# High-dimensional forecasting with known knowns and known unknowns††thanks:
This paper developed from Pesaran’s Deane-Stone Lecture at the National
Institute of Economic and Social Research, London. We greatly benefitted from
comments when earlier versions of this paper were presented at NIESR on 21
June 2023, the International Association of Applied Econometrics Conference,
Oslo 27 June 2023, and at Bayes Business School, City University 22 November
2023. We are also grateful for comments from Alex Chudik, George Kapetanios,
Alessio Sancetta, and Mahrad Sharifvaghefi. We particularly thank Hayun Song,
our research assistant, for his work in coding, empirical implementation, and
the tabulation of results, along with his invaluable research assistance.

M. Hashem Pesaran
University of Southern California, and Trinity College, Cambridge  Ron P.
Smith
Birkbeck, University of London

###### Abstract

Forecasts play a central role in decision making under uncertainty. After a
brief review of the general issues, this paper considers ways of using high-
dimensional data in forecasting. We consider selecting variables from a known
active set, known knowns, using Lasso and OCMT, and approximating unobserved
latent factors, known unknowns, by various means. This combines both sparse
and dense approaches. We demonstrate the various issues involved in variable
selection in a high-dimensional setting with an application to forecasting UK
inflation at different horizons over the period 2020q1-2023q1. This
application shows both the power of parsimonious models and the importance of
allowing for global variables.

JEL Classifications: C53, C55, E37, E52

Key Words: Forecasting, high-dimensional data, Lasso, OCMT, latent factors,
principal components

”All the business of war, indeed all the business of life, is endeavour to
find out what you don’t know by what you do; that’s what I called ‘guess what
was the other side of the hill’ ”

Duke of Wellington

##  1 Introduction

Forecasts play a central role in decision making under uncertainty. Good
forecasts are those that lead to good decisions, in the sense that the
expected payoff to the decision maker using the forecast is greater than it
would be otherwise.111The linkage between forecasting and decision making is
discussed in Granger and Pesaran (2000a, b), who argue in favour of a closer
link between the decision and forecast evaluation problems. Pesaran and
Skouras (2004) provide a more general survey of decision theoretic approaches
to forecast evaluation. In the case of inflation forecasts, which we consider
below, the Bank of England makes forecasts to help it set monetary policy to
keep inflation within a target range.222The letter of 26 June 2023 by Huw
Pill, Bank chief economist to the chair of the House of Commons Treasury
Committee sets out his assessment of the role played by the forecasts in the
policy process. The payoff is the variation of inflation around target.
However, it is not clear how one would quantify the contribution of the
forecast to the payoff in terms of a specific central bank loss function.

Since forecasts are designed to inform decisions, they are inherently linked
to policy making. However, there is an issue as to whether one should use the
same model for both forecasting and setting the policy instruments. Different
questions require different types of model to answer them. A policy model
might be quite large, while a forecasting model might be quite small. There is
also an issue of how transparent the model should be. It may be difficult to
interpret why a machine learning statistical model makes the predictions it
does and this can be a major disadvantage when policy requires communication
of a persuasive narrative.

In recent years forecasting has been influenced by the increasing availability
of high-dimensional data, improvements in computational power and advances in
econometrics and machine learning techniques. In some areas, such as
meteorology, this has resulted in improved forecasts, increasing the number of
hours ahead for which accurate predictions can be made. The improved forecasts
lead to better decision making as people change their behaviour in response to
the predictions and the effect of such responses on mortality from heat and
cold is examined in Shrader, Bakkensen, and Lemoine (2023). Despite advances
in data, computation and technique, the improvement in accuracy of weather
forecasts has not been matched by economic forecasts. This is a cause for
concern, since as emphasised in the classic analysis of Whittle (1983),
prediction and control are inherently linked and decisions over such elements
of economic management as monetary policy are dependent on a view of the
future.

Macroeconomic forecasting is challenging because lags in responses to policies
or shocks are long and variable and the economic system is responsive, events
prompting changes in the structure of the economy. Forecasting tends to be
relatively successful during normal times, but in times of crises and change,
in the face of large shocks or structural changes, when accurate predictions
are most needed, forecasters tend to fail. For instance, inflation was 5.4% in
December 2021. This was the last figure they had when, in February 2022, the
Bank forecast that inflation would peak at 7.1% in 2022q2, and fall to 5.5% in
2023q1, and be back within target at 2.6% in 2024q1. The 2023q1 actual was
10.2%, almost 5 percentage points higher than forecast. This burst in
inflation was a global phenomenon and other central banks made similar errors.

Economic forecasters may use purely statistical models or more structural
economic models, which include the policy variables and important economic
linkages. We will call these more structural models ‘policy models’, given the
way structural has a number of interpretations. The statistical models will
typically be conditional on information available at the time of the forecast,
which may be inaccurate: knowing where one is at the time of forecast,
nowcasting, is an important element. Policy models may also be conditional on
assumed future values. The Bank of England makes forecasts conditional on
market expectations of future interest rates, assumptions about future energy
prices and government announcements about future fiscal policy as well as
other measures. Wrong assumptions about those future values may cause problems
as it did for the Bank in August 2022, when it anticipated a rise in energy
prices but not the government response to them, so over-estimated inflation.
There is also a policy issue as to whether fiscal and monetary policy should
be determined independently by different institutions or jointly.

If both statistical and policy models are used, there is an issue as to how to
integrate them. In September 2018, the Bank of England Independent Evaluation
Office, IEO reported back to Court of the Bank on the implementation and
impact of the 2015 IEO review of forecasting performance. In response some
’non-structural’ models had been introduced as a source of challenge, and
outputs were routinely shown to the Monetary Policy Committee as a way of
cross checking the main forecast. Some but not all members found them helpful,
but there was no desire to develop more models of this sort. Internally they
had not been integrated into the forecast process as a source of challenge.”
Forecast averaging has been widely shown to improve forecast performance, but
forming averages on many variables may lack coherence and consistency.

Forecasts by central banks fulfill multiple purposes including as a means of
communication to influence expectations in the wider economy. This also makes
it difficult to choose a loss function to evaluate forecasts. For instance,
Bank of England forecasts for inflation at a 2 year horizon are always close
to the target of 2 per cent. Even were the Bank to think it unlikely that it
could get back to target within 2 years, it might feel that its credibility
might be damaged were it to admit that. An institutional issue is who ”owns”
the forecast. The Bank of England forecast is the responsibility of the 9
member Monetary Policy Committee (MPC); other central banks have different
systems. For instance the US Federal Reserve has a staff forecast not
necessarily endorsed by the decision makers.

There is an issue about the optimal amount of information to use: both with
respect to breadth, how many variables, and length, how long a run of data.
With respect to breadth, in principle, one should use information on as many
variables as possible and not just for the country being forecast, since in a
networked world foreign variables contain information. This is the information
that is used in the Global VAR (GVAR) whose use is surveyed in Chudik and
Pesaran (2016). While the use of many variables might imply a large model, in
practice quite parsimonious small models tend to be difficult to beat in
forecasting competitions.

With respect to length, in a May 2023 hearing, the Chair of the House of
Commons Treasury Select Committee asked the chief economist of the Bank of
England, Huw Pill: ”Are you saying that, despite the Bank of England having
been in existence for over 300 years, you look at only the last 30 years when
you think about what the risks are to inflation?”. Pill emphasised the
importance of the policy regime, which had been different in the past 30 years
of inflation targeting than in earlier high inflation periods. The 30 years up
to 2019 had also been different in terms of the absence of large real shocks,
like Covid-19 and the effects of the Russian invasion of Ukraine.

Whether it is statistical or policy, the model will typically be supplemented
by a judgemental input, justified by the argument that the forecaster has a
larger information set than the model. In evidence to the Treasury Select
Committee in September 2023, Sir Jon Cunliffe said: ”We start with the model.
All models are caricatures of real life. There is a suite of models; that is
the starting point. But then the MPC itself puts judgments that change the
model, and we have made some quite big judgments in the past about inflation
persistence and the like. Finally, when we have the best collective view of
the committee, which is our judgment on top of the model, the model keeps us
honest. It ensures that there is a general equilibrium and we cannot just move
things around.”

In short, macroeconomic forecasting faces important challenges. It depends on
how forecasts are announced and used in the decision making process. To deal
with a constantly changing economic environment, forecasts must continually
adapt to new data sets, statistical techniques and theory-based economic
insights, knowing that there are still key variables that might have been left
out, either due to difficulties in measurement, oversight or ignorance.
Forecasters must answer a range of difficult questions. What sample periods
and potential variables to consider? How to decide which variables to use for
forecasting, and whether to use the same sample periods for variable selection
and for forecasting? Should one use ensemble forecasting from forecasts
obtained either from different models or from the same model estimated over
different sample sizes or with different degrees of down-weighting? One must
only be humbled by the sheer extent of the uncertainty that these choices
entail. It is within this wider context that this paper tries to formalize
some elements of the problem of forecasting with high-dimensional data and
illustrates the various issues involved with an application to forecasting UK
inflation.

The rest of this paper is organized as follows. Section 2 sets out the high-
dimensional forecasting framework we will be considering. Section 3 considers
”known knowns”, selecting relevant variables from a known active set. Section
4 considers ”known unknowns” where there are known to be unobserved latent
variables. Section 5 presents the empirical application on forecasting UK
inflation. Section 6 contains some concluding comments.

##  2 The high-dimensional forecasting problem

Suppose the aim is to forecast a scalar target variable, denoted by $y_{T+h},$
at time $T$, for the future dates, $T+h$, $h=1,2,...,H$. Given the historical
observations the optimal forecast of the target variable, $y_{T+h},$ depends
on how the forecasts are used, namely the underlying decision problem. In
practice, specifying loss functions associated with decision problems is hard,
hence the tendency to fall back on mean squared error loss. Under this loss
function the optimal forecasts are given by conditional expectations,
$E\left(y_{T+h}\left|\mathcal{I}_{T}\right.\right)$, where $\mathcal{I}_{T}$
is the set of available information, and expectations are formed with respect
to the joint probability distribution of the target variable and the set of
potential predictors under consideration. But when the number of potential
predictors, say $K$, is large even this result is too general to be of much
use in practice.

The high-dimensional nature of the forecasting problem also presents a
challenge of its own when we come to multi-step ahead forecasting when
forecasts of the target variable are required for different horizons,
$h=1,2,...,H$. Many decision problems require having forecasts many periods
ahead, months, years and even decades ahead. Monetary policy is often
conducted over the business cycle, at least 2-3 years ahead of the policy
formulation. Climate change policy requires forecasts over many decades ahead.
In interpreting Pharaoh’s dreams, Joseph considered a two-period decision
problem whereby seven years of plenty are predicted to be followed by seven
years of drought. Multi-horizon forecasting is relatively straightforward when
the number of potential predictors is small and a complete system of
equations, such as a vector autoregression (VAR), can be used to generate
forecasts for different horizons from the same forecasting model in an
iterative manner. Such an iterated approach is not feasible, and might not
even be desirable, when the number of potential predictors is too large, since
future forecasts of predictors are also needed to generate forecasts of
$y_{T+h}$ for $h\geq 2$. This is why in high-dimensional set ups multi-period
ahead forecasts are typically formed using different models for different
horizons. This is known as the direct approach and avoids the need for forward
iteration by directly regressing the target variable $y_{t+h}$ on the
predictors at time $t$, thus possibly ending up with different models and/or
estimates for each $h$.333Marcellino, Stock, and Watson (2006) discuss the
pros and cons of iterated and direct approaches to forecasting when $K$ is
small, and the target variable and the predictors can be jointly modelled as
low-dimensional VARs or VARMAs. It is shown that if the underlying VAR model
is correctly specified then iterated forecasts, being coherent, are preferred
to direct forecasts. However, under misspecification direct forecasts could
perform better. Pesaran, Pick, and Timmermann (2011) reconsider the comparison
of iterated and direct forecasts to factor augmented VARs.

To be more specific, ignoring intercepts and factors which we introduce below,
suppose $y_{t}$ is the first element of the high-diemnsional vector
$\boldsymbol{w}_{t}$, assumed to follow the first order VAR model,

$\boldsymbol{w}_{t}=\boldsymbol{\Phi w}_{t-1}+\mathbf{u}_{t}.$ (1)

Higher order VARs can be written as first order VARs using the companion form.
The error vector, $\mathbf{u}_{t}$, satisfies the orthogonality condition
$E\left(\mathbf{u}_{t}\left|\mathcal{I}_{t-1}\right.\right)=\mathbf{0}$, where
$\mathcal{I}_{t-1}=(\boldsymbol{w}_{t-1},\boldsymbol{w}_{t-2},...)$. Then

$\boldsymbol{w}_{T+h}=\boldsymbol{\Phi}^{h}\boldsymbol{w}_{T}+\mathbf{u}_{h,T+h},$
(2)

where, except for $h=1,$ the overlapping observations cause the error in (2)
to have the moving average structure of order $h-1$:

$\mathbf{u}_{h,t+h}=\mathbf{u}_{t+h}+\boldsymbol{\Phi}\mathbf{u}_{t+h-1}+\boldsymbol{\Phi}^{2}\mathbf{u}_{t+h-2}+...+\boldsymbol{\Phi}^{h-1}\mathbf{u}_{t+1}.$

Under the VAR specification
$E\left(\mathbf{u}_{h,T+h}\left|\mathcal{I}_{T}\right.\right)=0$, for
$h=1,2,...$ and the optimal (in the mean squared error sense) $h$-step ahead
forecast of $\boldsymbol{w}_{T+h}$ is
$E\left(\boldsymbol{w}_{T+h}\left|\mathcal{I}_{T}\right.\right)=\boldsymbol{\Phi}^{h}\boldsymbol{w}_{T}$.
But given that in most forecasting applications the dimension of
$\boldsymbol{w}_{t}$ is large, it is not feasible to estimate
$\boldsymbol{\Phi}$ directly without imposing strong sparsity restrictions.
Instead we take the target variable, $y_{T+h}$, to be the first element of
$\boldsymbol{w}_{T+h}$ and consider the direct regression

$y_{t+h}=\boldsymbol{\phi}_{h}^{\prime}\boldsymbol{w}_{t}+u_{h,t+h\text{ }},$

where $\boldsymbol{\phi}_{h}^{\prime}$ is the first row of
$\boldsymbol{\Phi}^{h},$ and $u_{h,t+h}$ is the first element of
$\mathbf{u}_{h,t+h}$. We still face a high-dimensional problem since there are
a large number of potential covariates in $\boldsymbol{w}_{t}$. We consider
the implementation of the direct approach under two scenarios concerning the
potential predictors. First, when it is known that the target variable
$y_{t+h}$ is a sparse linear function of a large set of observed variables
$\boldsymbol{x}_{t}$ (a subset of $\boldsymbol{w}_{t}$) in the sense that
$y_{t+h}$ depends on a small number of covariates that are known to be a
subset of $\boldsymbol{w}_{t},$ known as the ‘active set’. The machine
learning literature focuses on this case, which we refer to as the case of
”known knowns”. Second, when $y_{t+h}$ could also depend on a few latent
(unobserved) factors, $\mathbf{f}_{t}$, not directly included in the active
set, which we call the case of ”known unknowns”.

Specifically, we suppose that for each $h$, $y_{t+h}$ can be approximated by
the following linear model, where the predictors are also elements of
$\boldsymbol{w}_{t}$ in the high-dimensional VAR (1),

$y_{t+h}=c_{h}+\mathbf{a}_{h}^{\prime}\mathbf{z}_{t}+\sum_{j=1}^{K}\beta_{jh}I(j\in
DGP)x_{jt}+\mathbf{\psi}_{h}^{\prime}\mathbf{f}_{t}+u_{h,t+h\text{ }},$ (3)

for $t=1,2,...,T-h$, where $c_{h}$ is the intercept, $\mathbf{z}_{t}$ is a
vector of small number, $p$, of pre-selected covariates included across all
horizons $h$. Obvious examples, include lagged values of the target variable
$(y_{t},y_{t-1},...)$. Other variables can also be included in
$\mathbf{z}_{t}$ on the basis of a priori theory or strong beliefs. The second
component of $y_{t+h}$ specifies the subset of variables in the active set
$\boldsymbol{x}_{t}=\left(x_{1t},x_{2t},....,x_{Kt}\right)^{\prime}$. $I(j\in
DGP)$ is an indicator variable which takes the value of unity if $x_{jt}$ is
included in the data generating process (DGP) for $y_{t+h}$ and zero
otherwise. It is only if $I(j\in DGP)=1$ that $\beta_{jh}$ will be identified.
The number of variables included in the DGP is given by
$k=\sum_{j=1}^{K}I(j\in DGP)$, which is supposed to be small and fixed as $T$
(and possibly $K$) become large. This assumption imposes sparsity on the
relationship between the target and the variables in the active set. In
addition, we allow for a small number of latent factors, $\mathbf{f}_{t},$
that represent other variables influencing $y_{t+h}$ that are not observed
directly, but known to be present - the known unknowns.

Giannone, Lenza, and Primiceri (2021) contrast sparse methods, that select a
few variables from the active set as predictors, such as Lasso and OCMT
discussed below, and dense methods, that select all the variables in the
active set but attach small weights to many of them, such as principal
components (PC), ridge regression and other shrinkage techniques. Rather than
having to choose between sparse and dense predictors, we consider approaches
that combine the two. We apply sparse selection methods to the variables in
the active set, and use dense shrinkage methods to approximate
$\mathbf{f}_{t}$ from a wider set of variables with $\boldsymbol{x}_{t}$
included as a subset. We first consider the selection problem, known knowns,
where we know the active set of potential covariates, and we then consider
known unknowns where there are unobserved factors.

##  3 Known knowns

In the case of known knowns, forecasts are obtained assuming that $y_{t+h}$ is
a linear function of $\boldsymbol{x}_{t}$

$y_{t+h}=c_{h}+\mathbf{a}_{h}^{\prime}\boldsymbol{z}_{t}+{\displaystyle\sum\limits_{j=1}^{K}}\beta_{hj}x_{jt}+u_{h,t+h}\text{,
for }t=1,2,...,T,$ (4)

subject to some penalty condition on $\\{\beta_{hj}\\}$. Some of the
covariates, $x_{jt}$, could be transformations of other covariates, such as
interaction terms. It is assumed that the model is correctly specified, in the
sense that, apart from $\boldsymbol{z}_{t}$, the variables that drive
$y_{t+h}$ are all included in the active set,
$\boldsymbol{x}_{t}=\left(x_{1t},x_{2t},....,x_{Kt}\right)^{\prime}$.

