metadata: authors: 0: author: Dieppe,Alistair Matthew
1: author: Neville,Francis
2: author: Kindberg-Hanlon,Gene
count: World
docna: 0: docna: New Approaches to the Identification of
            Low-Frequency Drivers : An Application to Technology Shocks
docty: Policy Research Working Paper
subtopic: Labor Markets,Macroeconomic Management,National Governance,Social Analysis,Quality of Life & Leisure,Youth and Governance,Government Policies,Public Finance Decentralization and
            Poverty Reduction,Macro-Fiscal Policy,Taxation & Subsidies,Public Sector Economics,Economic Adjustment and Lending
teratopic: Macroeconomics and Economic Growth,Social Development,Governance,Social Protections and Labor,Public Sector Development
abstracts: cdata!: This paper addresses the identification
            of low-frequency macroeconomic shocks, such as technology,
            in Structural Vector Autoregressions. Whilst identification
            issues with long-run restricted VARs are well documented,
            the recent attempt to overcome said issues using the
            Max-Share approach of Francis et al. (2014) and Barsky and
            Sims (2011) has its own shortcomings, primarily that they
            are vulnerable to bias from confounding non-technology
            shocks. A modification to the Max-Share approach and two
            further spectral methods are proposed to improve empirical
            identification. Performance directly hinges on whether these
            confounding shocks are of high or low frequency. Applied to
            US and emerging market data, spectral identifications are
            most robust across specifications, and non-technology shocks
            appear to be biasing traditional methods of identifying
            technology shocks. These findings also extend to the SVAR
            identification of dominant business-cycle shocks, which are
            shown will be a variance-weighted combination of shocks
            rather than a single structural driver.
repnme: repnme: New Approaches to the Identification of
            Low-Frequency Drivers : An Application to Technology Shocks
display_title: New Approaches to the Identification
            of Low-Frequency Drivers : An Application to Technology Shocks
listing_relative_url: /research/2019/10/31505127/new-approaches-identification-low-frequency-drivers-application-technology-shocks
topicv3: Political Economy of Reform m1326263 836,Treasury m1327009 1045,Monetary Policy m1327609 444,Taxation m1327615 1121,Labor Productivity m1504060 2756
publishtoextweb_dt: 2019-10-24T00:00:00Z
historic_topic: Macroeconomics and Economic Growth,Social Development,Governance,Social Protections and Labor,Public Sector Development
no_of_pages: 55
author_keys: 0: author_key: 1674011
1: author_key: 1676939
model_extractions: Policy Research Working Paper 9047 New Approaches to the Identification of Low-Frequency Drivers An Application to Technology _Alistair Dieppe Neville Francis Gene Kindberg-Hanlon_ Equitable Growth, Finance and Institutions Global Practice October 2019
Policy Research Working Paper 9047

# **Abstract**

This paper addresses the identification of low-frequency macroeconomic shocks, such as technology, in Structural Vector Autoregressions. Whilst identification issues with long-run restricted VARs are well documented, the recent attempt to overcome said issues using the MaxShare approach of Francis et al. (2014) and Barsky and Sims (2011) has its own shortcomings, primarily that they are vulnerable to bias from confounding non-technology shocks. A modification to the Max-Share approach and two further spectral methods are proposed to improve empirical identification. Performance directly hinges on whether these confounding shocks are of high or low frequency. Applied to US and emerging market data, spectral identifications are most robust across specifications, and non-technology shocks appear to be biasing traditional methods of identifying technology shocks. These findings also extend to the SVAR identification of dominant business-cycle shocks, which are shown will be a variance-weighted combination of shocks rather than a single structural driver.

This paper is a product of the Equitable Growth, Finance and Institutions Global Practice. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at gkindberghanlon@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
New Approaches to the Identification of Low-Frequency Drivers: An Application to Technology Shocks<sup>_∗_</sup> Alistair Dieppe<sup>_†_</sup> , Neville Francis<sup>_‡_</sup> , and Gene Kindberg-Hanlon<sup>_§_</sup>

**Keywords** : Productivity, Technology shocks, SVARs. **JEL classification** : C11, C30

> _∗_ We would like to thank seminar participants at UCSD and the Midwest Macro Conference, Atlanta 2019. We have also benefited from conversations with Michal Andrle, Calebe Figueiredo, James Hamilton, Simona Manu, Garey Ramey, Valerie Ramey, and Robert Vigfusson. We gratefully acknowledge financial support from the PHRD fund. The views expressed here are the authors, and do not reflect those of the World Bank, its Executive Directors, or the countries they represent.

