayed in Table 1, summarizing some descriptive statistics conditional on states, the former characterizes periods of sustained and stable stock prices and equity returns, higher output, inflation, and higher interest rates; the latter, the low finance state, defines periods of lower stock prices and crushes in equity returns.Footnote ¹³ By decomposing the S&P500 stock price index in a fundamental and a bubbly component, as explained in Section 2.1, stock prices in the high finance state contain a sizeable bubble component, which explains the higher stock price levels under this regime. Stock returns are also explained (in levels and standard deviations) by their bubble component, which grows at higher and more stable rates. The price-dividend ratio is substantially higher in the high finance regime.

As for the US historical evolution of regimes, Figure 2 reports the smoothed probabilities evaluated at the posterior mode for the Markov chains emerging under the selected model in 1960–2019 quarterly sample. The top panel displays states’ probabilities for the variance, capturing high (dark gray area), medium (light gray area), and low (white area) shocks’ sizes. The bottom panel displays probabilities for the monetary-financial states, where the gray areas identify the high finance state and the white areas the low finance state. The red line tracks the evolution of the monetary policy rate over time to enhance regimes’ interpretation.

Figure 2.

United States: variance and monetary-financial regimes. Notes: The figure shows the states’ smoothed probabilities at the posterior mode from the best-fit benchmark model, $3\xi ^v2\xi ^c_{r,q}$ . The upper plot displays the smoothed probabilities for the variance states. The bottom plot displays the smoothed probabilities for states emerging on the monetary policy and the asset price equations. The series for the interest rate is shown below the latter.

Variance and coefficient regime switches can be interpreted in terms of historical events. Regimes on shocks’ variances (low, medium, and high) detect high volatility during the two oil price turmoils in the 70s [as in Bianchi and Ilut (Reference Bianchi and Ilut2017)], the reserve targeting period of the early 80s, and the first stages of the global financial crisis in 2008–09. The medium variance regime materialized around the second half of the 80s, covering the stock market crash in 1987 and the recession due to the Gulf War I of the early 90s. The remaining periods, thus the pre-70s, the financial deepening era of the early 90s until 2008, and the post-2009, reflected conditions of relative stability and low variations in shocks (low variance state).

Monetary-financial states emerge as a result of the interaction of monetary and financial phenomena.Footnote ¹⁴ Consistent with Jarrow and Kwok (Reference Jarrow and Kwok2021), and Fusari et al. (Reference Fusari, Jarrow and Lamichhanne2022), high finance regimes closely track market performance. In contrast, switches towards low finance states are centered around stock price turning points and occurred in the late 60s, during the first half of the 80s (Volker era), the dot-com bubble (2001), and the 2015–16 stock market sell-off, including the Chinese stock market turbulence, the Greek debt default and the effects of the end of quantitative easing in the US.

2.4. Impulse responses

We evaluate the regime-specific conditional dynamics generated by an unexpected increase in the federal funds rate considering regime-dependent impulse response functions (IRFs).Footnote ¹⁵ To enhance the interpretation of results concerning the transmission dynamics across regimes, we condition the IRFs to a 1% contractionary monetary shock in all states.Footnote ¹⁶

In Figure 3, impulse responses are regime-dependent and display different channels of monetary transmission, depending on whether the high or low finance state is in place. Specifically, a 1% interest rate shock triggers a drop in real GDP and real dividends, which worsens expectations of future profitability and reduces stock price evaluations, as predicted by the traditional theory.Footnote ¹⁷ The high finance state, where the bubbly component of stock prices is higher, features more amplification and less policy effectiveness in stabilizing target variables, leading to more robust and more persistent drops in real activity and dividends and a further increase in real interest rates.

Figure 3.

Monetary policy shock. Impulse responses. Notes: The figure shows the impulse responses to a 1% monetary policy shock. The difference across regimes only reflects nonlinearities in the monetary policy and asset price equations, driven by $\xi ^{mf}$ . Shaded areas denote 68% credibility sets.

