2. Empirical methodology

We use the LP method described in Jordà (Reference Jordà2005) because it can easily handle state-dependent models.Footnote ¹⁰ Calculating impulse responses using the LP method requires estimation of a series of regressions for each horizon, h, and for each variable of interest. A simple linear version of the LP methodology starts with a sequence of forecasting equations given by

(1) \begin{equation} x_{t+s}=\alpha ^{s}+\sum _{i=1}^{p}\Gamma _{i}^{s+1}x_{t-i}+u_{t+s}^{s} \ \ \ \ \ \ \ \ \ \ s=0,1,\ldots,h, \end{equation}

where $x_{t}$ is a vector of model variables that are forecast $s$ steps ahead for $h$ different forecast horizons, and $p$ is the number of lags in the system identified using an appropriate information criterion. The parameter $\alpha ^{s}$ is an $n \times 1$ vector of constants and $\Gamma _{i}^{s+1}$ denotes an $n\times n$ square matrix of parameters corresponding to the $i$ th lag of $x_{t-i}$ , in the $s$ step ahead forecasting equation. Of particular interest is $\Gamma _{i}^{s+1}$ which is a sequence of coefficients for the $i$ th lag of $x_{t-i}$ estimated locally for $h$ different regression equations over the forecast horizon. It is also important to note that the error term $u_{t+s}^{s}$ is a moving average of the forecast errors from time $t$ to $t+s$ . The IRFs are given by

(2) \begin{equation} \widehat{IR}(t,s,q_{i})= \widehat{\Gamma }_{1}^{s}q_{i} \ \ \ \ \ \ \ \ \ \ s=0,1,\ldots,h, \end{equation}

where $\Gamma _{1}^0=I$ and $q_{i}$ is an $n\times 1$ column vector of an $n\times n$ experimental matrix, $Q$ , that contains the mapping from the structural shock for the $i$ th element of $x_{t}$ to the experimental shocks as defined below.Footnote ¹¹ We construct this mapping matrix following Jordà (Reference Jordà2005), which essentially follows methods used in the traditional VAR literature and begins by estimating a linear VAR and applying a Cholesky decomposition to the variance–covariance matrix. Under this approach, one defines $q_{i}$ as a vector of experimental shocks by

(3) \begin{equation} q_{i}=A_{0}^{-1}\Omega _{\varepsilon }\varphi _{i}, \end{equation}

where $A_{0}$ is a coefficient matrix from a structural form VAR, $\Omega _{\varepsilon }$ is the diagonal variance–covariance matrix associated with the structural shocks, and $\varphi _{i}$ is a column vector with a one in the $i$ th position and zeros elsewhere.Footnote ¹² With this formulation, $q_{i}$ can be interpreted by noting that the term $\Omega _{\varepsilon }\varphi _{i}$ gives a vector with a one standard error shock for the $i$ th variable only in the $i$ th position and with zeros elsewhere. So by multiplying by $A_{0}^{-1}$ , $q_{i}$ can be interpreted as a vector of experimental shocks that arise from a one standard deviation structural shock in the $i$ th variable. This means the IRFs given by (2) show how the vector of variables $x_{t}$ responds to a one standard deviation shock in the $i$ th structural variable at various forecast horizons.

One important property of the LP method is that it is flexible enough to handle state-dependent models easily. In particular, for a state-dependent model with state indicator $I_{t-1}\in \{0,1\}$ , (1) is extended to the sequence of forecast equations given by

(4) \begin{equation} x_{t+s}=I_{t-1}\Bigg [\alpha _{A}^{s}+\sum _{i=1}^{p}\Gamma _{i,A}^{s+1}x_{t-i}\Bigg ]+(1-I_{t-1})\Bigg [\alpha _{B}^{s}+\sum _{i=1}^{p}\Gamma _{i,B}^{s+1}x_{t-i}\Bigg ]+u_{T,t+s}^{s} \quad s=0,1,\ldots,h, \end{equation}

where the notation is mostly the same as in (1), except now there is a subscript $j\in \{A,B\}$ applied to the various parameter coefficients to denote the above-trend leverage and the below-trend leverage states, respectively. These states are described more thoroughly below. Another difference is that there is now an additional subscript $T$ applied to the error term to distinguish it from the nonthreshold model. For our application, the threshold dummy variable, $I_{t-1} \in \{0,1\}$ , indicates the state of the economy in terms of BD leverage before the monetary policy shock hits. This means that $I_{t-1}$ takes a value of $1$ in the above-trend leverage state and $0$ otherwise. Computation of the IRF for the threshold model is generalized to

