# Local Projection Inference in High Dimensions

Robert Adamek Present Address: Department of Economics and Business Economics,
Aarhus University, Fuglesangs All´e 4, 8210 Aarhus V, Denmark. Department of
Quantitative Economics, Maastricht University, The Netherlands  Stephan
Smeekes  Department of Quantitative Economics, Maastricht University, The
Netherlands  Ines Wilms  Department of Quantitative Economics, Maastricht
University, The Netherlands

###### Abstract

In this paper, we estimate impulse responses by local projections in high-
dimensional settings. We use the desparsified (de-biased) lasso to estimate
the high-dimensional local projections, while leaving the impulse response
parameter of interest unpenalized. We establish the uniform asymptotic
normality of the proposed estimator under general conditions. Finally, we
demonstrate small sample performance through a simulation study and consider
two canonical applications in macroeconomic research on monetary policy and
government spending.

Keywords: high-dimensional data, honest inference, impulse responses, lasso,
time series

##  1 Introduction

In this paper, we develop a simple approach for conducting valid inference on
impulse responses in high-dimensional settings. We do so by pairing the
estimation framework of local projections (LPs) with the inference framework
of the desparsified (or debiased) lasso. Since their introduction by Jordà
(2005), LPs have been widely used in macroeconomic research for studying the
dynamic propagation of shocks through impulse response analysis, see e.g.
Angrist et al. (2018), Ramey and Zubairy (2018), Ramey (2016) and Stock and
Watson (2018). As such, they have become an increasingly used alternative to
structural Vector AutoRegressions (SVAR) pioneered by Sims (1980), see for
instance Ramey (2016) or Kilian and Lütkepohl (2017) for an overview on this
work.

The distinct feature of LPs compared to SVARs is that only univariate
regressions have to be estimated, as compared to estimating the whole system
needed for the SVAR. This is particularly useful where system estimation is
complicated, such as in the presence of nonlinearities or when the dimension
of the system grows. Plagborg-Møller and Wolf (2021) established the
equivalence between SVARs and LPs in population, showing that the underlying
impulse response estimands are the same for both, and emphasizing that
identification strategies typically used for SVARs can equivalently be
implemented with LPs, and vice versa. As such, the choice between SVAR or LP
estimation of impulse responses becomes a finite-sample consideration, where
the dimensionality of the problem may play a large role in this choice.

We consider high-dimensional local projections (HDLPs) in a general time
series framework where the number of regressors can grow faster than the
sample size. Impulse response analysis quickly becomes high-dimensional in
macroeconomic research. Even when considering impulse response analysis with
few variables, the number of regressors in LPs or SVARs is often large due to
the common practices of including many lags to control for autocorrelation
(see e.g., Bernanke and Mihov, 1998, Romer and Romer, 2004 and Sims and Zha,
2006) or to robustify against (near) unit roots via lag augmentation as in
Montiel Olea and Plagborg-Møller (2021). Similarly, additional regressors are
often introduced to capture seasonal patterns in impulse responses (see e.g.,
quarter-dependent coefficients used in Blanchard and Perotti, 2002), or to
permit nonlinearities to produce state-dependent impulse responses (Koop et
al., 1996; Ramey and Zubairy, 2018).

Additionally, one might be interested in impulse response analysis with many
variables. Motivations of the latter include, amongst others, avoidance of
contaminated measurements of policy innovations or informational deficiency
with impulse response estimates that are distorted by omitted variable bias,
the ability to use several observable measures of difficult-to-measure
variables in theoretical models, or opportunities for researchers and policy
makers to investigate the impact of shocks on a larger, possibly more
disaggregated set of variables they care about (see e.g., Bernanke et al.,
2005, Forni and Gambetti, 2014, Stock and Watson, 2016 and Kilian and
Lütkepohl, 2017, Ch. 16).

Regular estimation approaches, however, become infeasible for high-dimensional
LPs or SVARs. Traditional approaches to handle the high-dimensionality include
modelling commonalities between variables, such as through the factor-
augmented VAR (Bernanke et al., 2005), dynamic factor model (Forni et al.,
2009; Stock and Watson, 2016) or – in the case of a panel structure – global
VAR (Chudik and Pesaran, 2016); as well as through Bayesian shrinkage methods
(Bańbura et al., 2010; Chan, 2020). Recently, sparse shrinkage methods such as
the lasso have gained considerable popularity in econometrics for policy
evaluation (Belloni et al., 2014) as well as (macroeconomic) time series
analysis (Kock et al., 2020). Their use in structural impulse response
analysis however has only very scarcely been explored so far.

While a multitude of methods and theoretical results now exist for the
estimation of sparse VAR models – see e.g., Basu and Michailidis (2015); Kock
and Callot (2015); Masini et al. (2022) and the references cited therein –
inference on impulse responses is complicated by two issues. First, sparse
estimation techniques such as the lasso perform model selection, which induces
issues with non-uniformity of limit results if this selection is ignored (Leeb
and Pötscher, 2005). Second, while several methods such as orthogonalization
(Belloni et al., 2014) and debiasing, or desparsifying the lasso (van de Geer
et al., 2014; Javanmard and Montanari, 2014) have been proposed to yield
uniformly valid (or ‘honest’) post-selection inference, the impulse response
parameters are complex nonlinear functions of _all_ VAR parameters. This
severely limits the applicability of existing post-selection inference methods
which are typically designed for (relatively) low-dimensional parameters of
interest that can directly be estimated. Indeed, to our knowledge, impulse
response analysis in sparse HD-SVARs is only considered in Krampe et al.
(2022), who construct a complex multi-step algorithm to overcome these
complications. Instead, by casting the problem in the LP framework, we reduce
the impulse response parameter(s) to a (directly estimable) low-dimensional
object in the presence of high-dimensional nuisance parameters, which makes
the standard post-selection tools available.

In this paper, we develop HDLP inference for impulse responses based on the
desparsified lasso of van de Geer et al. (2014) and its time series extension
in Adamek et al. (2022b). We tailor the approach of Adamek et al. (2022b)
specifically to high-dimensional LPs consisting of a small number of
parameters of interest – typically the dynamic response of certain variable to
a shock at a given horizon – and many controls. In particular, we modify their
approach by leaving the parameter of interest unpenalized during the
estimation procedure to ensure it does not suffer from penalization bias. Such
a setting is of more general relevance for treatment effect models consisting
of a small number of variables whose effects are of interest combined with a
large set of controls. We theoretically show that the combination of few
unpenalized parameters with many penalized ones does not affect the asymptotic
behaviour of the desparsified lasso; it remains asymptotically normal.
Importantly, we find considerable gains in terms of better coverage rates over
the standard desparsified lasso in finite samples.

We consider two canonical macroeconomic applications and demonstrate the
performance of the proposed desparsified-lasso based for HDLPs in recovering
structural impulse responses. Specifically, we first extend the work of
Bernanke et al. (2005) on macroeconomic responses to a shock in monetary
policy to a HDLP setting which additionally allows for differing responses
during the zero lower bound state. Second, we consider the work by Ramey and
Zubairy (2018) on state-dependent impulse responses to a shock in government
spending and extend it to a HDLP specification with more lags as well as a
more complex state-dependent HDLP with interaction between different state
variables. In addition, an implementation of our work is available in the R
package desla (Adamek et al., 2022a), which offers both the methods of Adamek
et al. (2022b) as well as the high-dimensional local projections proposed
here.

Section 2 introduces the HDLP specification with the corresponding
desparsified lasso-based estimation and inference procedure. We present our
main result on the asymptotic normality of the desparsified lasso. Section 3
contains a simulation study that examines the small sample performance of the
proposed desparsified lasso. Section 4 presents the results on the two
macroeconomic applications. Section 5 concludes.

A word on notation. For any $N$ dimensional vector $\boldsymbol{x}$,
$\left\|\boldsymbol{x}\right\|_{r}=\left(\sum\limits_{i=1}^{N}\left|x_{i}\right|^{r}\right)^{1/r}$
denotes the $l_{r}$-norm, with the familiar convention that
$\left\lVert\boldsymbol{x}\right\rVert_{0}=\sum_{i}1(\left\lvert
x_{i}\right\rvert>0)$. Depending on the context, $\sim$ denotes equivalence in
order of magnitude of sequences, or equivalence in distribution.

##  2 High-dimensional Local Projections

Consider the local projection regression

$y_{t+h}=\phi_{h}x_{t}+\boldsymbol{\eta}_{h}^{\prime}\boldsymbol{x}_{s,t}+\sum_{k=1}^{K}\boldsymbol{\delta}_{h,k}^{\prime}\boldsymbol{z}_{t-k}+u_{h,t},\quad
h=0,1,\ldots,h_{\max},$ (1)

where $\phi_{h},\boldsymbol{\eta}_{h},\boldsymbol{\delta}_{h,k}$ are the
projection parameters, the set of variables
$\boldsymbol{z}_{t}=\left(\boldsymbol{x}_{s,t}^{\prime},x_{t},y_{t},\boldsymbol{x}_{f,t}^{\prime}\right)^{\prime}$
consist of the shock variable $x_{t}$, the response $y_{t}$ and the vectors of
control variables $\boldsymbol{x}_{s,t}\in\mathds{R}^{n_{s}}$,
$\boldsymbol{x}_{f,t}\in\mathds{R}^{n_{f}}$, and $u_{h,t}$ is the projection
error.111We omit the constant in our definition, since we generally demean the
data when using the desparsified lasso. Our single parameter of interest is
$\phi_{h}$, the dynamic response at horizon $h$ of $y_{t}$ after an impulse in
$x_{t}$. The LP impulse response function of $y_{t}$ with respect to $x_{t}$
is given by $\\{\phi_{h}\\}_{h\geq 0}$ and can be obtained by estimating eq. 1
for $h=0,1,\dots,h_{\max}$. We propose a desparsified lasso estimator to
obtain point-wise confidence intervals for $\phi_{h}$ at each horizon $h$.

For structural interpretation purposes, the partition of variables in
$\boldsymbol{z}_{t}$ requires care. For simplicity and given the setup of our
applications, we focus here on (partially) recursively identified impulse
responses. Plagborg-Møller and Wolf (2021) provide detailed discussions on how
other identification schemes, such as sign restrictions, can be imposed while
estimating eq. 1. In a partially recursively identified system, the variable
$x_{t}$ can be thought of as being predetermined with respect to $y_{t}$, the
“slow” variables in $\boldsymbol{x}_{s,t}$ as being predetermined with respect
to $\\{x_{t},y_{t}\\}$ and finally the “fast” variables in
$\boldsymbol{x}_{f,t}$ can react to the shock variable $x_{t}$ within the same
observation period and should only enter the equation with a lag. To permit
causal interpretation of the impulse responses, structural assumptions about
the Data Generating Process (DGP) are needed. Example 1 considers
identification via a Structural Vector Moving Average model (taken from
Assumption 3 in Plagborg-Møller and Wolf, 2021) as an example.