Many standard forecasting techniques result from the particular choice of the
penalty function. Shrinkage estimators such as ridge or some Bayesian
forecasts can be derived using the $\ell_{2}$ norm $\sum_{i=1}^{K}$
$\beta_{hj}^{2}<C_{h}<\infty.$ Lasso (least absolute shrinkage and selection
operator) follows when the $\ell_{1}$ norm is used
$\sum_{i=1}^{K}\left|\beta_{hj}\right|<C_{h}<\infty.$ The difference is shown
in Figure 1 below in Tibshirani (1996) where the $\ell_{1}$ norm yields corner
solutions with many of the coefficients, $\beta_{hj}$, estimated to be zero.
In contrast, the use of $\ell_{2}$ norm yields non-zero estimates for all the
coefficients with many very close to zero.

Figure 1: Estimation for the Lasso (left) and ridge (right) regression

There are a very large number of penalized techniques. Elastic net
regularisation has both an $\ell_{1}$ and an $\ell_{2}$ penalty, with ridge
regression and Lasso being special cases. There are also a large number of
variants of Lasso, including adaptive Lasso, grouped Lasso, double Lasso. We
will focus on Lasso itself, which we use in our empirical application and
which we will compare to OCMT as an alternative procedure which is based on
inferential rather than penalized procedures.

###  3.1 Lasso

Lasso estimates $\boldsymbol{\beta}_{h}$ by solving the following optimization
problem:

$\min_{\boldsymbol{\beta}_{h}}\left\\{{\displaystyle\sum\limits_{t=1}^{T}}\left(y_{t+h}-c_{h}-\boldsymbol{\beta}_{h}^{\prime}\boldsymbol{x}_{t}\right)^{2}+\lambda_{hT}{\displaystyle\sum\limits_{i=1}^{K}}{\left|\beta_{hi}\right|}\right\\}\text{
},$ (5)

where
$\boldsymbol{\beta}_{h}=(\beta_{h1},\beta_{h2},...,\beta_{hK})^{\prime}$,
$\boldsymbol{x}_{Kt}=\left(x_{1t},x_{2t},....,x_{Kt}\right)^{\prime}$ for a
given choice of the ”tuning” parameter, $\lambda_{hT}$. The variable selection
consistency of Lasso has been investigated by Zhao and Yu (2006), Meinshausen
and Bühlmann (2006) and more recently by Lahiri (2021). The key condition is
the so-called ”Irrepresentable Condition, IRC” that places restrictions on the
magnitudes of the correlations between the signals ($\mathbf{X}_{1h},$
standardized) and the rest of the covariates ($\mathbf{X}_{2h}$,
standardized), taken as given (deterministic). The IRC is:

$\text{IRC:
}\left\|(T^{-1}\mathbf{X}_{2h}^{\prime}\mathbf{X}_{1h})\left(T^{-1}\mathbf{X}_{1h}^{\prime}\mathbf{X}_{1h}\right)^{-1}sign(\boldsymbol{\beta}_{h}^{0})\right\|_{\infty}<1,$
(6)

where
$\boldsymbol{\beta}_{h}^{0}=(\beta_{1h}^{0},\beta_{2h}^{0},...,\beta_{k_{h}h}^{0})^{\prime}$
denotes the vector of true signal coefficients.444The number of signals,
$k_{h}$, could vary with $h$. The IRC condition is met for pure noise
variables, but need not hold for proxy variables, noise variables that are
correlated with the true signals.

To appreciate the significance of the IRC, suppose the DGP contains $x_{1t}$
and $x_{2t}$ and the rest of the covariates in the active set are
$x_{3t},x_{4t},...,x_{Kt}.$ Denote the sample correlation coefficient between
$x_{1t}$ and $x_{2t}$ by $\hat{\rho}$ ($\hat{\rho}^{2}<1$) and the sample
correlation coefficient of $x_{1t}$ and $x_{2t}$ with the rest of the
covariates in the active set by $\hat{\rho}_{1s},\hat{\rho}_{2s},$ for
$s=3,4,...,K.$ Then, dropping the subscript $h,$ the IRC for the $s^{th}$
covariate is given by

$\left|\left[sign\left(\beta_{01}\right),sign\left(\beta_{02}\right)\right]^{\prime}\left(\begin{array}[c]{cc}1&\hat{\rho}\\\
\hat{\rho}&1\end{array}\right)^{-1}\left(\begin{array}[c]{c}\hat{\rho}_{1s}\\\
\hat{\rho}_{2s}\end{array}\right)\right|<1$

which yields

$\left|sign\left(\beta_{01}\right)\left(\hat{\rho}_{1s}-\hat{\rho}\hat{\rho}_{2s}\right)+sign\left(\beta_{02}\right)\left(\hat{\rho}_{2s}-\hat{\rho}\hat{\rho}_{1s}\right)\right|<1-\hat{\rho}^{2}$

for $s=3,4,...,K.$ In this example there are two cases to consider: A:
$sign\left(\beta_{02}\right)=sign\left(\beta_{01}\right);$ and B:
$sign\left(\beta_{02}\right)=-sign\left(\beta_{01}\right).$ For case A
$\sup_{s}\left|\hat{\rho}_{1s}+\hat{\rho}_{2s}\right|<1+\hat{\rho}$, and for
case B $\sup_{s}\left|\hat{\rho}_{1s}-\hat{\rho}_{2s}\right|<1-\hat{\rho}$.
Since the signs of the coefficients are unknown, for all possible values of
$\hat{\rho}$, $\hat{\rho}_{1s}$ and $\hat{\rho}_{2s}$, we can ensure the IRC
condition is met if
$\left|\hat{\rho}\right|+\sup_{s}\left|\hat{\rho}_{1s}\right|+\sup_{s}\left|\hat{\rho}_{2s}\right|<1.$
This example shows the importance of the correlations between the true
covariates in the DGP as well as between the true covariates and the other
members of the active set that do not belong to the DGP. The IRC is quite a
stringent condition and it is not just when one has proxies in the active set
that are highly correlated with the true covariates that Lasso will tend to
choose too many variables. In practice one cannot check the IRC condition
since one does not know which variables are the true signals.

In addition to the IRC it is also required that

$\text{MinC:
min}_{j=1,2,...,k}\left|\beta_{jh}^{0}\right|>(2T)^{-1}\lambda_{hT}\left|\left(T^{-1}\mathbf{X}_{1h}^{\prime}\mathbf{X}_{1h}\right)^{-1}sign(\mathbf{\beta}_{h}^{0})\right|_{j}\text{,
}$ $\text{Penalty Condition : }T^{-1}\lambda_{hT}=o(1).$

The penalty condition, which follows from MinC, says that the penalty has to
rise with $T$, but not too fast and not too slowly. The expansion rate of
$\lambda_{hT}$ depends on the magnitude and the sign of $\beta_{jh}^{0}$, and
the correlations of signals with the proxy variables. Lahiri (2021) shows that
the penalty condition can be relaxed to
$\lim_{T\rightarrow\infty}T^{-1}\lambda_{hT}<\lim\inf_{T\rightarrow\infty}d_{hT},$
where

$d_{hT}=2\min_{j}\left|\beta_{jh}^{0}\right|/\left|\left(T^{-1}\mathbf{X}_{1h}^{\prime}\mathbf{X}_{1h}\right)^{-1}sign(\mathbf{\beta}_{h}^{0})\right|_{j}.$

The above conditions do not restrict the choice of $\lambda_{hT}$ very much,
hence the recourse to cross-validation to determine it. In practice,
$\lambda_{hT}$ is calibrated using $M$-fold cross-validation techniques. The
observations, $t=1,2,...,T$, are partitioned into $M$ disjoint subsets
(folds), of size approximately $m=T/M.$ Then $M-1$ subsets are used for
training and one for evaluation. This is repeated with each fold being used in
turn for evaluation. $M$ is typically set to $5$ or $10$. Cross-validation
methods are often justified in machine learning literature under strong
assumptions, such as independence and parameter stability across the sub-
samples used in cross-validation. These assumptions are rarely met in the case
of economic time series data, an issue that is discussed further in the
context of the empirical example in Section 5.

###  3.2 OCMT

The need for cross-validation is avoided in the procedure proposed by Chudik,
Kapetanios, and Pesaran (2018), (CKP). This is the ”One Covariate at a time
Multiple Testing” (OCMT) procedure, where covariates are selected one at a
time, using $t-$statistic for testing the significance of the variables in the
active set, individually.555Since $t$-ratios are invariant to scale, no pre-
standardization of the covariates in the active set is required. This is in
contrast to Lasso which is typically implemented after in-sample
standardization of the covariates. Ideas from the multiple testing literature
are used to control the false discovery rate, and ensure the selected
covariates encompass the true covariates (signals) with probability tending to
unity, under certain regularity conditions. Like Lasso, OCMT has no difficulty
in dealing with (pure) noise variables, and is very effective at eliminating
them. Also, like Lasso, it requires some $min$ condition such as
$\left|\beta_{jh}^{0}\right|>>\sqrt{\frac{k\log(K)}{T}}$, for
$j=1,2,...,k$.666To simplify the exposition we have dropped explicit reference
to the forecast horizon, $h$. But in practice, and as we shall see from the
empirical applications below, the number and identity of selected signals
could differ with $h$. But because it considers a single variable at a time,
OCMT does not require the IRC condition to hold and is not affected by the
correlation between the members of the DGP as Lasso is. Instead it requires
the number of pseudo signals, say $k_{T}^{\ast}$, to rise not faster than
$\sqrt{T}$. Chudik, Pesaran, and Sharifvaghefi (2023), discussed below, is
primarily concerned with parameter instability, but Section 4 of that paper
has a detailed comparison of the assumptions required for Lasso and OCMT under
parameter stability.

OCMT’s condition on $k_{T}^{\ast}$ has been recently relaxed by Sharifvaghefi
(2023) who allows $k_{T}^{\ast}\rightarrow\infty$, with $T$. He considers the
following DGP

$y_{t+h}=c_{h}+\mathbf{a}_{h}^{\prime}\boldsymbol{z}_{t}+\sum_{j=1}^{K}\beta_{jh}I(j\in
DGP)x_{jt}+u_{h,t+h}\text{ ,}$ (7)

where as before $\boldsymbol{z}_{t}$ is a known vector of pre-selected
variables, and it is assumed that the $k$ signals are contained in the known
active set $\mathcal{S}_{K,t}=\left\\{x_{jt},j=1,2,...,K\right\\}$. Note that
for now the DGP in (7) does not include the additional latent factors,
$\mathbf{f}_{t}$, introduced in (3). Without loss of generality, consider the
extreme case where there are no noise variables and all proxy or pseudo signal
variables ($x_{jt}$, for $j=k+1,k+2,...,K)$ are correlated with the signals,
$\mathbf{x}_{1t}=(x_{1t,}x_{2t},...,x_{kt})^{\prime}$. In this case
$k_{T}^{\ast}$ rises with $K$ and OCMT is no longer applicable. However, in
this case by construction the signals, $\mathbf{x}_{1t}$, become latent
factors for the proxy variables and we have

$x_{jt}=\phi_{j0}+\sum_{i=1}^{k}\phi_{ji}x_{it}+\varepsilon_{jt}=\phi_{j0}+\mathbf{\phi}_{j}^{\prime}\mathbf{x}_{1t}+\varepsilon_{jt}\text{,}$

for $j=k+1,k+2,....,K$. Although the identity of these common factors are
unknown, they can be approximated by the principal components of the variables
in the active set. Specifically, following Sharifvaghefi (2023), denote the
latent factors that result in non-zero correlations between the noise
variables in the active set and the signals by $\boldsymbol{\varkappa}_{t}$
and consider the factor model

$x_{jt}=\boldsymbol{\kappa}_{j}^{\prime}\boldsymbol{\varkappa}_{t}\mathbf{+}v_{jt}\text{,
for }j=1,2,...,K\text{ ,}$ (8)

where $\boldsymbol{\kappa}_{j}$, for $j=1,2,...,K$ are the factor loadings and
the errors, $v_{jt}$, are weakly cross-correlated and distributed
independently of the factors and their loadings. Under (8), the DGP, (7), can
be written equivalently as

$y_{t+h}=c_{h}+\mathbf{a}_{h}^{\prime}\boldsymbol{z}_{t}+\mathbf{b}_{h}^{\prime}\boldsymbol{\varkappa}_{t}+\sum_{j=1}^{K}\beta_{jh}I(j\in
DGP)v_{jt}+u_{h,t+h}\text{ },$ (9)

where $\mathbf{b}_{h}=\sum_{j=1}^{K}I(j\in
DGP)\beta_{jh}\boldsymbol{\kappa}_{j}$. When $\boldsymbol{\varkappa}_{t}$ and
$v_{jt}$ are known, the problem reduces to selecting $v_{jt}$ from
$\mathcal{S}_{K,t}^{v}=\left\\{v_{jt},j=1,2,...,K\right\\},$ conditional on
$\mathbf{z}_{t}$ and $\boldsymbol{\varkappa}_{t}$. Sharifvaghefi shows that
the OCMT selection can be carried out using the principal component estimators
of $\boldsymbol{\varkappa}_{t}$ and $\mathbf{v}_{jt}$ \- denoted by
$\boldsymbol{\hat{\varkappa}}_{t}$ and $\mathbf{\hat{v}}_{jt}$, if both $K$
and $T$ are large. He labels this procedure as generalized OCMT (GOCMT).

###  3.3 GOCMT

The GOCMT procedure simply augments the OCMT regressions with the PCs,
$\boldsymbol{\hat{\varkappa}}_{t}$, and considers the statistical significance
of $\mathbf{\hat{v}}_{jt}$ for each $j$, _one at the time_. Lasso-factor
models have also been considered by Fan, Ke, and Wang (2020) and Hansen and
Liao (2019). In practice, since
$\mathbf{x}_{j}=\mathbf{\hat{\Xi}\hat{\psi}}_{j}+\mathbf{\hat{v}}_{j}$, where
$\mathbf{\hat{\Xi}=}\left(\boldsymbol{\hat{\varkappa}}_{1},\boldsymbol{\hat{\varkappa}}_{2},...,\boldsymbol{\hat{\varkappa}}_{T}\right)^{\prime}$,
then
$\mathbf{M}_{\mathbf{\hat{\Xi}}}\mathbf{x}_{j}=\mathbf{M}_{\mathbf{\hat{\Xi}}}\mathbf{\hat{v}}_{j}$,
where
$\mathbf{M}_{\mathbf{\hat{\Xi}}}=\mathbf{I}_{T}-\mathbf{\hat{\Xi}}\left(\mathbf{\hat{\Xi}}^{\prime}\mathbf{\hat{\Xi}}\right)^{-1}\mathbf{\hat{\Xi}}^{\prime}$,
and GOCMT reduces to OCMT when $\mathbf{z}_{t}$ is augmented with
$\boldsymbol{\hat{\varkappa}}_{t}$, where the statistical significance of
$x_{jt}$ as a predictor of $y_{t+h}$ is evaluated for each $j$, one at a time.
Like OCMT, GOCMT allows for the multiple testing nature of the procedure ($K$
separate tests - with $K$ large) by increasing the level of significance with
$K$. The number of PCs, $dim(\boldsymbol{\hat{\varkappa}}_{t})$, can be
determined using one of the criteria suggested in the factor literature.

In the first stage, $K$ separate OLS regressions are computed, where the
variables in the active set are entered one at a time:

$y_{t+h}=c_{h}+\mathbf{a}_{h}^{\prime}\boldsymbol{z}_{t}+\mathbf{b}_{h}^{\prime}\boldsymbol{\hat{\varkappa}}_{t}+\phi_{jh}x_{jt}+e_{j,h,t+h}\text{,
}t=1,2,...,T\text{, for }j=1,2,...,K,$ (10)

Denote the $t$-ratio of $\phi_{jh}$ by $t_{\hat{\phi}_{j,\left(1\right)}}$.
Then variable $j$ is selected if

$\widehat{\mathcal{J}}_{j,(1)}=I\left[\left|t_{\hat{\phi}_{j,\left(1\right)}}\right|>c_{p}(K,\delta)\right],\text{
for }j=1,2,...,K\text{ },$ (11)

where $c_{p}(K,\delta)$ is a critical value function given by

$c_{p}(K,\delta)=\Phi^{-1}\left(1-\frac{p}{2K^{\delta}}\right),$ (12)

$p$ is the nominal size (usually set to $5\%$), $\Phi^{-1}(\cdot)$ is the
inverse of a standard normal distribution function and $\delta$ is a fixed
constant set in the interval $[1,1.5]$. In the second step a multivariate
regression of $y_{t+h}$ on $\boldsymbol{z}_{t}$ and all the selected
regressors is considered for inference and forecasting. Note that at the
moment it is assumed that $\boldsymbol{\varkappa}_{t}$ does not directly
affect $y_{t+h},$ it only enters through the $x_{jt}.$ It represents the
signals, the common factors correlated with the proxies, and provides a way of
filtering the correlations in the first step.

When the covariates are not highly correlated, OCMT applies irrespective of
whether $K$ is small or large relative to $T$, so long as
$T=\ominus\left(K^{c}\right)$, for any finite $c>0$. But to allow for highly
correlated covariates, GOCMT requires $K$ to be sufficiently large to enable
the identification of $\boldsymbol{\varkappa}_{t}$. In cases where $K$ is not
that large, it might be a good idea to augment the active set for the target
variable, $y_{t+h}$, $\mathcal{S}_{K,t}=\left\\{x_{jt},\text{
}j=1,2,...,K\right\\}$, with covariates for other variables determined
simultaneously with $y_{t+h}$, ending up with $\overline{\mathcal{K}}>K$
covariates for identification of $\boldsymbol{\varkappa}_{t}$. GOCMT does not
impose any restriction on the correlations between the variables other than
that they cannot be perfectly collinear.

###  3.4 High-dimensional variable selection in presence of parameter
instability

OCMT has also been recently generalized by Chudik, Pesaran, and Sharifvaghefi
(2023) to deal with parameter instability. Under parameter instability OCMT
correctly selects the covariates with non-zero average (over time) effects,
using the full sample. However the adverse effects of changing parameters on
the forecast may mean that while the full sample is the best to use for
selection, it need not be the best to use for estimating the forecasting
model. Instead, it may be better to use shorter windows or weight the
observations in the light of the evidence on break points and break sizes.

Determining the appropriate window or weighting for the observations before
estimation is a difficult problem and no fully satisfactory procedure seems to
be available. It is common in finance to use rolling windows of 60 or 120
months, but one problem with shorter windows is that if you have periods of
instability interspersed with periods of stability, like the Great Moderation,
estimates using a short window from the stable period may understate the
degree of uncertainty. This happened during the financial crisis when the
short windows used for estimation did not reflect past turbulence. Similarly,
the Bank of England estimating their models using the low inflation regime of
the past 30 years discounted the evidence from the high inflation regime of
the 1970s and 1980s.