(5) \begin{equation} \widehat{IR}_{j}(t,s,q_{i})=\widehat{\Gamma }_{1,j}^{s}q_{i} \ \ \ \ \ \ \ \ \ \ s=0,1,\ldots,h, \end{equation}

with normalizations of $\Gamma _{1,j}^0=I$ and $q_{i}$ is the same $n\times 1$ column vector that maps structural shocks to experimental shocks as was used for the non-threshold setting.

We can compute confidence bands using estimates of the standard deviations for the impulses. In doing this, we need to recognize that because the data-generating process is unknown, there could be a serial correlation in the error terms of (1) or (4) induced by the successive leads of the dependent variable. We address this issue by using Newey and West (Reference Newey and West1987)’s standard errors, which correct for heteroskedasticity and autocorrelation (HAC). Letting, $\widehat{\sum }_{s}$ be the estimated HAC corrected variance–covariance matrix of the coefficients $\widehat{\Gamma }_{1}^{s}$ , the confidence interval for each element of the IRF at horizon $s$ can be constructed by $\widehat{IR}(t,s,q_{i})\pm \sigma (q_{i}^{\prime}\widehat{\sum }_{s}q_{i})$ , where $\sigma$ is a $n\times 1$ column vector of ones.

Finally, it is useful to comment on some of the advantages of the LP method relative to methods that have been applied in other regime-switching studies. Unlike the regime-switching VAR models, the LP method does not require one to take a stance on how long a given state lasts. This is because the linear projection makes use of data for all states in formulating a projection, and the duration of any state is implicitly built into the data on which the projection is based. In a threshold VAR, it is assumed that the economy would stay at the same state over the horizon in which the impulse responses are calculated. In contrast, for the LP method, IRFs are a function of the sequence of estimated coefficients for data in all states over the forecast horizon. These coefficients capture the average effects of the shock locally for each period of interest, conditional on the state of the economy one begins the projection from, and because these coefficients are for all states, they reflect the probability of staying in the current state or switching to another state. This explains why, unlike in a VAR, the impulse responses for the LP method are not always smooth. Moreover, in the LP estimation process, the lagged control variables capture the natural transitions in the data between two states due to other factors in the economy, which is absent in a threshold VAR [Ramey and Zubairy (Reference Ramey and Zubairy2018)].Footnote ¹³

2.1 Defining the state variable

We present two baseline state-dependent models which differ only by the statistical method used to compute the trend in BD leverage. In one we use the Hodrick and Prescott (Reference Hodrick and Prescott1997) filter, popularly known as the HP filter, and in another we use methods described in Hamilton (Reference Hamilton2018). Both methods decompose a series into a trended and a nontrended series, and we define the state variable as separating the economy into below-trend BD leverage and above-trend BD leverage states.

The first step for both methods begins by computing a BD leverage series and we follow Adrian et al. (Reference Adrian, Etula and Muir2014) and Istiak and Serletis (Reference Istiak and Serletis2016) by using

(6) \begin{equation} \text{BD Leverage} = \frac{\text{Total financial assets}}{\text{Total financial assets $-$ Total financial liabilities}}. \end{equation}

The BD leverage variable is constructed using the total financial liabilities series (FL664190005.Q) and the total financial assets series (FL664090005.Q), which were obtained from the Flow of Funds account of the Board of Governors of the Federal Reserve System database. To interpret this leverage variable, note that higher values indicate larger liabilities and lower equity, while lower values indicate smaller liabilities and higher equity. Thus, larger values indicate higher amounts of leverage while lower values indicate smaller amounts of leverage. BDs adjust their leverage with the rise and fall of asset prices.Footnote ¹⁴ Note that both assets and liabilities are nonstationary series, so leverage is as well. To confirm it further, we conducted several unit root tests.Footnote ¹⁵ We consistently reject the null hypothesis of stationarity. However, the series are not highly persistent because the BDs’ balance sheets are continuously marked to market.