######  Example 1 (SVMA).

Consider the Structural Vector Moving Average (SVMA)

$\underset{n_{z}\times
1}{\boldsymbol{z}_{t}}=\boldsymbol{\mu}+\underset{n_{z}\times
n_{\epsilon}}{\boldsymbol{A}(L)}\underset{n_{\epsilon}\times
1}{\boldsymbol{\epsilon}_{t}},\leavevmode\nobreak\ \leavevmode\nobreak\
\boldsymbol{A}(L)=\sum_{k=0}^{\infty}\underset{n_{z}\times
n_{\epsilon}}{\boldsymbol{A}_{k}}L^{k},\leavevmode\nobreak\
\leavevmode\nobreak\
\boldsymbol{\epsilon}_{t}\overset{i.i.d.}{\sim}N(\boldsymbol{0},\boldsymbol{I}),$
(2)

where $\boldsymbol{\epsilon}_{t}$ is a vector of structural shocks, $L$ is the
lag operator, $\\{\boldsymbol{A}_{k}\\}_{k}$ is absolutely summable, and
$\boldsymbol{A}(x)$ has full row rank for all complex scalars $x$ on the unit
circle. Under this assumption, the errors $u_{h,t}$ from eq. 1 are generally
autocorrelated due to the leads of $y_{t}$ on the left-hand-side. It is
therefore important to estimate the LPs with a procedure that is robust to
autocorrelation, as will be discussed in Section 2.1.

While the LP in eq. 1 has a single parameter of interest, it can easily be
extended to allow for LPs with a small number of parameters of interest, see
Example 2.

######  Example 2 (State-Dependent LP).

Consider the nonlinear state-dependent model of Ramey and Zubairy (2018):

$\begin{split}y_{t+h}&=I_{t-1}\left[\alpha_{h}+\phi_{A,h}x_{t}+\sum_{k=1}^{K}\boldsymbol{\delta}_{A,h,k}^{\prime}\boldsymbol{z}_{t-k}\right]\\\
&\quad+(1-I_{t-1})\left[\phi_{B,h}x_{t}+\sum_{k=1}^{K}\boldsymbol{\delta}_{B,h,k}^{\prime}\boldsymbol{z}_{t-k}\right]+u_{h,t}.\end{split}$
(3)

Here, $\phi_{A,h}$ and $\phi_{B,h}$ are the two responses of $y_{t}$ after an
impulse in $x_{t}$, corresponding to two different states distinguished by the
dummy variable $I_{t}$.

To generally accommodate LP settings with a small number of parameters of
interest and a large number of controls, we will use the more general notation

${y_{t}}=\underset{1\times
S}{\boldsymbol{x}_{\mathcal{S},t}^{\prime}}\underset{S\times
1}{\boldsymbol{\beta}_{\mathcal{S}}}+\underset{1\times(N-S)}{\boldsymbol{x}_{-\mathcal{S},t}^{\prime}}\underset{(N-S)\times
1}{\boldsymbol{\beta}_{-\mathcal{S}}}+u_{t},\leavevmode\nobreak\
\leavevmode\nobreak\ t=1,\dots,T,$ (4)

and in matrix form

$\underset{T\times 1}{\boldsymbol{y}}=\underset{T\times
S}{\boldsymbol{X}_{\mathcal{S}}}\boldsymbol{\beta}_{\mathcal{S}}+\underset{T\times(N-S)}{\boldsymbol{X}_{-\mathcal{S}}}\boldsymbol{\beta}_{-\mathcal{S}}+\underset{T\times
1}{\boldsymbol{u}}.$

We hereby separate the parameter vector $\underset{N\times
1}{\boldsymbol{\beta}}=[\boldsymbol{\beta}_{\mathcal{S}}^{\prime},\boldsymbol{\beta}_{-\mathcal{S}}^{\prime}]^{\prime}$
into two groups. The first group consists of the low-dimensional set of
parameters of interest $\boldsymbol{\beta}_{\mathcal{S}}$, corresponding to
the variables $\boldsymbol{x}_{\mathcal{S},t}$, which will be unpenalized
during the estimation procedure to ensure they are not affected by
penalization bias. The second group consists of the large set of parameters
$\boldsymbol{\beta}_{-\mathcal{S}}$, corresponding to control variables
$\boldsymbol{x}_{-\mathcal{S},t}$, which will be penalized during the
estimation procedure. We hereby use $\mathcal{S}$ to denote the set that
indexes the $S$ variables of interest. Without loss of generality, we order
these variables first in $\underset{T\times
N}{\boldsymbol{X}}=[\boldsymbol{X}_{\mathcal{S}},\boldsymbol{X}_{-\mathcal{S}}]$.
Finally, since each horizon of the LPs is estimated separately, we suppress
the dependence on $h$ in our notation.

###  2.1 Local Projection Estimation

We start by estimating eq. 4 with the lasso to obtain an initial estimate of
$\boldsymbol{\beta}$, while leaving the parameters of interest
$\boldsymbol{\beta}_{\mathcal{S}}$ unpenalized. The first stage lasso
estimator is defined as

$\hat{\boldsymbol{\beta}}^{(L)}=[\hat{\boldsymbol{\beta}}_{\mathcal{S}}^{(L)\prime},\hat{\boldsymbol{\beta}}_{-\mathcal{S}}^{(L)\prime}]^{\prime}=\operatorname*{arg\,min}\limits_{\boldsymbol{\beta}^{*}\in\mathds{R}^{N}}\left\lVert\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}^{*}\right\rVert_{2}^{2}/T+2\lambda\left\lVert\boldsymbol{W}\boldsymbol{\beta}^{*}\right\rVert_{1},$
(5)

where $\underset{N\times N}{\boldsymbol{W}}$ is a diagonal matrix with
$\boldsymbol{W}_{i,i}=0$ for $i\in\mathcal{S}$ and 1 otherwise. The Frish-
Waugh-Lovell-style result of Yamada (2017) ensures that the penalized and
unpenalized estimates can be respectively obtained by

$\begin{split}\hat{\boldsymbol{\beta}}^{(L)}_{-\mathcal{S}}&=\operatorname*{arg\,min}\limits_{\boldsymbol{\beta}^{*}\in\mathds{R}^{(N-S)}}\left\lVert\boldsymbol{M}_{(\boldsymbol{X}_{\mathcal{S}})}\boldsymbol{y}-\boldsymbol{M}_{(\boldsymbol{X}_{\mathcal{S}})}\boldsymbol{X}_{-\mathcal{S}}\boldsymbol{\beta}^{*}\right\rVert_{2}^{2}/T+2\lambda\left\lVert\boldsymbol{\beta}^{*}\right\rVert_{1},\\\
\hat{\boldsymbol{\beta}}_{\mathcal{S}}^{(L)}&=\hat{\boldsymbol{\Sigma}}_{\mathcal{S}}^{-1}\boldsymbol{X}_{\mathcal{S}}^{\prime}(\boldsymbol{y}-\boldsymbol{X}_{-\mathcal{S}}\hat{\boldsymbol{\beta}}_{-\mathcal{S}}^{(L)})/T,\end{split}$

where the residual maker
$\boldsymbol{M}_{(\boldsymbol{X}_{\mathcal{S}})}:=(I-\boldsymbol{X}_{\mathcal{S}}(\boldsymbol{X}_{\mathcal{S}}^{\prime}\boldsymbol{X}_{\mathcal{S}})^{-1}\boldsymbol{X}_{\mathcal{S}}^{\prime})$,
and
$\hat{\boldsymbol{\Sigma}}_{\mathcal{S}}:=\frac{\boldsymbol{X}_{\mathcal{S}}^{\prime}\boldsymbol{X}_{\mathcal{S}}}{T}$.
We can therefore use regular lasso estimation on the transformed response
$\boldsymbol{M}_{(\boldsymbol{X}_{\mathcal{S}})}\boldsymbol{y}$ and predictors
$\boldsymbol{M}_{(\boldsymbol{X}_{\mathcal{S}})}\boldsymbol{X}_{-\mathcal{S}}$
to obtain $\hat{\boldsymbol{\beta}}^{(L)}_{-\mathcal{S}}$.

We next “desparsify” this estimate to allow for uniformly valid inference. The
desparsified lasso estimator is defined as

$\hat{\boldsymbol{\beta}}_{\mathcal{S}}:=\hat{\boldsymbol{\beta}}_{\mathcal{S}}^{(L)}+\hat{\boldsymbol{\Theta}}\boldsymbol{X}^{\prime}(\boldsymbol{y}-\boldsymbol{X}\hat{\boldsymbol{\beta}}^{(L)})/T,$
(6)

where $\underset{S\times N}{\hat{\boldsymbol{\Theta}}}$ is a submatrix of the
approximate inverse covariance matrix of
$\hat{\boldsymbol{\Sigma}}:=\frac{\boldsymbol{X}^{\prime}\boldsymbol{X}}{T}$.
We obtain $\hat{\boldsymbol{\Theta}}$ from so-called nodewise regressions,
which can be seen as a linear projection of $\boldsymbol{x}_{j}$ onto
$\boldsymbol{X}_{-j}$, where $\boldsymbol{x}_{j}$ is the $j$th column of
$\boldsymbol{X}$ and $\boldsymbol{X}_{-j}$ is $\boldsymbol{X}$ without the
$j$th column

$\boldsymbol{x}_{j}=\boldsymbol{X}_{-j}\boldsymbol{\gamma}_{j}+\boldsymbol{v}_{j},\text{
where
}\boldsymbol{\gamma}_{j}=\operatorname*{arg\,min}_{\boldsymbol{\gamma}_{j}^{*}\in\mathds{R}^{N-1}}\mathbb{E}\left\lVert\boldsymbol{x}_{j}-\boldsymbol{X}_{-j}\boldsymbol{\gamma}_{j}^{*}\right\rVert_{2}^{2}/T.$
(7)

The lasso estimates of the nodewise regressions are given by

$\hat{\boldsymbol{\gamma}}_{j}=\operatorname*{arg\,min}_{\boldsymbol{\gamma}_{j}^{*}\in\mathds{R}^{N-1}}\left\lVert\boldsymbol{x}_{j}-\boldsymbol{X}_{-j}\boldsymbol{\gamma}_{j}^{*}\right\rVert_{2}^{2}/T+2\lambda_{j}\left\lVert\boldsymbol{\gamma}_{j}^{*}\right\rVert_{1},$

