Optimal Macroeconomic Policy in Nonlinear Models: A VSTAR Perspective

1.1 Related Literature

The paper makes several contributions to the literature. The methodological part is related to the research on nonlinear time series modelling, which has long established methods for the specification, estimation and structural analysis of VSTAR, see Granger et al. (1993), Dijk et al. (2002) and Martin et al. (2013). The paper complements this literature, tackling a different task, the analysis of systematic policy. The methodological part also contributes to the macroeconomic literature on optimal decision analysis. As mentioned above, the predominant paradigm here is still the LQR, which applies to linear economic environments, see Ljungqvist and Sargent (2018). The approach adopted in the paper illustrates how to bring the same LQR techniques to nonlinear economic models.

The paper also contributes to three strands of applied econometrics. The first is the analysis of the wider effects of QE and the ZLB based on VAR or semi-structural models. Examples include Lenza et al. (2010), Kapetanios et al. (2012), Gambacorta et al. (2014), Dahlhaus et al. (2018). These works evaluate what the unconventional monetary policy undertaken by the Fed since the Great Recession managed to achieve relative to a scenario of no-intervention. The paper complements this literature evaluating what the actual QE and ZLB policy could have achieved, using as reference the framework from the optimal policy. The second is the applied literature on macroeconomic asymmetries and the transmission channels of QE and the ZLB, typified by the works of Guerrieri and Iacoviello (2017) and Sims and Wu (2020); Sims et al. (2023). The present paper shares with these works a similar emphasis on the dual source of nonlinearity, from the economy and the policy sectors, but for the purpose of empirical analysis. The third is the literature using VSTAR-type models to account for nonlinearity in macroeconomic time series, whose applications spans from business cycle, Terasvirta and Anderson (1992), international spillovers, Gefang and Strachan (2009), fiscal policy, Auerbach and Gorodnichenko (2012) (conventional) monetary policy, Dahlhaus (2017), and credit markets, Atanasova (2003). The paper contributes to this literature using a VSTAR for the analysis of QE and the ZLB.

There is a growing literature that develops dynamic stochastic general equilibrium (DSGE) models to study monetary policy when the short term interest rate is constrained by the zero lower bound and the Fed embarks on QE. This literature considers actual and/or optimal QE intervention targeting either private assets, through outright purchases or collateralized short-term loans, as for example in Del Negro et al. (2017), or long term government bonds, as for example in Sims and Wu (2021). The present paper contributes to this literature considering in a reduced form framework the design of QE programmes that simultaneously involve private securities and government bonds purchases. Being reduced form, the analysis is unable to explain the economic mechanisms that may explain why each type of QE is required and desired. It is also silent about the distortions that these purchases may induce on private sector decisions. The main advantage of taking a reduced form approach is that it facilitates the empirical analysis of the effects of conventional and unconventional monetary policy despite the large data instabilities occurred during both the build-up of QE and its first phase of unwinding since 2015, allowing to compare and contrast the effects of different types of asset purchases, how these policies interact with each other and quantify their outcomes in terms of stabilization gains.

This paper is also related to a recent literature that uses DSGE models to conduct counterfactual policy analysis in reduced-form or semi-structural frameworks that aim to be robust to the Lucas critique, see, Barnichon and Mesters (2023), Beraja (2023) and McKay and Wolf (2023). While the methodology in this paper does not explicitly aim to insulate policy analysis from the Lucas critique, it is implemented in a way that attempts to minimize the quantitative relevance of the critique. The methodologies in this recent literature have also limited applicability. Beraja (2023)’s approach only applies to DSGE models that can be linearized, thereby excluding nonlinearities induced by ZLB or leverage constraints that typically features DSGE models with QE. Barnichon and Mesters (2023)’s sufficient statistics cannot be used to compute the optimal policy if there is not enough empirical evidence to estimate the responses to all policy news shocks (The authors also discuss circumstances in which their optimal policy perturbation approach is not applicable to nonlinear models). The identification result of McKay and Wolf (2023) also relies on linearity and is not suited to study change in the steady state. In contrast, the approach proposed in this paper has the advantage of being applicable to a much broader set of nonlinearities and policy experiments, as demonstrated in the empirical application.

The paper is organized as follows. Section 2 describes the reduced-form VSTAR and the two VSTARX representations under A1 and A2. Section 3 illustrates the SDC method and derives the VSTAR under control. Sections 4 and 5 present the results from the estimated VSTAR and the analysis of optimal policy, respectively. Section 6 concludes with a summary. Supplementary material and robustness results are in Appendix.