For the HP filter method, we use a smoothing parameter of $\lambda = 1600$ to construct the trend, which is conventional for quarterly data as suggested by Hodrick and Prescott (Reference Hodrick and Prescott1997). Although the HP filter has been a standard approach for detrending series for years, it has many well-known issues. Recently, Hamilton (Reference Hamilton2018) has suggested an alternative that uses a simple regression of the future value of the series on current and lagged values of the series. We also use this approach for generating a cyclical BD leverage series in order to show that the asymmetric monetary policy implications are not connected to the detrending approach. We follow the recommendation in Hamilton (Reference Hamilton2018) of running a regression of eight quarters ahead future values of BD leverage on current and the next three lagged values of the BD leverage to generate the cyclical component of BD leverage.

Figure 1 shows the decomposition results for both the HP and Hamilton detrending procedures in a side-by-side comparison. In the upper left graph, BD leverage is plotted with the HP trend, while in the upper right graph, BD leverage is plotted with the Hamilton trend. Before commenting on the trended series, let’s first focus on the raw BD leverage series which is presented in both of the upper graphs. One might wonder whether BD leverage is correlated with economic expansions and recessions, and thus serves as a proxy for the business cycle. Figure 1 shows that, at least over the sample used for this study, these two things do not necessarily coincide. For instance, leverage was declining during the mid 1970s and 2008–09 recessions, somewhat flat during the early 1980s recession, but increasing during the 1991 and 2001 recessions. This means leverage should not be interpreted as a proxy for the business cycle. Recently, Tenreyro and Thwaites (Reference Tenreyro and Thwaites2016) used economic recessions and expansions as threshold indicators to show that monetary policy shocks are less effective in stimulating output during recessions. Our modeling of the economy’s state captures a different nonlinearity in the responses to monetary policy shocks.

Figure 1.

BD leverage and its decomposition into trended and cyclical series. The two top graphs plot BD leverage (black lines) and its trend (neon green lines) using both the HP filter (on the left) and Hamilton’s regression filter (on the right). The two bottom graphs plot the respective cyclical components of the BD leverage series, while the shaded regions show the NBER recession dates.

Another interesting feature of the raw BD leverage data is that when leverage gets far above the trend, there is often a sudden snap back toward the trend. Adrian and Shin (Reference Adrian and Shin2014) provided evidence that this sharp fall in leverage reflects the downward adjustments of the BDs balance sheets due to the high VaR. Since the BDs’ balance sheets are continuously marked to market, their responses to asset prices can be interpreted as follows. For leverage values below the trend, equity is relatively high, or VaR is relatively low, implying that BDs have some capacity to borrow through the “risk-taking channel” of monetary policy as described in Borio and Zhu (Reference Borio and Zhu2012) and Bruno and Shin (Reference Bruno and Shin2015). With the excess capacity, BDs expand their balance sheets by borrowing short-term from the REPO (repurchase agreements) market (Adrian and Shin, Reference Adrian and Shin2010). On the other hand, BDs’ VaR is relatively high when leverage is above the trend, which is defined as a situation when the value of assets relative to equity is higher than the average level. Adrian and Shin (Reference Adrian and Shin2010, Reference Adrian and Shin2014) show that in order to minimize the exposure to risks, BDs reduce their leverage by selling assets and bringing it back to desired values. Thus, the distinction between above-trend leverage and below-trend leverage is important and we use this in defining our state variable below. We will often use the notation A to denote above-trend leverage and B to denote below-trend leverage states.Footnote ¹⁶

Now compare the HP trend and the Hamilton trend by looking between the two upper row graphs in Figure 1. These graphs show that the HP trend is quite smooth, while the Hamilton trend is very jagged. The jaggedness of the Hamilton trend is not surprising because this method is designed mostly for the recovery of the cyclical part of a series. Another way to understand this is to note that the Hamilton approach runs a regression of a several step ahead value of the series on current and several lags of the series to generate the trend series. Because the current and lagged values are different at each date, the fitted value, i.e. the trend, is jagged. This is in contrast to the HP approach which uses the entire series to arrive at the trend value. As noted, we used the recommendation in Hamilton (Reference Hamilton2018) of running a regression of eight quarter ahead future values of BD leverage on current and the next three lagged values. If one were to increase the number of lags in the regression, a less jagged trend would be found. However, adding more lags also reduces the number of final detrended values as can also be seen in the plots where the Hamilton method results in both trends and detrended series that start 11 quarters later than the HP series.