While identifying the date of a break might not be difficult, identifying the
size of the break may be problematic if the break point is quite recent. If
there is a short time since the break, there is little data on which to
estimate the post-break coefficient with any degree of precision. If there is
a long time since the break, then using post break data is sensible. Pesaran,
Pick, and Pranovich (2013) examine optimal forecasts in the presence of
continuous and discrete structural breaks. These present quite different sorts
of challenges. With continuous breaks the parameters change often by small
amounts. With discrete breaks the parameters change rarely but by large
amounts. They propose weighting observations to obtain optimal forecasts in
the MSFE sense and derive optimal weights for one step ahead forecasts for the
two types of break. Under continuous breaks, their approach largely recovers
exponential smoothing weights. Under discrete breaks between two regimes the
optimal weights follow a step function that allocates constant weights within
regimes but different weights in different regimes. In practice, the time and
size of the break is uncertain and they investigate robust optimal weights.
Averaging forecasts with different weighting schemes, for instance with
exponential smoothing parameters between 0.96 and 0.99, may also be a way to
produce more robust forecasts.

##  4 Known unknowns

So far we have considered techniques (penalized regressions and OCMT) that
assume $y_{t+h}$ depends on $\mathbf{z}_{t}$ and a subset of a set of
covariates - the active set - which is assumed known. In contrast, shrinkage
type techniques such as PCs, (implicitly) assume that $y_{t+h}$ depends on
$\mathbf{z}_{t}$ and the $m\times 1$ vector of unknown factors
$\mathbf{f}_{t}$

$y_{t+h}=c_{h}+\mathbf{a}_{h}^{\prime}\boldsymbol{z}_{t}+\mathbf{\theta}_{h}^{\prime}\mathbf{f}_{t}+u_{h,t+h}.$

This is a simple example of techniques that in our terminology can be viewed
as belonging to a class of forecasting models based on known unknowns. The
uncertainty about $\mathbf{f}_{t}$ is resolved assuming it can be identified
from a known active set, such as
$\mathcal{S}_{K,t}=\left\\{x_{jt},j=1,2,...,K\right\\}$. Individual covariates
in $\mathcal{S}_{K,t}$ are not considered for selection (although a few could
be pre-selected and included in $\boldsymbol{z}_{t}$). To forecast $y_{t+h}$
one still requires to forecast the PCs and to allow for the uncertainty
regarding $m=\dim(\mathbf{f}_{t})$.

Factor augmented VARs (FAVAR), initially proposed by (Bernanke, Boivin, and
Eliasz, 2005, BBE), augment the standard VAR models with a set of unobserved
common factors. In the context of our set up, FAVAR can be viewed as a
generalized version of (3), where $\boldsymbol{y}_{t+h}$ is a vector and
$\boldsymbol{z}_{t}=\left\\{\boldsymbol{y}_{t},\boldsymbol{y}_{t-1},...,\boldsymbol{y}_{t-p}\right\\}$.
BBE argue that small VARs gave implausible impulse response functions, such as
the ”price puzzle”, which were interpreted as reflecting omitted variables.
One response was to add variables and use larger VARs, but this route rapidly
runs out of degrees of freedom, since Central Bankers monitor hundreds of
variables. The FAVAR was presented as a solution to this problem. Big Bayesian
VARs are an alternative solution.

The assumptions that underlie both penalized regression and PC shrinkage are
rather strong. The former assumes that $\mathbf{f}_{t}$ can affect $y_{t+h}$
only indirectly through $x_{jt}$, $j=1,2,...,K$, and the latter does not allow
for individual variable selection. Suppose that $\mathbf{f}_{t}$ also enters
(7) then the model can be written as (3) above, repeated here for convenience:

$y_{t+h}=c_{h}+\mathbf{a}_{h}^{\prime}\boldsymbol{z}_{t}+\sum_{j=1}^{K}\beta_{jh}I(j\in
DGP)x_{jt}+\mathbf{\psi}_{h}^{\prime}\mathbf{f}_{t}+u_{h,t+h}.$

The forecasting problem now involves both selection and shrinkage. The
$\mathbf{f}_{t}$ can be identified by, for instance, the PCs of the augmented
active set $x_{jt},$ $j=1,2...,K,K+1,...,\overline{\mathcal{K}}$ which can be
wider than the active set of covariates used to predict $y_{t}$. There is a
variety of other ways that the unobserved $\mathbf{f}_{t}$ could be estimated,
but we use PCs as an example, since they are widely used.

The missing factors are unlikely to be only specific to the target variable
under consideration. For example, if $y_{t+h}$ is UK inflation, the missing
factor(s) could be external to the UK. Inflation and output growth in other
countries could be equally relevant. Call this vector of global factors
$\mathbf{g}_{t}$. A natural extension is to introduce forecast equations for
other countries (entities) who have close trading relationships with UK and
use penalized panel regressions, where the panel dimension allows
identification of the known unknowns.

Suppose there are $N$ other units (countries) that are affected by observed
country specific covariates, $\mathbf{z}_{it},$ $i=1,2,...,N,$ and $x_{i,jt}$
for $j=1,2,...,K_{i},$ plus domestic latent factors $\mathbf{f}_{it}$, and
global latent factors, $\mathbf{g}_{t}$. The forecasting equations are now
generalized as

$\displaystyle y_{i,t+h}$
$\displaystyle=c_{h}+\mathbf{a}_{ih}^{\prime}\boldsymbol{z}_{it}+\sum_{j=1}^{K_{i}}\beta_{ijh}I(j\in
DGP_{i})x_{i,jt}+$
$\displaystyle\mathbf{\theta}_{ih}^{\prime}\mathbf{f}_{it}+\mathbf{\psi}_{ih}^{\prime}\mathbf{g}_{t}+u_{i,h,t+h}\text{,}$

where $k_{i}=\sum_{j=1}^{K_{i}}I(j\in DGP_{i})$ is small with $K_{i}$ quite
large, for $i=0,1,2,...,n$. For the country-specific covariates we postulate
that there is an augmented active set

$x_{i,jt}=\mathbf{\gamma}_{ij}^{\prime}\mathbf{f}_{it}+v_{i,jt},\text{
}j=1,2,...,K_{i}\text{, }K_{i+1},....,\overline{\mathcal{K}}_{i},$

where $\mathbf{f}_{it}$ are latent factors. The global factors are then
identified as the common components of the country-specific factors, namely

$\mathbf{f}_{it}=\boldsymbol{\Psi}_{i}\mathbf{g}_{t}+\boldsymbol{\xi}_{it}\text{
},$

for $i=1,2,..,N$, with $N$ large.

Variable selection for the target variable (say UK inflation) can now proceed
by applying GOCMT, with the UK model augmented with UK-specific PCs,
$\mathbf{\hat{f}}_{it}$ as well as the PC estimator of the global factor,
$\mathbf{g}_{t},$ that drives the country specific factors. This can be
extracted from $\mathbf{\hat{f}}_{it}$ as PCs of the country-specific PCs. In
addition to common factor dependence, countries are also linked through trade
and other more local features (culture, language). Such ”network” effects can
be captured by using ”starred” variables, to use the GVAR terminology. A
simple example would be$\ ($for $i=0,1,....,N)$

$\displaystyle y_{i,t+h}$
$\displaystyle=c_{ih}+\delta_{ih}y_{it}^{\ast}+\mathbf{a}_{ih}^{\prime}\boldsymbol{z}_{it}+\sum_{j=1}^{K_{i}}\beta_{ijh}I(j\in
DGP_{i})x_{i,jt}+$ (13)
$\displaystyle\mathbf{\theta}_{ih}^{\prime}\mathbf{f}_{it}+\mathbf{\psi}_{ih}^{\prime}\mathbf{g}_{t}+u_{i,h,t+h}\text{
,}$

where $i=0$ represents UK, and $y_{it}^{\ast}=\sum_{j=1}^{N}w_{ij}y_{jt},$
$w_{ij}$ (trade weights) measures the relative importance of country $j$ in
determination of country $i^{th}$ target variable. Similarly,
$\boldsymbol{z}_{it}^{\ast}=\sum_{j=1}^{N}w_{ij}^{\ast}\boldsymbol{z}_{jt}$
can also be added to the model if deemed necessary.

The network effects can be included either as an element of
$\boldsymbol{z}_{it}$ or could be made subject to variable selection. The
problem becomes much more complicated if we try to relate $y_{i,t+h}$
simultaneously to $y_{i,t+h}^{\ast}$. Further, for forecasting, following
Chudik, Grossman, and Pesaran (2016), one might also need to augment the UK
regressions with time series, forecasting models for the common factors.

Equation (13) allows for a number of different approaches to dimension
reduction.  As has been pointed out by Wainwright (2019) : “Much of high-
dimensional statistics involves constructing models of high-dimensional
phenomena that involve some implicit form of low-dimensional structure, and
then studying the statistical and computational gains afforded by exploiting
this structure”. Shrinkage methods, like PCs, assume a low dimensional factor
structure. The two selection procedures that we have considered, Lasso and
OCMT, exploit different aspects of the low-dimensional sparsity structure
assumed for the underlying data generating process. Lasso restricts the
magnitude of the correlations within and between the signals and the noise
variables. OCMT limits the rate at which the number of proxy variables rises
with the sample size. GOCMT relaxes this restriction by filtering out the
effects of latent factors that bind the proxies to the true signals before
implementing the OCMT procedure.

##  5 Forecasting UK inflation

###  5.1 Introduction

We apply the procedures proposed above to the problem of forecasting quarterly
UK inflation at horizons $h=1,2$ and $4$. Given this choice of target
variable, we need to choose the samples used for variable selection and
estimation, the pre-selected variables to include in $\boldsymbol{z}_{t},$ the
variables to include in the active set
$\mathcal{S}_{K,t}=\left\\{x_{jt},j=1,2,...,K\right\\}$, and the variable
selection techniques to use.

Since we have emphasised the importance of international network effects, we
need to use a quarterly dataset that includes a large number of countries to
estimate global factors and to allow the construction of the $y_{it}^{\ast}$
variables that appear in (13). The Global VAR (GVAR) data set provides such a
source. The publicly available data set compiled by Mohaddes and Raissi (2024)
covers 1979q1-2023q3. We are very grateful to them for extending the data.
While the latest GVAR dataset goes up to 2023q3 we only had access to data
till 2023q1 when we started the forecasting exercise, the results of which are
reported in this paper, but the data that we used matches that GVAR 2023
vintage which was released in January 2024.777GVAR Data 1979q1-2023q3 (2023
Vintage) is available at https://www.mohaddes.org/gvar, further material on
the GVAR is provided at https://sites.google.com/site/gvarmodelling/gvar-
toolbox

The data base includes quarterly macroeconomic data for 6 variables (log real
GDP, y, the rate of inflation, dp, short-term interest rate, r, long-term
interest rate, lr, the log deflated exchange rate, ep, and log real equity
prices, eq), for 33 economies as well as data on commodity prices (oil prices,
poil, agricultural raw material, pmat, and metals prices, pmetal). These 33
countries cover more than 90% of world GDP. The GVAR data was supplemented
with other specific UK data on money, wages, employment and vacancies, in the
construction of the active set discussed below.

In the light of the argument in Chudik, Pesaran, and Sharifvaghefi (2023), we
use the full sample beginning in 1979q1 for variable selection. There are
arguments for down-weighting earlier data for estimation when there have been
structural changes, as discussed by Pesaran, Pick, and Pranovich (2013).
However, the full sample was used both for variable selection and estimation
of the forecasting model in order to allow evidence from the earlier higher
inflation regime to inform both aspects.

The target variable is average annual UK inflation, labelled DPUK4, defined as
$\pi_{t+h}=100\times\log(p_{t+h}/p_{t+h-4}),$ where $p_{t}$ is the UK consumer
price index taken from the IMF International Financial Statistics. Two sets of
variables are considered for inclusion in $\boldsymbol{z}_{t}$. The first set,
which we label AR2, includes lags of the target variable $\pi_{t},\pi_{t-1}$
(or equivalently $\pi_{t}$ and $\Delta\pi_{t}).$ Given the importance we
attach to global variables and network effects, the second set, which we label
ARX2, also includes $\pi_{t}^{\ast}$, $\pi_{t-1}^{\ast}$ (or equivalently
$\pi_{t}^{\ast}$ and $\Delta\pi_{t}^{\ast})$ where $\pi_{t}^{\ast}$ is a
measure of UK specific foreign inflation constructed using UK trade weights
with the other countries.888Specifically,
$\pi_{t}^{\ast}=\sum_{j}w_{j}\pi_{jt},$ where $\pi_{jt}$ is the inflation rate
in country $j$ and $w_{j}$ is the trade weight of country $j$ with UK.

If there is a global factor in inflation, the inflation rates of different
countries will be highly correlated and tend to move together. Figure 2
demonstrates that this is in fact the case. It plots the inflation rates for
19 countries over the period 1979-2022. It is clear that they do move
together, reflecting a strong common factor. The dispersion is somewhat
greater in the high inflation 1980s. At times individual countries break away
from the herd with idiosyncratic bursts of inflation, like New Zealand in the
mid 1980s. But it is striking that inflation in every country increases from
2020.

Figure 2: Inflation Across Advanced Economies

To demonstrate the importance of the global factor for the UK, Figure 3 plots
$\pi_{t}$, and $\pi_{t}^{\ast},$ UK inflation and UK specific foreign
inflation. The two series move together, and from the mid 1990s they are very
close. This indicates that not only is one unlikely to be able to explain UK
inflation just by UK variables but that there are good reasons to include this
UK specific measure of foreign inflation in one of our specifications for
$\boldsymbol{z}_{t}.$ The GVAR estimates also indicate the higher sensitivity
of the UK to foreign variables than the US or euro area. This is not
surprising, they are larger, less open, economies.

Figure 3: UK and UK-specific global inflation

###  5.2 Active set

We now turn to the choice of the members of the active set,
$\mathcal{S}_{K,t}=\left\\{x_{jt},j=1,2,...,K\right\\},$ some of which may be
included in $\boldsymbol{z}_{t}.$ While our focus is on forecasting not on
building a coherent economic model, our choice of the covariates in the active
set is motivated by the large Phillips curve literature, which suggests
important roles for demand, supply and expectations variables. The demand and
supply variables in the active set are both domestic and foreign from both
product and labour markets. Expectations are captured by financial variables.
Interaction terms were not included, but the non-linearities that have been
investigated in the literature may be picked up by the latent foreign
variables. These are represented by $y_{it}^{\ast}$ terms, UK specific
measures of foreign inflation and output. Explicit measures of
$\mathbf{f}_{it}$ and $\mathbf{g}_{t}$ did not seem to be needed.

Accordingly, we consider 26 covariates ($x_{jt}$) listed in Table 1, and their
changes $\Delta x_{jt}=x_{jt}-x_{j,t-1}$, giving an active set with $K=52$
variables to select from. Whereas in a regression including current and lagged
values of a regressor (say $x_{t}$ and $x_{t-1})$ is equivalent to including
current and change (namely $x_{t}$ and $\Delta x_{t}$), in selection the two
specifications can result in different outcomes. Including $\Delta x_{t}$ is
better since, as compared to $x_{t-1}$, it is less correlated with the level
of the other variables in the active set. The 26 included covariates are
measured as four-quarter rates of change, changes or averages to match the
definition of the target variable. The rates of change are per cent per annum.

UK goods market demand indicators: rate of change of output, two measures of
the output gap: log output minus either a $P=8$ or $P=12$ quarter moving
average of log output, $Gap(y_{t},P)=y_{t}-P^{-1}\sum_{p=1}^{P}y_{t-p}$;

UK labour market demand indicators: rate of change of UK employment,
vacancies, and average weekly earnings and the change in unemployment;

UK financial indicators: annual averages of UK short and long interest rates,
the rate of change of money, UK M4, and of UK real equity prices;

Global cost pressures on the UK: rate of change of the price of oil, metals,
materials, UK import prices and deflated dollar exchange rate;

Foreign demand and supply variables: UK specific global measures, foreign
inflation, rate of change of foreign output and two measures of the foreign
output gap: log foreign output minus either an 8 or 12 quarter moving average
of log foreign output. In addition, large country variables were added: annual
average of US short and long interest rates, rates of change of US output and
prices, and of Chinese output.

Table 1: List of covariates for UK inflation forecasting Variable |  | Description
---|---|---
DPUK4 |  |  4 quarter UK rate of inflation,
DYUK4 |  |  4 quarter rate of change of UK real GDP
GAPUK8 |  | UK log real GDP relative to its 8 quarter moving average
GAPUK12 |  | UK log real GDP relative to its 12 quarter moving average
DEMUK4 |  | 4 quarter rate of change of UK employment
DVUK4 |  | 4 quarter rate of change of UK vacancies
DUUK |  | 4 quarter change in UK unemployment
DWUK4 |  | 4 quarter rate of change of average weekly earnings,
RUK4 |  | 4 quarter average UK short interest rate
LRUK4 |  | 4 quarter average UK long interest rate
DMUK4 |  | 4 quarter rate of change of UK M4 money
DEQUK4 |  | 4 quarter rate of change of UK real equity prices
DPOIL4 |  | 4 quarter rate of change of oil prices
DPMAT4 |  | 4 quarter rate of change of material prices
DPMETAL4 |  | 4 quarter rate of change of metal prices
DPMUK4 |  | 4 quarter rate of change of UK import prices
DEPUK4 |  | 4 quarter rate of change of UK deflated dollar exchange rate
DPSUK4 |  | UK specific measure of 4 quarter foreign inflation
DYSUK4 |  | 4 quarter rate of change of UK specific foreign real GDP
RUS4 |  | 4 quarter average US short interest rate
LRUS4 |  | 4 quarter average US long interest rate
DYCHINA4 |  | 4 quarter rate of change of Chinese real GDP
GAPSUK8 |  | UK specific log foreign real GDP relative to 8 quarter moving average
GAPSUK12 |  | UK specific log foreign real GDP relative to 12 quarter moving average
DYUS4 |  | 4 quarter rate of change of US real GDP
DPUS4 |  | 4 quarter US rate of inflation

###  5.3 Variable selection

####  5.3.1 Variable selection procedures

We consider Lasso, Lasso conditional on $\boldsymbol{z}_{t}$, and GOCMT
conditional on $\boldsymbol{z}_{t}$. With Lasso, the variables are
standardized in-sample before implementing variable selection. The Lasso
penalty parameter, $\lambda_{T},$ is estimated using 10 fold cross-validation
(CV), across subsets of the observations. As noted above, the assumptions
needed for standard CV procedures, for instance those used in the program
cv.glmnet, are not appropriate for time series. Time series show features such
as persistence and changing variance that are incompatible with those
assumptions. In the standard procedure the CV subsets (folds) are typically
chosen randomly. This is appropriate if the observations are independent draws
from a common distribution, but this is not the case with time series. Since
order matters in time series, we retain the time order of the data within each
subset. See Bergmeir, Hyndman, and Koo (2018) who provide Monte Carlo evidence
on various procedures suggested for the case of serially correlated data. We
use all the data, and do not leave gaps between subsets. In addition, the
standard procedure chooses the $\hat{\lambda}_{hT}$ that minimises the pooled
MSE over the ten subsets. But when variances differ substantially over subsets
pooling is not appropriate, instead we follow Chudik, Kapetanios, and Pesaran
(2018, CKP), and use the average of the $\hat{\lambda}_{hT}$ chosen in each
subset. Full details are provided in the online simulation appendix to CKP
(2018).