2.1 Reduced Form

Let y t = x t ′ u t ′ ′ be a n × 1 vector, with x _t denoting a p × 1 vector of variables describing the economy and u _t a m × 1 vector of policy instruments. The dynamic of y _t for t ≥ 1 is described by the VSTAR:

where y₀ is given; λ _j are vectors of constant coefficients of dimensions n × 1; Λ j L denote q lags of the vector y _t , i.e. Λ j L = ∑ k = 1 q Λ j k L k with each Λ _jk being a n × n matrix of coefficients and L the lag operator; v _t is a vector of independently and identically distributed reduced-form disturbances with covariance matrix Ω _t given by:

where Ω _j are symmetric matrices of coefficients, j = 1, …, 4. The terms g i s u ′ y t and l i s x ′ y t denote continuous (transition) functions, bounded between zero and one, capturing the effect of nonlinearity on the transmission mechanism of shocks in the mean, i = M, and covariance matrix, i = V, of the VSTAR in (1) and (2), s _x and s _u being selection vectors identifying transition variables from x _t and u _t , respectively. In particular, l i s x ′ y t links variations in the VSTAR coefficients to changes in the state of the economy, while g s u ′ y t links variations in the VSTAR coefficients to changes in the policy stance.

The VSTAR model in (1)-(2) provides a natural framework to study monetary policy in non-linear macroeconomic environments. The idea of considering dual nonlinearity, from the economy and the policy sectors, is not new. It features in recent works of Guerrieri and Iacoviello (2017) and Sims and Wu (2020); Sims et al. (2023) based on DSGE models where nonlinearity stems from the private sector, due to collateral constraints on borrowing, and from the policy sector, due to the ZLB. The specified VSTAR shares with these works a similar emphasis on the dual nonlinearity, with the purpose however of evaluating its significance for empirical analysis.

2.2 VSTARX Form

The VSTAR in (1)-(2) is a reduced form, suitable for econometric estimation and dynamic analysis like forecasting. The coefficients in the equations for the policy instruments describe the so-called actual policy rules. The coefficient in the equations for the economy vector describe how the economy evolves over time subject to the actual policy rule.

To quantify the impact of a change in policy, the coefficients in the equations for the instruments need to be replaced with those from another policy rule, either ad-hoc or optimal in some sense. This requires transforming the VSTAR into a VSTARX where the policy variables are treated as exogenous. As in a VAR, this transformation requires only a zero block factorization (sufficient condition) of the reduced-form covariance matrix, to separate the policy vector from the economy vector. Thus, there are (at least) two possible representations depending on whether the policy variables are ordered first or last.^[3]

To elucidate the structure of, and the differences between, these two VSTARX forms, consider partitioning the mean equation (1) to separate the economy from the policy instruments.^[4] The covariance matrix (2) can be conformably partitioned as:

Under A1, the economy reacts with a lag to a change in policy, requiring Ω _xut = 0 in (3).

Under A2, the economy reacts within the same period, thus requiring Ω _uxt = 0.

To start, consider applying A1 and A2 to the VAR obtained after all transition functions in the VSTAR are set equal to zero. This yields:

The above are two VARX models in which the vector of policy instruments u enters on the right side as an exogenous variable.^[5] These two VARX differ in terms of the interaction between the economy and the policy instruments, being with a lag under A1 and contemporaneous under A2. Using either of them as constraint for optimization makes no difference in terms dimensionality, since a change in the policy vector has a direct effect on the endogenous variables only through the time-invariant coefficients Γ ̃ 1 and Γ ̂ 1 under A1 and A2, respectively. Further, disturbances are homoscedastic in both models. Thus a change in policy has not effect on the covariance matrices Ω ̃ and Ω ̂ , implying that certainty-equivalent policy rules can be calculated using both A1 and A2.

Applying A1 to the VSTAR, yields:

where ϕ j = λ x j ′ 0 ′ ′ ; Φ j L = Λ x x j L Λ x u j L − Λ x u 1 j 0 0 , with (i) Λ x x j L and Λ x u j L denoting partitions of the responses of the economy variables, Λ x j L , to their own lagged values and to the policy instruments, respectively, and (ii) Λ x u 1 j L the responses of the economy variables to the first lag of the policy instruments; Γ j = Λ x u j 1 ′ 0 ′ ′ is the response of the economy variables to the first lag of the policy instruments; j = 1, …, 4; and e t = v x t ′ 0 ′ ′ ∼ i . i . d . 0 , Σ t .

Conversely, applying A2 to the VSTAR yields:

where ϕ j t = λ x j − G 12 t λ u j ′ 0 ′ ′ , Φ j t L = Λ x j L − G 12 t Λ u j L 0 , j = 1, …, 4, e t = v x t − G 12 t v u t ′ 0 ′ ′ ∼ i . i . d . 0 , Σ , and σ x 2 denotes the variance of v _xt − G_12tv _ut .

The nonlinear dynamic of y _t under both (4) and (6) describes a VSTAR with exogenous variables, i.e. a VSTARX, since the policy vector u _t enters on the right side of both models as an exogenous variable.^[6] Therefore, both (4) and (6) can be used to represent the nonlinear economy constraint faced by a regulator in charge of setting u _t . When the objective of the regulator is approximated by a quadratic function, the resulting problem is referred to as the NLQR problem.

It is useful to point out the differences between (4)-(5) and (6)-(7) due A1 and A2. The VSTARX in (4)-(5) corresponds to the equations for the economy vector x _t in the reduced-form VSTAR (1)-(2). In contrast, the VSTARX in (6)-(7) is obtained by extrapolating from the reduced-form covariance matrix (3) the contemporaneous response of the economy to policy and then incorporating this into the systematic part of x _t .