> _†_ World Bank adieppe@worldbank.org

> _‡_ University of North Carolina and World Bank nfrancis@unc.edu

> _§_ World Bank gkindberghanlon@worldbank.org
# **Contents**

|**1**|**Intr**|**oduction**<br>**3**|
|---|---|---|
|**2**|**Em**|**pirical Approaches**<br>**5**|
||2.1|Identifcations and their Drawbacks . . . . . . . . . . . . . . . . . . .<br>6|
|||2.1.1<br>Long-Run Restrictions . . . . . . . . . . . . . . . . . . . . . .<br>6|
|||2.1.2<br>Max-Share<br>. . . . . . . . . . . . . . . . . . . . . . . . . . . .<br>6|
||2.2|New Approaches<br>. . . . . . . . . . . . . . . . . . . . . . . . . . . . .<br>7|
|||2.2.1<br>Non-Accumulated Max-Share (NAMS) . . . . . . . . . . . . .<br>8|
|||2.2.2<br>Spectral Identifcation . . . . . . . . . . . . . . . . . . . . . .<br>9|
|||2.2.3<br>Limited-Horizon Spectral Identifcation<br>. . . . . . . . . . . .<br>11|
|**3**|**Mo**|**del Evaluation**<br>**11**|
||3.1|DSGE Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . .<br>11|
||3.2|Monte-Carlo DSGE Results . . . . . . . . . . . . . . . . . . . . . . .<br>12|
||3.3|An Illustrative Two Variable Data-Generating Process . . . . . . . .<br>14|
|||3.3.1<br>Motivating the Choice of Stochastic Processes . . . . . . . . .<br>16|
|||3.3.2<br>SVAR Performance . . . . . . . . . . . . . . . . . . . . . . . .<br>17|
||3.4|The Case of Unit Roots . . . . . . . . . . . . . . . . . . . . . . . . .<br>19|
||3.5|Summary of Identifcation Performance<br>. . . . . . . . . . . . . . . .<br>22|
|**4**|**SV**<br>|**AR Performance in the Framework of Chari, Kehoe, McGrattan**<br>|
||**(20**|**08)**<br>**23**|
|**5**|**Ap**|**plication to US Data**<br>**27**|
|**6**|**Ap**|**plications to Emerging Markets**<br>**31**|
|**7**|**Det**|**ecting Business-Cycle Shocks**<br>**32**|
|**8**|**Con**|**clusion**<br>**34**|
|**9**|**Ap**|**pendix**<br>**35**|
||9.1|What shock is Max-Share capturing?: Sources of bias<br>. . . . . . . .<br>35|
|||9.1.1<br>Low and high-frequency drivers of forecast errors . . . . . . .<br>35|
|||9.1.2<br>Solving the maximization problem . . . . . . . . . . . . . . .<br>36|
||9.2|DSGE Model Specifcation . . . . . . . . . . . . . . . . . . . . . . . .<br>39|
||9.3|Alternative simulations. . . . . . . . . . . . . . . . . . . . . . . . . .<br>44|
|||9.3.1<br>More persistent confounding shocks<br>. . . . . . . . . . . . . .<br>44|
|||9.3.2<br>Changing the target horizon for Max-Share and NAMS<br>. . .<br>45|
|||9.3.3<br>Where do long-run restrictions work well? . . . . . . . . . . .<br>46|
||9.4|CKM (2008) simulations . . . . . . . . . . . . . . . . . . . . . . . . .<br>47|
||9.5|US Data Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . .<br>50|

2
# **1 Introduction**

In this paper we revisit the use of Structural Vector Autoregressions (hereafter, SVAR) to identify shocks with highly persistent impacts, focusing on its specific application to the identification of technology shocks. We examine the recently proposed solutions to address known drawbacks of the traditional long-run restriction by Francis et al. (2014) and Barsky and Sims (2011), and unearth a key weakness of these methodologies that have so far been ignored, or more accurately not recognized. We suggest further modifications to the existing methods and point out the features of the data generating process that will result in the choice of one over another.

The use of SVARs to ‘look-through’ cyclical changes in productivity and isolate structural developments, or changes in technology, can be traced back to Blanchard and Quah (1989). In their approach, long-run restrictions are imposed in a structural VAR to separate the effects of temporary ‘demand’ and permanent ‘supply’ shocks on GDP. This methodology was later adapted by Gal ́ı (1999) to specifically identify technology shocks in a two-variable VAR containing log-differences of productivity and hours worked.<sup>1</sup> Approaches using long-run restrictions are advantageous on a number of grounds, requiring only a parsimonious selection of variables and imposing no restriction on the short-run impacts of technology shocks; there are, however, some controversy associated with the latter.<sup>2</sup> Long-run restrictions have been criticized on two main grounds, one economic and the other econometric: first, it is restrictive to assume that technology is the only shock that can affect productivity in the long-run;<sup>3</sup> and second, econometrically, that imposing long-run restrictions on a finite sample leads to biased and estimates.

It is the second strand of the literature we investigate more deeply in this paper, notably the pitfalls of using alternative medium-run restrictions to identify longrun shocks. The medium-run identification strategy is more robust to estimation on finite samples, identifying technology shocks as those which contribute the most to the forecast error variance decomposition (FEVD) of labor productivity at the 10-year horizon (Francis et al., 2014). We hereafter refer to this approach as the ‘Max-Share’ identification. This strategy has also been used to identify technology

> 1Whereas Christiano et al. (2004) challenge the inclusion of hours in first differences in Gal ́ı (1999), noting that the level of hours worked is likely to be a stationary variable, not requiring differencing.

> 2In a real-business cycle model, hours rise in response to a positive technology shock. In contrast, technology shocks cause total hours to fall in a New Keynesian model with sticky prices. The technology VAR literature is generally agnostic about the short-run impact of technology shocks, imposing only long or medium-run restrictions. However, Dedola and Neri (2007) have estimated a sign-restricted VAR based on impulse-response commonalities between an RBC and a New Keynesian model under a variety of parameterizations.

> 3Several strands of research have suggested other non-technology shocks that may also result in permanent effects on labor productivity. Mertens and Ravn (2013) find that changes in taxation can have long-running effects of productivity, which once controlled for, lead to different dynamics of macroeconomic variables in response to identified technology shocks. Fisher (2006) separates productivity shocks into those that specifically apply to investment goods (IST shocks), and neutral technology shocks that affect aggregate production. In addition, Francis and Ramey (2005) and Uhlig (2004) have debated the extent to which the presence of these additional shocks alters the literature’s findings on the impact of technology shocks on hours worked.

