1. Introduction

The price of risk plays a central role in the macroeconomy. It is determined by an interplay of preferences, expectations formation, and institutional characteristics (Woodford, 2019; Bordalo, Gennaioli, Shleifer, and Terry, 2019; He and Krishnamurthy, 2013; Brunnermeier and Sannikov, 2014). The price of risk has been shown to forecast future growth via credit spreads (Gilchrist and Zakrajˇsek, 2012), recessions via the term spread (Estrella and Mishkin, 1998), and crises via funding market spreads (Bernanke, 2018). Financial conditions shape the conditional distribution of real activity (Adrian, Boyarchenko, and Giannone, 2019) and help forecast the conditional mean and volatility of output and output gap growth many periods into the future (Adrian, Grinberg, Liang, and Malik, 2018; Adrian and Duarte, 2018). At the same time, the price of risk is a first-order determinant of financial conditions. The empirical linkages between the price of risk, financial conditions, and real outcomes suggest practical benefits from associating the price of risk with financial conditions within a parsimonious macroeconomic framework.

In this paper we show how the three-equation New Keynesian (NK) model can be modified to that effect, and we apply the resulting framework to analyze monetary and macroprudential policies. We use the NK model as our starting point because it continues to be the most common workhorse model for many macroeconomic debates: either directly in its three equation form, or because these key relationships are at the core of more quantitatively oriented setups. We modify the NK model in two ways. First, we introduce an endogenous “financial conditions” wedge in the IS equation. The dynamic response of this wedge to exogenous shocks depends on past financial conditions, current output, and expected output. Second, we introduce “financial vulnerabilities” by allowing the second moment of output to depend on state variables, including financial conditions. Financial vulnerabilities produce endogenous time-variation in second moments and generate, in certain states, a much higher sensitivity of macroeconomic variables to shocks than in the NK model. Together, the two new elements affect the transmission and amplification of financial shocks and shape the entire probability distribution of output. We label the model NKV for “New Keynesian Vulnerability”.

We show that the NKV model, when appropriately calibrated, replicates not only the same basic moments that the NK model does, but also the key empirical regularities that link financial and macroeconomic variables uncovered in the recent literature by Adrian, Boyarchenko, and Giannone (2019); Adrian, Grinberg, Liang, and Malik (2018); Adrian and Duarte (2018): (i) Loose financial conditions today predict a higher mean and lower volatility of the one- to four-quarter-ahead conditional output distribution, but lower mean and higher volatility at longer horizons (and vice-versa for tight financial conditions); (ii) Financial conditions predict the term structure of growth-at-risk (the low quantiles of the future conditional output distribution at different predictive horizons), even when controlling for macroeconomic variables; (iii) The term-structures of growth-at-risk cross: looser initial financial conditions lead to downside risks to output that are lower over the following two years, but higher over the following three to five years, than when initial financial conditions are tight; (iv) the volatility paradox (Brunnermeier and Sannikov, 2014): a compressed price of risk predicts low volatility of output in the short run, but increased volatility in the medium term; (v) Financial conditions are not good predictors of the conditional distribution for inflation. Statements (i)-(iv) also hold when output is replaced by the output gap, which implies that these empirical regularities are mostly cyclical phenomena.

The NKV model is a small-scale dynamic, stochastic, general equilibrium (DSGE) model that has the same number of shocks as the NK model, one more equation, and only five more parameters. Just as the three-equation NK model, it is intended more as an illustration of the economic forces at work than a state-of-the-art quantitative framework for forecasting and policy analysis. Nonetheless, the NKV model delivers (i)-(iv) not only qualitatively, but also quantitatively. More generally, we highlight dimensions along which our setup does considerably better than any linearized model and we show that it can also outperform non-linear, quantitative models recently used for policy analysis.

One of our main findings is that instrument rules that provide a good approximation to optimal policy in the NK model lead to real and financial instability in the NKV model, once financial conditions are accounted for.¹ This leads us to argue that relevant policy models need to explicitly account for real-financial interactions, and to propose a model which does so in a parsimonious fashion.

Existing alternatives to the three equation NK model fall into one of three main categories. The first comprises econometric models featuring financial conditions, such as the quantile regression based estimates proposed in Adrian, Boyarchenko, and Giannone (2019). The second consists of DSGE models in which spreads / risk premia are exogenous, either small-scale, such as Sims and Wu (2019), or more quantitatively oriented like Smets and Wouters (2007). The third includes medium and large scale DSGE models, in which spreads / financial conditions are endogenous (Bernanke, Gertler, and Gilchrist, 1999; Gertler and Karadi, 2011; Christiano, Motto, and Rostagno, 2014).

Because of the Lucas (1976) critique, the first two groups of models, while undeniably useful and informative, are not particularly well-suited for policy purposes. In the first case, much like in the context of Lucas’s original contribution, changes in the conduct of policy would be expected to affect the laws of motion of financial conditions, but the estimates provide little guidance as to how. In the second case, by construction, policy changes leave the dynamics of spreads and risk premia unaffected, i.e., in that sense, there are no real-financial interactions. Models in the third group tend to be considerably more complicated, and, as we show, have trouble accounting for the stylized facts (i)-(iv) listed above, even when the underlying non-linearities are allowed to play a role.² Furthermore, the interplay of the non-linearities tends to make the intuitive interpretation of results more challenging, which is perhaps one reason why none of those setups has been widely accepted as a “new synthesis”.

One of our contributions is to show that empirical performance along the “risk” dimension can be improved, while considerably decreasing the complexity of the resulting specification. In addition, the inclusion of financial conditions considerably enriches the propagation mechanism, which the real business cycle (RBC) model, and the NK model built on it, have been criticized for lacking.³ Broadly, the intuition underlying our results is simple: risk can endogenously build up in “good times”, with the magnitude of the resulting financial vulnerability only revealed during a subsequent downcycle.

