Finance and Economics Discussion Series: 2013-08 Screen Reader version ^{♣}

January 2, 2013

Keywords: Vector Error Correction Model, long-run restrictions, news shocks

Abstract:

Beaudry and Portier (2006) propose an identification scheme to study the effects of news shocks about future productivity in Vector Error Correction Models (VECM). This comment shows that their methodology does not have a unique solution, when applied to their VECMs with more than two variables.
The problem arises from the interplay of cointegration assumptions and long-run restrictions imposed by Beaudry and Portier (2006).

*JEL Classification:* G12, E32, E44

In a highly influential paper, Beaudry and Portier (2006) estimate Vector Error Correction Models (VECMs) on U.S. data and find that shocks generating a stock market boom but no contemporaneous movement in Total Factor Productivity () -- henceforth called " news" -- are closely related to shocks driving long-run variations in . Moreover, these news cause increases in consumption, investment, output and hours on impact and constitute an
important source of business cycle fluctuations. These results run counter to basic dynamic stochastic general equilibrium (DSGE) models and have sparked a new literature attempting to generate news-driven positive comovement among macroeconomic aggregates.^{1}

This comment shows that in the VECMs with more than two variables estimated by Beaudry and Portier (2006), their identification scheme fails to determine news. Yet, these
higher-dimension systems are crucial to quantify the business cycle effects of news.^{2} The identification problem arises from the interplay of two assumptions. First, the Beaudry-Portier identification scheme requires that one of the non-news shocks has no permanent impact on either or consumption. Second, the VECMs estimated by Beaudry and Portier (2006) impose that and consumption are
cointegrated. This means that and consumption have the same permanent component, which makes one of the two long-run restrictions redundant and leaves an infinity of candidate solutions
with very different implications for the business cycle. The results reported in Beaudry and Portier (2006) represent just one arbitrary choice among these solutions.^{3}

A potential way to address the identification problem is to drop the cointegration restriction between and consumption. We do so by applying Beaudry and Portier's (2006) restrictions, called the "BP restrictions" from hereon, on a vector autoregressive (VAR) system in levels that does not require any a priori assumptions about cointegration. The point estimates of the BP news shock responses in the level VAR resemble closely the results reported by Beaudry and Portier (2006) for their VECM systems. However, this identification is surrounded by a tremendous degree of uncertainty because the VAR estimates imply about a 50% chance that and consumption are cointegrated, in which case the BP restrictions fail to identify news. One can therefore not have any reasonable degree of confidence about the results obtained from the VAR in levels.

We also apply the BP restrictions to an alternative VAR system that, consistent with a large class of DSGE models, imposes absence of cointegration between and consumption. In this case, the identification problem disappears but the shock implied by the BP restrictions is largely unrelated to . In sum, dropping the cointegration restriction between and consumption fails to solve the identification problem or generates results that are difficult to interpret as news about future productivity.

The remainder of the comment proceeds as follows. Section 2 explains the identification problem arising with the BP restrictions. Section 3 applies the BP restrictions to VAR-based systems that do not impose cointegration between and consumption. Section 4 evaluates the BP restrictions in VAR systems with alternative cointegration assumptions. Section 5 concludes by briefly describing alternative identification strategies of news that do not depend on cointegration restrictions between and .

Beaudry and Portier (2006) estimate bivariate, three-variable and four-variable VECMs in , a real stock market price (), consumption (), hours () and investment (). These VECMs can be expressed in vector moving average form as

where is empty for the bivariate case; for the trivariate case; and or for the four-variable case. All variables are logged and detrended. The lag polynomial is inferred from the VECM parameter estimates; the vector contains the one-period ahead prediction errors and has variance covariance matrix .

Crucially, the VECM imposes a set of cointegration restrictions , where denotes the matrix of cointegrating vectors. As discussed by King, Plosser, Stock and Watson (1991) and Hamilton (1994), cointegration imposes restrictions on . In particular, since is stationary, and thus, is singular. This constrains the set of linearly independent restrictions that can be imposed on the VECM to identify structural shocks. The identification problem arising with the BP restrictions stems from these constraints.

Identification maps to structural shocks by , with and thus . Impulse responses to the structural shocks are then given by . Beaudry and Portier's (2006) original idea is that news about future do not have a contemporaneous effect on measured ; i.e. if the news innovation is the second element of , that the element of is zero. For the bivariate systems that Beaudry and Portier (2006) use as their baseline case, this restriction together with uniquely identifies news.

The identification problem arises in the three- and four-variate systems where one zero restriction is no longer sufficient to identify structural shocks. Beaudry and Portier's (2006) strategy consists of adding zero restrictions until identification is achieved. In the trivariate case, these additional restrictions are that one of the non-news shocks has no permanent effect on and ; so when this non-news shock is the third element of , the and elements of the long-run impact matrix are zero. In the four-variable case, the additional restrictions consist of the same two long-run restrictions plus the assumption that one of the other non-news shocks can only have a contemporaneous effect on , respectively ; so when this other non-news shock is the fourth element of , the , and elements of are zero.