There are three further differences, specific to the nonlinear nature of the VSTAR. The first refers to the response coefficient of the economy variables x _t to changes in the policy vector u _t . In (4) this reflects the nonlinearity in the mean of the VSTAR as it is given by Γ j = Λ x u j 1 ′ 0 ′ ′ , j = 1, …, 4. In (6), this is determined by the nonlinearity in the covariance matrix of the VSTAR as it is given by Γ _t in (7). The second refers to the coefficients ϕ , Φ and Γ. In (4) these are fixed, and it is their nonlinear combination that varies across states, reflecting the state-dependent nature of the VSTAR coefficients. In (6), these coefficients are time varying and written as ϕ _jt , Φ _jt and Γ _jt , j = 1, …, 4, since they are re-calculated to incorporate time variation from the reduced-form covariance matrix. The third difference refers to the structure of the covariance matrix of the disturbances in the two representations: this is time varying in (5), but constant in (7).

Both (4)-(5) and (6)-(7) can be employed as constraints in a NLQR problem. The VSTARX in (4)-(5) has however two main drawbacks. First, the NLQR solution becomes highly dimensional. This is because the coefficients in (4) depend on the lagged value of the policy vector, the variable set by the regulator. Thus changes in u _t affect on impact the entire system through Γ _j and g M s u ′ y t . Second, the solution no longer satisfies certainty equivalence, since changes in the policy vector also affect the covariance matrix in (5). The VSTARX in (6)-(7) yields a solution that has lower dimensionality compared to (4)-(5). This is because changes in the policy vector u_t+1 affect the economy through Γ _j but not g M s u ′ y t , which is pre-determined when policy choices are made. In addition, the VSTARX in (6)-(7) satisfies the certainty-equivalence principle, because the disturbances in (6) have constant covariance given by (7). In other words, the representation in (6)-(7) results in a low-dimension and certainty-equivalent NLQR.

The empirical evidence on optimal monetary policy analysis based VAR, finds that the two timing protocols of A1 and A2 in general do not result in large quantitative differences, see Polito and Wickens (2012). Given the methodological advantages highlighted above, the NLQR problem is thus specified and solved in the next section using the VSTARX based on A2 in (6)-(7). A further reason for preferring A2 is that a reduced-form policy experiment could miss the timing of the policy impact on the economy, as forward-looking agents in the private sector can be expected to incorporate information about the implementation of policy at the time of announcement, see Dahlhaus et al. (2018). As also argued by Stock and Watson (2001), A2 minimizes this shortcoming by at least allowing the economy to react within the same period of a change in policy.^[7]

3.1 Specification

The problem of setting optimal policy is specified as a closed-loop regulator determining the sequence of policy instruments u t t = 1 ∞ that minimizes the quadratic loss function

where E₀ denotes mathematical expectation conditional on information at t = 0; β ∈ 0,1 is a discount factor; y ̄ is a vector of targets; Q is a symmetric positive semidefinite matrix of coefficients. Given y₀, the regulator sets u t t = 1 ∞ subject to the constraints in (6)-(7).^[8]

The above NLQR problem has not closed-form solution. One option is to linearize (6) so that the NLQR problem effectively becomes a LQR and a closed-form solution can be calculated with linear dynamic programming, for example using Riccati equation iteration as in Ljungqvist and Sargent (2018). The alternative is to solve the problem numerically using the nonlinear model of the economy as it is or a higher-order approximation. Linearization would somehow be inconsistent with the aim of this paper, i.e. studying the significance of macroeconomic nonlinearity for optimal policy analysis. The main drawback of numerical methods is that they can be computationally intensive and there is not guarantee about their optimization performance in highly nonlinear models. This is because direct search methods can easily default on local minima while gradient methods can quickly loose efficiency once differentiation becomes either intractable or very high dimensional due to nonlinearity. Luukkonen et al. (1988) show how to employ Taylor-series expansion of the transition function of the VSTAR to derive a polynomial approximation that can be used to test for nonlinearity in the data. With this approach, however, the economy constraint of the NLQR would remain still highly nonlinear, even when using a first-order approximation of the transition function.

The SDC method extends Riccati equation iteration used in LQR to the nonlinear case. This is done by approximating the NLQR problem into a sequence of one-period LQR problems that can be solved independently. The SDC method has the advantage of being numerically simpler than many other nonlinear optimization techniques, as well as being closely related to the well-understood Riccati equation method used for LQR. The SDC method is widely applied in engineering and data science.^[9] The next subsections describe how to adapt it for optimal macroeconomic policy analysis with a VSTAR.

3.2 The SDC Method

Applying the SDC method to solve the NLQR problem of minimizing (8) subject to (6)-(7) requires two steps. The first consists of employing SDC factorization to transform the nonlinear model (6) into an affine structure with SDC matrices. As a result, the NLQR problem can be written in a form that is isomorphic to the LQR problem and it can be solved with linear dynamic programming techniques. The second step consists of solving the NLQR problem iterating on the resulting SDC solutions. This yields a feedback rule for the policy instruments with time-varying response coefficients that can be combined with the original constraint to derive the VSTAR under control.