3
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technology shocks just as long-run identifications can. We consider three alternative approaches to sharpen the identification of technology shocks in the presence of confounding shocks. The first is a modification to the Max-Share approach. This modification hinges on the choice of horizon used in estimation. We next propose a spectral identification approach whose rotation is in the frequency domain. Finally, we modify the spectral approach by substituting the long-run variance-covariance matrix with one that can be reasonably obtained in small samples.

When using Monte-Carlo simulations to validate our methodologies, we highlight that a key drawback of this exercise is that few DSGE calibrations contain nontechnology shocks with a material impact on the variance of productivity. That is, in order to generate non-trivial confounding shocks, we need to incorporate certain kinds of stochastic processes not usually assumed for these models, but which are present in the data. As an alternative, we employ a simple two-variable model that provides greater transparency around the data-generating process than a DSGE model.

In summary, this paper makes the following contributions to the literature. First, we demonstrate that the Max-Share identification is susceptible to bias from confounding low and high-frequency shocks. Second, we propose three alternative identification methodologies. The first, Non-Accumulated Max-Share (NAMS), shows a lower degree of bias in the presence of low-frequency confounding non-technology shocks. The second and third, labeled Spectral and Limited Spectral approaches, show less bias in the presence of non-technology high and medium frequency shocks other than technology. Finally, we show that both the Max-Share and our improved alternative identifications are more robust than long-run restrictions to the problems of both confounding non-technology shocks and lag-truncation bias identified by Chari et al. (2008). Therefore, SVAR techniques can still play an important role in testing the predictions of macroeconomic models.

The remainder of the paper is as follows. We first revisit existing technology identification methodologies, drawing attention to newly identified shortcomings of the Max-Share approach before outlining several proposed improvements. We then show the performance of the existing and new identifications in Monte Carlo simulations on DSGE-generated data and data generated by a simple 2-variable model. Finally, we take each specification to data for the United States and several major emerging and developing economies (EMDEs), finding that there are several sources of contamination in traditional VAR approaches that can be addressed with our new methodologies.

# **2 Empirical Approaches**

In this section, we briefly revisit the standard long-run and Max-Share identification of a technology shock. The potential for the Max-Share approach to be contaminated by other non-technology shocks is then explained, before we detail three approaches that can reduce contamination, depending on its source.

5
<!-- Start of picture text --> t JtJ<br>ee eeee<br><!-- End of picture text -->
(Francis et al., 2014) surmise that 10 years is longer than the period over which the business cycle occurs (typically assumed to be 2-8 years), but short enough to reduce challenges related to estimation on a finite sample. (Francis et al., 2014) impose this restriction in a VAR containing productivity, hours, consumption and investment as a share of GDP. The forecast error at horizon _k_ can be written: By defining an orthonormal matrix _A_ 0 with columns _α_ , and _e_ as a selection vector (size 1 _×_ n), we find the shock _j_ which maximizes the contribution to the total forecast error variance of variable _i_ at horizon _k_ The technology shock at this maximized value is then: _ε_<sup>_tech_</sup> _t_ = _α_<sup>_′_</sup> _chol_ (Σ _u_ )<sup>_−_1</sup> _ut_ . Following Uhlig (2003), identifying the structural shock that maximizes the contribution to the forecast error variance of productivity is solved by identifying the eigenvector associated with the maximum eigenvalue of _Vτ_ , where _Vτ_ is the FEVD of the target variable based on reduced-form shocks, and the denominator of _ω_ ( _α_ ).

# **2.2 New Approaches**

A drawback of the Max-Share approach is that in addition to capturing the long-run shock of interest it also captures aspects of other shocks in the data. We propose three alternative empirical approaches that offer reduced interference from these confounding shocks in different circumstances.

We first formally demonstrate the confounding nature of other shocks when using the Max-Share identification. We present the key equations of the eigenvalueeigenvector problem and refer the reader to Appendix Section 9.1 for details of the proof.

Max-Share involves setting up the Lagrangian for _Vτ_ : whose first order conditions reduce to solving for the eigenvector associated with the largest eigenvalue of _Vτ_ 7
For a simple two variable VAR with the true structural coefficients of the form: 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. Given reduced form MA coefficient counterparts, _B_ , the solution for _α_ will be of the general form:<sup>5</sup> In an attempt to isolate the low frequency technology shock, we can clearly see the potential contamination coming from the high frequency shock as _α_ depends on <u>11</u> the initial impact of the respective shocks of the target variable (1) (<sup>_A_</sup> _A_<sup>_lh_</sup> 12<sup>), and their</sup> relative persistence: <u>�</u> _kτk_ <u>=1</u><sup><u>(</u></sup><sup>_B_</sup> _<u>τ</u>_<sup>11</sup><sup>_Al_</sup> <u>11</u><sup>+</sup><sup>_B_</sup> _<u>τ</u>_<sup>12</sup><sup>_Al_</sup> <u>21</u><sup><u>)</u>Essentially,thederivedshockwillnot</sup> <u>�</u> _τ_ =1<sup>(</sup><sup>_B_</sup> _τ_<sup>11</sup><sup>_Ah_</sup> 12<sup>+</sup><sup>_B_</sup> _τ_<sup>12</sup><sup>_Ah_</sup> 22<sup>).</sup> be of a ‘pure’ form, but rather a combination of shocks with the ratio dependent on their importance in driving the forecast error variance at the chosen horizon. As the true form of _A_ 0 is unobserved, the extent of this contamination in empirical applications is also unknown. _In general, there is no closed form solution to the eigenvalue-eigenvector problem and the above expression is for an extremely special case where such a solution exists._<sup>6</sup> _However, we consider this illustrative as the more complicated solution will exhibit the features of this restrictive expression._

# **2.2.1 Non-Accumulated Max-Share (NAMS)**

The standard Max-Share approach takes cumulative forecast errors up to time _k_ : see equation 6. As indicated above, and expanded in the Appendix, we show that there may be instances where non-trivial proportions of the forecast error variance is driven by shocks of lower persistence than the shock of interest.