In a typical VAR, the additional zero restrictions, together with the zero impact restriction on and
, would be sufficient to uniquely identify all elements of
and thus news. Here, this is unfortunately not the case
because the three- and four-variable VECMs estimated by Beaudry and Portier (2006) are subject to two, respectively three cointegration restrictions; i.e.
is a
matrix, respectively a
matrix of linearly independent rows.^{5}
Since
, the rows of
and
are linearly dependent of each other. In fact, given the number of cointegrating relationships,
and
are just of rank 1, and only one linearly independent restriction
can be imposed on
. One of the two long-run zero restrictions is therefore redundant, leaving
and the shock that is supposed to capture news
under-identified.^{6}

Another, perhaps more intuitive way to understand the identification problem is to realize that the imposed cointegration relationships imply for and to share a common trend. But then, when a particular shock, the third element of in this case, is restricted to have zero long-run effect on , it automatically also has zero long run effect on .

The identification problem implies that there exists an infinity of solutions consistent with the BP restrictions. The results reported in Beaudry and Portier (2006) represent one particular solution but there is no economic justification for why this solution should be preferred over any of the other solutions. As we show in the Web-Appendix, some of these solutions are not correlated with the shock driving long-run movements in and generate very different impulse responses. In the context of the three- and four-variable VECMs estimated by Beaudry and Portier (2006), it is therefore impossible to draw any conclusions about news based on the BP restrictions.

A seemingly natural way to address the identification problem while keeping with the BP restrictions is to drop the cointegration restriction between and . Indeed, as Beaudry and Portier (2006) note themselves, the econometric evidence in favor of two versus one cointegration relationship between , and is not clear-cut, which leaves open the door that and do not share a common trend. Beaudry and Portier (2006) entertain this possibility in the NBER working paper version of their paper where they report results for one of their baseline bivariate systems estimated as a VAR in levels; i.e. with no cointegration restrictions imposed. However, they do not report any results for level VARs with more than two variables.

One important challenge with implementing the BP restrictions in a VAR in levels is that for the type of non-stationary variables involved in the estimation, there is no finite-valued solution for the long-run impact matrix of the different shocks. Hence, the long-run zero restrictions on which
Beaudry and Portier's (2006) identification scheme relies cannot be imposed exactly.^{7} We resolve this issue by first computing the linear combination of VAR
residuals that account for most of the forecast error variance (FEV) of , respectively , at a long but *finite* horizon of 400 quarters; and then using a projection-based procedure to implement the BP restrictions.^{8}

We estimate the three- and four-variable level VAR equivalents of Beaudry and Portier's (2006) VECMs using their original data with the number of lags set to four based on traditional information criteria and Portmanteau tests.^{9} The first row of Figure 1 reports the results for the four-variable level VAR in (
); the second row reports the results for the level VAR in (
). Very similar results obtain for the three-variable case and are therefore not reported. The red solid lines and the blue dashed lines display, respectively, the impulse
responses -- generated by the point estimates -- to the shock identified by the BP restrictions and the shock driving long-run variations in . The grey intervals represent a measure of
uncertainty about the identification implied by the BP restrictions, which will be discussed further below.

Figure 1 about here |

The impulse responses derived from the point estimates of both level VARs come surprisingly close to the results reported in Beaudry and Portier (2006) for their VECM systems.^{10} In particular, the shocks identified from the BP restrictions and the long-run shock lead to almost identical impulse responses and account for a large
fraction of movements in at longer-run frequencies and , and at business cycle frequencies.

At first sight, one could thus be led to conclude that dropping the cointegration assumption by estimating VARs in levels addresses the identification problem and resurrects the results reported in Beaudry and Portier (2006). However, the reported impulse responses reflect just the point
estimates of the level VARs. The problem is that when sampling confidence sets from the estimated level VARs, about 50% of all draws imply that and share a common trend.^{11} But then, as described in the previous section, the BP
restrictions do not identify news and one is left instead with an infinity of candidate solutions.

To illustrate this uncertainty about the BP identification, we take each draw that implies a common trend between and and compute all candidate solutions that are consistent with the BP restrictions and generate a positive impact response of .^{12} The grey envelopes in Figure 1 show the resulting range of impulse responses. Clearly, the range is very wide, encompassing the zero line for all variables and
frequently extending far beyond the displayed scale. Hence, one cannot have any confidence in the impulse responses generated from the BP restrictions when evaluating the level VARs at their point estimates.

In principle, the lack of identification found in the VECMs could be addressed by estimating level systems, that do not impose the common trend assumption on and . For example, the point estimates of the level VARs generate a unique solution. But draws generated from the level VARs place sufficient odds in favor of the common trend assumption, such that this approach does not successfully address the identification problem.

