2.1 Max-Share

The Max-Share approach to identification assumes that a particular shock is the dominant driver over a specific horizon. The shock of interest is that which drives the largest proportion of the forecast error variance at the chosen horizon (see Francis et al. (Reference Francis, Owyang, Roush and DiCecio2014)). For long-run applications, Francis et al. (Reference Francis, Owyang, Roush and DiCecio2014) surmises that 10 years is long enough to capture dominant long-run shocks.

The forecast error at horizon $k$ can be written:

(1) \begin{equation} y_{t+k} - \hat {y}_{t+k} = \sum _{\tau =0}^{k-1} D_\tau u_{t+k-\tau }, \end{equation}

where $D_\tau$ is the matrix of moving average coefficients. By defining $\tilde {A}_{0}$ as the Cholesky decomposition of the variance-covariance matrix $\Sigma _{u}$ and $\varphi _i$ as a selection vector (size $n \times 1$ ) with 1 in the $i$ -th position, we find the shock vector $\alpha$ which maximizes the contribution to the total forecast error variance of variable $i$ at horizon $k$ :

(2) \begin{equation} \max _{\alpha } f(\alpha ) = \frac {\varphi _i^\prime \left ( \sum _{\tau =0}^{k-1} D_\tau \tilde {A}_{0} \alpha (\tilde {A}_{0} \alpha )^\prime D_\tau ^\prime \right ) \varphi _i}{\varphi _i^\prime \left ( \sum _{\tau =0}^{k-1} D_\tau \Sigma _u D_\tau ^\prime \right ) \varphi _i} \quad \text{s.t.} \quad \alpha ^\prime \alpha =1. \end{equation}

The identified structural shock is defined by the impact vector $\tilde {A}_{0}\alpha$ . Following Uhlig (Reference Uhlig2003), the vector $\alpha$ that solves this maximization problem is the eigenvector associated with the maximum eigenvalue of the matrix $S_{i,k}$ . This matrix is derived by rearranging the numerator in (2) to isolate  $\alpha$ :

(3) \begin{equation} S_{i,k} = \sum _{\tau =0}^{k-1} \tilde {A}_{0}^\prime D_\tau ^\prime \varphi _i \varphi _i^\prime D_\tau \tilde {A}_{0}. \end{equation}

Note that the denominator in (2) represents the total forecast error variance of variable $i$ , which is a scalar constant independent of $\alpha$ and therefore does not affect the first-order conditions of the maximization.

2.1.1 First warning on the use of Max-Share

When employing Max-Share, using more lags of the reduced-form MA coefficients does not necessarily translate into a better estimate. Researchers have to be cognizant of the fact that as the horizon lengthens there are (cumulative) errors in the MA coefficients on which the identification rests. These errors can be sizeable as we argue in Lemma1:

Lemma 1. VARs can produce increasingly biased and uncertain impulse responses as the horizon increases. A simple argument illustrates this issue.

Consider the AR(1) model

\begin{equation*} y_t = \phi y_{t-1} + \varepsilon _t, \qquad |\phi | \lt 1. \end{equation*}

Let $\hat \phi$ be an estimator of $\phi$ such that $\hat \phi \xrightarrow {p} \phi$ and

\begin{equation*} \sqrt {T}\,(\hat \phi - \phi ) \xrightarrow {d} \mathcal{N} \big(0,\sigma _\phi ^2\big). \end{equation*}

Define the $j$ -step impulse response $\Psi _j = \phi ^j$ and its plug-in estimator $\hat \Psi _j = \hat \phi ^j$ for a fixed integer $j \ge 1$ . Then, by the delta method,

\begin{equation*} \sqrt {T}\,\big(\hat \Psi _j - \Psi _j\big) \xrightarrow {d} \mathcal{N}\!\left (0,\, \big(j\phi ^{j-1}\big)^2\,\sigma _\phi ^2\right ). \end{equation*}

Moreover, a second-order delta-method expansion yields the approximation

\begin{equation*} \mathbb{E}[\hat \Psi _j] \approx \phi ^j + \frac {1}{2}\,j(j-1)\,\phi ^{j-2}\,{\textrm {Var}}(\hat \phi ), \end{equation*}

and therefore the (approximate) bias satisfies, for $j \ge 2$ ,

\begin{equation*} \mathbb{E}[\hat \Psi _j] - \Psi _j \approx \frac {1}{2}\,j(j-1)\,\phi ^{j-2}\,{\textrm {Var}}(\hat \phi ). \end{equation*}

If $\hat \phi$ is additionally biased with $\mathbb{E}[\hat \phi ] = \phi + b$ and ${\textrm {Var}}(\hat \phi ) = v$ (with $b, v$ small), then

\begin{equation*} \mathbb{E}[\hat \Psi _j] - \Psi _j \approx j\phi ^{j-1} b + \frac {1}{2}\,j(j-1)\,\phi ^{j-2} v. \end{equation*}