Notwithstanding the drop in expected dividends, the fall in stock price is lower under the high finance (bubbly) state and vanishes after 1 year. Indeed, asset prices do not only co-move with dividends but also embed an additional premium resulting from their bubbly component, growing with real rates. By decomposing the stock price impulse response in a fundamental and a bubble component (equation 1), Figure 4 shows that the rise in the real interest rate, combined with the decline in dividends, generates a drop in the asset price fundamental component (in line with the conventional wisdom and economic theory). Under the high finance regime, real rates and dividends respond more, and the bubbly component of asset prices increases. This result is explained by the larger relative size of the bubble, as shown in Table 1, and by the more persistent response of real interest rates.

Figure 4.

Monetary policy shock. Impulse responses: stock price decomposition. Notes: The figure shows the impulse responses of asset prices, and their decomposition in a fundamental and bubbly component, to a 1% monetary policy shock. Differences across regimes only reflect nonlinearities in the monetary policy and asset price equations, driven by $\xi ^{mf}$ . Shaded areas denote 68% credibility sets.

Our evidence confirms the findings in Galì and Gambetti (Reference Galì and Gambetti2015), which—using a time-varying VAR—find that an exogenous monetary tightening inflates the bubble, amplifying financial instability, mainly after the 90s with the Great Moderation and the onset of financial deepening. We qualify that heightened monetary-financial interaction, resulting in high equity premia, real rates, and a high bubble share in the composition of asset prices, is the main determinant of their result. Policy ineffectiveness to dampen the emergence of rational bubbles depends on the (recurrent) joint determination of financial outcomes with the monetary policy behavior.

3.1. The market for bubbles

The equilibrium between demand (by young borrowers) and supply (by old borrowers) of the bubble in every period is $B_{t+1}=B_{t}+B_{t+1}^{N}$ , where $B_{t+1}\geq 0$ is the physical amount of the bubble supplied at $ t+1$ and $B_{t+1}^{N}$ represents newly issued bubbles. The bubble equilibrium equation can be expressed as

(4) \begin{equation} Q_{t+1}=R_{t+1}^{B}Q_{t}+Q_{t+1}^{N} \end{equation}

where $R_{t+1}^{B}=P_{t+1}^{B}/P_{t}^{B}$ is the real factor of return on the bubble, $P_{t+1}^{B}$ and $Q_{t}=P_{t}^{B}B_{t}$ are the real price and the real value of the bubble, respectively. The new bubble creation process is linked to the economy’s size through the parameter $\omega \gt 0$ : $q_{t}^{N}=\omega y=\omega Ak^{\alpha }$ , where $q_{t}^{N}=Q_{t}^{N}/g^t$ is the trend-less value of the new bubble and $y$ , $k$ are the stationary levels of output and capital, respectively; hence, $\omega$ measures the bubble-to-output ratio.Footnote ²⁰ If $\omega =0$ , the economy sets itself into a “no bubble” stationary state in which the level of capital $k_{\text{NB}}$ is generally lower than that of a “bubbly” economy ( $\omega \gt 0$ ).

We assume the bubble share follows a two-state Markov process $\xi ^{mf}_t$ , capturing the financial side of the monetary-financial interaction we evidence in the MS-VAR. The Markov process produces two states: a bubbly and a no-bubbly scenario.Footnote ²¹ The monetary policy counterpart will be discussed in the next section.

In order to understand the effects of $\bar{\omega }$ on the stationary value of capital $k$ , and hence on $y$ , take the equilibrium relationship between $k$ and the real interest factor $R$ in the credit market, $Ak^{\alpha -1}=f_{B}\left ( R\right )$ from equation (G27), and solve it for $k$ . Then, compute the following derivative:

\begin{align*} \frac {dk(R)}{d\bar {\omega }} & =\frac {gk(R)^{2-\alpha }}{\left ( 1-\alpha \right )AR} \left [ \frac {\phi r^k-\frac {\varepsilon _{R,\bar {\omega }}}{\bar {\omega }}\left ( \frac {r^{k}}{\mu \alpha }\left ( \phi \mu \alpha +\bar {\omega } \right ) +\left ( 1-\delta _{K}\right ) \phi \right ) }{\frac {g}{R}( \phi \mu \alpha +\bar {\omega })} +\frac {\left ( 1-\phi \right ) \eta }{ \eta +\beta }\mu \alpha \right ]\\[4pt] \varepsilon _{R,\bar {\omega } } & =\frac {dR/R}{\frac {d\bar {\omega }}{\bar {\omega }} }\gt 0,\,\,\,\,r^{k}=\alpha \mu Ak^{\alpha -1}\end{align*}