Alternatively, the identification can be addressed by estimating systems which impose that and have separate trends. Fisher (2010), for example, notes that DSGE models with neutral and investment-specific technology shocks imply that is not cointegrated with
, while sharing a common trend with and .^{13} These balanced growth assumptions are straightforward to implement by estimating
a stationary VAR in
, , and , respectively .^{14} Since is no longer cointegrated with , the BP restrictions imply a unique identification across all draws.

Figure 2 about here |

We estimate this stationary VAR specification with Beaudry and Portier's (2006) data and apply the BP restrictions. As shown in Figure 2, the resulting point estimates are very different from the ones reported in Beaudry and Portier (2006). In particular, the identified shock generates a
*drop* in that lasts for 10 years or more and accounts for only a very small fraction of future movements in . This makes it difficult to interpret the identified shock as news about future productivity.

This comment shows that the results reported in Beaudry and Portier (2006) are subject to an important identification problem. The problem arises from the interplay of long-run restrictions and cointegration assumptions that Beaudry and Portier (2006) impose with respect to and . Dropping the cointegration restriction between and by estimating a VAR in levels fails to address the identification problem because there is about a 50% probability that and share a common trend. Alternatively, imposing that and are not cointegrated by estimating a stationary VAR generate dynamics for that look very different from the ones reported in Beaudry-Portier (2006) and are difficult to interpret as news about future productivity.

The results raise the important question of how to identify news in alternative ways. One example is Beaudry and Lucke (2010) who invoke short- and long-run zero restrictions for non-news shocks that do not depend on cointegration between and . As Fisher (2010) shows, however, the implications for news coming out of this identification crucially depend on the number of cointegration relationships imposed.

Another strategy, recently proposed by Barsky and Sims (2011), is to identify news as the shock orthogonal to contemporaneous movements that accounts for the maximum share of unpredictable future movements in . This strategy, which is consistent with Beaudry and Portier's (2006) original idea that is driven by a contemporaneous component and a slowly diffusing news component, has the advantage that it does not rely on additional zero restrictions about other non-news shocks. As a result, it is robust to different assumptions about cointegration and can be applied to arbitrary vector moving-average systems. Interestingly, Barsky and Sims (2011) find that the news resulting from their identification accounts for a substantial share of and macroeconomic aggregates at medium- and long-run horizons. However, their news shock does not generate the type of joint increase in real macroeconomic aggregates on impact that Beaudry and Portier (2006) report and that generated a lot of interest in the literature.

News shocks and business cycles.

Are technology improvements contractionary?

-1448.

Do mood swings drive business cycles and is it rational?

NBER Working Papers, 1765

1, National Bureau of Economic Research, Inc.

Stock prices, news and economic fluctuations.

NBER Working Papers, 10548, National Bureau of Economic Research, Inc.

Stock prices, news, and economic fluctuations.

When can changes in expectations cause business cycle fluctuations in neo-classical settings?

Shocks.

Anticipated growth and business cycles in matching models.

When do long-run identifying restrictions give reliable results?

A quarterly, utilization-adjusted series on total factor productivity.

Working Paper Series, 2012-19, Federal Reserve Bank of San Francisco.

The dynamic effects of neutral and investment specific technology shocks.

A flexible finite-horizon alternative to long-run restrictions with an application to technology shocks.

Working Papers, 2005-024, Federal Reserve Bank of St. Louis.

Princeton, NJ: Princeton University Press.

Can news about the future drive the business cycle?

Small-sample confidence intervals for impulse response functions.

Stochastic trends and economic fluctuations.

*The American Economic Review* *81*(4), 819-40.

What's news in business cycles.

*Econometrica* *80*(6), 2733-2764.

This appendix derives the vector-moving average (VMA) representation for the VECM systems and shows that the matrix of (non-structural) long-run coefficients, , in equation (2) of the main paper, is singular when derived from the VECM systems estimated by Beaudry and Portier (2006). This relationship holds not only in population but also for any set of sample estimates of the underlying VECM coefficients. Moreover, has only rank 1, implying that only one (independent) long-run restriction can be imposed on . Since is assumed to be non-singular, the same properties hold for the sum of the structural VMA coefficients .

Let be a vector of non-stationary variables, which are cointegrated such that for some matrix of cointegrating vectors . There is then a VECM representation

For the sake of brevity it is assumed that there is only a first-order lag dependence in the VECM, which can be easily generalized to higher order cases. In addition, the notation abstracts from constants and other deterministic components of the data. (Our estimated VECM systems include a constant.)

The associated state-space representation is:

(3) |

And it follows the VMA representation:

As will be shown below, the matrix of long-run coefficients is singular because of the assumed cointegrating relationships. In particular we have , since measures the long-run effect of a shock on the cointegrating vectors, which are stationary and thus their long-run responses are zero (Hamilton, 1994). In the VECM systems used by Beaudry and Portier (2006), there are cointegrating relationships, and has columns, when the VECM has variables. Thus, has only rank 1.