As well as standard Lasso, for consistency with OCMT, we also generated Lasso
forecasts conditional on $\boldsymbol{z}_{t}$ by including a pre-selected set
of variables $\boldsymbol{z}_{t}$ in the optimization problem (5). This
generalized Lasso procedure solves the following optimization problem:

$\min_{\mathbf{a}_{h},\mathbf{\beta}_{h}}\left\\{{\displaystyle\sum\limits_{t=1}^{T}}\left(y_{t+h}-c_{h}-\mathbf{a}_{h}^{\prime}\boldsymbol{z}_{t}-\boldsymbol{\beta}_{h}^{\prime}\boldsymbol{x}_{t}\right)^{2}+\lambda_{T}{\displaystyle\sum\limits_{i=1}^{K}}{\left|\beta_{hi}\right|}\right\\}\text{
,}$ (14)

where the penalty is applied only to the variables in the active set,
$\boldsymbol{x}_{t},$ and not to the pre-selected variables,
$\boldsymbol{z}_{t}$. The above optimization problem can be solved in two
stages. In the first stage the common effects of $\boldsymbol{z}_{t}$ are
filtered out by regressing $y_{t+h}$ and $\boldsymbol{x}_{t}$ on the pre-
selected variables $\boldsymbol{z}_{t}$ and saving the residuals $e_{y.z}$ and
$e_{xj.z}$, $j=1,2,..,K$. In the second stage Lasso is applied to these
residuals. A proof that this two-step procedure solves the constrained
minimization problem in (14) is provided by Sharifvaghefi and reproduced in
the Appendix.

In the OCMT critical value function,
$c_{p}(K,\delta)=\Phi^{-1}\left(1-\frac{p}{2K^{\delta}}\right),$ we set
$p=0.05$ and $\delta=1$. With $K=52$ this means that we only retain variables
with t-ratios (in absolute value) exceeding $c_{0.05}(52,1)=3.3$. We also
experimented with setting $\delta=1.5,$ which yields, $c_{0.05}(52,1.5)=3.82$.
The results were reasonably robust and we focus on the baseline choice of
$\delta=1,$ also recommended by CKP.

We implement Lasso and OCMT conditional on two pre-selected sets of variables,
either an AR2 written as level and change $\mathbf{z}_{t}=$ ($\pi_{t}$,
$\Delta\pi_{t})^{\prime}$ or given the role of foreign inflation, shown above,
the AR2 augmented by the level and change of the UK specific measure of
foreign inflation, denoted ARX, $\mathbf{z}_{t}=$ ($\pi_{t}$,
$\Delta\pi_{t}$,$\pi_{t}^{\ast}$,$\Delta\pi_{t}^{\ast})^{\prime}$. As noted
above, for selection including current and change is better than including
current and lag. For comparative purposes we also generated forecasts with the
pre-selected variables only, namely the AR2 forecasts generated from the
regressions

$AR2:\text{ \ \ \ \
}\pi_{t+h}=c_{h}+a_{1}\pi_{t}+a_{2h}\Delta\pi_{t}+u_{h,t+h}\text{,}$

and the ARX forecasts generated from

$ARX:\text{ \ \ \ \ \ \ \ \
}\pi_{t+h}=c_{h}+a_{1h}\pi_{t}+a_{2h}\Delta\pi_{t}+a_{3h}\pi_{t}^{\ast}+a_{4h}\Delta\pi_{t}^{\ast}+u_{h,t+h}.$

Variable selection is carried out recursively, for each forecast horizon $h$
separately, using an expanding windows approach. All data samples start in
$1979q2$ and end in the quarter that forecasts are made. To forecast the
average inflation over the four quarters to $2020q1$ using a forecast horizon
of $h=4$, the sample used for selection and estimation ends in $2019q1$. The
end of the sample is then moved to $2019q2$ to forecast the average inflation
over the four quarters to $2020q2,$ and so on. Similarly, to forecast the
average inflation over the four quarters to $2020q1$ using $h=2$, the sample
ends in $2019q3$, and using $h=1$ the sample ends in $2019q4.$ These sequences
continue one quarter at a time until the models are selected and estimated to
forecast inflation over the four quarters to $2023q1$. Thus for $h=4,$ there
are $17$ samples used for variable selection, while for $h=2$ and $h=1$ there
are $15$ and $14$ such variable selection samples. This process of recursive
model selection and estimation means that the variables selected can change
from quarter to quarter, and for each forecast horizon, $h$.

Section S-3 of the online supplement list the variables selected by each of
the procedures, for each quarter and each forecast horizon. The main features
are summarised here.

####  5.3.2 Number of variables selected

Table 2 gives the minimum, maximum, and average number of variables selected
for the 3 forecast horizons and 5 variable selection procedures. Except for
AR2-OCMT, at $h=4,$ OCMT chooses fewer variables than Lasso. Lasso conditional
on the pre-selected variables selects a larger number of variables in total
than the standard Lasso without conditioning. Conditioning on pre-selected
variables is much more important for OCMT as compared to Lasso. This finding
is in line with the theoretical results obtained by Sharifvaghefi (2023) who
establishes the importance of conditioning on the latent factors when applied
to an active set with highly correlated covariates. The number of variables
Lasso selects falls with the forecast horizon,999This is possibly because, as
$h$ increases the $\mathbf{\beta}_{h}^{0}$ get smaller in (6), the IRC is more
likely to be satisfied and Lasso is less likely to falsely select additional
variables. whilst the number of variables selected by OCMT rises with the
horizon. These results show that Lasso and OCMT could select very different
models for forecasting.

Table 2: Number of variables selected by Lasso and OCMT including preselected

| Forecast horizon, $h$, in quarters
---|---
| Total number of pre-selected and selected variables
|  |  $\ h=1$ |  |  $\ h=2$ |  |  $h=4$
|  | Min | Max | Mean | .. | Min | Max | Mean | .. | Min | Max | Mean
Lasso |  | 7 | 12 | 8.1 |  | 5 | 9 | 6.1 |  | 3 | 6 | 5.2
AR2-Lasso |  | 5 | 11 | 8.2 |  | 9 | 16 | 13.5 |  | 8 | 11 | 9.5
AR2-OCMT |  | 2 | 3 | 2.2 |  | 4 | 5 | 4.5 |  | 5 | 14 | 6.2
ARX-Lasso |  | 8 | 16 | 12.4 |  | 12 | 19 | 16.3 |  | 2 | 15 | 9.9
ARX-OCMT |  | 4 | 4 | 4 |  | 5 | 5 | 5 |  | 5 | 8 | 5.8

_Note_ : The reported results are based on 14, 15 and 17 variable selection
samples for 1, 2 and 4 quarter ahead models, respectively. The AR2 and ARX
components include 2 and 4 pre-selected variables, respectively.

As expected, the number of variables selected by Lasso correlates with the
estimates of the penalty parameter $\hat{\lambda}_{hT}$, computed by cross-
validation. These values are summarized in Table 3. For all three Lasso
applications the mean of the estimated penalty parameter increases with the
forecast horizon, though more slowly for the specifications that include pre-
selected variables. For Lasso (without pre-selection) the number of variables
selected falls with the forecast horizon because of the increasing penalty
parameter. This is not as clear cut for the specifications including pre-
selected variables.

Table 3: Estimates of the Lasso penalty parameter computed by 10-fold cross-
validation procedure.

|  |  | Forecast horizon, $h$, in quarters |  |
---|---|---|---|---|---
| $\ \ \ h=1$ |  | $\ \ \ \ h=2$ |  | $\ \ \ \ h=4$
| Min | Max | Mean | .. | Min | Max | Mean | .. | Min | Max | Mean
Lasso | 0.07 | 0.12 | 0.11 |  | 0.22 | 0.31 | 0.26 |  | 0.33 | 0.47 | 0.41
AR2-Lasso | 0.08 | 0.10 | 0.09 |  | 0.09 | 0.14 | 0.11 |  | 0.18 | 0.27 | 0.22
ARX-Lasso | 0.04 | 0.08 | 0.06 |  | 0.07 | 0.12 | 0.09 |  | 0.15 | 0.33 | 0.22

_Note_ : The reported estimates are based on Lasso penalty estimates (obtained
from 10-fold cross-validation) for 14, 15 and 17 variable selection samples
for 1, 2, and 4 quarter ahead models, respectively.

####  5.3.3 OCMT: selected variables by horizon

OCMT selects only a few variables in addition to the pre-selected UK and UK-
specific foreign inflation ($\pi_{t},\Delta\pi_{t}$,$\pi_{t}^{\ast}$, and
$\Delta\pi_{t}^{\ast}$).101010The OCMT results reported here are based on the
critical value function given by (12) with $\delta=1$. Using the larger value
of $\delta=1.5$ reduces the number of selected variables for a few of sample
periods, but the outcomes are generally robust to the choice on $\delta$ on
the interval $[1,1.5]$. The selection results for OCMT with $\delta=1.5$ are
reported in the online supplement. For $h=1,$ the variables selected are given
in sub-section S-3.1.1 of the online supplement. In addition to the $2$ pre-
selected variables ($\pi_{t},\Delta\pi_{t}$), AR2-OCMT selects the rate of
change of wages (DWUK4) for samples ending in $2021q2$, $2021q4$ and $2022q1$,
and no other variables. ARX-OCMT does not select any additional variables for
any of the $14$ variable selection samples!

For $h=2,$ the variables selected are given in subsection S-3.1.2 of the
online supplement. AR2-OCMT selects the rates of change of money (DMUK4) and
exchange rate (DEPUK4) from samples ending in $2019q3$ to $2020q3$, then the
rate of change of wages is added till the sample ending in $2022q2$, from then
the rates of change of money (DMUK4) and wages (DWUK4) are selected. ARX-OCMT
selects just the rate of change of money (DMUK4) as an additional variable in
every sample for $h=2$.

For $h=4$, the variables selected are given in subsection S-3.1.3 of the
online supplement. AR2-OCMT chooses the same $3$ extra variables - the rate of
change of money (DMUK4) and exchange rate (DEPUK4) as well as the UK-specific
measure of foreign inflation (DPSUK4, $\pi_{t}^{\ast}$) - for every sample
ending from $2019q1$ to $2021q1$. Then, in the sample ending in $2021q2,$
AR2-OCMT chooses $12$ extra variables. The number of variables selected then
falls to $7$ in $2021q3$, $6$ in $2021q4$, $5$ in $2022q1$ and $4$ in
$2022q2-2022q4.$ These $4$ are the rate of change of money (DMUK4), of
material prices (DPMAT4), of wages (DWUK4), and $\pi_{t}^{\ast}$ (DPSUK4). The
number of variables selected falls to $3$ in the sample ending in $2023q1$
when the foreign inflation measure is no longer selected. ARX-OCMT chooses the
rate of change of employment (DEMUK4) and of money for samples ending in
$2019q1-2021q4$, then adds material prices in $2022q1$, and selects just the
rate of change of money for the last four samples.

####  5.3.4 Lasso: selected variables by horizon

Lasso selections for each sample and horizon are given in the online
supplement, subsection S-3.2. Lasso tends to select more variables than OCMT
so we give less detail. Table 4 lists the variables chosen by standard Lasso
at each horizon and the number of times they were chosen out of the maximum
number of possible samples: $14,$ for $h=1,$ $15$ for $h=2,$ and $17$ for
$h=4$. UK inflation, $\pi_{t}$ (DPUK4) is always chosen in every sample at
every horizon as is the UK measure of foreign inflation, $\pi_{t}^{\ast}$
(DPSUK4). The change in UK inflation, $\Delta\pi_{t},$ (DDPUK4) is chosen in
every sample in the case of models for $h=1,$ and $h=2,$ but never for $h=4.$
The change in foreign inflation $\Delta\pi_{t}^{\ast}$ (DDPSUK4) is chosen in
every sample at $h=1,$ in $3$ samples at $h=2$ but never at $h=4$. Thus Lasso
provides considerable support for the choice of pre-selected variables in
$\boldsymbol{z}_{t}$ that include foreign inflation as well as the two lagged
inflation variables.

Apart from these variables, the rate of change of wages and of money figure
strongly when using Lasso. The rate of change of wages (DWUK4) is chosen in
all the samples for $h=1$ and $h=2$ and $14$ of the $17$ samples for $h=4.$
The rate of change of money (DMUK) is chosen in $10$ of the $14$ samples for
$h=1,$ and in every sample for $h=2$ and $h=4.$ Money and wages are also
chosen by OCMT but the rate of change of the exchange rate selected by OCMT is
never chosen by Lasso.

When $h=1$ Lasso also always selects two other variables, namely the change in
long interest rates (DLRUK4), and import price inflation (DPMUK).

Table 4: The number of times covariates from the active set are selected by
Lasso at different forecast horizons

| Horizon | Selected covariates
---|---|---
| ${\small h=1}$ | ${\small h=2}$ | ${\small h=4}$ |
DPUK4 | 14 | 15 | 17 | 4 quarter UK rate of inflation
DPSUK4 | 14 | 15 | 17 | UK specific measure of 4 quarter foreign inflation
DWUK4 | 14 | 15 | 14 | 4 quarter rate of change of average weekly earnings
DDPUK4 | 14 | 15 | 0 | Change in 4 quarter UK rate of inflation
DPMUK | 14 | 8 | 0 | 4 quarter rate of change of UK import prices
DLRUK4 | 14 | 4 | 0 | Change in 4 quarter average UK long interest rate
DDPSUK4 | 14 | 3 | 0 | Change in UK specific measure of 4 quarter foreign inflation
DMUK | 10 | 15 | 17 | 4 quarter rate of change of UK M4 money
DRUK4 | 2 | 0 | 0 | Change in 4 quarter average UK short interest rate
DEMUK4 | 1 | 0 | 6 | 4 quarter rate of change of UK employment
DDPOIL4 | 1 | 0 | 0 | Change in 4 quarter rate of change of the oil price
DDEQUK4 | 1 | 0 | 0 | Change in 4 quarter rate of change of UK equity prices
DDYSUK4 | 1 | 0 | 0 | Change in 4 quarter rate of change of UK foreign real GDP
DGAPSUK12 | 1 | 0 | 0 | UK log foreign real GDP relative to 8 quarter moving average
DPUS4 | 0 | 2 | 2 | 4 quarter US rate of inflation

_Note_ : The number of variable selection samples are 14, 15 and 17 for $h=1$,
$2$, and $4$ quarter ahead models, respectively.

###  5.4 Forecasts

The point forecasts of inflation for $h=1,2$ and $h=4$ for the various
selection procedures are summarized in Section S-4 of the online supplement.
For each forecast horizon we have $13$ forecasts and their realizations for
the quarters $2020q1$ to $2023q1$ inclusive. These are summarised in Table 5
below. We use root mean square forecast error (RMSFE) as our forecast
evaluation criterion. Since $13$ forecast errors represent a very short
evaluation sample with considerable serial correlation, testing for the
significance of the loss differences using the Diebold and Mariano (1995) test
would not be reliable, and is not pursued here.

####  5.4.1 One quarter ahead forecasts

Figure 4 gives plots of actual inflation and forecasts one quarter ahead.
Section S-4.1 of the online supplement gives the point forecasts. For $h=1$,
ARX has the lowest RMSFE, the $\pi_{t}^{\ast}$ and $\Delta\pi_{t}^{\ast}$
improve forecast performance relative to the AR2. AR2-OCMT adds wage growth in
three periods. Lasso suffers from choosing too many variables relative to
OCMT. The forecasts are very similar, except Lasso predicted a large drop in
$2020q3$ with a subsequent rebound. This results from selecting an output gap
measure, when UK output dropped sharply in $2020q2$. This sharp drop and
rebound was also a feature of Lasso forecasts at other horizons. The Bank of
England over-estimated inflation in $2022q4$, correctly anticipating higher
energy prices but not anticipating the government energy price guarantees.

Figure 4: One quarter ahead forecasts.

####  5.4.2 Two quarter ahead forecasts

Figure 5 gives plots of actual inflation and forecasts two quarters ahead.
Section S-4.2 of the online supplement gives the values. For $h=2$, ARX again
has the lowest RMSFE. ARX-OCMT selects money growth in every period. Lasso
selects between 5 and 9 variables.

Figure 5: Plot of forecasts two quarters ahead

####  5.4.3 Four quarter ahead forecasts

Figure 6 gives plots of actual inflation and forecasts four quarters ahead.
Section S-4.3 of the online supplement gives the values. The case of $h=4$ is
the only one where the ARX does not have the lowest RMSFE. The lowest RMSFE is
obtained by AR2-OCMT. It does well by having a very high inflation forecast in
$2022Q2$. This corresponds to the selection of 12 extra variables in the
sample ending in $2021q2$. It then rejoins the pack in $2022q3$.

Figure 6: Plot of forecasts four quarters ahead

####  5.4.4 Summary

Table 5 brings together the RMSFE for each of the selection methods at the
different horizons. Both the variable selection and forecasting exercises
highlight the importance of taking account of persistence and foreign
inflation for UK inflation forecasting. Lasso selects $\pi_{t}$ and
$\pi_{t}^{\ast}$ in all three forecast horizon models. ARX which includes UK
and foreign inflation as pre-selected variables tends to perform best in
forecasting but in the present application the OCMT component does not seem to
add much once the pre-selected variables are included. However, Lasso performs
rather poorly when it is conditioned on the preselected variables.