4.1 Data

The VSTAR is estimated using aggregate monthly data for the United States from 1979:8 to 2018:10.^[14] In the benchmark specification, the endogenous vector comprises seven variables: an indicator of credit risk, the spread between the Baa corporate rate and the 10-year treasury rate (s); the inflation rate (p), measured as the annual change in the personal consumption expenditures deflator; the annual growth rate of industrial production (ip); the difference between the actual and the natural rate of unemployment, the unemployment gap (ug); Treasury securities held by the Fed as a proportion to GDP (TS); private securities held by the Fed as a proportion to GDP (PS); and the federal fund rate (R).^[15] The selected measure of credit risk is used as a proxy for financial conditions in the macroeconomy.^[16] Inflation and the unemployment gap are used as indicators of the Fed targets of price stability and maximum employment, respectively, as in Williamson (2015). Industrial production is used as a proxy for aggregate output, following Sack (2000). The sum of TS and PS equals total assets held in the Fed’s balance sheet. Thus the VSTAR includes conventional and QE monetary policy tools, and can account for variation in both the size of the Fed’s balance sheet and the composition of its assets portfolio.

As in Gambacorta et al. (2014), the total size of the Fed’s balance sheet is used to proxy QE. The distinction between TS and PS captures the effects of QE on aggregate demand through the portfolio-balance and the credit channels, respectively, highlighted by the economic theory. Kuttner (2018) supports this distinction on the ground that it characterizes the conduct of QE in the United States since 2008, which involved large purchases of PS in the immediate aftermath of the Great Recession, and of TS afterwards.^[17]

Figure 1 plots the data, with NBER recession periods highlighted in grey for reference. with NBER recessions highlighted in grey. Several clear irregularities are visible over the sample period, such as the large spikes in the credit risk, inflation, industrial production and unemployment gap; the asymmetric dynamics of the unemployment gap above and below zero; the large transitional changes in the two QE variables around the Great Recession; and the flat path of the federal fund rate while at the ZLB. It is the presence of these irregularities that motivates the use of a VSTAR.^[18]

4.2 Specification

Variables are ordered as y t ′ = [ s t p t i p t u g t T S t P S t R t ] . The transition functions are specified as follows. l i s x ′ y t is set to capture nonlinearity due to the amount of slack in the economy, measured by the unemployment gap. Thus s x ′ = [ 0 0 0 1 0 0 0 ] and l i s x ′ y t = l i u g t . This is similar to Ramey and Zubairy (2018), except that in their work the natural unemployment rate is constant, while it is time varying here following the definition of the unemployment gap of Williamson (2015).^[19] g i s u ′ y t is set to capture changes in the VSTAR parameters depending on whether or not the economy is near to the ZLB. Thus s u ′ = [ 0 0 0 0 0 0 1 ] and g i s u ′ y t = g i R t .

Transition across states due to nonlinearity in the economy is modelled using a logistic function l i u g t : − ∞ , + ∞ → 0,1 specified as: l i u g t = 1 + exp − γ l i u g t − c i − 1 , where γ _li > 0 denotes the speed of adjustment across states for the parameters of y _t , i = M, and Ω _t , i = V; c _i is a threshold parameter. Since the federal funds rate cannot be negative, transition across states due to nonlinearity in the policy vector is instead modelled with the incomplete gamma function g i R t : 0 , + ∞ → 0,1 described as: g i R t = 1 Γ γ g i ∫ 0 R t e − t t γ g i − 1 d t , where Γ γ g i is the gamma function with shape parameter γ _gi for y _t , i = M, and Ω _t , i = V; and the implied scale parameter has been set equal to one. Logistic functions are widely employed in the applied macroeconomic literature to capture nonlinearity stemming from the economy, see, e.g., Auerbach and Gorodnichenko (2012), Galvão and Owyang (2018), Ramey and Zubairy (2018). The incomplete gamma function is used to capture nonlinearity driven by a variable that is nonnegative, being for example volatility, as in Lanne and Saikkonen (2005), or the nominal rate of interest, as in Hurn et al. (2022). To discipline the economic interpretation, the threshold parameters are set equal to zero, i.e. c _i = 0, i = M, V. This means that nonlinearity stemming from the economy vector depends entirely on whether the actual rate of unemployment is above or below the natural rate, i.e. the amount of slack in the economy.

To ensure positive definiteness on the variance covariance matrix in (2) and on its determinant, each matrix of coefficients Ω _j is decomposed as

where D _j is a lower triangular matrix, j = 1, …, 4.

4.3 Estimation

Given a sample of t = 1, …, T observations and the i.i.d. assumption on the disturbances v _t , the log-likelihood function of the VSTAR in (1)-(2) conditional on the first q observations is formed as:

where Π is a vector collecting all VSTAR parameters in equations (1), (2) and (13). It is convenient to partition the vector of parameters as Π = [Π₁, Π₂], where Π₁ = {λ₁, Λ₁(L), λ₂, Λ₂(L), λ₃, Λ₃(L), λ₄, Λ₄(L)} includes the linear coefficients in the conditional mean (1), while Π₂ = {D₁, D₂, D₃, D₄, γ _lM , γ _lV , γ _gM , γ _gV } includes the coefficients pertinent to the covariance matrix and to the transition functions.