The lower two graphs of Figure 1 show the cyclical values of BD leverage for each of these methods. As one would expect, these series are very similar to each other. However, there are some differences. The HP method results in about 52% of the observations being above the trend while the Hamilton method results in about 60% being above trend. Overall, the two methods are in agreement about being above and below trend 66% of the time with most of the disagreements arising around switching points from periods above trend to below trend and vice versa. As one might expect, the HP method with all its persistence built into the structure tends to have greater persistence within a state, while the Hamilton filter results in faster transitions from one state to the next.

With this intuition in mind, we define the threshold dummy variable for both the HP filter and the Hamilton approach, $I_{t-1}$ , by

(7) \begin{equation} I_{t-1} = \begin{cases} 1 & \text{Leverage is above the trend,} \\ \\[-7pt] 0 & \text{Leverage is below the trend.} \end{cases} \end{equation}

As a robustness investigation, described later in the robustness section, we considered some alternative filtering assumptions. For instance, following the credit cycle literature, we choose a high smoothing parameter for the HP detrending because credit cycles are twice as long as business cycles.Footnote ¹⁷ Moreover, we also detrended the leverage series using the Baxter and King (Reference Baxter and King1999) bandpass filter, where we isolate frequencies between 4 and 64 quarters and between 8 and 32 quarters that are consistent, respectively, with credit cycle frequencies and the business cycle frequencies. The qualitative results do not change when using these alternative assumptions because the differences do not yield very different turning points. This is because our estimation technique relies on a dummy variable approach in defining the state of the economy. Hence, these differences in filtering assumptions do not yield very different turning points.

3. Data

In the baseline analysis, we use quarterly data from 1968:Q1 to 2021:Q4.Footnote ¹⁸ One potential concern for the two-sided HP filter with the full sample is the “well-known” endpoint problem. One of the many robustness checks discussed below uses an alternative sample to ensure that the endpoint problem does not pose an issue. For this check, we followed Alpanda and Zubairy (Reference Alpanda and Zubairy2019) by excluding twelve quarters from both the start and end of the sample. Another problem is that the FFR was stuck at the zero lower bound (ZLB) during the periods from 2008:Q4 to 2015:Q4 and from 2020:Q2 to 2021:Q4. Although we could drop the ZLB period data, excluding this period would greatly reduce the sample size. Furthermore, while the ZLB period raises the question of the efficacy of monetary policy during this period, Swanson and Williams (Reference Swanson and Williams2014) show that monetary policy remained effective in influencing the long-term interest rate throughout the ZLB period. In addition, other research shows that the Federal Reserve Bank’s announcements affect asset prices through their effects on financial market expectations of future monetary policy, rather than changes in the current FFR (Gürkaynak et al. Reference Gürkaynak, Sack and Swanson2005). To overcome the nearly zero values observations in the FFR during the ZLB period, we use the shadow FFR constructed by Wu and Xia (Reference Wu and Xia2016). However, as another robustness check, we show that the results are similar if we end the data interval at 2008:Q3, thus excluding the ZLB period.

Our baseline model has five variables. These include the three variables typically found in New Keynesian models, including an output measurement, an inflation measurement and a monetary policy variable. Here, we use the growth rate of real GDP (GDPC1) as our output variable. The inflation rate was computed from the consumer price index for all urban consumers (CPIAUCSL) compiled by the U.S. Bureau of Labor Statistics and the effective federal funds rate (FEDFUNDS) came from the Board of Governors of the Federal Reserve System. These data were obtained from the Federal Reserve Bank of St. Louis (FRED) database. The Wu and Xia shadow rate for the ZLB period was obtained from the Federal Reserve Bank of Atlanta. To these three variables, we add the growth rate of the BD leverage statistic described above and the VIX index. The VIX index, which is available only from 1990:Q1 onward, was obtained from the Chicago Board Options Exchange. For the pre-1990 period, we use the volatility index constructed by Bloom (Reference Bloom2009).Footnote ¹⁹ Finally, we consider several alternative economic specifications and variables for robustness checks.