Finally, we construct $\hat{\boldsymbol{\Theta}}:=\underset{S\times
S}{\hat{\boldsymbol{\Upsilon}}^{-2}}\underset{S\times
N}{\hat{\boldsymbol{\Gamma}}}$, where
$\hat{\boldsymbol{\Upsilon}}^{-2}:=\text{diag}(1/\hat{\tau}^{2}_{1},\dots
1/\hat{\tau}^{2}_{S})$, with
$\hat{\tau}_{j}^{2}=\left\lVert\hat{\boldsymbol{v}}_{j}\right\rVert_{2}^{2}/T+\lambda_{j}\left\lVert\hat{\boldsymbol{\gamma}}_{j}\right\rVert_{1},\hat{\boldsymbol{v}}_{j}=\boldsymbol{x}-\boldsymbol{X}\hat{\boldsymbol{\gamma}}_{j}$
and $\hat{\boldsymbol{\Gamma}}$ stacks the $\hat{\boldsymbol{\gamma}}_{j}$’s
row-wise with ones on the left diagonal section

$\hat{\boldsymbol{\Gamma}}:=\left[\begin{array}[]{ccccc}1&\dots&-\hat{\gamma}_{1,S}&\dots&-\hat{\gamma}_{1,N}\\\
\vdots&\ddots&\vdots&&\vdots\\\
-\hat{\gamma}_{S,1}&\dots&1&\dots&-\hat{\gamma}_{S,N}\end{array}\right].$

The specific structure of the LPs consisting of the same set of regressors at
each horizon $h$ allows for computational and efficiency gains in obtaining
the desparsified lasso. The initial lasso in eq. 5 should be obtained for each
horizon, but the population linear projection coefficients
$\boldsymbol{\gamma}_{j}$ in the nodewise regression eq. 7 do not change with
the horizon. It is therefore sufficient to compute the nodewise lasso
estimator once. This can be done best at horizon zero where most time points
are available for estimating the nodewise regression. We therefore partly
avoid the loss of efficiency at further horizons where the most recent
observation at each new horizon would be lost for estimation otherwise. As an
example, for the LP in eq. 1, we estimate

$x_{t}=\boldsymbol{\gamma}^{(0)\prime}\boldsymbol{x}_{s,t}+\sum\limits_{k=1}^{K}\boldsymbol{\gamma}^{(k)\prime}\boldsymbol{z}_{t-k}+v_{1,t},$
(8)

then store $\hat{\boldsymbol{\Theta}}$ constructed from
$\hat{\boldsymbol{\gamma}}_{1}=\left[\hat{\boldsymbol{\gamma}}^{(0)\prime},\dots,\hat{\boldsymbol{\gamma}}_{K}^{(K)\prime}\right]^{\prime}$,
and the residual vector $\hat{\boldsymbol{v}}_{1}$. The former can then be re-
used at each horizon in the desparsified lasso estimator in eq. 6, while the
latter can be re-used in the estimation of the long-run covariance matrix
which enters the asymptotic distribution of the desparsified lasso, see
Section 2.2.

Finally, regarding the choice of tuning parameter $\lambda$ in the initial
regression and $\lambda_{j}$ in the nodewise regressions, we follow the
approach of Adamek et al. (2022b) who propose an iterative plug-in procedure
that simulates the quantiles of an appropriate “empirical process” which
should be dominated by $\lambda$; see their Section 5.1 for details.

###  2.2 Local Projection Inference

In this section, we discuss the asymptotic properties of the desparsified
lasso estimator $\hat{\boldsymbol{\beta}}_{\mathcal{S}}$. The assumptions
needed for our analysis are listed in detail in Appendix A, and are discussed
briefly below. More detailed discussions can be found in Adamek et al.
(2022b), on whose approach our theory is built.

We need assumptions on the DGP (Assumption A.1), the sparsity of the parameter
(Assumption A.2), regularity conditions for high-dimensional settings
(Assumption A.3) and assumptions on the set of parameters on which inference
is conducted (Assumption A.4). Assumption A.1(a) requires that
$\boldsymbol{z}_{t}=\left(\boldsymbol{x}_{s,t}^{\prime},x_{t},y_{t},\boldsymbol{x}_{f,t}^{\prime}\right)^{\prime}$
the regressors are contemporaneously uncorrelated with the error term, and
that the error terms and the variables in $\boldsymbol{z}_{t}$ have finite
unconditional moments up to a certain order. Crucially, Assumption A.1(a)
allows for serial correlation in the error terms, which is a typical feature
of LP regressions. Assumption A.1(c) assumes $\boldsymbol{z}_{t}$ to be near-
epoch-dependent (NED). An NED process can be interpreted as a process that is
well-approximated by a mixing process, but does not have to be mixing itself.
Unlike mixing conditions, which may exclude even very simple autoregressive
processes, NED conditions permit very general forms of dependence often
encountered in macroeconomic research, see Adamek et al. (2022b) for a more
detailed discussion.

Comparing our DGP assumptions to the SVMA assumptions of Example 1, one can
verify that the moment condition is satisfied trivially under Gaussianity, and
the NED assumption can be shown to hold when the VMA coefficients
$\boldsymbol{A}_{k}$ converge to 0 sufficiently quickly; see Example 17.3 of
Davidson (2002). If one would relax the Gaussianity assumption, a trade-off
would occur between the number of moments and the allowed dependence – the
more moments, the more persistent the process may be.

Second, we impose a sparsity assumption on the parameter $\boldsymbol{\beta}$
(Assumption A.2), as typically required for lasso estimators. We derive our
results under weak sparsity, thereby recognizing that the true parameters are
likely not exactly zero. In particular, we assume that
$\left\lVert\boldsymbol{\beta}\right\rVert_{r}^{r}$ is sufficiently small for
some $0\leq r<1$. While for $r=0$ our results boil down to the requirement of
exact sparsity– which assumes few non-zero entries in the parameter vector –
for larger values of $r$, we allow for many non-zero entries in
$\boldsymbol{\beta}$ provided they are relatively small. The smaller $r$, the
more restrictive this assumption therefore is. We also require weak sparsity
in the nodewise regressions, which amounts to assuming that the inverse
population covariance matrix
$\boldsymbol{\Sigma}^{-1}:=\left(\mathbb{E}\boldsymbol{X}^{\prime}\boldsymbol{X}/T\right)^{-1}$
is sparse in terms of (induced) $r$-norm. Since eq. 1 has a similar regressor
structure as a VAR model, this can be shown to hold under conditions similar
to the sparse VAR in Example 6 of Adamek et al. (2022b).

Next, we require that the minimum eigenvalue of $\boldsymbol{\Sigma}$ is
bounded away from 0 (Assumption A.3). This assumption ensures that the
population covariance matrix of the regressors remains well-behaved with
increasing $N$. Again, thanks to the similarity of eq. 1 to a VAR, lower
bounds on the minimum eigenvalue can be derived using the results on page 6 of
Masini et al. (2022). Finally, we also require that the number of unpenalized
parameters of interest to be bounded (Assumption A.4). In LPs, this is
trivially satisfied, for instance $S=1$ in eq. 1, and $S=3$ in eq. 3.222While
the impulse responses are the two parameters of interest, we may also want to
leave the dummy parameter unpenalized.

Under these assumptions, Theorem 1 establishes the asymptotic normality of our
desparsified lasso estimator, which allows for valid asymptotic inference.

######  Theorem 1.

Under Assumptions A.1, A.2, A.3, A.4 and A.5, for the model in eq. 4, we have

$\sup_{\underset{z\in\mathds{R}}{\boldsymbol{\beta}\in\boldsymbol{B}_{N}(r,s_{r})}}\left|\mathbb{P}\left(\sqrt{T}\frac{\boldsymbol{r}^{\prime}(\hat{\boldsymbol{\beta}}_{S}-\boldsymbol{\beta}_{S})}{\sqrt{\boldsymbol{r}^{\prime}\hat{\boldsymbol{\Upsilon}}^{-2}\hat{\boldsymbol{\Omega}}\hat{\boldsymbol{\Upsilon}}^{-2}\boldsymbol{r}}}\leq
z\right)-\Phi(z)\right|=o_{p}(1),$

where $\Phi(\cdot)$ is the CDF of $N(0,1)$, $\boldsymbol{r}\in\mathds{R}^{S}$
is chosen to test hypotheses of the form
$\boldsymbol{r}^{\prime}\boldsymbol{\beta}_{S}=0$, and
$\hat{\boldsymbol{\Omega}}$ is a consistent long-run covariance estimator.

As in Adamek et al. (2022b), we use the Newey-West long run covariance
estimator

$\underset{S\times
S}{\hat{\boldsymbol{\Omega}}}:=\sum\limits_{l=1-Q_{T}}^{Q_{T}-1}\left(1-l/Q_{T}\right)\hat{\boldsymbol{\Xi}}(l),$

for some $Q_{T}<T$, where
$\hat{\boldsymbol{\Xi}}(l):=\frac{1}{T-l}\left.\sum_{t=l+1}^{T}\hat{\boldsymbol{w}}_{t}\hat{\boldsymbol{w}}_{t-l}^{\prime}\right.$,
$\hat{\boldsymbol{w}}_{t}:=\left(\hat{v}_{1,t},\dots,\hat{v}_{S,t}\right)^{\prime}\hat{u}_{t}$.
To obtain our asymptotic results, we generally require that the bandwidth
$Q_{T}$ grows as $T\to\infty$ at a sufficiently slow rate, see Assumption A.5
in Appendix A for details. Regarding the choice of $Q_{T}$ in finite samples,
we use the bandwidth estimator of Andrews (1991).

For the LP in eq. 1, Theorem 1 permits asymptotic confidence intervals of the
form

$CI_{h}(\alpha):=\left[\hat{\phi}_{h}\pm
z_{\alpha/2}\sqrt{\frac{\hat{\omega}_{h}/\hat{\tau}_{h}^{4}}{T}}\right],$ (9)

where $\hat{\omega}_{h}$ and $\hat{\tau}_{h}^{-2}$ are the scalar versions of
$\hat{\boldsymbol{\Omega}}_{h}$ and $\hat{\boldsymbol{\Upsilon}}^{-2}_{h}$
respectively, and $z_{\alpha/2}:=\Phi^{-1}(1-\alpha/2)$. Note the dependence
on $h$ in this result, since the residuals
$\hat{\boldsymbol{u}}=\hat{\boldsymbol{u}}_{h}$ are different at each horizon,
and therefore $\hat{\boldsymbol{\Omega}}=\hat{\boldsymbol{\Omega}}_{h}$ and
$\hat{\boldsymbol{\Upsilon}}^{-2}=\hat{\boldsymbol{\Upsilon}}_{h}^{-2}$ as
well.