The two key aspects of our narrative are an amplification mechanism that captures endogenous fluctuations in risk, and a persistent propagation of shocks, with financial conditions materially overshooting their long-run values for several years. Much like with the Calvo (1983) and Rotemberg (1982) pricing assumptions, which yield identical dynamics in the standard reduced-form NK model, the underlying amplification and propagation mechanisms of the NKV model are consistent with a variety of microfoundations. We show that the amplification we propose could arise either through endogenous heteroskedasticity of demand shocks, or because of occasionally binding Value-at-Risk (VaR) constraints of intermediaries (as in Adrian and Duarte, 2018). Similarly, we show that the persistence in propagation can be microfounded by either using diagnostic expectation formation, in line with Bordalo, Gennaioli, and Shleifer (2018), or introducing slow moving changes in leverage arising from a standard financial accelerator mechanism (Bernanke, Gertler, and Gilchrist, 1999). The fact that the NKV can be microfounded with explicit preferences, expectations, and so on, addresses the Lucas (1976) critique, a crucial step for credible policy analysis.⁴

How should monetary policy and macroprudential policy be conducted? First, monetary policy that aims at fully stabilizing the volatility of the output gap and inflation leads to explosive financial conditions. Second, a suitable combination of macroprudential and monetary policies can ensure efficiency if macroprudential policy is sufficiently potent to independently stabilize financial conditions. Third, when macroprudential tools are not perfectly effective – as is arguably the case in the real world – a Taylor rule augmented to include expected financial conditions increases welfare relative to a standard Taylor rule, effectively eliminating states of high vulnerability.

The remainder of the paper is organized as follows: Section 2 provides an overview of related literature, helping put our findings in context. Section 3 describes the model and studies its theoretical properties. Section 4 discusses how we solve, estimate and validate the model. We then study the interplay of monetary and macroprudential policies in Section 5, characterizing conditions under which the the separation principle provides a good guide to policy, and highlighting circumstances in which aggressive monetary policy can lead to financial and real instability. Section 6 uses the model to study alternative policy paths, further clarifying how the conduct of policy endogenously affects the dynamics of financial conditions, and characterizing second-best policies, which help minimize vulnerability when macroprudential tools are unavailable. Section 7 concludes.

2. Related Literature

Our paper is related to research that positions the financial sector at the heart of macroe-conomic fluctuations and the transmission mechanism. Woodford (2010), for example, incorporates credit conditions by augmenting a Keynesian IS-LM model with financial intermediary frictions, based on Curdia and Woodford (2010). In that setting, the additional friction gives rise to an extra state variable that can be mapped into credit spreads, and optimal policy is shown to explicitly depend on credit supply conditions, broadly in line with our findings.⁵ Financial frictions can arise when lenders face asymmetric information, in which case financial conditions have the propensity to improve the net worth of borrowers, and, through a financial accelerator effect, increase credit for households and businesses. Macrofinancial linkages can also arise because financial intermediaries respond endogenously to looser financial conditions, with institutional constraints providing further amplification.⁶ As discussed, the specification of the NKV is consistent with several of these narratives, with low rates and a low price of risk boosting current growth while simultaneously making the economy more vulnerable to future shocks and future financial instability.⁷

In the behavioral literature, diagnostic expectations of investors can give rise to extrapolative forecasting heuristics and lead to the neglect of tail risk when recent news has been good, generating predictable dynamics of credit spreads (Bordalo, Gennaioli, and Shleifer, 2018). Extrapolative beliefs in the stock market can amplify technology shocks, giving rise to booms and busts in stock prices and the real economy, with deviations from rational expectations potentially playing a more powerful role during times of low interest rates (Adam and Merkel, 2019). Extrapolative beliefs in credit markets can also create a feedback loop, because investors will refinance maturing debt on more favorable terms when defaults have been low, reducing risks in the short run, even if underlying cash flow fundamentals are weakening (Greenwood, Hanson, and Jin, 2019). As discussed, our specification of the process governing the dynamics of financial conditions is closely related to some of these behavioral theories of expectation formation and we manage to incorporate these without sacrificing tractability, allowing our model to be readily applied to study different policy questions.

Our paper is also related to those studying how monetary and macroprudential policy could reduce risks to financial stability. In particular, we revisit the separation principle in which monetary policy should focus on price stability and real activity, while macroprudential policies should be directed to reduce vulnerabilities consistent with an acceptable level of financial stability risk.⁸ Svensson (2017) estimates the costs and benefits of using monetary policy to prevent a severe recession in a model where the costs are related to higher unemployment, while the benefits are associated with a lower probability of a future recession on account of reduced household borrowing. In that model, the costs of using monetary policy to reduce household credit are much higher than the benefits because tighter policy lowers the probability of a severe recession by only a small amount and does not markedly reduce its severity.

Our model instead accounts for the fact that risk is endogenous to monetary policy, and that monetary policy that ignores financial vulnerabilities will lead to booms and busts with greater amplitude. In a similar vein, Filardo and Rungcharoenkitkul (2016) incorporate a financial cycle in which booms and busts are recurring, and show that in their setup monetary policy can constrain the accumulation of imbalances and significantly lessen the duration and costs associated with crises.⁹ Arguably, however, our approach is both more parsimonious and more easily portable to larger models, including those typically used in central banks for policy purposes.