5. Monetary policy and credit risk

Heretofore the focus has been on what the IRFs and FEVD tell us about the monetary policy effects on economic variables across the leverage cycle. We have seen evidence that the monetary policy transmission mechanism changes depending on the state of the leverage cycle and that during above-trend leverage states, BDs reduce their leverage, which is associated with a rising VIX index in response to expansionary monetary policy and may, in part, explain why stimulative monetary policy is less effective during this phase of the leverage cycle. In this section, we provide further evidence of the strains on BDs by looking at interest rates that BDs are routinely involved with. Here, we focus on the TED spread, which is defined as the difference between the yield on 3-month London Interbank Offered Rate loans (LIBOR) and the U.S. 3-month Treasury rate.Footnote ²⁷ BDs are routine participants in the LIBOR market and an increase in the TED spread is one of the most commonly followed measures of funding conditions that financial intermediaries take into consideration when they decide on their risk exposure.Footnote ²⁸ In normal times, the TED spread is very small, and money flows freely and cheaply between banks because of low default risks.Footnote ²⁹ However, in bad times, the interbank loan rate (LIBOR) rises as banks demand more interest to make up for the rising risk of default. Therefore, a lower TED spread reflects higher liquidity in the market, while a higher spread indicates less willingness to lend by major banks. Here, we show that the TED spread and hence the transmission mechanism of monetary policy depends on the state of BD leverage.Footnote ³⁰

We investigate these connections using a threshold regression model given by

(10) \begin{equation} TED \ Spread_{t}=I_{t-1}\Bigg [\alpha ^{A}+\sum _{i=0}^n\beta _i^{A}\Delta FFR_{t-i}\Bigg ]+(1-I_{t-1})\Bigg [\alpha ^{B}+\sum _{i=0}^n\beta _i^{B}\Delta FFR_{t-i}\Bigg ]+\nu _{t}, \end{equation}

where $\alpha ^{j}$ and $\beta _i^{j}$ are constant and coefficient notations, respectively, and the index $i=0,\ldots n$ is used to indicate lags for the independent variables. We investigate models that have $n=0,\ldots 4$ lags. In addition, we use the index $j\in \{A,B\}$ to distinguish above and below-trend leverage states. Changes in the stance of monetary policy are measured by $\Delta FFR_{t}$ where for date $t$ , $\Delta FFR_{t}$ is the difference between FFR at quarter $t$ and quarter $t-1$ , and $\nu _{t}$ is the error term. The indicator series $I_{t-1}\in \{0,1\}$ are defined as before by using (7), and either the HP filter or the Hamilton filter is used to determine whether BD leverage is above or below trend. For comparison purposes, we also estimate a linear version of (10) which does not include the switching structure. The regressions are run using data from 1986:Q1 to 2021:Q4, and the data for the $TED \ Spread$ (TEDRATE) is obtained from the FRED website. This data interval starts somewhat later than IRF and FEVD analysis because the TED spread data is not available prior to 1986:Q1.

Table 2 shows the results of these exercises. The table is organized into three vertical panels, where the first panel shows the results for various linear models which do not distinguish the above- and below-trend leverage states, while the next two panels show various HP-filtered and Hamilton-filtered state-dependent models. For both linear- and state-dependent models, we included a range of lags, with the first model having no lags ( $n=0$ ) and the next four models incrementally adding additional lags. The table is also organized into three horizontal panels with the top panel (rows beginning with “ $\alpha$ or $\alpha ^B$ ” and ending with “ $\beta _4$ or $\beta _4^B$ ”) having either the linear model coefficient results or the below-trend leverage state coefficient results for the state-dependent models, while the middle panel (rows beginning with “ $\alpha ^A$ ” and ending with “ $\beta _4^A$ ”) has only the above-trend leverage state coefficient results for the state-dependent models. The third horizontal panel (rows beginning with $R^2$ and ending with the Bayesian Information Criterion which is abbreviated as $BIC$ ) reports common regression information, such as $R^2$ , at the bottom of the table. All standard errors are computed using the Newey-West procedure with four lags.Footnote ³¹

Table 2.

Ted spread regression results

Notes: Values in parenthesis are standard errors while ^***, ^* and ^* denote the significance at the 1%, 5%, and 10% level, respectively.

To interpret the results in Table 2, it is important to start by understanding how $\Delta FFR_{t}$ is computed. As noted earlier, $\Delta FFR_{t}$ is the difference between $FFR$ in quarter $t$ and quarter $t-1$ . This means positive values indicate contractionary monetary policy, and negative values indicate expansionary monetary policy. Because the real-world cases where highly leveraged states produced counter-intuitive economic outcomes occurred during the expansionary policy, we will use negative values for the independent variable in our discussion below.