To establish the optimal lag length and estimate the model parameters, an algorithm similar to that used by Auerbach and Gorodnichenko (2012) is adapted to the VSTAR in equations (1), (2) and (13).^[20] The algorithm exploits the possibility of writing the model in a linear form conditional on a subset of the parameters, thereby estimating the remaining parameters analytically. Thus, estimation involves repeating until convergence two sequential steps. The first consists of determining a draw of Π₂. Conditional on Π₂, the VSTAR is linear in the remaining parameters over the four states. In the second step the remaining parameters in Π₁ can be determined analytically via generalized least squares and the log-likelihood can be evaluated from (14). The estimation is undertaken including up to 12 lags and the model fit evaluated using information criteria such as, Akaike (AIC), Hannan (HIC) and Schwarz (SIC), in both standard and normalised form. The model with one lag is the preferred specification according to almost any of the information criteria considered.^[21] In Appendix E in Supplementary Material it is also shown that the proposed VSTAR specification yields a better fit of the data compared to several other VSTAR specifications often used in the applied macroeconomic literature.

Figure 2 plots the transition probabilities estimated from the VSTAR for the mean (top panel) and the covariance matrix (bottom panel) parameters. The black lines show the probability of the economy being, l i u g t ≃ 1 , or not being, l i u g t ≃ 0 , in a slack state. The red lines the probability of policy being away from, g i R t ≃ 1 , or near to, g i R t ≃ 0 , the ZLB. The top panel shows four sharp shifts in the mean coefficients linked to the economy vector occurring around recessions.^[22] After 2001, nonlinearity from the policy sector becomes significant too, in particular during 2009–2015, as the federal funds rate reaches the ZLB. The bottom panel shows shifts in the coefficients of the covariance matrix around recessions, whereas changes driven by R _t reflect predominantly the downward trend in the federal funds rate observable in Figure 1. The results in Figure 2 highlight the flexibility of the estimated VSTAR, given that it can simultaneously capture gradual changes in the mean coefficients linked with changes in the amount of slack in the economy and the more sudden variation in the volatility coefficients.^[23]

4.4 Impulse Response Functions

IRF analysis with VSTAR is typically based on the algorithm of Koop et al. (1996), with shocks identified through Cholesky factorization of the covariance matrix Ω _t . In a small-scale VAR the Cholesky decomposition yields structural shocks that can be easily reconciled with an economic interpretation. This is not necessarily the case in larger (and nonlinear) models like the estimated VSTAR. For this reason, IRF analysis is carried out in this paper by adapting the algorithm of Koop et al. (1996) to include sign restrictions.^[24] To formulate a benchmark, the restrictions are set to identify a credit-type shock that on impact yields macroeconomic dynamics similar to those observed in the United States after 2008, leaving entirely unrestricted the policy variables. This shock is such that both the spread and the unemployment gap increase while inflation and industrial production decrease at the one month horizon.^[25] To evaluate the different responses of the policy instruments during periods when either conventional monetary policy or QE is active, the IRFs are computed using as starting values for y _t the averages over two sub-samples of equal size. The first is from 1994:6 to 2001:5. This is referred to as the normal times, covering a period when the federal funds rate is well above the ZLB. The second includes observations for the ZLB period, from 2009:1 to 2015:12.

Figure 3 displays the responses of the three policy instruments plus the total size of the Fed’s balance sheet (TS + PS) to the identified credit shock over 24 months. In each panel, the black lines denote the responses during normal times, the red lines the responses in the ZLB period and the shaded areas denote two standard deviations confidence bands. The response of the policy instruments mimics that of expansionary monetary policy intervention at different periods of the United States’ economic history. In normal times the shock leads to a reduction of the federal funds rate, while QE shows little dynamics. During the ZLB the monetary expansion occurs through QE. It is remarkable how the path of the IRFs for the two QE variables reproduces changes in the Fed assets’ portfolio similar to those observed after QE1-QE2, as visible from the initial expansion of PS holding, subsequently supported by a large increase of TS. These dynamics of the Fed’s balance sheet, consisting of a large initial purchase of PS followed by TS, are also visible from the evolution of TS and PS around the Great Recession in Figure 1.^[26]

The IRF analysis indicates that the estimated VSTAR returns a plausible description of the monetary policy instrument responses to macroeconomic shocks at different times of the United States’s monetary history, thereby providing a reliable tool for the analysis of optimal policy in the next section. In Appendix F in Supplementary Material it is shown that these results are robust to different specifications of the VSTAR, obtained by replacing industrial production with either GDP or public debt, and different types of economic disturbances, such as demand and supply shocks.