3. The NKV Model

3.1. A risk-augmented IS curve. We consider an economy with a large number of identical households seeking to maximize the expected value of their utility,

where β ∈ (0,1) is the discount factor, σ > 0 is the inverse of the elasticity of intertemporal substitution, and ϕ > 0 is the inverse of the Frisch elasticity of labor supply. Period utility depends on the level of consumption C_t, hours worked N_t, and a random variable ξ_t > 0. The variable ξ_t will act as a demand shifter and can be interpreted in several ways. It can represent a preference shock or a change of measure (i.e., a belief shock). We will consider both exogenous and endogenous versions of ξ_t; what is necessary for our derivations to be correct is that households take it as given. Financial markets are complete. Each household faces the flow budget constraint

where P_t is the price level, D_t is the nominal value of the households’s end-of-period portfolio of financial assets, F_t-1 is the beginning-of-period total financial wealth of the household, W_t is the nominal wage, and T_t is the lump sum component of income (that can include government transfers and dividends from firms). We note that households are not restricted to holding only riskless bonds as the variable D_t refers to the value of a portfolio of any number of state-contingent assets (including a riskless bond, or financial instruments allowing it to be replicated).

The absence of arbitrage implies that there exist a stochastic discount factor Q_t that prices all assets. For example, the household’s financial wealth satisfies D_t = E_t[Q_t+1+F_t+1]. In equilibrium, the optimality conditions of the representative household implies that

where Π_t+1 = P_t+1/P_t is gross inflation between t and t + 1.

The one-period risk-free nominal interest rate faced by the household is $i_t^{int}\equiv 1/E_t\left[Q_{t+1}\right]-1$, which is known at time t. It differs from the interest rate ĩt set by the central bank according to

where $${\widetilde{spr}}_t$$ an endogenous spread linked to the current state of economic and financial conditions. The first-order optimality condition of the representative household —the Euler equation— for its holdings of the riskless nominal bond is

Appendix B shows that equation 3 can be approximated by

In equation 4, $$y_t^{gap}$$ is the output gap, i_t is the deviation of the central bank nominal rate ĩ_t from its steady-state value, π_t is inflation, spr_t is the deviation of the spread $${\widetilde{spr}}_t$$ from its steady state value, and g_t+1 is the deviation of log (ξ_t+1/ξ_t) from its steady state value. Equation 4 is a standard log-linear first-order Taylor approximation, with the exception that we have preserved some of the potential non-linearities in the last term, $\frac{1}{\sigma }{E}_t\left[g_{t+1}\right]$, by not approximating it as linear function of state or other primitive variables. If, for example, E_t [g_t+1] is a non-linear function of, say, the output gap, then 4 is non-linear. However, if ξ_t is exogenous, then E_t [g_t+1] is also exogenous and equation 4 becomes a true linear approximation.

We decompose E_t [g_t+1] into two multiplicative terms

where ${\epsilon }_t^{ygap}$ is a random variable with zero conditional mean and V(X_t-1) is a function of time t — 1 state variables denoted by X_t-1.¹⁰ Using this decomposition in equation 5, we arrive at the “risk-adjusted” IS curve

There are two new elements in equation 5 relative to the standard textbook version of the IS curve. First, there is a “financial conditions term” $-\frac{1}{\sigma }sp{r}_t$ that arises because there is a spread between the interest rate $i_t^{\mathrm{int}}$ that households can access and the interest rate ĩ_t set by the central bank, as captured in equation 2.

The second element is the time varying stochastic volatility V(X_t-1) in the disturbance term $-\frac{1}{\sigma }V\left(X_{t-1}\right){\epsilon }_t^{ygap}$. It arises from the demand shifter ξ_t in the representative household’s utility. The demand shifter has stochastic volatility V(X_t-1) without loss of generality. We assume that the exogenous disturbance ξ_t is such that ${\epsilon }_t^{ygap}$ is an i.i.a. N (0,1) stochastic process. The decomposition of the term is without loss of generality.

Intuitively, given the assumed positive relationship between spreads and financial conditions, higher spreads and tighter financial conditions are associated with lower contemporaneous values of the output gap and, ceteris paribus, push down on demand and current activity. In the next section, we propose two alternative microfoundations for how the spread spr_t is determined. The first is rooted in the diagnostic expectations approach, which nests rational expectations as a special case (Bordalo, Gennaioli, and Shleifer, 2018). The second uses the financial accelerator model of Bernanke, Gertler, and Gilchrist (1999).

There is another way of deriving Equation 5 that is directly motivated by financial intermediation frictions. In particular, Adrian and Duarte (2018) focus on the role of occasionally binding Value-at-Risk (VaR) constraints of financial intermediaries in a general equilibrium model and arrive at an equivalent specification of the dynamic IS curve. We refer the reader to Adrian and Duarte (2018) for full details of the derivation (see, in particular, Equations 6 and 7 on p.8), but stress that our proposed IS curve extension is compatible with at least two alternative microfoundations.

3.2. Microfoundations. We now offer microfoundations for the dynamics of the spr_t wedge and the vulnerability function V(X_t-1) appearing in equation 5, which hold irrespective of the preferred method of deriving that aggregate relationship. In the former case, there too will be several alternative ways of arriving at the specification endogenously linking financial and real conditions. The one we focus on first starts by assuming that households form beliefs according to diagnostic expectations (Bordalo, Gennaioli, and Shleifer (2018); Bordalo, Gennaioli, Shleifer, and Terry (2019) and Greenwood, Hanson, and Jin (2019)).