In particular, even if $b \lt 0$ , the impulse response estimator can be upward biased when

\begin{equation*} \frac {1}{2}(j-1)v \gt -\phi \, b. \end{equation*}

3.1 Demonstrating the Impacts of Confounding Shocks: Second Warning to the Use of Max-Share

The prevailing approach in the literature is to use Max-Share to identify long-run shocks. For a simple two-variable VAR consisting of labor productivity ( $p$ ) and hours, ( $n$ ), with the true structural form:

\begin{equation*} A_{0}\mathbf{\epsilon } = \begin{bmatrix} A_{lp} &\quad A_{hp} \\ A_{ln} &\quad A_{hn} \\ \end{bmatrix} \begin{bmatrix} \epsilon ^{l} \\ \epsilon ^{h} \\ \end{bmatrix}, \end{equation*}

where we use $l$ and $h$ to characterize the first shock as a low-frequency (technology shock) and the second as a high-frequency (business cycle) shock. In the simplest case, at horizon $k=0$ , the solution’s impact on $p$ and $n$ will match the impacts using a solution based on the Cholesky decomposition of $\Sigma _{u}$ .

\begin{align*} \tilde {A}_{0}\tilde {q}(0) = \begin{bmatrix} \sqrt {A^2_{lp}+A^2_{hp}}\\[8pt] \frac {A_{lp}A_{ln}+A_{hp}A_{hn}}{\sqrt {A^2_{lp}+A^2_{hp}}} \end{bmatrix} \end{align*}

The impact of the identified shock on productivity will be a combination of the low- and high-frequency structural shocks, in proportion to their respective impacts. The impact on hours will depend on the relative impact on productivity and the relative impact on hours of each shock.

Whilst intuitive, the above expression is an oversimplification of the effects of confounding shocks. This is because we have only demonstrated its effects on the impact impulse responses. Often, however, the practitioner is interested in longer-run applications and thus wants to identify low-frequency shocks operating at long(er) horizons. In these cases, the solution becomes even more complex (Appendix A). Nevertheless, the identified dominant shock at longer horizons will be a hybrid shock, capturing elements of the underlying structural drivers in proportion to their initial impacts and persistence.

3.2 Establishing a More Robust Estimator

To reduce the influence of confounding shocks, we need to be able to filter out these competing shocks from the maximization to the greatest extent possible so that the true shock is as dominant as possible at the horizon of interest. To this end, we appeal to a truncated-kernel spectral density. We interpret the difference between our (time-domain) Max-Share approach and its truncated-kernel spectral counterpart as one plausible measure of the extent to which confounding shocks impact Max-Share. However, we italicize more robust in anticipation of a future cautionary caveat.

3.2.1 Spectral identification

The spectral approach to identifying dominant structural shocks operates in the frequency domain by maximizing the share of variance of a specific target variable explained within a prespecified range of frequencies.

To implement this methodology, consider the Wold representation

\begin{equation*} Y_t = \sum _{j=0}^{\infty } D_j u_{t-j}, \qquad \mathbb{E}[u_t u^{\prime}_t] = \Sigma _u. \end{equation*}

and define the autocovariance matrices

\begin{equation*} \Gamma (\tau ) = \mathbb{E}[Y_t Y^{\prime}_{t-\tau }], \qquad \tau \in \mathbb{Z}. \end{equation*}

The spectral density matrix is given by

(4) \begin{equation} S_{YY}(\omega ) = \frac {1}{2\pi } D\big(e^{-i\omega }\big) \Sigma _u D\big(e^{i\omega }\big)^{\prime}, \qquad D\big(e^{-i\omega }\big) = \sum _{j=0}^{\infty } D_j e^{-i\omega j}, \end{equation}

and satisfies the Fourier relationship

(5) \begin{equation} \Gamma (\tau ) = \int _{-\pi }^{\pi } e^{i\omega \tau } S_{YY}(\omega )\, d\omega , \qquad S_{YY}(\omega ) = \frac {1}{2\pi } \sum _{\tau =-\infty }^{\infty } \Gamma (\tau ) e^{-i\omega \tau }. \end{equation}

To identify the shock, we define $\tilde {A}_{0}$ as the Cholesky decomposition of $\Sigma _{u}$ and $\varphi _i$ as a selection vector for the target variable $i$ . We seek the structural shock vector $\alpha$ (where $\alpha ^{\prime}\alpha =1$ ) that maximizes the variance of variable $i$ within the frequency band $[\underline {\omega },\bar {\omega }]$ :

(6) \begin{equation} \max _{\alpha } f(\alpha ) = \frac {\varphi^{\prime} _i \left ( \int _{\underline {\omega }}^{\bar {\omega }} D(e^{-i\omega }) \tilde {A}_{0} \alpha (\tilde {A}^{\prime}_{0} \alpha ) D(e^{i\omega })^{\prime} \, d\omega \right ) \varphi _i}{\varphi^{\prime} _i \left ( \int _{-\pi }^{\pi } S_{YY}(\omega ) \, d\omega \right ) \varphi _i}. \end{equation}