The same holds also in sample, for any point estimates of , and -- provided that is stable. This can be verified by computing the partitioned inverse of :

(4) |

The standard formulas for the inverse of a partitioned matrix imply in this case . Further, it follows that

(5) | ||

And Sylvester's determinant theorem yields: |
||

(6) |

Furthermore, it is straightforward to show that for any point estimates of , and . To see this, notice that

When has columns and is a matrix, it follows that has rank 1.

A..2 Long-run shocks to TFP in the VECM

This section shows how to implement the identification of long-run shocks to in the VECM systems. Throughout, a one-to-one mapping is assumed between forecast errors and structural shocks , which must obviously satisfy .

For the VECMs considered by Beaudry and Portier (2006), there is a single common trend driving the permanent component of all variables, since there are cointegrating relationships when the system has variables. For the sake of convenience, the shock driving this trend will be referred to as long-run shocks to , while it should be understood that the same shock also accounts for all long-run movements in , and potential other variables, denoted . This section describes how to construct these long-run shocks from the reduced form parameters of the VECM.

Consider the matrix of structural long-run responses , and let the first column of be the responses of forecast errors to the long-run shock. Since no other shock is issued to have a permanent effect on any of the VECM's variables, it follows that

where denotes the column vector of long-run responses of to the long-run shock.

A singular-value decomposition of yields

where and are conformably partitioned, unitary matrices, and .

Without loss of generality, can be written as the product of and another matrix . As will be seen next, the long-run restriction requires that is (block-) triangular:

(9) |

The restriction follows from (7) and (8), since it ensures that

(10) |

where denotes an arbitrary column vector.

factorizes . A factorization of that satisfies the long-run restriction (7) is the Choleski factorization. The first column of -- the column associated with the long-run shock -- is then given by the first column of

chol | (11) |

and the long-run shocks are the first element of

(12) |

where the remaining column of , and thus also the remaining elements of , reflect an arbitrary permutation of the remaining shocks, without structural interpretation. For future use, the long-run shocks will be denoted .

B. Multiple BP shock candidates

The BP scheme for identifying news shocks hinges on two long-restrictions, namely that one of the non-news shocks has zero effect on and in the long-run. But as shown above, the matrix of long-run responses in the VECM's VMA representation is singular, with a rank of 1, and one of these long-run restrictions is superfluous, and news shocks are not uniquely identified by the BP scheme. This section describes how to compute the set of candidate shocks in the VECM systems, that are all consistent with the BP restrictions.

As an illustration, we reestimate Beaudry and Portier's (2006) four-variable VECMs with their original data and apply the procedure described here to obtain all possible impulse vectors that respect the BP restrictions and generate a positive impact response of the stock market. The results are reported in Section B.2 below.

B..1 The entire set of solutions the BP scheme

To recap, the BP restrictions for the four-variable case are

- There is a measurement error shock, which affects only the fourth variable in on impact; depending on the VECM specification this variable is either or . The shock is denoted .
- The "news shock", denoted is orthogonal to on impact.
- There is a pure demand shock, denoted , which has no permanent effect on and . (As argued above, this shock has thus no permanent effect on any of the VECM variables.)

A candidate vector of structural shocks can simply be constructed by applying a series of projections using the forecast errors and long-run shocks (see Appendix A.2) as follows:

- is the standardized residual in a regression of the fourth VECM residual, onto the other three residuals.
- A "news shock" candidate can then be constructed as any linear combination of the VECM residuals, which is orthogonal to the forecast error for , , and the measurement error shocks . As will be shown below, it is then always possible to construct with the desired properties. Because of the two orthogonality restrictions, only linear combinations in and need to be considered when constructing the news shock candidate. Specifically, we use a Givens rotation to construct and compute the news shock candidate as the standardized residual in regressing onto and . Different news shock candidates are thus indexed by the angle , denoted (Only the half circle is considered, since the sign of the shock is determined by the restriction that it generates a positive stock market response on impact.)
- For a given it is straightforward to compute a demand shock candidate, , which has no permanent effect on the VECM variables. To ensure this long-run restriction, the demand shock must be orthogonal to , as constructed in Appendix A.2, since is the sole driver of the permanent component in . In addition, the demand shock has to be orthogonal to and . In sum, the demand shock candidate can be constructed as any linear combination of the VECM residuals which is orthogonal to , and . Since there are only four VECM residuals and there are three orthogonality constraints, any linear combination of the VECM residuals yields the same projection residual (up to scale and sign) -- unless this linear combination should lie in the span of the three orthogonality restrictions, which is easy to check.

For a given candidate vector of shocks the corresponding candidate matrix is equal to the covariance matrix , which satisfies the BP restrictions by construction. All these computations hold both for population and sample moments.

For the trivariate VECMs, the procedure is identical, except for the absence of . The set of BP candidate shocks is then described by any linear combination of the VECM residuals that is orthogonal to on impact. Again, up to scale and sign, candidate shocks can be computed by projecting of any linear combination of the residuals of and , denoted and , off .

