Without loss of generality (i.e., remaining agnostic about the method of estimation and structural identification), consider an arbitrary full impulse response path matrix:

and let and denote respectively the vectorized forms of and its estimate. Then, as usual, we have the following asymptotic result:

where denotes the -vector of diagonal elements of the estimated covariance matrix , and denotes the -quantile of the standard Gaussian distribution.

Note that these bands capture the unconditional uncertainty of the impulse response coefficients at each horizon step . While these results are both useful and valid, this approach to confidence interval computation ignores the collinearity among impulse response coefficients, and ignores the temporal ordering of the impulse response path. The latter is a particularly important omission as a given value of the impulse response path affects the future trajectory of the path, but not conversely.

Lütkepohl describes analytic estimation of the covariance matrix for results obtained from traditional VAR estimation and provides the necessary equations for its computation.

Let and . Then we have the following asymptotic results:

and .

The covariance of the impulse responses, , described in Equation (46.23) is the sum of two components:

The nominal % confidence interval for a specific impulse response coefficient at horizon step satisfies:

where is the -th element of .

where is the covariance of the .

where is the -th element of .

Jordà (2005) warns that the innovation process of the local-projection regression has a moving average component of order . Although the presence of the moving average doesn’t affect estimator consistency, Jordà (2005) recommends the use of a Newey-West estimator for at each horizon step to improve the accuracy of the confidence interval estimate.

Since the residuals from a local-projection regression exhibit a moving average component of order , for sequential estimation, Jordà (2005) recommends use of a Newey-West estimator of the residual covariance matrix at each horizon step .

where we replace the estimated in Equation (46.26) with , and is a transformation matrix that depends on the lag coefficients of the LP (Jordà 2009, p. 642).

which we obtain using the LDL decomposition of

where is a lower triangular matrix with 1’s along the main diagonal; is a diagonal matrix, and is the critical value of .

Note that, even with joint estimation, we can compute the classical confidence bands by ignoring serial correlation and collinearity by deriving the band using only the diagonal elements of .

From “Impulse Response Confidence Intervals” we see that the impulse response path is a nonlinear function of the vectorized VAR coefficients , and the vectorized residual covariance matrix , which we may denote as

The method of MC integration is based on drawing from the posterior distribution, , where:

The error bands are obtained as the and percentiles of the simulated posterior distribution of the impulse response coefficients. Extending the result to structural shocks is straightforward and only requires scaling by an appropriate .

Suppose there exists a data generating process (DGP) we're interested in estimating. Without loss of generality, we can assume such a DGP is parametric with a particular model specification , an associated core dataset , some parameter space , and an auxiliary data/parameter space to complete the specification. We can succinctly describe such a DGP as a tuple

Then, , , and .

Then, the general principle governing bootstrap methods is that distribution of can be approximated by generating a large number, say , of different outcome paths ( with ) for , where the superscript indicates that the object is derived from simulation. In particular, the outcomes are obtained by simulating from (via repeated applications of ) to generate independent outcomes of , and for each such draw, estimate and compute the corresponding in the same way was derived from the original data. is then estimated as the empirical distribution of .

Note that the final bootstrap estimator of , typically denoted as , is derived as the average of the individual bootstrap estimates . In other words:

In traditional regression settings, the bootstrap procedure often used to obtain is the residual based bootstrap. Assuming a simple autoregressive DGP of the form above, the algorithm is summarized below:

The block of blocks bootstrap is a non-parametric method designed to accommodate weakly dependent stationary observations. While there are several implementations of this methodology, each addressing important practical aspects, here we only outline the basic idea. In particular, data is bootstrapped by selecting blocks of length randomly with replacement from . Again, assuming a simple autoregressive DGP shown earlier, the algorithm is described below:

It is sometimes necessary to obtain bootstrapped versions of bootstrap statistics. For instance, the object of interest may be the distribution of . This is typically needed in deriving the variance of a bootstrap statistic. In this case, for each , the distribution is approximated by generating a large number, say , of different outcome paths ( with ) for , where the superscript indicates that the object is derived from a second stage bootstrap simulation. In particular, the outcomes are obtained by simulating from each first stage bootstrap DGP in exactly the same way that was simulated from the original DGP. is estimated as the empirical distribution of

Performing a full double bootstrap algorithm is extremely computationally demanding. A single bootstrap algorithm typically requires the computation of test statistics -- a single statistic performed on the original data and an additional statistics obtained in the bootstrap loop. On the other hand, a full double bootstrap requires statistics, with the additional statistics computed in the inner loop of the double bootstrap algorithm. For instance, setting and , the double bootstrap will require the computation of no less than 499,501 test statistics.