Analogous to the Max-Share approach, the vector $\alpha$ that solves this maximization corresponds to the eigenvector associated with the largest eigenvalue of the matrix $V_{i}(\underline {\omega },\bar {\omega })$ , which isolates the contribution of the shock to the target variable:

\begin{equation*} V_{i}(\underline {\omega },\bar {\omega }) = \int _{\underline {\omega }}^{\bar {\omega }} \text{Re}\left ( \tilde {A}^{\prime}_{0} D(e^{i\omega })^{\prime} \varphi _i \varphi ^{\prime}_i D(e^{-i\omega }) \tilde {A}_{0} \right ) \, d\omega . \end{equation*}

3.2.2 Limited-horizon spectral identification

One criticism that can be leveled against the spectral identification approach is that the long-run VAR representation used to calculate the spectrum of the endogenous variables may be biased when estimated on a short sample of data and with limited lags. A “windowed” selection of autocorrelations is often used to estimate the spectrum of a series. The same principle can be used in the spectral VAR identification process, effectively applying a truncated kernel estimation approach. For example, the infinite-MA representation, $D(e^{-i\tau \omega })$ , can be replaced with a limited horizon series of impulse response coefficients.

(7) \begin{equation} D^{k}(e^{-i\omega }) = \sum ^{k-1}_{j=0}D_{j}e^{-ij\omega }. \end{equation}

The same procedure used to identify the shock of interest in the general spectral approach can be straightforwardly applied here as well. Additionally, applying this truncation provides a direct comparison to Max-Share. That is, both Max-Share and Limited-Spectral methods will use the same horizon $k$ in their identification step, making the only difference between the two approaches the additional filtering of the Limited-Spectral approach at specific frequencies. It is this feature of the Limited-Spectral method to which we appeal when interpreting anything extraneous coming from Max-Share as potentially reflecting confounding shocks that influence the data at frequencies within the excluded domain.

3.3 Quantifying Confounding Shocks

In population and abstracting from spectral estimation error, the Limited-Spectral identification can be interpreted as a frequency-restricted analogue of the Max-Share procedure, emphasizing variation within a prespecified band. In estimation, differences between the two approaches may therefore be informative about confounding dynamics but can also arise from finite-sample spectral estimation error or frequency leakage. Accordingly, we interpret such differences as a diagnostic rather than as a definitive measure of confounding shocks.

The forecast error variance matrix with respect to which the maximization is performed can be written, for the Max-Share identification, as

(8) \begin{equation} V^{\text{Max-Share}}_k = \sum _{\tau =0}^{k-1} D^\tau \Sigma _u D^{\tau \prime }, \end{equation}

and for the Limited-Spectral identification, as

(9) \begin{equation} V^{\text{L-Spectral}}_k = \int _{\underline {\omega }}^{\bar {\omega }} \text{Re}\left (\sum _{\tau =0}^{k-1} D^\tau \left (e^{i\tau \omega }\right )\, \Sigma _u\, D^\tau \left (e^{-i\tau \omega }\right )^{\prime }\right )d\omega . \end{equation}

More formally, Equation 9 corresponds to a finite-sample discrete-frequency approximation to the band-integrated spectral density obtained by replacing the infinite-horizon moving-average representation with its truncated counterpart. This construction is equivalent to applying a windowed spectral estimator that emphasizes variation at the selected frequencies while maintaining comparability with the finite-horizon Max-Share criterion. Differences between the two identified impulse responses therefore reflect the exclusion of variation outside the targeted frequency band and may capture the influence of dynamics operating at excluded frequencies. For example, comparing Max-Share applied at long horizons with the Limited-Spectral identification at low frequencies can be informative about the role of higher-frequency shocks in shaping long-horizon dynamics.

From this point forward, we focus exclusively on the Limited-Spectral approach in the frequency domain. For brevity, we occasionally refer to this method simply as the spectral approach.

4.1 Two-Variable Model and Identification

We work with a simple two-variable DGP in which labor productivity $L_t$ and hours $N_t$ are driven by a technology shock $\epsilon ^z_t$ and a non-technology shock $\epsilon ^b_t$ :

(10) \begin{align} L_t &= z^l_t + b_t, \end{align}

(11) \begin{align} N_t &= 0.7 N_{t-1} - 0.3 N_{t-2} - 0.3 z^l_t + 0.3 b_t, \end{align}

(12) \begin{align} z^l_t &= \rho _z z^l_{t-1} + z^g_t, \end{align}

(13) \begin{align} z^g_t &= \rho _{z^g} z^g_{t-1} + \epsilon ^z_t, \end{align}

(14) \begin{align} b_t &= \rho _{b,1} b_{t-1} + \rho _{b,2} b_{t-2} + \epsilon ^b_t. \end{align}

The technology innovation $\epsilon ^z_t$ drives a highly persistent level component $z^l_t$ and, through $z^g_t$ , a growth component that can itself be persistent. The non-technology disturbance $b_t$ is a stationary but cyclical AR(2) process that acts as a confounding driver of $L_t$ and $N_t$ .