Table 5: Root mean square forecast errors by forecast horizons, h=1,2 and 4
over the period 2020q1-2023q1

Forecast Source |  | h=1 |  | h=2 |  | h=4
---|---|---|---|---|---|---
AR2 |  | $0.9141$ |  | $1.6039$ |  | $3.2524$
ARX |  | $\mathbf{0.7884}$ |  | $\mathbf{1.3813}$ |  | $2.9883$
Lasso |  | $0.9696$ |  | $1.4109$ |  | $3.0131$
AR2–Lasso |  | $0.9617$ |  | $2.8719$ |  | $4.3440$
ARX–Lasso |  | $0.8750$ |  | $2.9231$ |  | $4.5021$
AR2–OCMT |  | $0.9233$ |  | $1.6800$ |  | $\mathbf{2.4643}$
ARX–OCMT |  | $\mathbf{0.7884}$ |  | $1.4217$ |  | $3.2470$
Bank of England |  | $1.3288$ |  | $1.6959$ |  | $3.0042$

_Note_ : The RMSFE figures are taken from online supplement Tables S-4.1,
S-4.2, and S-4.3. The least value for RMSFE for each forecast horizon is shown
in bold.

###  5.5 Contemporaneous drivers

Our forecasts of $\pi_{t+h}$ are based on variables observed at time $t$, and
do not depend on any conditioning. But, as noted above with respect to the
Bank of England, it is common to condition on contemporaneous values of
variables which are considered as proximate causes of the variable to be
forecast. Even if such causal variables can be identified, however, it does
not mean that they help with forecasting - often such causal variables are
themselves difficult to forecast. The Bank of England over-estimated inflation
in 2022q4, correctly anticipating higher energy prices but not anticipating
the government energy price guarantees. This illustrates the dangers of
conditioning on variables that cannot be forecast. Understanding does not
necessarily translate into better forecasts. For example, knowing the causes
of earthquakes does not necessarily help in predicting them in a timely
manner.

This point can be illustrated by including contemporaneous changes in oil
prices, in the UK inflation equation (over the period pre Covid-19 and the
full sample) for the case $h=1$. For both samples ${\footnotesize\Delta
poil}_{t+1}$ are highly statistically significant, but their lagged values
${\footnotesize\Delta poil}_{t}$ are not.

Table 6: Contemporaneous and lagged effects of oil price changes on UK inflation $(\pi_{t+1})$ Covariates |  | 1979q2–2019q4 |  | 1979q2–2022q4
---|---|---|---|---
$\pi_{t}$ |  | 0.936 | 0.936 |  | 0.957 | 0.962
|  | (13.40) | (12.97) |  | (13.70) | (13.23)
$\pi_{t-1}$ |  | –0.134 | –0.136 |  | –0.145 | –0.151
|  | (–2.05) | (–2.02) |  | (–2.20) | (–2.21)
$\pi_{t}^{\ast}$ |  | 0.480 | 0.512 |  | 0.600 | 0.524
|  | (4.14) | (3.31) |  | (5.50) | (4.39)
$\pi_{t-1}^{\ast}$ |  | –0.381 | –0.316 |  | –0.504 | –0.432
|  | (–3.28) | (–2.52) |  | (–4.64) | (–3.61)
$\Delta poil_{t+1}$ |  | 0.013 | $\cdot$ |  | 0.012 | $\cdot$
|  | (3.40) | $\cdot$ |  | (3.91) | $\cdot$
$\Delta poil_{t}$ |  | $\cdot$ | 0.006 |  | $\cdot$ | 0.005
|  | $\cdot$ | (1.47) |  | $\cdot$ | (1.37)
$\bar{R}^{2}$ |  | 0.953 | 0.950 |  | 0.952 | 0.948
SER |  | 0.612 | 0.631 |  | 0.626 | 0.651

##  6 Conclusion

High dimensional data are not a panacea; the data must have some predictive
content which might come from spatial or temporal sequential patterns.
Forecasting is particularly challenging either if there are unknown unknowns
(factors that are not even thought about) or if there are known factors that
are falsely believed to be important. When there are new global factors, like
Covid-19, or when a relevant variable has shown little variation over the
sample period, forecasting their effect is going to be problematic.

Many forecasting problems require a hierarchical structure where latent
factors at local and global levels are explicitly taken into account. This is
particularly relevant for macro forecasting in an increasingly inter-connected
world. It is important that we allow for global factors in national
forecasting exercises - and GVAR was an attempt in this direction.

A number of key methodological issues were illustrated with a simple approach
to forecasting UK inflation which has become a topic of public discussion.
This example showed both the power of parsimonious models and the importance
of global factors. There remain many challenges. How to allow for regime
change and parameter instability in the case of high-dimensional data
analysis? How to choose data samples? Our recent research suggests that it is
best to use long time series samples for variable selections, but consider
carefully what sample to use for forecasting. Given a set of selected
variables, parameter estimation can be based on different window sizes, or
down-weighting. Should we use ensemble or forecast averaging? Forecast
averaging will only work if the covariates used to forecast the target
variable are driven by strong common factors, otherwise one will be averaging
over noise.

There are some more general lessons. Econometric and statistical models must
not become a straightjacket. Forecasters should be open minded about factors
not included in their model, and acknowledge that forecasts are likely to be
wrong if unexpected shocks hit.

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## Appendix

The Lasso procedure with a set of preselected variables111111We are grateful
to Dr. Mahrad Sharifvaghefi for providing the proofs in this appendix.

Let $\mathbf{y}=\left(y_{1},y_{2},\cdots,y_{T}\right)^{\prime}$ be the vector
of observations for the target variable. Suppose we have a vector of pre-
selected covariates denoted by
$\mathbf{z}_{t}=\left(z_{1t},z_{2t},\cdots,z_{mt}\right)^{\prime}$.
Additionally, there is a vector of covariates denoted by
$\mathbf{x}_{t}=\left(x_{1t},x_{2t},\cdots,x_{nt}\right)^{\prime}$, from which
we aim to select the relevant ones for the target variable using the Lasso
procedure. We can further stack the observations for $\mathbf{z}_{t}$ and
$\mathbf{x}_{t}$ in matrices
$\mathbf{Z}=\left(\mathbf{z}_{1},\mathbf{z}_{2},\cdots,\mathbf{z}_{T}\right)^{\prime}$
and
$\mathbf{X}=\left(\mathbf{x}_{1},\mathbf{x}_{2},\cdots,\mathbf{x}_{T}\right)^{\prime}$,
respectively. For a given value of the tuning parameter, $\lambda$, the Lasso
problem can be written as:

$\left(\hat{\boldsymbol{\delta}}(\lambda),\hat{\boldsymbol{\beta}}(\lambda)\right)^{\prime}=\text{argmin}_{\mathbf{b}_{z},\mathbf{b}}\left\\{\left(\mathbf{y}-\mathbf{Z}\mathbf{b}_{z}-\mathbf{X}\mathbf{b}\right)^{\prime}\left(\mathbf{y}-\mathbf{Z}\mathbf{b}_{z}-\mathbf{X}\mathbf{b}\right)+\lambda\|\mathbf{b}\|_{1}\right\\}.$

Partition $\mathbf{X}=\left(\mathbf{X}_{1},\mathbf{X}_{2}\right)$ where
$\mathbf{X}_{1}$ is the matrix of covariates with the corresponding vector of
estimated coefficients, $\boldsymbol{\hat{\beta}}_{1}(\lambda)$, different
from zero and $\mathbf{X}_{2}$ is the matrix of covariates with the
corresponding vector of estimated coefficients,
$\boldsymbol{\hat{\beta}}_{2}(\lambda)$, equal to zero. So,
$\mathbf{X}\boldsymbol{\hat{\beta}}(\lambda)=\mathbf{X}_{1}\boldsymbol{\hat{\beta}}_{1}(\lambda)$.
By the first order conditions we have:

$\mathbf{X}_{1}^{\prime}\left(\mathbf{y}-\mathbf{Z}\hat{\boldsymbol{\delta}}(\lambda)-\mathbf{X}_{1}\hat{\boldsymbol{\beta}}_{1}(\lambda)\right)-\lambda\text{sign}\left(\hat{\boldsymbol{\beta}}_{1}(\lambda)\right)=0,$
(A.1)
$\mathbf{Z}^{\prime}\left(\mathbf{y}-\mathbf{Z}\hat{\boldsymbol{\delta}}(\lambda)-\mathbf{X}_{1}\hat{\boldsymbol{\beta}}_{1}(\lambda)\right)=0.$
(A.2)

and

$-\lambda\mathbf{1}\leq\mathbf{X}_{2}^{\prime}\left(\mathbf{y}-\mathbf{Z}\hat{\boldsymbol{\delta}}(\lambda)-\mathbf{X}_{1}\hat{\boldsymbol{\beta}}_{1}(\lambda)\right)\leq\lambda\mathbf{1},$
(A.3)

Where $\mathbf{1}$ represents a vector of ones. We can further conclude from
Equation (A.2) that:

$\hat{\boldsymbol{\delta}}(\lambda)=\left(\mathbf{Z}^{\prime}\mathbf{Z}\right)^{-1}\mathbf{Z}^{\prime}\left(\mathbf{y}-\mathbf{X}_{1}\hat{\boldsymbol{\beta}}_{1}(\lambda)\right).$
(A.4)

By substituting $\hat{\boldsymbol{\delta}}(\lambda)$ from (A.4) into (A.1), we
have

$\mathbf{X}_{1}^{\prime}\left(\mathbf{y}-\mathbf{X}_{1}\hat{\boldsymbol{\beta}}_{1}(\lambda)-\mathbf{Z}\left(\mathbf{Z}^{\prime}\mathbf{Z}\right)^{-1}\mathbf{Z}^{\prime}\left(\mathbf{y}-\mathbf{X}_{1}\hat{\boldsymbol{\beta}}_{1}(\lambda)\right)\right)-\lambda\text{sign}\left(\hat{\boldsymbol{\beta}}_{1}(\lambda)\right)=0.$

We can further write this as:

$\mathbf{X}_{1}^{\prime}\left(\mathbf{I}-\mathbf{Z}\left(\mathbf{Z}^{\prime}\mathbf{Z}\right)^{-1}\mathbf{Z}^{\prime}\right)\left(\mathbf{y}-\mathbf{X}_{1}\hat{\boldsymbol{\beta}}_{1}(\lambda)\right)-\lambda\text{sign}\left(\hat{\boldsymbol{\beta}}_{1}(\lambda)\right)=0.$

Therefore,

$\tilde{\mathbf{X}}_{1}^{\prime}\left(\tilde{\mathbf{y}}-\tilde{\mathbf{X}}_{1}\hat{\boldsymbol{\beta}}_{1}(\lambda)\right)-\lambda\text{sign}\left(\hat{\boldsymbol{\beta}}_{1}(\lambda)\right)=0,$
(A.5)

where $\tilde{\mathbf{X}}_{1}=\mathbf{M}_{Z}\mathbf{X}_{1}$,
$\tilde{\mathbf{y}}=\mathbf{M}_{Z}\mathbf{y}$ and
$\mathbf{M}_{Z}=\mathbf{I}-\mathbf{Z}\left(\mathbf{Z}^{\prime}\mathbf{Z}\right)^{-1}\mathbf{Z}^{\prime}$.

Similarly, by substituting $\hat{\boldsymbol{\delta}}(\lambda)$ from (A.4)
into (A.3), we have

$-\lambda\mathbf{1}\leq\tilde{\mathbf{X}}_{2}^{\prime}\left(\tilde{\mathbf{y}}-\tilde{\mathbf{X}}_{1}\hat{\boldsymbol{\beta}}_{1}(\lambda)\right)\leq\lambda\mathbf{1}.$
(A.6)

Note that (A.5) and (A.6) are the first order conditions of the following
Lasso problem:

$\hat{\boldsymbol{\beta}}(\lambda)=\text{argmin}_{\mathbf{b}}\left\\{\left(\tilde{\mathbf{y}}-\tilde{\mathbf{X}}\mathbf{b}\right)^{\prime}\left(\tilde{\mathbf{y}}-\tilde{\mathbf{X}}\mathbf{b}\right)+\lambda||\mathbf{b}||_{1}\right\\}.$
(A.7)

Therefore, we can first obtain the estimator of the vector coefficients for
$\mathbf{X}$, $\hat{\boldsymbol{\beta}}(\lambda)$, by solving the Lasso
problem given by (A.7) and then estimate the vector of coefficients for
$\mathbf{Z}$, $\hat{\boldsymbol{\delta}}(\lambda)$, by using Equation (A.4).

Online Supplement

to

High-dimensional forecasting with known knowns and known unknowns

by

M. Hashem Pesaran

University of Southern California, USA and Trinity College, Cambridge, UK

Ron P. Smith

Birkbeck, University of London, United Kingdom

##  Appendix S-1 Introduction

This online supplement presents the details of the methods used to forecast UK
inflation, the variables selected and the results obtained. For this purpose
the framework set out in the paper can be summarised as:

$\pi_{t+h}=c_{h}+\mathbf{a}_{h}^{\prime}\mathbf{z}_{t}+\sum_{j=1}^{K}\beta_{jh}I(j\in
DGP)x_{jt}+u_{h,t+h}.$ (S.1)

The target variable is average annual UK inflation, labelled DPUK4, defined as
$\pi_{t+h}=100\times\log(p_{t+h}/p_{t+h-4}),$ where $p_{t}$ is the UK consumer
price index taken from the IMF International Financial Statistics.121212In
terms of the GVAR data, which report $dp_{t}=\log(p_{t}/p_{t-1})$ where
$p_{t}$ is the CPI, $\pi_{t}$ is defined by
$\pi_{t}=100\times(dp_{t}+dp_{t-1}+dp_{t-2}+dp_{t-3})$. $\mathbf{z}_{t}$ is
the vector of pre-selected variables which we set to ($\pi_{t},\Delta\pi_{t})$
or ($\pi_{t},\Delta\pi_{t}$,$\pi_{t}^{\ast}$, and $\Delta\pi_{t}^{\ast}$). The
active set, $x_{jt},$ $j=1,2,...,K,$ is the list of variables from which
selection is made.

The active set consists of the 26 variables detailed in Table 1 of the paper
plus their first differences (in total $K=52$). As noted in the paper, whereas
including a current and lagged value is equivalent to including a current
value and its change in estimation, it is not in the selection. We denote
$\mathcal{A}_{m}^{h}$ as the active set corresponding to model $m$ among the 9
models that are examined. For each horizon, selection proceeds recursively
using an expanding window starting in 1979q2 initially ending in 2019q1, for
$h=4,$ 2019q3 for $h=2,$ and 2019q4 for $h=4.$ The window then expands one
quarter at a time to 2023q1. We end up with $13$ point forecasts for the
quarters $2020q1-2023q1$. Since the model is re-estimated for each forecast
period and horizon, the variables selected may change by quarter and horizon.

##  Appendix S-2 Models

The first two models set all the $\beta_{jh}=0$ and differ in what is included
in $\mathbf{z}_{t}.$

The $AR2$ model is:

$\pi_{t+h}=c_{h}+a_{1h}\pi_{t}+a_{2h}\Delta\pi_{t}+u_{h,t+h}.$

The $ARX$ adds UK-specific foreign inflation, $\pi_{t}^{\ast}$ and its
change$:$

$\pi_{t+h}=c_{h}+a_{1h}\pi_{t}+a_{2h}\Delta\pi_{t}+a_{3h}\pi_{t}^{\ast}+a_{4h}\Delta\pi_{t}^{\ast}+u_{h,t+h}$

The next model sets $\mathbf{a}_{h}^{\prime}=0$ and just selects using Lasso,
which performs a variable selection from the whole active set of $52$
variables inclusive of $\pi_{t},\Delta\pi_{t}$,$\pi_{t}^{\ast}$, and
$\Delta\pi_{t}^{\ast}$. All covariates are standardized before solving the
following minimization problem:

$\min_{\boldsymbol{\beta}\in\mathbb{R}^{|\mathcal{A}_{\text{Lasso}}^{h}|}}\frac{1}{2T}\sum_{t=1}^{T}\left(\pi_{t+h}-c_{h}-\sum_{j\in\mathcal{A}_{\text{Lasso}}^{h}}\tilde{x}_{tj}\beta_{j}\right)^{2}+\lambda\sum_{j\in\mathcal{A}_{\text{Lasso}}^{h}}\left\|\beta_{j}\right\|_{1},$

where $\tilde{x}_{tj}=(x_{jt}-\bar{x}_{jT})/s_{jT}$,
$\bar{x}_{jT}=T^{-1}\sum_{t=1}^{T}x_{jt}$ and
$s_{jT}=T^{-1}\sum_{t=1}^{T}\left(x_{jt}-\bar{x}_{jT}\right)$.
$|\mathcal{A}_{\text{Lasso}}^{h}|$ represents the cardinality of the active
set, and $||\cdot||_{1}$ is the $\ell_{1}$-norm. The regularization parameter
$\hat{\lambda}_{hT}$ is chosen via 10-fold cross-validation (CV). As discussed
in the paper we adapt the standard CV procedures to take account of the
persistence and changing variance of time series data. In the standard
procedure the CV subsets (folds) are typically chosen randomly, but since
these are time series where order matters, we retain the order of the time
series data in construction of subsets using all the data, not dropping
observations between subsets. In addition, the standard 10-fold procedure
chooses the $\hat{\lambda}_{hT}$ that minimises the pooled MSE over the ten
subsets. But when variances differ substantially over subsets, pooling is not
appropriate, instead we follow Chudik, Kapetanios, and Pesaran (2018, CKP),
and use the average of the $\hat{\lambda}_{hT}$ chosen in each subset. Full
details are provided in the online simulation appendix to CKP (2018).

We considered a number of Lasso models where we did not apply selection to the
AR2 or ARX variables which were included in the $\boldsymbol{z}_{t}$. Then
Lasso was applied to the active set having removed the effect of
$\boldsymbol{z}_{t}.$ This gives models AR2-LASSO; ARX-LASSO. We now consider
GOCMT-based Models. Like OCMT, GOCMT allows for the multiple testing nature of
the procedure ($K$ separate tests - with $K$ large) by increasing the level of
significance with $K$. But it also includes the principal component of the
standardised values of the active set, $\boldsymbol{\hat{\varkappa}}_{t},$ to
allow for the correlations between the variables. In the first stage, $K$
separate OLS regressions are computed entering the variables from the active
set one at a time, together with any $\boldsymbol{z}_{t}$ and
$\boldsymbol{\hat{\varkappa}}_{t}$:

$y_{t+h}=c_{h}+\mathbf{a}_{h}^{\prime}\boldsymbol{z}_{t}+\mathbf{b}_{h}^{\prime}\boldsymbol{\hat{\varkappa}}_{t}+\phi_{jh}x_{jt}+e_{j,h,t+h}\text{,
}t=1,2,...,T\text{, for }j=1,2,...,K,$

Denote the $t$-ratio of $\phi_{jh}$ by $t_{\hat{\phi}_{j,\left(1\right)}}$.
Then variable $j$ is selected if

$\widehat{\mathcal{J}}_{j,(1)}=I\left[\left|t_{\hat{\phi}_{j,\left(1\right)}}\right|>c_{p}(K)\right],\text{
for }j=1,2,...,K,$

where the critical value is given by

$c_{p}(K,\delta)=\Phi^{-1}\left(1-\frac{p}{2K^{\delta}}\right).$

$p$ is the nominal size set to $5\%$, $\Phi^{-1}(\cdot)$ is the inverse of a
standard normal distribution function and $\delta$ is a fixed constant set
close to $1$. We experiment with two values for $\delta,$ $\delta=1$ and
$\delta=1.5$.