The first methodology we propose will sharpen the identification of shocks with long-run impacts by reducing the weight given to less persistent processes than technology. For example, a high volatility AR(1) process with a small AR coefficient is also a predominantly low-frequency process, but has no meaningful lasting impact at the 10-year horizon. As an alternative to Max-Share, we propose finding the maximum of the square of the impulse response functions at a particular horizon _k_ . Here we aim to find the shock which has the maximum effects at say period 40, ignoring all the previous horizon shocks. At this horizon, it may be expected that the effects of lower-persistence shocks will have dissipated. The NAMS approach is implemented in a similar way to the Max-Share approach, by solving:

> 5To satisfy the unit length restriction we will need to further normalize this by the length of the eigenvector.

> 6For example, assuming the variance-covariance matrix is commutative at each horizon

8
Kurmann and Sims (2017) also advocate reducing the impact of less persistent shocks in the Max-Share approach. In the Barsky and Sims (2011) identification, the forecast error variance under consideration is ’double-weighted’ - the maximization is applied to the summed forecast error variance from periods 1 to _k_ (max<sup>�</sup><sup>_k_</sup> _i_ =0<sup>_ωi_(</sup><sup>_α_),comparedtotheFrancisetal.(2014)approachofmax</sup><sup>_ωk_(</sup><sup>_α_)).</sup> Kurmann and Sims (2017) propose returning to the original identification of Francis et al. (2014), which maximizes a single forecast error variance at horizon _k_ , finding that it helps sharpen the identification of technology news shocks. Our NAMS approach takes this to its logical conclusion, further reducing the distortion caused by transitory shocks.<sup>7</sup>

# **2.2.2 Spectral Identification**

The NAMS approach deals with contamination from low-persistence, low-frequency processes. However, the Max-Share approach may also be contaminated by driving processes that occur at business-cycle and higher frequencies. Where the amplitude of these shocks are (coincidentally) high at the chosen target range ( _k_ ), the NAMS approach may also be biased.

We investigate the use of identification in the frequency domain, which can maximize the share of variance explained only at frequencies that are of interest, excluding those that are not.

Identifying technology shocks through restrictions that explain the majority of low (long-term) frequency volatility of productivity is a novel approach. However, this methodology has in the past been used to assess the types of shocks which drive the business cycle. For example, Angeletos et al. (2018) find that a single shock drives the majority of the variance of a range of macroeconomic variables at business cycle frequencies. DiCecio and Owyang (2010) use the spectral approach to conduct an identification of technology shocks that is similar to what we propose in this article but do not evaluate its performance.<sup>8</sup> We effectively apply a band-pass filter to the reduced form coefficients of a VAR containing macroeconomic variables, identifying the spectral density of the variables within a particular frequency band. We then identify the technology shock by maximizing the variance of productivity explained at the desired frequency.

The spectral density of series Y at frequency _ω_ can be written as a Fourier transform of its auto and cross covariances ( _γ_ ):

> 7Uhlig (2004) suggests an identification in which non-technology shocks will have no effect after 10 years, such that technology shocks can be identified by a restriction such that only the technology shock has an impact on productivity at that horizon. Our approach differs in that we do not exclude other shocks from having an effect at this horizon, and instead look for the shock that _dominates_ at 10-years.

> 8In addition, Christiano et al. (2006) find that by using a spectral estimator at frequency 0, Longrun restriction estimates prove less biased following a MCMC assessment of VAR performance.

9
Therefore, once _γ_ ( _τ_ ) in known, the spectrum, _SY Y_ ( _ω_ ), can be straightforwardly calculated. The reverse also holds. That is, knowing the spectrum, _SY Y_ ( _ω_ ), leads to an easy computation of _γ_ ( _τ_ ) by just using the inverse Fourier transform, Setting _τ_ = 0, gives the variance of the time series _Y_ .

This means that the variance of _Y_ is the sum of the spectrum over all frequencies, _−π < ω < π_ . This further indicates that the spectrum decomposes the variance of _Y_ into components from non-overlapping frequencies. Therefore, similar to the MaxShare identification, spectral analysis allows us to gauge the importance of cycles at different frequencies to the variance of the series of interest. And importantly, we can remove unwanted frequencies from the maximization problem.

To employ this methodology we first need to uncover a VAR representation of the spectral density of Y. We start by writing the Wold representation of the VAR (assuming it is invertible): By post-multiplying _Yt_ by _Yt−τ_ and summing across its lags (of _τ_ periods), the series of auto and cross covariances is: Then, by writing _D_ � _e_<sup>_−iτω_�</sup> = � _I −_ � _B_ 1 _Le_<sup>_−iω_</sup> + _B_ 2 _L_<sup>2</sup> _e_<sup>_−i_2</sup><sup>_ω_</sup> + _. . . BpL_<sup>_p_</sup> _e_<sup>_−ipω_��</sup><sup>_−_1</sup> , the spectral density of _Y_ can be written as a function of the reduced-form VAR coTo assess the spectral density within a frequency band, the spectral power can be summed between _ω_ = [ _<u>ω</u> , ω_ ̄] As in the case of the Max-Share approach, the shock which maximizes the contribution to the variance of productivity over this band is the eigenvector associated with the largest eigenvalue of the matrix _SY Y_ ( _band_ ). To identify technology, the band of interest is restricted to frequencies below (longer-than) 10 years, to exclude business-cycle frequencies.