In the baseline implementation we simulate $T = 2{,}000$ usable observations and discard an initial burn-in of $500$ periods to eliminate the influence of initial conditions. The structural shocks are i.i.d. Gaussian,

\begin{equation*} \epsilon ^z_t \sim \mathcal{N}\big(0,\sigma _z^2\big), \qquad \epsilon ^b_t \sim \mathcal{N}\big(0,\sigma _b^2\big), \end{equation*}

with $\sigma _z = 1$ throughout. The remaining parameters $(\rho _z,\rho _{z^g},\rho _{b,1},\rho _{b,2},\sigma _b)$ are calibrated so that the technology component is highly persistent while the non-technology component is less persistent but relatively volatile. In the baseline stationary case we set

\begin{equation*} \rho _z = 0.9,\qquad \rho _{z^g} = 0, \qquad \sigma _z = 1, \end{equation*}

and parameterize the confounding disturbance $b_t$ as a business-cycle AR(2) process with complex roots. Specifically, we choose

\begin{equation*} \rho _{b,1} = 2R\cos (\lambda ),\qquad \rho _{b,2} = -R^2,\qquad R=0.85,\qquad \lambda = 2\pi /8, \qquad \sigma _b = 2. \end{equation*}

This calibration concentrates the spectral mass of the confounding shock at cyclical frequencies (roughly an 8-quarter cycle), while the technology component remains highly persistent and contributes disproportionately at very low frequencies. The resulting DGP therefore isolates the case of business-cycle confounding: the non-technology shock generates substantial medium-frequency variation that can contaminate variance-maximizing identification in finite samples.

Given a realization of $(L_t,N_t)$ , we estimate a bivariate VAR of order two,

\begin{equation*} Y_t = A_1 Y_{t-1} + A_2 Y_{t-2} + u_t, \qquad Y_t = \begin{pmatrix}L_t \\ N_t\end{pmatrix}, \end{equation*}

using Gaussian maximum likelihood. Let $\Sigma _u = \textrm{Var}(u_t)$ denote the reduced-form covariance matrix, and let $A_{\tilde {}}$ be its lower-triangular Cholesky factor, $\Sigma _u = A_{\tilde {}} A_{\tilde {}}^{\prime}$ .

From the estimated VAR we construct the moving-average (MA) representation

\begin{equation*} Y_{t+h} = \sum _{\tau =0}^\infty D_\tau u_{t+h-\tau }, \end{equation*}

where $D_0 = I_2$ and, for $h\geq 1$ , the MA coefficients satisfy the VAR(2) recursion

\begin{equation*} D_h = A_1 D_{h-1} + A_2 D_{h-2}. \end{equation*}

We work with a finite horizon $H=20$ and use $\{D_h\}_{h=0}^{H-1}$ both to form impulse responses and to construct the identification criteria.

Max-Share identification. Let $e_1 = (1,0)^{\prime}$ select labor productivity. For a given impact direction $a\in \mathbb{R}^2$ , the implied response at horizon $h$ is $D_h A_{\tilde {}} a$ , and the contribution of the associated shock to the forecast error variance of $L_t$ over horizons $0,\ldots ,H-1$ is summarized by

\begin{equation*} a^{\prime} H^{MS} a, \qquad H^{MS} = \sum _{h=0}^{H-1} (D_h A_{\tilde {}})^{\prime} (e_1 e^{\prime}_1) (D_h A_{\tilde {}}). \end{equation*}

The Max-Share shock is defined as the unit vector $a$ that maximizes this quadratic form. In the implementation this is obtained as the leading eigenvector $\alpha _{MS}$ of the symmetrized matrix $\tfrac {1}{2}(H^{MS} + (H^{MS})^{\prime})$ , normalized to unit Euclidean length. The corresponding impact vector is $A_0^{MS} = A_{\tilde {}} \alpha _{MS}$ and the Max-Share impulse responses are

\begin{equation*} \text{IRF}^{MS}_h = D_h A_0^{MS}, \quad h=0,\ldots ,H-1. \end{equation*}

Limited-spectral identification. The Limited-Spectral estimator maximizes the contribution of $L_t$ at low frequencies. For a frequency band $[-\omega _B,\omega _B]$ we construct the spectral objective

\begin{equation*} H^{LS} = \frac {1}{J} \sum _{j=1}^J \left ( D(e^{-i\omega _j}) A_{\tilde {}} \right )^{\prime} (e_1 e^{\prime}_1) \left ( D(e^{-i\omega _j}) A_{\tilde {}} \right ), \end{equation*}

where $\omega _j$ are $J$ equally spaced grid points in $[-\omega _B,\omega _B]$ and

\begin{equation*} D(e^{-i\omega }) = \sum _{h=0}^{H-1} D_h e^{-i\omega h} \end{equation*}

is the finite-horizon frequency-domain representation of the MA. In the implementation we use $J=500$ grid points and set the low-frequency band width to $\omega _B = 2\pi /10$ , corresponding to cycles longer than roughly ten periods. As in the Max-Share case, the Limited-Spectral shock is the leading eigenvector $\alpha _{LS}$ of $\tfrac {1}{2}(H^{LS} + (H^{LS})^{\prime})$ ; the associated impact vector is $A_0^{LS} = A_{\tilde {}} \alpha _{LS}$ and the Limited-Spectral IRFs are

\begin{equation*} \text{IRF}^{LS}_h = D_h A_0^{LS}, \quad h=0,\ldots ,H-1. \end{equation*}