The principal component $\boldsymbol{\hat{\varkappa}}_{t}$ is used just to
control for correlations across the variables in the active set at the
selection stage, it is not included in the final forecast regression. The
AR2-OCMT model includes the AR2 in the $\boldsymbol{z}_{t}$ and applies GOCMT
to the rest of the active set, similarly for ARX-OCMT.

Thus we have a set of 9 models: (1) AR2, (2) ARX, (3) LASSO, (4) AR2-LASSO,
(5) ARX-LASSO, (6) AR2-OCMT ($\delta=1$), (7) ARX-OCMT ($\delta=1$), (8)
AR2-OCMT ($\delta=1.5$), (9) ARX-OCMT ($\delta=1.5$).

##  Appendix S-3 Variable Selection

This section sets out the variables chosen by each selection method for each
horizon by the quarter in which the sample ended. The abbreviations for the
variables are given in Table 1, in the paper.

###  S-3.1 Variables selected by OCMT

####  S-3.1.1 For h=1 quarter ahead models

This sub-section sets out the variables chosen by OCMT for $h=1$.

AR2-OCMT selects DWUK for samples ending in 2021q2, 2021q4 and 2022q1 and no
other additional variables. ARX-OCMT ($\delta=1$), AR2-OCMT ($\delta=1.5$),
and ARX-OCMT ($\delta=1.5$) do not select any additional variables.

####  S-3.1.2 For h=2 quarter ahead models

This sub-section sets out the variables chosen by OCMT for $h=2$.

Table S.1: Selected Variables by AR2-OCMT ($\delta=1$) 2q Ahead # |  | Ending Year
---|---|---
|  | 2019q3 | 2019q4 | 2020q1 | 2020q2 | 2020q3 | 2020q4
1 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4
2 |  | DEPUK4 | DEPUK4 | DEPUK4 | DEPUK4 | DEPUK4 | DEPUK4
3 |  | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | DWUK4
|  | 2021q1 | 2021q2 | 2021q3 | 2021q4 | 2022q1 | 2022q2
1 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4
2 |  | DEPUK4 | DEPUK4 | DEPUK4 | DEPUK4 | DEPUK4 | DEPUK4
3 |  | DWUK4 | DWUK4 | DWUK4 | DWUK4 | DWUK4 | DWUK4
|  | 2022q3 | 2022q4 | 2023q1 |  |  |
1 |  | DMUK4 | DMUK4 | DMUK4 |  |  |
2 |  | DWUK4 | DWUK4 | DWUK4 |  |  |
|  | $\delta=1$

ARX-OCMT $\delta=1$ selects just DMUK4 for every sample.

Table S.2: Selected Variables by AR2-OCMT ($\delta=1.5$) 2q Ahead # |  | Ending Year
---|---|---
|  | 2019q3 | 2019q4 | 2020q1 | 2020q2 | 2020q3 | 2020q4
1 |  | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | DMUK4
|  | 2021q1 | 2021q2 | 2021q3 | 2021q4 | 2022q1 | 2022q2
1 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4
2 |  | $\cdot$ | DWUK4 | DWUK4 | DWUK4 | DWUK4 | DWUK4
|  | 2022q3 | 2022q4 | 2023q1 |  |  |
1 |  | DWUK4 | $\cdot$ | $\cdot$ |  |  |
2 |  | $\cdot$ | $\cdot$ | $\cdot$ |  |  |
|  | $\delta=1.5$

ARX-OCMT $\delta=1.5$ selects DMUK4 for 2020q4-2022q2.

####  S-3.1.3 For h=4 quarter ahead models

This sub-section sets out the variables chosen by OCMT selection method for
$h=4$.

Table S.3: Selected Variables by AR2-OCMT ($\delta=1$) 4q Ahead # |  | Ending Year
---|---|---
|  | 2019q1 | 2019q2 | 2019q3 | 2019q4 | 2020q1 | 2020q2
1 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4
2 |  | DEPUK4 | DEPUK4 | DEPUK4 | DEPUK4 | DEPUK4 | DEPUK4
3 |  | DPSUK4 | DPSUK4 | DPSUK4 | DPSUK4 | DPSUK4 | DPSUK4
|  | 2020q3 | 2020q4 | 2021q1 | 2021q2 | 2021q3 | 2021q4
1 |  | DMUK4 | DMUK4 | DMUK4 | DDYUK4 | DEMUK4 | LRUK4
2 |  | DEPUK4 | DEPUK4 | DEPUK4 | DGAPUK8 | LRUK4 | DMUK4
3 |  | DPSUK4 | DPSUK4 | DPSUK4 | DGAPUK12 | DMUK4 | DPMAT4
4 |  | $\cdot$ | $\cdot$ | $\cdot$ | DEMUK4 | DPMAT4 | DPSUK4
5 |  | $\cdot$ | $\cdot$ | $\cdot$ | RUK4 | DPSUK4 | LRUS4
6 |  | $\cdot$ | $\cdot$ | $\cdot$ | LRUK4 | LRUS4 | DWUK4
7 |  | $\cdot$ | $\cdot$ | $\cdot$ | DMUK4 | DWUK4 | $\cdot$
8 |  | $\cdot$ | $\cdot$ | $\cdot$ | DPMAT4 | $\cdot$ | $\cdot$
9 |  | $\cdot$ | $\cdot$ | $\cdot$ | DPSUK4 | $\cdot$ | $\cdot$
10 |  | $\cdot$ | $\cdot$ | $\cdot$ | RUS4 | $\cdot$ | $\cdot$
11 |  | $\cdot$ | $\cdot$ | $\cdot$ | LRUS4 | $\cdot$ | $\cdot$
12 |  | $\cdot$ | $\cdot$ | $\cdot$ | DWUK4 | $\cdot$ | $\cdot$
|  | 2022q1 | 2022q2 | 2022q3 | 2022q4 | 2023q1 |
1 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 |
2 |  | DPMAT4 | DPMAT4 | DPMAT4 | DPMAT4 | DPMAT4 |
3 |  | DPSUK4 | DPSUK4 | DPSUK4 | DPSUK4 | DWUK4 |
4 |  | LRUS4 | DWUK4 | DWUK4 | DWUK4 | $\cdot$ |
5 |  | DWUK4 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ |
|  | $\delta=1$
Table S.4: Selected Variables by ARX-OCMT ($\delta=1$) 4q Ahead # |  | Ending Year
---|---|---
|  | 2019q1 | 2019q2 | 2019q3 | 2019q4 | 2020q1 | 2020q2
1 |  | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4
2 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4
|  | 2020q3 | 2020q4 | 2021q1 | 2021q2 | 2021q3 | 2021q4
1 |  | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4
2 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4
|  | 2022q1 | 2022q2 | 2022q3 | 2022q4 | 2023q1 |
1 |  | DEMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 |
2 |  | DMUK4 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ |
3 |  | DPMAT4 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ |
|  | $\delta=1$
Table S.5: Selected Variables by AR2-OCMT ($\delta=1.5$) 4q Ahead # |  | Ending Year
---|---|---
|  | 2019q1 | 2019q2 | 2019q3 | 2019q4 | 2020q1 | 2020q2
1 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4
|  | 2020q3 | 2020q4 | 2021q1 | 2021q2 | 2021q3 | 2021q4
1 |  | DMUK4 | DMUK4 | DMUK4 | DGAPUK8 | DMUK4 | DMUK4
2 |  | $\cdot$ | $\cdot$ | $\cdot$ | DGAPUK12 | DPSUK4 | DPMAT4
3 |  | $\cdot$ | $\cdot$ | $\cdot$ | DEMUK4 | LRUS4 | DPSUK4
4 |  | $\cdot$ | $\cdot$ | $\cdot$ | LRUK4 | $\cdot$ | $\cdot$
5 |  | $\cdot$ | $\cdot$ | $\cdot$ | DMUK4 | $\cdot$ | $\cdot$
6 |  | $\cdot$ | $\cdot$ | $\cdot$ | DPMAT4 | $\cdot$ | $\cdot$
7 |  | $\cdot$ | $\cdot$ | $\cdot$ | DPSUK4 | $\cdot$ | $\cdot$
8 |  | $\cdot$ | $\cdot$ | $\cdot$ | LRUS4 | $\cdot$ | $\cdot$
|  | 2022q1 | 2022q2 | 2022q3 | 2022q4 | 2023q1 |
1 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 |
2 |  | DPMAT4 | DPMAT4 | DPMAT4 | DWUK4 | DWUK4 |
3 |  | DPSUK4 | DWUK4 | DWUK4 | $\cdot$ | $\cdot$ |
4 |  | DWUK4 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ |
|  | $\delta=1.5$
Table S.6: Selected Variables by ARX-OCMT ($\delta=1.5$) 4q AheadA # |  | Ending Year
---|---|---
|  | 2019q1 | 2019q2 | 2019q3 | 2019q4 | 2020q1 | 2020q2
1 |  | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4
2 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4
|  | 2020q3 | 2020q4 | 2021q1 | 2021q2 | 2021q3 | 2021q4
1 |  | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4
2 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4
|  | 2022q1 | 2022q2 | 2022q3 | 2022q4 | 2023q1 |
1 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 |
|  | $\delta=1.5$

###  S-3.2 Variables selected by Lasso

####  S-3.2.1 For h=1 quarter ahead models

This sub-section sets out the variables chosen by Lasso for $h=1$.

Table S.7: Selected Variables by Lasso 1q Ahead

|  | Ending Year
---|---|---
# |  | 2019q4 | 2020q1 | 2020q2 | 2020q3 | 2020q4 | 2021q1 | 2021q2
1 |  | DPUK4 | DPUK4 | DPUK4 | DPUK4 | DPUK4 | DPUK4 | DPUK4
2 |  | DDPUK4 | DDPUK4 | DDPUK4 | DDPUK4 | DDPUK4 | DDPUK4 | DDPUK4
3 |  | DLRUK4 | DLRUK4 | DRUK4 | DLRUK4 | DLRUK4 | DEMUK4 | DLRUK4
4 |  | DMUK4 | DMUK4 | DLRUK4 | DMUK4 | DMUK4 | DRUK4 | DMUK4
5 |  | DPMUK4 | DPMUK4 | DMUK4 | DPMUK4 | DPMUK4 | DLRUK4 | DPMUK4
6 |  | DPSUK4 | DPSUK4 | DDPOIL4 | DPSUK4 | DPSUK4 | DMUK4 | DPSUK4
7 |  | DDPSUK4 | DDPSUK4 | DPMUK4 | DDPSUK4 | DWUK4 | DDEQUK4 | DDPSUK4
8 |  | DWUK4 | DWUK4 | DPSUK4 | DWUK4 | $\cdot$ | DPMUK4 | DWUK4
9 |  | $\cdot$ | $\cdot$ | DDPSUK4 | $\cdot$ | $\cdot$ | DPSUK4 | $\cdot$
10 |  | $\cdot$ | $\cdot$ | DDYSUK4 | $\cdot$ | $\cdot$ | DDPSUK4 | $\cdot$
11 |  | $\cdot$ | $\cdot$ | DGAPSUK12 | $\cdot$ | $\cdot$ | DWUK4 | $\cdot$
12 |  | $\cdot$ | $\cdot$ | DWUK4 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
$\lambda$ |  | 0.1094 | 0.1093 | 0.0736 | 0.1098 | 0.1101 | 0.0741 | 0.1099
|  |  |  |  |  |  |  |
# |  | 2021q3 | 2021q4 | 2022q1 | 2022q2 | 2022q3 | 2022q4 | 2023q1
1 |  | DPUK4 | DPUK4 | DPUK4 | DPUK4 | DPUK4 | DPUK4 | DPUK4
2 |  | DDPUK4 | DDPUK4 | DDPUK4 | DDPUK4 | DDPUK4 | DDPUK4 | DDPUK4
3 |  | DLRUK4 | DLRUK4 | DLRUK4 | DLRUK4 | DLRUK4 | DLRUK4 | DLRUK4
4 |  | DMUK4 | DMUK4 | DMUK4 | DPMUK4 | DPMUK4 | DPMUK4 | DPMUK4
5 |  | DPMUK4 | DPMUK4 | DPMUK4 | DPSUK4 | DPSUK4 | DPSUK4 | DPSUK4
6 |  | DPSUK4 | DPSUK4 | DPSUK4 | DDPSUK4 | DDPSUK4 | DDPSUK4 | DDPSUK4
7 |  | DDPSUK4 | DDPSUK4 | DDPSUK4 | DWUK4 | DWUK4 | DWUK4 | DWUK4
8 |  | DWUK4 | DWUK4 | DWUK4 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
$\lambda$ |  | 0.1096 | 0.1091 | 0.1091 | 0.1104 | 0.1128 | 0.1159 | 0.1184

_Note_ : ${\footnotesize\hat{\lambda}}_{hT}$ is the estimate of the Lasso
penalty parameter computed by 10-fold cross-validation.

Table S.8: Selected Variables by AR2-Lasso 1q Ahead

|  | Ending Year
---|---|---
# |  | 2019q4 | 2020q1 | 2020q2 | 2020q3 | 2020q4 | 2021q1 | 2021q2
1 |  | DEMUK4 | DMUK4 | DEMUK4 | DEMUK4 | DMUK4 | DPSUK4 | DMUK4
2 |  | DMUK4 | DPSUK4 | DLRUK4 | DDWUK4 | DPSUK4 | DDPSUK4 | DDEQUK4
3 |  | DPSUK4 | DDPSUK4 | DMUK4 | DLRUK4 | DDPSUK4 | DWUK4 | DPSUK4
4 |  | DDPSUK4 | DGAPSUK8 | DPMUK4 | DMUK4 | DWUK4 | $\cdot$ | DDPSUK4
5 |  | DGAPSUK8 | DGAPSUK12 | DPSUK4 | DDEQUK4 | $\cdot$ | $\cdot$ | DWUK4
6 |  | DGAPSUK12 | DWUK4 | DDPSUK4 | DPMUK4 | $\cdot$ | $\cdot$ | $\cdot$
7 |  | DWUK4 | $\cdot$ | DGAPSUK8 | DPSUK4 | $\cdot$ | $\cdot$ | $\cdot$
8 |  | $\cdot$ | $\cdot$ | DGAPSUK12 | DDPSUK4 | $\cdot$ | $\cdot$ | $\cdot$
9 |  | $\cdot$ | $\cdot$ | DWUK4 | DWUK4 | $\cdot$ | $\cdot$ | $\cdot$
$\hat{\lambda}_{hT}$ |  | 0.0883 | 0.0913 | 0.0823 | 0.0767 | 0.0899 | 0.0955 | 0.0919
|  |  |  |  |  |  |  |
# |  | 2021q3 | 2021q4 | 2022q1 | 2022q2 | 2022q3 | 2022q4 | 2023q1
1 |  | DMUK4 | DLRUK4 | DLRUK4 | DLRUK4 | DLRUK4 | DLRUK4 | DLRUK4
2 |  | DDEQUK4 | DMUK4 | DMUK4 | DDEQUK4 | DMUK4 | DDEQUK4 | DDEQUK4
3 |  | DPSUK4 | DDEQUK4 | DDEQUK4 | DPMUK4 | DDEQUK4 | DPMUK4 | DPMUK4
4 |  | DDPSUK4 | DPMUK4 | DPMUK4 | DPSUK4 | DPMUK4 | DPSUK4 | DPSUK4
5 |  | DWUK4 | DPSUK4 | DPSUK4 | DDPSUK4 | DPSUK4 | DDPSUK4 | DDPSUK4
6 |  | $\cdot$ | DDPSUK4 | DDPSUK4 | DWUK4 | DDPSUK4 | DWUK4 | DWUK4
7 |  | $\cdot$ | DWUK4 | DWUK4 | $\cdot$ | DWUK4 | $\cdot$ | $\cdot$
$\hat{\lambda}_{hT}$ |  | 0.0952 | 0.0927 | 0.0878 | 0.0874 | 0.0787 | 0.0904 | 0.0840

_Note_ : ${\footnotesize\hat{\lambda}}_{hT}$ is the estimate of the Lasso
penalty parameter computed by 10-fold cross-validation.