5.1 Gain from Optimization

What would have been the gain in terms of macroeconomic stabilization if QE had been optimally coordinated with interest rate policy? The answer to this question is first quantified considering macroeconomic dynamics in the 10 years from November 2007, one year before the Fed announcement of QE1, to November 2017, the date from which the Fed’s balance sheet is allowed to shrink gradually.^[29] Accordingly, the policy analysis is based on data that exclude the pre-2008 policy regime change, thereby limiting the quantitative impact of the Lucas critique on the results. Following Sack (2000), the gain from monetary policy optimization is evaluated comparing two scenarios. The first, referred to as the actual policy, is based on the observed dynamics of the economy and policy instruments over the considered sub-period. The second, the optimal policy, is constructed assuming that the Fed chooses the optimal combination of conventional and QE policy in each month and this optimal policy is implemented given the true state of the economy in that month. The economy response to the optimal policy in any given month is computed including the actual shocks occurring in that month, using the VSTAR in (12). In addition, the inflation target in the loss function (15) is set as p ̄ = 1.55 , corresponding to the average inflation rate from November 2007 onward. As a further reference, a no-policy scenario is also simulated.^[30] This is based on the implied economy and policy dynamics obtained if no-change in QE had been implemented since November 2007.

Following Dennis and Söderström (2006), two measures of the optimization gain are computed. The first is the (percentage) change in the loss due to implementation of a new policy, V′, relative to the loss measured from another policy, V, i.e. G = 100 × [1 − V′/V]. The second is the unemployment-equivalent compensation. To clarify this, consider the unemployment gap term in (15). This can be written as υ u g u g t 2 = υ u g u t − u t ̄ 2 , where u _t and u ̄ t denote the actual and the natural rate of unemployment, respectively. The unemployment-equivalent compensation is defined as the permanent deviation u ̂ ≠ 0 of the actual unemployment rate from the natural rate that results in a change of the loss by 1 − β ∑ t = 0 ∞ β t υ u g u ̂ 2 such that V equals V′. It follows that u ̂ = V − V ′ / υ u g . Thus G is a direct measure of the macroeconomic stabilization differential between two policies, whereas u ̂ quantifies the implied permanent change in the rate of unemployment.

Table 1 presents the results. Columns two to six report the volatility of the five items in the loss function (15); column seven the implied losses; whereas the last two columns report the calculated measures of the optimization gain. For robustness, six different specifications of the weights in (15) are considered. The first (Baseline) is the benchmark which gives equal weight to each item in (15). The remaining five evaluate the effects of lower weight to the stabilization of either inflation (II), unemployment (III), all policy instruments (IV), the two QE instruments (V), or the federal funds rate (VI). For each set of policy weights the loss is computed as the undiscounted weighted average of the five terms in (15).^[31] The stabilization gains and unemployment-equivalent compensations are measured first to compare monetary policy optimization relative to the actual policy undertaken by the Fed, and then the actual policy relative to the no-QE scenario.

Two main results emerge from Table 1. First, policy optimization would have increased the stabilization gain during 2007–2017. On average across different weight specifications, optimal policy would have reduced macroeconomic volatility by about 2 % relative to the actual policy. In terms of unemployment-equivalent compensation, policy optimization would have resulted on average in a further reduction of the unemployment rate of about 0.33 % compared to the actual policy. Second, the gains from switching from the actual to the optimal policy are not as large as those achieved by the actual implementation of QE relative to the no-QE policy scenario. On average across the different weight specifications, macroeconomic volatility and the unemployment rate would have been about 12 and 1 % higher, respectively, had QE not been implemented during 2007–2017.

Looking across the different specifications of the weights, it can be observed that the gains from either the actual or the optimal policy increase when the Fed targets unemployment stabilization more aggressively than inflation stabilization (compare case II with either I or III). This is because unemployment is relative more volatile than inflation during 2007–2017, in turn implying that assigning a higher weight to this variable brings larger gains. Reducing the weight attached to any policy instrument in the objective function produces higher stabilization gains (compare case I with cases IV, V and VI). This is because the further reduction in the volatility of inflation and unemployment more than compensates the increase in the volatility of the instruments.

5.2 Impulse Response Functions

How the economy and the policy instruments respond to shocks at the ZLB once changes to QE and the federal funds rate are optimally coordinated? To answer this question, the IRFs are calculated from the VSTAR under the optimal policy in (12) assuming that the economy and the policy instruments (y _t ) start from the ZLB period.^[32] The preferences of the Fed are set as under the baseline calibration and the inflation target is set equal to the ZLB sample average of 1.29 %.

Figure 4 shows the responses from the VSTAR under the optimal policy rules (blue) to a credit shock during the ZLB period. For comparison, the corresponding responses from the estimated VSTAR, which reflect the dynamics under the actual Fed monetary policy are also reported (red).^[33] According to these results, when the federal funds rate is constrained by the ZLB, the Fed responds under the optimal policy to a credit shock by increasing the size of the balance sheet more sharply than under the actual policy, at least in the short run (first 3 months). At the same time there is significant portfolio rebalancing towards Treasury securities, since TS holdings increase while PS holdings decrease in the short run horizon. The response of the Fed’s asset portfolio under the optimal policy show less persistence than those from the actual policy. This is because the optimal policy dampens very rapidly the impact of the shock on inflation and unemployment.

