## Abstract

Carriero, Clark, and Marcellino (2018, CCM2018) used a large BVAR model with a factor structure to stochastic volatility to produce an estimate of time-varying macroeconomic and financial uncertainty and assess the effects of uncertainty on the economy. The results in CCM2018 were based on an estimation algorithm that has recently been shown to be incorrect by Bognanni (2022) and fixed by Carriero et al. (2022). In this corrigendum we use the algorithm correction of Carriero et al. (2022) to correct the estimates of CCM2018. Although the correction has some impact on the original results, the changes are small and the key findings of CCM2018 are upheld.

## I. Introduction

TO make tractable the estimation of the large model of Carriero, Clark, and Marcellino (2018, hereafter denoted CCM2018), we used an equation-by-equation approach to the vector autoregression (VAR) based on a triangularization of the conditional posterior distribution of the coefficient vector developed in Carriero, Clark, and Marcellino (2019, hereafter CCM2019). However, Bognanni (2022) recently identified a conceptual problem with the triangular algorithm of CCM2019: the triangularization does not deliver the intended posterior of the VAR's coefficients. The same problem afflicts the estimation algorithm used in CCM2018.

In response, Carriero et al. (2022) have developed a corrected triangular algorithm for Bayesian VARs that does yield the intended posterior. This new algorithm permits an equation-by-equation approach to the VAR and offers the same basic computational advantages of the original triangular algorithm. In addition, the new algorithm can be used to properly estimate the uncertainty model of CCM2018.

In this corrigendum, we provide corrected versions of the published results of CCM2018. Drawing from Carriero et al. (2022), Section II briefly explains the problem with the original triangular algorithm and the correction. Section III presents corrected versions of the results of CCM2018. Although the correction has some impact on results, these impacts are small, and the key findings of CCM2018 are upheld.

## II. Original Algorithm and Correction

*iid*$N(0,\Phi \nu )$ and independent among themselves, so that $\Phi \nu =diag(\varphi 1,\u2026,\varphi n)$. These shocks are also independent of the conditional errors $\epsilon t$. The reduced-form error covariance matrix is $\Sigma t=A-1\Lambda tA-1'$.

However, as follows from the results in Bognanni (2022), drawing the VAR's coefficients in this way does not deliver the intended posterior distribution of the coefficient matrix. That is, drawing the coefficients as was done in CCM2018 does not actually sample from the density (7). As explained in more detail in Carriero et al. (2022), the actual density associated with the original algorithm is missing a term, involving the information about $\pi (j)$ contained in the most recent observations of the dependent variables of equations $j+1,\u2026,n$.

with $y\u02dct=Ayt$ a vector with generic $j$th element $y\u02dcj,t=yj,t+aj,1y1,t+\cdots +aj,j-1yj-1,t$.

With this recursive system (10), it is evident that the coefficients $\pi (j)$ of equation $j$ influence not only equation $j$ but also the following equations $j+1,\u2026,n$, which is yet another way of seeing that these equations have some extra information about $\pi (j)$ that the old algorithm missed. Importantly, though, it remains true that the previous equations $1,\u2026,j-1$ have no information about the coefficients of equation $j$. With coefficient priors $\pi (j)\u223cN(\mu \u0332\pi (j),\Omega \u0332\pi (j))$, $j=1,\u2026,n$, that are independent across equations (as is the case in all common VAR implementations), the first $j-1$ elements in the quadratic term above do not contain $\pi (j)$. It follows that the conditional distribution $p(\pi (j)|\pi (-j),A,\beta ,f1:T,m1:T,h1:T,y1:T)$ can be obtained using the subsystem composed of the last $n-j+1$ equations of (10).

where $zj+l,t=y\u02dcj+l,t-\u2211i\u2260j,i=1j+laj+l,ixt'\pi (i)$, for $l=0,\u2026,n-j$, and $ai,i=1$.

As documented in Carriero et al. (2022), this approach preserves the gains in computational complexity described in CCM2019. Although the use of additional information (data) for all but the $n$th equation makes this algorithm empirically slower than that originally used in the paper, in application the computational time is comparable. Accordingly, in this note, we use this approach to sampling the VAR's coefficients to correct and update the results of CCM2018.^{1}

## III. Corrected Results

In general, the correction of the estimation algorithm has proven to make it somewhat more difficult to disentangle measures of macroeconomic and financial uncertainty. Abstracting from algorithm considerations, some challenges are to be expected, given the comovement of forecast error variances across the variables of the model, the countercyclicality of uncertainty, the nonlinear features of the model, and the large size of the model. The algorithm correction seems to have made these challenges steeper, for reasons not easy to pinpoint. For example, with some of the loose prior settings of CCM2018, estimates with the new algorithm showed more issues with mixing and convergence of the MCMC chain.

Accordingly, to be able to reliably estimate the model with the corrected algorithm, we have made two changes relative to the settings of CCM2018. First, we have tightened a few prior settings. We lowered the hyperparameter $\theta 3$ governing shrinkage of the factor coefficients in the VAR's conditional mean from the paper's uninformative setting of 1,000 to a more modestly informative setting of 1. We also lowered the prior variance on the elements of the $A$ matrix from the paper's largely uninformative setting of 10 to a more modestly informative setting of 1. Second, we have shortened the estimation sample, so that it starts in January 1985 instead of July 1960 as in the paper. With the shorter sample, there are fewer concerns with potential sample instabilities owing to various structural shifts in the economy, monetary policy in particular. Some other work in the uncertainty literature (Baker, Bloom, & Davis, 2016, and Basu & Bundick, 2017) also focuses on samples starting in the mid-1980s. Since some other studies on uncertainty, such as Alessandri and Mumtaz (2019) and Shin and Zhong (2020), started estimation in the 1970s, we have repeated our analysis with a sample starting in 1975, finding results qualitatively very similar to those reported below.