B..2 Application to the BP-VECMs

The first row of Figure 1 reports the results for Beaudry and Portier's (2006) four-variable VECM in (
).^{15} The second row of
Figure 1 reports equivalent results for the four-variable VECM in

(
). Results for the trivariate VECM in (
) are very similar and are available upon request. The blue solid lines replicate impulse responses for the long-run shock reported in Figure 8 of Beaudry and Portier (2006). The grey intervals show the range of candidate solutions consistent with the BP restrictions. Finally, Example 1 (dash-dotted black lines) and Example 2 (dotted red lines) display the impulse
responses for two particular solutions. Example 1 corresponds to the solution that fits the impulse response of to the long-run shock best in a least square sense; Example 2 corresponds to
the solution that generates a near-zero response of at the 40 quarter forecast horizon.

Consistent with the BP restrictions, none of the candidate solutions affect TFP on impact. Likewise but not shown here, none of the corresponding non-news shocks in the third position of have a permanent effect on either or ; and none of the corresponding non-news shocks in the fourth position of have a contemporaneous effect on , and . This confirms numerically that there is an infinity of candidate solutions satisfying the BP restrictions.

The grey intervals and the two examples show that the candidate solutions have very different implications. As Example 1 shows, there exists a solution that appears very close to the impulse responses reported in Figure 8 of Beaudry and Portier (2006). By contrast, as Example 2 shows, another solution that is equally consistent with the BP restrictions generates almost no reaction in but a persistent drop in consumption and hours, respectively investment.

Given the very different results across rotations, it should not come as a surprise that the range of correlation coefficients between the shocks satisfying the BP restrictions and the long-run shock is wide for both VECMs, ranging from about -0.50 to 0.99. Likewise, as Table 1 shows, the forecast error variance (FEV) shares of the different variables attributable to shocks consistent with the BP restrictions extends from basically 0% to above 80% for certain forecast horizons.

1 | 4 | 8 | 16 | 40 | 120 | |

VECM with TFP | 0.00 - 0.00 | 0.78 - 1.56 | 0.94 - 9.09 | 1.47 - 9.03 | 2.04 - 28.04 | 2.53 - 84.41 |

VECM with SP | 0.00 - 95.28 | 0.12 - 88.05 | 0.10 - 83.00 | 1.90 - 86.52 | 5.45 - 89.77 | 12.50 - 81.08 |

VECM with C | 0.00 - 95.43 | 1.50 - 88.29 | 1.73 - 83.25 | 0.94 - 82.65 | 0.46 - 88.02 | 0.34 - 95.64 |

VECM with H | 0.00 - 17.01 | 1.05 - 54.30 | 1.26 - 71.22 | 1.22 - 77.12 | 2.38 - 80.04 | 6.51 - 75.81 |

VECM with TFP | 0.00 - 0.00 | 0.82 - 1.24 | 0.91 - 8.84 | 1.29 - 8.78 | 2.87 - 18.38 | 2.41 - 81.92 |

VECM with SP | 0.00 - 94.66 | 0.06 - 85.92 | 0.05 - 82.25 | 1.51 - 86.33 | 6.42 - 88.54 | 12.67 - 80.48 |

VECM with C | 0.00 - 94.52 | 1.54 - 84.44 | 1.76 - 79.27 | 0.97 - 80.10 | 0.41 - 87.31 | 0.32 - 95.69 |

VECM with I | 0.00 - 15.66 | 0.83 - 38.15 | 1.05 - 36.71 | 0.95 - 36.39 | 0.78 - 47.15 | 1.30 - 72.98 |

Each of these candidate solutions also implies different responses to the "demand shock", . As required, all of these solutions have zero effect on and , and -- by virtue of the assumed common trend in all variables -- neither on and . This is illustrated in Figure 2, which depicts the set of impulse responses the demand shock in each VECM at very long horizons. These results provide a computational consistency check, that the BP restrictions indeed hold for the entire range of shock responses shown in Figure 1.

Note: The top row depicts estimates generated by a VECM in , , and . Bottom row shows estimates from VECM in , , and . Both VECMs are estimated with 5 lags and 3 cointegrating vectors, identical to what has been used by Beaudry and Portier (2006, "BP"). In each panel, the grey shaded area depicts the set of all impulse responses to the "demand" shock consistent with the BP restrictions. By construction, this shock has no long-run effect on either or , and by virtue of the assumed cointegrating relationships, neither on .C. BP restrictions in the stationary VAR

This section describes the identification of BP shocks in the stationary VAR. The implementation is fairly similar to the VECM case described in Appendix B above. The major difference is that there is now a unique solution for the BP identification, since the stationary VAR allows for distinct trends in and .