To illustrate how the optimal policy changes in response to different macroeconomic conditions, Figure 5 shows the responses under the actual (red) and optimal (blue) policy to a supply-type shock during the ZLB.^[34] The federal funds rate remains constrained by the ZLB under both policies. The responses of QE are however different, being expansionary under the actual policy and contractionary under the optimal. The optimal QE policy response to the supply-type shock shows little rebalancing of the Fed portfolio contrary to what is observed from the responses to the credit shock displayed above.

In summary, QE under the optimal policy results in increase of the size of the balance sheet and significant short-run portfolio rebalancing toward TS in response to a credit-type shock. The responses to the supply-type shock are expansionary under the actual policy and contractionary under the optimal, with little evidence of portfolio rebalancing.

5.3 Counterfactual Monetary Policy

What would have been the dynamics of the economy and policy instruments had the federal funds rate and QE been coordinated optimally since 2008? To answer this question, the VSTAR under the optimal policy is used for three counterfactual experiments. The simulations are carried out starting from November 2007, using the specification of the Fed preferences with equal weights.

Each counterfactual simulation employs a different value for the inflation target, 2, 3 and 4 %. These do not reflect any specific view on the actual preferences of the Fed, but are selected for gauging indication on the sensitivity of the results to the choice of the inflation target.^[35] It is important to clarify the role of the inflation target in these simulations. All other things equal, the optimal policy would require expansionary (contractionary) QE whenever actual inflation is below (above) target. As the unemployment gap is positive during most of the post-2007 period, the response of monetary policy to unemployment stabilization is stronger the lower is actual inflation relative to the target.

Figure 6 illustrates the results from these counterfactuals. Black lines (A) show the actual policy instruments dynamics as observed from data; whereas blue (O2), red (O3) and green (O4) lines denote the dynamics of the same instruments simulated from the optimal VSTAR under the three alternative inflation targets of 2, 3 and 4 %, respectively. The simulations are carried out assuming that in each period the Fed implements the optimal policy given the observed state of the economy.

Three main results can be observed from Figure 6. First, the dynamics of the economy are fairly similar across the three counterfactual simulations. Compared to the data, the main differences are that credit risk would have been lower and inflation higher during 2008–2013 under the optimal policy regardless of the choice of the inflation target. The dynamics of industrial production and the unemployment gap are remarkably similar to those observed from the data, under any of the three counterfactual simulations. Second, the path of the federal funds rate appears close to that observed from the data under any of the counterfactual simulations, particularly when using the 4 % inflation target. Third, the dynamics of the Fed’s balance sheet are qualitatively similar to the observed ones under any of the three counterfactual scenarios. The most noticeable exception is the period 2009–2015 period, when the size of the simulated Fed’s balance sheet is larger than the actual one. This is due to the more rapid increase in TS purchases under any of the counterfactual relative to the observed purchases.

Table 2 gives a breakdown of the duration of the ZLB period under the actual and optimal policy, based on the counterfactual analysis. In the data, the federal funds rate never reaches zero and the identification of the periods in which the ZLB is binding is somewhat arbitrary, depending on the chosen threshold below which the policy rate is deemed to be at the ZLB. To guard against this, the number of ZLB periods in Table 2 is counted considering four different thresholds, corresponding to months in which the federal funds rate is below to either 25, 50, 75 or 100 basis points. According to the results in the table, the actual duration of the ZLB ranges between 85 and 104 months, depending on the selected interest rate threshold. The duration of the ZLB from the counterfactual experiments is closer to the observed duration as the inflation target and the cut-off rate for the ZLB increase. It is worth highlighting that the estimated duration of the ZLB period is considerably longer than that otherwise predicted by conventional evaluations of monetary policy. For example, Williamson (2015) estimates that the ZLB should have terminated around the beginning of 2011 (26 months) had monetary policy during the Great Recession being conducted according to a standard Taylor rule.

The counterfactual simulations can be used to evaluate different segments of the QE intervention undertaken by the Fed since November 2007. The Fed embarked in three large-scale asset purchase programs, commonly known as QE1, QE2, QE3 and the Maturity Extension Program (MEP).^[36] Table 3 reports how TS holding, PS holding and the total size of the Fed’s balance sheet change during each of these programs under the actual policy and the three counterfactual simulations. Under the actual QE policy, the total increase in the balance sheet was about 17.96 % of GDP. The size of the change of the Fed’s balance sheet as a percentage of GDP under the optimal policy is close to the actual one, ranging between 17.98 and 19.33 % depending on the inflation target. Comparing across programs, QE1 should have been almost a third larger, regardless of the inflation target. Consequently, this would have resulted in QE2 and QE3 programs of smaller size under the optimal policy compared to the actual. It would have also resulted in a reduction of the size of the balance sheet (relative to GDP) during the MEP under the optimal policy about four times larger than the actual reduction observed in the data.