In the remainder of this corrigendum, we provide results for the 1985–2014 sample corresponding to those in CCM2018, but using the corrected algorithm for VAR estimation described above. In general, the corrected results are qualitatively the same as those provided in CCM2018.

Table 2 (we preserve the numbering of the paper for ease of reference) provides correlations of our updated estimates of macroeconomic and financial uncertainty shocks with some well-known macro shocks. In most cases, the uncertainty shocks continue to show little correlation with “known” macroeconomic shocks. For example, the correlations of uncertainty shocks with productivity shocks are small and insignificant in these updated estimates, as they were in the paper's reported results. However, with the shorter sample and updates, there are a few instances of small, significant correlations of the uncertainty shocks with “known” macroeconomic shocks. For example, the monetary policy shocks have a small, statistically significant correlation with the shock to financial uncertainty. Some of the shift in these results seems to be due just to the shortening of the sample; in a few cases, with the sample starting in 1985, the uncertainty shocks of the paper's original estimates show similarly significant correlations with “known” macroeconomic shocks.

. | Macro . | Financial . |
---|---|---|

Known . | Uncertainty . | Uncertainty . |

Shock . | Shock . | Shock . |

Productivity: Fernald TFP | $-$0.065 | 0.137 |

(1985:Q1–2014:Q2) | (0.406) | (0.164) |

Oil supply: Hamilton (2003) | 0.144 | 0.150 |

(1985:Q1–2014:Q2) | (0.039) | (0.009) |

Oil supply: Kilian (2008) | $-$0.123 | 0.064 |

(1985:Q1–2004:Q3) | (0.236) | (0.651) |

Monetary policy: Gürkaynak et al. (2005) | $-$0.054 | 0.159 |

(1990:Q1–2004:Q4) | (0.570) | (0.029) |

Monetary policy: Coibion et al. (2016) | $-$0.143 | $-$0.332 |

(1985:Q1–2008:Q4) | (0.173) | (0.000) |

Fiscal policy: Ramey (2011) | 0.076 | 0.093 |

(1985:Q1–2008:Q4) | (0.343) | (0.036) |

Fiscal policy: Mertens and Ravn (2012) | 0.079 | $-$0.033 |

(1985:Q1–2006:Q4) | (0.101) | (0.248) |

. | Macro . | Financial . |
---|---|---|

Known . | Uncertainty . | Uncertainty . |

Shock . | Shock . | Shock . |

Productivity: Fernald TFP | $-$0.065 | 0.137 |

(1985:Q1–2014:Q2) | (0.406) | (0.164) |

Oil supply: Hamilton (2003) | 0.144 | 0.150 |

(1985:Q1–2014:Q2) | (0.039) | (0.009) |

Oil supply: Kilian (2008) | $-$0.123 | 0.064 |

(1985:Q1–2004:Q3) | (0.236) | (0.651) |

Monetary policy: Gürkaynak et al. (2005) | $-$0.054 | 0.159 |

(1990:Q1–2004:Q4) | (0.570) | (0.029) |

Monetary policy: Coibion et al. (2016) | $-$0.143 | $-$0.332 |

(1985:Q1–2008:Q4) | (0.173) | (0.000) |

Fiscal policy: Ramey (2011) | 0.076 | 0.093 |

(1985:Q1–2008:Q4) | (0.343) | (0.036) |

Fiscal policy: Mertens and Ravn (2012) | 0.079 | $-$0.033 |

(1985:Q1–2006:Q4) | (0.101) | (0.248) |

The table provides the correlations of the orthogonalized shocks to uncertainty (measured as the posterior medians of $C\Phi -1ut$, where $C\Phi $ denotes the Choleski decomposition of $\Phi $) with selected macroeconomic shocks. The monthly shocks from the model are averaged to the quarterly frequency. The Fernald TFP shocks are updates of estimates in Basu, Fernald, and Kimball (2006). Entries in parentheses provide the sample period of the correlation estimate (column 1) and the $p$-values of $t$-statistics of the coefficient obtained by regressing the uncertainty shock on the macroeconomic shock (and a constant). The variances underlying the $t$-statistics are computed with the prewhitened quadratic spectral estimator of Andrews and Monaghan (1992).

## Note

^{1}

See Carriero et al. (2022) for an implementation of computations that makes use of a data-matrix type of notation that is easy to implement and computationally efficient in programming languages such as Matlab.

## REFERENCES

*Journal of Monetary Economics*

*Econometrica*

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*Econometrica*

*American Economic Review*

*Econometrica*

*Journal of Econometrics*

*Journal of Econometrics*

*Journal of Econometrics*

*International Journal of Central Banking*

*Journal of Econometrics*

*American Economic Journal: Economic Policy*

*Quarterly Journal of Economics*

*Journal of Business and Economics Statistics*

## Author notes

The views expressed here are solely our own and do not necessarily reflect the views of the Federal Reserve Bank of Cleveland or the Federal Reserve System. We are grateful to Mark Bognanni for identifying a problem in the triangular estimation algorithm of Carriero, Clark, and Marcellino (2019), later corrected in Carriero et al. (2022).