##  3 Simulations

We perform a simulation study to analyse the finite sample performance of the
desparsified lasso estimator. Consider the following DGP

$\left(\begin{array}[]{c}y_{t}\\\
\boldsymbol{x}_{s,t}\end{array}\right)=\underset{P\times
1}{\boldsymbol{x}_{t}}=\sum\limits_{k=1}^{4}\boldsymbol{A}_{k}\boldsymbol{x}_{t-k}+\boldsymbol{\epsilon}_{t},\quad\boldsymbol{\epsilon}_{t}\overset{iid}{\sim}N(\boldsymbol{0},\boldsymbol{I}),$
(10)

where the autoregressive parameter matrices $\boldsymbol{A}_{k}$ are tapered
Toeplitz matrices for which we consider two different settings. In the first
setting, we take $(\boldsymbol{A}_{k})_{i,j}=\rho_{k}^{\left\lvert
i-j\right\rvert+1}$ if $\left\lvert i-j\right\rvert<P/2$ and 0 otherwise, and
$(\rho_{1},\rho_{2},\rho_{3},\rho_{4})=(0.2,0.15,0.1,0.05)$. In the second
setting, we simply switch the signs of all entries in the second and fourth
autoregressive parameter matrices. We consider different values for the number
of variables $P=\\{20,40,100\\}$ and time series length
$T=\\{100,200,500\\}$.333The number of regressors in the model is $N=5\times
P$, due to the included lags.

Equation 10 can be seen as a simple structural VAR with structural shocks
$\boldsymbol{\epsilon}_{t}$. The true impulse responses are obtained by
inverting the lag polynomial, rewriting it as
$\boldsymbol{x}_{t}=\sum\limits_{k=0}^{\infty}\boldsymbol{B}_{k}\boldsymbol{\epsilon}_{t-k}=\boldsymbol{B}(L)\boldsymbol{\epsilon}_{t},$
where
$\boldsymbol{B}(L)=\boldsymbol{A}(L)^{-1}=\left(\boldsymbol{I}-\sum\limits_{k=1}^{4}\boldsymbol{A}_{k}L^{k}\right)^{-1}$
and $\boldsymbol{B}_{0}=\boldsymbol{I}$. We analyse the response of
$y_{t}=x_{1,t}$ to the first shock $\epsilon_{1,t}$ using local projections

$y_{t+h}=\phi_{h}y_{t}+\boldsymbol{x}_{s,t}^{\prime}\boldsymbol{\eta}_{h}+\sum_{k=1}^{K}(y_{t-k},\boldsymbol{x}_{s,t-k}^{\prime})\boldsymbol{\delta}_{h,k}+u_{h,t},\quad
h=1,\dots,h_{\max},$

and obtain estimates of the impulse responses
$\hat{\phi}_{1},\dots,\hat{\phi}_{h_{\max}}$ with $h_{\max}=10$ and
$K=4$.444At horizon 0 the response is $(\boldsymbol{B}_{0})_{1,1}=1$ by our
identification scheme, and thus does not need to be estimated. We investigate
whether the 95% point-wise confidence intervals corresponding to the impulse
response parameters at horizons 0 to 10 cover their true values at the correct
rates. Results are reported for 1000 replications. Figure 1 presents the
coverage rates for the proposed desparsified lasso (blue), which leaves the
impulse response parameters of interest unpenalized, in comparison to the
standard desparsified lasso (red). Figure C.1 in Appendix C includes the mean
interval widths.

Figure 1: Coverage rates of the standard desparsified lasso (red) and the
proposed desparsified lasso with $\phi_{h}$ unpenalized (blue). Dashed lines
indicate results for the second sign-switching DGP.

The proposed desparsified lasso estimator attains good coverage rates close to
the nominal level of 95% for all combinations of $P$ and $T$. For both DGPs,
coverage rates are usually the lowest around 70% to 90% for horizon 1, but
revert to the nominal level for larger horizons. Good coverage at further
horizons is to be expected, since the local projection coefficients will tend
to become more sparse in stationary models. Importantly, considerable gains
are obtained compared to the standard desparsified lasso. Especially at
shorter horizons – which are usually of most interest in practice – coverage
rates of the latter suffer with drops to 25% in the first DGP or even more
severe ones in the second DGP. We also see in Figure C.1 of Appendix C that
the interval widths are broadly similar or even smaller than the ones for the
standard desparsified lasso, meaning we obtain better coverage without losing
power.

The poor coverage rates of the standard desparsified lasso occur since the
parameters of interest are not selected in the initial lasso regression. A
simple alteration of the desparsified lasso that leaves this parameter
unpenalized thus brings the coverage rates much closer to the nominal level.
Note that the standard desparsified lasso has coverage exceeding our proposed
estimator at further horizons; this is because the true impulse response
becomes close to zero, where the bias towards zero of the lasso is beneficial.
This does not run counter to our conclusion, as we believe more uniform
coverage over different parameter values is desirable.

##  4 Structural Impulse Responses Estimated by HDLPs

We apply the desparsified lasso for HDLPs to two canonical macroeconomic
applications. In Section 4.1, we build on the work of Bernanke et al. (2005)
on monetary policy analysis, in Section 4.2 we build on the work by Ramey and
Zubairy (2018) on government spending. All analyses are performed in R (R Core
Team, 2022) using the package desla (Adamek et al., 2022a).

###  4.1 Impulse Responses to a Shock in Monetary Policy

We consider the work of Bernanke et al. (2005), wherein they estimate impulse
responses to a monetary policy shock identified by the federal funds rate in a
factor-augmented SVAR. In this paper, we use HDLPs to perform the impulse
response analysis and expand the dataset used in the original paper to the
more recent and larger FRED-MD database (McCracken and Ng, 2016), thereby
considering monthly data from January 1960 to December 2019 on 122
macroeconomic variables. For a full description of the data, see Appendix D.

We estimate the HDLP in eq. 1, where as response variables $y_{t}$ we take the
federal funds rate (FFR), industrial production (IP) and the consumer price
index (CPI). As shock variable of interest $x_{t}$ we consider the federal
funds rate (FFR). Furthermore, we use the identification strategy of Bernanke
et al. (2005), which consists of classifying variables as “slow” in
$\boldsymbol{x}_{s,t}$ or “fast” in $\boldsymbol{x}_{f,t}$, according to
whether these variables can respond within the same month to an unexpected
change in the FFR.555As discussed before, this strategy is equivalent to
recursive (partial) identification in an SVAR, as shown in Example 1 of
Plagborg-Møller and Wolf (2021). In fact, this application can also be seen as
a high-dimensional extension of Christiano et al. (2005). For variables in
FRED-MD that can be directly matched to their counterparts in the DRI/McGraw
Hill Basic Economics Database, we use the same slow/fast classification as
Bernanke et al. (2005). For new variables in FRED-MD, we apply the same
general classification rule; i.e., variables in the categories Prices, Output
& Income, Labour Market, and Consumption are classified as slow, whereas those
in the categories Interest & Exchange Rates, Money & Credit, Stock Market, and
Housing as fast.666One exception in the Category Prices is the series
“OILPRICEx”, which we classify as fast. We apply the transformation codes of
Bernanke et al. (2005) and McCracken and Ng (2016) (for new variables) to
remove stochastic trends. Using $K=13$ lags. the resulting HDLP consists of
$N=1654$ regressors, while the time series length is $T=707$.777The exact
number of variables in the model depends on which variable is taken as the
response, and the effective number of observations depends on the horizon.
These numbers are for the model where FFR is the response, at horizon 1.

We additionally extend the analysis to a nonlinear state-dependent HDLP,
inspired by recent work concerning the effectiveness of the FFR as a monetary
policy instrument at the zero lower bound (ZLB) (Gambacorta et al., 2014;
Inoue and Rossi, 2021; Wu and Xia, 2016). Specifically, we estimate the model
in Example 2, taking as $I_{t-1}$ the indicator function when the lag of FFR
is at or below 0.25%, thereby following the choice of Wu and Xia (2016). As
such, we obtain different impulse response estimates $\hat{\phi}_{A,h}$ and
$\hat{\phi}_{B,h}$ depending on the state of the economy. In this state-
dependent HDLP, we effectively double the number of regressors compared to the
original HDLP, with $N=3309$.

We compare the HDLP impulse responses obtained from the desparsified lasso to
the ones obtained from a 3-factor FAVAR as used in Bernanke et al. (2005).
Additional details about the FAVAR estimation are provided in Appendix E.
Figure 2 shows the impulse responses of FFR, IP, and CPI to a shock in the FFR
of a size such that the FFR has unit response at horizon 0. Since we use the
first difference of IP and CPI in the model, we cumulate the impulse response
which then corresponds to the response in the original (level)
variables.888Cumulative impulse response functions for the HDLP are obtained
by cumulating the dependent variable, i.e. taking
$\sum_{\ell=0}^{h}y_{t+\ell}$ on the LHS of eq. 1. For the FAVAR, we take the
cumulative sum of the regular impulse responses from previous horizons. To
ensure that these impulse responses are comparable, we scale the FAVAR impulse
responses by the response of the FFR at horizon 0.  The comparable figure in
Bernanke et al. (2005) is Figure 1.

Figure 2: Estimated impulse responses with 95% confidence intervals of FFR,
IP, and CPI to an identified monetary policy shock by the FFR for the HDLP and
FAVAR specifications. Black impulse responses with grey shaded confidence
intervals correspond to the linear model, blue with dotted confidence
intervals to the ZLB state, red with solid-bordered confidence intervals to
the above ZLB state.

The impulse responses from the FAVAR (bottom panel Figure 2) and linear HDLP
(black line in top panel) are generally similar. The impulse response of the
FFR peaks at horizon 1 and then steadily declines to zero, thereby being in
line with Figure 1 in Bernanke et al. (2005). For the HDLP, the response stays
at a higher level for a longer period than for the FAVAR. Next, the response
of IP is also in line with Bernanke et al. (2005), with the largest drop in
response around horizon 20, before eventually returning to zero. Notably, the
response obtained with the HDLP specification is considerably smaller than the
one for the FAVAR. Finally for CPI, in contrast to the results of Bernanke et
al. (2005), we see a pronounced price puzzle for both the FAVAR and HDLP
specifications, though the HDLP response is smaller in magnitude and turns
negative around horizon 20.