Spectral intuition. Both criteria admit a frequency-domain interpretation. Let $S_{YY}(\omega )$ denote the reduced-form spectral density of $Y_t$ and $S_{LL}(\omega )$ its $(1,1)$ element. For a given direction $a$ , the contribution of the associated shock to the spectrum of $L_t$ at frequency $\omega$ is proportional to

\begin{equation*} g(\omega ,a) = \left ( D(e^{-i\omega }) A_{\tilde {}} a \right )^{\prime} e_1 e^{\prime}_1 \left ( D(e^{-i\omega }) A_{\tilde {}} a \right ). \end{equation*}

The Max-Share criterion aggregates $g(\omega ,a)$ over the full frequency range, reflecting the fact that the finite-horizon forecast error variance can be written as an integral over $[-\pi ,\pi ]$ . In contrast, the Limited-Spectral criterion integrates $g(\omega ,a)$ only over the low-frequency band $[-\omega _B,\omega _B]$ . In our calibration the confounding disturbance $b_t$ is concentrated at business-cycle frequencies (around an 8-quarter cycle), while the technology component is highly persistent and therefore contributes disproportionately near zero frequency. As a result, restricting attention to cycles longer than roughly ten quarters shifts the maximizer away from the cyclical confounder and toward the truly low-frequency technology-driven component.

True IRFs and shock correlations. Because the DGP is known, we can compute the true IRFs of $(L_t,N_t)$ to a one-time technology innovation directly from the structural model. Setting $\epsilon ^z_0 = 1$ and $\epsilon ^z_t = \epsilon ^b_t = 0$ for $t\geq 1$ , and starting from zero initial conditions, we iterate

\begin{align*} z^g_t &= \rho _{z^g} z^g_{t-1} + \epsilon ^z_t, \\ z^l_t &= \rho _z z^l_{t-1} + z^g_t, \\ b_t &= \rho _{b,1} b_{t-1} + \rho _{b,2} b_{t-2}, \\ L_t &= z^l_t + b_t, \\ N_t &= 0.7 N_{t-1} - 0.3 N_{t-2} - 0.3 z^l_t + 0.3 b_t, \end{align*}

for $t=0,\ldots ,H-1$ . This yields the benchmark IRFs $\text{IRF}^{true}_h = (L_h,N_h)^{\prime}$ .

To evaluate identification performance, we also compare the estimated shocks with the true structural shocks $(\epsilon ^z_t,\epsilon ^b_t)$ . From the estimated VAR we obtain the reduced-form residuals $u_t$ and define the candidate structural shocks

\begin{equation*} \hat s^{MS}_t = \alpha ^{\prime}_{MS} A_{\tilde {}}^{-1} u_t, \qquad \hat s^{LS}_t = \alpha ^{\prime}_{LS} A_{\tilde {}}^{-1} u_t. \end{equation*}

Because the sign of an eigenvector is arbitrary, we fix the sign of each estimator so that its IRF for productivity is aligned with the true productivity IRF.Footnote ² We then report correlations with the true shocks in absolute value,

\begin{equation*} \big |\textrm{corr}\big(\hat s^{MS}_t,\epsilon ^z_t\big)\big |,\quad \big |\textrm{corr}\big(\hat s^{LS}_t,\epsilon ^z_t\big)\big |,\quad \big |\textrm{corr}\big(\hat s^{MS}_t,\epsilon ^b_t\big)\big |,\quad \big |\textrm{corr}\big(\hat s^{LS}_t,\epsilon ^b_t\big)\big |, \end{equation*}

as well as pointwise absolute IRF bias, $|\text{IRF}^{\cdot }_h - \text{IRF}^{true}_h|$ .

4.2 Monte Carlo Evaluation

We assess the finite-sample performance of the Max-Share and Limited-Spectral estimators through Monte Carlo experiments. For each of $R=500$ replications we:

1. 1. Draw sequences of $(\epsilon ^z_t,\epsilon ^b_t)$ and simulate the structural model for $T+500$ periods using the baseline calibration described above, discarding the first $500$ observations.

2. 2. Estimate a VAR(2) in levels on the remaining $T=300$ observations of $(L_t,N_t)$ .Footnote ³

3. 3. Construct the MA coefficients $\{D_h\}_{h=0}^{H-1}$ , compute the Max-Share and Limited-Spectral impact vectors $A_0^{MS}$ and $A_0^{LS}$ , and obtain the corresponding IRFs $\text{IRF}^{MS}_h$ and $\text{IRF}^{LS}_h$ .

