**4. Conclusions**

In this paper, we have analyzed the effects and transmission of conventional and unconventional monetary policy in the USA. For that purpose, we have proposed a medium- to large-scale model that allows parameters to drift and residual variances to change over time. Our main results remain qualitatively unaffected when considering an alternative measure for banking sector assets, including investment growth as a further transmission channel and using different Cholesky orderings in the estimation stage of the model. These can be summarized as follows:

First, we discuss the monetary policy shock. The rate cut has positive and rather persistent effects on output growth. These are driven by an expansion of asset and deposit growth of the banking sector and thus by a broad credit/bank lending channel. By contrast and in line with previous findings (see, e.g., (Ludvigson et al. 2002)), the wealth channel appears less important for the transmission of conventional monetary policy in the USA. A forecast error variance decomposition lends further support to these findings. More importantly though, we find a pronounced and distinct pattern of monetary policy effectiveness over time. More specifically, our results point to comparably modest effects on output growth in response to a hypothetical and unexpected lowering of the policy rate during the period of the global financial crisis. In this sense, our results corroborate the findings of a recent strand of the literature stating that monetary policy is weak in recessions associated with either high economic uncertainty or more generally financial crises; see, e.g., (Aastveit et al. 2017; Bech et al. 2014; Hubrich and Tetlow 2015; Tenreyro and Thwaites 2016). There is less empirical work on the effectiveness of monetary policy in the aftermath of the global financial crisis, a period in which the main U.S. policy rate was effectively zero. Our results show the strongest responsiveness of the economy to a hypothetical monetary policy shock during that period. From the perspective of a policymaker, this seems less relevant in practical terms, since obviously, the policy rate cannot enter negative territory. However, it is rather the fact that the policy rate has not changed for an extended time than the level at which the policy rate stood that drives this result. If changes in the policy rate are rare, volatility associated with a monetary policy shock is low, and a deviation from the commitment can provide a particularly strong boost to output growth. Note, however, that a central bank's loss function typically consists of other additional targets such as price stabilization, and hence, our finding does not directly translate into a policy recommendation to deviate from a commitment. Still, it suggests that effects of a correction of the monetary policy stance after an extended period of unchanged monetary policy might have large macroeconomic effects.

Second, and looking at the term spread shock, we find positive, but short-lived effects on output and consumer price growth. These work mainly through the consumer wealth channel and via steering inflation, while there is less evidence of impetus via banks' asset and deposit growth. Effects of the term spread shock show also a distinct pattern over time. More specifically, we find that the term spread shock impacts most strongly the output growth during the period of the global financial crisis and less so in its aftermath. Taken at face value, this result implies that the effectiveness of the Fed's unconventional monetary policy measures has abated since the early programs. Smaller effects in the most recent period stem from a decrease in stimulus of consumer wealth and a smaller responsiveness of inflation. These might be attributed to an implicit signaling channel, which is particularly effective when financial markets are impaired and economic conditions are characterized by high uncertainty (Engen et al. 2015). In addition, we show that effects of quantitative easing on investment growth have diminished over time providing, thereby less stimulus for overall GDP growth.

**Author Contributions:** Conceptualization, M.F. and F.H.; methodology, F.H.; software, F.H.; validation, M.F., F.H.; formal analysis, M.F. and F.H.; investigation, M.F.; resources, M.F.; data curation, M.F.; writing—original draft preparation, M.F.; writing—review and editing, F.H.; visualization, M.F.; supervision, M.F. and F.H.; project administration, M.F.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Structural Identification**

To implement the sign restrictions technically, note that Equation (8) can be written as:

$$\mathbf{A}\_{t}\mathbf{y}\_{t} = \mathbf{c}\_{t} + \sum\_{j=1}^{p} \mathbf{B}\_{jt}\mathbf{y}\_{t-j} + \mathbf{A}\_{t}^{0.5}\mathbf{v}\_{t\prime} \tag{A1}$$

where **Λ** = **Λ**0.5 *t* **Λ**0.5 *t* and *vt* ∼ N (**0**, *<sup>I</sup>m*) is a standard normal vector error term. Multiplication from the left by **Λ**−0.5 *t*yields:

$$\bar{A}\_{t}y\_{t} = \bar{\mathfrak{c}}\_{t} + \sum\_{j=1}^{p} \mathfrak{B}\_{jt} y\_{t-j} + \mathfrak{v}\_{t} \tag{A2}$$

with *A* ˜ *t* = **Λ**−0.5 *t At*, ˜*ct* = **Λ**−0.5 *t ct* and *B*˜ *jt* = **Λ**−0.5 *t <sup>B</sup>jt*.