Diagnostic expectations refer to a belief formation mechanism based on Kahneman and Tversky’s representativeness heuristic. The expectations formation process is forward looking and depends on the underlying shocks and structure of the economy, making it immune to the Lucas critique.¹¹ Intuitively, diagnostic expectations capture over-reaction to news and deliver extrapolation and neglect of risk in a unified framework (see also Bordalo, Gennaioli, and Shleifer, 2018, for additional motivation as well as careful testing of the underlying hypothesis). The severity of judging by “representativeness” rather than by the true probability distribution is indexed by a single variable θ ≥ 0, where θ = 0 recovers rational expectations as a special case. Bordalo, Gennaioli, and Shleifer (2018) show that diagnostic expectations, when combined with a simple contracting framework between firms and households, can rationalize several empirical regularities about the connection of the entire distribution of asset prices and economic outcomes to expectations formation. Bordalo, Gennaioli, Porta, and Shleifer (2019) show that diagnostic expectations can also account for the empirical shape of the relation between the expectation of analysts and the actual performance of firms, as well as for the relation between analysts’ expectations and the predictability of stock returns.

Under the diagnostic expectations microfoundation, the random variable ξ_t that enters the household utility is a change of measure that converts true probabilities into diagnostic ones. In other words, diagnostic expectations make the agents behave as if the probability density that households use to compute expectations is distorted. The distortion is endogenously determined and depends on expectations of future equilibrium variables, but households take it as given (they cannot choose their beliefs). In equation 5, the distortion induced by diagnostic expectations relative to rational expectations is $-\frac{1}{\sigma }V\left(X_{t-1}\right){\epsilon }_t^{ygap}$. We show in Appendix C that if the fundamentals of the economy, denoted by ε_t, follow an AR(1) process

where 0 < b < 1 and v_t is a sequence of i.i.d. normal random variables, then the shock in the risk-adjusted IS curve, ${ϵ}_t^{ygap}$, is a constant multiple of vt, and hence also i.i.d. normal. More importantly, we show in the appendix that V(X_t-1) maps exactly to θ, the severity of judging by “representativeness”. We will later pick the shape of the function V (·) to determine how state variables influence the degree of distortion in beliefs induced by diagnostic expectations. Similar to Bordalo, Gennaioli, Kwon, and Shleifer (2020), we will obtain time-varying amplification (or “excess volatility”) arising from diagnostic expectations.

As alluded to above, the spread spr_t can also be microfounded using diagnostic expectations. In the baseline NK model, there are no financial frictions. Households can finance firms with debt, equity, or a combination of both. Since the Modigliani-Miller theorem holds in our setup, how firms are financed is irrelevant. Bordalo, Gennaioli, and Shleifer (2018) show that under diagnostic expectations, a simple real contracting model in which investment depends on the perceived probability with which each firm repays its debt in the following period gives the following ARMA(1,1) equilibrium law of motion for spreads¹²

The coefficients σ₀ and σ₁ denote the average level of spreads and their sensitivity to expectations of future fundamentals, respectively. The distortions of diagnostic expectations affect the perceived safety of different firms. Intuitively, in good times, households are optimistic, spreads are compressed and firms issue debt, but when times turn sour, households become pessimistic, spreads rise and firms cut debt issuance and investment.¹³

To sum up, if households have diagnostic expectations and finance firms, we can microfound the financial conditions spread spr_t and the stochastic volatility V(X_t-1) in the risk-adjusted IS curve 5. For our purposes, it is important that monetary policy can change neither the expectations formation process nor the shape of the debt contract between firms and investors, so the central bank takes these elements as given.

Letting

and adopting a simplifying assumption that financial conditions η_t are primarily pinned down by spreads, and hence approximately equal up to some constant scaling factor, we arrive at

which can be rearranged as

where

Under suitable convergence criteria (see Appendix C), equation (7) implies that financial conditions can be approximated by

where, importantly, the terms omitted to arrive at this approximate relationship do not in any way depend on the conduct of monetary policy. Since we have already established that

where ${\epsilon }_t\equiv E_t{g}_{t+1}^{\xi }/\sigma$, therefore

and the process for financial conditions can be rearranged and expressed as

where the coefficients in Equation 25 are given by,

Here σ₁ < 0 implies that θ_y > 0,θ_η > 0, which we impose when estimating the model.

Equation 8 would also approximately hold if, instead of relying on the diagnostic expectation model, we used a standard financial accelerator setup (Appendix C). Accordingly, and in line with the extended IS curve specification, our law of motion for financial conditions is compatible with at least two microfoundations, and can obtain under both diagnostic and rational expectations.

3.3. Links between financial conditions and the price of risk. We now highlight the relationship linking financial conditions η_t and the pricing kernel’s conditional volatility, i.e., the price of risk. As explained in Section 3.1 above, we can think of the household block as being entirely standard, with the resulting consumption-based log-SDF ${\tilde{m}}_t$ given by

where, as done previously, we have exploited the assumption of a CRRA utility function, goods-market clearing c_t ≡ y_t and $y_t^{nat}\equiv 0implying{y}_t^{gap}=y_t$.¹⁴ An important feature of our four-equation specification is that the state variable is X_t = (η_t-1, η_t-2) and expanding the model by adding in a definition of the log-SDF would enlarge the set of states to $X_t^1=\{{\eta }_{t-1},{\eta }_{t-2},y_{t-1}^{gap}\}$. The equilibrium solution for the log-SDF is of the form

where the a_i’s and b_m can be found by solving the linear, homoskedastic model, and where they are, respectively, the elements of A and B characterizing how the stochastic discount factor loads on the state variables and shock ${ϵ}_t\equiv V\left(X_t\right){\epsilon }_t^{ygap}$.¹⁵ It follows that the conditional mean and variance of m_t are given by

and

Expressed alternatively, after plugging in the definition of V (·, ·, ·), the conditional volatility of the log pricing kernel m_t+1 can be expressed as¹⁶

where x⁺ ≡ max{x, 0} and F_t-1 is the time t — 1 information set.¹⁷ This expression establishes that in our simple NKV model, the pricing kernel’s conditional volatility is piecewise-affine in η and the IS curve wedge ε^ygap.¹⁸