The sign of $\frac{dk\left ( R\right ) }{d\bar{\omega } }$ defines two different regions for the effect on capital of an increase in the bubble share. We are in crowding-in (crowding-out) when an increase (decrease) in the bubble share ends up stimulating (dampening) the accumulation of capital:

\begin{equation*} \bar {\omega } \phi r^{k}-\left [ \left ( r^{k}+1-\delta _{K}\right ) \phi +\frac {q^{N}}{k}\right ] \varepsilon _{R,\bar {\omega } }\gtrless 0 \end{equation*}

Therefore, the direction in which the bubble share $\bar{\omega }$ in the (locally unique) stationary state affects capital depends on three different channels acting on the three terms of the above inequality:

• Collateral channel: an increase in the bubbly asset (higher $\bar{\omega }$ ) slackens the collateral constraint allowing borrowers to demand more funds and invest more. This yields a positive effect on $k$ and $y$ (crowding-in), through the term $\bar{\omega } \phi r^{k}$ ;

• Price channel: a higher $\bar{\omega }$ increases the cost of borrowed funds, leading borrowers to demand fewer funds and to invest less. This result produces a negative effect on $k$ and $y$ (crowding-out), via the term $\varepsilon _{R,\bar{\omega }}$ ;

• Asset allocation channel: a higher $\bar{\omega }$ increases the quantity of the bubbly asset to be purchased, crowding out productive investment expenditures. This effect increases with the total return on capital, and it yields a negative effect on $k$ and $y$ (crowding-out), through the term $\left (r^{k}+1-\delta _{K}\right ) \phi +\frac{q^{N}}{k}$ .

The final effect of an exogenous increase of $\bar{\omega }$ on $k$ (i.e., whether the economy is in a crowding-in or a crowding-out regime) is determined by the relative size of these three channels.

3.2. Monetary policy

In order to simulate an exogenous shock to the policy rate, we assume the following monetary rule:Footnote ²²

(5) \begin{align} \widehat{\left ( 1+i_{t}\right ) } &=\rho _{i,\xi ^{mf}_t}\widehat{\left ( 1+i_{t-1}\right ) }+\left ( 1-\rho _{i,\xi ^{mf}_t}\right ) \left ( \delta _{\pi,\xi ^{mf}_t}\pi _{t}+\delta _{q,\xi ^{mf}_t}\hat{q}_{t}\right ) +e_{t}^{i} \\ &e_{t}^{i}\sim i.i.d.\left ( 0;\,\sigma _{e^i}^{2}\right ) \notag \end{align}

where $\widehat{\left ( 1+i_{t}\right ) }$ is the deviation of the monetary policy factor from its steady-state value, $\delta _{\pi }\gt 1$ and $\delta _{q}\geq 0$ denote the policy reaction parameters to inflation and the bubble ( $\delta _{q}\gt 0$ indicates a LAW policy), and $\rho _{i}$ captures the persistence of the policy rate.Footnote ²³ We set the policy reaction to inflation and the bubble and the policy persistence as functions of the two-state Markov process, $\xi ^{mf}_t\in \{\text{H-fin},\text{L-fin}\}$ , which jointly affects the steady-state bubble-to-output ratio. This process evolves according to a transition matrix, $H^{mf}$ , whose calibration is given by the VAR estimates, and which delivers monetary-financial regimes.