It can be shown that left multiplying Equation (A2) with an *m* × *m*-dimensional orthonormal matrix *R* with *RR* = *Im* leaves the likelihood function untouched. This implies that impulse responses are set-identified. To implement the sign restrictions approach, we simply draw *R* using the algorithm outlined in Rubio-Ramírez et al. (2010) until the impulse response functions satisfy a given set of sign restrictions to be chosen by the researcher. This has to be done for each draw from the posterior, which in our application boils down to 500 draws randomly taken from the full set of 15,000 posterior draws. To speed up computation, we do not search for each point in time a new rotation matrix. Instead, we look for new rotation matrices after 10 quarters and check whether the restrictions are fulfilled throughout the sample. These leaves us with 11 time periods for which we look for new rotation matrices. For each of these time points, we recovered 250–300 rotation matrices that fulfilled our restrictions. There was no visible time pattern over the amount of sign restrictions recovered throughout our sample period.

To impose the additional restriction that the short-term interest rate reacts sluggishly with respect to an unconventional monetary policy shock, we construct the following deterministic rotation matrix (Baumeister and Benati 2013):

$$\mathbf{S} = \begin{pmatrix} I\_{m-2} & \mathbf{0}\_{m-2 \times 2} \\ \mathbf{0}\_{2 \times m-2} & \mathbf{U} \end{pmatrix} \tag{A3}$$

with:

$$\mathbf{U} = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}. \tag{A4}$$

The rotation angle is defined as:

$$\vartheta = \tan^{-1}([\bar{A}\_t \mathbf{R}']\_{\bar{\imath}\jmath} / [\bar{A}\_t \mathbf{R}']\_{\bar{\imath}\imath}). \tag{A5}$$

Here, the notation [*A*˜*tR*]*ij* selects the *i*, *j*-th element of the impact matrix, corresponding to the contemporaneous response of variable the short-term interest rate (variable *i*) to an unconventional monetary policy shock (variable *j*). Multiplying the impact matrix with *U* from the right yields a new impact matrix that satisfies the set of sign restrictions specified in Section 2.4 and the zero impact restriction described above.

Since we assume that the central bank is constrained by the zero lower bound, we zero-out the structural coefficients of the monetary policy rule for the first eight quarters after the shock hit the economy. This procedure, however, is subject to the Lucas critique because economic agents are not allowed to change their behavior accordingly. However, the findings in Baumeister and Benati (2013) sugges<sup>t</sup> that the differences between the results obtained by manipulating the structural coefficients or by manipulating the historical structural shocks to keep the interest rate at the zero lower bound are quite similar. Moreover, manipulating the structural shocks gives rise to additional shortcomings like the fact that this approach ignores the impact of agents expectations about future changes in the policy rate. In addition, the systematic component of monetary policy implies that the short-term interest rate reacts to different shocks. However, the unsystematic part, by construction, offsets this behavior, and the corresponding shocks would no longer originate from a white noise process.

#### **Appendix B. A Brief Sketch of the Markov Chain Monte Carlo Algorithm**

Since we impose a Cholesky structure on the model a priori and estimate the system equation-by-equation, our Markov chain Monte Carlo (MCMC) algorithm consists of the following three steps:


Step 1 is a standard application of Gibbs sampling in state-space models. In Step 2, we draw the parameters of the corresponding state equations conditional on the states. Step 3 is described in more detail in the Appendix. Finally, note that we sample the parameters of the different equations simultaneously.