4. 4. Recover the identified shocks $(\hat s^{MS}_t,\hat s^{LS}_t)$ from the VAR residuals, align their sign with the true productivity IRF, and compute the absolute correlations with $(\epsilon ^z_t,\epsilon ^b_t)$ as well as the absolute IRF bias relative to $\text{IRF}^{true}_h$ .

Figure 1 summarizes the Monte Carlo distribution of IRFs and biases, while Table 1 reports correlations between the estimated shocks and the true structural innovations. In this business-cycle-confounding environment, the distinction between Max-Share and Limited-Spectral is visible both in the recovered shocks and in the induced impulse responses.

Table 1.

Monte Carlo correlation of VAR-identified shocks with true shocks

Notes: Entries report the median absolute correlation between the estimated shock $\hat s_t$ and the true structural shocks $(\epsilon ^z_t,\epsilon ^b_t)$ across $R=500$ Monte Carlo replications of the two-variable model. Brackets show the 5th and 95th percentiles of the Monte Carlo distribution.

Figure 1.

Monte Carlo impulse responses and bias in the two-variable model.

Notes: Each panel summarizes $R=500$ Monte Carlo replications. Top-left: IRF of productivity $L$ to a technology shock. Top-right: IRF of hours $N$ . Shaded regions denote 5th–95th percentiles across replications. Dashed lines give median IRFs under Max-Share (MS) and Limited-Spectral (LS) identification. Bottom panels show the median absolute deviation from the true IRF for each horizon.

First, Table 1 shows that Limited-Spectral produces a substantially cleaner estimate of the technology shock. The median absolute correlation with the true technology innovation is

\begin{equation*} \big |\textrm{corr}\big(\hat s^{MS}_t,\epsilon ^z_t\big)\big |_{\textrm{median}} = 0.578 \;\; [0.408, 0.725], \qquad \big |\textrm{corr}\big(\hat s^{LS}_t,\epsilon ^z_t\big)\big |_{\textrm{median}} = 0.813 \;\; [0.697, 0.872]. \end{equation*}

Conversely, Max-Share loads much more strongly on the confounding non-technology shock:

\begin{equation*} \big |\textrm{corr}\big(\hat s^{MS}_t,\epsilon ^b_t\big)\big |_{\textrm{median}} = 0.806 \;\; [0.658, 0.894], \qquad \big |\textrm{corr}\big(\hat s^{LS}_t,\epsilon ^b_t\big)\big |_{\textrm{median}} = 0.534 \;\; [0.380, 0.698]. \end{equation*}

These statistics align directly with the design of the experiment: because $b_t$ is concentrated at business-cycle frequencies, unrestricted variance maximization can inadvertently select directions that capture substantial cyclical variance even when the object of interest is the persistent technology component. The Limited-Spectral restriction attenuates this mechanism by excluding variance outside the low-frequency band.

Second, Figure 1 shows how this contamination translates into impulse-response distortions: even though technology dominates asymptotically at very low frequencies, finite-horizon variance maximization remains sensitive to cyclical components when identification is performed using truncated impulse responses. The top panels plot the true technology IRFs (black) against the Monte Carlo distribution of identified IRFs under Max-Share (red) and Limited-Spectral (blue). The estimated responses exhibit pronounced cyclical dynamics – reflecting the business-cycle component embedded in the confounding shock – and therefore deviate from the monotone, highly persistent technology-driven path implied by the DGP. The bottom panels quantify these deviations: the median absolute IRF bias is largest at short horizons and declines with the horizon as the transitory cyclical component dissipates, with Limited-Spectral systematically, particularly at short horizons, reducing the bias relative to Max-Share.

Overall, the Monte Carlo evidence illustrates the central message of this section in its cleanest form. When confounding disturbances have substantial power at cyclical frequencies, time-domain variance maximization tends to recover a hybrid shock that is highly correlated with the confounder and generates biased IRFs. Imposing a low-frequency restriction in the objective function materially improves identification of the technology innovation and reduces IRF bias, though it does not eliminate bias entirely in finite samples. This improvement relies on spectral separation between the target and confounding shocks; when multiple disturbances overlap within the low-frequency band itself, frequency restriction need not alter the variance-maximizing solution – a case we document empirically below.

4.3 Interpretation and Link to the Business-Cycle Application

The two-variable simulations are deliberately stylized, but they capture a key feature of the environment where Limited-Spectral is most informative: one component is highly persistent and dominates very low-frequency movements in productivity, while confounding shocks are concentrated at higher (here, business-cycle) frequencies. By construction, the Limited-Spectral objective assigns weight only to a narrow low-frequency band in which the technology shock has relatively greater power, whereas the Max-Share criterion aggregates contributions from all frequencies and is therefore more exposed to variance generated by cyclical confounding disturbances.