The fact that η_t depends indirectly on interest rates, via the output gap in Equation (8) and the IS curve in Equation (5), is the “risk-taking channel” of monetary policy. The “vulnerability channel,” in contrast, is present because lower interest rates directly impact the price of risk and V(X_t), that is, the conditional volatility of output. It follows that when making monetary policy decisions, the policymaker has to consider not only the output-inflation tradeoff, but also an intertemporal risk-return tradeoff introduced by the “vulnerability channel.” While easier monetary policy leads to lower volatility, thus allowing short-term risk-taking, we will show that the lower short-run volatility is also associated with larger medium-term risk.¹⁹

4. Model Solution and Estimation We now discuss the strategy adopted to solve the model and estimate its parameter values.

4.1. Solution. The NKV comprises the following four equations

Clearly, the model is linear except for the presence of the vulnerability function V (·).

To solve the model, consider first an alternative linear specification, in which Equation 10 is replaced by

The solution to this simplified model is of the following form,

where $Y_t=[y_t^{gap},{\pi }_t,{\eta }_t,i_t{]}^{\prime }$ and A, B are constant matrices. We also know that the solution to this simplified linear model has the certainty equivalence property, which means that neither the A nor B matrix are affected by the volatility of ${\widetilde{ϵ}}_t^{gap}$. This observation allows us to substitute in ${\widetilde{ϵ}}_t^{gap}\equiv V\left(X_t\right){\epsilon }_t^{ygap}$ and use

as an approximation to the NKV model solution, effectively exploiting the independence of the A and B matrices from the vulnerability function V(·). Such an approach has two main advantages. First, the one-step-ahead conditional distributions remain tractably normal, allowing for quick, analytical evaluation of conditional moments. Importantly, however, neither the k-step ahead conditional distributions (with k > 1) nor the ergodic distributions are restricted to be Gaussian. Second, the evolution of the conditional mean is not affected by the specification of the vulnerability function, allowing us to split the estimation process into two steps, effectively separating the estimation of the “financial conditions” channel and the “financial vulnerabilities” channel (discussed in more detail in Section 4.3).

The main drawback of this first approach is that impulse responses and simulations cannot be directly obtained using standard model solution software. Accordingly, we additionally use pruned, second- and third-order perturbation approximations to the solution of the non-linear system (10) – (13). These can be readily obtained using Dynare (see also Adjemian, Bastani, Juillard, Karame, Maih, Mihoubi, Perendia, Pfeifer, Ratto, and Villemot, 2011), which can also be used to generate impulse responses, run simulations, and which provides a convenient cross-check on solution accuracy.²⁰

4.2. Data. We use the log-difference between real GDP and the Congressional Budget Office’s estimate of potential as a measure of the output gap. We use annual core personal consumption expenditures (PCE) inflation and the National Financial Conditions Index (NFCI) compiled by the Federal Reserve Bank of Chicago. The NFCI aggregates 105 financial market, money market, credit supply, and shadow bank indicators to compute a single index using the filtering methodology of Stock and Watson (1998). The NFCI data start in 1973, and our estimation period is 1973 to 2017.

4.3. Parameter Estimation. Parameters of the NKV model can be split into two groups: the ones shared with the three-equation New Keynesian workhorse, and additional coefficients related to the evolution of financial conditions η_t and vulnerability V(X). We now discuss these in turn.

4.3.1. NK coefficients. For the first group of coefficients, common to both the NK and NKV models, we simply adopt the values proposed in Chapter 3 of the Gal´ı (2008) textbook (and reproduced in Table 1 for completeness). We also retain all structural parameter relationships, i.e., we have,

^{Table 1.}

New Keynesian Parameter Values

^{Table 1.}

New Keynesian Parameter Values

α	β	ε	φ	φπ	φy	σ	θ
1/3	0.99	6	1	1.5	0.125	1	2/3

View Table

^{Table 1.}

New Keynesian Parameter Values

α	β	ε	φ	φπ	φy	σ	θ
1/3	0.99	6	1	1.5	0.125	1	2/3

While this way of proceeding severely limits the degrees freedom we have in specifying the model, it also ensures that the NKV nests the NK model as a special case.²¹ This is important as it helps clarify that the differences in model properties that we document subsequently are not driven by different assumptions about the “standard” NK building blocks, but instead are due to the changes in propagation and amplification arising on account of endogenous risk.

4.3.2. Additional coefficients., We follow a procedure similar in spirit to that in King and Rebelo (1999), and find parameters that help our model match key moments of the data.

As shown in Table 3, we focused on four moments: autocorrelations of first differences of the output gap and its conditional mean, as well as their correlations with financial conditions.