When examining the impulse responses in the state-dependent model, we see
considerably different responses between the states. Above the ZLB, responses
are similar to the linear HDLP specification, while the responses are much
larger in magnitude and more uncertain at the ZLB. As we only have 85 time
points at the ZLB in our sample and very little variation in the FFR, it is
not surprising to see much more erratic estimates during this state. The
response of the FFR is not significantly different from zero past horizon 0,
and the response of IP becomes large and positive after horizon 15, unlike the
linear model which estimates a negative response at the same horizon.
Regarding CPI, we see a reduced price puzzle above the ZLB compared to the
linear HDLP specification, and correspondingly a pronounced price puzzle at
the ZLB. We find these results to be in line with the idea that the FFR
becomes an ineffective monetary policy instrument at or near the ZLB.

###  4.2 Impulse Responses to a Shock in Government Spending

We next build on the work of Ramey and Zubairy (2018), who estimate impulse
responses to a shock in US government spending identified based on military
spending news. In this paper, we first augment the authors’ main LP
specification with more lags as a robustness check, before considering an
extended state-dependent HDLP specification with interacting states related to
unemployment and recession conditions. We use the quarterly data provided by
the authors at econweb.ucsd.edu/$\sim$vramey which covers the period 1889Q1 to
2015Q4.

We estimate the non-linear state-dependent HDLP in Example 2 for different
responses $y_{t}$, namely real per capita GDP and government spending, while
the shock variable of interest $x_{t}$ is the military spending news shock. We
include taxes as controls in $\boldsymbol{x}_{f,t}$, while no variables are
included in $\boldsymbol{x}_{s,t}$.999While Ramey and Zubairy (2018) do not
include taxes in their main analysis, they mention in footnote 11 that their
results were robust to their inclusion. We prefer to include them as
additional controls, following the VAR of Blanchard and Perotti (2002). The
state dummy variable $I_{t}$ distinguishes between low and high unemployment
states, with $I_{t}=1$ when unemployment in period $t$ is larger than 6.5%.
For full details on the construction of the shock variable, and the treatment
of the other variables, see section II.B of Ramey and Zubairy (2018). We use
$K=40$ lags, such that the resulting HDLP consists of $N=464$ regressors,
while the time series length is $T=161$.101010As in Section 4.1, these
correspond to the model estimated at horizon 1.

Figure 3 shows the impulse responses to a shock in government spending. Note
that the military spending news variable is scaled by GDP such that the
impulse responses can be seen as a reaction to a shock of size 1% of GDP. The
comparable figure in Ramey and Zubairy (2018) is Figure 5. The general shape
and magnitude of these impulse responses are similar to the LP model with only
4 lags used in Ramey and Zubairy (2018). In our analysis, the peak of the
responses in the high unemployment state occurs 2-3 quarters earlier, and
experiences a sharper drop after horizon 15. We also find that the impulse
response is not significant at horizon 0 in the linear and low-unemployment
state.

Figure 3: Estimated impulse responses with 95% confidence intervals of
government spending and GDP to a government spending shock, in a model with 40
lags. The left column shows the linear HDLP specification (black), and the
right column compares the high-unemployment state (blue with dotted intervals)
and the low-unemployment state (red with solid-bordered intervals).

Next to allowing the inclusion of more lags, the HDLP framework can easily be
extended to allow for $S$ states. The LP is then given by

$y_{t+h}=\sum\limits_{i=1}^{S}\alpha_{i}I_{i,t-1}+\sum_{i=1}^{S}I_{i,t-1}\left[\phi_{i,h}x_{t}+\sum_{k=1}^{K}\boldsymbol{\delta}_{i,h,k}^{\prime}\boldsymbol{z}_{t-k}\right]+u_{h,t}.$

In addition to the unemployment state dummy $I_{t}^{(U)}$, we consider
recession dummy $I_{t}^{(R)}$.111111As in Ramey and Zubairy (2018), we follow
the NBER classification of recession periods. Letting them interact results in
$S=4$ distinct state dummies:

$I_{1,t}=I_{t}^{(U)}I_{t}^{(R)},\leavevmode\nobreak\
I_{2,t}=I_{t}^{(U)}(1-I_{t}^{(R)}),\leavevmode\nobreak\
I_{3,t}=(1-I_{t}^{(U)})I_{t}^{(R)},\leavevmode\nobreak\
I_{4,t}=(1-I_{t}^{(U)})(1-I_{t}^{(R)}).$

Figure 4 displays the impulse response functions for the non-linear HDLP with
four states. The impulse responses out of recessions resemble those from the
state-dependent model in Figure 3, both in shape and magnitude, but we see a
different pattern in the recession state. During a recession with low
unemployment, there is a negative impact response for both government spending
and GDP, which becomes insignificant after a few periods. In the high
unemployment state, in contrast, we see a large, positive, and persistent
effect on both variables.

Figure 4: Estimated Impulse responses with 95% confidence intervals of
government spending and GDP to a government spending shock. Blue curves with
dotted intervals indicate the impulse response in the high unemployment state,
red curves with solid-bordered intervals in the low unemployment state.

##  5 Conclusion

In this paper, we propose a modified version of the desparsified lasso to
estimate local projections in high dimensions. The modification is simple,
namely leaving the (small number of) dynamic impulse response parameter(s) of
interest unpenalized in the initial lasso regression, yet provides
considerable improvements in finite sample performance in terms of better
coverage. The modified desparsified lasso estimator still comes with desirable
asymptotic normality and uniformly valid asymptotic inference. Finally, we
show how this method performs on canonical macroeconomic applications where we
use the high-dimensional local projections framework to estimate structural
impulse responses.

## Acknowledgements

The first and second author were financially supported by the Dutch Research
Council (NWO) under grant number 452-17-010. Previous versions of this paper
were presented at seminars at Aarhus University and ESSEC Business School, the
NESG 2022 conference and the 2022 Maastricht Workshop on Dimensionality
Reduction and Inference in High-Dimensional Time Series. We gratefully
acknowledge the comments by participants at these seminars and conferences. In
addition, we thank Otilia Boldea and Lenard Lieb for helpful discussions. All
remaining errors are our own.

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##  Appendix A Assumptions

In the appendix, we use the following additional notation. For a matrix
$\boldsymbol{A}$, we let
$\left\lVert\boldsymbol{A}\right\rVert_{r}=\max_{\left\lVert\boldsymbol{x}\right\rVert=1}\left\lVert\boldsymbol{A}\boldsymbol{x}\right\rVert_{r}$
for any $r\in[0,\infty]$ and
$\left\lVert\boldsymbol{A}\right\rVert_{\max}=\max\limits_{i,j}\left|a_{i,j}\right|$.
$\Lambda_{\min}(\boldsymbol{A})$ denotes the minimum eigenvalue of
$\boldsymbol{A}$. We frequently make use of arbitrary positive finite
constants $C$ (or its sub-indexed version $C_{i}$) whose values may change
from line to line throughout the paper, but they are always independent of the
time and cross-sectional dimension. We say a sequence $\eta_{T}$ is of size
$-x$ if $\eta_{T}=O\left(T^{-x-\varepsilon}\right)$ for some $\varepsilon>0$.

######  Assumption A.1.

There exist some constants $\bar{m}>m>2$, and
$d\geq\max\\{1,(\bar{m}/m-1)/(\bar{m}-2)\\}$ such that

  1. (a)

$(\boldsymbol{x}_{t}^{\prime},u_{t})^{\prime}$ is a mean zero process with
$\mathbb{E}\boldsymbol{x}_{t}u_{t}=\boldsymbol{0}$ and
$\max\limits_{1\leq j\leq N}\mathbb{E}\left\lvert
x_{j,t}\right\rvert^{2\bar{m}}\leq C$, $\mathbb{E}\left\lvert
u_{t}\right\rvert^{2\bar{m}}\leq C$,

  2. (b)

$\max\limits_{1\leq j\leq S}\mathbb{E}\left\lvert
v_{j,t}\right\rvert^{2\bar{m}}\leq C$,

  3. (c)

Let $\\{\boldsymbol{\epsilon}_{T,t}\\}$ denote a $k(T)$-dimensional triangular
array that is $\alpha$-mixing of size $-d/(1/m-1/\bar{m})$ with
$\sigma\text{-field}$
$\mathcal{F}^{\boldsymbol{\epsilon}}_{t}:=\sigma\left\\{\boldsymbol{\epsilon}_{T,t},\boldsymbol{\epsilon}_{T,t-1},\dots\right\\}$
such that $(\boldsymbol{x}_{t}^{\prime},u_{t})^{\prime}$ is
$\mathcal{F}^{\boldsymbol{\epsilon}}_{t}$-measurable. The processes $u_{t}$
and $x_{j,t}$ are $L_{2m}$-near-epoch-dependent (NED) of size $-d$ on
$\boldsymbol{\epsilon}_{T,t}$ with positive bounded constants, uniformly over
$j=1,\ldots,N$.

######  Assumption A.2.

For some $0\leq r<1$ and sparsity level $s_{r}$, define the $N$-dimensional
sparse compact parameter space

$\boldsymbol{B}_{N}(r,s_{r}):=\left\\{\boldsymbol{\beta}\in\mathds{R}^{N}:\left\lVert\boldsymbol{\beta}\right\rVert_{r}^{r}\leq
s_{r},\;\left\lVert\boldsymbol{\beta}\right\rVert_{\max}\leq C,\,\exists
C<\infty\right\\}.$

Then (a) $\boldsymbol{\beta}\in\boldsymbol{B}_{N}(r,s_{r})$ and (b)
$\boldsymbol{\gamma}_{j}\in\boldsymbol{B}_{N-1}(r,s_{r,j})$ for all
$j\in\mathcal{S}$.

######  Assumption A.3.

Let $\Lambda_{\min}(\boldsymbol{\Sigma})$ denote the smallest eigenvalue of
$\boldsymbol{\Sigma}=\mathbb{E}\boldsymbol{X}^{\prime}\boldsymbol{X}/T$, and
$\Lambda_{\min}(\boldsymbol{\Omega}_{N,T})$ the smallest eigenvalue of
${\boldsymbol{\Omega}}_{T}:=\mathbb{E}\left[\frac{1}{T}\left(\sum\limits_{t=1}^{T}\boldsymbol{w}_{t}\right)\left(\sum\limits_{t=1}^{T}\boldsymbol{w}^{\prime}_{t}\right)\right]$
Assume that $1/C\leq\Lambda_{\min}(\boldsymbol{\Sigma})\leq C$, and
$1/C\leq\Lambda_{\min}(\boldsymbol{\Omega}_{N,T})\leq C$.

######  Assumption A.4.

$S\leq C$, where $S$ is the cardinality of $\mathcal{S}$.