10
### **2.2.3 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. In practice, 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. For example, the infinite-MA representation, _D_ ( _e_<sup>_iω_</sup> ) can be replaced with a limited horizon series of impulse response coefficients.

One proposal is to use coefficients extending to 10 years of data, as in the original Max-Share approach:

# **3 Model Evaluation**

Traditionally, SVARs seeking to extract an unobservable shock, such as technology, have been evaluated on their performance using simulated data from a DSGE model (Barsky and Sims (2011), Francis et al. (2014), Chari et al. (2009), Erceg et al. (2005)). This is an intuitive evaluation, given the true underlying shock is known by construction. In this section, we argue that many of these tests have been performed using DSGE models which fail to test for the influence of material confounding shocks needed to replicate key features of US macroeconomic data. We later show that existing methodologies perform poorly when these features are included in simulated data from a simple 2-variable model that provides greater transparency over the data-generating process compared to a DSGE model.

## **3.1 DSGE Monte Carlo simulations**

Each VAR specification is tested on a standard medium-scale New Keynesian model, of the type proposed by Christiano et al. (2005), and used by Francis et al. (2014) and Barsky and Sims (2011) to evaluate their respective implementations of the Max-Share identifications.<sup>9</sup> This model contains features such as persistent consumption habits, investment adjustment costs, capital utilization, and partial price and wage indexation.

The model can be written as the following planner’s problem (full specification available in Appendix Section 9.2):

> 9In the Barsky and Sims (2011) ‘news’ implementation of the Max-Share identification, an additional ‘news’ shock is added to the standard medium-scale NK model, which affects technology with a lag of one period.

11
Subject to investment adjustment costs, affected by shocks _Zt_ : Monetary policy follows an inertial Taylor rule: and a government sector accounts for a stochastic proportion of output ( _ω_<sup>_g_</sup> ) Total output is a function of costly capital capacity utilization _ut_ : Four shocks drive the model: namely, technology, investment-adjustment costs, government spending and monetary policy shocks: The dynamics of the monetary policy shock ( _εi,t_ ) are governed by the monetary policy inertia parameter ( _ρi,t_ ).

# **3.2 Monte-Carlo DSGE Results**

A standard shock calibration is used to test each VAR specification (Table 1) based on (Barsky and Sims, 2011). However, we argue that this test may not be reflective of the types of factors that can influence the VAR results when they are applied to real data.

**Table 1:** Shock processes in NK model

|Identifcation|**Baseline**|
|---|---|
|_ρA_|0.95|
|_σA_|0.66|
|_ρ_<sup>1</sup><br>_Z_|0.8|
|_σZ_|0.15|
|_ρg_|0.8|
|_σg_|0.15|
|_ρi_|.8|
|_σi_|0.15|

12
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.

13
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a small modification to this model is made in order to cover the case of unit roots with a stochastic growth process, which appears to be present in the US data.

# **3.3.1 Motivating the Choice of Stochastic Processes**

We choose our shock processes to examine two plausible scenarios in the detection of technology shocks:

1. There are confounding low-frequency but less persistent processes in the data other than technology;

2. There are high or business cycle frequency processes in the data.

As described earlier, our NAMS approach has been designed to deal with the first case, while the spectral identifications are targeted at the second. Before we turn to the simulations, we briefly describe the specifications that can generate the two described processes. This includes covering the frequency domain properties of AR(1) and AR(2) processes.

Our choice of driving processes is motivated by the following well-known spectral density factoid. Consider the white noise process _εt_ , with variance _γ_ (0) = _σ_<sup>2</sup> and autocovariance function _γ_ ( _h_ ) = 0 for _h_ = 0.

Therefore, the spectrum is Now consider the AR(1) process _vt_ = _ρvt−_ 1 + _ε_ with autocovariance function _γ_ ( _h_ ) = _σ_<sup>2</sup> _ρ_<sup>_|h|_</sup> /(1- _ρ_<sup>2</sup> ). The associated spectrum is,

Notice when _ρ >_ 0 it is obvious that the spectrum is dominated by low-frequency shocks, and in the case of negative autocorrelation, _ρ <_ 0, the spectrum is dominated by high frequency components. This simple factoid shows that a specification of a simple AR(1) process for _b_ allows us to generate a confounding low-frequency process.<sup>13</sup> A business cycle frequency shock requires an AR(2) process.

We replicate a sinusoidal business-cycle shock process with a specific frequency ( _f_ ) using the following AR(2) process: Here we set _f_ such that the shock process for _b_ has a periodicity of 8-quarters (2 years).

In our application, to ensure the cyclical process degrades over time (and avoids a unit root), we multiply both coefficients ( _ρb,_ 1 and _ρb,_ 2) by 0.9. We also calibrate

> 13A negatively-signed AR coefficient would allow us to include a confounding high frequency shock. For a more detailed discussion of the data-generating processes behind a range of spectral densities, see Medel (2014).

16
the variance of the shocks to be different for a clear distinction between the driving processes. In both cases, the shock standard deviations are calibrated so that _ε_<sup>_z_</sup> , the technology shock, explains the majority (just over 50%) of the FEVD at period 40. This allows us to demonstrate that even where the shock of interest is the dominant shock, the application of the Max-Share identification will still result in biased estimates.

The two calibrations are shown in Table 3.

