The results below are based on 100 simulations of the DSGE model, generating 250 observations after a burn-in of 100 periods. Each VAR is estimated via a Gibbs sampling procedure with flat priors, saving 1000 draws following a 500-period burnin. The same procedure is used for all simulations throughout the document.

A four-variable VAR is estimated using the level of productivity, total hours worked, and the share of investment and consumption in GDP. For the long-run identification, log-differenced productivity is used, as is standard.

Each VAR specification makes a relatively small error relative to the true DSGE IRF of the technology shock to labor productivity, although the long-run specification diverges over time. This can be seen in the bias of the median IRF for each identification relative to the true DSGE IRF (Figure 2). In addition, the shocks uncovered by the VARs have a high correlation with the true underlying technology shock (Table 2). In the case of the long-run identification, given that the technology shock is not a true unit-root process, it is not surprising that it performs poorly. However, as we will show in subsequent simulations, the long-run specification is also outperformed in a variety of DSGE and non-DSGE settings where technology is a unit-root process. In addition, the NAMS specification has a modestly higher IRF bias than the Max-Share and Spectral specifications. This is in part due to the high persistence of the IRFs of non-technology shocks on labor productivity in the DSGE model.<sup>10</sup> This is a challenge to the NAMS framework, which aims to isolate processes that are still in the data at this horizon, assuming that business cycle factors should have faded at this point. This is one motivation to switch to a simpler and more transparent framework for testing the VARs than by using DSGE models.

**Table 2:** Correlation of VAR-identified technology shocks with true DSGE-generated shock

|Identifcation|**Max-Share**|**Long-Run**|**Spectral**|**Limited Spectral**|**NAMS**|
|---|---|---|---|---|---|
|**Correlation**|0.94|0.62|0.93|0.93|0.93|
||(0.86, 0.98)|(0.19, 0.85)|(0.85, 0.97)|(0.84, 0.97)|(0.77, 0.97)|

Note: 5th and 95th percentile values shown in brackets The high performance of the VARs is likely to be in part due to the shock processes driving the DSGE model. The highly persistent technology shock in this and other similar DSGE models tends to drive the vast majority of the variance of labor productivity. In this calibration, over 99% of the volatility of labor productivity is driven by _εa_ , the technology shock.<sup>11</sup> Therefore, several of the methodologies find a similar result given how insignificant the non-technology driving processes are in the variation of productivity. In (Francis et al., 2014), the technology shock was a true unit-root process, which dominates the variance of technology by even more than the persistent AR(1) chosen here.

> 10As shown in Appendix section 9.2, up to 25% of the standard deviation of the initial non-technology shocks are still present in the labor productivity data at period 40.

> 11Based on resimulating the model over 100,000 periods one shock at a time.

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