######  Assumption A.5.

Define $\lambda_{\min}:=\min\\{\lambda,\min\limits_{j}\lambda_{j}\\}$,
$\lambda_{\max}:=\max\\{\lambda,\max\limits_{j}\lambda_{j}\\}$, and
$s_{r,\max}:=\max\\{s_{r},\max\limits_{j}s_{r,j}\\}$. Then
$Q_{T}T^{-\frac{1}{2/d+2m/(m-2)}}\to 0$ for some $Q_{T}\to\infty$,
$\lambda\sim\lambda_{\max}\sim\lambda_{\min}$, and

$\begin{split}0<r<1:\quad&(\ln\ln
T)^{-1}s_{r,\max}^{1/r}\left[\frac{N^{\left(\frac{2}{d}+\frac{2}{m-1}\right)}}{\sqrt{T}}\right]^{\frac{1}{r\left(\frac{1}{d}+\frac{m}{m-1}\right)}}\leq\lambda\\\
&\leq\ \ln\ln T\left[Q_{T}^{2}\sqrt{T}s_{r,\max}\right]^{-1/(2-r)},\\\
r=0:\quad&(\ln\ln T)^{-1}\frac{N^{1/m}}{\sqrt{T}}\leq\lambda\leq\ln\ln
T\left[Q_{T}^{2}\sqrt{T}s_{0,\max}\right]^{-1/2}.\end{split}$

These bounds are feasible when
$Q_{T}^{r}s_{r,\max}N^{\left(2-r\right)\left(\frac{d+m-1}{dm+m-1}\right)}T^{\frac{1}{4}\left(r-\frac{d(m-1)(2-r)}{dm+m-1}\right)}\to
0$, and additionally when $Q_{T}^{2}s_{0,\max}\frac{N^{2/m}}{\sqrt{T}}\to 0$
if $r=0$.

##  Appendix B Proofs

######  Lemma B.1.

Take a vector
$\left\\{\boldsymbol{z}\in\mathds{R}^{N}:\left\lVert\boldsymbol{z}_{S^{c}}\right\rVert_{1}\leq
3\left\lVert\boldsymbol{z}_{S}\right\rVert_{1}\right\\}$, an index set $S$
with cardinality $\left\lvert S\right\rvert$ and define the subscript operator
$\boldsymbol{z}_{S}:=\\{\boldsymbol{z}^{*}\in\mathds{R}^{N}:z^{*}_{j}=z_{j}\mathds{1}_{\\{j\in
S\\}}\\}$. Under Assumption A.3, on the set
$\mathcal{CC}_{T}(S):=\left\\{\left\lVert\hat{\boldsymbol{\Sigma}}-\boldsymbol{\Sigma}\right\rVert_{\max}\leq
C/\left\lvert S\right\rvert\right\\}$,

$\|\boldsymbol{z}_{S}\|_{1}\leq\sqrt{\frac{2|S|\boldsymbol{z}^{\prime}\hat{{\boldsymbol{\Sigma}}}\boldsymbol{z}}{\Lambda_{\min}}}$

Proof: By Lemma 6.17 and Corollary 6.8 in Bühlmann and van De Geer (2011),
this result holds for the “$\boldsymbol{\Sigma}$-compatibility condition”. By
Lemma 6.23 in Bühlmann and van De Geer (2011), this compatibility condition
holds under Assumption A.3, and the result follows from the fact that
$\Lambda_{\min}$ bounds the compatibility constant from below. $\square$

######  Lemma B.2.

Let $P$ be an index set with cardinality $\left\lvert P\right\rvert$ such that
$\mathcal{S}\subseteq P$. Define the set
$\mathcal{E}_{T}(z)=\left\\{\max\limits_{j\leq N,s\leq
T}\left[\left|\sum\limits_{t=1}^{s}u_{t}x_{j,t}\right|\right]\leq z\right\\}$.
Under Assumption A.3, on
$\mathcal{E}_{T}(T\frac{\lambda}{4})\bigcap\mathcal{CC}_{T}(P)$:

$\begin{split}\frac{\|\boldsymbol{X}(\hat{\boldsymbol{\beta}}^{(L)}-{\boldsymbol{\beta}})\|_{2}^{2}}{T}+\frac{\lambda}{4}\|\hat{\boldsymbol{\beta}}^{(L)}-{\boldsymbol{\beta}}\|_{1}\leq&C_{1}\lambda^{2}|P|+C_{2}\lambda\|{\boldsymbol{\beta}}_{P^{c}}\|_{1}.\\\
\end{split}$

Proof: The proof closely follows the proof of Lemma A.6 in Adamek et al.
(2022b), based on Theorem 2.2 of van de Geer (2016), with small modifications
to handle the penalty matrix $\boldsymbol{W}$. From the Lasso optimization
problem in eq. 5, we have the Karush-Kuhn-Tucker (KKT) conditions
$\frac{\boldsymbol{X}^{\prime}({\boldsymbol{y}}-\boldsymbol{X}\hat{\boldsymbol{\beta}}^{(L)})}{T}=\boldsymbol{W}\hat{\kappa}^{*}\lambda,$
where $\hat{\kappa}^{*}=(\boldsymbol{0}_{1\times
H},\hat{\kappa}^{\prime})^{\prime}$, and $\hat{\kappa}$ is the subdifferential
of $\|\hat{\boldsymbol{\beta}}^{(L)}_{-\mathcal{S}}\|_{1}$. This then leads to
the inequality

$\begin{split}\frac{({\boldsymbol{\beta}}-\hat{\boldsymbol{\beta}}^{(L)})^{\prime}\boldsymbol{X}^{\prime}({\boldsymbol{y}}-\boldsymbol{X}\hat{\boldsymbol{\beta}}^{(L)})}{T}&=({\boldsymbol{\beta}}-\hat{\boldsymbol{\beta}}^{(L)})^{\prime}\boldsymbol{W}\hat{\kappa}^{*}\lambda={{\boldsymbol{\beta}}}^{\prime}\boldsymbol{W}\hat{\kappa}^{*}\lambda-\hat{\boldsymbol{\beta}}^{(L)\prime}\boldsymbol{W}\hat{\kappa}^{*}\lambda\\\
&\leq\lambda\|\boldsymbol{W}^{\prime}\boldsymbol{\beta}\|_{1}-\lambda\|\boldsymbol{W}^{\prime}\hat{\boldsymbol{\beta}}^{(L)}\|_{1}.\end{split}$

To deal with the matrix $\boldsymbol{W}$ in these expressions, we use the
bound
$\left\lVert\boldsymbol{W}^{\prime}(\hat{\boldsymbol{\beta}}_{S}^{(L)}-\boldsymbol{\beta}_{S})\right\rVert_{1}\leq\left\lVert\hat{\boldsymbol{\beta}}_{S}^{(L)}-\boldsymbol{\beta}_{S}\right\rVert_{1}$
and the property that
$\left\lVert\boldsymbol{W}^{\prime}\boldsymbol{z}_{S^{c}}\right\rVert_{1}=\left\lVert\boldsymbol{z}_{S^{c}}\right\rVert_{1}$
for $\boldsymbol{z}\in\mathds{R}^{N}$, since $\mathcal{S}\subseteq S$, and
those elements set to zero by $\boldsymbol{W}$ are already zero due to
subscript $S^{c}$ operator. Using these additional arguments, we can show that

$\frac{\left\lVert\boldsymbol{X}(\hat{\boldsymbol{\beta}}^{(L)}-\boldsymbol{\beta})\right\rVert_{2}^{2}}{T}\leq\frac{5\lambda}{4}\left\lVert\hat{\boldsymbol{\beta}}_{S}^{(L)}-\boldsymbol{\beta}_{S}\right\rVert_{1}-\frac{3\lambda}{4}\left\lVert\hat{\boldsymbol{\beta}}_{S^{c}}^{(L)}-\boldsymbol{\beta}_{S^{c}}\right\rVert_{1}+2\lambda\left\lVert\boldsymbol{\beta}_{S^{c}}\right\rVert_{1},$

and the remainder of the proof is analogous to the proof of Lemma A.6 in
Adamek et al. (2022b), using Lemma B.1 instead of Lemma A.5 in Adamek et al.
(2022b). $\square$

######  Lemma B.3.

Under Assumptions A.1, A.2, A.3 and A.4, for any

$\begin{split}0<r<1:\quad&\lambda\geq
C\ln(\ln(T))^{\frac{d+m-1}{r(dm+m-1)}}\left[s_{r}\left(\frac{N^{\left(\frac{2}{d}+\frac{2}{m-1}\right)}}{\sqrt{T}}\right)^{\frac{1}{\left(\frac{1}{d}+\frac{m}{m-1}\right)}}\right]^{\frac{1}{r}}\\\
r=0:\quad&s_{0}\leq
C\ln(\ln(T))^{-\frac{d+m-1}{dm+m-1}}\left[\frac{\sqrt{T}}{N^{\left(\frac{2}{d}+\frac{2}{m-1}\right)}}\right]^{\frac{1}{\left(\frac{1}{d}+\frac{m}{m-1}\right)}},\\\
&\lambda\geq C{\ln(\ln(T))}^{1/m}\frac{N^{1/m}}{\sqrt{T}}\end{split}$ (B.1)

and when $N,T$ are sufficiently large, the following holds with probability at
least $1-\upsilon^{\mathcal{CC}}_{N,T}-\upsilon^{\mathcal{E}}_{N,T}$:

$\text{(i)}\quad\frac{1}{T}\left\lVert\boldsymbol{X}(\hat{\boldsymbol{\beta}}^{(L)}-{\boldsymbol{\beta}})\right\rVert_{2}^{2}\leq
C\lambda^{2-r}s_{r},\qquad\text{(ii)}\quad\left\lVert\hat{\boldsymbol{\beta}}^{(L)}-{\boldsymbol{\beta}}\right\rVert_{1}\leq
C\lambda^{1-r}s_{r},$

where $\upsilon^{\mathcal{CC}}_{N,T}$ and $\upsilon^{\mathcal{E}}_{N,T}$ are
sequences that converge to 0 for large $T,N$ (defined formally in the proof).