**Table 3:** Shock process calibration for two-variable simulations

|Parameter<br>**Case 1: **|**Low-frequency AR(1)**|**Case 2: Business frequency**|
|---|---|---|
|_ρb,_1|0.3|1.27|
|_ρb,_2|0.0|-0.7|
|_σb_|2|0.7|
|_σz_|1|1|

# **3.3.2 SVAR Performance**

**Case 1: low-frequency confounding shocks** : In the presence of an additional low-frequency, albeit less persistent shock, both the Spectral and NAMS identifications outperform the traditional Max-Share and long-run restriction approaches. The NAMS approach is least affected by the confounding shock, consistent with its intended purpose.

As predicted, the IRF of the Max-Share identification is biased upwards by the higher-variance shock _b_ (Figure 4), even though the targeted shock _z_ explains the majority of the forecast error variance at the 10-year horizon. In the presence of confounding low-frequency shocks (the less persistent AR(1) process), the NAMS approach shows the least IRF bias and shows minimal bias in the estimation of the FEVD share of technology in productivity (Figure 5). This is to be expected: by design, the NAMS approach gives minimal weight to low-persistence processes. The end result is a very high correlation between the estimated NAMS technology shocks and the true underlying shocks (Table 4).

The Spectral approaches also show less IRF bias than the Max-Share approach. This might seem unintuitive at first, as the confounding shock is also a low-frequency process, like the targeted technology shock. However, the lower persistence of the variable _b_ relative to _z_ lowers its contribution to the variance at low frequencies, reducing the bias. Referring to equation 30, it is clear that the contribution of the process _b_ to the variance of productivity at low frequencies will be increasing in the size of the persistence parameter _ρ_ .

**Case 2: business-frequency confounding shocks** : In the presence of the business-cycle frequency confounding shock, the Spectral VAR specifications have a clear advantage. While there is some evidence of contamination in the IRF (Figure 4), it is significantly below the contamination in the Max-Share, NAMS and longrun approaches. The shocks uncovered by the Spectral specifications also have a

17
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# **4 SVAR Performance in the Framework of Chari, Kehoe, McGrattan (2008)**

So far, we have shown that the NAMS, Spectral and Limited Spectral identifications either perform as well as, or exceed the performance of the Max-Share and longrun identification SVARs. However, our framework is a simple non-structural twovariable system designed for ease of interpretation. In this section, we use the model of Chari et al. (2008) to show that our methodologies outperform the long-run identification SVAR, and perform comparably with the Max-Share identification in their RBC DSGE framework. Chari et al. (2008) use this model to demonstrate the difficulties faced by long-run restricted SVARs in capturing technology shocks as other non-technology confounding shocks drive an increasing share of the variance in the model. We therefore test our proposed identification methodologies against this known DGP with which long-run restrictions are shown to have difficulty.

In this model, there is no advantage of using the new specifications above the advantages offered by the Max-Share identification, as the technology shock has a unit root, and has no impact on the _growth_ of productivity. However, we demonstrate that compared to the long-run restriction, all other specifications show lower degrees of bias. Therefore, in contrast to the findings of CKM, using the right identifications, SVARs can prove useful in identifying the impacts of technology shocks in a range of DGPs.

Consumers’ utility functions are given by where _c_ is consumption, _l_ is per capita labor, _β_ is the discount rate and _γ_ the population growth rate. Consumers maximize utility subject to the budget constraint _ct_ + (1 + _τx_ )[(1 + _γ_ ) _kt_ +1 _−_ (1 _− δ_ ) _kt_ ] = (1 _− τlt_ ) _wtlt_ + _rtkt_ + _Tt_ . Where _τx_ is a tax on investment, _k_ is the capital stock, _δ_ the depreciation rate, _w_ wages, _r_ the rental rate on capital and _T_ a lump sum transfer. Firms face a resource constraint _ct_ + (1 + _γ_ ) _kt_ +1 = _yt_ + (1 _− δ_ ) _kt_ .

The technology and non-technology shocks evolve according to This RBC model contains a unit root for technology, but also a highly persistent non-technology shock _τ_ , where _ρl_ is 0.95 in the standard calibration. Chari et al. (2008) note that two sources of bias exist for the long-run SVAR methodology: non-technology shocks will increase IRF bias as they drive a larger proportion of the variance of output; and a lag-truncation bias, where limited VAR lags result in a bias due to the true specification of the VAR having an _∞_ -representation.

23
<!-- Start of picture text --> 400 Productivity IRF, Long-Run, Alpha=0<br>5 50<br>(s)<br>@ ee<br>3 «0<br>oO<br>Qa<br>S 50 we<br>e ae<br>-100 =<br>100 90 80 70 60 50 40 30<br>Output variability due to tech. shocks (%)<br><!-- End of picture text --> <!-- Start of picture text --> 100 Productivity IRF, Max-Share<br>5 50<br>2 ee<br>© ae<br>oQ peeeecncenncnnenenenennnemnnnneae— coeeeeeenenenesenen<br>1) GE<br>© a<br>a Sa,vce,<br>2 50<br>5 —<br>-100<br>100 90 80 70 60 50 40 30<br>Output variability due to tech. shocks (%)<br><!-- End of picture text --> <!-- Start of picture text --> 400 Productivity IRF, Limited Spectral<br>5 50<br>2 aoa<br>2 mec ececrenmee te<br>8 emcee ereeee ene<br>5ioa 0 ——|ae<br>o 50 Oe<br>2 me<br>lo ee,<br>-100<br>100 90 80 70 60 50 40 30<br>Output variability due to tech. shocks (%)<br><!-- End of picture text -->

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Each VAR is estimated using 4 lags. The long-run SVAR is estimated with log hours specified as _ht − αht−_ 1, where _α_ determines the degree of quasi-differencing, as in CKM. This allows the VAR to be estimated with total hours in both levels and a highly quasi-differenced form.<sup>16</sup> All other SVAR specifications use hours in levels.