The BP news shock is constructed by projecting a linear combination of off the measurement error shock , the demand shock and the forecast error in . As before, is given by projecting off . (The construction of the demand shock will be described further below.) Let these three innovations be stacked in a vector

and notice that is entirely spanned by . Since has four elements and has three elements, the residuals of projecting any linear combination off are perfectly correlated (provided the linear combination is not perfectly spanned by ). For example, we can project off to construct the BP shock (up to sign and scale). The sign of the news shock is then determined by the condition that and the scale is identified from .What remains to be shown is the construction of the demand shock , which in turn will depend on constructing two shocks, that drive the permanent components of and ; denoted and . These two shocks can be constructed using the conventional procedure of Blanchard and Quah (1989) for long-run identification. Notice that these two shocks have no structural interpretation in this context, they are merely sufficient statistics for implementing the long-run restrictions on the demand shock. Specifically, the long-run restrictions amount to require that is orthogonal to and .

The long-run "innovations" and , are constructed by factorizing the long-run variance of , denoted as follows:

chol | ||

In this implementation, accounts entirely for fluctuations in the permanent component of , as well as for some of the permanent component in , while explains fluctuations in the stochastic trend in , which are orthogonal to trend movements in .

Given , and , the demand shock can be constructed as the standardized residual from projecting any linear combination of onto . Using similar reasoning as before, any linear combination yields the same standardized residuals (except for the degenerate cases where the linear combination is completely spanned and the residuals are all zero).

As before, the matrix of impact coefficients is identical to the matrix of covariances between VAR residuals and structural shocks, and these relationships hold in population as well as for sample moments.

D. BP restrictions in the level VAR

Our implementation of the BP restrictions in the level VAR is very similar to the procedure for the stationary VAR outlined in Appendix C. For given shocks , and , the news shock can be estimated as the projection residual between any linear combination of the VAR's forecast errors, , and the above-mentioned three shocks. As before, the measurement error shock , can be obtained by projecting the fourth VAR residual off the other three VAR residuals.

The only special feature of our implementation for the level VAR, is the identification of the long-run shocks. Since point estimates of the level VAR typically imply explosive behavior, the sum of the estimated VMA coefficients does not converge to a finite number, and long-run shocks cannot be constructed as in Blanchard and Quah (1989) by factorizing the long-run variance (see also Appendix D).

We follow Francis et al. (2012) and identify the long-run shocks based on their explanatory power for variations in and at long but *finite* horizons. Specifically, we construct
, to explain as much as possible of the forecast-error variance of at lags, and similarly for
and .

For this method it is convenient to express the identification in terms of an orthonormal matrix ( ). and not in terms of the matrix of impact coefficients , where both are assumed to be related via the Cholesky decomposition of the VAR's forecast error variance, chol.

We seek the column of , associated with a long-run shock to . Denoting this column , it solves the following variance maximization problem

subject to |

where are the coefficients of the VAR's vector moving average representation, , selects from the vector of variables in the VAR. Shocks are constructed using where spans the null space of such that is orthonormal.

The procedure is analogous for , using instead of a vector , which selects from the vector of VAR variables.

A similar procedures is also used to identify news shocks as defined by Barsky and Sims (2011) and Beaudry et al. (2011). There are just two differences: First, both procedures uses different forecast horizons. Beaudry et al. consider forecast horizons of of 40, 80 or 120 leads; and our paper reports results for 120 leads. Barsky and Sims average over the forecast error variances at leads one to 40. Second, both approaches impose the additional requirement that the maximizing shock vector is orthogonal to a vector which selects from the set of VAR variables; in the present context, this requirement amounts to the first element of being zero.

As a necessary condition, and must not be perfectly correlated, to obtain a unique solution to the projection-based procedure described in Appendix C. When both long-run shocks are perfectly correlated, the orthogonal complement to the space spanned by is not anymore one-dimensional. (A similar issue would arise, if one of the two long-run shocks were perfectly correlated with , the measurement error shock to the fourth variable.)

When simulating confidence sets for the level VARs, we found that for about 50% of the draws, and are so highly collinear, that their variance covariance matrix is ill-conditioned. As a consequence, the variance-covariance matrix of is ill-conditioned. In these cases, we treat and as perfectly correlated, such that and share the same common trend. The news shocks are then underidentified, and an infinite number of solutions can be traced out, using a procedure analagously to what is described in Appendix B.

This appendix provides the following supplemental results: Figure 3 reports impulse-responses to the BP shocks in the level VARs. The results are identical to those shown in Figure 1 of the paper, except that Figure 3 displays the results at full scale. Table 2 reports the shares of forecast error variances explained by the BP shocks at different horizons in the level VARs, and Table 3 reports the analogous results for the stationary VARs.