Table S.9: Selected Variables by ARX-Lasso 1q Ahead

|  | Ending Year
---|---|---
# |  | 2019q4 | 2020q1 | 2020q2 | 2020q3 | 2020q4 | 2021q1 | 2021q2
1 |  | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4
2 |  | DVUK4 | DVUK4 | DVUK4 | DVUK4 | DDWUK4 | DVUK4 | DVUK4
3 |  | DDWUK4 | DDWUK4 | DDWUK4 | DDWUK4 | DLRUK4 | DDWUK4 | DDWUK4
4 |  | DLRUK4 | DLRUK4 | DLRUK4 | DLRUK4 | DMUK4 | DLRUK4 | DLRUK4
5 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DDEQUK4 | DMUK4 | DMUK4
6 |  | DYCHINA4 | DYCHINA4 | DDEQUK4 | DDEQUK4 | $\cdot$ | DDEQUK4 | DDEQUK4
7 |  | DGAPSUK8 | DGAPSUK8 | DGAPSUK8 | $\cdot$ | $\cdot$ | DPMUK4 | DPUS4
8 |  | DGAPSUK12 | DGAPSUK12 | DGAPSUK12 | $\cdot$ | $\cdot$ | DPUS4 | DWUK4
9 |  | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | DWUK4 | $\cdot$
$\hat{\lambda}_{hT}$ |  | 0.0659 | 0.0678 | 0.0665 | 0.0702 | 0.0796 | 0.0562 | 0.0639
|  |  |  |  |  |  |  |
# |  | 2021q3 | 2021q4 | 2022q1 | 2022q2 | 2022q3 | 2022q4 | 2023q1
1 |  | DEMUK4 | DEMUK4 | DDYUK4 | DDYUK4 | DDYUK4 | DEMUK4 | DEMUK4
2 |  | DLRUK4 | DLRUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DDVUK4 | DDVUK4
3 |  | DMUK4 | DMUK4 | DDVUK4 | DDVUK4 | DDVUK4 | DDWUK4 | DLRUK4
4 |  | DDEQUK4 | DDEQUK4 | DDUUK | DDUUK | DDUUK | DLRUK4 | DMUK4
5 |  | $\cdot$ | DWUK4 | DLRUK4 | DLRUK4 | DLRUK4 | DMUK4 | DDEQUK4
6 |  | $\cdot$ | $\cdot$ | DMUK4 | DMUK4 | DMUK4 | DDEQUK4 | DPMUK4
7 |  | $\cdot$ | $\cdot$ | DDEQUK4 | DDEQUK4 | DDEQUK4 | DDPMAT4 | DYCHINA4
8 |  | $\cdot$ | $\cdot$ | DDPMAT4 | DDPMAT4 | DDPMAT4 | DPMUK4 | DWUK4
9 |  | $\cdot$ | $\cdot$ | DPMUK4 | DPMUK4 | DPMUK4 | DYCHINA4 | $\cdot$
10 |  | $\cdot$ | $\cdot$ | DYCHINA4 | DYCHINA4 | DYCHINA4 | DDPUS4 | $\cdot$
11 |  | $\cdot$ | $\cdot$ | DPUS4 | DDPUS4 | DDPUS4 | DWUK4 | $\cdot$
12 |  | $\cdot$ | $\cdot$ | DWUK4 | DWUK4 | DWUK4 | $\cdot$ | $\cdot$
$\hat{\lambda}_{hT}$ |  | 0.0752 | 0.0714 | 0.0422 | 0.0457 | 0.0456 | 0.0550 | 0.0624

_Note_ : ${\footnotesize\hat{\lambda}}_{hT}$ is the estimate of the Lasso
penalty parameter computed by 10-fold cross-validation.

####  S-3.2.2 For h=2 quarter ahead models

This sub-section sets out the variables chosen by Lasso for $h=2$.

Table S.10: Selected Variables by Lasso 2q Ahead

|  | Ending Year
---|---|---
# |  | 2019q3 | 2019q4 | 2020q1 | 2020q2 | 2020q3 | 2020q4 | 2021q1 | 2021q2
1 |  | DPUK4 | DPUK4 | DPUK4 | DPUK4 | DPUK4 | DPUK4 | DPUK4 | DPUK4
2 |  | DDPUK4 | DDPUK4 | DDPUK4 | DDPUK4 | DDPUK4 | DDPUK4 | DDPUK4 | DDPUK4
3 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4
4 |  | DPSUK4 | DPSUK4 | DPSUK4 | DPSUK4 | DPSUK4 | DPSUK4 | DPSUK4 | DPMUK4
5 |  | DWUK4 | DWUK4 | DWUK4 | DWUK4 | DWUK4 | DWUK4 | DWUK4 | DPSUK4
6 |  | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | DWUK4
$\hat{\lambda}_{hT}$ |  | 0.3113 | 0.3114 | 0.2494 | 0.2813 | 0.3133 | 0.2521 | 0.2529 | 0.2521
|  |  |  |  |  |  |  |  |
# |  | 2021q3 | 2021q4 | 2022q1 | 2022q2 | 2022q3 | 2022q4 | 2023q1 |
1 |  | DPUK4 | DPUK4 | DPUK4 | DPUK4 | DPUK4 | DPUK4 | DPUK4 |
2 |  | DDPUK4 | DDPUK4 | DDPUK4 | DDPUK4 | DDPUK4 | DDPUK4 | DDPUK4 |
3 |  | DLRUK4 | DMUK4 | DMUK4 | DMUK4 | DLRUK4 | DLRUK4 | DLRUK4 |
4 |  | DMUK4 | DPMUK4 | DPMUK4 | DPMUK4 | DMUK4 | DMUK4 | DMUK4 |
5 |  | DPMUK4 | DPSUK4 | DPSUK4 | DPSUK4 | DPMUK4 | DPMUK4 | DPMUK4 |
6 |  | DPSUK4 | DWUK4 | DWUK4 | DWUK4 | DPSUK4 | DPSUK4 | DPSUK4 |
7 |  | DWUK4 | $\cdot$ | $\cdot$ | $\cdot$ | DDPSUK4 | DDPSUK4 | DDPSUK4 |
8 |  | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | DWUK4 | DPUS4 | DPUS4 |
9 |  | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | DWUK4 | DWUK4 |
$\hat{\lambda}_{hT}$ |  | 0.2196 | 0.2490 | 0.2475 | 0.3112 | 0.2221 | 0.2297 | 0.2366 |

_Note_ : ${\footnotesize\hat{\lambda}}_{hT}$ is the estimate of the Lasso
penalty parameter computed by 10-fold cross-validation.

Table S.11: Selected Variables by AR2-LASSO 2q Ahead

|  | Ending Year
---|---|---
# |  | 2019q3 | 2019q4 | 2020q1 | 2020q2 | 2020q3 | 2020q4
1 |  | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4
2 |  | DVUK4 | DVUK4 | DVUK4 | DVUK4 | DVUK4 | DVUK4
3 |  | DDWUK4 | DDWUK4 | DDWUK4 | DDWUK4 | DDWUK4 | DDUUK
4 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DDWUK4
5 |  | DPMAT4 | DPMUK4 | DPMAT4 | DPMAT4 | DPMAT4 | DLRUK4
6 |  | DPMUK4 | DDPMUK4 | DPMUK4 | DPMUK4 | DPMUK4 | DMUK4
7 |  | DDPMUK4 | DPSUK4 | DDPMUK4 | DDPMUK4 | DDPMUK4 | DPMAT4
8 |  | DEPUK4 | DDPSUK4 | DEPUK4 | DEPUK4 | DEPUK4 | DDPMETAL4
9 |  | DPSUK4 | DDYCHINA4 | DPSUK4 | DPSUK4 | DPSUK4 | DDPMUK4
10 |  | DDPSUK4 | DGAPSUK8 | DDPSUK4 | DDPSUK4 | DDPSUK4 | DEPUK4
11 |  | DYCHINA4 | DWUK4 | DDYCHINA4 | DDYCHINA4 | DDYCHINA4 | DPSUK4
12 |  | DDYCHINA4 | $\cdot$ | DGAPSUK8 | DGAPSUK8 | DGAPSUK8 | DDPSUK4
13 |  | DGAPSUK8 | $\cdot$ | DWUK4 | DWUK4 | DWUK4 | DGAPSUK8
14 |  | DWUK4 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | DWUK4
$\hat{\lambda}_{hT}$ |  | 0.0966 | 0.1298 | 0.1020 | 0.1084 | 0.0955 | 0.0941
|  |  |  |  |  |  |
# |  | 2021q1 | 2021q2 | 2021q3 | 2021q4 | 2022q1 | 2022q2
1 |  | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4
2 |  | DVUK4 | DVUK4 | DVUK4 | DDWUK4 | DDUUK4 | DDUUK4
3 |  | DDUUK | DDUUK | DDUUK | DLRUK4 | DLRUK4 | DLRUK4
4 |  | DDWUK4 | DDWUK4 | DDWUK4 | DMUK4 | DMUK4 | DMUK4
5 |  | DLRUK4 | DLRUK4 | DLRUK4 | DPMAT4 | DPMAT4 | DPMAT4
6 |  | DMUK4 | DMUK4 | DMUK4 | DPMUK4 | DPMUK4 | DPMUK4
7 |  | DPMAT4 | DPMAT4 | DPMAT4 | DDPMUK4 | DDPMUK4 | DDPMUK4
8 |  | DDPMETAL4 | DDPMETAL4 | DDPMUK4 | DPSUK4 | DPSUK4 | DPSUK4
9 |  | DDPMUK4 | DDPMUK4 | DEPUK4 | DDPSUK4 | DDPSUK4 | DDPSUK4
10 |  | DEPUK4 | DEPUK4 | DPSUK4 | DWUK4 | DWUK4 | DWUK4
11 |  | DPSUK4 | DPSUK4 | DDPSUK4 | $\cdot$ | $\cdot$ | $\cdot$
12 |  | DDPSUK4 | DDPSUK4 | DWUK4 | $\cdot$ | $\cdot$ | $\cdot$
13 |  | DWUK4 | DWUK4 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
$\hat{\lambda}_{hT}$ |  | 0.1005 | 0.0955 | 0.1043 | 0.1164 | 0.1127 | 0.1237
|  |  |  |  |  |  |
# |  | 2022q3 | 2022q4 | 2023q1 |  |  |
1 |  | DEMUK4 | DEMUK4 | DDUUK4 |  |  |
2 |  | DDUUK4 | DDUUK4 | DMUK4 |  |  |
3 |  | DLRUK4 | DLRUK4 | DPMUK4 |  |  |
4 |  | DMUK4 | DMUK4 | DDPMUK4 |  |  |
5 |  | DPMAT4 | DPMAT4 | DPSUK4 |  |  |
6 |  | DPMUK4 | DPMUK4 | DDPSUK4 |  |  |
7 |  | DDPMUK4 | DDPMUK4 | DWUK4 |  |  |
8 |  | DPSUK4 | DPSUK4 | $\cdot$ |  |  |
9 |  | DDPSUK4 | DDPSUK4 | $\cdot$ |  |  |
10 |  | DWUK4 | DWUK4 | $\cdot$ |  |  |
$\hat{\lambda}_{hT}$ |  | 0.1232 | 0.1252 | 0.1406 |  |  |

_Note_ : ${\footnotesize\hat{\lambda}}_{hT}$ is the estimate of the Lasso
penalty parameter computed by 10-fold cross-validation.

Table S.12: Selected Variables by ARX-Lasso 2q Ahead

# |  | Ending Year
---|---|---
|  | 2019q3 | 2019q4 | 2020q1 | 2020q2 | 2020q3 | 2020q4
1 |  | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4
2 |  | DVUK4 | DVUK4 | DVUK4 | DVUK4 | DVUK4 | DVUK4
3 |  | DDWUK4 | DDWUK4 | DDWUK4 | DDWUK4 | DDWUK4 | DDUUK4
4 |  | DLRUK4 | DLRUK4 | DLRUK4 | DLRUK4 | DLRUK4 | DDWUK4
5 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DLRUK4
6 |  | DPMAT4 | DPMAT4 | DPMAT4 | DPMAT4 | DDPMUK4 | DMUK4
7 |  | DDPMAT4 | DDPMUK4 | DDPMUK4 | DDPMUK4 | DEPUK4 | DPMAT4
8 |  | DPMUK4 | DEPUK4 | DEPUK4 | DEPUK4 | DYCHINA4 | DDPMETAL4
9 |  | DDPMUK4 | DYCHINA4 | DYCHINA4 | DYCHINA4 | DGAPSUK8 | DDPMUK4
10 |  | DEPUK4 | DDYCHINA4 | DDYCHINA4 | DDYCHINA4 | DDPUS4 | DEPUK4
11 |  | DYCHINA4 | DGAPSUK8 | DGAPSUK8 | DGAPSUK8 | $\cdot$ | DYCHINA4
12 |  | DDYCHINA4 | DDPUS4 | DDPUS4 | DDPUS4 | $\cdot$ | DGAPSUK8
13 |  | DGAPSUK8 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | DPUS4
14 |  | DDPUS4 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | DDPUS4
15 |  | DWUK4 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | DWUK4
$\hat{\lambda}_{hT}$ |  | 0.0757 | 0.0925 | 0.0920 | 0.0920 | 0.1036 | 0.0713
|  |  |  |  |  |  |
# |  | 2021q1 | 2021q2 | 2021q3 | 2021q4 | 2022q1 | 2022q2
1 |  | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4
2 |  | DVUK4 | DVUK4 | DVUK4 | DVUK4 | DDUUK | DDUUK
3 |  | DDUUK | DDUUK | DDUUK | DDUUK | DDWUK4 | DLRUK4
4 |  | DDWUK4 | DDWUK4 | DDWUK4 | DDWUK4 | DLRUK4 | DMUK4
5 |  | DLRUK4 | DLRUK4 | DLRUK4 | DLRUK4 | DMUK4 | DPMUK4
6 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DPMAT4 | DDPMUK4
7 |  | DPMAT4 | DPMAT4 | DPMAT4 | DPMAT4 | DPMUK4 | DYCHINA4
8 |  | DDPMETAL4 | DDPMETAL4 | DDPMETAL4 | DDPMETAL4 | DDPMUK4 | DWUK4
9 |  | DDPMUK4 | DDPMUK4 | DDPMUK4 | DDPMUK4 | DEPUK4 | $\cdot$
10 |  | DEPUK4 | DEPUK4 | DEPUK4 | DEPUK4 | DYCHINA4 | $\cdot$
11 |  | DYCHINA4 | DYCHINA4 | DYCHINA4 | DYCHINA4 | DDPUS4 | $\cdot$
12 |  | DPUS4 | DPUS4 | DPUS4 | DPUS4 | DWUK4 | $\cdot$
13 |  | DDPUS4 | DDPUS4 | DDPUS4 | DWUK4 | $\cdot$ | $\cdot$
14 |  | DWUK4 | DWUK4 | DWUK4 | $\cdot$ | $\cdot$ | $\cdot$
$\hat{\lambda}_{hT}$ |  | 0.0699 | 0.0666 | 0.0739 | 0.0921 | 0.0964 | 0.1092
|  |  |  |  |  |  |
# |  | 2022q3 | 2022q4 | 2023q1 |  |  |
1 |  | DEMUK4 | DEMUK4 | DEMUK4 |  |  |
2 |  | DDUUK | DDUUK | DDUUK |  |  |
3 |  | DDWUK4 | DDWUK4 | DDWUK4 |  |  |
4 |  | RUK4 | RUK4 | DLRUK4 |  |  |
5 |  | DLRUK4 | DLRUK4 | DMUK4 |  |  |
6 |  | DMUK4 | DMUK4 | DPMUK4 |  |  |
7 |  | DPMUK4 | DDPMAT4 | DDPMUK4 |  |  |
8 |  | DDPMUK4 | DPMUK4 | DYCHINA4 |  |  |
9 |  | DYCHINA4 | DDPMUK4 | DWUK4 |  |  |
10 |  | DDPUS4 | DYCHINA4 | $\cdot$ |  |  |
11 |  | DWUK4 | DDPUS4 | $\cdot$ |  |  |
12 |  | $\cdot$ | DWUK4 | $\cdot$ |  |  |
$\hat{\lambda}_{hT}$ |  | 0.1046 | 0.0974 | 0.1191 |  |  |

_Note_ : ${\footnotesize\hat{\lambda}}_{hT}$ is the estimate of the Lasso
penalty parameter computed by 10-fold cross-validation.

####  S-3.2.3 For h=4 quarter ahead models

This sub-section sets out the variables chosen by Lasso for $h=4$.

Table S.13: Selected Variables by Lasso 4q Ahead

# |  | Ending Year
---|---|---
|  | 2019q1 | 2019q2 | 2019q3 | 2019q4 | 2020q1 | 2020q2
1 |  | DPUK4 | DPUK4 | DPUK4 | DPUK4 | DPUK4 | DPUK4
2 |  | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4
3 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4
4 |  | DPMAT4 | DPMAT4 | DPMAT4 | DPMAT4 | DPMAT4 | DPMAT4
5 |  | DPSUK4 | DPSUK4 | DPSUK4 | DPSUK4 | DPSUK4 | DPSUK4
6 |  | DWUK4 | DWUK4 | DWUK4 | DWUK4 | DWUK4 | DWUK4
$\hat{\lambda}_{hT}$ |  | 0.3345 | 0.3346 | 0.3823 | 0.3824 | 0.3829 | 0.3610
|  |  |  |  |  |  |
|  | 2020q3 | 2020q4 | 2021q1 | 2021q2 | 2021q3 | 2021q4
1 |  | DPUK4 | DPUK4 | DPUK4 | DPUK4 | DPUK4 | DPUK4
2 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4
3 |  | DPMAT4 | DPMAT4 | DPMAT4 | DPMAT4 | DPSUK4 | DPMAT4
4 |  | DPSUK4 | DPSUK4 | DPSUK4 | DPSUK4 | $\cdot$ | DPSUK4
5 |  | DWUK4 | $\cdot$ | DWUK4 | $\cdot$ | $\cdot$ | DWUK4
$\hat{\lambda}_{hT}$ |  | 0.3874 | 0.3897 | 0.4401 | 0.4394 | 0.4613 | 0.4296
|  |  |  |  |  |  |
|  | 2022q1 | 2022q2 | 2022q3 | 2022q4 | 2023q1 |
1 |  | DPUK4 | DPUK4 | DPUK4 | DPUK4 | DPUK4 |
2 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 |
3 |  | DPMAT4 | DPMAT4 | DPMAT4 | DPMAT4 | DPMAT4 |
4 |  | DPSUK4 | DPSUK4 | DPSUK4 | DPSUK4 | DPSUK4 |
5 |  | DWUK4 | DWUK4 | DWUK4 | DPUS4 | DPUS4 |
6 |  | $\cdot$ | $\cdot$ | $\cdot$ | DWUK4 | DWUK4 |
$\hat{\lambda}_{hT}$ |  | 0.4506 | 0.4632 | 0.4669 | 0.4163 | 0.4240 |

_Note_ : ${\footnotesize\hat{\lambda}}_{hT}$ is the estimate of the Lasso
penalty parameter computed by 10-fold cross-validation.