Proof: Let $P_{\lambda}:=\\{j:\left\lvert\beta_{j}\right\rvert>\lambda\\}$,
and $P_{\mathcal{S},\lambda}:=\mathcal{S}\bigcup P_{\lambda}$, such that by
construction, $\mathcal{S}\subseteq P_{\mathcal{S},\lambda}$. It then follows
that by Assumption A.3 and Lemma B.2, we have on the set
$\mathcal{E}_{T}(T\frac{\lambda}{4})\bigcap\mathcal{CC}_{T}(P_{\mathcal{S},\lambda})$

$\begin{split}\left\lVert\boldsymbol{X}(\hat{\boldsymbol{\beta}}^{(L)}-\boldsymbol{\beta})\right\rVert_{2}^{2}/T+\frac{\lambda}{4}\|\hat{\boldsymbol{\beta}}^{(L)}-{\boldsymbol{\beta}}\|_{1}\leq&C_{1}\lambda^{2}|P_{\mathcal{S},\lambda}|+C_{2}\lambda\|{\boldsymbol{\beta}}_{P_{\mathcal{S},\lambda}^{c}}\|_{1}.\\\
\end{split}$

It follows directly from Assumption A.2 that

$\begin{split}|P_{\mathcal{S},\lambda}|&\leq
H+\sum\limits_{j=1}^{N}\mathds{1}_{\\{\left\lvert\beta_{j}\right\rvert>\lambda\\}}\left(\frac{\left\lvert\beta_{j}\right\rvert}{\lambda}\right)^{r}\leq
H+\lambda^{-r}\sum\limits_{j=1}^{N}\left\lvert\beta_{l}\right\rvert^{r}=H+\lambda^{-r}s_{r}.\\\
\end{split}$

Note that since $H\leq C$ by Assumption A.4 and both $\lambda^{-r}$ and
$s_{r}$ asymptotically grow as $N,T\to\infty$, for sufficiently large $N$ and
$T$, $H+\lambda^{-r}s_{r}\leq C\lambda^{-r}s_{r}$. Following the proof of
Lemma A.7 in Adamek et al. (2022b), we can also show that
$\left\lVert\boldsymbol{\beta}_{P_{\mathcal{S},\lambda}^{c}}\right\rVert_{1}\leq\lambda^{1-r}s_{r}$,
and obtain the error bound

$\displaystyle\frac{\|\boldsymbol{X}(\hat{\boldsymbol{\beta}}^{(L)}-{\boldsymbol{\beta}})\|_{2}^{2}}{T}+\lambda\|\hat{\boldsymbol{\beta}}^{(L)}-{\boldsymbol{\beta}}\|_{1}$
$\displaystyle\leq
C_{1}\lambda^{2-r}s_{r}+C_{2}\lambda^{2-r}s_{r}=C\lambda^{2-r}s_{r}.$

For the set $\mathcal{CC}_{T}(P_{\mathcal{S},\lambda})$, under Assumptions
A.1, A.2 and A.3, we can apply Lemma A.3 in Adamek et al. (2022b) with
$\eta_{T}=1/\ln(\ln(T))^{\frac{dm+m-1}{d+m-1}}$ to show that
$\mathbb{P}\left(\mathcal{CC}_{T}(P_{\lambda})\right)\geq
1-3\left[1/\ln(\ln(T))\right]^{\frac{dm+m-1}{d+m-1}}:=1-\upsilon^{\mathcal{CC}}_{N,T}\to
1$ as $N,T\to\infty$. Note that the condition $\eta_{T}\leq\frac{N^{2}}{e}$ is
satisfied for sufficiently large $N,T$, and we also use this in the proof of
Lemma A.3 in Adamek et al. (2022b) to plug in $|P_{\mathcal{S},\lambda}|\leq
H+\lambda^{-r}s_{r}\leq C\lambda^{-r}s_{r}$.

Regarding the set $\mathcal{E}_{T}(T\frac{\lambda}{4})$, by Assumption A.1 and
Lemma A.4 in Adamek et al. (2022b),
$\mathbb{P}\left(\mathcal{E}_{T}(T\lambda/4)\right)\geq
1-CN(\sqrt{T}\lambda)^{-m}:=1-\upsilon^{\mathcal{E}}_{N,T}$ By the union
bound,

$\mathbb{P}\left(\mathcal{E}_{T}(T\lambda/4)\cap\mathcal{CC}_{T}(P_{\mathcal{S},\lambda})\right)\geq
1-\upsilon^{\mathcal{E}}_{N,T}-\upsilon^{\mathcal{CC}}_{N,T},$

and therefore the error bound holds with this probability as well. With the
error bound, items (i) and (ii) follow straightforwardly. $\square$

Proof of Theorem 1: This result follows from Corollary 2 of Adamek et al.
(2022b), with some small differences summarized here. First, in Adamek et al.
(2022b), the tested hypotheses involve the full vector of parameters
$\boldsymbol{\beta}$, whereas our result only holds for
$\boldsymbol{\beta}_{\mathcal{S}}$. However, the matrix $\boldsymbol{R}_{N}$
in Adamek et al. (2022b) is restricted to have only $H$ nonzero columns such
that $\left\\{j:\sum\limits_{p=1}^{P}\left\lvert
r_{p,j}\right\rvert>0\right\\}=\mathcal{S}$. The setup here is therefore
equivalent to taking $\boldsymbol{R}_{N}=\left(\underset{S\times
S}{\boldsymbol{I}},\underset{S\times(N-S)}{\boldsymbol{0}}\right)$, and the
elements we omit here are equal to zero.

Second, parts of the proof which require bounds on
$\left\lVert\hat{\boldsymbol{\beta}}^{(L)}-\boldsymbol{\beta}\right\rVert_{1}$
or $\left\lVert\hat{\boldsymbol{u}}^{(L)}-\boldsymbol{u}\right\rVert_{2}$ need
to be addressed by the new lasso error bound in Lemma B.3, rather than by
Corollary 1 in Adamek et al. (2022b). Specifically, in Lemma B.8 of Adamek et
al. (2022b) we need the bound
$\left\lVert\hat{\boldsymbol{\beta}}^{(L)}-\boldsymbol{\beta}\right\rVert_{1}\leq
C\lambda^{1-r}s_{r}$, and in Lemma B.13 of Adamek et al. (2022b) we need
$\left\lVert\hat{\boldsymbol{u}}-\boldsymbol{u}\right\rVert_{2}=\left\lVert\boldsymbol{X}\left(\hat{\boldsymbol{\beta}}^{(L)}-\boldsymbol{\beta}\right)\right\rVert_{2}\leq
C\sqrt{T\lambda^{2-r}s_{r}}$. Note that since the new lasso error bound is the
same as for the regular lasso, no changes to these proofs are necessary.
$\square$

##  Appendix C Simulations: Extra Figures

Figure C.1: Interval widths of the standard desparsified lasso (red) and the
proposed desparsified lasso with $\phi_{h}$ unpenalized (blue). Dashed lines
indicate results for the sign-switching DGP.

##  Appendix D Data Description

Table D.1: Definition of transformation codes T | Transformation
---|---
1 | $f(x_{t})=x_{t}$
2 | $f(x_{t})=x_{t}-x_{t-1}$
3 | $f(x_{t})=(x_{t}-x_{t-1})-(x_{t-1}-x_{t-2})$
4 | $f(x_{t})=\log(x_{t})$
5 | $f(x_{t})=\log(x_{t})-\log(x_{t-1})$
6 | $f(x_{t})=(\log(x_{t})-\log(x_{t-1}))-(\log(x_{t-1})-\log(x_{t-2}))$

In the following tables, the data transformation codes follow the definitions
in Table D.1. Codes marked with an asterisk indicate a transformation that is
different from the default provided by McCracken and Ng (2016) where we use
the transformations in Bernanke et al. (2005) instead. The F/S column
indicates whether we treat the variable as fast or slow respectively for our
identification scheme.

Table D.2: Output & Income

FRED Description DRI/McGraw T F/S 1 RPI Real Personal Income GMPYQ 5 S 2
W875RX1 Real personal income ex transfer receipts GMYXPQ 5 S 3 INDPRO IP Index
IP 5 S 4 IPFPNSS IP: Final Products and Nonindustrial Supplies IPP 5 S 5
IPFINAL IP: Final Products (Market Group) IPF 5 S 6 IPCONGD IP: Consumer Goods
IPC 5 S 7 IPDCONGD IP: Durable Consumer Goods IPCD 5 S 8 IPNCONGD IP:
Nondurable Consumer Goods IPCN 5 S 9 IPBUSEQ IP: Business Equipment IPE 5 S 10
IPMAT IP: Materials IPM 5 S 11 IPDMAT IP: Durable Materials IPMD 5 S 12 IPNMAT
IP: Nondurable Materials IPMND 5 S 13 IPMANSICS IP: Manufacturing (SIC) IPMFG
5 S 14 IPB51222S IP: Residential Utilities IPUT 5 S 15 IPFUELS IP: Fuels - 5 S
17 CUMFNS Capacity Utilization: Manufacturing IPXMCA 1* S

Table D.3: Money & Credit

FRED Description DRI/McGraw T F/S 1 M1SL M1 Money Stock FM1 5* F 2 M2SL M2
Money Stock FM2 5* F 3 M2REAL Real M2 Money Stock FM2DQ 5 F 4 BOGMBASE St.
Louis Adjusted Monetary Base FMFBA 5* F 5 TOTRESNS Total Reserves of
Depository Institutions FMRRA 5* F 6 NONBORRES Reserves Of Depository
Institutions FMRNBA 5* F 7 BUSLOANS Commercial and Industrial Loans FCLNQ 5* F
8 REALLN Real Estate Loans at All Commercial Banks - 6 F 9 NONREVSL Total
Nonrevolving Credit CCINRV 5* F 10 CONSPI Nonrevolving consumer credit to
Personal Income - 2 F 12 DTCOLNVHFNM Consumer Motor Vehicle Loans Outstanding
- 6 F 13 DTCTHFNM Total Consumer Loans and Leases Outstanding - 6 F 14 INVEST
Securities in Bank Credit at All Commercial Banks - 6 F

Table D.4: Labour Market

FRED Description DRI/McGraw T F/S 1 HWI Help-Wanted Index for United States
LHEL 5* S 2 HWIURATIO Ratio of Help Wanted/No. Unemployed LHELX 4* S 3 CLF16OV
Civilian Labor Force LHEM 5 S 4 CE16OV Civilian Employment LHNAG 5 S 5 UNRATE
Civilian Unemployment Rate LHUR 1* S 6 UEMPMEAN Average Duration of
Unemployment (Weeks) LHU680 1* S 7 UEMPLT5 Civilians Unemployed - Less Than 5
Weeks LHU5 1* S 8 UEMP5TO14 Civilians Unemployed for 5-14 Weeks LHU14 1* S 9
UEMP15OV Civilians Unemployed - 15 Weeks & Over LHU15 1* S 10 UEMP15T26
Civilians Unemployed for 15-26 Weeks LHU26 1* S 11 UEMP27OV Civilians
Unemployed for 27 Weeks and Over - 5 S 12 CLAIMSx Initial Claims - 5 S 13
PAYEMS All Employees: Total nonfarm LPNAG 5 S 14 USGOOD All Employees: Goods-
Producing Industries LPGD 5 S 15 CES1021000001 All Employees: Mining and
Logging: Mining LPMI 5 S 16 USCONS All Employees: Construction LPCC 5 S 17
MANEMP All Employees: Manufacturing LPEM 5 S 18 DMANEMP All Employees: Durable
goods LPED 5 S 19 NDMANEMP All Employees: Nondurable goods LPEN 5 S 20 SRVPRD
All Employees: Service-Providing Industries LPSP 5 S 21 USTPU All Employees:
Trade, Transportation & Utilities LPTU 5 S 22 USWTRADE All Employees:
Wholesale Trade LPT 5 S 23 USTRADE All Employees: Retail Trade - 5 S 24 USFIRE
All Employees: Financial Activities LPFR 5 S 25 USGOVT All Employees:
Government LPGOV 5 S 26 CES0600000007 Avg Weekly Hours : Goods-Producing - 1 S
27 AWOTMAN Avg Weekly Overtime Hours : Manufacturing LPMOSA 1* S 28 AWHMAN Avg
Weekly Hours : Manufacturing LPHRM 1 S 30 CES0600000008 Avg Hourly Earnings :
Goods-Producing - 6 S 31 CES2000000008 Avg Hourly Earnings : Construction
LEHCC 5* S 32 CES3000000008 Avg Hourly Earnings : Manufacturing LEHM 5* S