3.3. Solution method

We solve the model using the efficient perturbation methods applied to Markov-switching models elaborated by Maih (Reference Maih2015) and Foerster et al. (Reference Foerster, Rubio-Ramirez, Waggoner and Zha2016).Footnote ²⁴ A detailed description of the solution method is reported in Appendix H. The model’s first-order approximated solution can be written in the following form:

(6) \begin{gather} \Upsilon _t=T_{\xi _t^{mf}}(\Upsilon _{t-1},\chi,\epsilon _t)\\ T_{\xi _t^{mf}}=T_{\xi _t^{mf}}\left (z_{\xi _t^{mf}}\right )+DT_{\xi _t^{mf}}\left (z_{\xi _t^{mf}}\right )\left (z_t-z_{\xi _t^{mf}}\right ) \notag \end{gather}

where $\Upsilon _t$ is the vector of model variables, $T_{\xi _t^{mf}}$ the Taylor first-order expansion, $\chi$ defines the perturbation parameter, $\epsilon _t$ the vector of structural shocks, and $DT$ the matrix of first-order derivatives. The expansion point, $z_{\xi _t^{mf}}=(\Upsilon,0,0)$ , is the vector of the steady states of the state variables, $z_t=(\Upsilon _{t-1},\sigma,\epsilon _t)$ . As in Aruoba et al. (Reference Aruoba, Cuba-Borda and Shorfeide2018), where both a targeted-inflation steady state and a deflationary steady state are considered, we expand around two steady states per regime.

3.4. Model-based impulse responses

We compute the impulse response functions to the monetary policy shock. We calibrate the model at the annual frequency by fixing $\beta =0.96$ , $\alpha =0.33$ , $\gamma _b=1$ , $g_s=1.02$ , $\gamma _s=0.2$ , $\mu =1-1/42$ , $\delta _k=0.1$ , $\beta _{\text{beq}}=42$ , $\phi =0.03$ , $A_2=0.3$ , $\sigma _i=0.01$ . The transition probabilities are fixed at the annual counterparts of the MS-VAR quarterly estimates: the probability of moving from the high to the low finance regime is thus 0.1834 and that from the low to the high 0.4205.

The Markov-switching parameters, $\omega$ , $\rho _i$ , $\delta _\pi$ , and $\delta _q$ , are estimated using a novel regime-dependent impulse response matching algorithm, mapping model-implied impulse responses to VAR results. More specifically, we match four regime-dependent VAR impulse responses: output, inflation, interest rate, and the bubble component of asset prices. Table 2 reports the results.

Table 2.

Model calibration based on impulse response-matching

The estimated values are generated by matching the regime-dependent impulse responses of output, inflation, interest rate, and the asset price bubble component. Weights are based on their confidence intervals.

The estimation leads to a regime of high finance corresponding to a bubbly economy with a more persistent and less reactive monetary policy rule for inflation and displaying a LAW monetary policy rule ( $\delta _q\gt 0$ ). Under this regime, the stationary level of the bubble is high enough to locate the economy in a crowding-in mode, with higher stationary levels of real rates, capital, and output. Instead, the L-fin regime corresponds to a no-bubbly economy, with policy rates targeting more actively inflation and lower interest rate persistence. This result provides further insight into Galì and Gambetti (Reference Galì and Gambetti2015)’s main conclusion.

Given the above calibration for the two states, Figure 5 shows the model-based impulse responses in the two regimes compared to the VAR counterparts. Following a positive shock to the nominal interest rate, the model predicts a recession/deflation and an increase in the real rate.Footnote ²⁵ More importantly, in a bubble economy (H-fin regime) the contractionary monetary policy inflates bubbles’ valuation, in line with our empirical evidence in Section 2.

Figure 5.

Monetary policy shock. Regime-dependent impulse responses. Notes: The figure shows the model-implied versus VAR-based impulse responses of selected variables (output, inflation, nominal interest rate, bubble, and the real interest rate) to a 1% monetary policy shock. Regimes affect the steady state parameter $\omega$ and the monetary policy rule, namely $\rho _i$ , $\delta _{\pi }$ and $\delta _q$ . The impulse response matching is based on the following grid: $\omega (\text{H-fin})\in [0,0.018]$ , $\omega (\text{L-fin})\in [0,0.01]$ , $\rho _i(\text{H-fin})\in [0.3,0.6]$ , $\rho _i(\text{L-fin})\in [0.1,0.3]$ , $\delta _q(\text{H-fin})\in [0,0.2]$ , $\delta _q(\text{L-fin})\in [0,2]$ , $\delta _\pi (\text{H-fin})\in [0.8,2]$ , $\delta _\pi (\text{L-fin})\in [1.25,2]$ . The corresponding loss function, computed on the distance between VAR and model-based impulse responses at the estimates, is equal to 0.1438. Shaded areas denote 68% credibility sets.