Several attempts have been proposed to reduce this computational burden, among the most popular of which is the fast double bootstrap (FDB) algorithm of Davidson and MacKinnon (2002) and general iterated variants proposed in Davidson and Trokic (2020). The principle governing the success of these algorithms relies on setting . For instance, the FDB has proved particularly useful in approximating full double bootstrap algorithms with roughly less than the twice the computational burden of the traditional bootstrap: to be precise.

To derive bootstrap confidence intervals for traditional VAR impulse response paths, let , , denote, respectively, some general impulse response coefficient, estimated using the original data and the DGP in Equation (46.1), and the associated bootstrap estimator so that .

The choice of bootstrap algorithm to derive is not particularly important. Traditionally, a procedure like the residual bootstrap is widely used for traditional VAR impulse response analysis. While modern approaches do suggest the viability of the block of blocks bootstraping, To follow the original literature, however, EViews offers only the residual bootstrap methods described above ( “Residual Based Bootstrap”) for the case of impulse responses derived via traditional VARs.

Let denote a given significance level so that the confidence interval of interest has coverage . Then, the following bootstrap confidence intervals have been considered in the impulse response literature. See Lütkepohl (2005) for further details.

where and are respectively the and quantiles of the empirical distribution of .

An alternative to the standard percentile CI is the Hall (1992) confidence interval. The idea behind this contribution is that the distribution of is asymptotically approximated by the distribution of . Accordingly, the bootstrap CI is derived as:

where and are respectively the and quantiles of the empirical distribution of . See Hall (1986) for details.

Another possibility is to asymptotically approximate the distribution of the studentized statistic , formalized as:

where and are respectively the and quantiles of the empirical distribution of where:

Here, can be estimated from the individual bootstrap estimates . In particular:

where .

Naturally, extending the argument above, can be estimated from an additional double bootstrap estimates . In this regard, the re-sampling mechanism is obtained by augmenting the residual based bootstrap algorithm to the double bootstrap approach ( “Double Bootstrap”). In particular:

where .

Alternately, we may employ the less computationally demanding fast double bootstrap approximation. Davidson and Trokic (2020) offer an algorithm to derive confidence intervals based on the FDB. The idea relies on the inverse relationship between -values and confidence intervals. In particular, any -value can be converted into a confidence bound through inversion. Thus, the FDB CI algorithm first derives the FDB -values for a studentized test statistic that rejects in the lower and upper tails as follows:

where is the usual indicator function and is the bootstrap -value derived as follows:

and is the empirical -quantile of the second stage bootstrap statistics . Note here that the latter are obtained by setting , thereby achieving a significant computational speedup.

Next, the algorithm inverts to obtain the lower and upper confidence bounds by solving the equation:

where is the empirical -quantile of the first stage bootstrap statistics .

One can also rely on bootstrap methods to derive confidence intervals for impulse responses derived via local projections. All that is really necessary is a bootstrap version of the data, namely , for . Once generated, we can apply local projection methods outlined in “Local Projection” in order to estimate accordingly.

While bootstrap methods outlined for traditional VARs are in principle valid for local projection, few papers formally consider the subject of bootstrapping impulse responses computed via local projections. Prominent amongst those that do, Kilian and Kim (2011) argue for using the block of blocks bootstrap to simulate .

The principles governing the computation of bootstrap CIs for VAR impulse responses remain valid for VEC impulse responses as well. Since every VAR() has an equivalent representation as a VEC() process, one can simply convert bootstrap VAR estimates into their VEC equivalents, and proceed to derive the VEC impulse responses using the latter coefficients. Alternatively, one can start from a VEC representation, obtain bootstrap coefficient estimates, and then integrate the VEC model to its VAR equivalent. The converted bootstrap coefficient estimates can then be used to derive bootstrap CIs for the corresponding VAR.