Table S.14: Selected Variables by AR2-Lasso 4q Ahead

# |  | Ending Year
---|---|---
|  | 2019q1 | 2019q2 | 2019q3 | 2019q4 | 2020q1 | 2020q2
1 |  | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4
2 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4
3 |  | DPMAT4 | DPMAT4 | DPMAT4 | DPMAT4 | DPMAT4 | DPMAT4
4 |  | DDPMUK4 | DDPMUK4 | DDPMUK4 | DDPMUK4 | DDPMUK4 | DDPMUK4
5 |  | DPSUK4 | DEPUK4 | DEPUK4 | DEPUK4 | DEPUK4 | DEPUK4
6 |  | DDPSUK4 | DPSUK4 | DPSUK4 | DPSUK4 | DPSUK4 | DPSUK4
7 |  | DGAPSUK8 | DDPSUK4 | DDPSUK4 | DDPSUK4 | DDPSUK4 | DDPSUK4
8 |  | $\cdot$ | DGAPSUK8 | DGAPSUK8 | DDYCHINA4 | DDYCHINA4 | DDYCHINA4
9 |  | $\cdot$ | $\cdot$ | $\cdot$ | DGAPSUK8 | DGAPSUK8 | DGAPSUK8
$\hat{\lambda}_{hT}$ |  | 0.2357 | 0.2233 | 0.2237 | 0.1999 | 0.1998 | 0.2014
|  |  |  |  |  |  |
|  | 2020q3 | 2020q4 | 2021q1 | 2021q2 | 2021q3 | 2021q4
1 |  | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4
2 |  | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4 | DMUK4
3 |  | DPMAT4 | DPMAT4 | DPMAT4 | DPMAT4 | DPMAT4 | DPMAT4
4 |  | DDPMUK4 | DDPMUK4 | DDPMUK4 | DDPMUK4 | DDPMUK4 | DDPMUK4
5 |  | DEPUK4 | DEPUK4 | DEPUK4 | DPSUK4 | DPSUK4 | DPSUK4
6 |  | DPSUK4 | DPSUK4 | DPSUK4 | DDPSUK4 | DDPSUK4 | DDPSUK4
7 |  | DDPSUK4 | DDPSUK4 | DDPSUK4 | $\cdot$ | $\cdot$ | $\cdot$
8 |  | DDYCHINA4 | DDYCHINA4 | DGAPSUK8 | $\cdot$ | $\cdot$ | $\cdot$
9 |  | DGAPSUK8 | DGAPSUK8 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
$\hat{\lambda}_{hT}$ |  | 0.1901 | 0.1902 | 0.1780 | 0.2158 | 0.2170 | 0.2106
|  |  |  |  |  |  |
|  | 2022q1 | 2022q2 | 2022q3 | 2022q4 | 2023q1 |
1 |  | DEMUK4 | DMUK4 | DMUK4 | DDUUK | DDUUK |
2 |  | DMUK4 | DPMAT4 | DPMAT4 | DMUK4 | DMUK4 |
3 |  | DPMAT4 | DDPMUK4 | DDPMUK4 | DPMAT4 | DPMAT4 |
4 |  | DDPMUK4 | DPSUK4 | DPSUK4 | DDPMUK4 | DDPMUK4 |
5 |  | DPSUK4 | DDPSUK4 | DDPSUK4 | DPSUK4 | DPSUK4 |
6 |  | DDPSUK4 | DDYSUK4 | DWUK4 | DDPSUK4 | DDPSUK4 |
7 |  | $\cdot$ | DWUK4 | $\cdot$ | DYUS4 | DWUK4 |
8 |  | $\cdot$ | $\cdot$ | $\cdot$ | DWUK4 | $\cdot$ |
$\hat{\lambda}_{hT}$ |  | 0.2735 | 0.2513 | 0.2736 | 0.2493 | 0.2522 |

_Note_ : ${\footnotesize\hat{\lambda}}_{hT}$ is the estimate of the Lasso
penalty parameter computed by 10-fold cross-validation.

Table S.15: Selected Variables by ARX-Lasso 4q Ahead

# |  | Ending Year
---|---|---
|  | 2019q1 | 2019q2 | 2019q3 | 2019q4 | 2020q1 | 2020q2
1 |  | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4
2 |  | DMUK4 | DMUK4 | DVUK4 | DMUK4 | DMUK4 | DVUK4
3 |  | DPMAT4 | DPMAT4 | DMUK4 | DPMAT4 | DPMAT4 | DMUK4
4 |  | DGAPSUK8 | DDPMUK4 | DPMAT4 | DDPMUK4 | DDPMUK4 | DPMAT4
5 |  | $\cdot$ | DEPUK4 | DDPMUK4 | DEPUK4 | DEPUK4 | DDPMUK4
6 |  | $\cdot$ | DGAPSUK8 | DEPUK4 | DGAPSUK8 | DGAPSUK8 | DEPUK4
7 |  | $\cdot$ | $\cdot$ | DGAPSUK8 | $\cdot$ | $\cdot$ | DDYCHINA4
8 |  | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | DGAPSUK8
9 |  | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | DPUS4
10 |  | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | DDPUS4
$\hat{\lambda}_{hT}$ |  | 0.2349 | 0.2238 | 0.2020 | 0.2139 | 0.2127 | 0.1579
|  |  |  |  |  |  |
|  | 2020q3 | 2020q4 | 2021q1 | 2021q2 | 2021q3 | 2021q4
1 |  | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4 | DEMUK4
2 |  | DVUK4 | DVUK4 | DVUK4 | DMUK4 | DVUK4 | DVUK4
3 |  | DMUK4 | DMUK4 | DMUK4 | DPMAT4 | DMUK4 | DMUK4
4 |  | DPMAT4 | DPMAT4 | DPMAT4 | DDPMUK4 | DPMAT4 | DPMAT4
5 |  | DDPMUK4 | DDPMUK4 | DDPMUK4 | $\cdot$ | DDPMUK4 | DDPMUK4
6 |  | DEPUK4 | DEPUK4 | DEPUK4 | $\cdot$ | DEPUK4 | $\cdot$
7 |  | DDYCHINA4 | DDYCHINA4 | DGAPSUK8 | $\cdot$ | DPUS4 | $\cdot$
8 |  | DGAPSUK8 | DGAPSUK8 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
9 |  | DPUS4 | DPUS4 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
10 |  | DDPUS4 | DDPUS4 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
$\hat{\lambda}_{hT}$ |  | 0.1471 | 0.1583 | 0.1923 | 0.2166 | 0.1847 | 0.1930
|  |  |  |  |  |  |
|  | 2022q1 | 2022q2 | 2022q3 | 2022q4 | 2023q1 |
1 |  | DEMUK4 | DEMUK4 | DMUK4 | DMUK4 | DMUK4 |
2 |  | DVUK4 | DMUK4 | DPMAT4 | DPMAT4 | DPMAT4 |
3 |  | DMUK4 | DPMAT4 | $\cdot$ | $\cdot$ | $\cdot$ |
4 |  | DPMAT4 | DDPMUK4 | $\cdot$ | $\cdot$ | $\cdot$ |
5 |  | DDPMUK4 | DDYSUK4 | $\cdot$ | $\cdot$ | $\cdot$ |
$\hat{\lambda}_{hT}$ |  | 0.2017 | 0.2330 | 0.2932 | 0.2719 | 0.3251 |

_Note_ : ${\footnotesize\hat{\lambda}}_{hT}$ is the estimate of the Lasso
penalty parameter computed by 10-fold cross-validation.

##  Appendix S-4 Realizations and Forecasts

###  S-4.1 h=1 quarter ahead forecasts

Table S.16: One Quarter Ahead Forecast Results

Models |  | Actual |  | AR2 |  | ARX |  | Lasso |  | Lasso-AR2
---|---|---|---|---|---|---|---|---|---|---
2020q1 |  | 1.6808 |  | 1.4930 |  | 1.5005 |  | 1.4411 |  | 1.1366
2020q2 |  | 0.7819 |  | 1.9051 |  | 1.8708 |  | 1.7274 |  | 1.2898
2020q3 |  | 0.7678 |  | 0.7901 |  | 0.3358 |  | –1.6184 |  | –1.7781
2020q4 |  | 0.7147 |  | 1.0091 |  | 0.9193 |  | 1.0883 |  | 0.9987
2021q1 |  | 0.9211 |  | 0.9482 |  | 0.8014 |  | 1.2458 |  | 1.0556
2021q2 |  | 2.0533 |  | 1.1989 |  | 1.3852 |  | 1.7410 |  | 1.4759
2021q3 |  | 2.6215 |  | 2.4556 |  | 2.7557 |  | 3.1343 |  | 3.0982
2021q4 |  | 4.3095 |  | 2.8118 |  | 2.9337 |  | 3.1241 |  | 3.0738
2022q1 |  | 5.3960 |  | 4.6293 |  | 4.8813 |  | 4.8406 |  | 5.0132
2022q2 |  | 7.6259 |  | 5.4458 |  | 5.7835 |  | 5.8914 |  | 6.0747
2022q3 |  | 8.3877 |  | 7.8385 |  | 8.1711 |  | 8.0610 |  | 8.3580
2022q4 |  | 8.9856 |  | 8.0723 |  | 8.2705 |  | 8.6325 |  | 8.7409
2023q1 |  | 8.6111 |  | 8.5777 |  | 8.3988 |  | 8.7458 |  | 9.0455
2023q2 |  |  |  | 7.9164 |  | 7.2945 |  | 7.6445 |  | 7.8217
RMSFE |  |  |  | 0.9141 |  | 0.7884 |  | 0.9696 |  | 0.9617
|  | Lasso-ARX |  | OCMT-AR2 |  | OCMT-ARX |  | OCMT-AR2 |  | OCMT-ARX
|  |  |  | ${\footnotesize\delta=1}$ |  | ${\footnotesize\delta=1}$ |  | ${\footnotesize\delta=1.5}$ |  | ${\footnotesize\delta=1.5}$
2020q1 |  | 1.3148 |  | 1.4930 |  | 1.5005 |  | 1.4930 |  | 1.5005
2020q2 |  | 1.8793 |  | 1.9051 |  | 1.8708 |  | 1.9051 |  | 1.8708
2020q3 |  | –0.9434 |  | 0.7901 |  | 0.3358 |  | 0.7901 |  | 0.3358
2020q4 |  | 1.4325 |  | 1.0091 |  | 0.9193 |  | 1.0091 |  | 0.9193
2021q1 |  | 1.1040 |  | 0.9482 |  | 0.8014 |  | 0.9482 |  | 0.8014
2021q2 |  | 1.7652 |  | 1.1989 |  | 1.3852 |  | 1.1989 |  | 1.3852
2021q3 |  | 2.6691 |  | 3.0457 |  | 2.7557 |  | 2.4556 |  | 2.7557
2021q4 |  | 3.0872 |  | 2.8118 |  | 2.9337 |  | 2.8118 |  | 2.9337
2022q1 |  | 4.9254 |  | 4.5468 |  | 4.8813 |  | 4.6293 |  | 4.8813
2022q2 |  | 5.9782 |  | 5.4612 |  | 5.7835 |  | 5.4458 |  | 5.7835
2022q3 |  | 8.6872 |  | 7.8385 |  | 8.1711 |  | 7.8385 |  | 8.1711
2022q4 |  | 9.1017 |  | 8.0723 |  | 8.2705 |  | 8.0723 |  | 8.2705
2023q1 |  | 9.3315 |  | 8.5777 |  | 8.3988 |  | 8.5777 |  | 8.3988
2023q2 |  | 7.6670 |  | 7.9164 |  | 7.2945 |  | 7.9164 |  | 7.2945
RMSFE |  | 0.8750 |  | 0.9233 |  | 0.7884 |  | 0.9141 |  | 0.7884

_Notes_ : The ”Actual” column gives the realized inflation rate for quarter
$t,$ the other columns the forecasts for each model. RMSFE is the Root Mean
Square Forecast Error. Lasso models are estimated using the code of Chudik,
Kapetanios, and Pesaran (2018).

Figure S.1: One Quarter Ahead Forecast Plots

###  S-4.2 h=2 quarters ahead forecasts

Table S.17: Two Quarters Ahead Forecast Results

Models |  | Actual |  | AR2 |  | ARX |  | Lasso |  | Lasso-AR2
---|---|---|---|---|---|---|---|---|---|---
2020q1 |  | 1.6808 |  | 1.9854 |  | 1.6773 |  | 1.7494 |  | 1.8994
2020q2 |  | 0.7819 |  | 1.6115 |  | 1.5979 |  | 1.4380 |  | 0.8428
2020q3 |  | 0.7678 |  | 2.0705 |  | 1.9977 |  | 1.8217 |  | 2.0103
2020q4 |  | 0.7147 |  | 0.8941 |  | 0.1859 |  | 0.5350 |  | –4.7147
2021q1 |  | 0.9211 |  | 1.2182 |  | 1.0423 |  | 1.3422 |  | 7.7188
2021q2 |  | 2.0533 |  | 1.1590 |  | 0.9049 |  | 1.5683 |  | 1.6895
2021q3 |  | 2.6215 |  | 1.4199 |  | 1.6725 |  | 1.8149 |  | 1.9367
2021q4 |  | 4.3095 |  | 2.6839 |  | 3.0931 |  | 3.6585 |  | 2.0096
2022q1 |  | 5.3960 |  | 2.9218 |  | 3.1026 |  | 3.3691 |  | 2.2883
2022q2 |  | 7.6259 |  | 4.7324 |  | 5.1079 |  | 4.7079 |  | 4.9191
2022q3 |  | 8.3877 |  | 5.3660 |  | 5.8973 |  | 5.6290 |  | 5.9572
2022q4 |  | 8.9856 |  | 7.7515 |  | 8.2645 |  | 7.4422 |  | 7.9237
2023q1 |  | 8.6111 |  | 7.7117 |  | 8.0551 |  | 8.1659 |  | 8.4389
2023q2 |  |  |  | 8.1422 |  | 7.8532 |  | 7.9158 |  | 7.4686
2023q3 |  |  |  | 7.3340 |  | 6.2890 |  | 6.1918 |  | 5.7513
RMSFE |  |  |  | 1.6039 |  | 1.3813 |  | 1.4109 |  | 2.8719
|  | Lasso-ARX |  | OCMT-AR2 |  | OCMT-ARX |  | OCMT-AR2 |  | OCMT-ARX
|  |  |  | $\delta=1$ |  | $\delta=1$ |  | $\delta=1.5$ |  | $\delta=1.5$
2020q1 |  | 1.8915 |  | 1.7638 |  | 1.4778 |  | 1.9854 |  | 1.6773
2020q2 |  | 1.0312 |  | 1.4532 |  | 1.5288 |  | 1.6115 |  | 1.5979
2020q3 |  | 2.2963 |  | 1.9038 |  | 1.8882 |  | 2.0705 |  | 1.9977
2020q4 |  | –3.5975 |  | 1.1311 |  | 0.3795 |  | 0.8941 |  | 0.1859
2021q1 |  | 8.4157 |  | 1.6797 |  | 1.5628 |  | 1.2182 |  | 1.0423
2021q2 |  | 2.1084 |  | 1.5683 |  | 1.3765 |  | 1.6344 |  | 1.3765
2021q3 |  | 1.9160 |  | 1.8149 |  | 2.1371 |  | 1.8554 |  | 2.1371
2021q4 |  | 1.9221 |  | 3.3262 |  | 3.3351 |  | 3.5873 |  | 3.3351
2022q1 |  | 2.2122 |  | 3.0279 |  | 3.0481 |  | 3.1239 |  | 3.0481
2022q2 |  | 4.0165 |  | 4.4689 |  | 4.9381 |  | 4.4676 |  | 4.9381
2022q3 |  | 6.3907 |  | 5.2246 |  | 5.6954 |  | 5.2253 |  | 5.6954
2022q4 |  | 8.4832 |  | 7.0244 |  | 7.8871 |  | 6.9260 |  | 7.8871
2023q1 |  | 9.1288 |  | 6.8038 |  | 7.5799 |  | 6.9298 |  | 8.0551
2023q2 |  | 8.0515 |  | 7.2864 |  | 7.4267 |  | 8.1422 |  | 7.8532
2023q3 |  | 5.8963 |  | 6.3556 |  | 5.6673 |  | 7.3340 |  | 6.2890
RMSFE |  | 2.9231 |  | 1.6880 |  | 1.4217 |  | 1.6662 |  | 1.4037

___Notes_ : The ”Actual” column gives the realized inflation rate at period
$t,$ the other columns the forecasts for each model. RMSFE is the Root Mean
Square Forecast Error. Lasso models are estimated using the code of Chudik,
Kapetanios, and Pesaran (2018).

Figure S.2: Two Quarters Ahead Forecast Plots

###  S-4.3 h=4 quarters ahead forecasts

Table S.18: Four Quarters Ahead Forecast Results

Models |  | Actual |  | AR2 |  | ARX |  | Lasso |  | Lasso-AR2
---|---|---|---|---|---|---|---|---|---|---
2020q1 |  | 1.6808 |  | 2.3062 |  | 1.6093 |  | 1.6462 |  | 1.8149
2020q2 |  | 0.7819 |  | 2.3417 |  | 2.2183 |  | 1.8204 |  | 1.9359
2020q3 |  | 0.7678 |  | 2.2620 |  | 1.7780 |  | 1.5496 |  | 1.4110
2020q4 |  | 0.7147 |  | 2.0491 |  | 2.0380 |  | 1.5778 |  | 0.8652
2021q1 |  | 0.9211 |  | 2.1876 |  | 2.0822 |  | 1.5795 |  | 1.4660
2021q2 |  | 2.0533 |  | 1.6754 |  | 0.5406 |  | 0.5873 |  | –7.2149
2021q3 |  | 2.6215 |  | 1.6277 |  | 1.3694 |  | 2.0021 |  | 8.3843
2021q4 |  | 4.3095 |  | 1.5822 |  | 1.1737 |  | 2.3696 |  | 1.1065
2022q1 |  | 5.3960 |  | 1.6856 |  | 2.1203 |  | 2.7836 |  | 1.4914
2022q2 |  | 7.6259 |  | 2.3353 |  | 2.9872 |  | 3.3813 |  | 2.8290
2022q3 |  | 8.3877 |  | 2.6974 |  | 2.9742 |  | 2.5152 |  | 2.8806
2022q4 |  | 8.9856 |  | 3.6744 |  | 4.2203 |  | 3.4427 |  | 3.5490
2023q1 |  | 8.6111 |  | 4.3493 |  | 5.1543 |  | 4.2743 |  | 4.6459
2023q2 |  |  |  | 5.7823 |  | 6.6329 |  | 4.7789 |  | 5.1198
2023q3 |  |  |  | 6.2075 |  | 6.7753 |  | 4.7462 |  | 5.4047
2023q4 |  |  |  | 6.6574 |  | 6.2714 |  | 4.8914 |  | 4.0569
2024q1 |  |  |  | 6.3149 |  | 4.6141 |  | 3.7130 |  | 2.6772
RMSFE |  |  |  | 3.2524 |  | 2.9883 |  | 3.0131 |  | 4.3440
|  | Lasso-ARX |  | OCMT-AR2 |  | OCMT-ARX |  | OCMT-AR2 |  | OCMT-ARX
|  |  |  | $\delta=1$ |  | $\delta=1$ |  | $\delta=1.5$ |  | $\delta=1.5$
2020q1 |  | 1.7270 |  | 1.6437 |  | 1.3401 |  | 1.7873 |  | 1.3401