The economic mechanism underlying these results is as follows: the monetary policy shock has a direct impact on the real interest rate, due to the nominal rigidity. In addition to the demand channel acting through the Euler equation and favoring future consumption, the increase in the real interest rate generates two additional effects: the demand for credit is reduced due to a price channel (the increased cost of borrowed funds); the value of the bubble rises, and an allocation channel redirects more resources toward the purchase of the bubble. These recessive channels more than compensate for the effect of the increase in the bubble size on credit demand due to the slackening of the collateral constraint, which allows borrowers to demand more funds (collateral channel). The outcome is a recession/deflation coupled with a rise in the bubble value. The policy reaction to the fall in inflation tends to mitigate the recessionary effects of the shock by reducing the amplitude of the downturn of output and inflation.

The recession/deflation and the increase in the real rate (and hence in the bubble component) are magnified when $\rho _{i}\gt 0$ and/or $\delta _{q}\gt 0$ , namely under the high finance regime. The reason is that, following the exogenous interest rate shock, the policy reaction is dampened by the higher policy persistence and by the response to the bubble. Under LAW, the signal sent from the fall of inflation, which calls for a reduction of the real rate, is partially offset by the increase in the bubble component. The response of the central bank results in an effort that is too feeble in stabilizing the real interest rate. This result explains more persistence in the nominal interest rate response, higher real rates, and more severe recession/deflation compared to the L-finance regime. In a no-bubbly economy, interest rates respond to inflation only (and more than in H-fin), dampening the initial effects of the shock.

3.5. Policy credibility and uncertainty scenarios

Our empirical and theoretical results are obtained under the hypothesis that stock market bubbles exist and can be measured, at least indirectly. A recent macro-finance literature [e.g., Jarrow (Reference Jarrow2016), Jarrow et al. (Reference Jarrow, Kchia and Protter2011a, Reference Jarrow, Kchia and Protterb), Jarrow and Kwok (Reference Jarrow and Kwok2021), Fusari et al. (Reference Fusari, Jarrow and Lamichhanne2022)] provides several convincing contributions on the existence of bubbles. However, the proliferating bubble detection methods testify that measurement issues are far from being solved. Uncertainty about the empirical reliability of specific bubble component’s evaluation methods is a pervading limitation our analysis shares with previous contributions. To widen its validity domain, we test the robustness of our results against alternative assumptions regarding the monetary-financial regimes and the agents’ capacity to observe them.

In the first set of counterfactual scenarios, we assess the role of agents’ beliefs on macroeconomic dynamics under the two regimes, by comparing the benchmark impulse responses to scenarios built by varying the probability distribution of regimes. Being part of their information set, they may depend on policy credibility. A second set of counterfactuals is used to consider the role of uncertainty around the realization of regimes, hence the existence and valuation of a bubble. In this case, we depart from the standard rational expectations assumption that agents can observe the true transition matrix for monetary-financial regime shifts and assume that they are uncertain about how long any observed regime shift will last and hence must learn about its duration. We do so by comparing the benchmark impulse responses to those built under learning, passing from the perception of a short-lived to a long-lived high finance regime. To complement the analysis, we evaluate the same counterfactuals for the case of a monetary authority setting the interest rate disregarding bubbles, even when they exist.

More specifically, in the first set of scenarios in Figure 6, the transition matrix is modified such that agents attach different probabilities to the occurrence of a regime change. We label as “H-fin credible” the scenario where we set to zero the probability of going from H-fin to L-fin and increase by 50% that of transitioning from L-fin to H-fin (from 0.1834 to 0.2751). We label as “L-fin credible” the scenario where we set to 0.01 the probability to go from L-fin to H-fin and increase by 50% that of transitioning from H-fin to L-fin (from 0.4205 to 0.6307).

Figure 6.

Impulse responses to monetary policy shock. Counterfactual scenarios on policy credibility. Notes: The figure compares the impulse responses extracted from the estimated model to those from counterfactual scenarios defined as follows: an H-fin credible scenario increases the probability of switching from L-fin to H-fin and decreases the probability from H-fin to L-fin; an L-fin credible scenario increases the probability to switch from H-fin to L-fin and decrease the probability from L-fin to H-fin.