^{Table 2.}

Additional, Non-NK Parameter Values

^{Table 2.}

Additional, Non-NK Parameter Values

γη	γη	γηη	θη	θy
0.01	1.97	-1.01	0.31	0.08

View Table

^{Table 2.}

Additional, Non-NK Parameter Values

γη	γη	γηη	θη	θy
0.01	1.97	-1.01	0.31	0.08

^{Table 3.}

Fit to Targeted Moments

^{Table 3.}

Fit to Targeted Moments

	$corr(\mathrm{\Delta }{y}_t^{gap},\mathrm{\Delta }{y}_{-1t}^{gap})$	$corr(E_t\mathrm{\Delta }{y}_{t+1}^{gap},E_{t-1}\mathrm{\Delta }{y}_t^{gap})$	$corr(\mathrm{\Delta }{y}_t^{gap},{\eta }_t)$	$corr(E_t\mathrm{\Delta }{y}_{t+1}^{gap},{\eta }_t)$
Data	0.33	0.82	-0.43	-0.82
NKV	0.42	0.75	-0.44	-0.36
VAR	0.30	0.77	-0.45	-0.81

View Table

^{Table 3.}

Fit to Targeted Moments

	$corr(\mathrm{\Delta }{y}_t^{gap},\mathrm{\Delta }{y}_{-1t}^{gap})$	$corr(E_t\mathrm{\Delta }{y}_{t+1}^{gap},E_{t-1}\mathrm{\Delta }{y}_t^{gap})$	$corr(\mathrm{\Delta }{y}_t^{gap},{\eta }_t)$	$corr(E_t\mathrm{\Delta }{y}_{t+1}^{gap},{\eta }_t)$
Data	0.33	0.82	-0.43	-0.82
NKV	0.42	0.75	-0.44	-0.36
VAR	0.30	0.77	-0.45	-0.81

Table 2 shows the values of the estimated parameters. We find these values by minimizing the unweighted sum of squared deviations of the ergodic moments from their estimated empirical targets. The signs and magnitudes of all the estimated parameters confirm our narrative and are compatible with the microfoundations we propose. They confirm the dependence of the output gap on financial conditions (γ_η ≠ = 0) and of financial conditions on the current and expected realizations of the output gap (θ_y, θ_η ≠ = 0). In addition, the autoregressive part of the financial conditions process evaluated at the estimated parameter values can be written as

indicating that the underlying process is both persistent (it will display cyclical dynamics) and sensitive to its past changes, with the latter feeding almost one-for-one into current FCI values.²²

The values of the moments implied by the estimated parameters are shown in the second row of Table 3. As a relevant benchmark, we also show in the third line of the table the moments obtained by fitting a vector auto-regression (VAR) of the form:

This VAR corresponds to the laws of motion for η_t and the output gap $y_t^{gap}$ in the NKV model after the model has been solved. The coefficients of a_η ≡ [a_ηη,a_ηη−1,a_ηε] and a_yη ≡ [a_yη ,a_yη−1, a_yε] are functions of the underlying structural parameters. The structural relationships in Equations (10) – (13) place complicated, non-linear restrictions on the coefficients of the a_η and a_y vectors. On the other hand, the VAR does not impose any of these restrictions. Hence, the fit of the VAR will always be better than the fit in the second row of Table 2. Comparing the three rows of the table, we see that the non-linear restrictions imposed by the NKV model do meaningfully discipline the model, yet the model-implied moments remain close to their unconstrained equivalents and the data.

In a similar fashion, we calibrate V (·) so that the NKV model delivers two empirical features displayed in Panel a of Figure 1:²³ i) the conditional median and volatility of output gap growth correlate negatively, and ii) the negative bivariate relationship is characterized by a high R² (0.91). The combination of these facts is important because it ensures the skewed dynamics of output gap growth quantiles discussed in Adrian, Duarte, Liang, and Zabczyk (2020), and also generates a smooth conditional 95^th quantile, and a more volatile conditional 5^th quantile.²⁴ In other words, a model that delivers (i) and (ii) immediately generates a conditional output gap distribution that replicates its empirical counterpart.

Output Gap Growth Conditional Median and Volatility

Citation: IMF Working Papers 2020, 236; 10.5089/9781513561066.001.A001

Note: Panel (a) shows estimates of the conditional median and conditional volatility of output gap growth one quarter ahead. Panel (b) shows the conditional median and volatility simulated from the NKV model.

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Output Gap Growth Conditional Median and Volatility

Citation: IMF Working Papers 2020, 236; 10.5089/9781513561066.001.A001

Note: Panel (a) shows estimates of the conditional median and conditional volatility of output gap growth one quarter ahead. Panel (b) shows the conditional median and volatility simulated from the NKV model.

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Output Gap Growth Conditional Median and Volatility

Citation: IMF Working Papers 2020, 236; 10.5089/9781513561066.001.A001

Note: Panel (a) shows estimates of the conditional median and conditional volatility of output gap growth one quarter ahead. Panel (b) shows the conditional median and volatility simulated from the NKV model.

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To have the 95th quantile constant, the vulnerability function would need to satisfy

where we have exploited the fact that the elements of $\overrightarrow{A}\left(1\right)$ corresponding to non-state variables will equal zero, and where $\overrightarrow{A_X}\left(1\right)$ denotes the truncation of $\overrightarrow{A}\left(1\right)$ to elements of the state vector. In words, under this specification, the (one step ahead) conditional output gap growth distribution is normal and fully described by conditional first and second moments that vary systematically with the state variables in a fashion which gives rise to the negative correlation documented in Figure 1.