#### **Appendix C. Sampling Log-Volatilities**

To simulate the full history of log-volatilities for the *i*-th equation *hTi* = (*hi*1, ... , *hiT*), we use the algorithm outlined in Kastner and Frühwirth-Schnatter (2013). This algorithm samples *hTi* , all without a loop. This is achieved by rewriting *hTi* in terms of a multivariate normal distribution. Moreover, the parameters of the state equation in Equation (7) are sampled through simple Metropolis–Hastings (MH) or Gibbs sampling steps. To achieve a higher degree of sampling efficiency, we sample the corresponding parameters from the centered parameterization in Equation (7) and a non-centered variant given by:

$$
\tilde{h}\_{it} = \rho\_i \tilde{h}\_{it-1} + \varepsilon\_{it\prime} \,\, \epsilon\_{it} \sim \mathcal{N}(0, 1). \tag{A6}
$$

To simplify the exposition, we illustrate the algorithm for the case when *i* = 2, ... , *m*. For *i* = 1, the same steps apply with only minor modifications. Let us begin by rewriting Equation (4) as:

$$
\varepsilon\_{it} = \varepsilon\_{it} - \sum\_{s=1}^{i-1} a\_{is,t} y\_{st} - \sum\_{j=1}^{p} b\_{ij,t} y\_{t-j} = \lambda\_{it}^{0.5} \varepsilon. \tag{A7}
$$

Squaring and taking logarithms yield:

$$e\_{it}^2 = h\_{it} + \ln(u\_{it}^2). \tag{A8}$$

Since ln(*u*2*it*) follows a *χ*<sup>2</sup>(1) distribution, we use a mixture of Gaussian distribution to render Equation (A8) conditionally Gaussian,

$$
\ln(u\_{it}^2)|r\_{it} \sim \mathcal{N}(m\_{it}, \mathbf{s}\_{it}^2),
\tag{A9}
$$

where *rit* is an indicator controlling the mixture component to use at time twith *rit* ∈ {1, ... , <sup>10</sup>}. *mit* and *s*2*it*define the mean and the variance of the mixture components employed.

The mixture indicators allow us to rewrite Equation (A8) as a linear Gaussian state space model:

$$
\epsilon\_{it}^2 = m\_{i\tau,t} + h\_{it} + \mathfrak{J}\_{it\prime} \; \mathfrak{J}\_{it} \sim \mathcal{N}(0, \mathfrak{s}\_{i\tau,t}^2). \tag{A10}
$$

The algorithm then consists of the following steps.

1. Sample *<sup>h</sup>i*,<sup>−</sup><sup>1</sup>|*rit*, *μi*, *ρi*, *σih*, **Ψ***it* or ˜ *hij*,<sup>−</sup><sup>1</sup>|*rij*, *ρi*, *σih*, Ψ*it*, all without a loop (AWOL). Here, **Ψ***it* = (*cit*, *ais*,*t*, ... , *aii*−1,*t*, *bi*1,*t*, ... , *<sup>b</sup>ip*,*<sup>t</sup>*) is a vector of stacked coefficients and *hi*,<sup>−</sup><sup>1</sup> = (*hi*2, ... , *hiT*). Following Rue (2001), *hi*,<sup>−</sup><sup>1</sup> can be written in terms of a multivariate normal distribution:

$$h\_{i,-1} \sim \mathcal{N}(\boldsymbol{\Omega}\_{h\_i}^{-1}\boldsymbol{c}\_i, \boldsymbol{\Omega}\_{h\_i}^{-1}).\tag{A11}$$

Similarly, the normal distribution corresponding to the non-centered parameterization is given by:

$$
\tilde{\mathbf{h}}\_{i\_\*-1} \sim \mathcal{N} \left( \tilde{\mathbf{D}}\_{h\_i}^{-1} \tilde{\mathbf{c}}\_i, \tilde{\mathbf{D}}\_{h\_i}^{-1} \right). \tag{A12}
$$

The corresponding posterior moments are:

$$
\boldsymbol{\Omega}\_{k\_{i}} = \begin{pmatrix}
\frac{1}{\sigma\_{ij,2}^{2}} + \frac{1}{\sigma\_{ih}^{2}} & \frac{-\rho\_{i}}{\sigma\_{ih}^{2}} & 0 & \cdots & 0 \\
0 & -\frac{\rho\_{i}}{\sigma\_{ih}^{2}} & \ddots & \ddots & 0 \\
\vdots & \ddots & \ddots & \frac{1}{\sigma\_{i\_{l},T-1}^{2}} + \frac{1+\rho\_{i}}{\sigma\_{ih}^{2}} & \frac{-\tau\_{ij}}{\sigma\_{ih}^{2}} \\
0 & \cdots & 0 & -\frac{\rho\_{i}}{\sigma\_{ih}^{2}} & \frac{1}{\sigma\_{i\_{l},T}^{2}} + \frac{1}{\sigma\_{ih}^{2}}
\end{pmatrix} \tag{A13}
$$