Note: The results shown in this figure are identical to Figure 1 of the main paper, except for the scale of each plot. The top row depicts estimates generated by the VAR in , , and . The bottom row shows estimates from the VAR in in , , and . Each VAR has 4 lags. In each panel, the solid red line shows point estimates for impulse responses to a news shock identified by the BP restrictions, and the dashed blue line depicts estimates for the long-run shock to . While the BP shocks are uniquely identified when evaluating the level VAR at its point estimates, this is not the case in 45% (upper panels) and 58% (lower panels), respectivelym, of the draws generated by bootstraps of the level VAR, since the estimated trends in and are perfectly correlated (up to machine accuracy) for these draws. The grey shaded areas depict the set of impulse responses consistent with the BP restrictions across all these draws. This area also comprises also any confidence set of impulse responses generated from the bootstrap draws, where the BP shocks are just identified.1 | 4 | 8 | 16 | 40 | 120 | |

VAR with TFP | 0.00 | 0.29 | 4.85 | 8.63 | 2.89 | 76.25 |

VAR with TFP (Range) | (0.00 - 99.08) | (0.02 - 97.58) | (0.13 - 98.47) | (0.06 - 98.44) | (0.06 - 99.16) | (0.22 - 99.99) |

VAR with C | 70.24 | 82.35 | 85.37 | 84.95 | 88.53 | 96.46 |

VAR with (Range) | (0.00 - 99.28) | (0.06 - 98.69) | (0.02 - 98.53) | (0.05 - 98.70) | (0.16 - 99.18) | (0.51 - 100.00) |

VAR with SP | 43.39 | 43.23 | 44.26 | 51.92 | 73.33 | 91.08 |

VAR with SP (Range) | (0.00 - 99.76) | (0.01 - 99.75) | (0.06 - 99.29) | (0.03 - 99.62) | (0.07 - 99.93) | (0.42 - 100.00) |

VAR with H | 14.64 | 46.76 | 68.85 | 82.18 | 90.07 | 94.28 |

VAR with H (Range) | (0.00 - 86.73) | (0.16 - 84.87) | (0.18 - 94.73) | (0.12 - 98.96) | (0.14 - 99.91) | (0.10 - 100.00) |

VAR with TFP | 0.00 | 0.24 | 3.39 | 6.29 | 1.99 | 63.44 |

VAR with TFP (Range) | (0.00 - 99.55) | (0.01 - 99.21) | (0.01 - 96.75) | (0.01 - 96.54) | (0.11 - 98.86) | (0.43 - 99.96) |

VAR with C | 53.26 | 67.86 | 72.42 | 74.24 | 79.47 | 89.43 |

VAR with C (Range) | (0.00 - 95.56) | (0.04 - 96.82) | (0.09 - 97.51) | (0.03 - 98.78) | (0.11 - 99.52) | (0.31 - 100.00) |

VAR with SP | 59.11 | 53.94 | 54.37 | 60.25 | 78.62 | 94.74 |

VAR with SP (Range) | (0.00 - 99.90) | (0.01 - 99.38) | (0.02 - 99.80) | (0.05 - 99.97) | (0.09 - 100.00) | (0.24 - 100.00) |

VAR with I | 10.02 | 31.71 | 42.06 | 48.08 | 61.17 | 86.73 |

VAR with I (Range) | (0.00 - 74.91) | (0.01 - 80.80) | (0.00 - 90.66) | (0.01 - 95.07) | (0.05 - 98.75) | (1.18 - 100.00) |

Note: Shares of forecast error variances explained by shocks identified from the Beaudry-Portier restrictions in the level VARs at different forecast horizons. Results are generated from VARs with 4 lags. Since the restrictions leave the solution underidentified in about 50% of the bootstraps, the entire range of potential solutions is reported in parentheses.