The situation will be more challenging in the business-cycle application below, where multiple structural shocks have substantial spectral mass in the same range of frequencies. When both demand and supply shocks contribute appreciably to the variance of output and inflation at business-cycle frequencies (for example, 6–32 quarters – in our empirical applications, we map calendar frequencies into angular frequencies as $\omega = 2\pi / T$ , so that business-cycle fluctuations over 6–32 quarters correspond to $\omega \in [2\pi /32,\; 2\pi /6]$ ), neither time- nor frequency-domain variance-maximizing identification can cleanly isolate a single structural innovation. In that case, both Max-Share and Limited-Spectral procedures will recover hybrids whose interpretation requires additional economic restrictions or sign constraints.

The two-variable simulations thus play a dual role. They provide a transparent benchmark in which the gains from Limited-Spectral identification are easiest to see, and they highlight the conditions under which such gains are likely to be realized in practice: a dominant low-frequency driver and confounding shocks that matter relatively more at shorter horizons. In the empirical sections that follow, we use these insights to guide the choice of frequency band and horizon and to interpret the resulting shocks in environments where low-frequency dominance may fail or where the shocks of interest are inherently multi-frequency.

5.1 Confounding the Identification of Business-Cycle Frequency Shocks

Recently, some influential papers have employed the spectral approach to Max-Share, as presented here, to identify the driving forces of business cycles. We note that the same logic of confounding shocks could also apply in the case of overlapping business cycle frequency shocks. For example, Giannone et al. (Reference Giannone, Lenza and Reichlin2019) find a shock that maximizes the variance of a target variable at business-cycle frequencies will encompass a linear combination of structural shocks. We formalize this statement, finding that the shock will encompass a combination of structural shocks broadly in proportion to their relative contribution to the variance over the select frequency band. We propose that the same logic outlined in Appendix A also holds in the case of targeting the shock that maximizes the variance within a business-cycle frequency band.

Take, for example, two structural business-cycle frequency shocks, $\epsilon _{BC1}$ and $\epsilon _{BC2}$ , affecting the (same) two endogenous variables in the previous example, but now at business-cycle frequencies. As in the example for low-frequency confounding shocks, the identified shock will be a combination of the shocks. In this case, those that drive business cycle frequency variation of the target variable.

This will make it difficult to interpret IRFs of the identified shock in question, given they will represent a combination of drivers with only an a priori understanding of their relative weights. Furthermore, the variance of many macroeconomic drivers in the business-cycle frequency domain are more prone to contamination than in the low-frequency range. For example, at business-cycle frequencies, a wider array of shocks influence real and nominal variables. The competing effects of supply-side and demand-side influences may therefore be difficult to disentangle.

5.2 Demonstrating the Potential for Confounding Shocks Affecting the Spectral Approach

In this section, we use an example from the business cycle anatomy literature to demonstrate the potential for confounding shocks to pervert the results at higher frequencies. We first replicate the empirical findings of Angeletos et al. (Reference Angeletos, Collard and Dellas2020). We then proceed to use a simple theoretical construct to show the possibility that confounding shocks could distort conclusions about the main macroeconomic drivers at business-cycle frequencies.

5.3 Identifying the Business-Cycle Anatomy with US Data: the Dichotomy Revealed

The Spectral approach to Max-Share has been recently applied to finding dominant business-cycle frequency shocks. For example, Angeletos et al. (Reference Angeletos, Collard and Dellas2020) find a single primary driver of multiple macroeconomic variables at business cycle frequencies.

Using the Spectral method, Angeletos et al. (Reference Angeletos, Collard and Dellas2020) argue that dominant business-cycle drivers of real variables such as GDP, unemployment, and investment have very little impact on inflation, thereby ruling out inflationary demand shocks of the standard New Keynesian variety as key drivers of the business cycle. We estimate the same VARs used to evaluate the identification of technology in the US data and confirm this finding. Figure 2 presents the impulse responses. By targeting the shock that maximizes the variance of GDP at business cycle frequencies (6–32 quarters), the shock found to boost output by 1% in the short run causes inflation to rise by just 0.05 percentage points at its peak. In contrast, the shock that maximizes the variance of inflation at business cycle frequencies has a more substantial impact on inflation (−0.3 percentage points) but a smaller impact on output. These findings confirm the business cycle dichotomy highlighted in Angeletos et al. (Reference Angeletos, Collard and Dellas2020).

Figure 2.

Targeting output and inflation at business-cycle frequencies.

Note: $16^{th}$ and $86^{th}$ percentile error bands. The first row shows impulse responses to the shock that maximizes the variance of output at business-cycle frequencies (6–32 quarters). The second targets inflation. The VAR consists of US data on labor productivity, hours-worked, the investment to GDP ratio, the consumption to GDP ratio, the consumption expenditure deflator (inflation), and the 10-year US treasury yield.