As noted above, the counterfactual simulations under the optimal policy entail only small changes from the actual data. To provide some indication of the extent of these small changes, Table 4 reports in the top row the mean of the absolute deviation from the actual data of the macroeconomic forecasts from the Survey of Professional Forecasters of the Federal Reserve Bank of Philadelphia during 2007:11-2018:10, whereas the remaining rows report the corresponding deviations for the simulated data used for the results in Figure 6 and Table 1.^[37] The table does not include TS and PS since the Survey covers all macroeconomic data used in the empirical analysis except for the balance sheet. The results in Table 4 provide some reassurance with regard to the closeness of the simulated data to the actual data. The counterfactual simulations involve smaller deviations from the actual data compared to the professional forecasts, for inflation, industrial production and unemployment, whereas the deviations for the spread the federal funds rate are marginally larger.

In summary, according to this analysis the scale of total QE intervention observed in the data appears to be close to that prescribed by the optimal QE policy under conventional calibration of the Fed preferences. The main differences between actual and optimal QE are in terms of timing and composition, since the optimal QE would have entailed earlier increases in the Fed’s holding of TS compared to the actual policy. This would have resulted in lower spread and higher inflation in the aftermath of the Great Recession. Overall the counterfactual simulations indicate that the observed duration of the ZLB is in line with that otherwise prescribed by the optimal policy under most calibrations.

5.4 Alternative Optimization

The results presented above are the outcomes of two important requirements of the proposed optimization methodology: transformation of the VSTAR into a VSTARX using A2 and optimization with the SDC method. To evaluate the performance of the proposed optimization methodology, the results are compared with those from three possible alter- native approaches.^[38] The first, labelled as LIN, is based on the linearized-equivalent of the VSTAR, i.e. the VAR model estimated over the same sample period, whose performance against the VSTAR is highlighted in Appendix E in Supplementary Material. As it is a reduced-form model, for the purpose of this exercise the VAR is transformed into a VARX using A2. The optimization problem then boils down to a LQR which is solved with standard Riccati equation iteration. The second and third approach both employ the VSTARX obtained from the estimated VSTAR under A1. For this reason optimization can only be undertaken with nonlinear methods. In the second, labelled as MPC, this is pursued using a method that mimics the logic of the model predictive control algorithm, whose application in a number of nonlinear decision problems in economics has been recently illustrated by Grüne et al. (2015). This consists of positing a time-varying coefficients feedback rule u t = k t ( M P C ) + K t ( M P C ) y t − 1 in conjunction with the system of equation (4) and then using the quasi-Newton method to find the coefficients k t ( M P C ) and K t ( M P C ) such that the one period ahead forecast of y _t minimizes the loss (15). In the third approach, labelled as DNS, optimization is pursued using direct numerical search over the policy rule coefficients. This consists of using the time-varying coefficients feedback rule u t = k t ( D N S ) + K t ( D N S ) x t − 1 to derive estimates of u ̂ t , and then forming y t = x t ′ u ̂ t ′ ′ . Again the quasi-Newton method is employed to find the coefficients k t ( D N S ) and K t ( D N S ) that minimize the loss (15). The results from LIN highlight the effects of ironing out (through linearization) all the nonlinear features in the data prior to optimization. The results from MPC and DNS highlight instead the effects of doing optimization under the alternative transformation protocol, A1, therefore not being able to use dynamic programming but having to rely upon numerical search. For the sake of comparison, the stabilization gains are re-computed under each method and the quality of the performance is evaluated considering three dimensions: feasibility (ability to find an optimal solution), efficiency (ability to find the solution with the lowest loss) and computational speed (time required to find a solution).

Table 5 gives a summary of the outcomes of these numerical experiments, considering optimization for the post-2007 period and the full sample. For each approach and sample, the optimization is carried using the protocol and each of the six weight specifications described in Section 5.1. The first eighth columns of Table 5 report the average across all specifications of each argument in the loss function (15), the value of the loss V, the stabilization gain G and the unemployment-equivalent compensation u ̂ , respectively. The final column reports an estimate of the speed of optimization (in seconds) obtained under the specification of the loss function with equal weights from 10 optimization trials under each approach. To facilitate comparison, the results from the actual and optimal policy under the SDC method are also reported.

According to the results in Table 5, each optimization approach can deliver a feasible solution, i.e. a solution that leads to a loss lower than under the actual policy. Regarding efficiency, the results from these numerical experiments suggest that SDC and LIN provide superior performance compared to the two numerical methods, as indicated from the lower losses (and therefore stabilization gains and unemployment compensations) that these two approaches deliver regardless of the sample choice. The comparison between SDC and LIN needs careful interpretation. This is because the analysis of optimal policy with LIN is based on the linear VAR model, which gives a very poor fit of the data dynamics compared to the VSTAR which is used by SDC, as shown in Appendix in Supplementary Material, Table E.2. Further, the performance of LIN relative to SDC deteriorates as the simulation horizon is extended to cover the whole sample, thereby exacerbating the importance of nonliterary in the data. This points towards the importance of accounting for nonlinearity in the quantitative analysis of monetary policy, highlighting the difference it makes using a VAR rather than a nonlinear model like the VSTAR. Finally, optimization with SDC and LIN is way much faster than with the two numerical methods. This is not surprising since iteration until convergence of the Riccati equation in SDC and LIN is known to be faster than numerical search.