Table D.5: Consumption & Orders

FRED Description DRI/McGraw T F/S 1 HOUST Housing Starts: Total New Privately
Owned HSFR 4 F 2 HOUSTNE Housing Starts, Northeast HSNE 4 F 3 HOUSTMW Housing
Starts, Midwest HSMW 4 F 4 HOUSTS Housing Starts, South HSSOU 4 F 5 HOUSTW
Housing Starts, West HSWST 4 F 6 PERMIT New Private Housing Permits (SAAR) - 4
F 7 PERMITNE New Private Housing Permits, Northeast (SAAR) - 4 F 8 PERMITMW
New Private Housing Permits, Midwest (SAAR) - 4 F 9 PERMITS New Private
Housing Permits, South (SAAR) - 4 F 10 PERMITW New Private Housing Permits,
West (SAAR) - 4 F

Table D.6: Orders & Inventories

FRED Description DRI/McGraw T F/S 1 DPCERA3M086SBEA Real personal consumption
expenditures GMCQ 5 S 2 CMRMTSPLx Real Manu. and Trade Industries Sales - 5 F
3 RETAILx Retail and Food Services Sales - 5 F 9 AMDMNOx New Orders for
Durable Goods - 5 F 11 AMDMUOx Unfilled Orders for Durable Goods - 5 F 12
BUSINVx Total Business Inventories - 5 F 13 ISRATIOx Total Business:
Inventories to Sales Ratio - 2 F

Table D.7: Interest rate & Exchange rates

FRED Description DRI/McGraw T F/S 1 FEDFUNDS Effective Federal Funds Rate FYFF
1* F 2 CP3Mx 3-Month AA Financial Commercial Paper Rate - 2 F 3 TB3MS 3-Month
Treasury Bill FYGM3 1* F 4 TB6MS 6-Month Treasury Bill FYGM6 1* F 5 GS1 1-Year
Treasury Rate FYGT1 1* F 6 GS5 5-Year Treasury Rate FYGT5 1* F 7 GS10 10-Year
Treasury Rate FYGT10 1* F 8 AAA Moody’s Seasoned Aaa Corporate Bond Yield
FYAAAC 1* F 9 BAA Moody’s Seasoned Baa Corporate Bond Yield FYBAAC 1* F 10
COMPAPFFx 3-Month Commercial Paper Minus FEDFUNDS - 1 F 11 TB3SMFFM 3-Month
Treasury C Minus FEDFUNDS SFYGM3 1 F 12 TB6SMFFM 6-Month Treasury C Minus
FEDFUNDS SFYGM6 1 F 13 T1YFFM 1-Year Treasury C Minus FEDFUNDS SFYGT1 1 F 14
T5YFFM 5-Year Treasury C Minus FEDFUNDS SFYGT5 1 F 15 T10YFFM 10-Year Treasury
C Minus FEDFUNDS SFYGT10 1 F 16 AAAFFM Moody’s Aaa Corporate Bond Minus
FEDFUNDS SFYAAAC 1 F 17 BAAFFM Moody’s Baa Corporate Bond Minus FEDFUNDS
SFYBAAC 1 F 19 EXSZUSx Switzerland / U.S. Foreign Exchange Rate EXRSW 5 F 20
EXJPUSx Japan / U.S. Foreign Exchange Rate EXRJAN 5 F 21 EXUSUKx U.S. / U.K.
Foreign Exchange Rate EXRUK 5 F 22 EXCAUSx Canada / U.S. Foreign Exchange Rate
EXRCAN 5 F

Table D.8: Prices

FRED Description DRI/McGraw T F/S 1 WPSFD49207 PPI: Finished Goods PWFSA 5* S
2 WPSFD49502 PPI: Finished Consumer Goods PWFCSA 5* S 3 WPSID61 PPI:
Intermediate Materials PWIMSA 5* S 4 WPSID62 PPI: Crude Materials PWCMSA 5* S
5 OILPRICEx Crude Oil, spliced WTI and Cushing - 6 F 6 PPICMM PPI: Metals and
metal products - 6 S 8 CPIAUCSL CPI : All Items PUNEW 5* S 9 CPIAPPSL CPI :
Apparel PU83 5* S 10 CPITRNSL CPI : Transportation PU84 5* S 11 CPIMEDSL CPI :
Medical Care PU85 5* S 12 CUSR0000SAC CPI : Commodities PUC 5* S 13
CUSR0000SAD CPI : Durables PUCD 5* S 14 CUSR0000SAS CPI : Services PUS 5* S 15
CPIULFSL CPI : All Items Less Food PUXF 5* S 16 CUSR0000SA0L2 CPI : All items
less shelter PUXHS 5* S 17 CUSR0000SA0L5 CPI : All items less medical care
PUXM 5* S 18 PCEPI Personal Cons. Expend.: Chain Index - 6 S 19
DDURRG3M086SBEA Personal Cons. Exp: Durable goods GMCDQ 5* S 20
DNDGRG3M086SBEA Personal Cons. Exp: Nondurable goods GMCNQ 5* S 21
DSERRG3M086SBEA Personal Cons. Exp: Services GMCSQ 5* S

Table D.9: Stock Market

FRED Description DRI/McGraw T F/S 1 S&P 500 S&P’s Common Stock Price Index:
Composite FSPCOM 5 F 2 S&P: indust S&P’s Common Stock Price Index: Industrials
FSPIN 5 F 3 S&P div yield S&P’s Composite Common Stock: Dividend Yield FSDXP
1* F 4 S&P PE ratio S&P’s Composite Common Stock: Price-Earnings Ratio FSPXE
1* F

##  Appendix E FAVAR Implementation

We closely follow the method described in Bernanke et al. (2005) to estimate
the FAVAR. The most important difference concerns the scaling of all impulse
responses by the response of the FFR at horizon 0. That means the shock to
which the variables are responding is of such a size that the FFR rises by one
on impact, as opposed to a size of one standard deviation, as is done by
Bernanke et al. (2005). We implement this change to ensures that the scale of
the responses from the FAVAR are comparable to those from our HDLP
specification.

Let $\underset{720\times 122}{\boldsymbol{X}_{\text{all}}}$ denote the matrix
containing all variables, and $\underset{720\times
67}{\boldsymbol{X}_{\text{slow}}}$ the one containing only the “slow”
variables which we assume to not react within the same period to a monetary
policy shock. We then estimate the factors $\underset{720\times
3}{\hat{\boldsymbol{C}}}$ and $\underset{720\times
3}{\hat{\boldsymbol{C}}^{*}}$ as the first 3 principal components of
$\boldsymbol{X}_{\text{all}}$ and $\boldsymbol{X}_{\text{slow}}$ respectively.
We regress $\hat{\boldsymbol{C}}$ onto $\hat{\boldsymbol{C}}^{*}$ and
$\underset{720\times 1}{\boldsymbol{R}_{s}}$, where $\boldsymbol{R}_{s}$ is
the FFR ($\boldsymbol{R}$) scaled such that it has mean 0 and variance 1, and
estimate by OLS

$\hat{\boldsymbol{C}}=\left[\hat{\boldsymbol{C}}^{*},\boldsymbol{R}_{s}\right]\underset{4\times
3}{\hat{\boldsymbol{\beta}}}+\hat{\boldsymbol{u}}.$

We then take $\underset{1\times 3}{\hat{\boldsymbol{b}}_{R}}$ as the last row
of $\hat{\boldsymbol{\beta}}$, and let $\underset{720\times
3}{\hat{\boldsymbol{F}}}=\hat{\boldsymbol{C}}-\boldsymbol{R}_{s}\boldsymbol{b}_{R}$.
Next, we use the vars package in R to estimate a 13 lag VAR with the four
variables in $[\hat{\boldsymbol{F}},\boldsymbol{R}]$ (ordering the FFR last),
and obtain the impulse responses of the factors and $\boldsymbol{R}$ using a
recursive identification scheme with Cholesky decomposition, and store them in
a matrix $\underset{(h_{\max}+1)\times 4}{\boldsymbol{IR_{\text{fac}}}}$.
These impulse responses are all scaled such that we use the unit shock
identification, similar to that of local projections. To obtain the impulse
responses of the individual variables in $\boldsymbol{X}_{\text{all}}$, we
regress $\boldsymbol{X}_{\text{all}}$ onto $\hat{\boldsymbol{F}}$, and
estimate by OLS

$\boldsymbol{X}_{\text{all}}=\hat{\boldsymbol{F}}\underset{3\times
122}{\hat{\boldsymbol{\Lambda}}}+\hat{\boldsymbol{v}}.$

The matrix of impulse responses for all variables is then obtained by

$\underset{(h_{\max}+1)\times
122}{\boldsymbol{IR}}=\boldsymbol{IR_{\text{fac}}}\left[\begin{array}[]{c}\hat{\boldsymbol{\Lambda}}\\\
\boldsymbol{0}\end{array}\right].$

Finally, we cumulate the impulse responses for the variables which were taken
in differences to obtain the response of the level variable.

For inference, we use a simple residual bootstrap. We draw with replacement
from the VAR residuals, and use them to construct the bootstrap series
$\hat{\boldsymbol{F}}^{*}$ and $\boldsymbol{R}^{*}$. We then perform the same
steps, i.e. estimate the VAR on the bootstrap data, obtain (unit-shock scaled)
impulse responses for the factors, compute the (cumulated) implied impulse
responses of all variables. We repeat this process $B=499$ times and use the
2.5% and 97.5% quantiles to construct confidence intervals.