and:

$$\mathbf{c}\_{i} = \begin{pmatrix} \frac{1}{s\_{r\_{ij,2}}^2} (\bar{y}\_{ij,2}^2 - m\_{r\_{ij,2}}) + \frac{\mu\_i (1 - \rho\_i)}{\sigma\_{ik}^2} \\ \vdots \\ \frac{1}{s\_{r\_{ij,T}}^2} (\bar{y}\_{ij,T}^2 - m\_{r\_{ij,T}}) + \frac{\mu\_i (1 - \rho\_i)}{\sigma\_{ik}^2} \end{pmatrix} . \tag{A14}$$

Multiplying by *σ*2*ih* yields the moments for the non-centered parameterization: **Ω**˜ *i* = *<sup>σ</sup>*2*ih***Ω***hij* and *c*˜*ij* = *<sup>σ</sup>*2*ih<sup>c</sup>ij*. Finally, the initial states of *hTi* , *hi*1 and ˜*hi*1 are obtained from their respective stationary distributions.


$$c\_{it}^2 - h\_{it} = \mathfrak{f}\_{it\prime} \; \mathfrak{f}\_{it} \sim \mathcal{N}(m\_{ir,t\prime}s\_{it}^2). \tag{A15}$$

This allows us to compute the posterior probabilities that *rit* = *j*, which are given by:

$$p(r\_{it} = c | \bullet) \propto p(r\_{it} = c) \frac{1}{s\_{ik}} \exp\left(-\frac{\left(\frac{\tilde{\mathbf{c}}\_{it} - m\_{ik}}{2s\_{r\_{it}}^2}\right)}{2s\_{r\_{it}}^2}\right),\tag{A16}$$

where *p*(*rit*= *c*|•) are the unnormalized weights associated with the *c*-th mixture component.

The algorithm simply draws the parameters under both parametrizations and decides ex-post which of the parametrizations to use. This choice depends on the relationship between the variances of Equations (7) and (A8). For more information, see Kastner and Frühwirth-Schnatter (2013) and Kastner (2013).

The sampled log-volatilities are shown in Figure A1.

**Figure A1.** Stochastic volatility over time. Notes: Posterior mean of residual variance over time.

Reduced form volatility of the short-term interest rate and the term spread has increased considerably in the run-up of the global financial crisis, a period during which the Fed has aggressively lowered interest rates. Volatility has spiked around mid-2008 and hence in the midst of the crisis. While the crisis peak of residual variance associated with the short-term interest rate marked also the peak over our sample period, volatility of the term spread peaked in the early 1990s.

The middle panel of Figure A1 shows the volatilities for variables related to the real side of the economy. Residual variance associated with real GDP growth was elevated in the early 2000s and peaked around the same time as the financial variables discussed above. During the early 2000s, the so-called "dot-com bubble" burst, causing the slowing down of the U.S. economy. Stochastic volatility of wealth, which is strongly anchored on movements in stock market prices, naturally was also elevated during that period. In contrast to the volatility of real GDP, residual variance of wealth was pronounced for a longer period during the global financial crisis. Residual variance of CPI inflation started to rise more considerably from the beginning of the 2000s until 2008, a period that was characterized by sound growth in price dynamics in the USA. Residual variance peaked in the aftermath of the crisis and hence a little later than that associated with real GDP growth, when CPI inflation reverted from negative to positive territory.

Last, the bottom panel of Figure A1 shows residual variance for variables related to the banking sector. Residual variance of asset growth of commercial banks was elevated during the early 2000s and the global financial crisis, where it peaked around the same time as residual variance of real GDP growth, short-term interest rates and the term spread. Since 2009, estimated volatility has declined and is considerably smaller in the most recent period in our sample compared to its peak value. Residual variance associated with bank deposits and net interest margins show a slightly different pattern. Bank deposit volatility increased gradually from the beginning of 2004 until 2009, after which it gradually started to decline until the end of our sample period. Volatility associated with net interest margins spiked around 1997 and peaked in late 2009. That is, for both variables, banking deposits and net interest margins, volatility spikes during the global financial crisis occurred slightly later than those of the other variables considered in this study.