- The long-run IRF for productivity is ‘confidently wrong’ (also demonstrated by Chari et al. (2008)) as the non-technology shock generates over 50% of the variance of output in the model in the quasi-differenced long-run specification. The specification with hours in levels has the largest bias and confidence intervals of the remaining specifications. However, the Max-Share and our new approaches _correctly_ display uncertainty in the identification of technology shocks, via wider error bands, as the non-technology shock variance increases. See Christiano et al. (2007) for a discussion of large confidence intervals relative to the size of the bias in the context of Chari et al. (2008).

- Our alternative methodologies show lower bias than the long-run identification as the non-technology shock increases in size. However, the Max-Share identification is marginally more efficient than the spectral approach, showing slightly less bias at all horizons. The NAMS identification becomes more biased as the highly persistent non-technology shock grows larger in importance, given that the non-technology shock is calibrated to be highly persistent by CKM (0.95 AR coefficient), and therefore continues to have a material effect on labor productivity even at the 10-year horizon.

In the second exercise, we examine the robustness of each methodology to lagtruncation bias. This is the bias caused by estimating the VARs using a finite number of AR coefficients when the true DSGE-generated data has an infinitelag order. There are two main findings from running each method on 100,000 simulated data points from the RBC model and varying the number of lags used in the estimation.

- At low lag levels, alternate methods show lower initial bias relative to the long-run specification. Further out, at long horizons, the new specifications show a similar bias to the long-run identifications.

- The NAMS specification continues to show more bias than the Spectral and Max-Share specification on impact, due to bias stemming from the highpersistence non-technology shock.

> 16Two specifications are used, with hours in levels ( _α_ = 0) and hours with a high-degree of quasidifferencing ( _α_ = 0 _._ 99).

25
<!-- Start of picture text --> 12 Productivity IRF, Long-Run Alpha=0<br>0.8 AE<br>0.6<br>—--Truth<br>0.4 —4<br>40<br>20<br>0.2 —30<br>—B50<br>100<br>0<br>10 20 30 40 50 60<br>Periods<br><!-- End of picture text --> <!-- Start of picture text --> 42 Productivity IRF, Max-Share<br>08F<br>0.6<br>—--Truth<br>0.4 —4<br>40<br>20<br>0.2 —30<br>—B50<br>100<br>0<br>10 20 30 40 50 60<br>Periods<br><!-- End of picture text --> <!-- Start of picture text --> 42 Productivity IRF, Limited Spectral<br>1 ao<br>0.8 LEO<br>0.6<br>—--Truth<br>0.4 —4<br>40<br>20<br>0.2 —30<br>—B50<br>100<br>0<br>10 20 30 40 50 60<br>Periods<br><!-- End of picture text --> <!-- Start of picture text --> 12 Productivity IRF, Long-Run Alpha=0.99<br>0.8<br>0.6<br>---Truth<br>—4<br>0.4 10<br>20<br>0.2 —30<br>—50<br>100<br>0<br>10 20 30 40 50 60<br>Periods<br><!-- End of picture text --> <!-- Start of picture text --> 42 Productivity IRF, Spectral<br>0.8 ,<br>7<br>0.6<br>---Truth<br>—4<br>0.4 10<br>20<br>0.2 —30<br>—50<br>100<br>0<br>10 20 30 40 50 60<br>Periods<br><!-- End of picture text --> <!-- Start of picture text --> 42 Productivity IRF, NAMS<br>—<br>0.8F UA<br>0.6%<br>---Truth<br>—4<br>0.4 10<br>20<br>0.2 —30<br>—50<br>100<br>0<br>10 20 30 40 50 60<br>Periods<br><!-- End of picture text -->
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consumer durables), PCE inflation, and the yield on the US 10-year treasury. The VAR is estimated between 1953:Q2 and 2018:Q3, with the starting date constrained by the availability of the 10-year treasury yield data. All data are from the FRED database produced by the Federal Reserve Bank of St. Louis (see Appendix 9.5 for further details).

While the IRFs provide qualitatively similar results (Figure 11), the forecast error variance decomposition (FEVD) shows important differences between the identifications. The Max-Share and Spectral identifications, which are both susceptible to low-frequency contamination, have the highest explained variance at each horizon (Table 6). The NAMS approach, which is less biased by this type of contamination as shown in section 3.3.2, has the lowest. This suggests that the Max-Share and Spectral approaches are also capturing confounding low-frequency shocks in the data.

There appear to be few confounding business frequency shocks in the productivity levels data. In part, this is observable in the spectral densities shown earlier, which show an overwhelming domination of the variance of productivity at low frequencies (Figure 1). This can also be seen by the similarities between the MaxShare and limited spectral FEVD identifications. One way of thinking about the relationship between the two methodologies is that the limited spectral approach is simply a Fourier transform of the Max-Share identification, with the additional feature of excluding certain frequencies. The forecast error variance matrix with which the maximization takes place for each can be written as: and Therefore, medium and high-frequency volatility must be a sufficiently small proportion of the variance at this horizon so that the maximization problems are essentially equivalent. Under the standard Spectral identification (using the longrun representation of VAR coefficients), confounding low-frequency shocks will be given less weight relative to highly persistent shocks such as technology, providing an explanation of the lower proportion of explained variance in the standard Spectral relative to the limited Spectral identification.