5.4 Dissecting the Business-Cycle Anatomy with a Simple New Keynesian Model

Can the dichotomous empirical finding be an artifact of the variance-maximizing method in general, rather than a feature of the true DGP which we hope that our spectral restrictions would capture? For a reasonable parameterization of the model, we argue that an alternative candidate explanation for the absence of significant inflationary impact from the main driver of output at business cycle frequencies is the presence of competing demand and supply-side influences. The combination of these competing forces dulls the response of inflation. Demand-side drivers of output will cause inflation to rise, while supply-side drivers of output can cause output to rise while reducing inflation (as with technology and markup shocks, for example). The presence of both types of shock in driving output will result in a hybrid IRF to inflation, consisting of a mix of the effects of both shocks.

A simple New Keynesian model demonstrates how the Spectral methodology can produce these results, even in the case where inflationary demand shocks drive the majority of the variance of output:

(15) \begin{align} y_{t} = \frac {-1}{\sigma }\left (R_{t}-E_{t}[\pi _{t+1}]\right ) +E_{t}[y_{t+1}]+\eta _{t} \\[-30pt] \nonumber \end{align}

(16) \begin{align} \pi _{t} = \kappa MC_{t} +\beta E_{t}[\pi _{t+1}] \\[-25pt] \nonumber \end{align}

(17) \begin{align} MC_{t} = (\sigma +\chi )y_{t}-(1+\chi )a_{t} \\[-25pt] \nonumber \end{align}

(18) \begin{align} R_{t} = (\phi _{t})y_{t}+\phi _{\pi }\pi _{t}, \end{align}

where $y_{t}$ is output, $R_{t}$ is the nominal interest rate, $\pi _{t}$ is inflation, and MC is marginal costs. $\eta _t$ represents a demand (preferences) shock, while $a_{t}$ reflects a supply-side technology shock. $\sigma$ is the intertemporal elasticity of substitution, $\chi$ is the Frisch elasticity of labor supply. $\kappa$ is the slope of the Phillips curve and is a function of the probability of not being able to reset prices each period ( $\theta$ ) and the discount rate $(\beta ) \, : \, \kappa = (1-\theta )(1-\beta \theta )/\theta$ .

In this standard framework, there are two key drivers of fluctuations in output and inflation. Let us suppose that these are transitory drivers, with sizable contributions to the business cycle. These shocks therefore follow auto-regressive processes with a coefficient of 0.7 on the shock’s auto-regressive coefficients ( $a_{t} = 0.7a_{t-1}+\epsilon _{a}$ and $\eta _{t} = 0.7\eta _{t-1}+\epsilon _{\eta }$ ). Elasticities are set to 1 for simplicity ( $\sigma$ and $\chi$ ) while the inflation aversion coefficient is set to 1.5 and the output gap coefficient is set to 0.5 in the policy-rate equation. Shock variances are set at $\sigma ^{2}_{a} = 0.004$ and $\sigma ^{2}_{\eta } = 0.002$ , which results in the supply shock explaining roughly three-quarters of the variance of inflation and the demand shock explaining roughly three-quarters of the variance of output.

Simulating the data for 250 periods and then separately identifying the dominant business-cycle driver of output and inflation in a two-variable VAR using the Spectral approach, as in Angeletos et al. (Reference Angeletos, Collard and Dellas2020), produces the postulated result, presented in Figure 3. Despite the presence of clearly separate driving processes, the IRFs for inflation end up as a mixture between the supply and demand shock IRFs. This provides the appearance of a subdued response of inflation to a demand shock, despite the demand shock dominating the variance of output in the true DGP. In addition, the response of output to the supply shock is subdued due to the offsetting impact of the demand shock.Footnote ⁴

Figure 3.

Targeting output and inflation at business-cycle frequencies in a two shock New Keynesian model.

Note: $16^{th}$ and $86^{th}$ percentile error bands. The first row shows impulse responses to the shock, which maximizes the variance of output produced by the simple New Keynesian model at business-cycle frequencies (6–32 quarters). The second targets inflation. Blue lines indicate the true IRF of the demand shock (top row) and supply shock (bottom row). The shaded area and red line indicate the VAR-estimated IRF median and 68% confidence bands.

The Spectral method will erroneously conclude that a dominant structural demand shock has little effect on inflation. Practitioners will need to have an a priori understanding of the range of potential confounding shocks in these exercises and interpret the results accordingly.

Additional restrictions within the band. One might ask whether hybridization at business-cycle frequencies could be reduced by imposing additional moment conditions – such as covariance restrictions, sign restrictions, or exclusion restrictions – within the 6–32 quarter band. While such restrictions can in principle help separate demand and supply shocks (see Francis and Kindberg-Hanlon (Reference Francis and Kindberg-Hanlon2022), they go beyond variance maximization and introduce auxiliary identifying structure. Our objective in this section is instead to show that variance-based identification alone, even when targeted to the business-cycle band, cannot recover pure structural shocks when the underlying shocks have overlapping spectral support. This limitation is thus inherent to the informational content of the data, not to the particular estimator we use.