1 | 4 | 8 | 16 | 40 | 120 | |

TFP | 0.00 | 1.15 | 11.34 | 12.21 | 5.15 | 24.64 |

TFP | (0.00 - 0.00) | (0.52 - 6.94) | (3.98 - 27.20) | (4.52 - 36.54) | (3.79 - 40.14) | (5.93 - 66.27) |

C | 81.81 | 87.63 | 83.20 | 79.69 | 75.86 | 76.25 |

C | (50.99 - 97.66) | (64.76 - 95.13) | (60.10 - 92.69) | (56.00 - 91.38) | (49.39 - 92.08) | (45.03 - 92.74) |

SP | 28.34 | 26.25 | 21.60 | 25.94 | 45.08 | 57.35 |

SP | (3.89 - 90.18) | (4.40 - 81.52) | (3.71 - 71.62) | (4.53 - 69.78) | (12.08 - 79.89) | (24.61 - 84.33) |

H | 15.76 | 51.14 | 63.22 | 64.94 | 64.51 | 60.38 |

H | (7.67 - 27.50) | (33.15 - 66.25) | (39.68 - 79.52) | (41.03 - 83.46) | (40.11 - 84.32) | (33.53 - 83.12) |

TFP | 0.00 | 0.67 | 8.31 | 10.41 | 6.63 | 1.25 |

TFP | (0.00 - 0.00) | (0.41 - 5.98) | (2.31 - 22.82) | (2.85 - 30.85) | (2.51 - 35.91) | (1.69 - 47.34) |

C | 89.55 | 85.04 | 78.64 | 74.27 | 63.75 | 43.72 |

C | (50.50 - 96.63) | (48.97 - 91.64) | (43.31 - 89.72) | (42.58 - 89.78) | (37.11 - 90.05) | (25.49 - 91.50) |

SP | 13.49 | 11.48 | 8.20 | 6.29 | 5.60 | 6.17 |

SP | (1.72 - 88.66) | (1.97 - 82.49) | (2.29 - 70.79) | (2.41 - 67.68) | (2.09 - 67.15) | (3.16 - 66.81) |

I | 0.47 | 15.07 | 18.05 | 18.03 | 21.62 | 27.97 |

I | (0.04 - 4.57) | (4.98 - 32.61) | (4.70 - 44.22) | (5.61 - 45.39) | (9.08 - 49.15) | (15.63 - 61.45) |

Note: Shares of forecast error variances at different horizons explained by the shocks identified by BP restrictions in the stationary VAR. Results are generated from VARs with 4 lags. 80 percent confidence sets reported in parentheses below the point estimates.

* The views expressed in this paper do not necessarily represent the views of the Federal Reserve System or the Federal Open Market Committee. Return to Text

1. See for example Beaudry and Portier (2007), DenHaan and Kaltenbrunner (2009), Jaimovich and Rebelo (2009), or Schmitt-Grohe and Uribe (2012). Return to Text

2. An equally important reason to work with systems in more than two variables is robustness. If the economy is complicated even in simple ways, then the type of bivariate systems that Beaudry and Portier (2006) use for their baseline analysis is likely to
generate inaccurate answers. See Faust and Leeper (1997) for an example in another context. Return to Text

3. The replication files posted on the AER website do not include code showing how news were computed by Beaudry and Portier (2006). In private
communication, we learned from the authors that their computations relied on a numerical solver that arbitrarily returned one from the infinite number of viable solutions. Return to Text

4. A Web-Appendix provides details of all derivations and computations. Return to Text

5. See Footnote 8 and the notes to Figures 9 and 10 in Beaudry and Portier (2006) for the number of cointegration restrictions imposed. The notes to the Figures also state that 4-variable VECMs with 3 (or 4) cointegration restrictions correspond to VARs in
levels. However, this seems to be a simple mistake. As Beaudry and Portier (2006) write themselves on page 1296, a VECM is equivalent to a VAR in levels only if the matrix of cointegrating vectors
is of full rank (also see Hamilton, 1994, chapter 19). Return to Text

6. Technically, the and the
equation of
on which the long-run restrictions are imposed are the same. This leaves the system short of one equation to identify
. Nothing would change about this identification problem if Beaudry and Portier (2006) had imposed cointegrating restrictions only on and but not on any of the other variables (i.e. if
was a row-vector with non-zero entries only in the positions related to and ). Return to Text

7. Formally, let the VAR in levels be defined as
. Then, the vector-moving average representation in
(1) can be recovered as
. Non-stationarity of the variables in
implies that the roots of
lie strictly inside the unit circle. In this case, the long-run impact matrix
does not converge to a finite-valued solution. Return to Text

8. Details of the procedure, which to our knowledge is new, are provided in the Web-Appendix. Our approach of first computing shocks that account for most of the FEV at long but finite horizons is reminiscent of Francis, Owyang, Roush and DiCecio's (2012)
method of imposing long-run restrictions. While approximately, the thus identified shocks account for more than 95% of movements in , respectively , at the 400 quarters horizon. Return to Text

9. The measure from Beaudry and Portier (2006) that we use is the Solow Residual adjusted with BLS's capacity utilization index. See Section III.B
of their paper. Results would be very similar if we instead used a quarterly interpolation of the measure in Basu, Fernald and Kimball (2006), as provided by Fernald (2012). Return to Text

10. See Figure 9 in the AER paper and Figure 20 in the NBER working paper. Return to Text

11. Specifically, for about 50% of the draws in each level VAR, the two shocks driving the long-term components of and -- as identified by maximizing the FEV share over 400 quarters -- are so highly collinear that their variance-covariance matrix is ill-conditioned. In these cases, the estimated trends in and cannot be reliably distinguished from each other, which is a key prerequisite for
unique identification under the BP restrictions. Further details are described in the web-appendix. Return to Text

12. More specifically, for each draw that implies cointegration between and , we apply Givens rotations to obtain all possible impulse vectors consistent with the BP restrictions. Any rotation with a negative impact response of is
eliminated so as not to include simple 180 degree rotations of candidate solutions. See the Web-Appendix for details. We could have instead eliminated rotations with a negative long-run effect on . None of the conclusions would have changed. Return to Text

13. Other possible causes for absence of cointegration between and are (permanent) changes in distortionary tax rates or labor force participation. Return to Text

14. Equivalently, the balanced growth assumptions can be implemented in Beaudry and Portier's (2006) VECMs by requiring the matrix of cointegrating vectors
to contain only 1s and 0s in the
appropriate positions. Return to Text

15. The measure from Beaudry and Portier (2006) that we use is the one adjusted with BLS' capacity utilization index. See Section III.B in their
paper. Return to Text