The credibility of announcing a different policy can significantly affect the efficacy of the current policy stance. When agents expect the high finance policy ( $\rho _i\gt 0$ , $\delta _q\gt 0$ ) to last longer (H-fin credible), the effect of a monetary policy shock is exacerbated both in terms of deeper recession/deflation and bubble’s inflation. In the long run, the increase in asset valuation boosts the borrowing capacity and favors a faster exit from the crisis. Suppose, instead, that in response to an announced shift toward a stabilizing regime (low finance), agents assign a higher probability to switch to the low finance regime (L-fin credible). In that case, they also assign a higher probability to the belief that the policy rule will be more stabilizing in the future. Hence, they may start disinvesting in the bubbly asset in anticipation, effectively raising the probability of switching to the no-bubble scenario. This result makes the deflationary and recessive effects of the monetary policy shock smoother. In a nutshell, in a bubbly world, the impact of monetary tightening depends on how agents form their beliefs on the credibility of the monetary policy and the state of the bubble.

Figure 7.

Impulse responses of the bubble to a monetary policy shock in high finance regime. Counterfactual scenarios on regime uncertainty. Notes: The figure compares the benchmark impulse response of the bubble to the monetary policy shock in the high finance regime to those from two counterfactual scenarios defined as follows: an H-fin long scenario where the probability of going from H-fin to H-fin long-lasting is 0.99, and the probability of going from H-fin temporary to the L-fin regime is 0.01; an H-fin short scenario where the probability of going from H-fin temporary to H-fin long-lasting is 0.01, and the probability of going from H-fin temporary to the L-fin regime is 0.4205. The left panel performs the counterfactual scenarios with a monetary policy rule targeting the bubble, as from the impulse response matching, $\delta _b=2$ . The right panel assumes the monetary authority does not respond to the bubble, $\delta _b=0$ .

In the second set of counterfactual scenarios, we model the idea that agents can still observe the regime in place at each point in time. However, they face uncertainty about the nature of deviations from the low finance regime. Namely, they are unsure if the central bank is engaging in a short or long-lasting high finance regime. This simulation introduces a learning mechanism on the persistence of regime changes [Bianchi et al. (Reference Bianchi, Lettau and Ludvigson2022)], by expanding the number of regimes to account for short- or long-lasting perceived high finance regimes.Footnote ²⁶ We use three states: a low finance regime, a short-lasting, and a long-lasting high finance regime. Uncertainty arises because the two high finance regimes share the same parameter values governed by $\xi ^{mf}$ , which agents recognize as belonging to a high finance regime. However, they cannot precisely infer whether it is a short- or long-lasting high finance regime. This is something they learn after switching to the new regime. After the switch to high finance, agents initially perceive it as a temporary deviation while adjusting only afterward to persistent changes.

In Figure 7, we compare the bubble impulse response to a monetary tightening in the high finance regime (blue line), under the estimated VAR-based calibration, to those when it is perceived as temporary (purple dotted) and long lasting (black dashed). In the first case, we assume a transition matrix where the probability of going from the temporary to the long-lasting high finance regime is 0.01, and the probability of going from short-lasting high finance to the low finance regime is 0.4205. We denote this as the “H-fin short scenario”. In the second case, we assume the probability of going from H-fin temporary to H-fin long-lasting is 0.99, and the probability of going from H-fin temporary to the L-fin regime is 0.01. We denote this scenario as an “H-fin long scenario”.

Learning about the persistence of the high finance regime implies that bubbles in the model respond to shifts to the high finance regime by initially underreacting to the monetary policy shock (short-lasting scenario) compared to the benchmark high finance response. Then, when they realize the shift is more persistent, they tend to overreact. We run the same counterfactual scenarios on a model where the central bank does not target the financial cycle and disregards bubble dynamics ( $\delta _b=0$ ). Both in the benchmark VAR-based version of the transition matrix and in the counterfactual scenarios on the duration of the high finance regime, the bubble response to a monetary policy shock is lower and less persistent, thus reinforcing our argument against the LAW policy.