As Equation 15 makes clear, the precise specification required to ensure a constant 95^th conditional quantile – and hence also a conditional mean – volatility relationship in line with Panel a of Figure 1 – would imply that for some realizations of X_t, V (X_t) would need to become negative. To rule out that possibility, which is inconsistent with V(·)’s interpretation as conditional volatility, we modify the formula in Equation 15 and specify vulnerability as²⁵

where

As shown in Panel b of Figure 1 the resulting specification closely replicates both the negative slope and high R2 of the bivariate regression estimated on conditional moments and reproduced in Panel a.²⁶

We conclude by dealing with a potential criticism: one could reasonably conjecture that our reliance on the vulnerability function provides a solution to an issue that doesn’t arise in more “quantitatively oriented” setups. Figure 2 illustrates why we don’t believe that to be the case, however. First, Panel (a) uses a linear NK model to reiterate the point that any linear or linearized model, irrespective of how “quantitatively relevant” these may be, will not generate a trade-off, because any such model implies a constant conditional volatility. Turning to non-linear specifications spanning the NK model (Panel b), the Bernanke, Gertler, and Gilchrist (1999) model (Panel c) and the Gertler and Karadi (2011) model (Panel d) we also see that popular non-linear models either generate a wrongly signed or much weaker relationship between the conditional moments (the R2 in the data for output growth exceeds 0.5).²⁷

Output Gap Growth Conditional Median and Volatility in Selected DSGE Models

Citation: IMF Working Papers 2020, 236; 10.5089/9781513561066.001.A001

Note: Panels (a) – (d) depict scatter plots of the conditional median (y-axis) and conditional volatility (x-axis) of output growth one quarter ahead in four different models. The dashed grey line shows the slope of the estimated bivariate regression in the data (it is the same in all four panels), while the solid grey line shows the corresponding bivariate regression estimated on data simulated from the models (on a sample of identical length). The specification of the solid grey line, including the corresponding R2, is provided at the bottom of each figure.Since all model solutions were approximated using second order perturbation, which does not allow for analytical expressions for moments, therefore conditional moments were estimated by averaging over 100k simulations, relying on finite sample approximations to the law of large numbers. The magnitude of the corresponding sampling errors can be inferred from Panel (a), where the model is linear and so the true underlying conditional volatilities are constant and lie on a vertical line.

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Output Gap Growth Conditional Median and Volatility in Selected DSGE Models

Citation: IMF Working Papers 2020, 236; 10.5089/9781513561066.001.A001

Note: Panels (a) – (d) depict scatter plots of the conditional median (y-axis) and conditional volatility (x-axis) of output growth one quarter ahead in four different models. The dashed grey line shows the slope of the estimated bivariate regression in the data (it is the same in all four panels), while the solid grey line shows the corresponding bivariate regression estimated on data simulated from the models (on a sample of identical length). The specification of the solid grey line, including the corresponding R2, is provided at the bottom of each figure.Since all model solutions were approximated using second order perturbation, which does not allow for analytical expressions for moments, therefore conditional moments were estimated by averaging over 100k simulations, relying on finite sample approximations to the law of large numbers. The magnitude of the corresponding sampling errors can be inferred from Panel (a), where the model is linear and so the true underlying conditional volatilities are constant and lie on a vertical line.

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Output Gap Growth Conditional Median and Volatility in Selected DSGE Models

Citation: IMF Working Papers 2020, 236; 10.5089/9781513561066.001.A001

Note: Panels (a) – (d) depict scatter plots of the conditional median (y-axis) and conditional volatility (x-axis) of output growth one quarter ahead in four different models. The dashed grey line shows the slope of the estimated bivariate regression in the data (it is the same in all four panels), while the solid grey line shows the corresponding bivariate regression estimated on data simulated from the models (on a sample of identical length). The specification of the solid grey line, including the corresponding R2, is provided at the bottom of each figure.Since all model solutions were approximated using second order perturbation, which does not allow for analytical expressions for moments, therefore conditional moments were estimated by averaging over 100k simulations, relying on finite sample approximations to the law of large numbers. The magnitude of the corresponding sampling errors can be inferred from Panel (a), where the model is linear and so the true underlying conditional volatilities are constant and lie on a vertical line.

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4.4. The Volatility Paradox. An important feature of the data – and the NKV model – is the volatility paradox (Brunnermeier and Sannikov, 2014), which refers to the observation that future risk builds during good times, when contemporaneous risk is low and growth is high. When η is low, indicating loose financial conditions, volatility is low in the short term. But this effect eventually reverts because risk-taking increases during good times, and the economy becomes more vulnerable to shocks as risks continue to build. This is shown in Figure 3 which depicts the elasticity of the conditional mean and conditional volatility of the output gap to η at projection horizons of up to 20 quarters.

The Volatility Paradox

Citation: IMF Working Papers 2020, 236; 10.5089/9781513561066.001.A001

Note: Elasticity of the conditional output gap median and volatility with respect to changes in η. Panel (a) shows estimates of the elasticity, while Panel ( b) shows estimates based on data simulated from the NKV model.

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The Volatility Paradox

Citation: IMF Working Papers 2020, 236; 10.5089/9781513561066.001.A001

Note: Elasticity of the conditional output gap median and volatility with respect to changes in η. Panel (a) shows estimates of the elasticity, while Panel ( b) shows estimates based on data simulated from the NKV model.

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The Volatility Paradox

Citation: IMF Working Papers 2020, 236; 10.5089/9781513561066.001.A001

Note: Elasticity of the conditional output gap median and volatility with respect to changes in η. Panel (a) shows estimates of the elasticity, while Panel ( b) shows estimates based on data simulated from the NKV model.