Looking at Table 2, note that the coefficients in the linear model are mostly insignificant, with only the first lag in the two lag model showing a marginal 10% level of significance. This is not surprising because as we will see, it is the state of the leverage cycle that is important.Footnote ³² Next, looking at the state-dependent models, we see similarities between the models using the two detrending methods. Both detrending methods show a few significant positive coefficients for the below-trend leverage states and consistently show significant negative coefficients for the no-lag coefficient in the above-trend leverage states. The positive coefficients in the below-trend leverage states mean that expansionary monetary policy, which has negative values for $\Delta FFR_{t}$ , results in TED spread decreasing. This means expansionary monetary policy can be interpreted as relieving credit stress.

On the other hand, the negative coefficients in the above-trend leverage states mean expansionary monetary policy widens the difference between the LIBOR rate and the Treasury Bill rate, thus aggravating credit stress. This situation likely arises because financial institutions, which are nervous about lending to each other in the REPO market, demand higher collateral because of the risk of default on a loan or a higher market price (LIBOR rate) for taking on the risk. Geanakoplos (Reference Geanakoplos2010) and Gorton and Metrick (Reference Gorton and Metrick2012) show that when collateral requirements increase, leverage falls. As a result, the risk-bearing capacity of the financial system can be diminished, which eventually reinforces further withdrawal of credit and deleveraging. The credit risks can, therefore, be deemed as the empirical link between monetary policy and BDs’ decisions on leverage, which in turn, have macroeconomic implications because increasing credit risks reduce the credit supply to the real sector. Our robustness checks using alternative economic specifications of the baseline model in Section 4.3.6 show that when the leverage is above the trend, credit to the private sector and investment decline significantly following an interest rate cut (seen in Figures A.12 and A.13 in the online appendix). Finally, looking at the bottom panel of Table 2, we see that the $F$ -statistics testing equality of the coefficients in the above-trend leverage and below-trend leverage states are mostly significant, indicating the switching model is appropriate. This means that these different responses to expansionary monetary policy in the two states are important characteristics for measuring credit risk in financial markets.

The results from the IRFs in combination with the results from the TED spread regressions show that there are different monetary policy transmission mechanisms in the above- and below-trend leverage states. Because of the similarities of the below-trend leverage state with the single state results, it is reasonable to infer that the transmission mechanism follows a traditional path in the below-trend leverage state. In particular, lower interest rates stimulate output through the traditional aggregate demand channel where the lower interest rates encourage higher spending on investment goods. On the other hand, the above-trend leverage case is consistent with two opposing demand effects and the dynamics can be thought of in terms of the relative strengths of the two effects. First, lower interest rates stimulate demand through the traditional aggregate demand channel in which the lower interest rates encourage higher investment spending. Second, the lower interest rates induce a chain of risk issues in the REPO markets which cut aggregate demand. In particular, as seen in both the impulse response of the VIX index and the TED spread results, an interest rate cut raises risk in the REPO market which induces BD to reduce VaR by reducing borrowing, resulting in a pullback in aggregate demand. The net above-trend leverage effect is then the sum of these two opposing forces. The IRFs from the HP filter shows that there is virtually no net increase in demand from the interest rate cut. However, the IRF from the Hamilton filter paints a darker picture in that the risk channel dominates the investment spending channel and net demand actually decreases in the above-trend state. The differences in the effects of monetary policy on economic activities between the above-trend state and below-trend state are distinct when considering alternative economic specifications of the baseline model (seen in Figures A.11, A.12 and A.13 in the online appendix).

Recognizing that monetary stimulus works differently in the above-trend leverage case is important to policymakers because of the longer lag associated with monetary policy. In general, monetary policymakers know that demand stimulus from interest rate cuts take time to entice increases in investment spending, so they tend to be patient for the effects of policy to kick in. However, because of the muted or negative effects of policy in the above-trend leverage state, patience will not help if there is no net stimulus to be seen. Because policy is at best ineffective and at worst destructive in the above-trend leverage state, policymakers may wish to avoid this state by taking preemptive actions to prevent excess leverage by BD. Similar recommendations have been made by Geanakoplos (Reference Geanakoplos2010) and Smets (Reference Smets2018).