28
**Table 6: Proportion of labor productivity FEVD explained at horizon t in each**

|Identifcation|**Max-Share**|**Long-Run**|**Spectral**|**Limited Spectral**|**NAMS**|
|---|---|---|---|---|---|
|**Levels: t=1**|0.45|NA|0.31|0.44|0.29|
|**Levels: t=20**|0.89|NA|0.81|0.89|0.69|
|**Levels: t=40**|0.93|NA|0.84|0.92|0.83|
|**Diferences: t=1**|0.99|0.16|0.36|0.35|0.13|
|**Diferences: t=20**|0.40|0.56|0.92|0.92|0.11|
|**Diferences: t=40**|0.32|0.64|0.94|0.94|0.13|

The growth rate of US productivity exhibits a wide range of frequencies driving its spectral density, unlike the level, suggesting an estimation of the VAR with productivity in log-differences will also be informative.

# **_Spectral identifications best capture technology shocks with persistent effects on productivity growth_**

When estimating the VARs in differences (Figure 12), the Spectral approaches show a more persistent IRF than when estimated on the level of productivity, and a similar share of forecast error variance explained of productivity. This demonstrates similar circumstances to the model-based scenario which simulated data in the case of technology taking the form of a unit-root plus stochastic trend growth specification (Section (3.4). The differenced Spectral VARs are therefore likely producing the least biased IRFs for the response of productivity to technology.

# **_Only the Spectral identifications produce consistent estimates when estimating on data in levels and differences_**

Comparing the impulse responses across the two specifications (productivity estimated in levels versus differences), shows that only the spectral estimators produce consistent impulse responses in each case (the long-run identification is always estimated in differences). In response to a positive technology shock, productivity, consumption, and investment rise, while hours, inflation, and the long rate fall. The confidence bands around the spectral estimates are also tighter than their counterparts. While the Max-Share impulses are mainly consistent, we see differences in the impulse of hours and long rates: both hours and the long rate are negative with productivity in levels and positive when estimated in differences.

NAMS produces contradictory or non-informative impulse responses across all variables except for consumption and investment when estimated on labor productivity in differences. Additionally, the accompanying wide confidence bands of the difference specification precludes us from drawing any concrete conclusions and again manifest themselves in the very low explained FEVD (Table 6). This is to be expected, given the impact of a technology shock on productivity _growth_ after 10 years is understandably small and uncertain.

29
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unsurprising to the extent that the news literature uses utilization-adjusted technology rather than labor productivity, which aims to remove some business cycle variation from the data. In addition, the maximized shock is considered orthogonal from period 1 technology innovations.<sup>17</sup> A second key difference is that our shock is not orthogonalized to contemporaneous innovations in productivity.

# **6 Applications to Emerging Markets**

In the United States, data is of relatively good quality, and volatility is relatively low. In emerging markets and developing economies (EMDEs), macroeconomic variables of interest are more volatile, and also often considered to be subject to larger business cycle fluctuations than advanced economies (Neumeyer and Perri (2005), Garc ́ıa-Cicco et al. (2010)). EMDEs are therefore ideal candidates to demonstrate differences across the proposed SVAR identifications.

EMDE data availability is limited. Few EMDEs publish quarterly data on hoursworked or employment before 2000. Instead, we prioritize data _span_ over data frequency, given our identifications focus on the persistence of shocks. Using annual data from the World Bank’s World Development Indicators (WDI), The Conference Board, and the Penn World Table 9.1, we estimate four-variable VARs consisting of productivity (output per employee), employment, the share of consumption in output, and the share of investment in output.

For illustrative purposes, we show results for three major EMDEs below, with time horizons restricted by data availability.

- Brazil - 1988-2017

- Indonesia - 1961-2017

- South Africa - 1961-2017

The first key finding is that the technology shock in the Max-Share identification always captures the largest proportion of forecast error variance in these EMDEs, particularly in the early stages of the estimation (Figure 13). This is consistent with previous simulation results showing that volatile but less persistent shocks can bias the results of the Max-Share identification. Both the Spectral and NAMS approaches are less influenced by these shocks and demonstrate a lower share of explained forecast-error variance initially (note that ’periods’ are now expressed in years rather than quarters). We note that in some cases the long-run restriction displays a lower share of FEVD than the other approaches (notably in Brazil), but in simulations has shown a tendency to underestimate the FEVD share relative to the true shock (see Figure 7).

> 17The news literature follows a similar procedure to our identification. The notable differences are that utilization-adjusted technology of Basu et al. (2006) is used in place of labor productivity in the VAR, and the shock that maximizes the FEVD of technology over the long-run is orthogonalized from contemporaneous innovations in technology. In a similar approach, Beaudry and Portier (2006) applies the long-run restriction identification and orthogonalizes the news shock from contemporaneous innovations in technology.

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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.

# **8 Conclusion**

In summary, this paper documents the biases that can be introduced into the longrun and increasingly widely used Max-Share SVAR identification methodologies by the presence of confounding shocks. We show theoretically why this is the case, and why Monte Carlo DSGE simulations have previously failed to account for this issue given the overwhelming dominance of technology shocks in such models. Three new SVAR identifications are proposed to deal with confounding shocks with different frequency domain properties.

Using a simple two-variable model, we show that our new NAMS approach is less susceptible to confounding low-frequency shocks when trying to identify highly persistent technology shocks. And we propose two types of SVAR identifications in the frequency domain which show significantly reduced estimation bias in the presence of higher-frequency confounding shocks relative to technology. We demonstrate that the US productivity data is likely to be affected by confounding shocks that affect the level of productivity temporarily and that US productivity is likely to be a unit root process driven by stochastic technology growth shocks. We propose that the Spectral identifications are the most robust to the features of the data, although the NAMS approach has an advantage in being less affected by the confounding transitory (low-frequency) shocks to productivity. Finally, we show that similar issues affect EMDE data, which is often more volatile than advanced economy data.

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