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While the elasticity of the conditional volatility to financial conditions in the near term is negative, but becomes positive as the projection horizon lengthens, the elasticity of the conditional mean starts positive and then, after around 10 quarters, falls and becomes negative. Even though the extent of the changes in conditional volatility implied by the NKV falls somewhat short of those in the data, we reiterate that the blue line would be exactly flat in any linear (or linearized) model, and so the NKV arguably outperforms those. This was, however, one of the main empirical tensions when specifying the NKV: while we succeeded in identifying specifications in which movements in conditional volatility were more pronounced, these typically implied excessive degrees of unconditional volatility and counterfactually volatile simulations. Investigating whether more quantitatively oriented, and possibly more complex, specifications can succeed in addressing these tensions would, in our view, constitute an interesting extension of our work.²⁸

4.5. The Dynamics and Correlates of Financial Vulnerability. Since the estimation process described previously offered no guarantee that the resulting financial vulnerability function would be economically interpretable or linked to the price of risk therefore this section briefly takes both of these issues head on. To that effect, to obtain a model-implied time series of vulnerability we evaluated the specification in Equation 16 at the historical values of financial conditions and the output gap, with the result plotted as a solid red line in Figure 4. Since we motivated the NKV extension by highlighting the links between the price of risk and spreads, and by stressing the importance of the risk taking channel of monetary policy, we therefore compared our estimate of vulnerability to the term spread (Estrella and Mishkin, 1998) and the real federal funds rate, which are the dashed blue lines in Panels a and b of Figure 4 respectively.

Vulnerability, Spreads and the Risk-Taking Channel

Citation: IMF Working Papers 2020, 236; 10.5089/9781513561066.001.A001

Note: The term spread is the FRED series T10Y3M which is the difference between the 10-Year Treasury Constant Maturity Yield and the 3-Month Treasury Constant Maturity Yield. Data for 1971Q1 to 1981Q4 are (minus) the termspr variable from the data set accompanying Gilchrist and Zakrajˇsek (2012) (the spread series is a quarterly average, expressed in annualized percentage points and not seasonally adjusted). The real federal funds rate (plotted on an inverse scale) is the effective federal funds rate (FEDFUNDS) minus 12-month core PCE inflation (PCEPILFE).

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Vulnerability, Spreads and the Risk-Taking Channel

Citation: IMF Working Papers 2020, 236; 10.5089/9781513561066.001.A001

Note: The term spread is the FRED series T10Y3M which is the difference between the 10-Year Treasury Constant Maturity Yield and the 3-Month Treasury Constant Maturity Yield. Data for 1971Q1 to 1981Q4 are (minus) the termspr variable from the data set accompanying Gilchrist and Zakrajˇsek (2012) (the spread series is a quarterly average, expressed in annualized percentage points and not seasonally adjusted). The real federal funds rate (plotted on an inverse scale) is the effective federal funds rate (FEDFUNDS) minus 12-month core PCE inflation (PCEPILFE).

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Vulnerability, Spreads and the Risk-Taking Channel

Citation: IMF Working Papers 2020, 236; 10.5089/9781513561066.001.A001

Note: The term spread is the FRED series T10Y3M which is the difference between the 10-Year Treasury Constant Maturity Yield and the 3-Month Treasury Constant Maturity Yield. Data for 1971Q1 to 1981Q4 are (minus) the termspr variable from the data set accompanying Gilchrist and Zakrajˇsek (2012) (the spread series is a quarterly average, expressed in annualized percentage points and not seasonally adjusted). The real federal funds rate (plotted on an inverse scale) is the effective federal funds rate (FEDFUNDS) minus 12-month core PCE inflation (PCEPILFE).

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The charts show that vulnerability closely co-moves with both series, with the corresponding correlation coefficients equal to 0.53 and -0.47 respectively. In other words, according to our model, contemporaneous vulnerability tends to be low when the term spread is compressed or when the real effective interest rate is high. As highlighted in Sections 4.4 and 3.3, there is more to the NKV than the contemporaneous correlation, however, with the model also suggesting that vulnerability tends to slowly build over time, i.e., periods of low contemporaneous vulnerability are also the ones in which policy makers ought to carefully consider future risks.²⁹

4.6. Fan Charts. We conclude this section by assessing the degree of amplification built into the model, implicitly testing whether the specification of vulnerability doesn’t imply unrealistic fan chart dynamics. To verify that, Figure 5 compares fan charts from a linear version of the model – in which vulnerability plays no role – to their equivalents generated using a second-order accurate approximation to the solution. In both cases the model is initialized in a synthetic, “high vulnerability” state defined as one in which initial conditions were an average of those in the top percentile of most vulnerable states from a long simulation.

Fan Charts in a High Vulnerability State

Citation: IMF Working Papers 2020, 236; 10.5089/9781513561066.001.A001

Note: The purple fan charts are from a version of the NKV in which the non-linearity associated with vulnerability is effectively disabled by relying on a linear approximation to the solution. These are compared to fan charts generated using a second-order approximation to the solution (red lines), in which endogenous amplification is allowed to play a role. Both models were initialized in a ‘high vulnerability’ state, defined as one in which initial conditions were an average of those in 10,000 states (1%) with highest vulnerability from a 1,000,000 period simulation.

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Fan Charts in a High Vulnerability State

Citation: IMF Working Papers 2020, 236; 10.5089/9781513561066.001.A001

Note: The purple fan charts are from a version of the NKV in which the non-linearity associated with vulnerability is effectively disabled by relying on a linear approximation to the solution. These are compared to fan charts generated using a second-order approximation to the solution (red lines), in which endogenous amplification is allowed to play a role. Both models were initialized in a ‘high vulnerability’ state, defined as one in which initial conditions were an average of those in 10,000 states (1%) with highest vulnerability from a 1,000,000 period simulation.

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Fan Charts in a High Vulnerability State

Citation: IMF Working Papers 2020, 236; 10.5089/9781513561066.001.A001

Note: The purple fan charts are from a version of the NKV in which the non-linearity associated with vulnerability is effectively disabled by relying on a linear approximation to the solution. These are compared to fan charts generated using a second-order approximation to the solution (red lines), in which endogenous amplification is allowed to play a role. Both models were initialized in a ‘high vulnerability’ state, defined as one in which initial conditions were an average of those in 10,000 states (1%) with highest vulnerability from a 1,000,000 period simulation.

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